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Influence Maximization with ?-Almost Submodular Threshold Functions Qiang Li??, Wei Chen?, Xiaoming Sun??, Jialin Zhang?? ? CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences ? University of Chinese Academy of Sciences ? Microsoft Research {liqiang01,sunxiaoming,zhangjialin}@ict.ac.cn [email protected] Abstract Influence maximization is the problem of selecting k nodes in a social network to maximize their influence spread. The problem has been extensively studied but most works focus on the submodular influence diffusion models. In this paper, motivated by empirical evidences, we explore influence maximization in the nonsubmodular regime. In particular, we study the general threshold model in which a fraction of nodes have non-submodular threshold functions, but their threshold functions are closely upper- and lower-bounded by some submodular functions (we call them ?-almost submodular). We first show a strong hardness result: there is no 1/n?/c approximation for influence maximization (unless P = NP) for all networks with up to n? ?-almost submodular nodes, where ? is in (0,1) and c is a parameter depending on ?. This indicates that influence maximization is still hard to approximate even though threshold functions are close to submodular. We then provide (1 ? ?)` (1 ? 1/e) approximation algorithms when the number of ?-almost submodular nodes is `. Finally, we conduct experiments on a number of real-world datasets, and the results demonstrate that our approximation algorithms outperform other baseline algorithms. 1 Introduction Influence maximization, proposed by Kempe, Kleinberg, and Tardos [1], considers the problem of selecting k seed nodes in a social network that maximizes the spread of influence under predefined diffusion model. This problem has many applications including viral marketing [2, 3], media advertising [4] and rumors spreading [5] etc., and many aspects of the problem has been extensively studied. Most existing algorithms for influence maximization, typically under the independent cascade (IC) model and the linear threshold (LT) model [1], utilize the submodularity of the influence spread as a set function on the set of seed nodes, because it permits a (1 ? 1/e)-approximation solution by the greedy scheme [1, 6, 7], following the foundational work on submodular function maximization [8]. One important result concerning submodularity in the influence model is by Mossel and Roch [9], who prove that in the general threshold model, the global influence spread function is submodular when all local threshold functions at all nodes are submodular. This result implies that ?local" submodularity ensures the submodularity of ?global" influence spread. Although influence maximization under submodular diffusion models is dominant in the research literature, in real networks, non-submodularity of influence spread function has been observed. Backstrom et al. [10] study the communities of two networks LiveJournal and DBLP and draw 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. pictures of the impulse that a person joins a community against the number of his friends already in this community. The curve is concave overall, except that a drop is observed in first two nodes. Yang et al. [11] track emotion contagion under Flickr and find that the probability that an individual becomes happy is superlinear to the number of his happy friends with higher PageRank scores. These are all instances of non-submodular influence spread functions. Influence maximization under many non-submodular diffusion models are proved to be hard to approximate. For example, in the diffusions of rumors, innovations, or riot behaviors, the individual in a social network is activated only when the number of her neighbors already adopting the behavior exceeds her threshold. It has been shown that the influence maximization problem based on this fixed threshold model cannot be approximated within a ratio of n1?? for any ? > 0 [1]. Meanwhile Chen [12] proves that the seed minimization problem, to activate the whole network with minimum size of 1?? seed set, is also inapproximable, in particular, within a ratio of O(2log n ). In this paper we give the first attempt on the influence maximization under the non-submodular diffusion models. We study the general threshold model in which a fraction of nodes have nonsubmodular threshold functions, but their threshold functions are closely upper- and lower-bounded by some submodular functions (we call them ?-almost submodular). Such a model bears conceptual similarity to the empirical finding in [10, 11]: both studies show that the influence curve is only slightly non-concave, and Yang et al. [11] further shows that different roles have different curves ? some are submodular while others are not, and ordinary users usually have behaviors close to ? submodular while opinion leaders may not. We first show a strong hardness result: there is no 1/n c ? approximation for influence maximization (unless P = NP) for all networks with up to n ?-almost submodular nodes, where ? is in (0, 1) and c is a parameter depending on ?. On the other hand, we propose constant approximation algorithms for networks where the number of ?-almost submodular nodes is a small constant. The positive results imply that non-submodular problem can be partly solved as long as there are only a few non-submodular nodes and the threshold function is not far away from submodularity. Finally, we conduct experiments on real datasets to empirically verify our algorithms. Empirical results on real datasets show that our approximation algorithms outperform other baseline algorithms. Related Work. Influence maximization has been well studied over the past years [13, 6, 7, 14, 15]. In particular, Leskovec et al. [6] propose a lazy-forward optimization that avoids unnecessary computation of expected size. Chen et al. [7, 14] propose scalable heuristic algorithms that handle network of million edges. Based on the technique of Reverse Reachable Set, Borgs et al. [16] reduce the running time of greedy algorithms to near-linear under the IC model [1]. Tang et al. [17] implement the near-linear algorithm and process Twitter network with million edges. Subsequently, Tang et al. [18] and Nguyen et al. [19] further improve the efficiency of algorithms. These works all utilize the submodularity to accelerate approximation algorithms. Seed minimization, as the dual problem of influence maximization, is to find a small seed set such that expected influence coverage exceeds a desired threshold. Chen [12] provide some strong negative results on seed minimization problem under fixed threshold model, which is a special case of general threshold model where its threshold function has breaking points. Goyal et al. [20] propose a greedy algorithm with a bicriteria approximation. Recently, Zhang et al. [21] study the probabilistic variant of seed minimization problem. Due to the limitation of independent cascade and linear threshold model, general threshold model has been proposed [1, 9]. Not much is known about the general threshold model, other than it is NP-hard to approximate [1]. One special case which receives many attention is k-complex contagion where a node becomes active if at least k of its neighbours have been activated [22, 23, 24]. Gao et al. [25] make one step further of k-complex contagion model by considering the threshold comes from a probability distribution. Optimization of non-submodular function is another interesting direction. Du et al. [26] introduce two techniques ? restricted submodularity and shifted submodularity ? to analyze greedy approximation of non-submodular functions. Recently, Horel et al.[27] study the problem of maximizing a set function that is very close to submodular. They assume that function values can be obtained from an oracle and focused on its query complexity. In our study, the local threshold functions are close to submodular and our target is to study its effect on the global influence spread function, which is the result of complex cascading behavior derived from the local threshold functions. 2 2 Preliminaries For a set function f : 2V ? R, we say that it is monotone if f (S) ? f (T ) for all S ? T ? V ; we say that it is submodular if f (S ? {v}) ? f (S) ? f (T ? {v}) ? f (T ), for all S ? T ? V and v ? V \ T . For a directed graph G = (V, E), we use N in (v) to denote the in-neighbors of v in G. We now formally define the general threshold model used in the paper. Definition 1 (General Threshold Model [1]). In the general threshold model, for a social graph in G = (V, E), every node v ? V has a threshold function fv : 2N (v) ? [0, 1]. The function fv (?) should be monotone and fv (?) = 0. Initially at time 0, each node v ? V is in the inactive state and chooses ?v uniformly at random from the interval [0, 1]. A seed set S0 is also selected, and their states are set to be active. Afterwards, the influence propagates in discrete time steps. At time step t ? 1, node v becomes active if fv (St?1 ? N in (v)) ? ?v , where St?1 is the set of active nodes by time step t ? 1. The process ends when no new node becomes active in a step. General threshold model is one of the most important models in the influence maximization problem. Usually we focus on two properties of threshold function ? submodularity and supermodularity. Submodularity can be understood as diminishing marginal returns when adding more nodes to the seed set. In contrast, supermodularity means increasing marginal returns. Given a seed set S, let ?(S) denote the expected number of activated nodes after the process of influence propagation terminates. Submodularity is the key property that guarantees the performance of greedy algorithms [9]. In this paper, we would like to study the influence maximization with nearly submodular threshold function ? ?-almost submodular function, or in short ?-AS. Definition 2 (?-Almost Submodular (?-AS)). A set function f : 2V ? R is ?-almost submodular if there exists a submodular function f sub defined on 2V and for any subset S ? V , f sub (S) ? f (S) ? (1 ? ?)f sub (S). Here ? is a small positive number. The definition of ?-almost submodular here is equivalent to "Approximate submodularity" defined in [27]. For an ?-almost submodular threshold function fv , define its upper and lower submodular bound as f v and f v . Hence by definition, we have f v = (1 ? ?)f v . Given the definition of ?-almost submodular function, we then model the almost submodular graph. In this paper, we consider the influence maximization problem based on this kind of graphs. Definition 3 ((?, ?)-Almost Submodular Graph). Given fixed parameters ?, ? ? [0, 1], we say that a graph with n (n = |V |) nodes is a (?, ?)-Almost Submodular Graph (under the general threshold model), if there are at most n? nodes in the graph with ?-almost submodular threshold functions while other nodes have submodular threshold functions. Definition 4 (?-ASIM). Given a graph containing ?-almost submodular nodes and an input k, Influence Maximization problem on graph with ?-Almost Submodular nodes (?-ASIM) is the problem to find k seed nodes such that the influence spread invoked by the k nodes is maximized. 3 Inapproximability of ?-ASIM In this section we show that it is in general hard to approximate the influence maximization problem even if there are only sublinear number of nodes with ?-almost submodular threshold functions. The main reason is that even a small number of nodes with ?-almost submodular threshold functions fv (?) would cause the global influence spread function far from submodularity, making the maximization problem very difficult. The theorem below shows our hardness result. ? Theorem 1. For any small ? > 0 and any ? ? (0, 1), there is no 1/n c -approximation influence 2 maximization algorithm for all (?, ?)-almost submodular graphs where c = 3 + 3/ log 2?? , unless P=NP. We first construct a probabilistic-AND gate gadget by amplifying the non-submodularity through a binary tree. Then we prove the lower bound of approximation ratio by the reduction from set cover problem. Due to page limits, we only sketch the main technique. The full proof can be found in the supplementary material. 3 Here we construct a basic gadget with input s1 , s2 and output t (see Figure 1a). We assume that node t has two in-neighbours s1 , s2 and the threshold function g(?) of t is ?-almost submodular: g(S) = |S|/2, when |S| = 0 or 2; 1?? 2 , when |S| = 1. t d t .. . s1 .. . .. . s2 .. . .. . s1 .. . .. . .. . s2 (b) Tree gadget T? (a) Basic gadget Figure 1: Diagrams of gadegts Let Pa (v) be the activation probability of node v in this case. This simple gadget is obviously far away from the AND gate, and our next step is to construct a more complex gadget with input node s1 , s2 . We hope that the output node t is active only when both s1 , s2 are active, and if only one of s1 and s2 is active, the probability that node t becomes active is close to 0. We call it a probabilistic-AND gate. The main idea is to amplify the gap between submodularity and non-submodularity by binary tree (figure 1b). In this gadget T? with a complete binary tree, node t is the root of a full binary tree and each node holds a directed edge to its parent. For each leaf node v in the tree, both s1 , s2 hold the directed edges towards it. The threshold function for each node in the tree is g(?) defined above while ? is the index of gadget T? . The depth of the tree is parameter d which will be determined later. We use vi to denote a node of depth i (t is in depth 1). It is obviously that Pa (t) = 1 if both s1 , s2 are activated, and Pa (t) = 0 if neither s1 or s2 is activated. Thus, we would like to prove, in case when only one of s1 , s2 is activated, the activation probability becomes smaller for inner nodes in the tree. d Lemma 2. For gadget T? with depth d, the probability of activating output node t is less than ( 2?? 2 ) when only one of s1 , s2 is activated. Proof. In this case, for leaf node vd , we have Pa (vd ) = 1?? 2 . Apparently, the probability of becoming activated for nodes with depth d are independent with each other. Given a basic gadget, if each of the two children nodes is activated with an independent probability p, then the parent node will be activated with probability p2 ? g(2) + 2p(1 ? p) ? g(1) + (1 ? p)2 ? g(0) = p2 + 2p(1 ? p) 1?? = p(1 ? ?(1 ? p)). 2 So we have Pa (vi ) ? Pa (vi+1 )(1 ? ?(1 ? Pa (vi+1 ))). Since Pa (vd ) = 1?? 2 < 1/2, and Pa (vi ) ? Pa (vi+1 ) from above, we have pa (vi ) < 1/2 for all i, and thus we can rewrite the recurrence as Pa (vi ) ? Pa (vi+1 )(1 ? ?/2). Hence for the gadget with depth d, the probability that node t becomes 2?? d?1 d < ( 2?? activated is Pa (t) = Pa (v1 ) ? 1?? 2 ( 2 ) 2 ) . Lemma 2 shows that gadget T? is indeed a probabilistic-AND gate with two input nodes, and the probability that t is activated when only one of s1 and s2 is activated approaches 0 exponentially fast with the depth d. We say a gadget T? works well if output node t stay inactive when only one of the input nodes is activated. By the similar method we construct multi-input-AND gates based on 2-input-AND gates. Finally, we show that if the influence maximization problem can be approximated beyond the ratio shown above, we can solve the set cover problem in polynomial time. The main idea is as follows. For any set cover instance, we will put all elements to be the input of our multi-input-probabilistic-AND gate, and connect the output with a large number of additional nodes. Thus, if k sets can cover all elements, all of those addition nodes will be activated, on contrast, if at least one of the elements cannot be covered, almost all of the additional nodes will remain inactive. 4 4 Approximation Algorithms In the previous section, we show that influence maximization is hard to approximate when the number of ?-almost submodular nodes is sublinear but still a non-constant number. In this section, we discuss the situation where only small number of nodes hold ?-almost submodular threshold functions. We firstly provide a greedy algorithm for small number of non-submodular nodes which may not be ?-almost submodular, then, we restrict to the case of ?-almost submodular nodes. 4.1 Approximation Algorithm with Small Number of Non-submodular Nodes In the case of ` (` < k) non-submodular nodes, we provide an approximate algorithm as follows. We first add these non-submodular nodes into the seed set, and then generate the rest of the seed set by the classical greedy algorithm. The proof of Theorem 3 can be found in the supplementary material. Theorem 3. Given a graph of n nodes where all nodes have submodular threshold functions except k?` ` < k nodes, for influence maximization of k seeds with greedy scheme we can obtain a (1 ? e? k )approximation ratio. 4.2 Approximation Algorithm of ?-ASIM In this section, we consider the case when all non-submodular nodes have ?-almost submodular threshold functions, and provide an approximation algorithm that allows more than k ?-almost submodular nodes, with the approximation ratio close to 1 ? 1/e when ? is small. The main idea is based on the mapping between probability spaces. Given a graph containing nodes with ?-almost submodular threshold functions, we simply set each node?s threshold function to its submodular lower bound and then run classical greedy algorithm A on this graph (Algorithm 1). Algorithm 1 takes the lower bounds of ?-almost submodular threshold functions as input parameters. The following theorem analyzes the performance of Algorithm 1. Algorithm 1 Galg-L algorithm for Influence Maximization Input: G = (V, E), A, {fv }, {f v }, k Output: Seed set S 1: set S = ? 2: replace each nodes v?s threshold function fv with f v 3: run algorithm A on G with {f v } and obtain S 4: return S Theorem 4. Given a graph G = (V, E), under the general threshold model, assuming that ` nodes have ?-almost submodular threshold functions and the other nodes have submodular threshold functions. Then the greedy algorithm Galg-L has approximation ratio of (1 ? 1e )(1 ? ?)` . Proof. Let Ve be the set of nodes with ?-almost submodular threshold functions. Without loss of generality, we assume Ve = {v1 , v2 , . . . , v` }. Now consider two general threshold models M , M with different threshold functions. Both models hold threshold functions {fv } for v ? V ? Ve . For node v in Ve , M , M hold {f v } and {f v } respectively. In any threshold model, after we sample each node?s threshold ?v , the diffusion process becomes deterministic. A graph with threshold functions {fv } and sampled thresholds {?v } is called a possible world of the threshold model, which is similar to the live-edge graph in the independent cascade model. An instance of threshold model?s possible world can be written as {?v1 , ?v2 , . . . , ?vn ; fv1 , fv2 , . . . , fvn }. Here we build a one-to-one mapping from all M ?s possible worlds with ?v ? 1 ? ? (v ? Ve ) to all M ?s possible worlds: {?v1 , . . . , ?vn ; fv1 , . . . , fvn } ? { ?v` fv` fv1 ?v1 ,..., , ?v . . . , ?vn ; ,..., , fv`+1 , . . . , fvn }. 1?? 1 ? ? `+1 1?? 1?? The above corresponding relation shows this one-to-one mapping between M and M . For any instance of M ?s possible world with ?v ? 1 ? ? (v ? Ve ), we amplify the threshold of node v in ?v 1 Ve to 1?? . At the same time, we amplify the corresponding threshold function by a factor of 1?? . 5 Obviously, this amplification process will not effect the influence process under this possible world, because for each v ? Ve , both its threshold value and the its threshold function are amplified by the same factor 1/(1 ? ?). Furthermore, the amplified possible world is an instance of M . R ~ ~ ~ ~ Expected influence can be computed by ?(S) = ??[0,1] n D(?; f , S)d? ~ , where D(?; f , S) is the ~ ~ f~}. We refer M , M ?s expected deterministic influence size of seed set S under possible world {?; n ~ influence size functions as ?, ?. We define ? ? [0, 1] as the vector of n nodes threshold, and ?~e ? [0, 1]` , ?~0 ? [0, 1]n?` are the threshold vectors of Ve and V ? Ve . Besides, the threshold functions of Ve and V ? Ve will be represented as f~e , f~0 . A possible world is symbolized as {?~e , ?~0 ; f~e , f~0 }. For any seed set S, we have ?(S) = ? R ~ R??[0,1] n ~ f~, S)d~ D(?; ? R ~e ?[0,1??]` ? R ` = (1 ? ?) = (1 ? ?) ~0 ?[0,1]n?` ? ~e ? ` R 1?? ~e ? R 1?? ` R ?[0,1]` R ?[0,1]` D((?~e , ?~0 ); f~, S)d?~e d?~0 ~ ~0 ~ n?` D((?e , ? ); f , S)d ~0 ?[0,1] ? ~e ? 1?? d?~0 ~e f~e ? ~0 ~0 ~e ~0 ?[0,1]n?` D(( 1?? , ? ); ( 1?? , f ), S)d ? ? 1?? d?~0 ~ f~e ~0 = (1 ? ?) ??[0,1] ~ n D(?; ( 1?? , f ), S)d? ~ ` = (1 ? ?) ?(S). The third equality utilizes our one-to-one mapping, in particular D((?~e , ?~0 ); f~, S) = ~e ~0 ~e f~e ? ? D(( 1?? , ? ); ( 1?? , f~0 ), S) for 1?? ? [0, 1]` , because they follow the same deterministic propagation process. Hence given a seed set S, the respective influence sizes in model M , M satisfy the relation ?(S) ? (1 ? ?)` ?(S). Let ? be the expected influence size function of the original model, and assume that the optimal ? ? solution for ?, ?, ? are S , S ? , S ? respectly. Apparently, ?(S ) ? ?(S ? ) since for every node v, ? ? f v ? fv . According to the previous analysis, we have ?(S ? ) ? ?(S ) ? (1 ? ?)` ?(S ). Hence for A output S of the greedy algorithm for optimizing ?, we have approximation ratio 1 1 1 ? ?(S A ) ? ?(S A ) ? (1 ? )?(S ? ) ? (1 ? )(1 ? ?)` ?(S ) ? (1 ? )(1 ? ?)` ?(S ? ). e e e The theorem holds. If we replace threshold functions by their upper bound and run the greedy algorithm, we obtain Galg-U. With similar analysis, Galg-U also holds approximation ratio of (1 ? 1e )(1 ? ?)` on graphs with ` ?-almost submodular nodes. The novel technique used to prove approximation ratio is similar to the sandwich approximation in [28]. But their approximation ratio relies on instance-dependent influence sizes, while we utilize mapping of probabilistic space to provide instance-independent approximation ratio. 5 Experiments In addition to the theoretical analysis, we are curious about the performance of greedy algorithms Galg-U, Galg-L on real networks with non-submodular nodes. Our experiments run on a machine with two 2.4GHz Intel(R) Xeon(R) E5-2620 CPUs, 4 processors (24 cores), 128GB memory and 64bit Ubuntu 14.04.1. All algorithms tested in this paper are written in C++ and compiled with g++ 4.8.4. Some algorithms are implemented with multi-thread to decrease the running time. 5.1 Experiment setup Datasets. We conduct experiments on three real networks. The first network is NetHEPT, an academic collaboration network extracted from "High Energy Physics - Theory" section of arXiv (http://www.arXiv.org) used by many works [7, 14, 15, 19, 20]. NetHEPT is an undirected network with 15233 nodes and 31376 edges, each node represents an author and each edge represents that two authors collaborated on a paper. The second one is Flixster, an American movie rating social site. Each node represents a user, and directed edge (u, v) means v rated the same movie shortly after u 6 did. We select topic 3 with 29357 nodes and 174939 directed edges here. The last one is the DBLP dataset, which is a larger collaboration network mined from the computer science bibliography site DBLP with 654628 nodes and 1990259 undirected edges [14]. We process its edges in the same way as the NetHEPT dataset. Propagation Models. We adapt general threshold model in this paper. Our Galg-U,Galg-L are designed on submodular upper and lower bounds, respectively. Since directly applying greedy scheme on graphs with submodular threshold function is time-consuming, we assign the submodular threshold function and submodular upper bound of ?-AS function as linear function here: fv (S) = |S|/d(v), where d(v) is the in-degree of v. This makes the corresponding model an instance of the linearthreshold model, and thus the greedy algorithm can be accelerated with Reverse Reachable Set (RRset) technique [17]. We construct two different ?-almost submodular threshold functions in this paper: (1) a power function |S| ? d(v) ? |S| 1 1 with ? satisfying d(v) = d(v) (1 ? ?); (2) fv (S) = d(v) (1 ? ?) for |S| ? 2 and |S|/d(v) otherwise. The former ?-almost submodular function is a supermodular function. The supermodular phenomenon has been observed in Flickr [11]. The second ?-almost submodular function is just dropping down the original threshold function for the first several nodes, which is consistent with the phenomenon observed in LiveJournal [10]. We call them ?-AS-1 and ?-AS-2 functions respectively. Algorithms. We test our approximation Algorithm 1 and other baseline algorithms on the graphs with ?-almost submodular nodes. ? TIM-U, TIM-L: Tang et al. [17] proposed a greedy algorithm TIM+ accelerated with Reverse Reachable Set (RRset). The running time of TIM+ is O(k(m + n) log n) on graphs with n nodes and m edges. RRset can be sampled in live-edge graph of IC model, and with some extension we can sample RRset under Triggering model [1]. LT model also belongs to Triggering model, but General Threshold model with non-submodular threshold functions does not fall into the category of Triggering model. Thus TIM+ can not be directly applied on original graphs with non-submodular nodes. In our experiments, we choose ?-AS-1 and ?-AS-2 thresholds to ensure that TIM+ can run with their upper or lower bound. We then run Algorithm 1 with TIM+ as input. Algorithm Galg-L based on TIM+ can be written in short as TIM-L. By using the upper bound we obtain TIM-U. ? Greedy: We can still apply the naive greedy scheme on graph with ?-almost submodular nodes and generate results without theoretical guarantee. The naive greedy algorithm is time consuming, with running time is O(k(m + n)n). ? High-degree: High-degree outputs seed set according to the decreasing order of the out-degree. ? PageRank: PageRank is widely used to discover nodes with high influence. The insight of PageRank is that important nodes point to important nodes. In this paper, The transition probability on edge e = (u, v) is 1/d(u). We set restart probability as 0.15 and use the power method to compute the PageRank values. Finally PageRank outputs nodes with top PageRank values. ? Random: Random simply selects seeds randomly from node set. Experiment methods. The datasets provide the structure of network, and we first assume each node holds linear threshold function as described above. In each experiment, we randomly sample some nodes with in-degree greater than 2, and assign those nodes with our ?-almost submodular functions, ?-AS-1 or ?-AS-2. Since the naive greedy algorithm is quite time-consuming, we just run it on NetHEPT. 5.2 Experiment results Results on NetHEPT. Our first set of experiments focuses on the NetHEPT dataset with the aim of comparing TIM-U, TIM-L and Greedy. TIM-U, TIM-L have theoretical guarantee, but the approximation ratio is low when the graph contains a considerable number of ?-AS nodes. Figure 2 shows the influence size of each method, varying from 1 to 100 seeds. Figure 2a and 2b are results conducted on constructed graph with ?-AS-1 nodes. Observe that TIM-U, TIM-L slightly outperform Greedy in all cases. Compared with results of 3000 ?-AS nodes, influence of output seeds drops obviously in graph with 10000 ?-AS nodes. But the ratio that TIM-U, TIM-L exceed PageRank 7 (a) 3000 ?-AS-1 nodes (b) 10000 ?-AS-1 nodes (c) 3000 ?-AS-2 nodes (d) 10000 ?-AS-2 nodes Figure 2: Results of IM on NetHEPT with ? = 0.2 increases with rising fraction of ?-AS nodes. In particular, ?-AS-1 is indeed supermodular, TIM-U, TIM-L beats Greedy even when many nodes have supermodular threshold functions. We remark that TIM-U, TIM-L and Greedy outperform other baseline algorithms significantly. When k = 100, TIM-U is 6.1% better compared with PageRank and 27.2% better compared with Highdegree. When conducted with ?-AS-2 function, Figure 2c and 2d report that TIM-U, TIM-L and Greedy still perform extremely well. Influence size conducted on graphs with ?-AS-2 function is better than those with ?-AS-1 function. This is what we expect: supermodular function is harder to handle among the class of ?-almost submodular functions. Another thing to notice is that TIM-U, TIM-L can output seeds on NetHEPT within seconds, while it takes weeks to run the naive greedy algorithm. With RRsets technique, TIM+ dramatically reduces the running time. The ?-almost submodular functions selected here ensure that TIM+ can be invoked. Since TIM-U, TIM-L match the performance of Greedy while TIM-U, TIM-L are scalable, we do not run Greedy in the following larger datasets. Results on Flixster. Figure 3 shows the results of experiments conducted on Flixster with (a) 3000 ?-AS-1 nodes (b) 10000 ?-AS-1 nodes (c) 3000 ?-AS-2 nodes (d) 10000 ?-AS-2 nodes Figure 3: Results of IM on Flixster with ? = 0.2 ? = 0.2. We further evaluate algorithms by Flixster with ? = 0.4 (see Figure 4). Observe that TIM-U, TIM-L outperform other heuristic algorithms in all cases. Compared with PageRank, 30%, 46.3%, 26%, 29.7% improvement are observed in the four experiments in Figure 3. TIM-U performs closely to TIM-L consistently. The improvement is larger than that in NetHEPT. The extra improvement might due to more complex network structure. The average degree is 5.95 in Flixster, compared to 2.05 in NetHEPT. In dense network, nodes may be activated by multiple influence chains, which makes influence propagates further from seeds. Baseline algorithms only pay attention to the structure of the network, hence they are defeated by TIM-U, TIM-L that focus on influence spread. The more ?-AS nodes in network, the more improvement is obtained. When we set ? as 0.4, Figure 4 shows that TIM-U is 37.6%, 74.2%, 28%, 35.6% better than PageRank respectively. Notice that the gap between the performances of TIM-U and PageRank increases as ? increases. In Flixster dataset, we observe that TIM-U,TIM-L hold greater advantage in case of larger number of ?-AS nodes and larger ?. Results on DBLP. For DBLP dataset, the results are shown in Figure 5. TIM-U and TIM-L are still the best algorithms according to performance. But PageRank and High-degree also performs well, just about 2.6% behind TIM-U and TIM-L. DBLP network has many nodes with large degree, which correspond to those active scientists. Once such active authors are activated, the influence will increase significantly. This may partly explain why TIM-U,TIM-L perform similarly to PageRank. 8 (a) 3000 ?-AS-1 nodes (b) 10000 ?-AS-1 nodes (c) 3000 ?-AS-2 nodes (d) 10000 ?-AS-2 nodes Figure 4: Results of IM on Flixster with ? = 0.4 (a) 3000 ?-AS-1 nodes (b) 10000 ?-AS-1 nodes (c) 3000 ?-AS-2 nodes (d) 10000 ?-AS-2 nodes Figure 5: Results of IM on DBLP with ? = 0.2 6 Conclusion and Future Work In this paper, we study the influence maximization problem on propagation models with nonsubmodular threshold functions, which are different from most of existing studies where the threshold functions and the influence spread function are both submodular. We investigate the problem by studying a special case ? the ?-almost submodular threshold function. We first show that influence maximization problem is still hard to approximate even when the number of ?-almost submodular nodes is sub-linear. Next we provide a greedy algorithm based on submodular lower bounds of threshold function to handle the graph with small number of ?-almost submodular nodes and show its theoretical guarantee. We further conduct experiments on real networks and compare our algorithms with other baselines to evaluate our algorithms in practice. Experimental results show that our algorithms not only have good theoretical guarantees on graph with small number of ?-almost submodular nodes, they also perform well on graph with a fairly large fraction of ?-almost submodular nodes. Our study mainly focuses on handling ?-almost submodular threshold functions. One future direction is to investigate models with arbitrary non-submodular threshold functions. Another issue is that the greedy algorithms we propose are slow when the submodular upper bound or lower bound of threshold function do not correspond to the Triggering model. It remains open whether we could utilize RRset or other techniques to accelerate our algorithms under this circumstance. How to accelerate the naive greedy process with arbitrary submodular threshold functions is another interesting direction. Acknowledgments This work was supported in part by the National Natural Science Foundation of China Grant 61433014, 61502449, 61602440, the 973 Program of China Grants No. 2016YFB1000201. 9 References [1] David Kempe, Jon Kleinberg, and ?va Tardos. Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD, pages 137?146. ACM, 2003. [2] Mani Subramani and Balaji Rajagopalan. Knowledge-sharing and influence in online social networks via viral marketing. Communications of The ACM, 46(12):300?307, 2003. [3] Wei Chen, Fu Li, Tian Lin, and Aviad Rubinstein. Combining traditional marketing and viral marketing with amphibious influence maximization. In ACM Conference on Economics and Computation, 2015. [4] Cigdem Aslay, Wei Lu, Francesco Bonchi, Amit Goyal, and Laks V S Lakshmanan. Viral marketing meets social advertising: ad allocation with minimum regret. In Proceedings of The VLDB Endowment, pages 814?825, 2015. [5] Biao Wang, Ge Chen, Luoyi Fu, Li Song, Xinbing Wang, and Xue Liu. Drimux: Dynamic rumor influence minimization with user experience in social networks. In AAAI?16, pages 791?797, 2016. [6] Jure Leskovec, Andreas Krause, Carlos Guestrin, Christos Faloutsos, Jeanne M Vanbriesen, and Natalie Glance. Cost-effective outbreak detection in networks. In ACM Knowledge Discovery and Data Mining, pages 420?429, 2007. [7] Wei Chen, Yajun Wang, and Siyu Yang. Efficient influence maximization in social networks. In Proceedings of the 15th ACM SIGKDD. ACM, 2009. [8] G. Nemhauser, L. Wolsey, and M. Fisher. An analysis of the approximations for maximizing submodular set functions. Mathematical Programming, 14:265?294, 1978. [9] Elchanan Mossel and Sebastien Roch. On the submodularity of influence in social networks. In STOC?07, pages 128?134, 2007. [10] Lars Backstrom, Dan Huttenlocher, Jon Kleinberg, and Xiangyang Lan. Group formation in large social networks: membership, growth, and evolution. In KDD?06, pages 44?54. ACM, 2006. [11] Yang Yang, Jia Jia, Boya Wu, and Jie Tang. Social role-aware emotion contagion in image social networks. In AAAI, pages 65?71, 2016. [12] Ning Chen. On the approximability of influence in social networks. In SODA?08, 2008. [13] Shishir Bharathi, David Kempe, and Mahyar Salek. Competitive influence maximization in social networks. In International Workshop on Web and Internet Economics, pages 306?311. Springer, 2007. [14] Wei Chen, Chi Wang, and Yajun Wang. Scalable influence maximization for prevalent viral marketing in large-scale social networks. In KDD?10, 2010. [15] Amit Goyal, Wei Lu, and Laks V. S. Lakshmanan. SIMPATH: An Efficient Algorithm for Influence Maximization under the Linear Threshold Model. In ICDM?11, pages 211?220, 2011. [16] Christian Borgs, Michael Brautbar, Jennifer Chayes, and Brendan Lucier. Maximizing social influence in nearly optimal time. In SODA?14, pages 946?957. ACM-SIAM, 2014. [17] Youze Tang, Xiaokui Xiao, and Yanchen Shi. Influence maximization: near-optimal time complexity meets practical efficiency. In SIGMOD?14, 2014. [18] Youze Tang, Yanchen Shi, and Xiaokui Xiao. Influence maximization in near-linear time: A martingale approach. In SIGMOD?15, pages 1539?1554. ACM, 2015. [19] H. T. Nguyen, M. T. Thai, and T. N. Dinh. Stop-and-stare: Optimal sampling algorithms for viral marketing in billion-scale networks. In SIGMOD?16, pages 695?710. ACM, 2016. 10 [20] Amit Goyal, Francesco Bonchi, Laks VS Lakshmanan, and Suresh Venkatasubramanian. On minimizing budget and time in influence propagation over social networks. Social Network Analysis and Mining, pages 1?14, 2012. [21] Peng Zhang, Wei Chen, Xiaoming Sun, Yajun Wang, and Jialin Zhang. Minimizing seed set selection with probabilistic coverage guarantee in a social network. In KDD?14, pages 1306?1315, 2014. [22] Golnaz Ghasemiesfeh, Roozbeh Ebrahimi, and Jie Gao. Complex contagion and the weakness of long ties in social networks: revisited. In ACM Conference on Electronic Commerce, 2013. [23] Roozbeh Ebrahimi, Jie Gao, Golnaz Ghasemiesfeh, and Grant Schoenebeck. Complex contagions in kleinberg?s small world model. In ITCS?15, 2015. [24] Wei Chen, Qiang Li, Xiaoming Sun, and Jialin Zhang. The routing of complex contagion in kleinberg?s small-world networks. In International Computing and Combinatorics Conference, pages 307?318, 2016. [25] Jie Gao, Golnaz Ghasemiesfeh, Grant Schoenebeck, and Fang-Yi Yu. General threshold model for social cascades: Analysis and simulations. In ACM Conference on Economics and Computation, 2016. [26] Ding-Zhu Du, Ronald L Graham, Panos M Pardalos, Peng-Jun Wan, Weili Wu, and Wenbo Zhao. Analysis of greedy approximations with nonsubmodular potential functions. In SODA?08, pages 167?175, 2008. [27] Thibaut Horel and Yaron Singer. Maximization of approximately submodular functions. In NIPS?16, pages 3045?3053, 2016. [28] Wei Lu, Wei Chen, and Laks VS Lakshmanan. From competition to complementarity: comparative influence diffusion and maximization. Proceedings of the VLDB Endowment, 9(2):60?71, 2015. 11
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InfoGAIL: Interpretable Imitation Learning from Visual Demonstrations Yunzhu Li MIT [email protected] Jiaming Song Stanford University [email protected] Stefano Ermon Stanford University [email protected] Abstract The goal of imitation learning is to mimic expert behavior without access to an explicit reward signal. Expert demonstrations provided by humans, however, often show significant variability due to latent factors that are typically not explicitly modeled. In this paper, we propose a new algorithm that can infer the latent structure of expert demonstrations in an unsupervised way. Our method, built on top of Generative Adversarial Imitation Learning, can not only imitate complex behaviors, but also learn interpretable and meaningful representations of complex behavioral data, including visual demonstrations. In the driving domain, we show that a model learned from human demonstrations is able to both accurately reproduce a variety of behaviors and accurately anticipate human actions using raw visual inputs. Compared with various baselines, our method can better capture the latent structure underlying expert demonstrations, often recovering semantically meaningful factors of variation in the data. 1 Introduction A key limitation of reinforcement learning (RL) is that it involves the optimization of a predefined reward function or reinforcement signal [1?6]. Explicitly defining a reward function is straightforward in some cases, e.g., in games such as Go or chess. However, designing an appropriate reward function can be difficult in more complex and less well-specified environments, e.g., for autonomous driving where there is a need to balance safety, comfort, and efficiency. Imitation learning methods have the potential to close this gap by learning how to perform tasks directly from expert demonstrations, and has succeeded in a wide range of problems [7?11]. Among them, Generative Adversarial Imitation Learning (GAIL, [12]) is a model-free imitation learning method that is highly effective and scales to relatively high dimensional environments. The training process of GAIL can be thought of as building a generative model, which is a stochastic policy that when coupled with a fixed simulation environment, produces similar behaviors to the expert demonstrations. Similarity is achieved by jointly training a discriminator to distinguish expert trajectories from ones produced by the learned policy, as in GANs [13]. In imitation learning, example demonstrations are typically provided by human experts. These demonstrations can show significant variability. For example, they might be collected from multiple experts, each employing a different policy. External latent factors of variation that are not explicitly captured by the simulation environment can also significantly affect the observed behavior. For example, expert demonstrations might be collected from users with different skills and habits. The goal of this paper is to develop an imitation learning framework that is able to automatically discover and disentangle the latent factors of variation underlying expert demonstrations. Analogous to the goal of uncovering style, shape, and color in generative modeling of images [14], we aim to automatically learn similar interpretable concepts from human demonstrations through an unsupervised manner. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. We propose a new method for learning a latent variable generative model that can produce trajectories in a dynamic environment, i.e., sequences of state-actions pairs in a Markov Decision Process. Not only can the model accurately reproduce expert behavior, but also empirically learns a latent space of the observations that is semantically meaningful. Our approach is an extension of GAIL, where the objective is augmented with a mutual information term between the latent variables and the observed state-action pairs. We first illustrate the core concepts in a synthetic 2D example and then demonstrate an application in autonomous driving, where we learn to imitate complex driving behaviors while recovering semantically meaningful structure, without any supervision beyond the expert trajectories. 1 Remarkably, our method performs directly on raw visual inputs, using raw pixels as the only source of perceptual information. The code for reproducing the experiments are available at https://github.com/ermongroup/InfoGAIL. In particular, the contributions of this paper are threefold: 1. We extend GAIL with a component which approximately maximizes the mutual information between latent space and trajectories, similar to InfoGAN [14], resulting in a policy where low-level actions can be controlled through more abstract, high-level latent variables. 2. We extend GAIL to use raw pixels as input and produce human-like behaviors in complex high-dimensional dynamic environments. 3. We demonstrate an application to autonomous highway driving using the TORCS driving simulator [15]. We first demonstrate that the learned policy is able to correctly navigate the track without collisions. Then, we show that our model learns to reproduce different kinds of human-like driving behaviors by exploring the latent variable space. 2 Background 2.1 Preliminaries We use the tuple (S, A, P, r , ?0 , ) to define an infinite-horizon, discounted Markov decision process (MDP), where S represents the state space, A represents the action space, P : S ?A?S ! R denotes the transition probability distribution, r : S ! R denotes the reward function, ?0 : S ! R is the distribution of the initial state s0 , and 2 (0, 1) is the discount factor. Let ? denote a stochastic policy ? : S ? A ! [0, 1], and ?E denote the expert policy to which we only have access to demonstrations. The expert demonstrations ?E are a set of trajectories generated using policy ?E , each of which consists of a sequence of state-action pairs. We use an expectation with respect P1to a policy ? to denote an expectation with respect to the trajectories it generates: E? [f (s, a)] , E[ t=0 t f (st , at )], where s0 ? ?0 , at ? ?(at |st ), st+1 ? P (st+1 |at , st ). 2.2 Imitation learning The goal of imitation learning is to learn how to perform a task directly from expert demonstrations, without any access to the reinforcement signal r. Typically, there are two approaches to imitation learning: 1) behavior cloning (BC), which learns a policy through supervised learning over the stateaction pairs from the expert trajectories [16]; and 2) apprenticeship learning (AL), which assumes the expert policy is optimal under some unknown reward and learns a policy by recovering the reward and solving the corresponding planning problem. BC tends to have poor generalization properties due to compounding errors and covariate shift [17, 18]. AL, on the other hand, has the advantage of learning a reward function that can be used to score trajectories [19?21], but is typically expensive to run because it requires solving a reinforcement learning (RL) problem inside a learning loop. 2.3 Generative Adversarial Imitation Learning Recent work on AL has adopted a different approach by learning a policy without directly estimating the corresponding reward function. In particular, Generative Adversarial Imitation Learning (GAIL, [12]) is a recent AL method inspired by Generative Adversarial Networks (GAN, [13]). In the GAIL framework, the agent imitates the behavior of an expert policy ?E by matching the generated state-action distribution with the expert?s distribution, where the optimum is achieved when the 1 A video showing the experimental results is available at https://youtu.be/YtNPBAW6h5k. 2 distance between these two distributions is minimized as measured by Jensen-Shannon divergence. The formal GAIL objective is denoted as min ? max D2(0,1)S?A E? [log D(s, a)] + E?E [log(1 D(s, a))] H(?) (1) where ? is the policy that we wish to imitate ?E with, D is a discriminative classifier which tries to distinguish state-action pairs from the trajectories generated by ? and ?E , and H(?) , E? [ log ?(a|s)] is the -discounted causal entropy of the policy ?? [22]. Instead of directly learning a reward function, GAIL relies on the discriminator to guide ? into imitating the expert policy. GAIL is model-free: it requires interaction with the environment to generate rollouts, but it does not need to construct a model for the environment. Unlike GANs, GAIL considers the environment/simulator as a black box, and thus the objective is not differentiable end-to-end. Hence, optimization of GAIL objective requires RL techniques based on Monte-Carlo estimation of policy gradients. Optimization over the GAIL objective is performed by alternating between a gradient step to increase (1) with respect to the discriminator parameters, and a Trust Region Policy Optimization (TRPO, [2]) step to decrease (1) with respect to ?. 3 Interpretable Imitation Learning through Visual Inputs Demonstrations are typically collected from human experts. The resulting trajectories can show significant variability among different individuals due to internal latent factors of variation, such as levels of expertise and preferences for different strategies. Even the same individual might make different decisions while encountering the same situation, potentially resulting in demonstrations generated from multiple near-optimal but distinct policies. In this section, we propose an approach that can 1) discover and disentangle salient latent factors of variation underlying expert demonstrations without supervision, 2) learn policies that produce trajectories which correspond to these latent factors, and 3) use visual inputs as the only external perceptual information. 0 1 Formally, we assume that the expert policy is a mixture of experts ?E = {?E , ?E , . . . }, and we define the generative process of the expert trajectory ?E as: s0 ? ?0 , c ? p(c), ? ? p(?|c), at ? ?(at |st ), st+1 ? P (st+1 |at , st ), where c is a discrete latent variable that selects a specific policy ? from the mixture of expert policies through p(?|c) (which is unknown and needs to be learned), and p(c) is the prior distribution of c (which is assumed to be known before training). Similar to the GAIL setting, we consider the apprenticeship learning problem as the dual of an occupancy measure matching problem, and treat the trajectory ?E as a set of state-action pairs. Instead of learning a policy solely based on the current state, we extend it to include an explicit dependence on the latent variable c. The objective is to recover a policy ?(a|s, c) as an approximation of ?E ; when c is samples from the prior p(c), the trajectories ? generated by the conditional policy ?(a|s, c) should be similar to the expert trajectories ?E , as measured by a discriminative classifier. 3.1 Interpretable Imitation Learning Learning from demonstrations generated by a mixture of experts is challenging as we have no access to the policies employed by the individual experts. We have to proceed in an unsupervised way, similar to clustering. The original Generative Adversarial Imitation Learning method would fail as it assumes all the demonstrations come from a single expert, and there is no incentive in separating and disentangling variations observed in the data. A method that can automatically disentangle the demonstrations in a meaningful way is thus needed. The way we address this problem is to introduce a latent variable c into our policy function, ?(a|s, c). Without further constraints over c, applying GAIL directly to this ?(a|s, c) could simply ignore c and fail to separate different types of behaviors present in the expert trajectories 2 . To incentivize the model to use c as much as possible, we utilize an information-theoretic regularization enforcing that there should be high mutual information between c and the state-action pairs in the generated trajectory. This concept was introduced by InfoGAN [14], where latent codes are utilized to discover the salient semantic features of the data distribution and guide the generating process. In particular, the regularization seeks to maximize the mutual information between latent codes and trajectories, 2 For a fair comparison, we consider this form as our GAIL baseline in the experiments below. 3 denoted as I(c; ? ),which is hard to maximize directly as it requires access to the posterior P (c|? ). Hence we introduce a variational lower bound, LI (?, Q), of the mutual information I(c; ? )3 : LI (?, Q) = Ec?p(c),a??(?|s,c) [log Q(c|? )] + H(c) (2) ? I(c; ? ) where Q(c|? ) is an approximation of the true posterior P (c|? ). The objective under this regularization, which we call Information Maximizing Generative Adversarial Imitation Learning (InfoGAIL), then becomes: min max E? [log D(s, a)] + E?E [log(1 ?,Q D D(s, a))] 1 LI (?, Q) 2 H(?) (3) where 1 > 0 is the hyperparameter for information maximization regularization term, and 2 > 0 is the hyperparameter for the casual entropy term. By introducing the latent code, InfoGAIL is able to identify the salient factors in the expert trajectories through mutual information maximization, and imitate the corresponding expert policy through generative adversarial training. This allows us to disentangle trajectories that may arise from a mixture of experts, such as different individuals performing the same task. To optimize the objective, we use a simplified posterior approximation Q(c|s, a), since directly working with entire trajectories ? would be too expensive, especially when the dimension of the observations is very high (such as images). We then parameterize policy ?, discriminator D and posterior approximation Q with weights ?, ! and respectively. We optimize LI (?? , Q ) with stochastic gradient methods, ?? using TRPO [2], and Q is updated using the Adam optimizer [23]. An outline for the optimization procedure is shown in Algorithm 1. Algorithm 1 InfoGAIL Input: Initial parameters of policy, discriminator and posterior approximation ?0 , !0 , trajectories ?E ? ?E containing state-action pairs. Output: Learned policy ?? for i = 0, 1, 2, ... do Sample a batch of latent codes: ci ? p(c) Sample trajectories: ?i ? ??i (ci ), with the latent code fixed during each rollout. Sample state-action pairs i ? ?i and E ? ?E with same batch size. Update !i to !i+1 by ascending with gradients ? ? D!i (s, a))] !i = E i [r!i log D!i (s, a)] + E E [r!i log(1 Update i to i+1 0; expert by descending with gradients ? = 1 E i [r i log Q i (c|s, a)] i Take a policy step from ?i to ?i+1 , using the TRPO update rule with the following objective: ? [log D! (s, a)] E 1 LI (??i , Q i+1 ) 2 H(??i ) i i+1 end for 3.2 Reward Augmentation In complex and less well-specified environments, imitation learning methods have the potential to perform better than reinforcement learning methods as they do not require manual specification of an appropriate reward function. However, if the expert is performing sub-optimally, then any policy trained under the recovered rewards will be also suboptimal; in other words, the imitation learning agent?s potential is bounded by the capabilities of the expert that produced the training data. In many cases, while it is very difficult to fully specify a suitable reward function for a given task, it is relatively straightforward to come up with constraints that we would like to enforce over the policy. This motivates the introduction of reward augmentation [8], a general framework to incorporate prior knowledge in imitation learning by providing additional incentives to the agent without interfering 3 [14] presents a proof for the lower bound. 4 with the imitation learning process. We achieve this by specifying a surrogate state-based reward ?(?? ) = Es??? [r(s)] that reflects our bias over the desired agent?s behavior: min max E?? [log D! (s, a)] + E?E [log(1 ?, ! D! (s, a))] 0 ?(?? ) 1 LI (?? , Q ) 2 H(?? ) (4) where 0 > 0 is a hyperparameter. This approach can be seen as a hybrid between imitation and reinforcement learning, where part of the reinforcement signal for the policy optimization is coming from the surrogate reward and part from the discriminator, i.e., from mimicking the expert. For example, in our autonomous driving experiment below we show that by providing the agent with a penalty if it collides with other cars or drives off the road, we are able to significantly improve the average rollout distance of the learned policy. 3.3 Improved Optimization While GAIL is successful in tasks with low-dimensional inputs (in [12], the largest observation has 376 continuous variables), few have explored tasks where the input dimension is very high (such as images - 110 ? 200 ? 3 pixels as in our driving experiments). In order to effectively learn a policy that relies solely on high-dimensional input, we make the following improvements over the original GAIL framework. It is well known that the traditional GAN objective suffers from vanishing gradient and mode collapse problems [24, 25]. We propose to use the Wasserstein GAN (WGAN [26]) technique to alleviate these problems and augment our objective function as follows: min max E?? [D! (s, a)] ?, ! E?E [D! (s, a)] 0 ?(?? ) 1 LI (?? , Q ) 2 H(?? ) (5) We note that this modification is especially important in our setting, where we want to model complex distributions over trajectories that can potentially have a large number of modes. We also use several variance reduction techniques, including baselines [27] and replay buffers [28]. Besides the baseline, we have three models to update in the InfoGAIL framework, which are represented as neural networks: the discriminator network D! (s, a), the policy network ?? (a|s, c), and the posterior estimation network Q (c|s, a). We update D! using RMSprop (as suggested in the original WGAN paper), and update Q and ?? using Adam and TRPO respectively. We include the detailed training procedure in Appendix C. To speed up training, we initialize our policy from behavior cloning, as in [12]. Note that the discriminator network D! and the posterior approximation network Q are treated as distinct networks, as opposed to the InfoGAN approach where they share the same network parameters until the final output layer. This is because the current WGAN training framework requires weight clipping and momentum-free optimization methods when training D! . These changes would interfere with the training of an expressive Q if D! and Q share the same network parameters. 4 Experiments We demonstrate the performance of our method by applying it first to a synthetic 2D example and then in a challenging driving domain where the agent is imitating driving behaviors from visual inputs. By conducting experiments on these two environments, we show that our learned policy ?? can 1) imitate expert behaviors using high-dimensional inputs with only a small number of expert demonstrations, 2) cluster expert behaviors into different and semantically meaningful categories, and 3) reproduce different categories of behaviors by setting the high-level latent variables appropriately. The driving experiments are conducted in the TORCS (The Open Source Racing Car Simulator, [15]) environment. The demonstrations are collected by manually driving along the race track, and show typical behaviors like staying within lanes, avoiding collisions and surpassing other cars. The policy accepts raw visual inputs as the only external inputs for the state, and produces a three-dimensional continuous action that consists of steering, acceleration, and braking. We assume that our policies are Gaussian distributions with fixed standard deviations, thus H(?) is constant. 5 (a) Expert (b) Behavior cloning (c) GAIL (d) Ours Figure 1: Learned trajectories in the synthetic 2D plane environment. Each color denotes one specific latent code. Behavior cloning deviates from the expert demonstrations due to compounding errors. GAIL does produce circular trajectories but fails to capture the latent structure for it assumes that the demonstrations are generated from a single expert, and tries to learn an average policy. Our method (InfoGAIL) successfully distinguishes expert behaviors and imitates each mode accordingly (colors are ordered in accordance to the expert for visualization purposes, but are not identifiable). 4.1 Learning to Distinguish Trajectories We demonstrate the effectiveness of InfoGAIL on a synthetic example. The environment is a 2D plane where the agent can move around freely at a constant velocity by selecting its direction pt at (discrete) time t. For the agent, the observations at time t are positions from t 4 to t. The (unlabeled) expert demonstrations contain three distinct modes, each generated with a stochastic expert policy that produces a circle-like trajectory (see Figure 1, panel a). The objective is to distinguish these three distinct modes and imitate the corresponding expert behavior. We consider three methods: behavior cloning, GAIL and InfoGAIL (details included in Appendix A). In particular, for all the experiments we assume the same architecture and that the latent code is a one-hot encoded vector with 3 dimensions and a uniform prior; only InfoGAIL regularizes the latent code. Figure 1 shows that the introduction of latent variables allows InfoGAIL to distinguish the three types of behavior and imitate each behavior successfully; the other two methods, however, fail to distinguish distinct modes. BC suffers from the compounding error problem and the learned policy tends to deviate from the expert trajectories; GAIL does learn to generate circular trajectories but it fails to separate different modes due to the lack of a mechanism that can explicitly account for the underlying structure. In the rest of Section 4, we show how InfoGAIL can infer the latent structure of human decisionmaking in a driving domain. In particular, our agent only relies on visual inputs to sense the environment. 4.2 Utilizing Raw Visual Inputs via Transfer Learning The high dimensional nature of visual inputs poses a significant challenges to learning a policy. Intuitively, the policy will have to simultaneously learn how to identify meaningful visual features, and how to leverage them to achieve the desired behavior using only a small number of expert demonstrations. Therefore, methods to mitigate the high sample complexity of the problem are crucial to success in this domain. In this paper, we take a transfer learning approach. Features extracted using a CNN pre-trained on ImageNet contain high-level information about the input images, which can be adapted to new vision tasks via transfer learning [29]. However, it is not yet clear whether these relatively high-level features can be directly applied to tasks where perception and action are tightly interconnected; we demonstrate that this is possible through our experiments. We perform transfer learning by exploiting features from a pre-trained neural network that effectively convert raw images into relatively highlevel information [30]. In particular, we use a Deep Residual Network [31] pre-trained on the ImageNet classification task [32] to obtain the visual features used as inputs for the policy network. 4.3 Network Structure Our policy accepts certain auxiliary information as internal input to serve as a short-term memory. This auxiliary information can be accessed along with the raw visual inputs. In our experiments, the auxiliary information for the policy at time t consists of the following: 1) velocity at time t, which is a three dimensional vector; 2) actions at time t 1 and t 2, which are both three dimensional vectors; 3) damage of the car, which is a real value. The auxiliary input has 10 dimensions in total. 6 Figure 2: Visualizing the training process of turn. Here we show the trajectories of InfoGAIL at different stages of training. Blue and red indicate policies under different latent codes, which correspond to ?turning from inner lane? and ?turning from outer lane? respectively. The rightmost figure shows the trajectories under latent codes [1, 0] (red), [0, 1] (blue), and [0.5, 0.5] (purple), which suggests that, to some extent, our method is able to generalize to cases previously unseen in the training data. For the policy network, input visual features are passed through two convolutional layers, and then combined with the auxiliary information vector and (in the case of InfoGAIL) the latent code c. We parameterize the baseline as a network with the same architecture except for the final layer, which is just a scalar output that indicates the expected accumulated future rewards. The discriminator D! accepts three elements as input: the input image, the auxiliary information, and the current action. The output is a score for the WGAN training objective, which is supposed to be lower for expert state-action pairs, and higher for generated ones. The posterior approximation network Q adopts the same architecture as the discriminator, except that the output is a softmax over the discrete latent variables or a factored Gaussian over continuous latent variables. We include details of our architecture in Appendix B. 4.4 Interpretable Imitation Learning from Visual Demonstrations In this experiment, we consider two subsets of human driving behaviors: turn, where the expert takes a turn using either the inside lane or the outside lane; and pass, where the expert passes another vehicle from either the left or the right. In both cases, the expert policy has two significant modes. Our goal is to have InfoGAIL capture these two separate modes from expert demonstrations in an unsupervised way. We use a discrete latent code, which is a one-hot encoded vector with two possible states. For both settings, there are 80 expert trajectories in total, with 100 frames in each trajectory; our prior for the latent code is a uniform discrete distribution over the two states. The performance of a learned policy is quantified with two metrics: the average distance is determined by the distance traveled by the agent before a collision (and is bounded by the length of the simulation horizon), and accuracy is defined as the classification accuracy of the expert state-action pairs according to the latent code inferred with Q . We add constant reward at every time step as reward augmentation, which is used to encourage the car to "stay alive" as long as possible and can be regarded as another way of reducing collision and off-lane driving (as these will lead to the termination of that episode). The average distance and sampled trajectories at different stages of training are shown in Figures 2 and 3 for turn and pass respectively. During the initial stages of training, the model does not distinguish the two modes and has a high chance of colliding and driving off-lane, due to the limitations of behavior cloning (which we used to initialize the policy). As training progresses, trajectories provided by the learned policy begin to diverge. Towards the end of training, the two types of trajectories are clearly distinguishable, with only a few exceptions. In turn, [0, 1] corresponds to using the inside lane, while [1, 0] corresponds to the outside lane. In pass, the two kinds of latent codes correspond to passing from right and left respectively. Meanwhile, the average distance of the rollouts steadily increases with more training. Learning the two modes separately requires accurate inference of the latent code. To examine the accuracy of posterior inference, we select state-action pairs from the expert trajectories (where the state is represented as a concatenation of raw image and auxiliary variables) and obtain the corresponding latent code through Q (c|s, a); see Table 1. Although we did not explicitly provide any label, our model is able to correctly distinguish over 81% of the state-action pairs in pass (and almost all the pairs in turn, confirming the clear separation between generated trajectories with different latent codes in Figure 2). 7 Figure 3: Experimental results for pass. Left: Trajectories of InfoGAIL at different stages of training (epoch 1 to 37). Blue and red indicate policies using different latent code values, which correspond to passing from right or left. Middle: Traveled distance denotes the absolute distance from the start position, averaged over 60 rollouts of the InfoGAIL policy trained at different epochs. Right: Trajectories of pass produced by an agent trained on the original GAIL objective. Compared to InfoGAIL, GAIL fails to distinguish between different modes. Table 1: Classification accuracies for pass. Method Accuracy Chance K-means PCA InfoGAIL (Ours) 50% 55.4% 61.7% 81.9% SVM CNN 85.8% 90.8% Table 2: Average rollout distances. Method Behavior Cloning GAIL InfoGAIL \ RB InfoGAIL \ RA InfoGAIL \ WGAN InfoGAIL (Ours) Human Avg. rollout distance 701.83 914.45 1031.13 1123.89 1177.72 1226.68 1203.51 For comparison, we also visualize the trajectories of pass for the original GAIL objective in Figure 3, where there is no mutual information regularization. GAIL learns the expert trajectories as a whole, and cannot distinguish the two modes in the expert policy. Interestingly, instead of learning two separate trajectories, GAIL tries to fit the left trajectory by swinging the car suddenly to the left after it has surpassed the other car from the right. We believe this reflects a limitation in the discriminators. Since D! (s, a) only requires state-action pairs as input, the policy is only required to match most of the state-action pairs; matching each rollout in a whole with expert trajectories is not necessary. InfoGAIL with discrete latent codes can alleviate this problem by forcing the model to learn separate trajectories. 4.5 Ablation Experiments We conduct a series of ablation experiments to demonstrate that our proposed improved optimization techniques in Section 3.2 and 3.3 are indeed crucial for learning an effective policy. Our policy drives a car on the race track along with other cars, whereas the human expert provides 20 trajectories with 500 frames each by trying to drive as fast as possible without collision. Reward augmentation is performed by adding a reward that encourages the car to drive faster. The performance of the policy is determined by the average distance. Here a longer average rollout distance indicates a better policy. In our ablation experiments, we selectively remove some of the improved optimization methods from Section 3.2 and 3.3 (we do not use any latent code in these experiments). InfoGAIL(Ours) includes all the optimization techniques; GAIL excludes all the techniques; InfoGAIL\WGAN switches the WGAN objective with the GAN objective; InfoGAIL\RA removes reward augmentation; InfoGAIL\RB removes the replay buffer and only samples from the most recent rollouts; Behavior Cloning is the behavior cloning method and Human is the expert policy. Table 2 shows the average rollout distances of different policies. Our method is able to outperform the expert with the help of reward augmentation; policies without reward augmentation or WGANs perform slightly worse than the expert; removing the replay buffer causes the performance to deteriorate significantly due to increased variance in gradient estimation. 8 5 Related work There are two major paradigms for vision-based driving systems [33]. Mediated perception is a two-step approach that first obtains scene information and then makes a driving decision [34?36]; behavior reflex, on the other hand, adopts a direct approach by mapping visual inputs to driving actions [37, 16]. Many of the current autonomous driving methods rely on the two-step approach, which requires hand-crafting features such as the detection of lane markings and cars [38, 33]. Our approach, on the other hand, attempts to learn these features directly from vision to actions. While mediated perception approaches are currently more prevalent, we believe that end-to-end learning methods are more scalable and may lead to better performance in the long run. [39] introduce an end-to-end imitation learning framework that learns to drive entirely from visual information, and test their approach on real-world scenarios. However, their method uses behavior cloning by performing supervised learning over the state-action pairs, which is well-known to generalize poorly to more sophisticated tasks, such as changing lanes or passing vehicles. With the use of GAIL, our method can learn to perform these sophisticated operations easily. [40] performs end-to-end visual imitation learning in TORCS through DAgger [18], querying the reference policies during training, which in many cases is difficult. Most imitation learning methods for end-to-end driving rely heavily on LIDAR-like inputs to obtain precise distance measurements [21, 41]. These inputs are not usually available to humans during driving. In particular, [41] applies GAIL to the task of modeling human driving behavior on highways. In contrast, our policy requires only raw visual information as external input, which in practice is all the information humans need in order to drive. [42] and [9] have also introduced a pre-trained deep neural network to achieve better performance in imitation learning with relatively few demonstrations. Specifically, they introduce a pre-trained model to learn dense, incremental reward functions that are suitable for performing downstream reinforcement learning tasks, such as real-world robotic experiments. This is different from our approach, in that transfer learning is performed over the critic instead of the policy. It would be interesting to combine that reward with our approach through reward augmentation. 6 Conclusion In this paper, we present a method to imitate complex behaviors while identifying salient latent factors of variation in the demonstrations. Discovering these latent factors does not require direct supervision beyond expert demonstrations, and the whole process can be trained directly with standard policy optimization algorithms. We also introduce several techniques to successfully perform imitation learning using visual inputs, including transfer learning and reward augmentation. Our experimental results in the TORCS simulator show that our methods can automatically distinguish certain behaviors in human driving, while learning a policy that can imitate and even outperform the human experts using visual information as the sole external input. We hope that our work can further inspire end-to-end learning approaches to autonomous driving under more realistic scenarios. Acknowledgements We thank Shengjia Zhao and Neal Jean for their assistance and advice. Toyota Research Institute (TRI) provided funds to assist the authors with their research but this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. This research was also supported by Intel Corporation, FLI and NSF grants 1651565, 1522054, 1733686. References [1] S. Levine and V. Koltun, ?Guided policy search.,? in ICML (3), pp. 1?9, 2013. [2] J. Schulman, S. Levine, P. Abbeel, M. I. Jordan, and P. Moritz, ?Trust region policy optimization.,? in ICML, pp. 1889?1897, 2015. [3] T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra, ?Continuous control with deep reinforcement learning,? arXiv preprint arXiv:1509.02971, 2015. 9 [4] J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. 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Variational Laws of Visual Attention for Dynamic Scenes Dario Zanca DINFO, University of Florence DIISM, University of Siena [email protected] Marco Gori DIISM, University of Siena [email protected] Abstract Computational models of visual attention are at the crossroad of disciplines like cognitive science, computational neuroscience, and computer vision. This paper proposes a model of attentional scanpath that is based on the principle that there are foundational laws that drive the emergence of visual attention. We devise variational laws of the eye-movement that rely on a generalized view of the Least Action Principle in physics. The potential energy captures details as well as peripheral visual features, while the kinetic energy corresponds with the classic interpretation in analytic mechanics. In addition, the Lagrangian contains a brightness invariance term, which characterizes significantly the scanpath trajectories. We obtain differential equations of visual attention as the stationary point of the generalized action, and we propose an algorithm to estimate the model parameters. Finally, we report experimental results to validate the model in tasks of saliency detection. 1 Introduction Eye movements in humans constitute an essential mechanism to disentangle the tremendous amount of information that reaches the retina every second. This mechanism in adults is very sophisticated. In fact, it involves both bottom-up processes, which depend on raw input features, and top-down processes, which include task dependent strategies [2; 3; 4]. It turns out that visual attention is interwound with high level cognitive processes, so as its deep understanding seems to be trapped into a sort of eggs-chicken dilemma. Does visual scene interpretation drive visual attention or the other way around? Which one ?was born? first? Interestingly, this dilemma seems to disappears in newborns: despite their lack of knowledge of the world, they exhibit mechanisms of attention to extract relevant information from what they see [5]. Moreover, there are evidences that the very first fixations are highly correlated among adult subjects who are presented with a new input [25]. This shows that they still share a common mechanism that drive early fixations, while scanpaths diverge later under top-down influences. Many attempts have been made in the direction of modeling visual attention. Based on the feature integration theory of attention [14], Koch and Ullman in [9] assume that human attention operates in the early representation, which is basically a set of feature maps. They assume that these maps are then combined in a central representation, namely the saliency map, which drives the attention mechanisms. The first complete implementation of this scheme was proposed by Itti et al. in [10]. In that paper, feature maps for color, intensity and orientation are extracted at different scales. Then center-surround differences and normalization are computed for each pixel. Finally, all this information is combined linearly in a centralized saliency map. Several other models have been proposed by the computer vision community, in particular to address the problem of refining saliency maps estimation. They usually differ in the definition of saliency, while they postulate a centralized control of the attention mechanism through the saliency map. For instance, it has been claimed that 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. the attention is driven according to a principle of information maximization [16] or by an opportune selection of surprising regions [17]. A detailed description of the state of the art is given in [8]. Machine learning approaches have been used to learn models of saliency. Judd et al. [1] collected 1003 images observed by 15 subjects and trained an SVM classifier with low-, middle-, and high-level features. More recently, automatic feature extraction methods with convolutional neural networks achieved top level performance on saliency estimation [26; 18]. Most of the referred papers share the idea that saliency is the product of a global computation. Some authors also provide scanpaths of image exploration, but to simulate them over the image, they all use the procedure defined by [9]. The winner-take-all algorithm is used to select the most salient location for the first fixation. Then three rules are introduced to select the next location: inhibition-of-return, similarity preference, and proximity preference. An attempt of introducing biological biases has been made by [6] to achieve more realistic saccades and improve performance. In this paper, we present a novel paradigm in which visual attention emerges from a few unifying functional principles. In particular, we assume that attention is driven by the curiosity for regions with many details, and by the need to achieve brightness invariance, which leads to fixation and motion tracking. These principles are given a mathematical expression by a variational approach based on a generalization of least action, whose stationary point leads to the correspondent Euler-Lagrange differential equations of the focus of attention. The theory herein proposed offers an intriguing model for capturing a mechanisms behind saccadic eye movements, as well as object tracking within the same framework. In order to compare our results with the state of the art in the literature, we have also computed the saliency map by counting the visits in each pixel over a given time window, both on static and dynamic scenes. It is worth mentioning that while many papers rely on models that are purposely designed to optimize the approximation of the saliency map, for the proposed approach such a computation is obtained as a byproduct of a model of scanpath. The paper is organized as follows. Section 2 provides a mathematical description of the model and the Euler-Lagrange equations of motion that describe attention dynamics. The technical details, including formal derivation of the motion equations, are postponed to the Appendix. In the Section 3 we describe the experimental setup and show performance of the model in a task of saliency detection on two popular dataset of images [12; 11] and one dataset of videos [27]. Some conclusions and critical analysis are finally drawn in Section 4. 2 The model In this section, we propose a model of visual attention that takes place in the earliest stage of vision, which we assume to be completely data driven. We begin discussing the driving principles. 2.1 Principles of visual attention The brightness signal b(t, x) can be thought of as a real-valued function b : R + ? R2 ? R (1) where t is the time and x = (x1 , x2 ) denotes the position. The scanpath over the visual input is defined as x : R + ? R2 (2) The scanpath x(t) will be also referred to as trajectory or observation. Three fundamental principles drive the model of attention. They lead to the introduction of the correspondent terms of the Lagrangian of the action. i) Boundedness of the trajectory Trajectory x(t) is bounded into a defined area (retina). This is modeled by a harmonic oscillator at the borders of the image which constraints the motion within the retina1 : X  2 2 V (x) = k (li ? xi ) ? [xi > li ] + (xi ) ? [xi < 0] (3) i=1,2 1 Here, we use Iverson?s notation, according to which if p is a proposition then [p] = 1 if p=true and [p] = 0 otherwise 2 where k is the elastic constant, li is the i-th dimension of the rectangle which represents the retina2 . ii) Curiosity driven principle Visual attention is attracted by regions with many details, that is where the magnitude of the gradient of the brightness is high. In addition to this local field, the role of peripheral information is included by processing a blurred version p(t, x) of the brightness b(t, x). The modulation of these two terms is given by C(t, x) = b2x cos2 (?t) + p2x sin2 (?t), (4) where bx and px denote the gradient w.r.t. x. Notice that the alternation of the local and peripheral fields has a fundamental role in avoiding trapping into regions with too many details. iii) brightness invariance Trajectories that exhibit brightness invariance are motivated by the need to perform fixation. Formally, we impose the constraint b? = bt + bx x? = 0. This is in fact the classic constraint that is widely used in computer vision for the estimation of the optical flow [20]. Its soft-satisfaction can be expressed by the associated term 2 B(t, x, x) ? = bt + bx x? . (5) Notice that, in the case of static images, bt = 0, and the term is fully satisfied for trajectory x(t) whose velocity x? is perpendicular to the gradient, i.e.when the focus is on the borders of the objects. This kind of behavior favors coherent fixation of objects. Interestingly, in case of static images, the model can conveniently be simplified by using the upper bound of the brightness as follows: B(t, x, x) ? = b? 2 (t, x) = (?bt + bx x) ? 2? ? x, x) ? 2b2t + 2b2x x? 2 := B(t, ? (6) This inequality comes from the parallelogram law of Hilbert spaces. As it will be seen the rest of the paper, this approximation significantly simplifies the motion equations. 2.2 Least Action Principle Visual attention scanpaths are modeled as the motion of a particle of mass m within a potential field. This makes it possible to construct the generalized action Z T S= L(t, x, x) ? dt (7) 0 where L = K ? U , where K is the kinetic energy K(x) ? = 1 mx? 2 2 (8) and U is a generalized potential energy defined as U (t, x, x) ? = V (x) ? ?C(t, x) + ?B(t, x, x). ? (9) Here, we assume that ?, ? > 0. Notice, in passing that while V and B get the usual sign of potentials, C comes with the flipped sign. This is due to the fact that, whenever it is large, it generates an attractive field. In addition, we notice that the brightness invariance term is not a truly potential, since it depends on both the position and the velocity. However, its generalized interpretation as a ?potential? comes from considering that it generates a force field. In order to discover the trajectory we look for a stationary point of the action in Eq. (7), which corresponds to the Euler-Lagrange equations d ?L ?L = , (10) dt ? x? i ?xi 2 A straightforward extension can be given for circular retina. 3 where i = 1, 2 for the two motion coordinates. The right-hand term in (10) can be written as ?L = ?Cx ? Vx ? ?Bx . ?x (11) d ?L d = m? x ? ? Bx? dt ? x? dt (12) Likewise we have so as the general motion equation turns out to be m? x?? d Bx? + Vx ? ?Cx + ?Bx = 0. dt (13) These are the general equations of visual attention. In the Appendix we give the technical details of the derivations. Throughout the paper, the proposed model is referred to as the EYe MOvement Laws (EYMOL). 2.3 Parameters estimation with simulated annealing Different choices of parameters lead to different behaviors of the system. In particular, weights can emphasize the contribution of curiosity or brightness invariance terms. To better control the system we use two different parameters for the curiosity term, namely ?b and ?p , to weight b and p contributions respectively. The best values for the three parameters (?b , ?p , ?) are estimated using the algorithm of simulated annealing (SA). This method allows to perform iterative improvements, starting from a known state i. At each step, the SA considers some neighbouring state j of the current state, and probabilistically moves to the new state j or stays on the current state i. For our specific problem, we limit our search to a parallelepiped-domain D of possible values, due to theoretical bounds and numerical3 issues. Distance between states i and j is proportional with a temperature T , which is initialized to 1 and decreases over time as Tk = ? ? Tk?1 , where k identifies the iteration step, and 0 << ? < 1. The iteration step is repeated until the system reaches a state that is good enough for the application, which in our case is to maximize the NSS similarity between human saliency maps and simulated saliency maps. Only a batch of a 100 images from CAT2000-TRAIN is used to perform the SA algorithm4 . This batch is created by randomly selecting 5 images from each of the 20 categories of the dataset. To start the SA, parameters are initialized in the middle point of the 3-dimensional parameters domain D. The process is repeated 5 times, on different sub-samples, to select 5 parameters configurations. Finally, those configurations together with the average configuration are tested on the whole dataset, to select the best one. Algorithm 1 In the psedo-code, P() is the acceptance probability and score() is computed as the average of NSS scores on the sample batch of 100 images. 1: procedure S IMULATEDA NNEALING 2: Select an initial state i ? D 3: T ?1 4: do 5: Generate random state j, neighbor of i 6: if P(score(i), score(j)) ? Random(0, 1) then 7: i?j 8: end if 9: T ???T 10: while T ? 0.01 11: end procedure 3 Too high values for ?b or ?p produce numerically unstable and unrealistic trajectories for the focus of attention. 4 Each step of the SA algorithm needs evaluation over all the selected images. Considering the whole dataset would be very expensive in terms of time. 4 Model version V1 (approx. br. inv.) V2 (exact br. inv.) MIT1003 AUC NSS 0.7996 (0.0002) 1.2784 (0.0003) 0.7990 (0.0003) 1.2865 (0.0039) CAT2000-TRAIN AUC NSS 0.8393 (0.0001) 1.8208 (0.0015) 0.8376 (0.0013) 1.8103 (0.0137) Table 1: Results on MIT1003 [1] and CAT2000-TRAIN [11] of the two different version of EYMOL. Between brackets is indicated the standard error. 3 Experiments To quantitative evaluate how well our model predicts human fixations, we defined an experimental setup for salient detection both in images and in video. We used images from MIT1003 [1], MIT300 [12] and CAT2000 [11], and video from SFU [27] eye-tracking database. Many of the design choices were common to both experiments; when they differ, it is explicitly specified. 3.1 Input pre-processing All input images are converted to gray-scale. Peripheral input p is implemented as a blurred versions of the brightness b. This blurred version is obtained by convolving the original gray-scale image with a Gaussian kernel. For the images only, an algorithm identifies the rectangular zone of the input image in which the totality of information is contained in order to compute li in (14). Finally both b and p are multiplied by a Gaussian blob centered in the middle of the frame in order to make brightness gradients smaller as we move toward periphery and produce a center bias. 3.2 Saliency maps computation Differently by many of the most popular methodologies in the state-of-the-art [10; 16; 1; 24; 18], the saliency map is not itself the central component of our model but it can be naturally calculated from the visual attention laws in (13). The output of the model is a trajectory determined by a system of two second ordered differential equations, provided with a set of initial conditions. Since numerical integration of (13) does not raise big numerical difficulties, we used standard functions of the python scientific library SciPy [21]. Saliency map is then calculated by summing up the most visited locations during a sufficiently large number of virtual observations. For images, we collected data by running the model 199 times, each run was randomly initialized almost at the center of the image and with a small random velocity, and integrated for a running time corresponding to 1 second of visual exploration. For videos, we collected data by running the model 100 times, each run was initialized almost at the center of the first frame of the clip and with a small random velocity. Model that have some blur and center bias on the saliency map can improve their score with respect to some metrics. A grid search over blur radius and center parameter ? have been used, in order to maximize AUC-Judd and NSS score on the training data of CAT2000 in the case of images, and on SFU in case of videos. 3.3 Saliency detection on images Two versions of the the model have been evaluated. The first version V1 implementing brightness invariance in the approximated form (6), the second version V2 implementing the brightness invariance in its exact form, as described in the Appendix. Model V1 and V2 have been compared on the MIT1003 and CAT2000-TRAIN datasets, since they provide public data about fixations. Parameters estimation have been conducted independently for the two models and the best configuration for each one is used in this comparison. Results are statistically equivalent (see Table2) and this proves that, in the case of static images, the approximation is very good and does not cause loss in the score. For further experiments we decided to use the approximated form V1 due to its simpler form of the equation that also reduces time of computation. Model V1 has been evaluated in two different dataset of eye-tracking data: MIT300 and CAT2000TEST. In this case, scores were officially provided by MIT Saliency Benchmark Team [15]. Description of the metrics used is provided in [13]. Table 2 and Table 3 shows the scores of our 5 Itti-Koch [10], implem. by [19] AIM [16] Judd Model [1] AWS [24] eDN [18] EYMOL AUC 0.75 0.77 0.81 0.74 0.82 0.77 SIM 0.44 0.40 0.42 0.43 0.44 0.46 MIT300 EMD CC 4.26 0.37 4.73 0.31 4.45 0.47 4.62 0.37 4.56 0.45 3.64 0.43 NSS 0.97 0.79 1.18 1.01 1.14 1.06 KL 1.03 1.18 1.12 1.07 1.14 1.53 Table 2: Results on MIT300 [12] provided by MIT Saliency Benchmark Team [15]. The models are sorted chronologically. In bold, the best results for each metric and benchmarks. Itti-Koch [10], implem. by [19] AIM [16] Judd Model [1] AWS [24] eDN [18] EYMOL AUC 0.77 0.76 0.84 0.76 0.85 0.83 SIM 0.48 0.44 0.46 0.49 0.52 0.61 CAT2000-TEST EMD CC 3.44 0.42 3.69 0.36 3.60 0.54 3.36 0.42 2.64 0.54 1.91 0.72 NSS 1.06 0.89 1.30 1.09 1.30 1.78 KL 0.92 1.13 0.94 0.94 0.97 1.67 Table 3: Results on CAT2000 [11] provided by MIT Saliency Benchmark Team [15]. The models are sorted chronologically. In bold, the best results for each metric and benchmarks. model compared with five other popular method [10; 16; 1; 24; 18], which have been selected to be representative of different approaches. Despite its simplicity, our model reaches best score in half of the cases and for different metrics. 3.4 Saliency detection on dynamic scenes We evaluated our model in a task of saliency detection with the dataset SFU [27]. The dataset contains 12 clips and fixations of 15 observers, each of them have watched twice every video. Table 4 provides a comparison with other four model. Also in this case, despite of its simplicity and even if it was not designed for the specific task, our model competes well with state-of-the-art models. Our model can be easily run in real-time to produce an attentive scanpath. In some favorable case, it shows evidences of tracking moving objects on the scene. Mean AUC Mean NSS EYMOL 0.817 1.015 SFU Eye-Tracking Database Surprise [17] Judd Model [1] 0.70 0.66 0.77 0.28 0.48 1.06 Itti-Koch [10] HEVC [28] 0.83 1.41 Table 4: Results on the video dataset SFU [27]. Scores are calculated as the mean of AUC and NSS metrics of all frames of each clip, and then averaged for the 12 clips. 4 Conclusions In this paper we investigated how human attention mechanisms emerge in the early stage of vision, which we assume completely data-driven. The proposed model consists of differential equations, which provide a real-time model of scanpath. These equations are derived in a generalized framework of least action, which nicely resembles related derivations of laws in physics. A remarkable novelty concerns the unified interpretation of curiosity-driven movements and the brightness invariance term for fixation and tracking, that are regarded as mechanisms that jointly contribute to optimize the acquisition of visual information. Experimental results on both image and video datasets of saliency are very promising, especially if we consider that the proposed theory offers a truly model of eye movements, whereas the computation of the saliency maps only arises as a byproduct. 6 In future work, we intend to investigate behavioural data, not only in terms of saliency maps, but also by comparing actual generated scanpaths with human data in order to discover temporal correlations. We aim at providing the integration of the presented model with a theory of feature extraction that is still expressed in terms of variational-based laws of learning [29]. Appendix: Euler-Lagrange equations In this section we explicitly compute the differential laws of visual attention that describe the visual attention scanpath, as the Euler-Lagrange equations of the action functional (7). First, we compute the partial derivatives of the different contributions w.r.t. x, in order to compute the exact contributions of (11). For the retina boundaries, X  Vx = k ? 2 (li ? xi ) ? [xi > li ] + 2xi ? [xi < 0] (14) i=1,2 The curiosity term (4) Cx =2cos2 (?t)bx ? bxx + 2sin2 (?t)px ? pxx (15) For the term of brightness invariance, ? 2 (bt + bx x) ? ?x = 2 (bt + bx x) ? (btx + bxx x) ? Bx = (16) (17) Since we assume b ? C 2 (t, x), by the Schwarz?s theorem5 , we have that btx = bxt , so that Bx = 2 (bt + bx x) ? (bxt + bxx x) ? ? ? = 2(b)(bx ) (18) (19) We proceed by computing the contribution in (12). Derivative w.r.t. x? of the brightness invariance term is ? 2 Bx? = (bt + bx x) ? (20) ? x? = 2 (bt + bx x) ? bx (21) ? = 2(b)(bx ) (22) So that, total derivative w.r.t. t can be write as   d Bx? =2 ?bbx + b? b? x dt (23) We observe that ?b ? ?b(t, x, x, ? x ?) is the only term which depends on second derivatives of x. Since we are interested in expressing EL in an explicit form for the variable x ?, we explore more closely its contribution d ?b(t, x, x, ? x ?) = b? (24) dt d = (bt + bx x) ? (25) dt =b? t + b? x ? x? + bx ? x ? (26) (27) Substituting it in (23) we have   d Bx? =2 (b? t + b? x ? x? + bx ? x ?)bx + b? b? x dt   =2 (b? t + b? x ? x)b ? x + b? b? x + 2(bx ? x ?)bx (28) (29) Schwarz?s theorem states that, if f : Rn ? R has continuous second partial derivatives at any given point in R , then ?i, j ? {1, ..., n} it holds fxi xj = fxj xi 5 n 7 So that, from (12) we get   d ?L = m? x ? 2? (b? t + b? x ? x)b ? x + b? b? x + (bx ? x ?)bx (30) dt ? x? Euler-Lagrange equations. Combining (11) and (30), we get Euler-Lagrange equation of attention   ? b? x ) + (bx ? x m? x ? 2? (b? t + b? x ? x)(b ? x ) + (b)( ?)bx = ?Cx ? Vx ? ?Bx (31) In order to obtain explicit form for the variable x ?, we re-write the equation as to move to the left all contributes which do not depend on that variable. ? b? x )) m? x ? 2?(bx ? x ?)bx =?Cx ? Vx ? ?Bx + 2?((b? t + b? x ? x)(b ? x ) + (b)( = ?Cx ? Vx + 2?(b? t + b? x ? x)(b ? x) | {z } (32) (33) A=(A1 ,A2 ) In matrix form, the equation is       m? x1 2?(bx1 x ?1 + bx2 x ?2 )bx1 A1 ? = m? x2 2?(bx1 x ?1 + bx2 x ?2 )bx2 A2 which gives us the system of two differential equations  m? x1 ? 2?(bx1 x ?1 + bx2 x ?2 )bx1 = A1 m? x2 ? 2?(bx1 x ?1 + bx2 x ?2 )bx2 = A2 Grouping by same variable,  x1 ? 2?(bx1 bx2 )? x2 (m ? 2?b2x1 )? ?2?(bx1 bx2 )? x2 x1 + (m ? 2?b2x2 )? = A1 = A2 (34) (35) (36) We define (m ? 2?b2x ) ?2?(bx bx ) 1 2 1 D= ?2?(bx1 bx2 ) (m ? 2?b2x2 ) (m ? 2?b2 ) A1 A1 ?2?(bx1 bx2 ) x 1 ,D = D1 = A2 (m ? 2?b2x2 ) 2 ?2?(bx1 bx2 ) A2 (37) (38) By the Cramer?s method we get differential equation of visual attention for the two spatial component, i.e. ? D1 ? ?x ? ? ?1 = D (39) ? ? D 2 ? ?x ?2 = D Notice that, this raise to a further condition over the parameter ?. In particular, in the case values of b(t, x) are normalized in the range [0, 1], it imposes to chose m D 6= 0 =? ? < (40) 4 In fact, D = (m ? 2?b2x1 )(m ? 2?b2x2 ) ? 4?2 (bx1 bx2 )2   = m m ? 2?(b2x1 + b2x1 ) (41) (42) For values of bx = 0, we have that D = m2 > 0 (43) D > 0. (44) so that ?t, we must impose 8 If ? > 0, then m ? 2?(b2x1 + b2x1 ) > 0 m ?< 2 2(bx1 + b2x1 ) m , so that the condition 4 m 0<?< 4 (45) (46) The quantity on the right reaches its minimum at (47) guarantees the well-posedness of the problem. References [1] Judd, T., Ehinger, K., Durand, F., Torralba, A.: Learning to Predict Where Humans Look. IEEE International Conference on Computer Vision (2009) [2] Itti, L., Koch, C.: Computational modelling of visual attention. Nature Reviews Neuroscience, vol 3, n 3, pp 194?203. (2001) [3] Connor, C.E., Egeth, H.E., Yantis, S.: Visual Attention: Bottom-Up Versus Top-Down. Current Biology, vol 14, n 19, pp R850?R852. (2004) [4] McMains, S., Kastner, S.: Interactions of Top-Down and Bottom-Up Mechanisms in Human Visual Cortex. Society for Neuroscience, vol 31, n 2, pp 587?597. (2011) [5] Hainline, L., Turkel, J., Abramov, I., Lemerise, E., Harris, C.M.: Characteristics of saccades in human infants. Vision Research, vol 24, n 12, pp 1771?1780. (1984) [6] Le Meur, O., Liu, Z.: Saccadic model of eye movements for free-viewing condition. Vision Research, vol 116, pp 152?164. (2015) [7] Gelfand, I.M., Fomin, S.V.: Calculus of Variation. Englewood : Prentice Hall (1993) [8] Borji, A., Itti, L.: State-of-the-Art in Visual Attention Modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 35, n 1. (2013) [9] Koch, C., Ullman, S.: Shifts in selective visual attention: towards the underlying neural circuitry. Springer Human Neurobiology, vol 4, n 4, pp 219-227. (1985) [10] Itti, L., Koch, C.: A Model of Saliency-Based Visual Attention for Rapid Scene Analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 20, n 11. (1998) [11] Borji, A., Itti, L.: CAT2000: A Large Scale Fixation Dataset for Boosting Saliency Research. arXiv:1505.03581. (2015) [12] Judd, T., Durand, F., Torralba, A.:: A Benchmark of Computational Models of Saliency to Predict Human Fixations. MIT Technical Report. (2012) [13] Bylinskii, Z., Judd, T., Oliva, A., Torralba, A.: What do different evaluation metrics tell us about saliency models? arXiv:1604.03605. (2016) [14] Treisman, A.M., Gelade, G.: A Feature Integration Theory of Attention. Cognitive Psychology, vol 12, pp 97-136. (1980) [15] Bylinskii, Z., Judd, T., Borji, A., Itti, L., Durand, F., Torralba, A.: MIT Saliency Benchmark. http://saliency.mit.edu/ [16] Bruce, N., Tsotsos, J.: Attention based on information maximization. J. Vis., vol 7, n 9. (2007) [17] Itti, L., Baldi, P.: Bayesian Surprise Attracts Human Attention. Vision Research, vol 49, n 10, pp 1295?1306. (2009) 9 [18] Vig, E., Dorr, M., Cox, D.: Large-Scale Optimization of Hierarchical Features for Saliency Prediction in Natural Images. IEEE Conference on Computer Vision and Pattern Recognition. (2014) [19] Harel, J.: A Saliency Implementation in MATLAB . http://www.klab.caltech.edu/ harel/share/gbvs.php [20] Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artificial Intelligence, vol 17, n 1, pp 185-203. (1981) [21] Jones, E., Travis, O., Peterson, P.: SciPy: Open source scientific tools for Python. http://www.scipy.org/. (2001) [22] Zhang, J., Sclaroff, S.: Saliency detection: a Boolean map approach . Proc. of the IEEE International Conference on Computer Vision. (2013) [23] Cornia, M., Baraldi, L., Serra, G., Cucchiara, R.: Predicting Human Eye Fixations via an LSTM-based Saliency Attentive Model. http://arxiv.org/abs/1611.09571. (2016) [24] Garcia-Diaz, A., Lebor?n, V., Fdez-Vida, X.R., Pardo, X.M.: On the relationship between optical variability, visual saliency, and eye fixations: Journal of Vision, vol 12, n 6, pp 17. (2012) [25] Tatler, B.W., Baddeley, R.J., Gilchrist, I.D.: Visual correlates of fixation selection: Effects of scale and time. Vision Research, vol 45, n 5, pp 643-659. (2005) [26] Kruthiventi, S.S.S., Ayush, K., Venkatesh, R.:DeepFix: arXiv:1510.02927. (2015) [27] Hadizadeh, H., Enriquez, M.J., Bajic, I.V.: Eye-Tracking Database for a Set of Standard Video Sequences. IEEE Transactions on Image Processing. (2012) [28] Xu, M., Jiang, L., Ye, Z., Wang, Z.: Learning to Detect Video Saliency With HEVC Features. IEEE Transactions on Image Processing. (2017) [29] Maggini, M., Rossi, A.: On-line Learning on Temporal Manifolds. AI*IA 2016 Advances in Artificial Intelligence Springer International Publishing, pp 321?333. (2016) 10
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Recursive Sampling for the Nystr?m Method Cameron Musco MIT EECS [email protected] Christopher Musco MIT EECS [email protected] Abstract We give the first algorithm for kernel Nystr?m approximation that runs in linear time in the number of training points and is provably accurate for all kernel matrices, without dependence on regularity or incoherence conditions. The algorithm projects the kernel onto a set of s landmark points sampled by their ridge leverage scores, requiring just O(ns) kernel evaluations and O(ns2 ) additional runtime. While leverage score sampling has long been known to give strong theoretical guarantees for Nystr?m approximation, by employing a fast recursive sampling scheme, our algorithm is the first to make the approach scalable. Empirically we show that it finds more accurate kernel approximations in less time than popular techniques such as classic Nystr?m approximation and the random Fourier features method. 1 Introduction The kernel method is a powerful for applying linear learning algorithms (SVMs, linear regression, etc.) to nonlinear problems. The key idea is to map data to a higher dimensional kernel feature space, where linear relationships correspond to nonlinear relationships in the original data. Typically this mapping is implicit. A kernel function is used to compute inner products in the high-dimensional kernel space, without ever actually mapping original data points to the space. Given n data points x1 , . . . , xn , the n ? n kernel matrix K is formed where Ki,j contains the highdimensional inner product between xi and xj , as computed by the kernel function. All computations required by a linear learning method are performed using the inner product information in K. Unfortunately, the transition from linear to nonlinear comes at a high cost. Just generating the entries of K requires ?(n2 ) time, which is prohibitive for large datasets. 1.1 Kernel approximation A large body of work seeks to accelerate kernel methods by finding a compressed, often low? to the true kernel matrix K. Techniques include random sampling and rank, approximation K embedding [AMS01, BBV06, ANW14], random Fourier feature methods for shift invariant kernels [RR07, RR09, LSS13], and incomplete Cholesky factorization [FS02, BJ02]. ? using a subset of One of the most popular techniques is the Nystr?m method, which constructs K ? ?landmark? data points [WS01]. Once s data points are selected, K (in factored form) takes just O(ns) kernel evaluations and O(s3 ) additional time to compute, requires O(ns) space to store, and ? takes O(ns2 ) time. can be manipulated quickly in downstream applications. E.g., inverting K The Nystr?m method performs well in practice [YLM+ 12, GM13, TRVR16], is widely implemented [HFH+ 09, PVG+ 11, IBM14], and is used in a number of applications under different names such as ?landmark isomap? [DST03] and ?landmark MDS? [Pla05]. In the classic variant, landmark points are selected uniformly at random. However, significant research seeks to improve performance via data31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. dependent sampling that selects landmarks which more closely approximate the full kernel matrix than uniformly sampled landmarks [SS00, DM05, ZTK08, BW09, KMT12, WZ13, GM13, LJS16]. Theoretical work has converged on leverage score based approaches, as they give the strongest provable guarantees for both kernel approximation [DMM08, GM13] and statistical performance in downstream applications [AM15, RCR15, Wan16]. Leverage scores capture how important an individual data point is in composing the span of the kernel matrix. Unfortunately, these scores are prohibitively expensive to compute. All known approximation schemes require ?(n2 ) time or only run quickly under strong conditions on K ? e.g. good conditioning or data ?incoherence? [DMIMW12, GM13, AM15, CLV16]. Hence, leverage score-based approaches remain largely in the domain of theory, with limited practical impact [KMT12, LBKL15, YPW15]. 1.2 Our contributions In this work, we close the gap between strong approximation bounds and efficiency: we present a new Nystr?m algorithm based on recursive leverage score sampling which achieves the ?best of both worlds?: it produces kernel approximations provably matching the high accuracy of leverage score methods while only requiring O(ns) kernel evaluations and O(ns2 ) runtime for s landmark points. Theoretically, this runtime is surprising. In the typical case when s ? n, the algorithm evaluates just a small subset of K, ignoring most of the kernel space inner products. Yet its performance guarantees hold for general kernels, requiring no assumptions on coherence or regularity. Empirically, the runtime?s linear dependence on n means that our method is the first leverage score algorithm that can compete with the most commonly implemented techniques, including the classic uniform sampling Nystr?m method and random Fourier features sampling [RR07]. Since our algorithm obtains higher quality samples, we show experimentally that it outperforms these methods on benchmark datasets ? it can obtain as accurate a kernel approximation in significantly less time. Our approximations also have lower rank, so they can be stored in less space and processed more quickly in downstream learning tasks. 1.3 Paper outline Our recursive sampling algorithm is built on top of a Nystr?m scheme of Alaoui and Mahoney that samples landmark points based on their ridge leverage scores [AM15]. After reviewing preliminaries in Section 2, in Section 3 we analyze this scheme, which we refer to as RLS-Nystr?m. To simplify prior work, which studies the statistical performance of RLS-Nystr?m for specific kernel learning tasks [AM15, RCR15, Wan16], we prove a strong, application independent approximation guarantee: ? is constructed with s = ?(d log d ) samples1 , where d = tr(K(K + I) 1 ) is for any , if K eff eff eff ? 2? . the so-called ? -effective dimensionality? of K, then with high probability, kK Kk In Appendix E, we show that this guarantee implies bounds on the statistical performance of RLSNystr?m for kernel ridge regression and canonical correlation analysis. We also use it to prove new results on the performance of RLS-Nystr?m for kernel rank-k PCA and k-means clustering ? in both cases just O(k log k) samples are required to obtain a solution with good accuracy. After affirming the favorable theoretical properties of RLS-Nystr?m, in Section 4 we show that its runtime can be significantly improved using a recursive sampling approach. Intuitively our algorithm is simple. We show how to approximate the kernel ridge leverage scores using a uniform sample of 12 of our input points. While the subsampled kernel matrix still has a prohibitive n2 /4 entries, we can recursively approximate it, using our same sampling algorithm. If our final Nystr?m approximation will use s landmarks, the recursive approximation only needs rank O(s), which lets us estimate the ridge leverage scores of the original kernel matrix in just O(ns2 ) time. ? ? Since n is cut in half ns2 ns2 2 at each level of recursion, our total runtime is O ns + 2 + 4 + ... = O(ns2 ), significantly improving upon the method of [AM15], which takes ?(n3 ) time in the worst case. Our approach builds on recent work on iterative sampling methods for approximate linear algebra [CLM+ 15, CMM17]. While the analysis in the kernel setting is technical, our final algorithm is 1 This is within a log factor of the best possible for any low-rank approximation with error . 2 simple and easy to implement. We present and test a parameter-free variation of Recursive RLSNystr?m in Section 5, confirming superior performance compared to existing methods. 2 Preliminaries Consider an input space X and a positive semidefinite kernel function K : X ? X ! R. Let F be an associated reproducing kernel Hilbert space and : X ! F be a (typically nonlinear) feature map such that for any x, y 2 X , K(x, y) = h (x), (y)iF . Given a set of n input points x1 , . . . , xn 2 X , define the kernel matrix K 2 Rn?n by Ki,j = K(xi , xj ). 0 It is often natural to consider the kernelized data matrix that generates K. Informally, let 2 Rn?d T be the matrix containing (x1 ), ..., (xn ) as its rows (note that d0 may be infinite). K = . While we use for intuition, in our formal proofs we replace it with any matrix B 2 Rn?n satisfying BBT = K (e.g. a Cholesky factor). Such a B is guaranteed to exist since K is positive semidefinite. We repeatedly use the singular value decomposition, which allows us to write any rank r matrix M 2 Rn?d as M = U?VT , where U 2 Rn?r and V 2 Rd?r have orthogonal columns (the left and right singular vectors of M), and ? 2 Rr?r is a positive diagonal matrix containing the singular + 1 T values: 1 (M) ... U . 2 (M) r (M) > 0. M?s pseudoinverse is given by M = V? 2.1 Nystr?m approximation The Nystr?m method selects a subset of ?landmark? points and uses them to construct a low-rank approximation to K. Given a matrix S 2 Rn?s that has a single entry in each column equal to 1 so that KS is a subset of s columns from K, the associated Nystr?m approximation is: ? = KS(ST KS)+ ST K. K (1) ? can be stored in O(ns) space by separately storing KS 2 Rn?s and (ST KS)+ 2 Rs?s . FurtherK more, the factors can be computed using just O(ns) evaluations of the kernel inner product to form KS and O(s3 ) time to compute (ST KS)+ . Typically s ? n so these costs are significantly lower than the cost to form and store the full kernel matrix K. We view Nystr?m approximation as a low-rank approximation to the dataset in feature space. ReT calling that K = , S selects s kernelized data points ST and we approximate using its 0 0 projection onto these points. Informally, let PS 2 Rd ?d be the orthogonal projection onto the row def T span of ST . We approximate by ? = PS . We can write PS = T S(ST S)+ ST . T 2 Since it is an orthogonal projection, PS PS = PS = PS , and so we can write: ? = ? ?T = K P2S T = T S(ST T S)+ ST T = KS(ST KS)+ ST K. This recovers the standard Nystr?m approximation (1). 3 The RLS-Nystr?m method We now introduce the RLS-Nystr?m method, which uses ridge leverage score sampling to select landmark data points, and discuss its strong approximation guarantees for any kernel matrix K. 3.1 Ridge leverage scores In classical Nystr?m approximation (1), S is formed by sampling data points uniformly at random. Uniform sampling can work in practice, but it only gives theoretical guarantees under strong regularity or incoherence assumptions on K [Git11]. It will fail for many natural kernel matrices where the relative ?importance? of points is not uniform across the dataset For example, imagine a dataset where points fall into several clusters, but one of the clusters is much larger than the rest. Uniform sampling will tend to oversample landmarks from the large cluster while undersampling or possibly missing smaller but still important clusters. Approximation of K and learning performance (e.g. classification accuracy) will decline as a result. 3 (a) Uniform landmark sampling. (b) Improved landmark sampling. Figure 1: Uniform sampling for Nystr?m approximation can oversample from denser parts of the dataset. A better Nystr?m scheme will select points that more equally cover the relevant data. To combat this issue, alternative methods compute a measure of point importance that is used to select landmarks. For example, one heuristic applies k-means clustering to the input and takes the cluster centers as landmarks [ZTK08]. A large body of theoretical work measures importance using variations on the statistical leverage scores. One natural variation is the ridge leverage score: Definition 1 (Ridge leverage scores [AM15]). For any > 0, the -ridge leverage score of data point xi with respect to the kernel matrix K is defined as def li (K) = K(K + I) 1 i,i (2) , where I is the n ? n identity matrix. For any B 2 Rn?n satisfying BBT = K, we can also write: li (K) = bTi (BT B + I) 1 (3) bi , where bTi 2 R1?n is the ith row of B. For conciseness we typically write li (K) as li . To check that (2) and (3) are equivalent note that bTi (BT B+ I) 1 bi = B(BT B + I) 1 BT i,i . Using the SVD to write B = U?VT and accordingly K = U?2 UT confirms that K(K+ I) 1 = B(BT B+ I) 1 BT = U?2 ?2 + I 1 UT . It is not hard to check (see [CLM+ 15]) that the ridge scores can be defined alternatively as: li = minn y2R 1 kbTi yT Bk22 + kyk22 . (4) This formulation provides better insight into these scores. Since BBT = K, any kernel algorithm effectively works with B?s rows as data points. The ridge scores reflect the relative importance of these rows. From (4) it?s clear that li ? 1 since we can set y to the ith standard basis vector. bi will have score ? 1 (i.e. is less important) when it?s possible to find a more ?spread out? y that uses other rows in B to approximately reconstruct bi ? in other words when the row is less unique. 3.2 Sum of ridge leverage scores As is standard in leverage score methods, we don?t directly select landmarks to be the points with the highest scores. Instead, we sample each point with probability proportional to li . Accordingly, the ? rank, is a random variable with expectation equal number of landmarks selected, which controls K?s to the sum of the -ridge leverage scores. To ensure compact kernel approximations, we want this sum to be small. Immediately from Definition 1, we have: Pn 1 Fact 2. ). i=1 li (K) = tr(K(K + I) def We denote deff = tr(K(K + I) 1 ). deff is a natural quantity, referred to as the ?effective dimension? or ?degrees of freedom? for a ridge regression problem on K with regularization [HTF02, Zha06]. deff increases monotonically as decreases. For any fixed it is essentially the smallest possible rank ? satisfying the approximation guarantee given by RLS-Nystr?m: kK Kk ? 2< . achievable for K 4 3.3 The basic sampling algorithm We can now introduce the RLS-Nystr?m method as Algorithm 1. We allow sampling each point by any probability greater than li , which is useful later when we compute the scores approximately. ? accuracy. It could cause us to take more samples, but Oversampling landmarks can only improve K?s we will always ensure that the sum of our approximate ridge leverage scores is not too large. Algorithm 1 RLS-N YSTR?M S AMPLING input: x1 , . . . , xn 2 X , kernel matrix K, ridge parameter > 0, failure probability 2 (0, 1/8) ?l > l for the -ridge leverage score of each x1 , . . . , xn 1: Compute an over-approximation, i n oi P ? ? 2: Set pi := min 1, l ? 16 log( l / ) . i i 3: Construct S 2 Rn?s by sampling x1 , . . . , xn each independently with probability pi . In other words, for each i add a column to S with a 1 in position i with probability pi . 4: return the Nystr?m factors KS 2 Rn?s and (ST KS)+ 2 Rs?s . 3.4 Accuracy bounds ? which spectrally approximates K up to a small additive We show that RLS-Nystr?m produces K error. This is the strongest type of approximation offered by any known Nystr?m method [GM13]. It ? is used in place of K in many learning applications [CMT10]. guarantees provable accuracy when K Theorem 3 (Spectral error approximation). For any > 0 and 2 (0, 1/8), Algorithm 1 returns P ? = KS(ST KS)+ ST K satisfies: S 2 Rn?s such that with probability 1 , s ? 2 i pi and K ? K K ? + I. K P When ridge scores are computed exactly, i pi = O deff log(deff / ) . (5) denotes the Loewner ordering: M N means that N M is positive semidefinite. Note that (5) ? 2? . immediately implies the well studied (see e.g [GM13]) spectral norm guarantee, kK Kk ? well approximates the top of K?s spectrum (i.e. any Intuitively, Theorem 3 guarantees that K eigenvalues > ) while losing information about smaller, less important eigenvalues. Due to space constraints, we defer the proof to Appendix A. It relies on the view of Nystr?m approximation as a low-rank projection of the kernelized data (see Section 2.1) and we use an intrinsic dimension matrix Bernstein bound to show accuracy of the sampled approximation. Often the regularization parameter is specified for a learning task, and for near optimal performance on this task, we set the approximation factor in Theorem 3 to ? . In this case we have: Corollary 4 (Tighter spectral error approximation). For any > 0 and run with ridge parameter ? returns S 2 R such that with probability 1 T + T ? ? ? + ? I. and K = KS(S KS) S K satisfies K K K n?s Proof. This follows from Theorem 3 by noting d?eff ? deff /? since (K+? I) 2 (0, 1/8),?Algorithm ?1 ,s=O 1 1 ? (K+ deff ? deff ? log I) 1 . ? can be used in place of K without sacrificing performance on Corollary 4 suffices to prove that K kernel ridge regression and canonical correlation tasks [AM15, Wan16]. We also use it to prove a projection-cost preservation guarantee (Theorem 12, Appendix B), which gives approximation bounds for kernel PCA and k-means clustering. Projection-cost preservation has proven a powerful concept in the matrix sketching literature [FSS13, CEM+ 15, CMM17, BWZ16, CW17] and we hope that extending the guarantee to kernels leads to applications beyond those considered in this work. Our results on downstream learning bounds that can be derived from Theorem 3 are summarized in Table 1. Details can be found in Appendices B and E. 5 ? obtained from RLS-Nystr?m (Algorithm 1). Table 1: Downstream guarantees for K Guarantee Theorem ? Space to store K (1 + ?) relative error risk bound Thm 16 ? ndeff ) O( ? Kernel k-means Clustering (1 + ?) relative error Thm 17 ? nk ) O( ? Rank k Kernel PCA (1 + ?) relative Frob norm error Thm 18 ? nk ) O( ? ? additive error Thm 19 Application Kernel Ridge Regression w/ param Kernel CCA w/ params ? 4 x, y x +nd y eff ? ndeff O( ? ) ? hides log factors in the failure probability, deff , and k. For conciseness, O(?) Recursive sampling for efficient RLS-Nystr?m Having established strong approximation guarantees for RLS-Nystr?m, it remains to provide an efficient implementation. Specifically, Step 1 of Algorithm 1 naively requires ?(n3 ) time. We show that significant acceleration is possible using a recursive sampling approach. 4.1 Ridge leverage score approximation via uniform sampling The key is to estimate the leverage scores by computing (3) approximately, using a uniform sample of the data points. To ensure accuracy, the sample must be large ? a constant fraction of the points. Our fast runtimes are achieved by recursively approximating this large sample. In Appendix F we prove: Lemma 5. For any B 2 Rn?n with BBT = K and S 2 Rn?s chosen by sampling each data point independently with probability 1/2, let ?li = bTi (BT SST B + I) 1 bi and pi = P min{1, 16?li log( i ?li / )} for any 2 (0, 1/8). Then with probability at least 1 : X X X 1) ?li li for all i 2) pi ? 64 li log( li / ). i i i The first condition ensures that the approximate scores ?li suffice for use in Algorithm 1. The second ensures that the Nystr?m approximation obtained will not have too many sampled landmarks. Naively computing ?li in Lemma 5 involves explicitly forming B, requiring ?(n2 ) time (e.g. ?(n3 ) via Cholesky decomposition). Fortunately, the following formula (proof in Appx. F) avoids this cost: Lemma 6. For any sampling matrix S 2 Rn?s , and any > 0: 1? T T T 1 ?l def = b (B SS B + I) b = K KS ST KS + I i i i 1 ST K ? i,i . It follows that we can compute all ?li for all i in O(ns2 ) time using just O(ns) kernel evaluations, to compute KS and the diagonal of K. 4.2 Recursive RLS-Nystr?m We apply Lemmas 5 and 6 to give an efficient recursive implementation of RLS-Nystr?m, Algorithm 2. We show that the output of this algorithm, S, is sampled according to approximate ridge leverage scores for K and thus satisfies the approximation guarantee of Theorem 3. Theorem 7 (Main Result). Let S 2 Rn?s be computed by Algorithm 2. With probability 1 3 , s ? 384 ? deff log(deff / ), S is sampled by overestimates of the -ridge leverage scores of K, and ? = KS(ST KS)+ ST K satisfies: thus by Theorem 3, the Nystr?m approximation K ? K ? + I. K K Algorithm 2 uses O(ns) kernel evaluations and O(ns2 ) computation time. 6 Algorithm 2 R ECURSIVE RLS-N YSTR?M. input: x1 , . . . , xm 2 X , kernel function K : X ? X ! R, ridge > 0, failure prob. 2 (0, 1/32) output: weighted sampling matrix S 2 Rm?s 1: if m ? 192 log(1/ ) then 2: return S := Im?m . 3: end if 4: Let S? be a random subset of {1, ..., m}, with each i included independently with probability 12 . ? ? = {xi1 , xi2 , ..., xi ? } for ij 2 S? be the data sample corresponding to S. . Let X |S| ? ? = [ei1 , ei2 , ..., ei ? ] be the sampling matrix corresponding to S. . Let S |S| ? ? 5: S := R ECURSIVE RLS-N YSTR?M(X, K, , /3). ? := S ? ? S. ? ? 6: S ? ? ? 1 3 T T ? ? ? ? ? 7: Set li := 2 K KS S KS + I S K for each i 2 {1, . . . , m} . i,i ?S ? T B + I) 1 BT )i,i . K denotes the kernel matrix for data. By Lemma 6, equals 32 (B(BT S points {x1 , . . . , xm } and kernel function K. P 8: Set pi := min{1, ? li ? 16 log( ?li / )} for each i 2 {1, . . . , m}. 9: Initially set weighted sampling matrix S to be empty. For each i 2 {1, . . . , m}, with probability pi , append the column p1pi ei onto S. 10: return S. p Note that in Algorithm 2 the columns of S are weighted by 1/ pi . The Nystr?m approximation ? = KS(ST KS)+ ST K is not effected by column weights (see derivation in Section 2.1). However, K ? is used in Step 6). the weighting is necessary when the output is used in recursive calls (i.e. when S We prove Theorem 7 via the following intermediate result: Theorem 8. For any inputs x1 , . . . , xm , K, > 0 and 2 (0, 1/32), let K be the kernel matrix for x1 , . . . , xm and kernel function K and let deff (K) be the effective dimension of K with parameter . With probability (1 3 ), R ECURSIVE RLS-N YSTR?M outputs S with s columns that satisfies: 1 T 3 T (B B + I) (BT SST B + I) (B B + I) for any B with BBT = K. (6) 2 2 def Additionally, s ? smax (deff (K), ) where smax (w, z) = 384 ? (w + 1) log ((w + 1)/z). The algorithm uses ? c1 msmax (deff (K), ) kernel evaluations and ? c2 msmax (deff (K), )2 additional computation time where c1 and c2 are fixed universal constants. Theorem 8 is proved via an inductive argument, given in Appendix C. Roughly, consider in Step 6 of ? := S ? instead of S ? ? S. ? By Lemma 5 and the formula in Lemma 6, the leverage Algorithm 2, setting S ? score approximations li computed in Step 7 would be good approximations to the true leverage scores, and S would satisfy Theorem 8 by a standard matrix Bernstein bound (see Lemma 9). ? := S, ? it will have n/2 columns in expectation, and the computation in Step 7 However, if we set S will be expensive ? requiring roughly O(n3 ) time. By recursively calling Algorithm 8 and applying ? satisfying with high probability: Theorem 8 inductively, we obtain S 1 T ? ?T ? S ?S ? T (S ? T B) + I) 3 (BS ?S ? T B + I). (B SS B + I) ((BS) 2 2 ? = S ??S ? in place of S ? in Step 7, the leverage score This guarantee ensures that when we use S estimates are changed only by a constant factor. Thus, sampling by these estimates, still gives us the ? and therefore S ? has just O(smax (d (K), )) columns, so Step 7 desired guarantee (6). Further, S eff can be performed very efficiently, within the stated runtime bounds. With Theorem 8 we can easily prove our main result, Theorem 7. Proof of Theorem 7. In our proof of Theorem 3 in Appendix A.1, we show that if 1 T 3 T (B B + I) (BT SST B + I) (B B + I) 2 2 7 for a weighted sampling matrix S, then even if we remove the weights from S so that it has all unit ? = KS(ST KS)+ ST K satisfies: entries (they don?t effect the Nystr?m approximation), K ? K ? + I. K K The runtime bounds also follow nearly directly from Theorem 8. In particular, we have established that O nsmax (deff (K), ) kernel evaluations and O nsmax (deff (K), )2 additional runtime are required by R ECURSIVE RLS-N YSTR?M. We only needed the upper bound to prove Theorem 8, but along the way actually show that in a successful run of R ECURSIVE RLS-N YSTR?M, S has ? deff (K) log deff (K)/ columns. Additionally, we may assume that deff (K) 1/2. If it is not, then it?s not hard to check (see proof of Lemma 20) that must be kKk. If this is the case, the ? satisfies K ? ? + I. guarantee of Theorem 7 is vacuous: any Nystr?m approximation K K K With deff (K) 1/2, deff (K) log deff (K)/ and thus s are ?(smax (deff (K), ) so we conclude that Theorem 7 uses O(ns) kernel evaluations and O(ns2 ) additional runtime. 5 Empirical evaluation We conclude with an empirical evaluation of our recursive RLS-Nystr?m method. We use a variant of Algorithm 2 where, instead of choosing a regularization parameter , the user sets a sample size s and is automatically determined such that s = ?(deff ? log(deff / )). This variant is practically appealing as it essentially yields the best possible approximation to K for a fixed sample budget. Pseudocode and proofs of correctness are included in Appendix D. 5.1 Performance of Recursive RLS-Nystr?m for kernel approximation We evaluate RLS-Nystr?m on the YearPredictionMSD, Covertype, Cod-RNA, and Adult datasets downloaded from the UCI ML Repository [Lic13] and [UKM06]. These datasets contain 515345, 581012, 331152, and 48842 data points respectively. We compare against the classic Nystr?m method with uniform sampling [WS01] and the random Fourier features method [RR07]. Due to the large size of the datasets, prior leverage score based Nystr?m approaches [DMIMW12, GM13, AM15], which require at least ?(n2 ) time are infeasible, and thus not included in our tests. We split categorical features into binary indicatory features and mean center and normalize features to have variance 1. We use a Gaussian kernel for all tests, with the width parameter selected via cross ? 2 is used to measure approximation error. validation on regression and classification tasks. kK Kk Since this quantity is prohibitively expensive to compute directly (it requires building the full kernel matrix K), the error is estimated using a random subset of 20,000 data points and repeated trials. 10 4 Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features 10 4 10 10 2 10 0 10 0 10 -2 10 0 1000 2000 3000 Samples (a) Adult 4000 5000 10 -2 0 500 1000 1500 0 1000 2000 3000 4000 Samples (b) Covertype 10 2 10 1 -2 10 -4 2000 Samples Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features 10 3 ? 2 ?K ? K? ? 2 ?K ? K? 10 0 10 -4 Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features 4 10 2 10 2 ? 2 ?K ? K? ? 2 ?K ? K? 10 4 (c) Cod-RNA 5000 10 0 0 1000 2000 3000 4000 5000 Samples (d) YearPredictionMSD Figure 2: For a given number of samples, Recursive RLS-Nystr?m yields approximations with lower ? 2 . Error is plotted on a logarithmic scale, averaged over 10 trials. error, measured by kK Kk Figure 2 confirms that Recursive RLS-Nystr?m consistently obtains substantially better kernel approximation error than the other methods. As we can see in Figure 3, with the exception of YearPredictionMSD, the better quality of the landmarks obtained with Recursive RLS-Nystr?m also translates into runtime improvements. While the cost per sample is higher for our method at O(nd + ns) time versus O(nd + s2 ) for uniform Nystr?m and O(nd) for random Fourier features, ? with a given accuracy. K ? will since RLS-Nystr?m requires fewer samples it more quickly obtains K also have lower rank, which can accelerate processing in downstream applications. 8 10 2 Recursive RLS-Nystrom Uniform Nystrom 10 -1 10 -2 10 -3 10 -4 10 0 10 -1 10 0 5 10 Recursive RLS-Nystrom Uniform Nystrom 10 0 10 -1 10 2 10 1 10 -2 -2 10 -3 15 Recursive RLS-Nystrom Uniform Nystrom 10 1 ? 2 ?K ? K? 10 1 ? 2 ?K ? K? ? 2 ?K ? K? 10 0 10 3 10 2 Recursive RLS-Nystrom Uniform Nystrom ? 2 ?K ? K? 10 1 10 -3 0 1 2 3 4 0 5 1 Runtime (sec.) (a) Adult 2 3 4 10 0 5 0 2 4 Runtime (sec.) Runtime (sec.) (b) Covertype 6 8 10 Runtime (sec.) (c) Cod-RNA (d) YearPredictionMSD Figure 3: Recursive RLS-Nystr?m obtains a fixed level of approximation faster than uniform sampling, only underperforming on YearPredictionMSD. Results for random Fourier features are not shown: while the method is faster, it never obtained high enough accuracy to be directly comparable. In Appendix G, we show that that runtime of RLS-Nystr?m can be further accelerated, via a heuristic approach that under-samples landmarks at each level of recursion. This approach brings the per sample cost down to approximately that of random Fourier features and uniform Nystr?m while nearly maintaining the same approximation quality. Results are shown in Figure 4. For datasets such as Covertype in which Recursive RLS-Nystr?m performs significantly better than uniform sampling, so does the accelerated method (see Figure 4b). However, the performance of the accelerated method does not degrade when leverage scores are relatively uniform ? it still offers the best runtime to approximation quality tradeoff (Figure 4c). We note further runtime optimizations may be possible. Subsequent work extends fast ridge leverage score methods to distributed and streaming environments [CLV17]. Empirical evaluation of these techniques could lead to even more scalable, high accuracy Nystr?m methods. Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features Accelerated Recursive RLS-Nystrom 2 Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features Acclerated Recursive RLS-Nystrom 1 0.5 0 0 500 1000 Samples 1500 2000 (a) Runtimes for Covertype. 10 2 10 0 10 10 3 ? 2 ?K ? K? ? 2 ?K ? K? Runtime (sec) 1.5 Recursive RLS-Nystrom Uniform Nystrom Random Fourier Features Accelerated Recursive RLS-Nystrom 10 4 10 2 10 1 -2 0 500 1000 1500 2000 Samples (b) Errors for Covertype. 10 0 0 1 2 3 4 5 Runtime (sec.) (c) Runtime/error tradeoff for YearPredictionMSD. Figure 4: Our accelerated Recursive RLS-Nystr?m, nearly matches the per sample runtime of random Fourier features and uniform Nystr?m while still providing much better approximation. 5.2 Additional Empirical Results In Appendix G we verify the usefulness of our kernel approximations in downstream learning tasks. While full kernel methods do not scale to our large datasets, Recursive RLS-Nystr?m does since its runtime depends linearly on n. For example, on YearPredictionMSD the method requires 307 sec. (averaged over 5 trials) to build a 2, 000 landmark Nystr?m approximation for 463,716 training points. Ridge regression using the approximate kernel then requires 208 sec. for a total of 515 sec. These runtime are comparable to those of the very fast random Fourier features method, which underperforms RLS-Nystr?m in terms of regression and classification accuracy. Acknowledgements We would like to thank Michael Mahoney for bringing the potential of ridge leverage scores to our attention and suggesting their possible approximation via iterative sampling schemes. We would also like to thank Michael Cohen for pointing out (and fixing) an error in our original manuscript and generally for his close collaboration in our work on leverage score sampling algorithms. 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Interpolated Policy Gradient: Merging On-Policy and Off-Policy Gradient Estimation for Deep Reinforcement Learning Shixiang Gu University of Cambridge Max Planck Institute [email protected] Richard E. Turner University of Cambridge [email protected] Timothy Lillicrap DeepMind [email protected] Bernhard Sch?lkopf Max Planck Institute [email protected] Zoubin Ghahramani University of Cambridge Uber AI Labs [email protected] Sergey Levine UC Berkeley [email protected] Abstract Off-policy model-free deep reinforcement learning methods using previously collected data can improve sample efficiency over on-policy policy gradient techniques. On the other hand, on-policy algorithms are often more stable and easier to use. This paper examines, both theoretically and empirically, approaches to merging on- and off-policy updates for deep reinforcement learning. Theoretical results show that off-policy updates with a value function estimator can be interpolated with on-policy policy gradient updates whilst still satisfying performance bounds. Our analysis uses control variate methods to produce a family of policy gradient algorithms, with several recently proposed algorithms being special cases of this family. We then provide an empirical comparison of these techniques with the remaining algorithmic details fixed, and show how different mixing of off-policy gradient estimates with on-policy samples contribute to improvements in empirical performance. The final algorithm provides a generalization and unification of existing deep policy gradient techniques, has theoretical guarantees on the bias introduced by off-policy updates, and improves on the state-of-the-art model-free deep RL methods on a number of OpenAI Gym continuous control benchmarks. 1 Introduction Reinforcement learning (RL) studies how an agent that interacts sequentially with an environment can learn from rewards to improve its behavior and optimize long-term returns. Recent research has demonstrated that deep networks can be successfully combined with RL techniques to solve difficult control problems. Some of these include robotic control (Schulman et al., 2016; Lillicrap et al., 2016; Levine et al., 2016), computer games (Mnih et al., 2015), and board games (Silver et al., 2016). One of the simplest ways to learn a neural network policy is to collect a batch of behavior wherein the policy is used to act in the world, and then compute and apply a policy gradient update from this data. This is referred to as on-policy learning because all of the updates are made using data that was collected from the trajectory distribution induced by the current policy of the agent. It is straightforward to compute unbiased on-policy gradients, and practical on-policy gradient algorithms tend to be stable and relatively easy to use. A major drawback of such methods is that they tend to be data inefficient, because they only look at each data point once. Off-policy algorithms based on Q-learning and actor-critic learning (Sutton et al., 1999) have also proven to be an effective approach 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. to deep RL such as in (Mnih et al., 2015) and (Lillicrap et al., 2016). Such methods reuse samples by storing them in a memory replay buffer and train a value function or Q-function with off-policy updates. This improves data efficiency, but often at a cost in stability and ease of use. Both on- and off-policy learning techniques have their own advantages. Most recent research has worked with on-policy algorithms or off-policy algorithms, and a few recent methods have sought to make use of both on- and off-policy data for learning (Gu et al., 2017; Wang et al., 2017; O?Donoghue et al., 2017). Such algorithms hope to gain advantages from both modes of learning, whilst avoiding their limitations. Broadly speaking, there have been two basic approaches in recently proposed algorithms that make use of both on- and off-policy data and updates. The first approach is to mix some ratio of on- and off-policy gradients or update steps in order to update a policy, as in the ACER and PGQ algorithms (Wang et al., 2017; O?Donoghue et al., 2017). In this case, there are no theoretical bounds on the error induced by incorporating off-policy updates. In the second approach, an off-policy Q critic is trained but is used as a control variate to reduce on-policy gradient variance, as in the Q-prop algorithm (Gu et al., 2017). This case does not introduce additional bias to the gradient estimator, but the policy updates do not use off-policy data. We seek to unify these two approaches using the method of control variates. We introduce a parameterized family of policy gradient methods that interpolate between on-policy and off-policy learning. Such methods are in general biased, but we show that the bias can be bounded.We show that a number of recent methods (Gu et al., 2017; Wang et al., 2017; O?Donoghue et al., 2017) can be viewed as special cases of this more general family. Furthermore, our empirical results show that in most cases, a mix of policy gradient and actor-critic updates achieves the best results, demonstrating the value of considering interpolated policy gradients. 2 Preliminaries A key component of our interpolated policy gradient method is the use of control variates to mix likelihood ratio gradients with deterministic gradient estimates obtained explicitly from a state-action critic. In this section, we summarize both likelihood ratio and deterministic gradient methods, as well as how control variates can be used to combine these two approaches. 2.1 On-Policy Likelihood Ratio Policy Gradient At time t, the RL agent in state st takes action at according to its policy ?(at |st ), the state transitions to st+1 , and the agent receives a reward r(st , at ). For a parametrized policy ?? , P the objective is to ? maximize the ?-discounted cumulative future return J(?) = J(?) = Es0 ,a0 ,????? [ t=0 ? t r(st , at )]. Monte Carlo policy gradient methods, such as REINFORCE (Williams, 1992) and TRPO (Schulman et al., 2015), use the likelihood ratio policy gradient of the RL objective, ? t , at ) ? b(st ))] = E?? ,? [?? log ?? (at |st )A(s ? t , at )], (1) ?? J(?) = E?? ,? [?? log ?? (at |st )(Q(s ? t , at ) = P?0 ? t0 ?t r(st0 , at0 ) is the Monte Carlo estimate of the ?critic? Q? (st , at ) = where Q(s t =t ? t , at )], and ?? = P? ? t p(st = s) are the unnormalized state visitation Est+1 ,at+1 ,?????|st ,at [Q(s t=0 frequencies, while b(st ) is known as the baseline, and serves to reduce the variance of the gradient estimate (Williams, 1992). If the baseline estimates the value function, V ? (st ) = Eat ??(?|st ) [Q? (st , at )], ? t ) is an estimate of the advantage function A? (st , at ) = Q? (st , at ) ? V ? (st ). Likelihood then A(s ratio policy gradient methods use unbiased gradient estimates (except for the technicality detailed by Thomas (2014)), but they often suffer from high variance and are sample-intensive. 2.2 Off-Policy Deterministic Policy Gradient Policy gradient methods with function approximation (Sutton et al., 1999), or actor-critic methods, are a family of policy gradient methods which first estimate the critic, or the value, of the policy by Qw ? Q? , and then greedily optimize the policy ?? with respect to Qw . While it is not necessary for such algorithms to be off-policy, we primarily analyze the off-policy variants, such as (Riedmiller, 2005; Degris et al., 2012; Heess et al., 2015; Lillicrap et al., 2016). For example, DDPG Lillicrap et al. (2016), which optimizes a continuous deterministic policy ?? (at |st ) = ?(at = ?? (st )), can be summarized by the following update equations, where Q0 denotes the target Q network and ? denotes 2 ? ? 6= ? ? 0 0 1 - CV No Yes No Examples REINFORCE (Williams, 1992),TRPO (Schulman et al., 2015) Q-Prop (Gu et al., 2017) DDPG (Silver et al., 2014; Lillicrap et al., 2016),SVG(0) (Heess et al., 2015) ?PGQ (O?Donoghue et al., 2017), ?ACER (Wang et al., 2017) Table 1: Prior policy gradient method objectives as special cases of IPG. some off-policy distribution, e.g. from experience replay (Lillicrap et al., 2016): yt = r(st , at ) + ?Q0 (st+1 , ?? (st+1 )) w ? arg min E? [(Qw (st , at ) ? yt )2 ], ? ? arg max E? [Qw (st , ?? (st ))]. (2) This provides the following deterministic policy gradient through the critic: ?? J(?) ? E?? [?? Qw (st , ?? (st ))]. (3) This policy gradient is generally biased due to the imperfect estimator Qw and off-policy state sampling from ?. Off-policy actor-critic algorithms therefore allow training the policy on off-policy samples, at the cost of introducing potentially unbounded bias into the gradient estimate. This usually makes off-policy algorithms less stable during learning, compared to on-policy algorithms using a large batch size for each update (Duan et al., 2016; Gu et al., 2017). 2.3 Off-Policy Control Variate Fitting The control variates method (Ross, 2006) is a general technique for variance reduction of a Monte Carlo estimator by exploiting a correlated variable for which we know more information such as analytical expectation. General control variates for RL include state-action baselines, and an example ? w , the first-order can be an off-policy fitted critic Qw . Q-Prop (Gu et al., 2017), for example, used Q Taylor expansion of Qw , as the control variates, and showed improvement in stability and sample efficiency of policy gradient methods. ?? here corresponds to the mean of the stochastic policy ?? . ? t , at ) ? Q ? w (st , at ))] + E?? [?? Qw (st , ?? (st ))]. ?? J(?) = E?? ,? [?? log ?? (at |st )(Q(s (4) The gradient estimator combines both likelihood ratio and deterministic policy gradients in Eq. 1 and 3. It has lower variance and stable gradient estimates and enables more sample-efficient learning. However, one limitation of Q-Prop is that it uses only on-policy samples for estimating the policy gradient. This ensures that the Q-Prop estimator remains unbiased, but limits the use of off-policy samples for further variance reduction. 3 Interpolated Policy Gradient ? Our proposed approach, interpolated policy gradient (IPG), mixes likelihood ratio gradient with Q, which provides unbiased but high-variance gradient estimation, and deterministic gradient through an off-policy fitted critic Qw , which provides low-variance but biased gradients. IPG directly interpolates the two terms from Eq. 1 and 3: ? t , at )] + ?E?? [?? Q ? ?w (st )], ?? J(?) ? (1 ? ?)E?? ,? [?? log ?? (at |st )A(s (5) ? w (st ) = where we generalized the deterministic policy gradient through the critic as ?? Q ? ?? E? [Qw (st , ?)]. This generalization is to make our analysis applicable with more general forms of the critic-based control variates, as discussed in the Appendix. This gradient estimator is biased from two sources: off-policy state sampling ?? , and inaccuracies in the critic Qw . However, as we show in Section 4, we can bound the biases for all the cases, and in some cases, the algorithm still guarantees monotonic convergence as in Kakade & Langford (2002); Schulman et al. (2015). 3.1 Control Variates for Interpolated Policy Gradient While IPG includes ? to trade off bias and variance directly, it contains a likelihood ratio gradient term, for which we can introduce a control variate (CV) Ross (2006) to further reduce the estimator variance. 3 ? ? (st ), The expression for the IPG with control variates is below, where A?w (st , at ) = Qw (st , at ) ? Q w ? t , at )] + ?E?? [?? Q ? ?w (st )] ?? J(?) ? (1 ? ?)E?? ,? [?? log ?? (at |st )A(s ? t , at ) ? A?w (st , at ))] = (1 ? ?)E?? ,? [?? log ?? (at |st )(A(s ? ? (st )] + ?E?? [?? Q ? ? (st )] + (1 ? ?)E?? [?? Q w w ? t , at ) ? A?w (st , at ))] + E?? [?? Q ? ?w (st )]. ? (1 ? ?)E?? ,? [?? log ?? (at |st )(A(s (6) The first approximation indicates the biased approximation from IPG, while the second approximation indicates replacing the ?? in the control variate correction term with ?? and merging with the last term. The second approximation is a design decision and introduces additional bias when ? 6= ? but it helps simplify the expression to be analyzed more easily, and the additional benefit from the variance reduction from the control variate could still outweigh this extra bias. The biases are analyzed in Section 4. The likelihood ratio gradient term is now proportional to the residual in on- and off-policy ? t , at ) ? A?w (st , at ), and therefore, we call this term residual likelihood ratio advantage estimates A(s gradient. Intuitively, if the off-policy critic estimate is accurate, this term has a low magnitude and the overall variance of the estimator is reduced. 3.2 Relationship to Prior Policy Gradient and Actor-Critic Methods Crucially, IPG allows interpolating a rich list of prior deep policy gradient methods using only three parameters: ?, ?, and the use of the control variate (CV). The connection is summarized in Table 1 and the algorithm is presented in Algorithm 1. Importantly, a wide range of prior work has only explored limiting cases of the spectrum, e.g. ? = 0, 1, with or without the control variate. Our work provides a thorough theoretical analysis of the biases, and in some cases performance guarantees, for each of the method in this spectrum and empirically demonstrates often the best performing algorithms are in the midst of the spectrum. Algorithm 1 Interpolated Policy Gradient input ?, ?, useCV 1: Initialize w for critic Qw , ? for stochastic policy ?? , and replay buffer R ? ?. 2: repeat 3: Roll-out ?? for E episodes, T time steps each, to collect a batch of data B = {s, a, r}1:T,1:E to R 4: Fit Qw using R and ?? , and fit baseline V? (st ) using B 5: Compute Monte Carlo advantage estimate A?t,e using B and V? 6: if useCV then 7: Compute critic-based advantage estimate A?t,e using B, Qw and ?? 8: Compute and center the learning signals lt,e = A?t,e ? A?t,e and set b = 1 9: else 10: Center the learning signals lt,e = A?t,e and set b = ? 11: end if 12: Multiply lt,e by (1 ? ?) 13: Sample D = s1:M from R Pand/or P B based on ? P b 1 ?? 14: Compute ?? J(?) ? ET e t ?? log ?? (at,e |st,e )lt,e + M m ?? Qw (sm ) 15: Update policy ?? using ?? J(?) 16: until ?? converges. 3.3 ? = 1: Actor-Critic methods Before presenting our theoretical analysis, an important special case to discuss is ? = 1, which corresponds to a deterministic actor-critic method. Several advantages of this special case include that the policy can be deterministic and the learning can be done completely off-policy, as it does not ? Prior work such as DDPG Lillicrap et al. (2016) have to estimate the on-policy Monte Carlo critic Q. and related Q-learning methods have proposed aggressive off-policy exploration strategy to exploit these properties of the algorithm. In this work, we compare alternatives such as using on-policy exploration and stochastic policy with classical DDPG algorithm designs, and show that in some domains the off-policy exploration can significantly deteriorate the performance. Theoretically, we confirm this empirical observation by showing that the bias from off-policy sampling in ? increases 4 monotonically with the total variation or KL divergence between ? and ?. Both the empirical and theoretical results indicate that well-designed actor-critic methods with an on-policy exploration strategy could be a more reliable alternative than with an on-policy exploration. 4 Theoretical Analysis In this section, we present a theoretical analysis of the bias in the interpolated policy gradient. This is crucial, since understanding the biases of the methods can improve our intuition about its performance and make it easier to design new algorithms in the future. Because IPG includes many prior methods as special cases, our analysis also applies to those methods and other intermediate cases. We first analyze a special case and derive results for general IPG. All proofs are in the Appendix. ? 6= ?, ? = 0: Policy Gradient with Control Variate and Off-Policy Sampling 4.1 This section provides an analysis of the special case of IPG with ? 6= ?, ? = 1, and the control variate. Plugging in to Eq. 6, we get an expression similar to Q-Prop in Eq. 4, ? t , at ) ? A? (st , at ))] + E?? [?? Q ? ? (st )], ?? J(?) ? E?? ,? [?? log ?? (at |st )(A(s w w (7) except that it also supports utilizing off-policy data for updating the policy. To analyze the bias for ? ? this gradient expression, we first introduce J(?, ? ), a local approximation to J(?), which has been used in prior theoretical work (Kakade & Langford, 2002; Schulman et al., 2015). The derivation and the bias from this approximation are discussed in the proof for Theorem 1 in the Appendix. ? ? J(?) = J(? ? ) + E?? ,? [A?? (st , at )] ? J(? ? ) + E??? ,? [A?? (st , at )] = J(?, ? ). (8) ? ? ? ? Note that J(?) = J(?, ? = ?) and ?? J(?) = ?? J(?, ? = ?). In practice, ? ? corresponds to policy ?k at iteration k and ? corresponds next policy ?k+1 after parameter update. Thus, this approximation is often sufficiently good. Next, we write the approximate objective for Eq. 7, ? ? ? ?) J??,?=0,CV (?, ? ? ) , J(? ? ) + E??? ,? [A?? (st , at ) ? A?w (st , at )] + E?? [A??,? w (st )] ? J(?, ? ? ? ? A??,? ? [Qw (st , ?)]. w (st ) = E? [Aw (st , ?)] = E? [Qw (st , ?)] ? E? (9) ? ? ? = ?) = J(?, ? = ?) = J(?), and ?? J??,?=0 (?, ? Note that J??,?=0 (?, ? ? = ?) equals Eq. 7. We can bound the absolute error between J??,?=0,CV (?, ? ? ) and J(?) by the following theorem, where max DKL (?i , ?j ) = maxs DKL (?i (?|s), ?j (?|s)) is the maximum KL divergence between ?i , ?j . ? ??,?? (s)|, then Theorem 1. If  = maxs |A??,? w (s)|, ? = maxs |A  q  q ? ?,?=0,CV max max ? (?, ? ? ) ? 2  DKL (? ? , ?) + ? DKL (?, ? ?) J(?) ? J (1 ? ?)2 1 Theorem 1 contains two terms: the second term confirms J??,?=0,CV is a local approximation around ? and deviates from J(?) as ? ? deviates, and the first term bounds the bias from off-policy sampling using the KL divergence between the policies ? ? and ?. This means that the algorithm fits well with policy gradient methods which constrain the KL divergence per policy update, such as covariant policy gradient (Bagnell & Schneider, 2003), natural policy gradient (Kakade & Langford, 2002), REPS (Peters et al., 2010), and trust-region policy optimization (TRPO) (Schulman et al., 2015). 4.1.1 Monotonic Policy Improvement Guarantee Some forms of on-policy policy gradient methods have theoretical guarantees on monotonic convergence Kakade & Langford (2002); Schulman et al. (2015). Such guarantees often correspond to stable empirical performance on challenging problems, even when some of the constraints are relaxed in practice (Schulman et al., 2015; Duan et al., 2016; Gu et al., 2017). We can show that a variant of IPG allows off-policy sampling while still guaranteeing monotonic convergence. The algorithm and the proof are provided in the appendix.This algorithm is usually impractical to implement; however, IPG with trust-region updates when ? 6= ?, ? = 1, CV = true approximates this monotonic algorithm, similar to how TRPO is an approximation to the theoretically monotonic algorithm proposed by Schulman et al. (2015). 5 4.2 General Bounds on the Interpolated Policy Gradient We can establish bias bounds for the general IPG algorithm, with and without the control variate, using Theorem 2. The additional term that contributes to the bias in the general case is ?, which represents the error between the advantage estimated by the off-policy critic and the true A? values. Theorem 2. If ? = maxs,a |A?? (s, a) ? A?? (s, a)|,  = maxs |A??,?? (s)|, ? = maxs |A??,?? (s)|, w ??,? w ??? ? J (?, ? ? ) , J(? ? ) + (1 ? ?)E??? ,? [A ] + ?E?? [A??,? w ] ? ? J??,?,CV (?, ? ? ) , J(? ? ) + (1 ? ?)E??? ,? [A??? ? A?w ] + E?? [A??,? w ]  q  q ?? ? max (? max (?, ? ? +2 D ? , ?) + ? D ? ) KL KL 1?? (1 ? ?)2 1  q  q ?? ? ?,?,CV max max ? (?, ? ? ) ?  DKL (? +2 ? , ?) + ? DKL (?, ? ?) J(?) ? J 1?? (1 ? ?)2 1 then, J(?) ? J??,? (?, ? ? ) ? This bound shows that the bias from directly mixing the deterministic policy gradient through ? comes from two terms: how well the critic Qw is approximating Q? , and how close the off-policy sampling policy is to the actor policy. We also show that the bias introduced is proportional to ? while the variance of the high variance likelihood ratio gradient term is proportional to (1 ? ?)2 , so ? allows directly trading off bias and variance. Theorem 2 fully bounds bias in the full spectrum of IPG methods; this enables us to analyze how biases arise and interact and help us design better algorithms. 5 Related Work An overarching aim of this paper is to help unify on-policy and off-policy policy gradient algorithms into a single conceptual framework. Our analysis examines how Q-Prop (Gu et al., 2017), PGQ (O?Donoghue et al., 2017), and ACER (Wang et al., 2017), which are all recent works that combine on-policy with off-policy learning, are connected to each other (see Table 1). IPG with 0 < ? < 1 and without the control variate relates closely to PGQ and ACER, but differ in the details. PGQ mixes in the Q-learning Bellman error objective, and ACER mixes parameter update steps rather than directly mixing gradients. And both PGQ and ACER come with numerous additional design details that make fair comparisons with methods like TRPO and Q-Prop difficult. We instead focus on the three minimal variables of IPG and explore their settings in relation to the closely related TRPO and Q-Prop methods, in order to theoretically and empirically understand in which situations we might expect gains from mixing on- and off-policy gradients. Asides from these more recent works, the use of off-policy samples with policy gradients has been a popular direction of research (Peshkin & Shelton, 2002; Jie & Abbeel, 2010; Degris et al., 2012; Levine & Koltun, 2013). Most of these methods rely on variants of importance sampling (IS) to correct for bias. The use of importance sampling ensures unbiased estimates, but at the cost of considerable variance, as quantified by the ESS measure used by Jie & Abbeel (2010). Ignoring importance weights produces bias but, as shown in our analysis, this bias can be bounded. Therefore, our IPG estimators have higher bias as the sampling distribution deviates from the policy, while IS methods have higher variance. Among these importance sampling methods, Levine & Koltun (2013) evaluates on tasks that are the most similar to our paper, but the focus is on using importance sampling to include demonstrations, rather than to speed up learning from scratch. Lastly, there are many methods that combine on- and off-policy data for policy evaluation (Precup, 2000; Mahmood et al., 2014; Munos et al., 2016), mostly through variants of importance sampling. Combining our methods with more sophisticated policy evaluation methods will likely lead to further improvements, as done in (Degris et al., 2012). A more detailed analysis of the effect of importance sampling on bias and variance is left to future work, where some of the relevant work includes Precup (2000); Jie & Abbeel (2010); Mahmood et al. (2014); Jiang & Li (2016); Thomas & Brunskill (2016). 6 Experiments In this section, we empirically show that the three parameters of IPG can interpolate different behaviors and often achieve superior performance versus prior methods that are limiting cases of this 6 (a) IPG with ? = 0 and the control variate. (b) IPG with ? = 1. Figure 1: (a) IPG-? = 0 vs Q-Prop on HalfCheetah-v1, with batch size 5000. IPG-?-rand30000, which uses 30000 random samples from the replay as samples from ?, outperforms Q-Prop in terms of learning speed. (b) IPG-?=1 vs other algorithms on Ant-v1. In this domain, on-policy IPG-?=1 with on-policy exploration significantly outperforms DDPG and IPG-?=1-OU, which use a heuristic OU (Ornstein?Uhlenbeck) process noise exploration strategy, and marginally outperforms Q-Prop. approach. Crucially, all methods share the same algorithmic structure as Algorithm 1, and we hold the rest of the experimental details fixed. All experiments were performed on MuJoCo domains in OpenAI Gym (Todorov et al., 2012; Brockman et al., 2016), with results presented for the average over three seeds. Additional experimental details are provided in the Appendix. 6.1 ? 6= ?, ? = 0, with the control variate We evaluate the performance of the special case of IPG discussed in Section 4.1. This case is of particular interest, since we can derive monotonic convergence results for a variant of this method under certain conditions, despite the presence of off-policy updates. Figure 1a shows the performance on the HalfCheetah-v1 domain, when the policy update batch size is 5000 transitions (i.e. 5 episodes). ?last? and ?rand? indicate if ? samples from the most recent transitions or uniformly from the experience replay. ?last05000? would be equivalent to Q-Prop given ? = 0. Comparing ?IPG-?rand05000? and ?Q-Prop? curves, we observe that by drawing the same number of samples randomly from the replay buffer for estimating the critic gradient, instead of using the on-policy samples, we get faster convergence. If we sample batches of size 30000 from the replay buffer, the performance further improves. However, as seen in the ?IPG-?-last30000? curve, if we instead use the 30000 most recent samples, the performance degrades. One possible explanation for this is that, while using random samples from the replay increases the bound on the bias according to Theorem 1, it also decorrelates the samples within the batch, providing more stable gradients. This is the original motivation for experience replay in the DQN method (Mnih et al., 2015), and we have shown that such decorrelated off-policy samples can similarly produce gains for policy gradient algorithms. See Table 2 for results on other domains. The results for this variant of IPG demonstrate that random sampling from the replay provides further improvement on top of Q-Prop. Note that these replay buffer samples are different from standard off-policy samples in DDPG or DQN algorithms, which often use aggressive heuristic exploration strategies. The samples used by IPG are sampled from prior policies that follow a conservative trust-region update, resulting in greater regularity but less exploration. In the next section, we show that in some cases, ensuring that the off-policy samples are not too off-policy is essential for good performance. 6.2 ? = ?, ? = 1 In this section, we empirically evaluate another special case of IPG, where ? = ?, indicating onpolicy sampling, and ? = 1, which reduces to a trust-region, on-policy variant of a deterministic actor-critic method. Although this algorithm performs actor-critic updates, the use of a trust region makes it more similar to TRPO or Q-Prop than DDPG. 7 IPG-?=0.2 IPG-cv-?=0.2 IPG-?=1 Q-Prop TRPO HalfCheetah-v1 ? = ? ? 6= ? 3356 3458 4216 4023 2962 4767 4178 4182 2889 N.A. Ant-v1 ? = ? ? 6= ? 4237 4415 3943 3421 3469 3780 3374 3479 1520 N.A. Walker-v1 ? = ? ? 6= ? 3047 1932 1896 1411 2704 805 2832 1692 1487 N.A. Humanoid-v1 ? = ? ? 6= ? 1231 920 1651 1613 1571 1530 1423 1519 615 N.A. Table 2: Comparisons on all domains with mini-batch size 10000 for Humanoid and 5000 otherwise. We compare the maximum of average test rewards in the first 10000 episodes (Humanoid requires more steps to fully converge; see the Appendix for learning curves). Results outperforming Q-Prop (or IPG-cv-?=0 with ? = ?) are boldface. The two columns show results with on-policy and off-policy samples for estimating the deterministic policy gradient. Results for all domains are shown in Table 2. Figure 1b shows the learning curves on Ant-v1. Although IPG-?=1 methods can be off-policy, the policy is updated every 5000 samples to keep it consistent with other IPG methods, while DDPG updates the policy on every step in the environment and makes other design choices Lillicrap et al. (2016). We see that, in this domain, standard DDPG becomes stuck with a mean reward of 1000, while IPG-?=1 improves monotonically, achieving a significantly better result. To investigate why this large discrepancy arises, we also ran IPG-?=1 with the same OU process exploration noise as DDPG, and observed large degradation in performance. This provides empirical support for Theorem 2. It is illuminating to contrast this result with the previous experiment, where the off-policy samples did not adversely alter the results. In the previous experiments, the samples came from Gaussian policies updated with trust-regions. The difference between ? and ? was therefore approximately bounded by the trust-regions. In the experiment with Brownian noise, the behaving policy uses temporally correlated noise, with potentially unbounded KL-divergence from the learned Gaussian policy. In this case, the off-policy samples result in excessive bias, wiping out the variance reduction benefits of off-policy sampling. In general, we observed that for the harder Ant-v1 and Walker-v1 domains, on-policy exploration is more effective, even when doing off-policy state sampling from a replay buffer. This results suggests the following lesson for designing off-policy actor-critic methods: for domains where exploration is difficult, it may be more effective to use on-policy exploration with bounded policy updates than to design heuristic exploration rules such as the OU process noise, due to the resulting reduction in bias. 6.3 General Cases of Interpolated Policy Gradient Table 2 shows the results for experiments where we compare IPG methods with varying values of ?; additional results are provided in the Appendix. ? 6= ? indicates that the method uses off-policy samples from the replay buffer, with the same batch size as the on-policy batch for fair comparison. We ran sweeps over ? = {0.2, 0.4, 0.6, 0.8} and found that ? = 0.2 consistently produce better performance than Q-Prop, TRPO or prior actor-critic methods. This is consistent with the results in PGQ (O?Donoghue et al., 2017) and ACER (Wang et al., 2017), which found that their equivalent of ? = 0.1 performed best on their benchmarks. Importantly, we compared all methods with the same algorithm designs (exploration, policy, etc.), since Q-Prop and TRPO are IPG-?=0 with and without the control variate. IPG-?=1 is a novel variant of the actor-critic method that differs from DDPG (Lillicrap et al., 2016) and SVG(0) (Heess et al., 2015) due to the use of a trust region. The results in Table 2 suggest that, in most cases, the best performing algorithm is one that interpolates between the policy-gradient and actor-critic variants, with intermediate values of ?. 7 Discussion In this paper, we introduced interpolated policy gradient methods, a family of policy gradient algorithms that allow mixing off-policy learning with on-policy learning while satisfying performance bounds. This family of algorithms unifies and interpolates on-policy likelihood ratio policy gradient and off-policy deterministic policy gradient, and includes a number of prior works as approximate limiting cases. Empirical results confirm that, in many cases, interpolated gradients have improved sample-efficiency and stability over the prior state-of-the-art methods, and the theoretical results provide intuition for analyzing the cases in which the different methods perform well or poorly. Our hope is that this detailed analysis of interpolated gradient methods can not only provide for more effective algorithms in practice, but also give useful insight for future algorithm design. 8 Acknowledgements This work is supported by generous sponsorship from Cambridge-T?bingen PhD Fellowship, NSERC, and Google Focused Research Award. References Bagnell, J Andrew and Schneider, Jeff. Covariant policy search. IJCAI, 2003. 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Dynamic Routing Between Capsules Sara Sabour Nicholas Frosst Geoffrey E. Hinton Google Brain Toronto {sasabour, frosst, geoffhinton}@google.com Abstract A capsule is a group of neurons whose activity vector represents the instantiation parameters of a specific type of entity such as an object or an object part. We use the length of the activity vector to represent the probability that the entity exists and its orientation to represent the instantiation parameters. Active capsules at one level make predictions, via transformation matrices, for the instantiation parameters of higher-level capsules. When multiple predictions agree, a higher level capsule becomes active. We show that a discrimininatively trained, multi-layer capsule system achieves state-of-the-art performance on MNIST and is considerably better than a convolutional net at recognizing highly overlapping digits. To achieve these results we use an iterative routing-by-agreement mechanism: A lower-level capsule prefers to send its output to higher level capsules whose activity vectors have a big scalar product with the prediction coming from the lower-level capsule. 1 Introduction Human vision ignores irrelevant details by using a carefully determined sequence of fixation points to ensure that only a tiny fraction of the optic array is ever processed at the highest resolution. Introspection is a poor guide to understanding how much of our knowledge of a scene comes from the sequence of fixations and how much we glean from a single fixation, but in this paper we will assume that a single fixation gives us much more than just a single identified object and its properties. We assume that our multi-layer visual system creates a parse tree-like structure on each fixation, and we ignore the issue of how these single-fixation parse trees are coordinated over multiple fixations. Parse trees are generally constructed on the fly by dynamically allocating memory. Following Hinton et al. [2000], however, we shall assume that, for a single fixation, a parse tree is carved out of a fixed multilayer neural network like a sculpture is carved from a rock. Each layer will be divided into many small groups of neurons called ?capsules? (Hinton et al. [2011]) and each node in the parse tree will correspond to an active capsule. Using an iterative routing process, each active capsule will choose a capsule in the layer above to be its parent in the tree. For the higher levels of a visual system, this iterative process will be solving the problem of assigning parts to wholes. The activities of the neurons within an active capsule represent the various properties of a particular entity that is present in the image. These properties can include many different types of instantiation parameter such as pose (position, size, orientation), deformation, velocity, albedo, hue, texture, etc. One very special property is the existence of the instantiated entity in the image. An obvious way to represent existence is by using a separate logistic unit whose output is the probability that the entity exists. In this paper we explore an interesting alternative which is to use the overall length of the vector of instantiation parameters to represent the existence of the entity and to force the orientation 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. of the vector to represent the properties of the entity1 . We ensure that the length of the vector output of a capsule cannot exceed 1 by applying a non-linearity that leaves the orientation of the vector unchanged but scales down its magnitude. The fact that the output of a capsule is a vector makes it possible to use a powerful dynamic routing mechanism to ensure that the output of the capsule gets sent to an appropriate parent in the layer above. Initially, the output is routed to all possible parents but is scaled down by coupling coefficients that sum to 1. For each possible parent, the capsule computes a ?prediction vector? by multiplying its own output by a weight matrix. If this prediction vector has a large scalar product with the output of a possible parent, there is top-down feedback which increases the coupling coefficient for that parent and decreasing it for other parents. This increases the contribution that the capsule makes to that parent thus further increasing the scalar product of the capsule?s prediction with the parent?s output. This type of ?routing-by-agreement? should be far more effective than the very primitive form of routing implemented by max-pooling, which allows neurons in one layer to ignore all but the most active feature detector in a local pool in the layer below. We demonstrate that our dynamic routing mechanism is an effective way to implement the ?explaining away? that is needed for segmenting highly overlapping objects. Convolutional neural networks (CNNs) use translated replicas of learned feature detectors. This allows them to translate knowledge about good weight values acquired at one position in an image to other positions. This has proven extremely helpful in image interpretation. Even though we are replacing the scalar-output feature detectors of CNNs with vector-output capsules and max-pooling with routing-by-agreement, we would still like to replicate learned knowledge across space. To achieve this, we make all but the last layer of capsules be convolutional. As with CNNs, we make higher-level capsules cover larger regions of the image. Unlike max-pooling however, we do not throw away information about the precise position of the entity within the region. For low level capsules, location information is ?place-coded? by which capsule is active. As we ascend the hierarchy, more and more of the positional information is ?rate-coded? in the real-valued components of the output vector of a capsule. This shift from place-coding to rate-coding combined with the fact that higher-level capsules represent more complex entities with more degrees of freedom suggests that the dimensionality of capsules should increase as we ascend the hierarchy. 2 How the vector inputs and outputs of a capsule are computed There are many possible ways to implement the general idea of capsules. The aim of this paper is not to explore this whole space but simply to show that one fairly straightforward implementation works well and that dynamic routing helps. We want the length of the output vector of a capsule to represent the probability that the entity represented by the capsule is present in the current input. We therefore use a non-linear "squashing" function to ensure that short vectors get shrunk to almost zero length and long vectors get shrunk to a length slightly below 1. We leave it to discriminative learning to make good use of this non-linearity. vj = ||sj ||2 sj 1 + ||sj ||2 ||sj || (1) where vj is the vector output of capsule j and sj is its total input. For all but the first layer of capsules, the total input to a capsule sj is a weighted sum over all ?prediction vectors? u ? j|i from the capsules in the layer below and is produced by multiplying the output ui of a capsule in the layer below by a weight matrix Wij X sj = cij u ? j|i , u ? j|i = Wij ui (2) i where the cij are coupling coefficients that are determined by the iterative dynamic routing process. The coupling coefficients between capsule i and all the capsules in the layer above sum to 1 and are determined by a ?routing softmax? whose initial logits bij are the log prior probabilities that capsule i 1 This makes biological sense as it does not use large activities to get accurate representations of things that probably don?t exist. 2 should be coupled to capsule j. exp(bij ) (3) cij = P k exp(bik ) The log priors can be learned discriminatively at the same time as all the other weights. They depend on the location and type of the two capsules but not on the current input image2 . The initial coupling coefficients are then iteratively refined by measuring the agreement between the current output vj of each capsule, j, in the layer above and the prediction u ? j|i made by capsule i. The agreement is simply the scalar product aij = vj .? uj|i . This agreement is treated as if it was a log likelihood and is added to the initial logit, bij before computing the new values for all the coupling coefficients linking capsule i to higher level capsules. In convolutional capsule layers, each capsule outputs a local grid of vectors to each type of capsule in the layer above using different transformation matrices for each member of the grid as well as for each type of capsule. Procedure 1 Routing algorithm. 1: procedure ROUTING(? uj|i , r, l) 2: for all capsule i in layer l and capsule j in layer (l + 1): bij ? 0. 3: for r iterations do 4: for all capsule i in layer l: ci ? softmax(b . softmax computes Eq. 3 P i) 5: for all capsule j in layer (l + 1): sj ? i cij u ? j|i 6: for all capsule j in layer (l + 1): vj ? squash(sj ) . squash computes Eq. 1 7: for all capsule i in layer l and capsule j in layer (l + 1): bij ? bij + u ? j|i .vj return vj 3 Margin loss for digit existence We are using the length of the instantiation vector to represent the probability that a capsule?s entity exists. We would like the top-level capsule for digit class k to have a long instantiation vector if and only if that digit is present in the image. To allow for multiple digits, we use a separate margin loss, Lk for each digit capsule, k: Lk = Tk max(0, m+ ? ||vk ||)2 + ? (1 ? Tk ) max(0, ||vk || ? m? )2 (4) where Tk = 1 iff a digit of class k is present3 and m+ = 0.9 and m? = 0.1. The ? down-weighting of the loss for absent digit classes stops the initial learning from shrinking the lengths of the activity vectors of all the digit capsules. We use ? = 0.5. The total loss is simply the sum of the losses of all digit capsules. 4 CapsNet architecture A simple CapsNet architecture is shown in Fig. 1. The architecture is shallow with only two convolutional layers and one fully connected layer. Conv1 has 256, 9 ? 9 convolution kernels with a stride of 1 and ReLU activation. This layer converts pixel intensities to the activities of local feature detectors that are then used as inputs to the primary capsules. The primary capsules are the lowest level of multi-dimensional entities and, from an inverse graphics perspective, activating the primary capsules corresponds to inverting the rendering process. This is a very different type of computation than piecing instantiated parts together to make familiar wholes, which is what capsules are designed to be good at. The second layer (PrimaryCapsules) is a convolutional capsule layer with 32 channels of convolutional 8D capsules (i.e. each primary capsule contains 8 convolutional units with a 9 ? 9 kernel and a stride of 2). Each primary capsule output sees the outputs of all 256 ? 81 Conv1 units whose receptive 2 For MNIST we found that it was sufficient to set all of these priors to be equal. We do not allow an image to contain two instances of the same digit class. We address this weakness of capsules in the discussion section. 3 3 Figure 1: A simple CapsNet with 3 layers. This model gives comparable results to deep convolutional networks (such as Chang and Chen [2015]). The length of the activity vector of each capsule in DigitCaps layer indicates presence of an instance of each class and is used to calculate the classification loss. Wij is a weight matrix between each ui , i ? (1, 32 ? 6 ? 6) in PrimaryCapsules and vj , j ? (1, 10). Figure 2: Decoder structure to reconstruct a digit from the DigitCaps layer representation. The euclidean distance between the image and the output of the Sigmoid layer is minimized during training. We use the true label as reconstruction target during training. fields overlap with the location of the center of the capsule. In total PrimaryCapsules has [32 ? 6 ? 6] capsule outputs (each output is an 8D vector) and each capsule in the [6 ? 6] grid is sharing their weights with each other. One can see PrimaryCapsules as a Convolution layer with Eq. 1 as its block non-linearity. The final Layer (DigitCaps) has one 16D capsule per digit class and each of these capsules receives input from all the capsules in the layer below. We have routing only between two consecutive capsule layers (e.g. PrimaryCapsules and DigitCaps). Since Conv1 output is 1D, there is no orientation in its space to agree on. Therefore, no routing is used between Conv1 and PrimaryCapsules. All the routing logits (bij ) are initialized to zero. Therefore, initially a capsule output (ui ) is sent to all parent capsules (v0 ...v9 ) with equal probability (cij ). Our implementation is in TensorFlow (Abadi et al. [2016]) and we use the Adam optimizer (Kingma and Ba [2014]) with its TensorFlow default parameters, including the exponentially decaying learning rate, to minimize the sum of the margin losses in Eq. 4. 4.1 Reconstruction as a regularization method We use an additional reconstruction loss to encourage the digit capsules to encode the instantiation parameters of the input digit. During training, we mask out all but the activity vector of the correct digit capsule. Then we use this activity vector to reconstruct the input image. The output of the digit capsule is fed into a decoder consisting of 3 fully connected layers that model the pixel intensities as described in Fig. 2. We minimize the sum of squared differences between the outputs of the logistic units and the pixel intensities. We scale down this reconstruction loss by 0.0005 so that it does not dominate the margin loss during training. As illustrated in Fig. 3 the reconstructions from the 16D output of the CapsNet are robust while keeping only important details. 4 Figure 3: Sample MNIST test reconstructions of a CapsNet with 3 routing iterations. (l, p, r) represents the label, the prediction and the reconstruction target respectively. The two rightmost columns show two reconstructions of a failure example and it explains how the model confuses a 5 and a 3 in this image. The other columns are from correct classifications and shows that model preserves many of the details while smoothing the noise. (l, p, r) (2, 2, 2) (5, 5, 5) (8, 8, 8) (9, 9, 9) (5, 3, 5) (5, 3, 3) Input Output Table 1: CapsNet classification test accuracy. The MNIST average and standard deviation results are reported from 3 trials. Method Routing Reconstruction MNIST (%) MultiMNIST (%) Baseline CapsNet CapsNet CapsNet CapsNet 5 1 1 3 3 0.39 0.34?0.032 0.29?0.011 0.35?0.036 0.25?0.005 no yes no yes 8.1 7.5 5.2 Capsules on MNIST Training is performed on 28 ? 28 MNIST (LeCun et al. [1998]) images that have been shifted by up to 2 pixels in each direction with zero padding. No other data augmentation/deformation is used. The dataset has 60K and 10K images for training and testing respectively. We test using a single model without any model averaging. Wan et al. [2013] achieves 0.21% test error with ensembling and augmenting the data with rotation and scaling. They achieve 0.39% without them. We get a low test error (0.25%) on a 3 layer network previously only achieved by deeper networks. Tab. 1 reports the test error rate on MNIST for different CapsNet setups and shows the importance of routing and reconstruction regularizer. Adding the reconstruction regularizer boosts the routing performance by enforcing the pose encoding in the capsule vector. The baseline is a standard CNN with three convolutional layers of 256, 256, 128 channels. Each has 5x5 kernels and stride of 1. The last convolutional layers are followed by two fully connected layers of size 328, 192. The last fully connected layer is connected with dropout to a 10 class softmax layer with cross entropy loss. The baseline is also trained on 2-pixel shifted MNIST with Adam optimizer. The baseline is designed to achieve the best performance on MNIST while keeping the computation cost as close as to CapsNet. In terms of number of parameters the baseline has 35.4M while CapsNet has 8.2M parameters and 6.8M parameters without the reconstruction subnetwork. 5.1 What the individual dimensions of a capsule represent Since we are passing the encoding of only one digit and zeroing out other digits, the dimensions of a digit capsule should learn to span the space of variations in the way digits of that class are instantiated. These variations include stroke thickness, skew and width. They also include digit-specific variations such as the length of the tail of a 2. We can see what the individual dimensions represent by making use of the decoder network. After computing the activity vector for the correct digit capsule, we can feed a perturbed version of this activity vector to the decoder network and see how the perturbation affects the reconstruction. Examples of these perturbations are shown in Fig. 4. We found that one dimension (out of 16) of the capsule almost always represents the width of the digit. While some dimensions represent combinations of global variations, there are other dimensions that represent 5 Figure 4: Dimension perturbations. Each row shows the reconstruction when one of the 16 dimensions in the DigitCaps representation is tweaked by intervals of 0.05 in the range [?0.25, 0.25]. Scale and thickness Localized part Stroke thickness Localized skew Width and translation Localized part variation in a localized part of the digit. For example, different dimensions are used for the length of the ascender of a 6 and the size of the loop. 5.2 Robustness to Affine Transformations Experiments show that each DigitCaps capsule learns a more robust representation for each class than a traditional convolutional network. Because there is natural variance in skew, rotation, style, etc in hand written digits, the trained CapsNet is moderately robust to small affine transformations of the training data. To test the robustness of CapsNet to affine transformations, we trained a CapsNet and a traditional convolutional network (with MaxPooling and DropOut) on a padded and translated MNIST training set, in which each example is an MNIST digit placed randomly on a black background of 40 ? 40 pixels. We then tested this network on the affNIST4 data set, in which each example is an MNIST digit with a random small affine transformation. Our models were never trained with affine transformations other than translation and any natural transformation seen in the standard MNIST. An under-trained CapsNet with early stopping which achieved 99.23% accuracy on the expanded MNIST test set achieved 79% accuracy on the affnist test set. A traditional convolutional model with a similar number of parameters which achieved similar accuracy (99.22%) on the expanded mnist test set only achieved 66% on the affnist test set. 6 Segmenting highly overlapping digits Dynamic routing can be viewed as a parallel attention mechanism that allows each capsule at one level to attend to some active capsules at the level below and to ignore others. This should allow the model to recognize multiple objects in the image even if objects overlap. Hinton et al. propose the task of segmenting and recognizing highly overlapping digits (Hinton et al. [2000] and others have tested their networks in a similar domain (Goodfellow et al. [2013], Ba et al. [2014], Greff et al. [2016]). The routing-by-agreement should make it possible to use a prior about the shape of objects to help segmentation and it should obviate the need to make higher-level segmentation decisions in the domain of pixels. 6.1 MultiMNIST dataset We generate the MultiMNIST training and test dataset by overlaying a digit on top of another digit from the same set (training or test) but different class. Each digit is shifted up to 4 pixels in each direction resulting in a 36 ? 36 image. Considering a digit in a 28 ? 28 image is bounded in a 20 ? 20 box, two digits bounding boxes on average have 80% overlap. For each digit in the MNIST dataset we generate 1K MultiMNIST examples. So the training set size is 60M and the test set size is 10M. 4 Available at http://www.cs.toronto.edu/~tijmen/affNIST/. 6 Figure 5: Sample reconstructions of a CapsNet with 3 routing iterations on MultiMNIST test dataset. The two reconstructed digits are overlayed in green and red as the lower image. The upper image shows the input image. L:(l1 , l2 ) represents the label for the two digits in the image and R:(r1 , r2 ) represents the two digits used for reconstruction. The two right most columns show two examples with wrong classification reconstructed from the label and from the prediction (P). In the (2, 8) example the model confuses 8 with a 7 and in (4, 9) it confuses 9 with 0. The other columns have correct classifications and show that the model accounts for all the pixels while being able to assign one pixel to two digits in extremely difficult scenarios (column 1 ? 4). Note that in dataset generation the pixel values are clipped at 1. The two columns with the (*) mark show reconstructions from a digit that is neither the label nor the prediction. These columns suggests that the model is not just finding the best fit for all the digits in the image including the ones that do not exist. Therefore in case of (5, 0) it cannot reconstruct a 7 because it knows that there is a 5 and 0 that fit best and account for all the pixels. Also, in case of (8, 1) the loop of 8 has not triggered 0 because it is already accounted for by 8. Therefore it will not assign one pixel to two digits if one of them does not have any other support. R:(2, 7) R:(6, 0) R:(6, 8) R:(7, 1) *R:(5, 7) *R:(2, 3) R:(2, 8) R:P:(2, 7) L:(2, 7) L:(6, 0) L:(6, 8) L:(7, 1) L:(5, 0) L:(4, 3) L:(2, 8) L:(2, 8) R:(8, 7) L:(8, 7) 6.2 R:(9, 4) L:(9, 4) R:(9, 5) L:(9, 5) R:(8, 4) L:(8, 4) *R:(0, 8) *R:(1, 6) L:(1, 8) L:(7, 6) R:(4, 9) L:(4, 9) R:P:(4, 0) L:(4, 9) MultiMNIST results Our 3 layer CapsNet model trained from scratch on MultiMNIST training data achieves higher test classification accuracy than our baseline convolutional model. We are achieving the same classification error rate of 5.0% on highly overlapping digit pairs as the sequential attention model of Ba et al. [2014] achieves on a much easier task that has far less overlap (80% overlap of the boxes around the two digits in our case vs < 4% for Ba et al. [2014]). On test images, which are composed of pairs of images from the test set, we treat the two most active digit capsules as the classification produced by the capsules network. During reconstruction we pick one digit at a time and use the activity vector of the chosen digit capsule to reconstruct the image of the chosen digit (we know this image because we used it to generate the composite image). The only difference with our MNIST model is that we increased the period of the decay step for the learning rate to be 10? larger because the training dataset is larger. The reconstructions illustrated in Fig. 5 show that CapsNet is able to segment the image into the two original digits. Since this segmentation is not at pixel level we observe that the model is able to deal correctly with the overlaps (a pixel is on in both digits) while accounting for all the pixels. The position and the style of each digit is encoded in DigitCaps. The decoder has learned to reconstruct a digit given the encoding. The fact that it is able to reconstruct digits regardless of the overlap shows that each digit capsule can pick up the style and position from the votes it is receiving from PrimaryCapsules layer. 7 Tab. 1 emphasizes the importance of capsules with routing on this task. As a baseline for the classification of CapsNet accuracy we trained a convolution network with two convolution layers and two fully connected layers on top of them. The first layer has 512 convolution kernels of size 9 ? 9 and stride 1. The second layer has 256 kernels of size 5 ? 5 and stride 1. After each convolution layer the model has a pooling layer of size 2 ? 2 and stride 2. The third layer is a 1024D fully connected layer. All three layers have ReLU non-linearities. The final layer of 10 units is fully connected. We use the TensorFlow default Adam optimizer (Kingma and Ba [2014]) to train a sigmoid cross entropy loss on the output of final layer. This model has 24.56M parameters which is 2 times more parameters than CapsNet with 11.36M parameters. We started with a smaller CNN (32 and 64 convolutional kernels of 5 ? 5 and stride of 1 and a 512D fully connected layer) and incrementally increased the width of the network until we reached the best test accuracy on a 10K subset of the MultiMNIST data. We also searched for the right decay step on the 10K validation set. We decode the two most active DigitCaps capsules one at a time and get two images. Then by assigning any pixel with non-zero intensity to each digit we get the segmentation results for each digit. 7 Other datasets We tested our capsule model on CIFAR10 and achieved 10.6% error with an ensemble of 7 models each of which is trained with 3 routing iterations on 24 ? 24 patches of the image. Each model has the same architecture as the simple model we used for MNIST except that there are three color channels and we used 64 different types of primary capsule. We also found that it helped to introduce a "none-of-the-above" category for the routing softmaxes, since we do not expect the final layer of ten capsules to explain everything in the image. 10.6% test error is about what standard convolutional nets achieved when they were first applied to CIFAR10 (Zeiler and Fergus [2013]). One drawback of Capsules which it shares with generative models is that it likes to account for everything in the image so it does better when it can model the clutter than when it just uses an additional ?orphan? category in the dynamic routing. In CIFAR-10, the backgrounds are much too varied to model in a reasonable sized net which helps to account for the poorer performance. We also tested the exact same architecture as we used for MNIST on smallNORB (LeCun et al. [2004]) and achieved 2.7% test error rate, which is on-par with the state-of-the-art (Cire?san et al. [2011]). The smallNORB dataset consists of 96x96 stereo grey-scale images. We resized the images to 48x48 and during training processed random 32x32 crops of them. We passed the central 32x32 patch during test. We also trained a smaller network on the small training set of SVHN (Netzer et al. [2011]) with only 73257 images. We reduced the number of first convolutional layer channels to 64, the primary capsule layer to 16 6D-capsules with 8D final capsule layer at the end and achieved 4.3% on the test set. 8 Discussion and previous work For thirty years, the state-of-the-art in speech recognition used hidden Markov models with Gaussian mixtures as output distributions. These models were easy to learn on small computers, but they had a representational limitation that was ultimately fatal: The one-of-n representations they use are exponentially inefficient compared with, say, a recurrent neural network that uses distributed representations. To double the amount of information that an HMM can remember about the string it has generated so far, we need to square the number of hidden nodes. For a recurrent net we only need to double the number of hidden neurons. Now that convolutional neural networks have become the dominant approach to object recognition, it makes sense to ask whether there are any exponential inefficiencies that may lead to their demise. A good candidate is the difficulty that convolutional nets have in generalizing to novel viewpoints. The ability to deal with translation is built in, but for the other dimensions of an affine transformation we have to chose between replicating feature detectors on a grid that grows exponentially with the number of dimensions, or increasing the size of the labelled training set in a similarly exponential way. Capsules (Hinton et al. [2011]) avoid these exponential inefficiencies by converting pixel intensities 8 into vectors of instantiation parameters of recognized fragments and then applying transformation matrices to the fragments to predict the instantiation parameters of larger fragments. Transformation matrices that learn to encode the intrinsic spatial relationship between a part and a whole constitute viewpoint invariant knowledge that automatically generalizes to novel viewpoints. Hinton et al. [2011] proposed transforming autoencoders to generate the instantiation parameters of the PrimaryCapsule layer and their system required transformation matrices to be supplied externally. We propose a complete system that also answers "how larger and more complex visual entities can be recognized by using agreements of the poses predicted by active, lower-level capsules". Capsules make a very strong representational assumption: At each location in the image, there is at most one instance of the type of entity that a capsule represents. This assumption, which was motivated by the perceptual phenomenon called "crowding" (Pelli et al. [2004]), eliminates the binding problem (Hinton [1981a]) and allows a capsule to use a distributed representation (its activity vector) to encode the instantiation parameters of the entity of that type at a given location. This distributed representation is exponentially more efficient than encoding the instantiation parameters by activating a point on a high-dimensional grid and with the right distributed representation, capsules can then take full advantage of the fact that spatial relationships can be modelled by matrix multiplies. Capsules use neural activities that vary as viewpoint varies rather than trying to eliminate viewpoint variation from the activities. This gives them an advantage over "normalization" methods like spatial transformer networks (Jaderberg et al. [2015]): They can deal with multiple different affine transformations of different objects or object parts at the same time. Capsules are also very good for dealing with segmentation, which is another of the toughest problems in vision, because the vector of instantiation parameters allows them to use routing-by-agreement, as we have demonstrated in this paper. The importance of dynamic routing procedure is also backed by biologically plausible models of invarient pattern recognition in the visual cortex. Hinton [1981b] proposes dynamic connections and canonical object based frames of reference to generate shape descriptions that can be used for object recognition. Olshausen et al. [1993] improves upon Hinton [1981b] dynamic connections and presents a biologically plausible, position and scale invariant model of object representations. Research on capsules is now at a similar stage to research on recurrent neural networks for speech recognition at the beginning of this century. There are fundamental representational reasons for believing that it is a better approach but it probably requires a lot more small insights before it can out-perform a highly developed technology. The fact that a simple capsules system already gives unparalleled performance at segmenting overlapping digits is an early indication that capsules are a direction worth exploring. Acknowledgement. Of the many who provided us with constructive comments, we are specially grateful to Robert Gens, Eric Langlois, Vincent Vanhoucke, Chris Williams, and the reviewers for their fruitful comments and corrections. 9 References Mart?n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. Jimmy Ba, Volodymyr Mnih, and Koray Kavukcuoglu. Multiple object recognition with visual attention. arXiv preprint arXiv:1412.7755, 2014. Jia-Ren Chang and Yong-Sheng Chen. Batch-normalized maxout network in network. arXiv preprint arXiv:1511.02583, 2015. Dan C Cire?san, Ueli Meier, Jonathan Masci, Luca M Gambardella, and J?rgen Schmidhuber. Highperformance neural networks for visual object classification. arXiv preprint arXiv:1102.0183, 2011. Ian J Goodfellow, Yaroslav Bulatov, Julian Ibarz, Sacha Arnoud, and Vinay Shet. Multi-digit number recognition from street view imagery using deep convolutional neural networks. arXiv preprint arXiv:1312.6082, 2013. Klaus Greff, Antti Rasmus, Mathias Berglund, Tele Hao, Harri Valpola, and J?rgen Schmidhuber. Tagger: Deep unsupervised perceptual grouping. In Advances in Neural Information Processing Systems, pages 4484?4492, 2016. Geoffrey E Hinton. Shape representation in parallel systems. In International Joint Conference on Artificial Intelligence Vol 2, 1981a. Geoffrey E Hinton. A parallel computation that assigns canonical object-based frames of reference. In Proceedings of the 7th international joint conference on Artificial intelligence-Volume 2, pages 683?685. Morgan Kaufmann Publishers Inc., 1981b. Geoffrey E Hinton, Zoubin Ghahramani, and Yee Whye Teh. Learning to parse images. In Advances in neural information processing systems, pages 463?469, 2000. Geoffrey E Hinton, Alex Krizhevsky, and Sida D Wang. Transforming auto-encoders. In International Conference on Artificial Neural Networks, pages 44?51. Springer, 2011. Max Jaderberg, Karen Simonyan, Andrew Zisserman, and Koray Kavukcuoglu. Spatial transformer networks. In Advances in Neural Information Processing Systems, pages 2017?2025, 2015. Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. Yann LeCun, Corinna Cortes, and Christopher JC Burges. The mnist database of handwritten digits, 1998. Yann LeCun, Fu Jie Huang, and Leon Bottou. Learning methods for generic object recognition with invariance to pose and lighting. In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages II?104. IEEE, 2004. Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. In NIPS workshop on deep learning and unsupervised feature learning, volume 2011, page 5, 2011. Bruno A Olshausen, Charles H Anderson, and David C Van Essen. A neurobiological model of visual attention and invariant pattern recognition based on dynamic routing of information. Journal of Neuroscience, 13(11):4700?4719, 1993. Denis G Pelli, Melanie Palomares, and Najib J Majaj. Crowding is unlike ordinary masking: Distinguishing feature integration from detection. Journal of vision, 4(12):12?12, 2004. Li Wan, Matthew D Zeiler, Sixin Zhang, Yann LeCun, and Rob Fergus. Regularization of neural networks using dropconnect. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pages 1058?1066, 2013. 10 Matthew D Zeiler and Rob Fergus. Stochastic pooling for regularization of deep convolutional neural networks. arXiv preprint arXiv:1301.3557, 2013. A How many routing iterations to use? In order to experimentally verify the convergence of the routing algorithm we plot the average change in the routing logits at each routing iteration. Fig. A.1 shows the average bij change after each routing iteration. Experimentally we observe that there is negligible change in the routing by 5 iteration from the start of training. Average change in the 2nd pass of the routing settles down after 500 epochs of training to 0.007 while at routing iteration 5 the logits only change by 1e ? 5 on average. Figure A.1: Average change of each routing logit (bij ) by each routing iteration. After 500 epochs of training on MNIST the average change is stabilized and as it shown in right figure it decreases almost linearly in log scale with more routing iterations. (b) Log scale of final differences. (a) During training. We observed that in general more routing iterations increases the network capacity and tends to overfit to the training dataset. Fig. A.2 shows a comparison of Capsule training loss on Cifar10 when trained with 1 iteration of routing vs 3 iteration of routing. Motivated by Fig. A.2 and Fig. A.1 we suggest 3 iteration of routing for all experiments. Figure A.2: Traning loss of CapsuleNet on cifar10 dataset. The batch size at each training step is 128. The CapsuleNet with 3 iteration of routing optimizes the loss faster and converges to a lower loss at the end. 11
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Incorporating Side Information by Adaptive Convolution Di Kang Debarun Dhar Antoni B. Chan Department of Computer Science City University of Hong Kong {dkang5-c, ddhar2-c}@my.cityu.edu.hk, [email protected] Abstract Computer vision tasks often have side information available that is helpful to solve the task. For example, for crowd counting, the camera perspective (e.g., camera angle and height) gives a clue about the appearance and scale of people in the scene. While side information has been shown to be useful for counting systems using traditional hand-crafted features, it has not been fully utilized in counting systems based on deep learning. In order to incorporate the available side information, we propose an adaptive convolutional neural network (ACNN), where the convolution filter weights adapt to the current scene context via the side information. In particular, we model the filter weights as a low-dimensional manifold within the high-dimensional space of filter weights. The filter weights are generated using a learned ?filter manifold? sub-network, whose input is the side information. With the help of side information and adaptive weights, the ACNN can disentangle the variations related to the side information, and extract discriminative features related to the current context (e.g. camera perspective, noise level, blur kernel parameters). We demonstrate the effectiveness of ACNN incorporating side information on 3 tasks: crowd counting, corrupted digit recognition, and image deblurring. Our experiments show that ACNN improves the performance compared to a plain CNN with a similar number of parameters. Since existing crowd counting datasets do not contain ground-truth side information, we collect a new dataset with the ground-truth camera angle and height as the side information. 1 Introduction Computer vision tasks often have side information available that is helpful to solve the task. Here we define ?side information? as auxiliary metadata that is associated with the main input, and that affects the appearance/properties of the main input. For example, the camera angle affects the appearance of a person in an image (see Fig. 1 top). Even within the same scene, a person?s appearance changes as they move along the ground-plane, due to changes in the relative angles to the camera sensor. Most deep learning methods ignore the side information, since if given enough data, a sufficiently large deep network should be able to learn internal representations that are invariant to the side information. In this paper, we explore how side information can be directly incorporated into deep networks so as to improve their effectiveness. Our motivating application is crowd counting in images, which is challenging due to complicated backgrounds, severe occlusion, low-resolution images, perspective distortion, and different appearances caused by different camera tilt angles. Recent methods are based on crowd density estimation [1], where each pixel in the crowd density map represents the fraction of people in that location, and the crowd count is obtained by integrating over a region in the density map. The current state-of-theart uses convolutional neural networks (CNN) to estimate the density maps [2?4]. Previous works have also shown that using side information, e.g., the scene perspective, helps to improve crowd counting accuracy [5, 6]. In particular, when extracting hand-crafted features (e.g., edge and texture statistics) [5?9] use scene perspective normalization, where a ?perspective weight? is applied at each 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. camera angle z -10? -20? -30? -40? -50? -65? filter manifold network FC, FC, ?, FC auxiliary input ? images filter weights + bias ? ?; ? 5x5 filter ? z=-10? z=-20? filter space ?5x5 z=-30? filter manifold ? convolution ? = ?(??? ?; ? ) z=-40? input maps output maps Figure 2: The adaptive convolutional layer Figure 1: (top) changes in people?s appearance due to camera with filter manifold network (FMN). The angle, and the corresponding changes in a convolution filter; (bot- FMN uses the auxiliary input to generate tom) the filter manifold as a function of the camera angle. Best the filter weights, which are then convolved with the input maps. viewed in color. z=-50? z=-65? pixel location during feature extraction, to adjust for the scale of the object at that location. To handle scale variations, typical CNN-based methods resize the input patch [2] based on the perspective weight, or extract features at different scales via multiple columns [3] or a pyramid of input patches [4]. However, incorporating other types of side information into the CNN is not as straightforward. As a result, all the difficulties due to various contexts, including different backgrounds, occlusion, perspective distortion and different appearances caused by different camera angles are entangled, which may introduce an extra burden on the CNNs during training. One simple solution is to add an extra image channel where each pixel holds the side information [10], which is equivalent to using 1st-layer filter bias terms that change with the side information. However, this may not be the most effective solution when the side information is a high-level property with a complex relationship with the image appearance (e.g., the camera angle). Our solution in this paper is to disentangle the context variations explicitly in the CNN by modifying the filter weights adaptively. We propose an adaptive CNN (ACNN) that uses side information (e.g., the perspective weight) as an auxiliary input to adapt the CNN to different scene contexts (e.g., appearance changes from high/low angle perspectives, and scale changes due to distance). Specifically, we consider the filter weights in each convolutional layer as points on a low-dimensional manifold, which is modeled using a sub-network where the side information is the input and the filter weights are the outputs. The filter manifold is estimated during training, resulting in different convolution filters for each scene context, which disentangles the context variations related to the side information. In the ACNN, the convolutional layers focus only on those features most suitable for the current context specified by the side information, as compared to traditional CNNs that use a fixed set of filters over all contexts. In other words, the feature extractors are tuned for each context. We test the effectiveness of ACNN at incorporating side information on 3 computer vision applications. First, we perform crowd counting from images using an ACNN with the camera parameters (perspective value, or camera tilt angle and height) as side information. Using the camera parameters as side information, ACNN can perform cross-scene counting without a fine-tuning stage. We collect a new dataset covering a wide range of angles and heights, containing people from different viewpoints. Second, we use ACNN for recognition of digit images that are corrupted with salt-and-pepper noise, where the noise level is the side information. Third, we apply ACNN to image deburring, where the blur kernel parameters are the side information. A single ACNN can be trained to deblur images for any setting of the kernel parameters. In contrast, using a standard CNN would require training a separate CNN for each combination of kernel parameters, which is costly if the set of parameter combinations is large. In our experiments, we show that ACNN can more effectively use the side information, as compared to traditional CNNs with similar number of parameters ? moving parameters from static layers to adaptive layers yields stronger learning capability and adaptability. The contributions of this paper are three-fold: 1) We propose a method to incorporate the side information directly into CNN by using an adaptive convolutional layer whose weights are generated via a filter manifold sub-network with side information as the input; 2) We test the efficacy of ACNN on a variety of computer vision applications, including crowd counting, corrupted digit recognition, and non-blind image deblurring, and show that ACNN is more effective than traditional CNNs with 2 similar number of parameters. 3) We collect a new crowd counting dataset covering a wide range of viewpoints and its corresponding side information, i.e. camera tilt angle and camera height. 2 Related work 2.1 Adapting neural networks The performance of a CNN is affected if the test set is not from the same data distribution as the training set [2]. A typical approach to adapting a CNN to new data is to select a pre-trained CNN model, e.g. AlexNet [11], VGG-net [12], or ResNet [13] trained on ImageNet, and then fine-tune the model weights for the specific task. [2] adopts a similar strategy ? train the model on the whole dataset and then fine-tune using a subset of image patches that are similar to the test scene. Another approach is to adapt the input data cube so that the extracted features and the subsequent classifier/regressor are better matched. [14] proposes a trainable ?Spatial Transformer? unit that applies an image transformation to register the input image to a standard form before the convolutional layer. The functional form of the image transformation must be known, and the transformation parameters are estimated from the image. Because it operates directly on the image, [14] is limited to 2D image transformations, which work well for 2D planar surfaces in an image (e.g., text on a flat surface), but cannot handle viewpoint changes of 3D objects (e.g. people). In contrast, our ACNN changes the feature extraction layers based on the current 3D viewpoint, and does not require the geometric transformation to be known. Most related to our work are dynamic convolution [15] and dynamic filter networks [16], which use the input image to dynamically generate the filter weights for convolution. However, their purpose for dynamically generating filters is quite different from ours. [15, 16] focus on image prediction tasks (e.g., predicting the next frame from the previous frames), and the dynamically-generated filters are mainly used to transfer a pixel value in the input image to a new position in the output image (e.g., predicting the movement of pixels between frames). These input-specific filters are suitable for low-level tasks, i.e. the input and the output are both in the same space (e.g., images). But for high-level tasks, dramatically changing features with respect to its input is not helpful for the end-goal of classification or regression. In contrast, our purpose is to include side information into supervised learning (regression and classification), by learning how the discriminative image features and corresponding filters change with respect to the side information. Hence, in our ACNN, the filter weights are generated from an auxiliary input corresponding to the side information. HyperNetworks [17] use relaxed weight-sharing between layers/blocks, where layer weights are generated from a low-dimensional linear manifold. This can improve the expressiveness of RNNs, by changing the weights over time, or reduce the number of learnable parameters in CNNs, by sharing weight bases across layers. Specifically, for CNNs, the weight manifold of the HyperNetwork is shared across layers, and the inputs/embedding vectors of the HyperNetwork are independently learned for every layer during training. The operation of ACNNs is orthogonal to HyperNetworks - in ACNN, the weight manifold is trained independently for each layer, and the input/side information is shared across layers. In addition, our goal is to incorporate the available side information to improve the performance of the CNN models, which is not considered in [17]. Finally, one advantage of [14?17] is that no extra information or label is needed. However, this also means they cannot effectively utilize the available side information, which is common in various computer vision tasks and has been shown to be helpful for traditional hand-crafted features [5]. 2.2 Crowd density maps [1] proposes the concept of an object density map whose integral over any region equals to the number of objects in that region. The spatial distribution of the objects is preserved in the density map, which also makes it useful for detection [18, 19] and tracking [20]. Most of the recent state-of-the-art object counting algorithms adopt the density estimation approach [2?4, 8, 21]. CNN-based methods [2?4] show strong cross-scene prediction capability, due to the learning capacity of CNNs. Specifically, [3] uses a multi-column CNN with different receptive field sizes in order to encourage different columns to capture features at different scales (without input scaling or explicit supervision), while [4] uses a pyramid of input patches, each sent to separate sub-network, to consider multiple scales. [2] introduces an extra fine-tuning stage so that the network can be better adapted to a new scene. In contrast to [2, 3], we propose to use the existing side information (e.g. perspective weight) as an input to adapt the convolutional layers to different scenes. With the adaptive convolutional layers, 3 only the discriminative features suitable for the current context are extracted. Our experiments show that moving parameters from static layers to adaptive layers yields stronger learning capability. 2.3 Image deconvolution Existing works [22?24] demonstrate that CNNs can be used for image deconvolution and restoration. With non-blind deblurring, the blur kernel is known and the goal is to recover the original image. [23] concatenate a deep deconvolution CNN and a denoising CNN to perform deblurring and artifact removal. However, [23] requires a separate network to be trained for each blur kernel family and kernel parameter. [24] trains a multi-layer perceptron to denoise images corrupted by additive white Gaussian (AWG) noise. They incorporate the side information (AWG standard deviation) by simply appending it to the vectorized image patch input. In this paper, we use the kernel parameter as an auxiliary input, and train a single ACNN for a blur kernel family (for all its parameter values), rather than for each parameter separately. During prediction, the ?filter-manifold network? uses the auxiliary input to generate the appropriate deblurring filters, without the need for additional training. 3 Adaptive CNN In this section, we introduce the adaptive convolutional layer and the ACNN. 3.1 Adaptive convolutional layer Consider a crowd image dataset containing different viewpoints of people, and we train a separate CNN to predict the density map for each viewpoint. For two similar viewpoints, we expect that the two trained CNNs have similar convolution filter weights, as a person?s appearance varies gradually with the viewpoint (see Fig. 1 top). Hence, as the viewpoint changes smoothly, the convolution filters weights also change smoothly, and thus sweep a low-dimensional manifold within the highdimensional space of filter weights (see Fig. 1 bottom). Following this idea, we use an adaptive convolutional layer, where the convolution filter weights are the outputs of a separate ?filter-manifold network? (FMN, see Fig. 2). In the FMN, the side information is an auxiliary input that feeds into fully-connected layers with increasing dimension (similar to the decoder stage of an auto-encoder) with the final layer outputting the convolution filter weights. The FMN output is reshaped into a 4D tensor of convolution filter weights (and bias), and convolved with the input image. Note that in contrast to the traditional convolutional layer, whose filter weights are fixed during the inference stage, the filter weights of an adaptive convolutional layer change with respect to the auxiliary input. Formally, the adaptive convolutional layer is given by h = f (x ? g(z; w)), where z is the auxiliary input, g(?; w) is the filter manifold network with tunable weights w, x is the input image, and f (?) is the activation function.1 Training the adaptive convolutional layer involves updating the FMN weights w, thus learning the filter manifold as a function of the auxiliary input. During inference, the FMN interpolates along the filter manifold using the auxiliary input, thus adapting the filter weights of the convolutional layer to the current context. Hence adaptation does not require fine-tuning or transfer learning. 3.2 Adaptive CNN for crowd counting We next introduce the ACNN for crowd counting. Density map estimation is not as high-level a task as recognition. Since the upper convolutional layers extract more abstract features, which are not that helpful according to both traditional [1, 5] and deep methods [2, 3], we will not use many convolutional layers. Fig. 3 shows our ACNN for density map estimation using two convolutional stages. The input is an image patch, while the output is the crowd density at the center of the patch. All the convolutional layers use the ReLU activation, and each convolutional layer is followed by a local response normalization layer [11] and a max pooling layer. The auxiliary input for the FMN is the perspective value for the image patch in the scene, or the camera tilt angle and camera height. For the fully-connected stage, we use multi-task learning to improve the training of the feature extractors [2, 25?27]. In particular, the main regression task predicts the crowd density value, while an auxiliary classification task predicts the number of people in the image patch. The adaptive convolutional layer has more parameters than a standard convolutional layer with the same number of filters and the same filter spatial size ? the extra parameters are in the layers of the 1 To reduce clutter, here we do not show the bias term for the convolution. 4 Layer CNN ACNN FMN1 ? 34,572 (832) conv1 1,664 (64) 0 (32) (1) (1) FMN1 FMN2 (10) (10) FMN2 ? 1,051,372 (25,632) (40) (40) FC4 FC5 conv2 102,464 (64) 0 (32) (832) (25632) (81) (15) input image FC1 2,654,720 (512) 1,327,616 (512) filter weights filter weights auxiliary patch (32x1x5x5)+32 FC2 41,553 (81) 41,553 (81) (32x32x5x5)+32 classification (1x33x33) task FC3 82 (1) 82 (1) FC4 419,985 (81) 210,033 (81) output FC5 1,312 (15) 1,312 (15) ? ? density total 3,221,780 2,666,540 conv2 conv1 Table 1: Comparison of number of parameters in (32x9x9) (32x17x17) each layer of the ACNN in Fig. 3 and an equivalent FC1 FC2 FC3 (512) (81) (1) CNN. The number in parenthesis is the number of Figure 3: The architecture of our ACNN with adap- convolution filters, or the number of outputs of the tive convolutional layers for crowd density estimation. FMN/fully-connected (FC) layer. auxiliary input: perspective value (1) FMN. However, since the filters themselves adapt to the scene context, an ACNN can be effective with fewer feature channels (from 64 to 32), and the parameter savings can be moved to the FMN (e.g. see Table 1). Hence, if side information is available, a standard CNN can be converted into an ACNN with a similar number of parameters, but with better learning capability. We verify this property in the experiments. Since most of the parameters of the FMN are in its last layer, the FMN has O(LF ) parameters, where F is the number of filter parameters in the convolution layer and L is the size of the last hidden layer of the FMN. Hence, for a large number of channels (e.g., 128 in, 512 out), the FMN will be extremely large. One way to handle more channels is to reduce the number of parameters in the FMN, by assuming that sub-blocks in the final weight matrix of the FMN form a manifold, which can be modeled by another FMN (i.e., an FMN-in-FMN). Here, the auxiliary inputs for the sub-block FMNs are generated from another network whose input is the original auxiliary input. 3.3 Adaptive CNN for image deconvolution Our ACNN for image deconvolution is based on the deconvolution CNN proposed in [23]. The ACNN uses the kernel blur parameter (e.g., radius of the disk kernel) as the side information, and consists of three adaptive convolutional layers (see Fig. 4). The ACNN uses 12 filter channels in the first 2 layers, which yields an architecture with similar number of parameters as the standard CNN with 38 filters in [23]. The ACNN consists of two long 1D adaptive convolutional layers: twelve 121?1 vertical 1D filters, followed by twelve 1?121 horizontal 1D filters. The result is passed through a 1?1 adaptive convolutional layer to fuse all the feature maps. The input is the blurred image and the output target is the original image. We use leaky ReLU activations [28] for the first two convolutional layers, and sigmoid activation for the last layer to produce a bounded output as image. Batch normalization layers [29] are used after the convolutional layers. During prediction, the FMN uses kernel parameter auxiliary input to generate the appropriate deblurring filters, without the need for additional training. Hence, the two advantages of using ACNN are: 1) only one network is needed for each blur kernel family, which is useful for kernels with too many parameter combinations to enumerate; 2) by interpolating along the filter manifold, ACNN can work on kernel parameters unseen in the training set. 4 Experiments To show their potential, we evaluate ACNNs on three tasks: crowd counting, digit recognition with salt-and-pepper noise, and image deconvolution (deblurring). In order to make fair comparisons, we compare our ACNN with standard CNNs using traditional convolutional layers, but increase the number of filter channels in the CNN so that they have similar total number of parameters as the ACNN. We also test a CNN with side information included as an extra input channel(s) (denoted as CNN-X), where the side information is replicated in each pixel of the extra channel, as in [10]. For ACNN, each adaptive convolution layer has its own FMN, which is a standard MLP with two hidden layers and a linear output layer. The size of the FMN output layer is the same as the number of filter parameters in its associated convolution layer, and the size of the last hidden layer (e.g., 40 in Fig. 3) was selected so that the ACNN and baseline CNN have roughly equal number of parameters. 5 auxiliary input: blurring kernel parameter Method MAE MESA [1] 1.70 (1) (1) (1) FMN1 FMN2 FMN3 Regression forest [21] 1.70 (4) (4) (4) (8) (8) (8) RR [8] 1.24 (4368) (36) (17486) CNN-patch+RR [2] 1.70 filter weights filter weights filter weights MCNN [3] 1.32 (12x3x121x1) (12x12x1x121) (3x12x1x1) +12 +12 CNN 1.26 CNN-X 1.20 CNN (normalized patch) 1.26 ACNN-v1 1.23 ACNN-v2 1.14 ACNN-v3 0.96 output image conv2 input image conv1 (3x184x184) (12x184x184) (12x184x184) (3x184x184) Table 2: Comparison of mean absolute error (MAE) Figure 4: ACNN for image deconvolution. The auxil- for counting with crowd density estimation methods iary input is the radius r of the disk blurring kernel. on the UCSD ?max? split. ? ? ? Method R1 R2 (unseen) R3 Avg. CNN 1.83 1.06 0.62 1.17 CNN-X 1.33 1.18 0.61 1.04 R2 (13.2-17.7) ACNN-v1 1.47 0.95 0.59 1.00 ACNN-v2 1.22 0.91 0.55 0.89 ACNN-v3 1.15 1.02 0.63 0.93 R3 (17.6-22.1) Table 3: Comparison of MAE on 3 bar regions on the Figure 5: UCSD dataset with 3 bar regions. The range UCSD ?max? split. of perspective values are shown in parentheses. R1 (6.7-13.2) 4.1 Crowd counting experiments For crowd counting, we use two crowd counting datasets: the popular UCSD crowd counting dataset, and our newly collected dataset with camera tilt angle and camera height as side information. 4.1.1 UCSD dataset Refer to Fig. 3 for the ACNN architecture used for the UCSD dataset. The image size is 238?158, and 33?33 patches are used. We test several variations of the ACNN: v1) only the first convolutional layer is adaptive, with 64 filters for both of the convolutional layers; v2) only the last convolutional layer is adaptive, with 64 filters for the first convolutional layer and 30 filters for its second convolutional layer; v3) all the convolutional layers are adaptive, with 32 filters for all layers, which provides maximum adaptability. The side information (auxiliary input) used for the FMN is the perspective value. For comparison, we also test a plain CNN and CNN-X with a similar architecture but using standard convolutional layers with 64 filters in each layer, and another plain CNN with input patch size normalization introduced in [2] (i.e., resizing larger patches for near-camera regions). The numbers of parameters are shown in Table 1. The count predictions in the region-of-interest (ROI) are evaluated using the mean absolute error (MAE) between the predicted count and the ground-truth. We first use the widely adopted protocol of ?max? split, which uses 160 frames (frames 601:5:1400) for training, and the remaining parts (frames 1:600, 1401:2000) for testing. The results are listed in Table 2. Our ACNN-v3, using two adaptive convolutional layers, offers maximum adaptability and has the lowest error (0.96 MAE), compared to the equivalent plain CNN and the reference methods. While CNN-X reduces the error compared to CNN, CNN-X still has larger error than ACNN. This demonstrates that the FMN of ACNN is better at incorporating the side information. In addition, using simple input patch size normalization does not improve the performance as effectively as ACNN. Examples of the learned filter manifolds are shown in Fig. 6. We also tested using 1 hidden layer in the FMN, and obtained worse errors for each version of ACNN (1.74, 1.15, and 1.20, respectively). Using only one hidden layer limits the ability to well model the filter manifold. In the next experiment we test the effect of the side information within the same scene. The ROI of UCSD is further divided into three bar regions of the same height (see Fig. 5). The models are trained only on R1 and R3 from the training set, and tested on all three regions of the test set separately. The results are listed in Table 3. After disentangling the variations due to perspective value, the performance on R1 has been significantly improved because the ACNN uses the context information to distinguish it from the other regions. Perspective values within R2 are completely unseen during training, but our ACNN still gives a comparable or slightly better performance than CNN, which demonstrates that the FMN can smoothly interpolate along the filter manifold. 6 Method MAE LBP+RR [2, 3] 23.97 MCNN [3] 8.80 CNN 8.72 CNN-X (AH) 9.05 CNN-X (AHP) 8.45 ACNN (AH) 8.35 Figure 6: Examples of learned filter manifolds for the 2nd convoluACNN (AHP) 8.00 tional layer. Each row shows one filter as a function of the auxiliary Table 4: Counting results on CityUHK-X, input (perspective weight), shown at the top. Both the amplitude the new counting dataset with side inforand patterns change, which shows the adaptability of the ACNN. mation. Image Predicted density map Image Predicted density map -20.4? , 6.1m 92.44 (1.57) -29.8? , 4.9m 18.22 (2.47) 6.7 ? ? ? ? ? ? 9.7 ? ? ? ? ? ? 12.6 ? ? ? ? ? ? 15.5 ? ? ? ? ? ? 18.5 ? ? ? ? ? ? 21.4 -39.8? , 6.7m -55.2? , 11.6m 28.99 (0.66) 21.71 (1.24) Figure 7: Examples of the predicted density map by our ACNN on the new CityUHK-X dataset. The extrinsic parameters and predicted count (absolute error in parenthesis) is shown above the images. 4.1.2 CityUHK-X: new crowd dataset with extrinsic camera parameters The new crowd dataset ?CityUHK-X? contains 55 scenes (3,191 images in total), covering a camera tilt angle range of [-10? , -65? ] and a height range of [2.2, 16.0] meters. The training set consists of 43 scenes (2,503 images; 78,592 people), and the test set comprises 12 scenes (688 images; 28,191 people). More information and demo images can be found in the supplemental. The resolution of the new dataset is 512?384, and 65?65 patches are used. The ACNN for this dataset contains three convolutional and max-pooling layers, resulting in the same output feature map size after the convolutional stage as in the ACNN for UCSD. The three adaptive convolutional layers use 40, 40 and 32 filters of size 5?5 each. The side information (auxiliary inputs) are camera tilt angle and camera height (denoted as ?AH?), and the camera tilt angle, camera height, and perspective value (denoted as ?AHP?). The baseline plain CNN and CNN-X use 64 filters of size 5?5 for all three convolutional layers. Results for ACNN, the plain CNN and CNN-X, and multi-column CNN (MCNN) [3] are presented in Table 4. The plain CNN and MCNN [3], which do not use side information, obtain similar results. Using side information with ACNN decreases the MAE, compared to the plain CNN and CNN-X, with more side information improving the results (AHP vs. AH). Fig. 7 presents example results. 4.2 Digit recognition with salt-and-pepper noise In this experiment, the task is to recognize handwritten digits that are corrupted with different levels of salt-and-pepper noise. The side information is the noise level. We use the MNIST handwritten digits dataset, which contains 60,000 training and 10,000 test examples. We randomly add salt-and-pepper noise (half salt and half pepper), on the MNIST images. Nine noise levels are used on the original MNIST training set from 0% to 80% with an interval of 10%, with the same number of images for each noise level, resulting in a training set of 540,000 samples. Separate validation and test sets, both containing 90,000 samples, are generated from the original MNIST test set. We test our ACNN with the noise level as the side information, as well as the plain CNN and CNN-X. We consider two architectures: two or four convolutional layers (2-conv or 4-conv) followed by 7 Architecture No. Conv. Filters Error Rate No. Parameters CNN 2-conv 32 + 32 8.66% 113,386 CNN-X 2-conv 32 + 32 8.49% (8.60%) 113,674 ACNN 2-conv 32 + 26 7.55% (7.64%) 105,712 CNN 4-conv 32 + 32 + 32 + 32 3.58% 131,882 CNN-X 4-conv 32 + 32 + 32 + 32 3.57% (3.64%) 132,170 ACNN 4-conv 32 + 32 + 32 + 26 2.92% (2.97%) 124,208 Table 5: Digit recognition with salt-and-pepper noise, where the noise level is the side information. The number of filters for each convolutional layer and total number of parameters are listed. In the Error Rate column, the parenthesis shows the error when using the estimated side information rather than the ground-truth. Arch-filters training set r r=3 r=5 r=7 r=9 r=11 all seen r unseen r blurred image ? 23.42 21.90 20.96 20.28 19.74 21.26 ? ? CNN [23] {3, 7, 11} +0.55 -0.25 +0.49 +0.69 +0.56 +0.41 +0.53 +0.22 CNN-X {3, 7, 11} +0.88 -0.70 +1.65 +0.47 +1.86 +0.83 +1.46 -0.12 ACNN {3, 7, 11} +0.77 +0.06 +1.17 +0.94 +1.28 +0.84 +1.07 +0.50 CNN-X (blind) {3, 7, 11} +0.77 -0.77 +1.23 +0.25 +0.98 +0.49 +0.99 -0.26 ACNN (blind) {3, 7, 11} +0.76 -0.04 +0.70 +0.80 +1.13 +0.67 +0.86 +0.38 CNN [23] {3, 5, 7, 9, 11} +0.28 +0.45 +0.62 +0.86 +0.59 +0.56 +0.56 ? CNN-X {3, 5, 7, 9, 11} +0.99 +1.38 +1.53 +1.60 +1.55 +1.41 +1.41 ? ACNN {3, 5, 7, 9, 11} +0.71 +0.92 +1.00 +1.28 +1.22 +1.03 +1.03 ? CNN-X (blind) {3, 5, 7, 9, 11} +0.91 +1.06 +0.81 +1.12 +1.24 +1.03 +1.03 ? ACNN (blind) {3, 5, 7, 9, 11} +0.66 +0.79 +0.64 +1.12 +1.04 +0.85 +0.85 ? Table 6: PSNRs for image deconvolution experiments. The PSNR for the blurred input image is in the first row, while the other rows are the change in PSNR relative to that of the blurred input image. Blind means the network takes estimated auxiliary value (disk radius) as the side information. two fully-connected (FC) layers.2 For ACNN, only the 1st convolutional layer is adaptive. All convolutional layers use 3?3 filters. All networks use the same configuration for the FC layers, one 128-neuron layer and one 10-neuron layer. ReLU activation is used for all layers, except the final output layer which uses soft-max. Max pooling is used after each convolutional layer for the 2-conv network, or after the 2nd and 4th convolutional layers for the 4-conv network. The classification error rates are listed in Table 5. Generally, adding side information as extra input channel (CNN-X) decreases the error, but the benefit diminishes as the baseline performance increases ? CNN-X 4-conv only decreases the error rate by 0.01% compared with CNN. Using ACNN to incorporate the side information can improve the performance more significantly. In particular, for ACNN 2-conv, the error rate decreases 0.94% (11% relatively) from 8.49% to 7.55%, while the error rate decreases 0.65% (18% relatively) from 3.57% to 2.92% for ACNN 4-conv. We also tested the ACNN when the noise level is unknown ? The noise level is estimated from the image, and then passed to the ACNN. To this end, a 4-layer CNN (2 conv. layers, 1 max-pooling layer and 2 FC layers) is trained to predict the noise level from the input image. The error rate increases slightly when using the estimated noise level (e.g., by 0.05% for the ACNN 4-conv, see Table 5). More detailed setting of the networks can be found in the supplemental. 4.3 Image deconvolution In the final experiment, we use ACNN for image deconvolution (deblurring) where the kernel blur parameter is the side information. We test on the Flickr8k [31] dataset, and randomly select 5000 images for training, 1400 images for validation, and another 1600 images for testing. The images were blurred uniformly using a disk kernel, and then corrupted with additive Gaussian noise (AWG) and JPEG compression as in [23], which is the current state-of-the-art for non-blind deconvolution using deep learning. We train the models with images blurred with different sets of kernel radii r ? {3, 5, 7, 9, 11}. The test set consists of images blurred with all r ? {3, 5, 7, 9, 11}. The evaluation is based on the peak signal-to-noise ratio (PSNR) between the deconvolved image and the original image, relative to the PSNR of the blurred image. The results are shown in Table 6 using different sets of radii for the training set. First, when trained on the full training set, ACNN almost doubles the increase in PSNR, compared to the CNN (+1.03dB vs. +0.56dB). Next, we consider a reduced training set with radii r ? {3, 7, 11}, and ACNN again doubles the increase in PSNR (+0.84dB vs. +0.41dB). The performance of ACNN on the unseen radii r ? {5, 9} is better than CNN, which demonstrates the capability of ACNN to interpolate along 2 On the clean MNIST dataset, the 2-conv and 4-conv CNN architectures achieve 0.81% and 0.69% error, while the current state-of-the-art is ?0.23% error [30]. 8 the filter manifold for unseen auxiliary inputs. Interestingly, CNN-X has higher PSNR than ACNN on seen radii, but lower PSNR on unseen radii. CNN-X cannot well handle interpolation between unseen aux inputs, which shows the advantage of explicitly modeling the filter manifold. We also test CNN-X and ACNN for blind deconvolution, where we estimate the kernel radius using manually-crafted features and random forest regression (see supplemental). For the blind task, the PSNR drops for CNN-X (0.38 on r ? {3, 5, 7, 9, 11} and 0.34 on r ? {3, 7, 11}) are larger than ACNN (0.18 and 0.17), which means CNN-X is more sensitive to the auxiliary input. Example learned filters are presented in Fig. 8, and Fig. 9 presents examples of deblurred images. Deconvolved images using CNN are overly-smoothed since it treats images blurred by all the kernels uniformly. In contrast, the ACNN result has more details and higher PSNR. parameter weights On this task, CNN-X performs better than ACNN on the seen radii, most likely because the relationship between the side information (disk radius) and the main input (sharp image) is not complicated and deblurring is a low-level task. Hence, incorporating the side information directly into the filtering calculations (as an extra channel) is a viable solution3 . In contrast, for the crowd counting and corrupted digit recognition tasks, the relationship between the side information (camera angle/height or noise level) and the main input is less straightforward and not deterministic, and hence the more complex FMN is required to properly adapt the filters. Thus, the adaptive convolutions are not universally applicable, and CNN-X could be used in some situations where there is a simple relationship between the auxiliary input and the desired filter output. 1.5 1.0 0.5 0.0 0.5 1.0 1.5 0 aux=3 aux=5 aux=7 aux=9 aux=11 20 40 60 80 100 1200 1-D filter parameters 20 40 60 80 100 120 Figure 8: Two examples of filter manifolds for image deconvolution. The y-axis is the filter weight, and x-axis is location. The auxiliary input is the disk kernel radius. Both the amplitude and the frequency can be adapted. (a) Original (target) (b) Blurred (input) PSNR=24.34 (c) CNN [23] PSNR=25.30 (d) ACNN PSNR=26.04 Figure 9: Image deconvolution example: (a) original image; (b) blurred image with disk radius of 7; deconvolved images using (c) CNN and (d) our ACNN. 5 Conclusion In this paper, we propose an adaptive convolutional neural network (ACNN), which employs the available side information as an auxiliary input to adapt the convolution filter weights. The ACNN can disentangle variations related to the side information, and extract features related to the current context. We apply ACNN to three computer vision applications: crowd counting using either the camera angle/height and perspective weight as side information, corrupted digit recognition using the noise level as side information, and image deconvolution using the kernel parameter as side information. The experiments show that ACNN can better incorporate high-level side information to improve performance, as compared to using simple methods such as including the side information as an extra input channel. The placement of the adaptive convolution layers is important, and should consider the relationship between the image content and the aux input, i.e., how the image contents changes with respect to the auxiliary input. For example, for counting, the auxiliary input indicates the amount of perspective distortion, which geometrically transforms the people?s appearances, and thus adapting the 2nd layer is more helpful since changes in object configuration are reflected in mid-level features. In contrast, salt-and-pepper-noise has a low-level (local) effect on the image, and thus adapting the first layer, corresponding to low-level features, is sufficient. 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Conic Scan-and-Cover algorithms for nonparametric topic modeling Mikhail Yurochkin Department of Statistics University of Michigan [email protected] Aritra Guha Department of Statistics University of Michigan [email protected] XuanLong Nguyen Department of Statistics University of Michigan [email protected] Abstract We propose new algorithms for topic modeling when the number of topics is unknown. Our approach relies on an analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Our algorithms are shown in practice to have accuracy comparable to a Gibbs sampler in terms of topic estimation, which requires the number of topics be given. Moreover, they are one of the fastest among several state of the art parametric techniques.1 Statistical consistency of our estimator is established under some conditions. 1 Introduction A well-known challenge associated with topic modeling inference can be succinctly summed up by the statement that sampling based approaches may be accurate but computationally very slow, e.g., Pritchard et al. (2000); Griffiths & Steyvers (2004), while the variational inference approaches are faster but their estimates may be inaccurate, e.g., Blei et al. (2003); Hoffman et al. (2013). For nonparametric topic inference, i.e., when the number of topics is a priori unknown, the problem becomes more acute. The Hierarchical Dirichlet Process model (Teh et al., 2006) is an elegant Bayesian nonparametric approach which allows for the number of topics to grow with data size, but its sampling based inference is much more inefficient compared to the parametric counterpart. As pointed out by Yurochkin & Nguyen (2016), the root of the inefficiency can be traced to the need for approximating the posterior distributions of the latent variables representing the topic labels ? these are not geometrically intrinsic as any permutation of the labels yields the same likelihood. A promising approach in addressing the aforementioned challenges is to take a convex geometric perspective, where topic learning and inference may be formulated as a convex geometric problem: the observed documents correspond to points randomly drawn from a topic polytope, a convex set whose vertices represent the topics to be inferred. This perspective has been adopted to establish posterior contraction behavior of the topic polytope in both theory and practice (Nguyen, 2015; Tang et al., 2014). A method for topic estimation that exploits convex geometry, the Geometric Dirichlet Means (GDM) algorithm, was proposed by Yurochkin & Nguyen (2016), which demonstrates attractive behaviors both in terms of running time and estimation accuracy. In this paper we shall continue to amplify this viewpoint to address nonparametric topic modeling, a setting in which the number of topics is unknown, as is the distribution inside the topic polytope (in some situations). We will propose algorithms for topic estimation by explicitly accounting for the concentration of mass and angular geometry of the topic polytope, typically a simplex in topic modeling applications. The geometric intuition is fairly clear: each vertex of the topic simplex can be identified by a ray emanating from its center (to be defined formally), while the concentration of mass can be quantified 1 Code is available at https://github.com/moonfolk/Geometric-Topic-Modeling. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. for the cones hinging on the apex positioned at the center. Such cones can be rotated around the center to scan for high density regions inside the topic simplex ? under mild conditions such cones can be constructed efficiently to recover both the number of vertices and their estimates. We also mention another fruitful approach, which casts topic estimation as a matrix factorization problem (Deerwester et al., 1990; Xu et al., 2003; Anandkumar et al., 2012; Arora et al., 2012). A notable recent algorithm coming from the matrix factorization perspective is RecoverKL (Arora et al., 2012), which solves non-negative matrix factorization (NMF) efficiently under assumptions on the existence of so-called anchor words. RecoverKL remains to be a parametric technique ? we will extend it to a nonparametric setting and show that the anchor word assumption appears to limit the number of topics one can efficiently learn. Our paper is organized as follows. In Section 2 we discuss recent developments in geometric topic modeling and introduce our approach; Sections 3 and 4 deliver the contributions outlined above; Section 5 demonstrates experimental performance; we conclude with a discussion in Section 6. 2 Geometric topic modeling Background and related work In this section we present the convex geometry of the Latent Dirichlet Allocation (LDA) model of Blei et al. (2003), along with related theoretical and algorithmic results that motivate our work. Let V be vocabulary size and ?V ?1 be the corresponding vocabulary probability simplex. Sample K topics (i.e., distributions on words) ?k ? DirV (?), k = 1, . . . , K, where ? ? RV+ . Next, sample M document-word probabilities pm residing in the topic simplex B := Conv(?1 , . . . , ?K ) (cf. Nguyen (2015)), by first generating P their barycentric coordinates (i.e., topic proportions) ?m ? DirK (?) and then setting pm := k ?k ?mk for m = 1, . . . , M and ? ? RK + . Finally, word counts of the m-th document can be sampled wm ? Mult(pm , Nm ), where Nm ? N is the number of words in document m. The above model is equivalent to the LDA when individual words to topic label assignments are marginalized out. Nguyen (2015) established posterior contraction rates of the topic simplex, provided that ?k ? 1 ?k and either number of topics K is known or topics are sufficiently separated in terms of the Euclidean distance. Yurochkin & Nguyen (2016) devised an estimate for B, taken to be a fixed unknown quantity, by formulating a geometric objective function, which is minimized when topic simplex B is close to the normalized documents w ?m := wm /Nm . They showed that the estimation of topic proportions ?m given B simply reduces to taking barycentric coordinates of the projection of w ?m onto B. To estimate B given K, they proposed a Geometric Dirichlet Means (GDM) algorithm, which operated by performing a k-means clustering on the normalized documents, followed by a geometric correction for the cluster centroids. The resulting algorithm is remarkably fast and accurate, supporting the potential of the geometric approach. The GDM is not applicable when K is unknown, but it provides a motivation which our approach is built on. The Conic Scan-and-Cover approach To enable the inference of B when K is not known, we need to investigate the concentration of mass inside the topic simplex. It suffices to focus on two types of geometric objects: cones and spheres, which provide the basis for a complete coverage of the simplex. To gain intuition of our procedure, which we call Conic Scan-and-Cover (CoSAC) approach, imagine someone standing at a center point of a triangular dark room trying to figure out all corners with a portable flashlight, which can produce a cone of light. A room corner can be identified with the direction of the farthest visible data objects. Once a corner is found, one can turn the flashlight to another direction to scan for the next ones. See Fig. 1a, where red denotes the scanned area. To make sure that all corners are detected, the cones of light have to be open to an appropriate range of angles so that enough data objects can be captured and removed from the room. To make sure no false corners are declared, we also need a suitable stopping criterion, by relying only on data points that lie beyond a certain spherical radius, see Fig. 1b. Hence, we need to be able to gauge the concentration of mass for suitable cones and spherical balls in ?V ?1 . This is the subject of the next section. 3 Geometric estimation of the topic simplex We start by representing B in terms of its convex and angular geometry. First, B is centered at a point denoted by Cp . The centered probability simplex is denoted by ?V0 ?1 := {x ? RV |x+Cp ? ?V ?1 }. 2 t! 0.4 0.4 0.3 0.3 0.4 c 0.3 (v3) 0.2 0.2 0.1 0.1 0.0 0.0 0.0 (v2) (v1) 0.1 (v2) (v1) 0.1 0.2 0.1 0.2 0.0 0.2 0.4 (a) An incomplete coverage using 3 cones (containing red points). c(v1) 0.2 0.2 0.4 (v1) c 0.1 1 (v3) 0.2 0.4 0.2 0.0 0.2 0.4 (b) Complete coverage using 3 cones (red) and a ball (yellow). c 0.3 0.4 1 c 0.2 0.0 0.2 0.4 (c) Cap ?c (v1 ) and cone S? (v1 ). Figure 1: Complete coverage of topic simplex by cones and a spherical ball for K = 3, V = 3. Then, write bk := ?k ? Cp ? ?V0 ?1 for k = 1, . . . , K and p?m := pm ? Cp ? ?V0 ?1 for m = 1, . . . , M . Note that re-centering leaves corresponding barycentric coordinates ?m ? ?K?1 ? := Conv{b1 , . . . , bK } can unchanged. Moreover, the extreme points of centered topic simplex B V now be represented by their directions vk ? R and corresponding radii Rk ? R+ such that bk = Rk vk for any k = 1, . . . , K. 3.1 Coverage of the topic simplex ? can be covered with The first step toward formulating a CoSAC approach is to show how B exactly K cones and one spherical ball positioned at Cp . A cone is defined as set S? (v) := {p ? ?V0 ?1 |dcos (v, p) < ?}, where we employ the angular distance (a.k.a. cosine distance) dcos (v, p) := 1 ? cos(v, p) and cos(v, p) is the cosine of angle ?(v, p) formed by vectors v and p. It is possible to choose ? so that the topic simplex can be covered with K S ? Moreover, each cone contains exactly one vertex. Suppose S? (vk ) ? B. exactly K cones, that is, The Conical coverage k=1 ? with r being the inradius. The incenter and inradius that Cp is the incenter of the topic simplex B, ? Let ai,k denote the distance between correspond to the maximum volume sphere contained in B. ? with amin ? ai,k ? amax for all i, k, and Rmax , Rmin such that the i-th and k-th vertex of B, Rmin ? Rk := kbk k2 ? Rmax ? k = 1, . . . , K. Then we can establish the following. ? and ? ? (?1 , ?2 ), where ?1 = 1 ? r/Rmax and ?2 = Proposition 1. For simplex B 2 max{(a2max )/(2Rmax ), max (1 ? cos(bi , bk )}, the cone S? (v) around any vertex direction i,k=1,...,K ? contains exactly one vertex. Moreover, complete coverage holds: v of B K S k=1 ? S? (vk ) ? B. We say there is an angular separation k ) ? 0 for any i, k = 1, . . . , K (i.e., the angles for  if cos(bi , b r all pairs are at least ?/2), then ? ? 1 ? Rmax , 1 6= ?. Thus, under angular separation, the range ? that allows for full coverage is nonempty independently of K. Our result is in agreement with that of Nguyen (2015), whose result suggested that topic simplex B can be consistently estimated without knowing K, provided there is a minimum edge length amin > 0. The notion of angular separation leads naturally to the Conic Scan-and-Cover algorithm. Before getting there, we show a series of results allowing us to further extend the range of admissible ?. The inclusion of a spherical ball centered at Cp allows us to expand substantially the range of ? for which conical coverage continues to hold. In particular, we can reduce the lower bound on ? in ? with cones using the Proposition 1, since we only need to cover the regions near the vertices of B following proposition. Fig. 1b provides an illustration. Proposition 2. Let B(Cp , R) = {? p ? RV |k? p ? Cp k2 ? R}, R > 0; ?1 , ?2 given in Prop. 1, and ? ? s   2 2 sin2 (b , b ) R R sin (b , b ) i j ? k i k k ? ?3 := 1 ? min min + cos(bi , bk ) 1 ? ,1 , (1) i,k R R2 3 then we have K S k=1 ? whenever ? ? (min{?1 , ?3 }, ?2 ). S? (vk ) ? B(Cp , R) ? B Notice that as R ? Rmax , the value of ?3 ? 0. Hence if R ? Rmin ? Rmax , the admissible range for ? in Prop. 2 results in a substantial strengthening from Prop. 1. It is worth noting that the above two geometric propositions do not require any distributional properties inside the simplex. Coverage leftovers In practice complete coverage may fail if ? and R are chosen outside of corresponding ranges suggested by the previous two propositions. In that case, it is useful to note that leftover regions will have a very low mass. Next we quantify the mass inside a cone that does contain a vertex, which allows us to reject a cone that has low mass, therefore not containing a vertex in it. Proposition 3. The cone S? (v1 ) whose axis is a topic direction v1 has mass P R1 ??1 ?1 (1 ? ?1 ) i6=1 ?i ?1 d?1 1?c 1 P = P(S? (v1 )) > P(?c (b1 )) = R 1 ? ?1 1 i6=1 ?i ?1 d? ? (1 ? ? ) 1 1 0 1 P PK PK PK PK   (2) c i6=1 ?i (1 ? c)?1 ?( i=1 ?i ) c i=1 ?i c2 ( i=1 ?i )( i=1 ?i + 1) P P P 1+ P + P + ??? , ( i6=1 ?i )?(?1 )?( i6=1 ?i ) ( i6=1 ?i + 1)( i6=1 ?i + 2) i6=1 ?i + 1 where ?c (b1 ) is the simplicial cap of S? (v1 ) which is composed of vertex b1 and a base parallel to ? and cutting adjacent edges of B ? in the ratio c : (1 ? c). the corresponding base of B See Fig. 1c for an illustration for the simplicial cap described in the proposition. Given the lower bound for the mass around a cone containing a vertex, we have arrived at the following guarantee. Proposition 4. For ? ? (0, 1), let c? be such that ? = min P(?c? (bk )) and let ?? be such that k s c? = 2 r2 1? 2 Rmax !?1 ! (sin(d) cot(arccos(1 ? ?? )) + cos(d)) , (3) where angle d ? min ?(bk , bk ? bi ). Then, as long as i,k  ??  ?? , max  a2max , max (1 ? cos(b , b ) , i k 2 i,k=1,...,K 2Rmax (4) the bound P(S? (vk )) ? ? holds for all k = 1, . . . , K. 3.2 CoSAC: Conic Scan-and-Cover algorithm Having laid out the geometric foundations, we are ready to present the Conic Scan-and-Cover (CoSAC) algorithm, which is a scanning procedure for detecting the presence of simplicial vertices based on data drawn randomly from the simplex. The idea is simple: iteratively pick the farthest point P 1 from the center estimate C?p := M m pm , say v, then construct a cone S? (v) for some suitably chosen ?, and remove all the data residing in this cone. Repeat until there is no data point left. Specifically, let A = {1, . . . , M } be the index set of the initially unseen data, then set v := argmax k? pm k2 and update A := A \ S? (v). The parameter ? needs to be sufficiently large to ensure p?m :m?A that the farthest point is a good estimate of a true vertex, and that the scan will be completed in exactly K iterations; ? needs to be not too large, so that S? (v) does not contain more than one vertex. The ? the condition of the existence of such ? is guaranteed by Prop.?1. In particular, for an equilateral B, Prop. 1 is satisfied as long as ? ? (1 ? 1/ K ? 1, 1 + 1/(K ? 1)). In our setting, K is unknown. A smaller ? would be a more robust choice, and accordingly the set A will likely remain non-empty after K iterations. See the illustration of Fig. 1a, where the blue regions correspond to A after K = 3 iterations of the scan. As a result, we proceed by adopting a stopping criteria based on Prop. 2: the procedure is stopped as soon as ? m ? A k? pm k2 < R, which allows us to complete the scan in K iterations (as in Fig. 1b for K = 3). The CoSAC algorithm is formally presented by Algorithm 1. Its running is illustrated in Fig. 2, where we show iterations 1, 26, 29, 30 of the algorithm by plotting norms of the centered documents 4 in the active set A and cone S? (v) against cosine distance to the chosen direction of a topic. Iteration 30 (right) satisfies stopping criteria and therefore CoSAC recovered correct K = 30. Note that this type of visual representation can be useful in practice to verify choices of ? and R. The following theorem establishes the consistency of the CoSAC procedure. Theorem 1. Suppose {?1 , . . . , ?K } are the true topics, incenter Cp is given, ?m ? DirK (?) and P ? pm := k ?k ?mk for m = 1, . . . , M and ? ? RK + . Let K be the estimated number of topics, {??1 , . . . , ??K? } be the output of Algorithm 1 trained with ? and R as in Prop. 2. Then ?  > 0, ( ) ! ? ? P min k?i ? ??j k >  , for any i ? {1, . . . , K} ? {K 6= K} ? 0 as M ? ?. ? j?{1,...,K} Remark We found the choices ? = 0.6 and R to be median of {k? p1 k2 , . . . , k? pM k2 } to be robust in practice and agreeing with our theoretical results. From Prop. 3 it follows that choosing R as median c 1?1/K K ( 1?c ) ? length is equivalent to choosing ? resulting in an edge cut ratio c such that 1 ? K?1 K?1 K/(K?1) 1/2, then c ? ( 2K ) , which, for any equilateral topic simplex B, is satisfied by setting ? ? (0.3, 1), provided that K ? 2000 based on the Eq. (3). 4 Document Conic Scan-and-Cover algorithm In the topic modeling problem, pm for m = 1, . . . , M are not given. Instead, under the bag-of-words assumption, we are given the frequencies of words in documents w1 , . . . , wM which provide a point estimate w ?m := wm /Nm for the pm . Clearly, if number of documents M ? ? and length of documents Nm ? ? ?m, we can use Algorithm 1 with the plug-in estimates w ?m in place of pm , P 1 since w ?m ? pm . Moreover, Cp will be estimated by C?p := M w ?m . In practice, M and Nm are finite, some of which may take relatively small values. Taking the topic direction to be the farthest point in the topic simplex, i.e., v = argmax kw ?m k2 , where w ?m := w ?m ? C?p ? ?V0 ?1 , may no w ?m :m?A longer yield a robust estimate, because the variance of this topic direction estimator can be quite high (in the Supplement we show that it is upper bounded with (1 ? 1/V )/Nm ). To obtain improved estimates, we propose a technique that we call ?mean-shifting?. Instead of taking the farthest point in the simplex, this technique is designed to shift the estimate of a topic to a high density region, where true topics are likely to be Pfound. Precisely, given a (current) cone S? (v), we re-position the cone by updating v := argmin m?S? (v) kw ?m k2 (1 ? cos(w ?m , v)). In other words, v we re-position the cone by centering it around the meanP direction of the cone weighted by the norms of the data points inside, which is simply given by v ? m?S? (v) w ?m / card(S? (v)). This results in reduced variance of the topic direction estimate, due to the averaging over data residing in the cone. The mean-shifting technique may be slightly modified and taken as a local update for a subsequent optimization which cycles through the entire set of documents and iteratively updates the cones. The optimization is with respect to the following weighted spherical k-means objective: min kvk k2 =1,k=1,...K K X X k=1 m?S k (vk ) kw ?m k2 (1 ? cos(vk , w ?m )), (5) where cones S k (vk ) = {m|dcos (vk , p?m ) < dcos (vl , p?i ) ?l = 6 k} yield a disjoint data partition K F S k (vk ) = {1, . . . , M } (this is different from S? (vk )). The rationale of spherical k-means k=1 optimization is to use full data for estimation of topic directions, hence further reducing the variance due to short documents. The connection between objective function (5) and topic simplex estimation is given in the Supplement. Finally, obtain topic norms Rk along the directions vk using maximum projection: Rk := max hvk , w ?m i. Our entire procedure is summarized in Algorithm 2. m:m?S k (vk ) Remark In Step 9 of the algorithm, cone S? (v) with a very low cardinality, i.e., card(S? (v)) < ?M , for some small constant ?, is discarded because this is likely an outlier region that does not actually contain a true vertex. The choice of ? is governed by results of Prop. 4. For small ?k = 1/K, ?k, 5 q ? we can choose d such that cos(d) = K+1 . Plugand for an equilateral B 2K  q ?1 q  q K?1 ? 1?? K+1 ging these values into Eq. (3) leads to c = 2 1 ? K12 ( ) + . 2K 2K 2 ? ? P(?c ) ? c(K?1)/K (K?1)(1?c) 1?(1??) Now, plugging in ? = 0.6 we obtain ? ? K ?1 for large K. Our approximations were based on large K to get a sense of ?, we now make a conservative choice ? = 0.001, so that (K)?1 > ? ?K < 1000. As a result, a topic is rejected if the corresponding cone contains less than 0.1% of the data. Finding anchor words using Conic Scan-and-Cover Another approach to reduce the noise is to consider the problem from a different viewpoint, where Algorithm 1 will prove itself useful. RecoverKL by Arora et al. (2012) can identify topics with diminishing errors (in number of documents M ), provided that topics contain anchor words. The problem of finding anchor words geometrically reduces to identifying rows of the word-to-word co-occurrence matrix that form a simplex containing other rows of the same matrix (cf. Arora et al. (2012) for details). An advantage of this approach is that noise in the word-to-word co-occurrence matrix goes to zero as M ? ? no matter the document lengths, hence we can use Algorithm 1 with "documents" being rows of the word-to-word co-occurrence matrix to learn anchor words nonparametrically and then run RecoverKL to obtain topic estimates. We will call this procedure cscRecoverKL. Algorithm 1 Conic Scan-and-Cover (CoSAC) Input: document generating distributions p1 , . . . , pM , angle threshold ?, norm threshold R Output: topics ?1 , . . . , ?k ?p = 1 P pm {find center}; p?m := pm ? C?p for m = 1, . . . , M {center the data} 1: C m M 2: A1 = {1, . . . , M } {initialize active set}; k = 1 {initialize topic count} 3: while ?m ? Ak : k? pm k2 > R do 4: vk = argmax k? pm k2 {find topic} p?m :m?Ak 5: S? (vk ) = {m : dcos (? pm , vk ) < ?} {find cone of near documents} 6: Ak = Ak \ S? (vk ) {update active set} 7: ?k = vk + C?p , k = k + 1 {compute topic} 8: end while topic v1 ? = 0.60 ? = 0.60 S? (v1) 0.10 ? = 0.60 S? (v26) ? = 0.60 S? (v29) topic v26 S? (v30) topic v29 topic v30 norm k? p i k2 0.08 R = 0.047 0.06 0.04 A2 0.02 0.0 0.2 0.4 0.6 0.8 cosine distance dcos(v1, p?i) A27 1.0 1.2 0.0 0.2 0.4 0.6 0.8 cosine distance dcos(v26, p?i) 1.0 A30 1.2 0.0 0.2 0.4 0.6 0.8 cosine distance dcos(v29, p?i) 1.0 A31 1.2 0.0 0.2 0.4 0.6 0.8 cosine distance dcos(v30, p?i) 1.0 1.2 Figure 2: Iterations 1, 26, 29, 30 of the Algorithm 1. Red are the documents in the cone S? (vk ); blue are the documents in the active set Ak+1 for next iteration. Yellow are documents k? pm k2 < R. 5 5.1 Experimental results Simulation experiments In the simulation studies we shall compare CoSAC (Algorithm 2) and cscRecoverKL based on Algorithm 1 both of which don?t have access to the true K, versus popular parametric topic modeling approaches (trained with true K): Stochastic Variational Inference (SVI), Collapsed Gibbs sampler, RecoverKL and GDM (more details in the Supplement). The comparisons are done on the basis of minimum-matching Euclidean distance, which quantifies distance between topic simplices (Tang et al., 2014), and running times (perplexity scores comparison is given in the Supplement). Lastly we will demonstrate the ability of CoSAC to recover correct number of topics for a varying K. 6 Algorithm 2 CoSAC for documents Input: normalized documents w ?1 , . . . , w ?M , angle threshold ?, norm threshold R, outlier threshold ? Output: topics ?1 , . . . , ?k ?p = 1 P w w ?m := w ?m ? C?p for m = 1, . . . , M {center the data} 1: C m ?m {find center}; M 2: A1 = {1, . . . , M } {initialize active set}; k = 1 {initialize topic count} 3: while ? m ? Ak : kw ?m k2 > R do 4: vk = argmax kw ?m k2 {initialize direction} w ?m :m?Ak 5: 6: 7: 8: 9: 10: 11: 12: 13: while vk not converged do {mean-shifting} S? (vkP ) = {m : dcos (w ?m , vk ) < ?} {find cone of near documents} vk = m?S? (vk ) w ?m / card(S? (vk )) {update direction} end while Ak = Ak \ S? (vk ) {update active set} if card(S? (vk )) > ?M then k = k + 1 {record topic direction} end while v1 , . . . , vk = weighted spherical k-means (v1 , . . . , vk , w ?1 , . . . , w ?M ) for l in {1, . . . , k} do Rl := max hvl , w ?m i {find topic length along direction vl } m:m?S l (vl ) ? 0.000 ? 0 2000 4000 ? 6000 8000 ? ? ? ? 50 10000 ? ? ? ?? ??? ? ? ?? ? ? ? ? 100 ? ? 150 200 ? ? 250 300 ? ? 0 2000 Length of documents Nm Number of documents M 30 20 ? ? ? ? cscRecoverKL CoSAC Bayes factor 10 300 200 0.3 0.2 ? ? ? ? ? Absolute topic number error ? ? ? ? ? cscRecoverKL RecoverKL CoSAC GDM ? Gibbs SVI 0 ? ? 100 ? ? Running time, sec ? ? ? cscRecoverKL RecoverKL CoSAC GDM Gibbs SVI 0 ? ? 0.1 ? ? Minimum Matching distance ? ? ? ?? ? cscRecoverKL RecoverKL CoSAC GDM Gibbs SVI 0.0 0.075 ? 0.050 ? ? 0.025 Minimum Matching distance 40 14: ?l = Rl vl + C?p {compute topic} 15: end for 4000 6000 8000 10 10000 20 30 40 50 True number of topics K Number of documents M 1600 1550 775 ? ? ? ? 50 LDA Gibbs HDP Gibbs CoSAC 100 Training time, sec ? ? ? ? ? 150 1500 ? 0 750 ? 700 ? Gibbs, M=1000 Gibbs, M=5000 CoSAC, M=1000 CoSAC, M=5000 Perplexity ? ? 725 ? Perplexity Gibbs, M=1000 Gibbs, M=5000 CoSAC, M=1000 CoSAC, M=5000 ? ? ? ? ? 675 0.04 0.06 ? 0.02 Minimum Matching distance Figure 3: Minimum matching Euclidean distance for (a) varying corpora size, (b) varying length of documents; (c) Running times for varying corpora size; (d) Estimation of number of topics. 0 50 100 Training time, sec ? ? ? 150 0 500 1000 1500 2000 Training time, min Figure 4: Gibbs sampler convergence analysis for (a) Minimum matching Euclidean distance for corpora sizes 1000 and 5000; (b) Perplexity for corpora sizes 1000 and 5000; (c) Perplexity for NYTimes data. Estimation of the LDA topics First we evaluate the ability of CoSAC and cscRecoverKL to estimate topics ?1 , . . . , ?K , fixing K = 15. Fig. 3(a) shows performance for the case of fewer M ? [100, 10000] but longer Nm = 500 documents (e.g. scientific articles, novels, legal documents). CoSAC demonstrates performance comparable in accuracy to Gibbs sampler and GDM. Next we consider larger corpora M = 30000 of shorter Nm ? [25, 300] documents (e.g. news articles, social media posts). Fig. 3(b) shows that this scenario is harder and CoSAC matches the performance of Gibbs sampler for Nm ? 75. Indeed across both experiments CoSAC only made mistakes in terms of K for the case of Nm = 25, when it was underestimating on average by 4 topics 7 and for Nm = 50 when it was off by around 1, which explains the earlier observation. Experiments with varying V and ? are given in the Supplement. It is worth noting that cscRecoverKL appears to be strictly better than its predecessor. This suggests that our procedure for selection of anchor words is more accurate in addition to being nonparametric. Running time A notable advantage of the CoSAC algorithm is its speed. In Fig. 3(c) we see that Gibbs, SVI, GDM and CoSAC all have linear complexity growth in M , but the slopes are very different and approximately are INm for SVI and Gibbs (where I is the number of iterations which has to be large enough for convergence), number of k-means iterations to converge for GDM and is of order K for the CoSAC procedure making it the fastest algorithm of all under consideration. Next we compare CoSAC to per iteration quality of the Gibbs sampler trained with 500 iterations for M = 1000 and M = 5000. Fig. 4(b) shows that Gibbs sampler, when true K is given, can achieve good perplexity score as fast as CoSAC and outperforms it as training continues, although Fig. 4(a) suggests that much longer training time is needed for Gibbs sampler to achieve good topic estimates and small estimation variance. Estimating number of topics Model selection in the LDA context is a quite challenging task and, to the best of our knowledge, there is no "go to" procedure. One of the possible approaches is based on refitting LDA with multiple choices of K and using Bayes Factor for model selection (Griffiths & Steyvers, 2004). Another option is to adopt the Hierarchical Dirichlet Process (HDP) model, but we should understand that it is not a procedure to estimate K of the LDA model, but rather a particular prior on the number of topics, that assumes K to grow with the data. A more recent suggestion is to slightly modify LDA and use Bayes moment matching (Hsu & Poupart, 2016), but, as can be seen from Figure 2 of their paper, estimation variance is high and the method is not very accurate (we tried it with true K = 15 and it took above 1 hour to fit and found 35 topics). Next we compare Bayes factor model selection versus CoSAC and cscRecoverKL for K ? [5, 50]. Fig. 3(d) shows that CoSAC consistently recovers exact number of topics in a wide range. We also observe that cscRecoverKL does not estimate K well (underestimates) in the higher range. This is expected because cscRecoverKL finds the number of anchor words, not topics. The former is decreasing when later is increasing. Attempting to fit RecoverKL with more topics than there are anchor words might lead to deteriorating performance and our modification can address this limitation of the RecoverKL method. 5.2 Real data analysis In this section we demonstrate CoSAC algorithm for topic modeling on one of the standard bag of words datasets ? NYTimes news articles. After preprocessing we obtained M ? 130, 000 documents over V = 5320 words. Bayes factor for the LDA selected the smallest model among K ? [80, 195], while CoSAC selected 159 topics. We think that disagreement between the two procedures is attributed to the misspecification of the LDA model when real data is in play, which affects Bayes factor, while CoSAC is largely based on the geometry of the topic simplex. The results are summarized in Table 1 ? CoSAC found 159 topics in less than 20min; cscRecoverKL estimated the number of anchor words in the data to be 27 leading to fewer topics. Fig. 4(c) compares CoSAC perplexity score to per iteration test perplexity of the LDA (1000 iterations) and HDP (100 iterations) Gibbs samplers. Text files with top 20 words of all topics are included in the Supplementary material. We note that CoSAC procedure recovered meaningful topics, contextually similar to LDA and HDP (e.g. elections, terrorist attacks, Enron scandal, etc.) and also recovered more specific topics about Mike Tyson, boxing and case of Timothy McVeigh which were present among HDP topics, but not LDA ones. We conclude that CoSAC is a practical procedure for topic modeling on large scale corpora able to find meaningful topics in a short amount of time. 6 Discussion We have analyzed the problem of estimating topic simplex without assuming number of vertices (i.e., topics) to be known. We showed that it is possible to cover topic simplex using two types of geometric shapes, cones and a sphere, leading to a class of Conic Scan-and-Cover algorithms. We 8 Table 1: Modeling topics of NYTimes articles K Perplexity Coherence cscRecoverKL HDP Gibbs LDA Gibbs CoSAC 27 221 ? 5 80 159 2603 1477 ? 1.6 1520 ? 1.5 1568 -238 ?442 ? 1.7 ?300 ? 0.7 -322 Time 37 min 35 hours 5.3 hours 19 min then proposed several geometric correction techniques to account for the noisy data. Our procedure is accurate in recovering the true number of topics, while remaining practical due to its computational speed. We think that angular geometric approach might allow for fast and elegant solutions to other clustering problems, although as of now it does not immediately offer a unifying problem solving framework like MCMC or variational inference. An interesting direction in a geometric framework is related to building models based on geometric quantities such as distances and angles. Acknowledgments This research is supported in part by grants NSF CAREER DMS-1351362, NSF CNS-1409303, a research gift from Adobe Research and a Margaret and Herman Sokol Faculty Award. 9 References Anandkumar, A., Foster, D. P., Hsu, D., Kakade, S. M., and Liu, Y. A spectral algorithm for Latent Dirichlet Allocation. NIPS, 2012. Arora, S., Ge, R., Halpern, Y., Mimno, D., Moitra, A., Sontag, D., Wu, Y., and Zhu, M. A practical algorithm for topic modeling with provable guarantees. arXiv preprint arXiv:1212.4777, 2012. Blei, D. M., Ng, A. Y., and Jordan, M. I. Latent Dirichlet Allocation. J. Mach. Learn. Res., 3:993?1022, March 2003. Deerwester, S., Dumais, S. T., Furnas, G. W., Landauer, T. K., and Harshman, R. Indexing by latent semantic analysis. Journal of the American Society for Information Science, 41(6):391, Sep 01 1990. Griffiths, Thomas L and Steyvers, Mark. Finding scientific topics. PNAS, 101(suppl. 1):5228?5235, 2004. Hoffman, Ma. D., Blei, D. M., Wang, C., and Paisley, J. Stochastic variational inference. J. Mach. Learn. Res., 14(1):1303?1347, May 2013. Hsu, Wei-Shou and Poupart, Pascal. Online bayesian moment matching for topic modeling with unknown number of topics. In Advances In Neural Information Processing Systems, pp. 4529?4537, 2016. Nguyen, XuanLong. Posterior contraction of the population polytope in finite admixture models. Bernoulli, 21 (1):618?646, 02 2015. Pritchard, Jonathan K, Stephens, Matthew, and Donnelly, Peter. Inference of population structure using multilocus genotype data. Genetics, 155(2):945?959, 2000. Tang, Jian, Meng, Zhaoshi, Nguyen, Xuanlong, Mei, Qiaozhu, and Zhang, Ming. Understanding the limiting factors of topic modeling via posterior contraction analysis. In Proceedings of The 31st International Conference on Machine Learning, pp. 190?198. ACM, 2014. Teh, Y. W., Jordan, M. I., Beal, M. J., and Blei, D. M. Hierarchical dirichlet processes. Journal of the american statistical association, 101(476), 2006. Xu, Wei, Liu, Xin, and Gong, Yihong. Document clustering based on non-negative matrix factorization. In Proceedings of the 26th Annual International ACM SIGIR Conference on Research and Development in Informaion Retrieval, SIGIR ?03, pp. 267?273. ACM, 2003. Yurochkin, Mikhail and Nguyen, XuanLong. Geometric dirichlet means algorithm for topic inference. In Advances in Neural Information Processing Systems, pp. 2505?2513, 2016. 10
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FALKON: An Optimal Large Scale Kernel Method Alessandro Rudi ? INRIA ? Sierra Project-team, ? Ecole Normale Sup?erieure, Paris Luigi Carratino University of Genoa Genova, Italy Lorenzo Rosasco University of Genoa, LCSL, IIT & MIT Abstract Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning. Our theoretical analysis shows that optimal ? statistical accuracy is achieved requiring essentially O(n) memory and O(n n) time. An extensive experimental analysis on large scale datasets shows that, even with a single machine, FALKON outperforms previous state of the art solutions, which exploit parallel/distributed architectures. 1 Introduction The goal in supervised learning is to learn from examples a function that predicts well new data. Nonparametric methods are often crucial since the functions to be learned can be non-linear and complex Kernel methods are probably the most popular among nonparametric learning methods, but despite excellent theoretical properties, they have limited applications in large scale learning because of time and memory requirements, typically at least quadratic in the number of data points. Overcoming these limitations has motivated a variety of practical approaches including gradient methods, as well accelerated, stochastic and preconditioned extensions, to improve time complexity [1, 2, 3, 4, 5, 6]. Random projections provide an approach to reduce memory requirements, popular methods including Nystr?om [7, 8], random features [9], and their numerous extensions. From a theoretical perspective a key question has become to characterize statistical and computational tradeoffs, that is if, or under which conditions, computational gains come at the expense of statistical accuracy. In particular, recent results considering least squares, show that there are large class of problems for which, by combining Nystr?om or random features approaches [10, 11, 12, 13, 14, 15] with ridge regression, it is possible to substantially reduce computations, while preserving the same optimal statistical accuracy of exact kernel ridge regression (KRR). While statistical lower bounds exist for this setting, there are no corresponding computational lower bounds. The state of the art approximation of KRR, for which optimal statistical bounds are known, typically requires complexities that are roughly O(n2 ) in time and memory (or possibly O(n) in memory, if kernel computations are made on the fly). In this paper, we propose and study FALKON, a new algorithm that, to the best of our knowledge, has the best known theoretical guarantees. At the same time FALKON provides an efficient approach to apply kernel methods on millions of points, and tested on a variety of large scale problems ? E-mail: [email protected]. This work was done when A.R. was working at Laboratory of Computational and Statistical Learning (Istituto Italiano di Tecnologia). 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. outperform previously proposed methods while utilizing only a fraction of computational resources. More precisely, we take a substantial step in provably reducing the computational requirements, ? showing that, up to logarithmic factors, a time/memory complexity of O(n n) and O(n) is sufficient for optimal statistical accuracy. Our new algorithm, exploits the idea of using Nystr?om methods to approximate the KRR problem, but also to efficiently compute a preconditioning to be used in conjugate gradient. To the best of our knowledge this is the first time all these ideas are combined and put to fruition. Our theoretical analysis derives optimal statistical rates both in a basic setting and under benign conditions for which fast rates are possible. The potential benefits of different sampling strategies are also analyzed. Most importantly, the empirical performances are thoroughly tested on available large scale data-sets. Our results show that, even on a single machine, FALKON can outperforms state of the art methods on most problems both in terms of time efficiency and prediction accuracy. In particular, our results suggest that FALKON could be a viable kernel alternative to deep fully connected neural networks for large scale problems. The rest of the paper is organized as follows. In Sect. 2 we give some background on kernel methods. In Sect. 3 we introduce FALKON, while in Sect. 4 we present and discuss the main technical results. Finally in Sect. 5 we present experimental results. 2 Statistical and Computational Trade-offs in Kernel Methods We consider the supervised learning problem of estimating a function from random noisy samples. In statistical learning theory, this can be formalized as the problem of solving Z inf E(f ), E(f ) = (f (x) ? y)2 d?(x, y), (1) f ?H (xi , yi )ni=1 given samples from ?, which is fixed but unknown and where, H is a space of candidate solutions. Ideally, a good empirical solution fb should have small excess risk R(fb ) = E(fb ) ? inf E(f ), f ?H (2) since this implies it will generalize/predict well new data. In this paper, we are interested in both computational and statistical aspects of the above problem. In particular, we investigate the computational resources needed to achieve optimal statistical accuracy, i.e. minimal excess risk. Our focus is on the most popular class of nonparametric methods, namely kernel methods. Kernel methods and ridge regression. Kernel methods consider a space H of functions f (x) = n X ?j K(x, xi ), (3) i=1 where K is a positive definite kernel 2 . The coefficients ?1 , . . . , ?n are typically derived from a convex optimization problem, that for the square loss is n 1X (f (xi ) ? yi )2 + ?kf k2H , fbn,? = argmin f ?H n i=1 (4) and defines the so called kernel ridge regression (KRR) estimator [16]. An advantage of least squares approaches is that they reduce computations to a linear system (Knn + ?nI) ? = yb, (5) where Knn is an n ? n matrix defined by (Knn )ij = K(xi , xj ) and yb = (y1 , . . . yn ). We next comment on computational and statistical properties of KRR. Computations. Solving Eq. (5) for large datasets is challenging. A direct approach requires O(n2 ) in space, to allocate Knn , O(n2 ) kernel evaluations, and O(n2 cK + n3 ) in time, to compute and invert Knn (cK is the kernel evaluation cost assumed constant and omitted throughout). Statistics. Under basic assumptions, KRR achieves an error R(fb?n ) = O(n?1/2 ), for ?n = n?1/2 , which is optimal in a minimax sense and can be improved only under more stringent assumptions [17, 18]. 2 K is positive definite, if the matrix with entries K(xi , xj ) is positive semidefinite ?x1 , . . . , xN , N ? N [16] 2 The question is then if it is possible to achieve the statistical properties of KRR, with less computations. Gradient methods and early stopping. A natural idea is to consider iterative solvers and in particular gradient methods, because of their simplicity and low iteration cost. A basic example is computing the coefficients in (3) by ?t = ?t?1 + ? [(Knn ?t?1 ? yb) + ?n?t?1 ] , (6) for a suitable step-size choice ? . Computations. In this case, if t is the number of iterations, gradient methods require O(n2 t) in time, O(n2 ) in memory and O(n2 ) in kernel evaluations, if the kernel matrix is stored. Note that, the kernel matrix can also be computed on the fly with only O(n) memory, but O(n2 t) kernel evaluations are required. We note that, beyond the above simple iteration, several variants have been considered including accelerated [1, 19] and stochastic extensions [20]. Statistics. The statistical properties of iterative approaches are well studied and also in the case where ? is set to zero, and regularization is performed by choosing a suitable stopping time [21]. In this ? latter case, the number of iterations can roughly be thought of 1/? and O( n) iterations are needed for basic gradient descent, O(n1/4 ) for accelerated methods and possible O(1) iterations/epochs for stochastic methods. Importantly, we note that unlike most optimization studies, here we are considering the number of iterations needed to solve (1), rather than (4). While the time complexity of these methods dramatically improves over KRR, and computations can be done in blocks, memory requirements (or number of kernel evaluations) still makes the application to large scale setting cumbersome. Randomization provides an approach to tackle this challenge. Random projections. The rough idea is to use random projections to compute Knn only approximately. The most popular examples in this class of approaches are Nystr?om [7, 8] and random features [9] methods. In the following we focus in particular on a basic Nystr?om approach based on considering functions of the form M X fe?,M (x) = ? ei K(x, x ei ), with {e x1 , . . . , x eM } ? {x1 , . . . , xn }, (7) i=1 defined considering only a subset of M training points sampled uniformly. In this case, there are only M coefficients that, following the approach in (4), can be derived considering the linear system > > H? e = z, where H = KnM KnM + ?nKM M , z = KnM y?. (8) Here KnM is the n ? M matrix with (KnM )ij = K(xi , x ej ) and KM M is the M ? M matrix with (KM M )ij = K(e xi , x ej ). This method consists in subsampling the columns of Knn and can be seen as a particular form of random projections. > Computations. Direct methods for solving (8) require O(nM 2 ) in time to form KnM KnM and O(M 3 ) for solving the linear system, and only O(nM ) kernel evaluations. The naive memory > requirement is O(nM ) to store KnM , however if KnM KnM is computed in blocks of dimension at most M ? M only O(M 2 ) memory is needed. Iterative approaches as in (6) can also be combined with random projections [22, 23, 24] to slightly reduce time requirements (see Table. 1, or Sect. F in the appendix, for more details). Statistics. The key point though, is that random projections allow to dramatically reduce memory requirements as soon as M  n and the question arises of whether this comes at expenses of statistical accuracy. Interestingly, recent results considering this question show that there are large ? ?n) suffices for the same optimal statistical accuracy of the classes of problems for which M = O( exact KRR [11, 12, 13]. In summary, in?this case the computations needed for optimal statistical accuracy are reduced from O(n2 ) to O(n n) kernel evaluations, but the best time complexity is basically O(n2 ). In the rest of the paper we discuss how this requirement can indeed be dramatically reduced. 3 FALKON Our approach is based on a novel combination of randomized projections with iterative solvers plus preconditioning. The main novelty is that we use random projections to approximate both the problem and the preconditioning. 3 Preliminaries: preconditioning and KRR. We begin recalling the basic idea behind preconditioning. The key quantity is the condition number, that for a linear system is the ratio between the largest and smallest singular values of the matrix defining the problem [25]. For example, for problem (5) the condition number is given by cond(Knn + ?nI) = (?max + ?n)/(?min + ?n), with ?max , ?min largest and smallest eigenvalues of Knn , respectively. The importance of the condition number is that it captures the time complexity of iteratively solving the corresponding linear system. For example, if a simple gradient descent (6) is used, the number of iterations needed for an  accurate solution of problem (5) is t = O(cond(Knn + ?nI) log(1/)). ? It is shown in [23] that in this case t = n log n are needed to achieve a solution with good statistical ? properties. Indeed, it can be shown that roughly t ? 1/? log( 1 ) are needed where ? = 1/ n and  = 1/n. The idea behind preconditioning is to use a suitable matrix B to define an equivalent linear system with better condition number. For (5), an ideal choice is B such that BB > = (Knn + ?nI)?1 (9) and B > (Knn + ?nI)B ? = B > y?. Clearly, if ?? solves the latter problem, ?? = B?? is a solution of problem (5). Using a preconditioner B as in (9) one iteration is sufficient, but computing the B is typically as hard as the original problem. The problem is to derive preconditioning such that (9) might hold only approximately, but that can be computed efficiently. Derivation of efficient preconditioners for the exact KRR problem (5) has been the subject of recent studies, [3, 4, 26, 5, 6]. In particular, [4, 26, 5, 6] consider random projections to approximately compute a preconditioner. Clearly, while preconditioning (5) leads to computational speed ups in terms of the number of iterations, requirements in terms of memory/kernel evaluation are the same as standard kernel ridge regression. The key idea to tackle this problem is to consider an efficient preconditioning approach for problem (8) rather than (5). Basic FALKON algorithm. We begin illustrating a basic version of our approach. The key ingredient is the following preconditioner for Eq. (8), n ?1 2 BB > = KM + ?nK , (10) M M M M which is itself based on a Nystr?om approximation3 . The above preconditioning is a natural approxi> mation of the ideal preconditioning of problem (8) that is BB > = (KnM KnM + ?nKM M )?1 and reduces to it if M = n. Our theoretical analysis, shows that M  n suffices for deriving optimal statistical rates. In its basic form FALKON is derived combining the above preconditioning and gradient descent, fb?,M,t (x) = M X ?t,i K(x, x ei ), with ?t = B?t and (11) i=1  ? > > B KnM (KnM (B?k?1 ) ? yb) + ?nKM M (B?k?1 ) , (12) n for t ? N, ?0 = 0 and 1 ? k ? t and a suitable chosen ? . In practice, a refined version of FALKON is preferable where a faster gradient iteration is used and additional care is taken in organizing computations. ?k = ?k?1 ? FALKON. The actual version of FALKON we propose is Alg. 1 (see Sect. A, Alg. 2 for the complete algorithm). It consists in solving the system B > HB? = B > z via conjugate gradient [25], since it is a fast gradient method and does not require to specify the step-size. Moreover, to compute B quickly, with reduced numerical errors, we consider the following strategy   1 1 B = ? T ?1 A?1 , T = chol(KM M ), A = chol T T > + ?I , (13) M n where chol() is the Cholesky decomposition (in Sect. A the strategy for non invertible KM M ). 3 For the sake of simplicity, here we assume KM M to be invertible and the Nystr?om centers selected with uniform sampling from the training set, see Sect. A and Alg. 2 in the appendix for the general algorithm. 4 Algorithm 1 MATLAB code for FALKON. It requires O(nM t + M 3 ) in time and O(M 2 ) in memory. See Sect. A and Alg. 2 in the appendixes for the complete algorithm. n?D n M ?D Input: Dataset X = (xi )n , y? = (yi )n xj )M , KernelMatrix i=1 ? R i=1 ? R , centers C = (? j=1 ? R computing the kernel matrix given two sets of points, regularization parameter ?, number of iterations t. Output: Nystr?om coefficients ?. function n = T = A = alpha = FALKON(X, C, Y, KernelMatrix, lambda, t) size(X,1); M = size(C,1); KMM = KernelMatrix(C,C); chol(KMM + eps*M*eye(M)); chol(T*T?/M + lambda*eye(M)); function w = KnM_times_vector(u, v) w = zeros(M,1); ms = ceil(linspace(0, n, ceil(n/M)+1)); for i=1:ceil(n/M) Kr = KernelMatrix( X(ms(i)+1:ms(i+1),:), C ); w = w + Kr?*(Kr*u + v(ms(i)+1:ms(i+1),:)); end end BHB = @(u) A?\(T?\(KnM_times_vector(T\(A\u), zeros(n,1))/n) + lambda*(A\u)); r = A?\(T?\KnM_times_vector(zeros(M,1), Y/n)); alpha = T\(A\conjgrad(BHB, r, t)); end Computations. in Alg. 1, B is never built explicitly and A, T are two upper-triangular matrices, so A?> u, A?1 u for a vector u costs M 2 , and the same for T . The cost of computing the preconditioner is only 43 M 3 floating point operations (consisting in two Cholesky decompositions and one product of two triangular matrices). Then FALKON requires O(nM t + M 3 ) in time and the same O(M 2 ) memory requirement of the basic Nystr?om method, if matrix/vector multiplications at each iteration are performed in blocks. This implies O(nM t) kernel evaluations are needed. The question remains to characterize M and the number of iterations needed for good statistical ? accuracy. Indeed, in the next section we show that roughly O(n n) computations and O(n) memory are sufficient for optimal accuracy. This implies that FALKON is currently the most efficient kernel method with the same optimal statistical accuracy of KRR, see Table 1. 4 Theoretical Analysis In this section, we characterize the generalization properties of FALKON showing it achieves the optimal generalization error of KRR, with dramatically reduced computations. This result is given in Thm. 3 and derived in two steps. First, we study the difference between the excess risk of FALKON and that of the basic Nystr?om (8), showing it depends on the condition number induced by the preconditioning, hence on M (see Thm.1). Deriving these results requires some care, since differently to standard optimization results, our goal is to solve (1) i.e. achieve small excess risk. e Second, we show that choosing M = O(1/?) allows to make?this difference as small as e?t/2 (see Thm.2). Finally, recalling that the basic Nystr?om for ? = 1/ n has essentially the same statistical properties of KRR [13], we answer the question posed at the end of the last section and show that roughly log n iterations are sufficient for optimal statistical accuracy. Following the discussion in the e ?n) in previous section this means that the computational requirements for optimal accuracy are O(n e time/kernel evaluations and O(n) in space. Later in this section faster rates under further regularity assumptions are also derived and the effect of different selection methods for the Nystr?om centers considered. 4.1 Main Result The first result is interesting in its own right since it corresponds to translating optimization guarantees into statistical results. In particular, we derive a relation the excess risk of the FALKON algorithm fb?,M,t from Alg. 1 and the Nystr?om estimator fe?,M from Eq. (8) with uniform sampling. 5 Algorithm train time SVM / KRR + direct method KRR + iterative [1, 2] Doubly stochastic [22] Pegasos / KRR + sgd [27] KRR + iter + precond [3, 28, 4, 5, 6] Divide & Conquer [29] Nystr?om, random features [7, 8, 9] Nystr?om + iterative [23, 24] Nystr?om + sgd [20] FALKON (see Thm. 3) kernel evaluations 3 2 n? 4 n2 ? n 2 n n n2 n2 n2 n2 n2 2 n ? n n n 2 n? n2 n n2 n?2 n ?n n ?n n ?n n?n n n memory 2 n n2 n n n n n n n n test time n n n n n n ? ?n ?n ?n n Table 1: Computational complexity required by different algorithms, for optimal generalization. Logarithmic terms are not showed. Theorem 1. Let n, M ? 3, t ? N, 0 < ? ? ?1 and ? ? (0, 1]. Assume there exists ? ? 1 such that K(x, x) ? ?2 for any x ? X. Then, the following inequality holds with probability 1 ? ? r 9?2 n 1/2 1/2 ??t b e 1+ log , R(f?,M,t ) ? R(f?,M ) + 4b ve ?n ? Pn 1/2 where vb2 = n1 i=1 yi2 and ? = log(1 + 2/(cond (B > HB) ? 1)), with cond (B > HB) the > condition number of B HB. Note that ?1 > 0 is a constant not depending on ?, n, M, ?, t. The additive term in the bound above decreases exponentially in the number of iterations. If the condition number of B > HB is smaller than a small universal constant (e.g. 17), then ? > 1/2 and t the additive term decreases as e? 2 . Next, theorems derive a condition on M that allows to control > cond (B HB), and derive such an exponential decay. Theorem 2. Under the same conditions of Thm. 1, if   8?2 14?2 log . M ?5 1+ ? ?? then the exponent ? in Thm. 1 satisfies ? ? 1/2. The above result gives the desired exponential bound showing that after log n iterations the excess risk of FALKON is controlled by that of the basic Nystr?om, more precisely    9?2 n b e e R(f?,M,t ) ? 2R(f?,M ) when t ? log R(f?,M ) + log 1 + log + log 16b v2 . ?n ? Finally, we derive an excess risk bound for FALKON. By the no-free-lunch theorem, this requires some conditions on the learning problem. We first consider a standard basic setting where we only assume it exists fH ? H such that E(fH ) = inf f ?H E(f ). Theorem 3. Let ? ? (0, 1]. Assume there exists ? ? 1 such that K(x, x) ? ?2 for any x ? X, and y ? [? a2 , a2 ], almost surely, a > 0. There exist n0 ? N such that for any n ? n0 , if 1 ?= ? , n M ? 75 ? n log 48?2 n , ? t ? 1 log(n) + 5 + 2 log(a + 3?), 2 then with probability 1 ? ?, c0 log2 24 ? ? . n In particular n0 , c0 do not depend on ?, M, n, t and c0 do not depend on ?. R(fb?,M,t ) ? The above result provides the desired bound, and all the constants are given in the appendix. The obtained learning rate is the same as the full KRR estimator and is known to be optimal in a minmax sense [17], hence not improvable. As mentioned before, the same bound is also achieved by the 6 basic Nystr?om method but with much worse time ? complexity. Indeed, as discussed before, using a simple iterative solver typically requires O( n log n) iterations, while we need?only O(log n). Considering the choice for M this leads to a computational time of O(nM t) = O(n n) for optimal generalization (omitting logarithmic terms). To the best of our knowledge FALKON currently provides the best time/space complexity to achieve the statistical accuracy of KRR. Beyond the basic setting considered above, in the next section we show that FALKON can achieve much faster rates under refined regularity assumptions and also consider the potential benefits of leverage score sampling. 4.2 Fast learning rates and Nystr?om with approximate leverage scores Considering fast rates and Nystr?om with more general sampling is considerably more technical and a heavier notation is needed. Our analysis apply to any approximation scheme (e.g. [30, 12, 31]) satisfying the definition of q-approximate leverage scores [13], satisfying q ?1 li (?) ? b li (?) ? qli (?), ? i ? {1, . . . , n}. Here ? > 0, li (?) = (Knn (Knn + ?nI)?1 )ii are the leverage scores and q ? 1 controls the quality of the approximation. In particular, given ?, the Nystr?om points are sampled independently from the dataset with probability pi ? b li (?). We need a few more definitions. Let Kx = K(x, ?) for any x ? X and H the reproducing kernel Hilbert space [32] of functions with inner product defined by H = span{Kx | x ? X} and closed with respect to the inner product h?, ?iH 0 0 defined by hK R x , Kx0 iH = K(x, x ), for all x, x ? X. Define C : H ? H to be the linear operator hf, CgiH = X f (x)g(x)d?X (x), for all f, g ? H. Finally define the following quantities, N? (?) = sup k(C + ?I)?1/2 Kx kH , N (?) = Tr(C(C + ?I)?1 ). x?X The latter quantity is known as degrees of freedom or effective dimension, can be seen as a measure of the size of H. The quantity N? (?) can be seen to provide a uniform bound on the leverage scores. 2 In particular note that N (?) ? N? (?) ? ?? [13]. We can now provide a refined version of Thm. 2. Theorem 4. Under the same conditions of Thm. 1, the exponent ? in Thm. 1 satisfies ? ? 1/2, when 2 1. either Nystr?om uniform sampling is used with M ? 70 [1 + N? (?)] log 8? ?? . 2 n 12?2 2 2. or Nystr?om q-approx. lev. scores [13] is used, with ? ? 19? n log 2? , n ? 405? log ? ,   8?2 M ? 215 2 + q 2 N (?) log . ?? We then recall the standard, albeit technical, assumptions leading to fast rates [17, 18]. The capacity condition requires the existence of ? ? (0, 1] and Q ? 0, such that N (?) ? Q2 ??? . Note that this condition is always satisfied with Q = ? and ? = 1. The source condition requires the existence of r ? [1/2, 1] and g ? H, such that fH = C r?1/2 g. Intuitively, the capacity condition measures the size of H, if ? is small then H is small and rates are faster. The source condition measures the regularity of fH , if r is big fH is regular and rates are faster. The case r = 1/2 and ? = D/(2s) (for a kernel with smoothness s and input space RD ) recovers the classic Sobolev condition. For further discussions on the interpretation of the conditions above see [17, 18, 11, 13]. We can then state our main result on fast rates Theorem 5. Let ? ? (0, 1]. Assume there exists ? ? 1 such that K(x, x) ? ?2 for any x ? X, and y ? [? a2 , a2 ], almost surely, with a > 0. There exist an n0 ? N such that for any n ? n0 the following holds. When 1 ? = n? 2r+? , t ? log(n) + 5 + 2 log(a + 3?2 ), 2 1. and either Nystr?om uniform sampling is used with M ? 70 [1 + N? (?)] log 8? ,   ?? 2 2. or Nystr?om q-approx. lev. scores [13] is used with M ? 220 2 + q 2 N (?) log 8? ?? , then with probability 1 ? ?, R(fb?,M,t ) ? c0 log2 2r 24 ? 2r+? n . ? where fb?,M,t is the FALKON estimator (Sect. 3, Alg. 1 and Sect. A, Alg. 2 in the appendix for the complete version). In particular n0 , c0 do not depend on ?, M, n, t and c0 do not depend on ?. 7 The above result shows that FALKON achieves the same fast rates as KRR, under the same conditions [17]. For r = 1/2, ? = 1, the rate in Thm. 3 is recovered. If ? < 1, r > 1/2, FALKON achieves a rate close to O(1/n). By selecting the Nystr?om points with uniform sampling, a bigger M could be needed for fast rates (albeit?always less than n). However, when approximate leverage scores are used M , smaller than n?/2  n is always enough for optimal generalization. This shows that FALKON with approximate leverage scores is the first algorithm to achieve fast rates with a computational ? ? complexity that is O(nN (?)) = O(n1+ 2r+? ) ? O(n1+ 2 ) in time. Main steps and novelties in the proof. The proof is long and technical and uses a variety of tools developed to analyze KRR. Our starting point is the analysis of the basic Nystr?om estimator given in [13]. The key novelty is the quantification of the approximations induced by the preconditioned iterative solver by relating its excess risk to the one of the basic Nystr?om estimator. A computational oracle inequality. First we prove that FALKON is equal to the exact Nystr?om estimator as the iterations go to infinity (Lemma 5). Then, in Lemma 8 (see also Lemma 6, 7) we show how optimization guarantees can be used to derive statistical results. More precisely, while optimization results in machine learning typically derives guarantees on empirical minimization problems, we show, using analytic and probabilistic tools, how these results can be turned into guarantees on the expected risks. Finally, in the proof of Thm. 1 we concentrate the terms of the inequality. The other key point is the study of the behavior of the condition number of B > HB with B given in (10). Controlling the condition number of B > HB. Let Cn , CM be the empirical P correlation operators in H n associated respectively to the training set and the Nystr?om points Cn = n1 i=1 Kxi ? Kxi , CM = P M 1 > ?> ? V (Cn + ?I)V A?1 ej ? Kx ej . In Lemma 1 we prove that B HB is equivalent to A j=1 Kx M for a suitable partial isometry V . Then in Lemma 2 we split it in two components B > HB = A?> V ? (CM + ?I)V A?1 + A?> V ? (Cn ? CM )V A?1 , (14) and prove that the first component is just the identity matrix. By denoting the second component with E, Eq. (14) implies that the condition number of B > HB is bounded by (1 + kEk)/(1 ? kEk), when kEk < 1. In Lemma 3 we prove that kEk is analytically bounded by a suitable distance between Cn ? CM and in Lemma 9, 10 we bound in probability such distance, when the Nystr?om centers are selected uniformly at random and with approximate leverage scores. Finally in Lemma 11, 12 we give a condition on M for the two kind of sampling, such that the condition number is controlled and the error term in the oracle inequality decays as e?t/2 , leading to Thm. 2, 4. 5 Experiments We present FALKON?s performance on a range of large scale datasets. As shown in Table 2, 3, FALKON achieves state of the art accuracy and typically outperforms previous approaches in all the considered large scale datasets including IMAGENET. This is remarkable considering FALKON required only a fraction of the competitor?s computational resources. Indeed we used a single machine equipped with two Intel Xeon E5-2630 v3, one NVIDIA Tesla K40c and 128 GB of RAM and a basic MATLAB FALKON implementation, while typically the results for competing algorithm have been performed on clusters of GPU workstations (accuracies, times and used architectures are cited from the corresponding papers). A minimal MATLAB implementation of FALKON is presented in Appendix G. The code necessary to reproduce the following experiments, plus a FALKON version that is able to use the GPU, is available on GitHub at https://github.com/LCSL/FALKON_paper . The error is measured with MSE, RMSE or relative error for regression problems, and with classification error (c-err) or AUC for the classification problems, to be consistent with the literature. For datasets which do not have a fixed test set, we set apart 20% of the data for testing. For all datasets, but YELP and IMAGENET, we normalize the features by their z-score. From now on we denote with n the cardinality of the dataset, d the dimensionality. MillionSongs [36] (Table 2, n = 4.6 ? 105 , d = 90, regression). We used a Gaussian kernel with ? = 6, ? = 10?6 and 104 Nystr?om centers. Moreover with 5 ? 104 center, FALKON achieves a 79.20 MSE, and 4.49 ? 10?3 rel. error in 630 sec. TIMIT (Table 2, n = 1.2 ? 106 , d = 440, multiclass classification). We used the same preprocessed dataset of [6] and Gaussian Kernel with ? = 15, ? = 10?9 and 105 Nystr?om centers. 8 Table 2: Architectures: ? cluster 128 EC2 r3.2xlarge machines, ? cluster 8 EC2 r3.8xlarge machines, o single machine with two Intel Xeon E5-2620, one Nvidia GTX Titan X GPU, 128GB RAM, ? cluster with IBM POWER8 12-core processor, 512 GB RAM, ? unknown platform. MillionSongs FALKON Prec. KRR [4] Hierarchical [33] D&C [29] Rand. Feat. [29] Nystr?om [29] ADMM R. F.[4] BCD R. F. [24] BCD Nystr?om [24] EigenPro [6] KRR [33] [24] Deep NN [34] Sparse Kernels [34] Ensemble [35] MSE Relative error 80.10 80.35 80.93 80.38 - ?3 YELP TIMIT Time(s) RMSE Time(m) c-err Time(h) 55 289? 293? 737? 772? 876? 958? - 0.833 0.949 0.861 0.854 - 20 42? 60? 500? - 32.3% 34.0% 33.7% 32.6% 33.5% 32.4% 30.9% 33.5% 1.5 1.7? 1.7? 3.9o 8.3? - 4.51 ? 10 4.58 ? 10?3 4.56 ? 10?3 5.01 ? 10?3 4.55 ? 10?3 - YELP (Table 2, n = 1.5 ? 106 , d = 6.52 ? 107 , regression). We used the same dataset of [24]. We extracted the 3-grams from the plain text with the same pipeline as [24], then we mapped them in a sparse binary vector which records if the 3-gram is present or not in the example. We used a linear kernel with 5 ? 104 Nystr?om centers. With 105 centers, we get a RMSE of 0.828 in 50 minutes. Table 3: Architectures: ? cluster with IBM POWER8 12-core cpu, 512 GB RAM, o single machine with two Intel Xeon E5-2620, one Nvidia GTX Titan X GPU, 128GB RAM, ? single machine [37] SUSY FALKON EigenPro [6] Hierarchical [33] Boosted Decision Tree [38] Neural Network [38] Deep Neural Network [38] Inception-V4 [39] HIGGS IMAGENET c-err AUC Time(m) AUC Time(h) c-err Time(h) 19.6% 19.8% 20.1% - 0.877 0.863 0.875 0.879 - 4 6o 40? 4680? - 0.833 0.810 0.816 0.885 - 3 78? - 20.7% 20.0% 4 - SUSY (Table 3, n = 5 ? 106 , d = 18, binary classification). We used a Gaussian kernel with ? = 4, ? = 10?6 and 104 Nystr?om centers. HIGGS (Table 3, n = 1.1 ? 106 , d = 28, binary classification). Each feature has been normalized subtracting its mean and dividing for its variance. We used a Gaussian kernel with diagonal matrix width learned with cross validation on a small validation set, ? = 10?8 and 105 Nystr?om centers. If we use a single ? = 5 we reach an AUC of 0.825. IMAGENET (Table 3, n = 1.3 ? 106 , d = 1536, multiclass classification). We report the top 1 c-err over the validation set of ILSVRC 2012 with a single crop. The features are obtained from the convolutional layers of pre-trained Inception-V4 [39]. We used Gaussian kernel with ? = 19, ? = 10?9 and 5 ? 104 Nystr?om centers. Note that with linear kernel we achieve c-err = 22.2%. Acknowledgments. The authors would like to thank Mikhail Belkin, Benjamin Recht and Siyuan Ma, Eric Fosler-Lussier, Shivaram Venkataraman, Stephen L. Tu, for providing their features of the TIMIT and YELP datasets, and NVIDIA Corporation for the donation of the Tesla K40c GPU used for this research. 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Structured Generative Adversarial Networks Zhijie Deng? , 2,3 Hao Zhang? , 2 Xiaodan Liang, 2 Luona Yang, 1,2 Shizhen Xu, 1 Jun Zhu? , 3 Eric P. Xing 1 Tsinghua University, 2 Carnegie Mellon University, 3 Petuum Inc. {dzj17,xsz12}@mails.tsinghua.edu.cn, {hao,xiaodan1,luonay1}@cs.cmu.edu, [email protected], [email protected] 1 Abstract We study the problem of conditional generative modeling based on designated semantics or structures. Existing models that build conditional generators either require massive labeled instances as supervision or are unable to accurately control the semantics of generated samples. We propose structured generative adversarial networks (SGANs) for semi-supervised conditional generative modeling. SGAN assumes the data x is generated conditioned on two independent latent variables: y that encodes the designated semantics, and z that contains other factors of variation. To ensure disentangled semantics in y and z, SGAN builds two collaborative games in the hidden space to minimize the reconstruction error of y and z, respectively. Training SGAN also involves solving two adversarial games that have their equilibrium concentrating at the true joint data distributions p(x, z) and p(x, y), avoiding distributing the probability mass diffusely over data space that MLE-based methods may suffer. We assess SGAN by evaluating its trained networks, and its performance on downstream tasks. We show that SGAN delivers a highly controllable generator, and disentangled representations; it also establishes start-of-the-art results across multiple datasets when applied for semi-supervised image classification (1.27%, 5.73%, 17.26% error rates on MNIST, SVHN and CIFAR-10 using 50, 1000 and 4000 labels, respectively). Benefiting from the separate modeling of y and z, SGAN can generate images with high visual quality and strictly following the designated semantic, and can be extended to a wide spectrum of applications, such as style transfer. 1 Introduction Deep generative models (DGMs) [12, 8, 26] have gained considerable research interest recently because of their high capacity of modeling complex data distributions and ease of training or inference. Among various DGMs, variational autoencoders (VAEs) and generative adversarial networks (GANs) can be trained unsupervisedly to map a random noise z ? N (0, 1) to the data distribution p(x), and have reported remarkable successes in many domains including image/text generation [17, 9, 3, 27], representation learning [27, 4], and posterior inference [12, 5]. They have also been extended to model the conditional distribution p(x|y), which involves training a neural network generator G that takes as inputs both the random noise z and a condition y, and generates samples that have desired properties specified by y. Obtaining such a conditional generator would be quite helpful for a wide spectrum of downstream applications, such as classification, where synthetic data from G can be used to augment the training set. However, training conditional generator is inherently difficult, because it requires not only a holistic characterization of the data distribution, but also fine-grained alignments between different modes of the distribution and different conditions. Previous works have tackled this problem by using a large amount of labeled data to guide the generator?s learning [32, 23, 25], which compromises the generator?s usefulness because obtaining the label information might be expensive. ? indicates equal contributions. ? indicates the corresponding author. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. In this paper, we investigate the problem of building conditional generative models under semisupervised settings, where we have access to only a small set of labeled data. The existing works [11, 15] have explored this direction based on DGMs, but the resulted conditional generators exhibit inadequate controllability, which we define as the generator?s ability to conditionally generate samples that have structures strictly agreeing with those specified by the condition ? a more controllable generator can better capture and respect the semantics of the condition. When supervision from labeled data is scarce, the controllability of a generative model is usually influenced by its ability to disentangle the designated semantics from other factors of variations (which we will term as disentanglability in the following text). In other words, the model has to first learn from a small set of labeled data what semantics or structures the condition y is essentially representing by trying to recognize y in the latent space. As a second step, when performing conditional generation, the semantics shall be exclusively captured and governed within y but not interweaved with other factors. Following this intuition, we build the structured generative adversarial network (SGAN) with enhanced controllability and disentanglability for semi-supervised generative modeling. SGAN separates the hidden space to two parts y and z, and learns a more structured generator distribution p(x|y, z) ? where the data are generated conditioned on two latent variables: y, which encodes the designated semantics, and z that contains other factors of variation. To impose the aforementioned exclusiveness constraint, SGAN first introduces two dedicated inference networks C and I to map x back to the hidden space as C : x ? y, I : x ? z, respectively. Then, SGAN enforces G to generate samples that when being mapped back to hidden space using C (or I), the inferred latent code and the generator condition are always matched, regardless of the variations of the other variable z (or y). To train SGAN, we draw inspirations from the recently proposed adversarially learned inference framework (ALI) [5], and build two adversarial games to drive I, G to match the true joint distributions p(x, z), and C, G to match the true joint distribution p(x, y). Thus, SGAN can be seen as a combination of two adversarial games and two collaborative games, where I, G combat each other to match joint distributions in the visible space, but I, C, G collaborate with each other to minimize a reconstruction error in the hidden space. We theoretically show that SGAN will converge to desired equilibrium if trained properly. To empirically evaluate SGAN, we first define a mutual predictability (MP) measure to evaluate the disentanglability of various DGMs, and show that in terms of MP, SGAN outperforms all existing models that are able to infer the latent code z across multiple image datasets. When classifying the generated images using a golden classifier, SGAN achieves the highest accuracy, confirming its improved controllability for conditional generation under semi-supervised settings. In the semisupervised image classification task, SGAN outperforms strong baselines, and establishes new state-of-the-art results on MNIST, SVHN and CIFAR-10 dataset. For controllable generation, SGAN can generate images with high visual quality in terms of both visual comparison and inception score, thanks to the disentangled latent space modeling. As SGAN is able to infer the unstructured code z, we further apply SGAN for style transfer, and obtain impressive results. 2 Related Work DGMs have drawn increasing interest from the community, and have been developed mainly toward two directions: VAE-based models [12, 11, 32] that learn the data distribution via maximum likelihood estimation (MLE), and GAN-based methods [19, 27, 21] that train a generator via adversarial learning. SGAN combines the best of MLE-based methods and GAN-based methods which we will discuss in detail in the next section. DGMs have also been applied for conditional generation, such as CGAN [19], CVAE [11]. DisVAE [32] is a successful extension of CVAE that generates images conditioned on text attributes. In parallel, CGAN has been developed to generate images conditioned on text [24, 23], bounding boxes, key points [25], locations [24], other images [10, 6, 31], or generate text conditioned on images [17]. All these models are trained using fully labeled data. A variety of techniques have been developed toward learning disentangled representations for generative modeling [3, 29]. InfoGAN [3] disentangles hidden dimensions on unlabeled data by mutual information regularization. However, the semantic of each disentangled dimension is uncontrollable because it is discovered after training rather than designated by user modeling. We establish some connections between SGAN and InfoGAN in the next section. There is also interest in developing DGMs for semi-supervised conditional generation, such as semisupervised CVAE [11], its many variants [16, 9, 18], ALI [5] and TripleGAN [15], among which the closest to us are [15, 9]. In [9], VAE is enhanced with a discriminator loss and an independency 2 constraint, and trained via joint MLE and discriminator loss minimization. By contrast, SGAN is an adversarial framework that is trained to match two joint distributions in the visible space, thus avoids MLE for visible variables. TripleGAN builds a three-player adversarial game to drive the generator to match the conditional distribution p(x|y), while SGAN models the conditional distribution p(x|y, z) instead. TripleGAN therefore lacks constraints to ensure the semantics of interest to be exclusively captured by y, and lacks a mechanism to perform posterior inference for z. 3 Structured Generative Adversarial Networks (SGAN) We build our model based on the generative adversarial networks (GANs) [8], a framework for learning DGMs using a two-player adversarial game. Specifically, given observed data {xi }N i=1 , GANs try to estimate a generator distribution pg (x) to match the true data distribution pdata (x), where pg (x) is modeled as a neural network G that transforms a noise variable z ? N (0, 1) ? = G(z). GANs assess the quality of x ? by introducing a neural network into generated data x discriminator D to judge whether a sample is from pdata (x) or the generator distribution pg (x). D is trained to distinguish generated samples from true samples while G is trained to fool D: min max L(D, G) = Ex?pdata (x) [log(D(x))] + Ez?p(z) [log(1 ? D(G(z)))], G D Goodfellow et al. [8] show the global optimum of the above problem is attained at pg = pdata . It is noted that the original GAN models the latent space using a single unstructured noise variable z. The semantics and structures that may be of our interest are entangled in z, and the generator transforms ? in a highly uncontrollable way ? it lacks both disentanglability and controllability. z into x We next describe SGAN, a generic extension to GANs that is enhanced with improved disentanglability and controllability for semi-supervised conditional generative modeling. Overview. We consider a semi-supervised setting, where we observe a large set of unlabeled data X = {xi }N i=1 . We are interested in both the observed sample x and some hidden structures y of ? that matches the true data x, and want to build a conditional generator that can generate data x distribution of x, while obey the structures specified in y (e.g. generate pictures of digits given 0-9). Besides the unlabeled x, we also have access to a small chunk of data Xl = {xlj , yjl }M j=1 where the structure y is jointly observed. Therefore, our model needs to characterize the joint distribution p(x, y) instead of the marginal p(x), for both fully and partially observed x. As the data generation process is intrinsically complex and usually determined by many factors beyond y, it is necessary to consider other factors that are irrelevant with y, and separate the hidden space into two parts (y, z), of which y encodes the designated semantics, and z includes any other factors of variation [3]. We make a mild assumption that y and z are independent from each other so that y could be disentangled from z. Our model thus needs to take into consideration the uncertainty of both (x, y) and z, i.e. characterizing the joint distribution p(x, y, z) while being able to disentangle y from z. Directly estimating p(x, y, z) is difficult, as (1) we have never observed z and only observed y for partial x; (2) y and z might be entangled at any time as the training proceeds. As an alternative, SGAN builds two inference networks I and C. The two inference networks define two distributions pi (z|x) and pc (y|x) that are trained to approximate the true posteriors p(z|x) and p(y|x) using two different adversarial games. The two games are unified via a shared generator x ? pg (x|y, z). Marginalizing out z or y obtains pg (x|z) and pg (x|y): Z Z pg (x|z) = p(y)pg (x|y, z)dy, pg (x|y) = p(z)pg (x|y, z)dz, (1) y z where p(y) and p(z) are appropriate known priors for y and z. As SGAN is able to perform posterior inference for both z and y given x (even for unlabeled data), we can directly imposes constraints [13] that enforce the structures of interest being exclusively captured by y, while those irreverent factors being encoded in z (as we will show later). Fig.1 illustrates the key components of SGAN, which we elaborate as follows. Generator pg (x|y, z). We assume the following generative process from y, z to x: z ? p(z), y ? p(y), x ? p(x|y, z), where p(z) is chosen as a non-informative prior, and p(y) as an appropriate prior that meets our modeling needs (e.g. a categorical distribution for digit class). We parametrize p(x|y, z) using a neural network generator G, which takes y and z as inputs, and outputs generated samples x ? pg (x|y, z) = G(y, z). G can be seen as a ?decoder? in VAE parlance, and its architecture depends on specific applications, such as a deconvolutional neural network for generating images [25, 21]. 3 ? ? ? ? ?(?, ?) ? ? ? ?(?, ?) ?* I(?) G(?, ?) ?# ?# (a) ? ? ? I(?) G(?, ?) ?# (b) ?* (c) ? ? G(?, ?) ?# (d) C(?) ?# (e) Figure 1: An overview of the SGAN model: (a) the generator pg (x|y, z); (b) the adversarial game Lxz ; (c) the adversarial game Lxy ; (d) the collaborative game Rz ; (e) the collaborative game Ry . Adversarial game Lxz . Following the adversarially learning inference (ALI) framework, we construct an adversarial game to match the distributions of joint pairs (x, z) drawn from the two different factorizations: pg (x, z) = p(z)pg (x|z), pi (x, z) = p(x)pi (z|x). Specifically, to draw samples from pg (x, z), we note the fact that we can first draw the tuple (x, y, z) following y ? p(y), z ? p(z), x ? pg (x|y, z), and then only taking (x, z) as needed. This implicitly performs the marginalization as in Eq. 1. On the other hand, we introduce an inference network I : x ? z to approximate the true posterior p(z|x). Obtaining (x, z) ? p(x)pi (z|x) with I is straightforward: x ? p(x), z ? pi (z|x) = I(x). Training G and I involves finding the Nash equilibrium for the following minimax game Lxz (we slightly abuse Lxz for both the minimax objective and a name for this adversarial game): min max Lxz = Ex?p(x) [log(Dxz (x, I(x)))] + Ez?p(z),y?p(y) [log(1 ? Dxz (G(y, z), z))], (2) I,G Dxz where we introduce Dxz as a critic network that is trained to distinguish pairs (x, z) ? pg (x, z) from those come from pi (x, z). This minimax objective reaches optimum if and only if the conditional distribution pg (x|z) characterized by G inverses the approximate posterior pi (z|x), implying pg (x, z) = pi (x, z) [4, 5]. As we have never observed z for x, as long as z is assumed to be independent from y, it is reasonable to just set the true joint distribution p(x, z) = p?g (x, z) = p?i (x, z), where we use p?g and p?i to denote the optimal distributions when Lxz reaches its equilibrium. Adversarial game Lxy . The second adversarial game is built to match the true joint data distribution p(x, y) that has been observed on Xl . We introduce the other critic network Dxy to discriminate (x, y) ? p(x, y) from (x, y) ? pg (x, y) = p(y)pg (x|y), and build the game Lxy as: min max Lxy = E(x,y)?p(x,y) [log(Dxy (x, y))] + Ey?p(y),z?p(z) [log(1 ? Dxy (G(y, z), y))]. G Dxy (3) Collaborative game Ry . Although training the adversarial game Lxy theoretically drives pg (x, y) to concentrate on the true data distribution p(x, y), it turns out to be very difficult to train Lxy to desired convergence, as (1) the joint distribution p(x, y) characterized by Xl might be biased due to its small data size; (2) there is little supervision from Xl to tell G what y essentially represents, and how to generate samples conditioned on y. As a result, G might lack controllability ? it might generate low-fidelity samples that are not aligned with their conditions, which will always be rejected by Dxy . A natural solution to these issues is to allow (learned) posterior inference of y to reconstruct y from generated x [5]. By minimizing the reconstruction error, we can backpropagate the gradient to G to enhance its controllability. Once pg (x|y) can generate high-fidelity samples that respect the structures y, we can reuse the generated samples (x, y) ? pg (x, y) as true samples in the first term of Lxy , to prevent Dxz from collapsing into a biased p(x, y) characterized by Xl . Intuitively, we introduce the second inference network C : x ? y which approximates the posterior p(y|x) as y ? pc (y|x) = C(x), e.g. C reduces to a N-way classifier if y is categorical. To train pc (y|x), we define a collaboration (reconstruction) game Ry in the hidden space of y: min Ry = ?E(x,y)?p(x,y) [log pc (y|x)] ? E(x,y)?pg (x,y) [log pc (y|x)], C,G (4) which aims to minimize the reconstruction error of y in terms of C and G, on both labeled data Xl and generated data (x, y) ? pg (x, y). On the one hand, minimizing the first term of Ry w.r.t. C guides C toward the true posterior p(y|x). On the other hand, minimizing the second term w.r.t. G enhances G with extra controllability ? it minimizes the chance that G could generate samples that would otherwise be falsely predicted by C. Note that we also minimize the second term w.r.t. C, which proves effective in semi-supervised learning settings that uses synthetic samples to augment the predictive power of C. In summary, minimizing Ry can be seen as a collaborative game between two players C and G that drives pg (x|y) to match p(x|y) and pc (y|x) to match the posterior p(y|x). 4 Collaborative games Rz . As SGAN allows posterior inference for both y and z, we can explicitly impose constraints Ry and Rz to separate y from z during training. To explain, we first note that optimizing the second term of Ry w.r.t G actually enforces the structure information to be fully persevered in y, because C is asked to recover the structure y from G(y, z), which is generated conditioned on y, regardless of the uncertainty of z (as z is marginalized out during sampling). Therefore, minimizing Ry indicates the constraint: following minC,G Ey?p(y) pc (y|G(y, z1 )), pc (y|G(y, z2 )) , ?z1 , z2 ? p(z), where a, b is some distance function between a and b (e.g. cross entropy if C is a N-way classifier). On the counter part, we also want to enforce any other unstructured information that is not of our interest to be fully captured in z, without being entangled with y. So we build the second collaborative game Rz as: min Rz = ?E(x,z)?pg (x,z) [log pi (z|x)] I,G (5) where I is required to recover z from those samples generated by G conditioned on z, i.e. reconstructing z in the hidden space. Similar to Ry , minimizing Rz indicates: minI,G Ez?p(z) pi (z|G(y1 , z)), pi (z|G(y2 , z)) , ?y1 , y2 ? p(y), and when we model I as a deterministic mapping [4], the k ? k distance between distributions is equal to the `-2 distance between the outputs of I. Theoretical Guarantees. We provide some theoretical results about the SGAN framework under the nonparametric assumption. The proofs of the theorems are deferred to the supplementary materials. Theorem 3.1 The global minimum of maxDxz Lxz is achieved if and only if p(x)pi (z|x) = ? = 21 . Similarly, the global minimum of maxDxy Lxy is achieved if p(z)pg (x|z). At that point Dxz ? and only if p(x, y) = p(y)pg (x|y). At that point Dxy = 12 . Theorem 3.2 There exists a generator G? (y, z) of which the conditional distributions pg (x|y) and pg (x|z) can both achieve equilibrium in their own minimax games Lxy and Lxz . Theorem 3.3 Minimizing Rz w.r.t. I will keep the equilibrium of the adversarial game Lxz . Similarly, minimizing Ry w.r.t. C will keep the equilibrium of the adversarial game Lxy unchanged. Algorithm 1 Training Structured Generative Adversarial Networks (SGAN). 1: Pretrain C by minimizing the first term of Eq. 4 w.r.t. C using Xl . 2: repeat 3: Sample a batch of x: xu ? p(x). 4: Sample batches of pairs (x, y): (xl , yl ) ? p(x, y), (xg , yg ) ? pg (x, y), (xc , yc ) ? pc (x, y). Obtain a batch (xm , ym ) by mixing data from (xl , yl ), (xg , yg ), (xc , yc ) with proper mixing portion. 5: 6: for k = 1 ? K do 7: Train Dxz by maximizing the first term of Lxz using xu and the second using xg . 8: Train Dxy by maximizing the first term of Lxy using (xm , ym ) and the second using (xg , yg ). 9: end for 10: Train I by minimizing Lxz using xu and Rz using xg . 11: Train C by minimizing Ry using (xm , ym ) (see text). 12: Train G by minimizing Lxy + Lxz + Ry + Rz using (xg , yg ). 13: until convergence. Training. SGAN is fully differentiable and can be trained end-to-end using stochastic gradient descent, following the strategy in [8] that alternatively trains the two critic networks Dxy , Dxz and the other networks G, I and C. Though minimizing Ry and Rz w.r.t. G will introduce slight bias, we find empirically it works well and contributes to disentangling y and z. The training procedures are summarized in Algorithm 1. Moreover, to guarantee that C could be properly trained without bias, we pretrain C by minimizing the first term of Ry until convergence, and do not minimize Ry w.r.t. C until G has started generating meaning samples (usually after several epochs of training). As the training proceeds, we gradually improve the portion of synthetic samples (x, y) ? pg (x, y) and (x, y) ? pc (x, y) in the stochastic batch, to help the training of Dxy and C (see Algorithm 1), and you can refer to our codes on GitHub for more details of the portion. We empirically found this mutual bootstrapping trick yields improved C and G. Discussion and connections. SGAN is essentially a combination of two adversarial games Lxy and Lxz , and two collaborative games Ry , Rz , where Lxy and Lxz are optimized to match the data distributions in the visible space, while Ry and Rz are trained to match the posteriors in the hidden space. It combines the best of GAN-based methods and MLE-based methods: on one hand, estimating 5 density in the visible space using GAN-based formulation avoids distributing the probability mass diffusely over data space [5], which MLE-based frameworks (e.g. VAE) suffer. One the other hand, incorporating reconstruction-based constraints in latent space helps enforce the disentanglement between structured information in y and unstructured ones in z, as we argued above. We also establish some connections between SGAN and some existing works [15, 27, 3]. We note the Lxy game in SGAN is connected to the TripleGAN framework [15] when its trade-off parameter ? = 0. We will empirically show that SGAN yields better controllability on G, and also improved performance on downstream tasks, due to the separate modeling of y and z. SGAN also connects to InfoGAN in the sense that the second term of Ry (Eq. 4) reduces to the mutual information penalty in InfoGAN under unsupervised settings. However, SGAN and InfoGAN have totally different aims and modeling techniques. SGAN builds a conditional generator that has the semantic of interest y as a fully controllable input (known before training); InfoGAN in contrast aims to disentangle some latent variables whose semantics are interpreted after training (by observation). Though extending InfoGAN to semi-supervised settings seems straightforward, successfully learning the joint distribution p(x, y) with very few labels is non-trivial: InfoGAN only maximizes the mutual information between y and G(y, z), bypassing p(y|x) or p(x, y), thus its direct extension to semi-supervised settings may fail due to lack of p(x, y). Moreover, SGAN has dedicated inference networks I and C, while the network Q(x) in InfoGAN shares parameters with the discriminator, which has been argued as problematic [15, 9] as it may compete with the discriminator and prevents its success in semisupervised settings. See our ablation study in section 4.2 and Fig.3. Finally, the first term in Ry is similar to the way Improved-GAN models the conditional p(y|x) for labeled data, but SGAN treats the generated data very differently ? Improved-GAN labels xg = G(z, y) as a new class y = K + 1, instead SGAN reuses xg and xc to mutually boost I, C and G, which is key to the success of semi-supervised learning (see section 4.2). 4 Evaluation We empirically evaluate SGAN through experiments on different datasets. We show that separately modeling z and y in the hidden space helps better disentangle the semantics of our interest from other irrelevant attributes, thus yields improved performance for both generative modeling (G) and posterior inference (C, I) (section 4.1 4.3). Under SGAN framework, the learned inference networks and generators can further benefit a lot of downstream applications, such as semi-supervised classification, controllable image generation and style transfer (section 4.2 4.3). Dataset and configurations. We evaluate SGAN on three image datasets: (1) MNIST [14]: we use the 60K training images as unlabeled data, and sample n ? {20, 50, 100} labels for semi-supervised learning following [12, 27], and evaluate on the 10K test images. (2) SVHN [20]: a standard train/test split is provided, where we sample n = 1000 labels from the training set for semi-supervised learning [27, 15, 5]. (3) CIFAR-10: a challenging dataset for conditional image generation that consists of 50K training and 10K test images from 10 object classes. We randomly sample n = 4000 labels [27, 28, 15] for semi-supervised learning. For all datasets, our semantic of interest is the digit/object class, so y is a 10-dim categorical variable. We use a 64-dim gaussian noise as z in MNIST and a 100-dim uniform noise as z in SVHN and CIFAR-10. Implementation. We implement SGAN using TensorFlow [1] and Theano [2] with distributed acceleration provided by Poseidon [33] which parallelizes line 7-8 and 10-12 of Algorithm. 1. The neural network architectures of C, G and Dxy mostly follow those used in TripleGAN [15] and we design I and Dxz according to [5] but with shallower structures to alleviate the training costs. Empirically SGAN needs 1.3-1.5x more training time than TripleGAN [15] without parallelization. It is noted that properly weighting the losses of the four games in SGAN during training may lead to performance improvement. However, we simply set them equal without heavy tuning1 . 4.1 Controllability and Disentanglability We evaluate the controllability and disentanglability of SGAN by assessing its generator network G and inference network I, respectively. Specifically, as SGAN is able to perform posterior inference for z, we define a novel quantitative measure based on z to compare its disentanglability to other DGMs: we first use the trained I (or the ?recognition network? in VAE-based models) to infer z for unseen x from test sets. Ideally, as z and y are modeled as independent, when I is trained to approach the true posterior of z, its output, when used as features, shall have weak predictability for y. Accordingly, we 1 The code is publicly available at https://github.com/thudzj/StructuredGAN. 6 use z as features to train a linear SVM classifier to predict the true y, and define the converged accuracy of this classifier as the mutual predictability (MP) measure, and expect lower MP for models that can better disentangle y from z. We conduct this experiment on all three sets, and report the averaged MP measure of five runs in Fig. 2, comparing the following DGMs (that are able to infer z): (1) ALI [5] and (2) VAE [12], trained without label information; (3) CVAE-full2 : the M2 model in [11] trained under the fully supervised setting; (4) SGAN trained under semi-supervised settings. We use 50, 1000 and 4000 labels for MNIST, SVHN and CIFAR-10 dataset under semi-supervised settings, respectively. ALI MP 0.9 VAE Clearly, SGAN demonstrates low MP when predicting y SGAN CVAE-full using z on three datasets. Using only 50 labels, SGAN 0.6 exhibits reasonable MP. In fact, on MNIST with only 20 labels as supervision, SGAN achieves 0.65 MP, outper0.3 forming other baselines by a large margin. The results clearly demonstrate SGAN?s ability to disentangle y and MNIST SVHN CIFAR-10 z, even when the supervision is very scarce. Figure 2: Comparisons of the MP measure On the other hand, better disentanglability also implies for different DGMs (lower is better). improved controllability of G, because less entangled y and z would be easier for G to recognize the designated semantics ? so G should be able to generate samples that are less deviated from y during conditional generation. To verify this, following [9], we use a pretrained gold-standard classifier (0.56% error on MNIST test set) to classify generated images, and use the condition y as ground truth to calculate the accuracy. We compare SGAN in Table 1 to CVAE-semi and TripleGAN [15], another strong baseline that is also designed for conditional generation under semi-supervised settings. We use n = 20, 50, 100 labels on MNIST, and observe a significantly higher accuracy for both TripleGAN and SGAN. For comparison, a generator trained by CVAE-full achieves 0.6% error. When there are fewer labels available, SGAN outperforms TripleGAN. The generator in SGAN can generate samples that consistently obey the conditions specified in y, even when there are only two images per class (n = 20) as supervision. These results verify our statements that disentangled semantics further enhance the controllability of the conditioned generator G. 4.2 Semi-supervised Classification # labeled samples It is natural to use SGAN for semi-supervised Model n = 20 n = 50 n = 100 prediction.With a little supervision, SGAN can CVAE-semi 33.05 10.72 5.66 deliver a conditional generator with reasonably 3.06 1.80 1.29 good controllability, with which, one can syn- TripleGAN SGAN 1.68 1.23 0.93 thesize samples from pg (x, y) to augment the training of C when minimizing Ry . Once C Table 1: Errors (%) of generated samples classified by a becomes more accurate, it tends to make less classifier with 0.56% test error. mistakes when inferring y from x. Moreover, as we are sampling (x, y) ? pc (x, y) to train Dxy during the maximization of Lxy , a more accurate C means more available labeled samples (by predicting y from unlabeled x using C) to lower the bias brought by the small set Xl , which in return can enhance G in the minimization phase of Lxy . Consequently, a mutual boosting cycle between G and C is formed. To empirically validate this, we deploy SGAN for semi-supervised classification on MNIST, SVHN and CIFAR-10, and compare the test errors of C to strong baselines in Table 2. To keep the comparisons fair, we adopt the same neural network architectures and hyper-parameter settings from [15], and report the averaged results of 10 runs with randomly sampled labels (every class has equal number of labels). We note that SGAN outperforms the current state-of-the-art methods across all datasets and settings. Especially, on MNIST when labeled instances are very scarce (n = 20), SGAN attains the highest accuracy (4.0% test error) with significantly lower variance, benefiting from the mutual boosting effects explained above. This is very critical for applications under low-shot or even one-shot settings where the small set Xl might not be a good representative for the data distribution p(x, y). 2 For CVAE-full, we use test images and ground truth labels together to infer z when calculating MP. We are unable to compare to semi-supervised CVAE as in CVAE inferring z for test images requires image labels as input, which is unfair to other methods. 7 Method Ladder [22] VAE [12] CatGAN [28] ALI [5] ImprovedGAN [27] TripleGAN [15] SGAN n = 20 16.77(?4.52) 5.40(?6.53) 4.0(?4.14) MNIST n = 50 2.21(?1.36) 1.59(?0.69) 1.29(?0.47) n = 100 0.89(?0.50) 3.33(?0.14) 1.39(?0.28) 0.93 (?0.07) 0.92(?0.58) 0.89(?0.11) SVHN n = 1000 36.02(?0.10) 7.3 8.11(?1.3) 5.83(?0.20) 5.73(?0.12) CIFAR-10 n = 4000 20.40(?0.47) 19.58(?0.58) 18.3 18.63(?2.32) 18.82(?0.32) 17.26(?0.69) Table 2: Comparisons of semi-supervised classification errors (%) on MNIST, SVHN and CIFAR-10 test sets. 4.3 Qualitative Results In this section we present qualitative results produced by SGAN?s generator under semi-supervised settings. Unless otherwise specified, we use 50, 1000 and 4000 labels on MNIST, SVHN, CIFAR-10 for the results. These results are randomly selected without cherry pick, and more results could be found in the supplementary materials. Controllable generation. To figure out how each module in SGAN contributes to the final results, we conduct an ablation study in Fig.3, where we plot images generated by SGAN with or without the terms Ry and Rz during training. As we have observed, our full model (a) w/o Ry , Rz (b) w/o Rz (c) Full model accurately disentangles y and z. Figure 3: Ablation study: conditional generation results by SGAN When there is no collaborative game (a) without Ry , Rz , (b) without Rz (c) full model. Each row has the involved, the generator easily col- same y while each column shares the same z. lapses to a biased conditional distribution defined by the classifier C that is trained only on a very small set of labeled data with insufficient supervision. For example, the generator cannot clearly distinguish the following digits: 0, 2, 3, 5, 8. Incorporating Ry into training significantly alleviate this issue ? an augmented C would resolve G?s confusion. However, it still makes mistakes in some confusing classes, such as 3 and 5. Ry and Rz connect the two adversarial games to form a mutual boosting cycle. The absence of any of them would break this cycle, consequently, SGAN would be under-constrained and may collapse to some local minima ? resulting in both a less accurate classifier C and a less controlled G. Visual quality. Next, we investigate whether a more disentangled y, z will result in higher visual quality on generated samples, as it makes sense that the conditioned generator G would be much easier to learn when its inputs y and z carry more orthogonal information. We conduct this experiment on CIFAR-10 that is consisted of natural images with more uncertainty besides the object categories. We compare several state-of-the-art generators in Fig 4 to SGAN without any advanced GAN training strategies (e.g. WGAN, gradient penalties) that are reported to possibly improve the visual quality. We find SGAN?s conditional generator does generate less blurred images with the main objects more salient, compared to TripleGAN and ImprovedGAN w/o minibatch discrimination (see supplementary). For a quantitative measure, we generate 50K images and compute the inception (a) CIFAR-10 data (b) TripleGAN (c) SGAN Figure 4: Visual comparison of generated images on CIFAR-10. For (b) and (c), each row shares the same y. 8 (a) (b) (c) (e) (f) (d) Figure 5: (a)-(c): image progression, (d)-(f): style transfer using SGAN. score [27] as 6.91(?0.07), compared to TripleGAN 5.08(?0.09) and Improved-GAN 3.87(?0.03) w/o minibatch discrimination, confirming the advantage of structured modeling for y and z. Image progression. To demonstrate that SGAN generalizes well instead of just memorizing the data, we generate images with interpolated z in Fig.5(a)-(c) [32]. Clearly, the images generated with progression are semantically consistent with y, and change smoothly from left to right. This verifies that SGAN correctly disentangles semantics, and learns accurate class-conditional distributions. Style transfer. We apply SGAN for style transfer [7, 30]. Specifically, as y is modeled as digit/object category on all three dataset, we suppose z shall encode any other information that are orthogonal to y (probably style information). To see whether I behaves properly, we use SGAN to transfer the unstructured information from z in Fig.5(d)-(f): given an image x (the leftmost image), we infer its unstructured code z. We generate images conditioned on z, but with different y. It is interesting to see that z encodes various aspects of the images, such as the shape, texture, orientation, background information, etc, as expected. Moreover, G can correctly transfer these information to other classes. 5 Conclusion We have presented SGAN for semi-supervised conditional generative modeling, which learns from a small set of labeled instances to disentangle the semantics of our interest from other elements in the latent space. We show that SGAN has improved disentanglability and controllability compared to baseline frameworks. SGAN?s design is beneficial to a lot of downstream applications: it establishes new state-of-the-art results on semi-supervised classification, and outperforms strong baseline in terms of the visual quality and inception score on controllable image generation. Acknowledgements Zhijie Deng and Jun Zhu are supported by NSF China (Nos. 61620106010, 61621136008, 61332007), the MIIT Grant of Int. Man. Comp. Stan (No. 2016ZXFB00001), Tsinghua Tiangong Institute for Intelligent Computing and the NVIDIA NVAIL Program. Hao Zhang is supported by the AFRL/DARPA project FA872105C0003. Xiaodan Liang is supported by award FA870215D0002. References [1] Mart?n Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for large-scale machine learning. In USENIX Symposium on Operating Systems Design and Implementation, 2016. 9 [2] James Bergstra, Olivier Breuleux, Fr?d?ric Bastien, Pascal Lamblin, Razvan Pascanu, Guillaume Desjardins, Joseph Turian, David Warde-Farley, and Yoshua Bengio. Theano: A cpu and gpu math compiler in python. pages 3?10, 2010. [3] Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. 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Triple generative adversarial nets. In Advances in Neural Information Processing Systems, 2017. [16] Chongxuan Li, Jun Zhu, Tianlin Shi, and Bo Zhang. Max-margin deep generative models. In Advances in Neural Information Processing Systems, pages 1837?1845, 2015. [17] Xiaodan Liang, Zhiting Hu, Hao Zhang, Chuang Gan, and Eric P Xing. Recurrent topic-transition gan for visual paragraph generation. arXiv preprint arXiv:1703.07022, 2017. [18] Lars Maal?e, Casper Kaae S?nderby, S?ren Kaae S?nderby, and Ole Winther. Auxiliary deep generative models. arXiv preprint arXiv:1602.05473, 2016. [19] Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014. [20] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. [21] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015. [22] Antti Rasmus, Mathias Berglund, Mikko Honkala, Harri Valpola, and Tapani Raiko. Semi-supervised learning with ladder networks. In Advances in Neural Information Processing Systems, pages 3546?3554, 2015. 10 [23] Scott Reed, Zeynep Akata, Xinchen Yan, Lajanugen Logeswaran, Bernt Schiele, and Honglak Lee. Generative adversarial text to image synthesis. In International Conference on Machine Learning, pages 1060?1069, 2016. [24] Scott Reed, A?ron van den Oord, Nal Kalchbrenner, Victor Bapst, Matt Botvinick, and Nando de Freitas. Generating interpretable images with controllable structure. In International Conference on Learning Representations, 2017. [25] Scott E Reed, Zeynep Akata, Santosh Mohan, Samuel Tenka, Bernt Schiele, and Honglak Lee. Learning what and where to draw. In Advances in Neural Information Processing Systems, pages 217?225, 2016. [26] Ruslan Salakhutdinov and Geoffrey Hinton. Deep boltzmann machines. In Artificial Intelligence and Statistics, pages 448?455, 2009. [27] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pages 2226?2234, 2016. [28] Jost Tobias Springenberg. Unsupervised and semi-supervised learning with categorical generative adversarial networks. arXiv preprint arXiv:1511.06390, 2015. [29] Luan Tran, Xi Yin, and Xiaoming Liu. Disentangled representation learning gan for pose-invariant face recognition. In Conference on Computer Vision and Pattern Recognition, 2017. [30] Hao Wang, Xiaodan Liang, Hao Zhang, Dit-Yan Yeung, and Eric P Xing. Zm-net: Real-time zero-shot image manipulation network. arXiv preprint arXiv:1703.07255, 2017. [31] Xiaolong Wang and Abhinav Gupta. Generative image modeling using style and structure adversarial networks. In European Conference on Computer Vision, pages 318?335. Springer, 2016. [32] Xinchen Yan, Jimei Yang, Kihyuk Sohn, and Honglak Lee. Attribute2image: Conditional image generation from visual attributes. In European Conference on Computer Vision, pages 776?791. Springer, 2016. [33] Hao Zhang, Zhiting Hu, Jinliang Wei, Pengtao Xie, Gunhee Kim, Qirong Ho, and Eric Xing. Poseidon: A system architecture for efficient gpu-based deep learning on multiple machines. arXiv preprint arXiv:1512.06216, 2015. 11
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Conservative Contextual Linear Bandits Abbas Kazerouni Stanford University [email protected] Mohammad Ghavamzadeh DeepMind [email protected] Yasin Abbasi-Yadkori Adobe Research [email protected] Benjamin Van Roy Stanford University [email protected] Abstract Safety is a desirable property that can immensely increase the applicability of learning algorithms in real-world decision-making problems. It is much easier for a company to deploy an algorithm that is safe, i.e., guaranteed to perform at least as well as a baseline. In this paper, we study the issue of safety in contextual linear bandits that have application in many different fields including personalized recommendation. We formulate a notion of safety for this class of algorithms. We develop a safe contextual linear bandit algorithm, called conservative linear UCB (CLUCB), that simultaneously minimizes its regret and satisfies the safety constraint, i.e., maintains its performance above a fixed percentage of the performance of a baseline strategy, uniformly over time. We prove an upper-bound on the regret of CLUCB and show that it can be decomposed into two terms: 1) an upper-bound for the regret of the standard linear UCB algorithm that grows with the time horizon and 2) a constant term that accounts for the loss of being conservative in order to satisfy the safety constraint. We empirically show that our algorithm is safe and validate our theoretical analysis. 1 Introduction Many problems in science and engineering can be formulated as decision-making problems under uncertainty. Although many learning algorithms have been developed to find a good policy/strategy for these problems, most of them do not provide any guarantee for the performance of their resulting policy during the initial exploratory phase. This is a major obstacle in using learning algorithms in many different fields, such as online marketing, health sciences, finance, and robotics. Therefore, developing learning algorithms with safety guarantees can immensely increase the applicability of learning in solving decision problems. A policy generated by a learning algorithm is considered to be safe, if it is guaranteed to perform at least as well as a baseline. The baseline can be either a baseline value or the performance of a baseline strategy. It is important to note that since the policy is learned from data, it is a random variable, and thus, the safety guarantees are in high probability. Safety can be studied in both offline and online scenarios. In the offline case, the algorithm learns the policy from a batch of data, usually generated by the current strategy or recent strategies of the company, and the question is whether the learned policy will perform as well as the current strategy or no worse than a baseline value, when it is deployed. This scenario has been recently studied heavily in both model-based (e.g., Petrik et al. [2016]) and model-free (e.g., Bottou et al. 2013; Thomas et al. 2015a,b; Swaminathan and Joachims 2015a,b) settings. In the model-based approach, we first use the batch of data and build a simulator that mimics the behavior of the dynamical system under study (hospital?s ER, financial market, robot), and then use this simulator to generate data and learn the policy. The main challenge here is to have guarantees on the performance of the learned policy, given the error in the simulator. This line of research is closely related to the area of robust learning and control. In the model-free approach, we learn the policy directly from the batch of data, without building a simulator. This line of research is related to off-policy evaluation and control. While the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. model-free approach is more suitable for problems in which we have access to a large batch of data, such as in online marketing, the model-based approach works better in problems in which data is harder to collect, but instead, we have good knowledge about the underlying dynamical system that allows us to build an accurate simulator. In the online scenario, the algorithm learns a policy while interacting with the real system. Although (reasonable) online algorithms will eventually learn a good or an optimal policy, there is no guarantee for their performance along the way (the performance of their intermediate policies), especially at the very beginning, when they perform a large amount of exploration. Thus, in order to guarantee safety in online algorithms, it is important to control their exploration and make it more conservative. Consider a manager that allows our learning algorithm runs together with her company?s current strategy (baseline policy), as long as it is safe, i.e., the loss incurred by letting a portion of the traffic handled by our algorithm (instead of by the baseline policy) does not exceed a certain threshold. Although we are confident that our algorithm will eventually perform at least as well as the baseline strategy, it should be able to remain alive (not terminated by the manager) long enough for this to happen. Therefore, we should make it more conservative (less exploratory) in a way not to violate the manager?s safety constraint. This setting has been studied in the multi-armed bandit (MAB) [Wu et al., 2016]. Wu et al. [2016] considered the baseline policy as a fixed arm in MAB, formulated safety using a constraint defined based on the performance of the baseline policy (mean of the baseline arm), and modified the UCB algorithm [Auer et al., 2002] to satisfy this constraint. In this paper, we study the notion of safety in contextual linear bandits, a setting that has application in many different fields including personalized recommendation. We first formulate safety in this setting, as a constraint that must hold uniformly in time, in Section 2. Our goal is to design learning algorithms that minimize regret under the constraint that at any given time, their expected sum of rewards should be above a fixed percentage of the expected sum of rewards of the baseline policy. This fixed percentage depends on the amount of risk that the manager is willing to take. In Section 3, we propose an algorithm, called conservative linear UCB (CLUCB), that satisfies the safety constraint. At each round, CLUCB plays the action suggested by the standard linear UCB (LUCB) algorithm (e.g., Dani et al. 2008; Rusmevichientong and Tsitsiklis 2010; Abbasi-Yadkori et al. 2011; Chu et al. 2011; Russo and Van Roy 2014), only if it satisfies the safety constraint for the worst choice of the parameter in the confidence set, and plays the action suggested by the baseline policy, otherwise. We prove an upper-bound for the regret of CLUCB, which can be decomposed ? into two terms. The first term is an upper-bound on the regret of LUCB that grows at the rate T log(T ). The second term is constant (does not grow with the horizon T ) and accounts for the loss of being conservative in order to satisfy the safety constraint. This improves over the regret bound derived in Wu et al. [2016] for the MAB setting, where the regret of being conservative grows with time. In Section 4, we show how CLUCB can be extended to the case that the reward of the baseline policy is unknown without a change in its rate of regret. Finally in Section 5, we report experimental results that show CLUCB behaves as expected in practice and validate our theoretical analysis. 2 Problem Formulation In this section, we first review the standard linear bandit setting and then introduce the conservative linear bandit formulation considered in this paper. 2.1 Linear Bandit In the linear bandit setting, at any time t, the agent is given a set of (possibly) infinitely many actions/options At , where each action a ? At is associated with a feature vector ?ta ? Rd . At each round t, the agent selects an action at ? At and observes a random reward yt generated as yt = h?? , ?tat i + ?t , (1) where ?? ? Rd is the unknown reward parameter, h?? , ?tat i = rat t is the expected reward of action at at time t, i.e., rat t = E[yt ], and ?t is a random noise such that Assumption 1 Each element ?t of the noise sequence {?t }? t=1 is conditionally ?-sub-Gaussian, i.e., E[e??t | a1:t , ?1:t?1 ] ? exp(? 2 ? 2 /2), ?? ? R. The sub-Gaussian assumption implies that E[?t | a1:t , ?1:t?1 ] = 0 and Var[?t | a1:t , ?1:t?1 ] ? ? 2 . 2 Note that the above formulation contains time-varying action sets and time-dependent feature vectors for each action, and thus, includes the linear contextual bandit setting. In linear contextual bandit, if we denote by xt , the state of the system at time t, the time-dependent feature vector ?ta for action a will be equal to ?(xt , a), the feature vector of state-action pair (xt , a). We also make the following standard assumption on the unknown parameter ?? and feature vectors: Assumption 2 There exist constants B, D ? 0 such that k?? k2 ? B, k?ta k2 ? D, and h?? , ?ta i ? [0, 1], for all t and all a ? At .   We define B = ? ? Rd : k?k2 ? B and F = ? ? Rd : k?k2 ? D, h?? , ?i ? [0, 1] to be the parameter space and feature space, respectively. Obviously, if the agent knows ?? , she will choose the optimal action a?t = arg maxa?At h?? , ?ta i at each round t. Since ?? is unknown, the agent?s goal is to maximize her cumulative expected rewards PT after T rounds, i.e., t=1 h?? , ?tat i, or equivalently, to minimize its (pseudo)-regret, i.e., RT = T X h?? , ?ta?t i ? t=1 T X h?? , ?tat i, (2) t=1 which is the difference between the cumulative expected rewards of the optimal and agent?s strategies. 2.2 Conservative Linear Bandit The conservative linear bandit setting is exactly the same as the linear bandit, except that there exists a baseline policy ?b (e.g., the company?s current strategy) that at each round t, selects action bt ? At and incurs the expected reward rbtt = h?? , ?tbt i. We assume that the expected rewards of the actions taken by the baseline policy, rbtt , are known (see Remark 1). We relax this assumption in Section 4 and extend our proposed algorithm to the case that the reward function of the baseline policy is not known in advance. Another difference between the conservative and standard linear bandit settings is the performance constraint, which is defined as follows: Definition 1 (Performance Constraint) At each round t, the difference between the performances of the baseline and the agent?s policies should remain below a pre-defined fraction ? ? (0, 1) of the baseline performance. This constraint may be written formally as t t t t t X X X X X i i i i ?t ? {1, . . . , T }, rbi ? rai ? ? rbi or equivalently as rai ? (1??) rbi i . (3) i=1 i=1 i=1 i=1 i=1 The parameter ? controls the level of conservatism of the agent. Small values show that only small losses are tolerated and the agent should be overly conservative, whereas large values indicate that the manager is willing to take risk and the agent can be more explorative. Here, given the value of ?, the agent should select her actions in a way to both minimize her regret (2) and to satisfy the performance constraint (3). In the next section, we propose a linear bandit algorithm to achieve this goal with high probability. Remark 1. Since the baseline policy is often our company?s strategy, it is reasonable to assume that a large amount of data generated by this policy is available, and thus, we have an accurate estimate of its reward function. If in addition to this accurate estimate, we have access to the actual data, we can use them in our algorithms. The reason we do not use the data generated by the actions suggested by the baseline policy in constructing the confidence sets of our algorithm in Section 3 is mainly to keep the analysis simple. However, when dealing with the more general case of unknown baseline reward in Section 4, we construct the confidence sets using all available data, including those generated by the baseline policy. It is important to note that having a good estimate of the baseline reward function does not necessarily mean that we know the unknown parameter ?? . This is because the data used for this estimate has been generated by the baseline policy, and thus, may only provide a good estimate of ?? in a limited subspace. 3 A Conservative Linear Bandit Algorithm In this section, we propose a linear bandit algorithm, called conservative linear upper confidence bound (CLUCB), whose pseudocode is shown in Algorithm 1. CLUCB is based on the optimism in the face of uncertainty principle, and given the value of ?, minimizes the regret (2) and satisfies the performance constraint (3) with high probability. At each round t, CLUCB uses the previous 3 Algorithm 1 CLUCB Input: ?, B, F Initialize: S0 = ?, z0 = 0 ? Rd , and C1 = B for t = 1, 2, 3, ? ? ? do Find (a0t , ?et ) ? arg max(a,?)?At ?Ct h?, ?ta i Compute Lt = min??Ct h?, zt?1 + ?ta0 i t P Pt if Lt + i?S c rbi i ? (1 ? ?) i=1 rbi i then t?1 Play at = a0t and observe reward yt defined by (1) c Set zt = zt?1 + ?tat , St = St?1 ? t, Stc = St?1 Given at and yt , construct the confidence set Ct+1 according to (5) else Play at = bt and observe reward yt defined by (1) c Set zt = zt?1 , St = St?1 , Stc = St?1 ? t, Ct+1 = Ct end if end for observations and builds a confidence set Ct that with high probability contains the unknown parameter ?? . It then selects the optimistic action a0t ? arg maxa?At max??Ct h?, ?ta i, which has the best performance among all the actions available in At , within the confidence set Ct . In order to make sure that the constraint (3) is satisfied, the algorithm plays the optimistic action a0t , only if it satisfies the constraint for the worst choice of the parameter ? ? Ct . To make this more precise, let St?1 be the set of rounds i < t at which CLUCB has played the optimistic action, i.e., ai = a0i . Similarly, c St?1 = {1, 2, ? ? ? , t ? 1} ? St?1 is the set of rounds j < t at which CLUCB has followed the baseline policy, i.e., aj = bj . In order to guarantee that it does not violate constraint (3), at each round t, CLUCB plays the optimistic action, i.e., at = a0t , only if zt?1 min h X ??Ct c i?St?1 rbi i t D z X}| { E i X + ?, ?iai + h?, ?ta0t i ? (1 ? ?) rbi i , i=1 i?St?1 and plays the conservative action, i.e., at = bt , otherwise. In the following, we describe how CLUCB constructs and updates its confidence sets Ct . 3.1 Construction of Confidence Sets CLUCB starts by the most general confidence set C1 = B and updates its confidence set only when it plays an optimistic action. This is mainly to simplify the analysis and is based on the idea that since the reward function of the baseline policy is known ahead of time, playing a baseline action does not provide any new information about the unknown parameter ?? . However, this can be easily changed to update the confidence set after each action. In fact, this is what we do in the algorithm proposed in Section 4. We follow the approach of Abbasi-Yadkori et al. [2011] to build confidence sets for ?? . Let St = {i1 , . . . , imt } be the set of rounds up to and including round t at which CLUCB has played the optimistic action. Note that we have defined mt = |St |. For a fixed value of ? > 0, let ?1 ?bt = (?t ?| + ?I) ?t Yt , (4) t i be the regularized least square estimate of ? at round t, where ?t = [?ia1i1 , . . . , ?amimt t ] and Yt = [yi1 , . . . , yimt ]> . For a fixed confidence parameter ? ? (0, 1), we construct the confidence set for the next round t + 1 as n o Ct+1 = ? ? Rd : k? ? ?bt kVt ? ?t+1 , (5) r where ?t+1 = ? d log ?  1+(mt +1)D 2 /? ?  + ? ?B, Vt = ?I + ?t ?> t , and the weighted norm is defined as kxkV = x> V x for any x ? Rd and any positive definite V ? Rd?d . Note that similar to the linear UCB algorithm (LUCB) in Abbasi-Yadkori et al. [2011], the sub-Gaussian parameter ? and the regularization parameter ? that appear in the definitions of ?t+1 and Vt should also be given to the CLUCB algorithm as input. The following proposition (Theorem 2 in Abbasi-Yadkori et al. 2011) shows that the confidence sets constructed by (5) contain the true parameter ?? with high probability. 4   Proposition 1 For the confidence set Ct defined by (5), we have P ?? ? Ct , ?t ? N ? 1 ? ?. As mentioned before, CLUCB ensures that performance constraint (3) holds for all ? ? Ct at all rounds t. As a result, if all the confidence sets hold (i.e., contain the true parameter ?? ), CLUCB is guaranteed to satisfy performance constraint (3). Proposition 1 indicates that this happens with probability at least 1 ? ?. It is worth noting that satisfying constraint (3) implies that CLUCB is at least as good as the baseline policy at all rounds. In this vein, Proposition 1 guarantees that, with probability at least 1 ? ?, CLUCB performs no worse than the baseline policy at all rounds. 3.2 Regret Analysis of CLUCB In this section, we prove a regret bound for the proposed CLUCB algorithm. Let ?tbt = rat ?t ? rbtt be the baseline gap at round t, i.e., the difference between the expected rewards of the optimal and baseline actions at round t. This quantity shows how sub-optimal the action suggested by the baseline policy is at round t. We make the following assumption on the performance of the baseline policy ?b . Assumption 3 There exist 0 ? ?l ? ?h and 0 < rl such that, at each round t, ?l ? ?tbt ? ?h and rl ? rbtt . (6) An obvious candidate for both ?h and rh is 1, as all the mean rewards are confined in [0, 1]. The reward lower-bound rl ensures that the baseline policy maintains a minimum level of performance at each round. Finally, ?l = 0 is a reasonable candidate for the lower-bound of the baseline gap. The following proposition shows that the regret of CLUCB can be decomposed into the regret of a linear UCB (LUCB) algorithm (e.g., Abbasi-Yadkori et al. 2011) and a regret caused by being conservative in order to satisfy the performance constraint (3). Proposition 2 The regret of CLUCB can be decomposed into two terms as follows: RT (CLUCB) ? RST (LUCB) + nT ?h , (7) where RST (LUCB) is the cumulative (pseudo)-regret of LUCB at rounds t ? ST and nT = |STc | = T ? mT is the number of rounds (in T rounds) at which CLUCB has played a conservative action. Proof: From the definition of regret (2), we have RT (CLUCB) = T X t=1 rat ?t ? T X t=1 rat t = X ?tb t X z t }| t { X t (rat ?t ? rat t ) + (ra?t ? rbt ) ? (ra?t ? rat t ) + nT ?h . (8) c t?ST t?ST t?ST The result follows from the fact that for t ? ST , CLUCB plays the exact same actions as LUCB, and thus, the first term in (8) represents LUCB?s regret for these rounds.  The regret bound of LUCB for the confidence set (5) can be derived from the results of AbbasiYadkori et al. [2011]. Let E be the event that ?? ? Ct , ?t ? N, which according to Proposition 1 holds w.p. at least 1 ? ?. The following proposition provides a bound on RST (LUCB). Since this proposition is a direct application of Thm. 3 in Abbasi-Yadkori et al. [2011], we omit its proof here. Proposition 3 On event E = {?? ? Ct , ?t ? N}, for any T ? N, we have s s      ? mT D 1 mT D RST (LUCB) ? 4 mT d log ? + ? B ? + ? 2 log( ) + d log 1 + d ? ?d     ? D = O d log T T . (9) ?? Now in order to bound the regret of CLUCB, we only need to find an upper-bound on nT , i.e., the number of times that CLUCB deviates from LUCB and selects the action suggested by the baseline policy. We prove an upper-bound on nT in Theorem 4, which is the main technical result of this section. Due to space constraint, we only provide a proof sketch for Theorem 4 in the paper and report its detailed proof in Appendix A. The proof requires several technical lemmas that have been proved in Appendix C. 5 Theorem 4 Let ? ? max(1, D2 ). Then, on event E, for any horizon T ? N, we have " ? (B ? + ?)2 nT ? 1 + 114d log ?rl (?l + ?rl ) 2 !#2 ? 62d(B ? + ?) ? . ?(?l + ?rl )  Proof Sketch: Let ? = max 1 ? t ? T | at 6= a0t be the last round that CLUCB takes an action suggested by the baseline policy. We first show that at round ? , the following holds: ? ? X rbtt X ? ?(m? ?1 + 1)?l + 2?? ??a0? V ?1 + 2 ?t ?tat V ?1 + 2?? ? t=1 t t?S? ?1 X t ? ?a0 + ?at ? t?S? ?1 . V??1 Next, using Lemmas 7 and 8 (reported in Appendix C), and the Cauchy-Schwartz inequality, we deduce that ? ? X ? rbtt ? ?(m? ?1 + 1)?l + 8d(B ? + ?) log  t=1 Since rbtt 2(m? ?1 + 1) ?  p (m? ?1 + 1). ? rl for all t, and ? = n? ?1 + m? ?1 + 1, it follows that ? ?rl n? ?1 ? ?(m? ?1 + 1)(?l + ?rl ) + 8d(B ? + ?) log  2(m? ?1 + 1) ?  p (m? ?1 + 1). (10) Note that n? ?1 and m? ?1 appear on the LHS and RHS of (10), respectively. The key point is that the RHS is positive only for a finite number of integers m? ?1 , and thus, it has a finite upper bound. Using Lemma 9 (reported and proved in Appendix C), we prove that ?rl n? ?1 ? 114d ? " ? + ?)2 ? log ?l + ?rl 2 (B !#2 ? 62d(B ? + ?) ? . ?(?l + ?rl ) Finally, the fact that nT = n? = n? ?1 + 1 completes the proof.  We now have all the necessary ingredients to derive a regret bound on the performance of the CLUCB algorithm. We report the regret bound of CLUCB in Theorem 5, whose proof is a direct consequence of the results of Propositions 2 and 3, and Theorem 4. Theorem 5 Let ? ? max(1, D2 ). With probability at least 1 ? ?, the CLUCB algorithm satisfies the performance constraint (3) for all t ? N, and has the regret bound   DT ? K?h , RT (CLUCB) = O d log T+ ?? ?rl (11) where K is a constant that only depends on the parameters of the problem as K = 1 + 114d ? " ? + ?)2 log ?l + ?rl 2 (B !#2 ? 62d(B ? + ?) ? . ?(?l + ?rl ) Remark 2. The first term in the regret bound (11) is the regret of LUCB, which grows at the rate ? T log(T ). The second term accounts for the loss incurred by being conservative in order to satisfy the performance constraint (3). Our results indicate that this loss does not grow with time (since CLUCB acts conservatively only in a finite number of rounds). This is a clear improvement over the regret bound reported in Wu et al. [2016] for the MAB setting, in which the regret of being conservative grows with time. Furthermore, the regret bound of Theorem 5 clearly indicates that CLUCB?s regret is larger for smaller values of ?. This perfectly matches the intuition that the agent must be more conservative, and thus, suffers higher regret for smaller values of ?. Theorem 5 also indicates that CLUCB?s regret is smaller for smaller values of ?h , because when the baseline policy ?b is close to optimal, the algorithm does not lose much by being conservative. 6 Algorithm 2 CLUCB2 Input: ?, rl , B, F Initialize: n ? 0, z ? 0, w ? 0, v ? 0 and C1 ? B for t = 1, 2, 3, ? ? ? do Let bt be the action suggested by ?b at round t e = arg max(a,?)?A ?C h?, ?t i Find (a0t , ?) a t t  Find Rt = max??Ct h?, v + ?tbt i & Lt = min??Ct h?, z + ?ta0 i + ? max min??Ct h?, wi, nrl t if Lt ? (1 ? ?)Rt then Play at = a0t and observe yt defined by (1) Set z ? z + ?ta0 and v ? v + ?tbt t else Play at = bt and observe yt defined by (1) Set w = w + ?tbt and n ? n + 1 end if Given at and yt , construct the confidence set Ct+1 according to (15) end for 4 Unknown Baseline Reward In this section, we consider the case where the expected rewards of the actions taken by the baseline policy, rbtt , are unknown at the beginning. We show how the CLUCB algorithm presented in Section 3 should be changed to handle this case, and present a new algorithm, called CLUCB2. We prove a regret bound for CLUCB2, which is at the same rate as that for CLUCB. This shows that the lack of knowledge about the reward function of the baseline policy does not hurt our algorithm in terms of the rate of the regret. The pseudocode of CLUCB2 is shown in Algorithm 2. The main difference with CLUCB is in the condition that should be checked at each round t to see whether we should play the optimistic action a0t or the conservative action bt . This condition should be selected in a way that CLUCB2 satisfies constraint (3). We may rewrite (3) as X X X  rai i + rat 0t + ? rbi i ? (1 ? ?) rbtt + rbi i . (12) c i?St?1 i?St?1 i?St?1 If we lower-bound the LHS and upper-bound the RHS of (12), we obtain X X X ?ibi + ?tbt i. min h?, ?iai + ?ta0t i + ? min h?, ?ibi i ? (1 ? ?) max h?, ??Ct ??Ct i?St?1 ??Ct c i?St?1 (13) i?St?1 Since each confidence set Ct is built in a way to contain the true parameter ?? with high probability, it is easy to see that (12) is satisfied whenever (13) is true. CLUCB2 uses both optimistic and conservative actions, and their corresponding rewards in building its confidence sets. Specifically for any t, we let ?t = [?1a1 , ?2a2 , ? ? ? , ?tat ], Yt = [y1 , y2 , ? ? ? , yt ]| , Vt = ?I + ?|t ?t , and define the least-square estimate after round t as ?1 ?bt = (?t ?|t + ?I) ?t Yt . Given Vt and ?bt , the confidence set for round t + 1 is constructed as n o Ct+1 = ? ? Ct : k? ? ?bt kVt ? ?t+1 , r where C1 = B and ?t = ? d log  1+tD 2 /? ?  (14) (15) ? + B ?. Similar to Proposition 1, we can easily ? prove that the confidence  sets built by (15) contain the true parameter ? with high probability, i.e., P ?? ? Ct , ?t ? N ? 1 ? ?. Remark 3. Note that unlike the CLUCB algorithm, here we build nested confidence sets, i.e., ? ? ? ? Ct+1 ? Ct ? Ct?1 ? ? ? ? , which is necessary for the proof of the algorithm. This can potentially increase the computational complexity of CLUCB2, but from a practical point of view, the confidence 7 Figure 1: Average per-step regret (over 1, 000 runs) of LUCB and CLUCB for different values of ?. sets become nested automatically after sufficient data has been observed. Therefore, the nested constraint in building the confidence sets can be relaxed after sufficiently large number of rounds. The following theorem guarantees that CLUCB2 satisfies the safety constraint (3) with high probability, while its regret has the same rate as that of CLUCB and is worse than that of LUCB only up to an additive constant. Theorem 6 Let ? ? max(1, D2 ) and ? ? 2/e. Then, with probability at least 1 ? ?, CLUCB2 algorithm satisfies the performance constraint (3) for all t ? N, and has the regret bound     K?h DT ? RT (CLUCB2) = O d log T+ 2 2 , (16) ?? ? rl where K is a constant that depends only on the parameters of the problem as " !#2 ? ? 10d(B ? + ?) + 1. K = 256d2 (B ? + ?)2 log ?rl (?)1/4 We report the proof of Theorem 6 in Appendix B. The proof follows the same steps as that of Theorem 5, with additional non-trivial technicalities that have been highlighted there. 5 Simulation Results In this section, we provide simulation results to illustrate the performance of the proposed CLUCB algorithm. We considered a time independent action set of 100 arms each having a time independent feature vectorliving in R4 space. These feature vectors and the parameter ?? are randomly drawn from N 0, I4 such that the mean reward associated to each arm is positive. The observation noise at each time step is also generated independently from N (0, 1), and the mean reward of the baseline policy at any time is taken to be the reward associated to the 10?th best action. We have taken ? = 1, ? = 0.001 and the results are averaged over 1,000 realizations. In Figure 1, we plot per-step regret (i.e., Rtt ) of LUCB and CLUCB for different values of ? over a horizon T = 40, 000. Figure 1 shows that per-step regret of CLUCB remains constant at the beginning (the conservative phase). This is because during this phase, CLUCB follows the baseline policy to make sure that the performance constraint (3) is satisfied. As expected, the length of the conservative phase decreases as ? is increased, since the performance constraint is relaxed for larger values of ?, and hence, CLUCB starts playing optimistic actions more quickly. After this initial conservative phase, CLUCB has learned enough about the optimal action and its performance starts converging to that of LUCB. On the other hand, Figure 1 shows that per-step regret of CLUCB at the first few periods remains much lower than that of LUCB. This is because LUCB plays agnostic to the safety constraint, and thus, may select very poor actions in its initial exploration phase. In regard to this, Figure 2(a) plots the percentage of the rounds, in the first 1, 000 rounds, at which the safety constraint (3) is violated by LUCB and CLUCB for different values of ?. According to this figure, 8 (a) (b) Figure 2: (a) Percentage of the rounds, in the first 1, 000 rounds, at which the safety constraint is violated by LUCB and CLUCB for different values of ?, (b) Per-step regret of LUCB and CLUCB for different values of ?, at round t = 40, 000. CLUCB satisfies the performance constraint for all values of ?, while LUCB fails in a significant number of rounds, specially for small values of ? (i.e., tight constraint). To better illustrate the effect of the performance constraint (3) on the regret of the algorithms, Figure 2(b) plots the per-step regret achieved by CLUCB at round t = 40, 000 for different values of ?, as well as that for LUCB. As expected from our analysis and is shown in Figure 1, the performance of CLUCB converges to that of LUCB after an initial conservative phase. Figure 2(b) confirms that the convergence happens more quickly for larger values of ?, where the constraint is more relaxed. 6 Conclusions In this paper, we studied the concept of safety in contextual linear bandits to address the challenges that arise in implementing such algorithms in practical situations such as personalized recommendation systems. Most of the existing linear bandit algorithms, such as LUCB [Abbasi-Yadkori et al., 2011], suffer from a large regret at their initial exploratory rounds. This unsafe behavior is not acceptable in many practical situations, where having a reasonable performance at any time is necessary for a learning algorithm to be considered reliable and to remain in production. To guarantee safe learning, we formulated a conservative linear bandit problem, where the performance of the learning algorithm (measured in terms of its cumulative rewards) at any time is constrained to be at least as good as a fraction of the performance of a baseline policy. We proposed a conservative version of LUCB algorithm, called CLUCB, to solve this constrained problem, and showed that it satisfies the safety constraint with high probability, while achieving a regret bound equivalent to that of LUCB up to an additive time-independent constant. We designed two versions of CLUCB that can be used depending on whether the reward function of the baseline policy is known or unknown, and showed that in each case, CLUCB acts conservatively (i.e., plays the action suggested by the baseline policy) only at a finite number of rounds, which depends on how suboptimal the baseline policy is. We reported simulation results that support our analysis and show the performance of the proposed CLUCB algorithm. 9 References Y. Abbasi-Yadkori, D. P?al, and C. Szepesv?ari. Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems, pages 2312?2320, 2011. P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite-time analysis of the multiarmed bandit problem. Machine Learning Journal, 47:235?256, 2002. L. Bottou, J. Peters, J. Quinonero-Candela, D. Charles, D. Chickering, E. Portugaly, D. Ray, P. Simard, and E. Snelson. Counterfactual reasoning and learning systems: The example of computational advertising. Journal of Machine Learning Research, 14:3207?3260, 2013. W. Chu, L. Li, L. Reyzin, and R. Schapire. Contextual bandits with linear payoff functions. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pages 208?214, 2011. V. Dani, T. Hayes, and S. Kakade. Stochastic linear optimization under bandit feedback. In COLT, pages 355?366, 2008. M. Petrik, M. Ghavamzadeh, and Y. Chow. Safe policy improvement by minimizing robust baseline regret. In Advances in Neural Information Processing Systems, pages 2298?2306, 2016. P. Rusmevichientong and J. Tsitsiklis. Linearly parameterized bandits. Mathematics of Operations Research, 35(2):395?411, 2010. D. Russo and B. Van Roy. Learning to optimize via posterior sampling. Mathematics of Operations Research, 39(4):1221?1243, 2014. A. Swaminathan and T. Joachims. Batch learning from logged bandit feedback through counterfactual risk minimization. Journal of Machine Learning Research, 16:1731?1755, 2015. A. Swaminathan and T. Joachims. Counterfactual risk minimization: Learning from logged bandit feedback. In Proceedings of The 32nd International Conference on Machine Learning, 2015. P. Thomas, G. Theocharous, and M. Ghavamzadeh. High confidence off-policy evaluation. 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Variational Memory Addressing in Generative Models J?rg Bornschein Andriy Mnih Daniel Zoran Danilo J. Rezende {bornschein, amnih, danielzoran, danilor}@google.com DeepMind, London, UK Abstract Aiming to augment generative models with external memory, we interpret the output of a memory module with stochastic addressing as a conditional mixture distribution, where a read operation corresponds to sampling a discrete memory address and retrieving the corresponding content from memory. This perspective allows us to apply variational inference to memory addressing, which enables effective training of the memory module by using the target information to guide memory lookups. Stochastic addressing is particularly well-suited for generative models as it naturally encourages multimodality which is a prominent aspect of most high-dimensional datasets. Treating the chosen address as a latent variable also allows us to quantify the amount of information gained with a memory lookup and measure the contribution of the memory module to the generative process. To illustrate the advantages of this approach we incorporate it into a variational autoencoder and apply the resulting model to the task of generative few-shot learning. The intuition behind this architecture is that the memory module can pick a relevant template from memory and the continuous part of the model can concentrate on modeling remaining variations. We demonstrate empirically that our model is able to identify and access the relevant memory contents even with hundreds of unseen Omniglot characters in memory. 1 Introduction Recent years have seen rapid developments in generative modelling. Much of the progress was driven by the use of powerful neural networks to parameterize conditional distributions composed to define the generative process (e.g., VAEs [1, 2], GANs [3]). In the Variational Autoencoder (VAE) framework for example, we typically define a generative model p(z), p? (x|z) and an approximate inference model q (z|x). All conditional distributions are parameterized by multilayered perceptrons (MLPs) which, in the simplest case, output the mean and the diagonal variance of a Normal distribution given the conditioning variables. We then optimize a variational lower bound to learn the generative model for x. Considering recent progress, we now have the theory and the tools to train powerful, potentially non-factorial parametric conditional distributions p(x|y) that generalize well with respect to x (normalizing flows [4], inverse autoregressive flows [5], etc.). Another line of work which has been gaining popularity recently is memory augmented neural networks [6, 7, 8]. In this family of models the network is augmented with a memory buffer which allows read and write operations and is persistent in time. Such models usually handle input and output to the memory buffer using differentiable ?soft? write/read operations to allow back-propagating gradients during training. Here we propose a memory-augmented generative model that uses a discrete latent variable a acting as an address into the memory buffer M. This stochastic perspective allows us to introduce a variational approximation over the addressing variable which takes advantage of target information 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. z m p(z) p(m) p(a) m x z p(x|z, m(z)) x a m z p(x|m, z) x p(x|ma , z) Figure 1: Left: Sketch of typical SOTA generative latent variable model with memory. Red edges indicate approximate inference distributions q(?|?). The KL(q||p) cost to identify a specific memory entry might be substantial, even though the cost of accessing a memory entry should be in the order of log |M|. Middle & Right: We combine a top-level categorical distribution p(a) and a conditional variational autoencoder with a Gaussian p(z|m). when retrieving contents from memory during training. We compute the sampling distribution over the addresses based on a learned similarity measure between the memory contents at each address and the target. The memory contents ma at the selected address a serve as a context for a continuous latent variable z, which together with ma is used to generate the target observation. We therefore interpret memory as a non-parametric conditional mixture distribution. It is non-parametric in the sense that we can change the content and the size of the memory from one evaluation of the model to another without having to relearn the model parameters. And since the retrieved content ma is dependent on the stochastic variable a, which is part of the generative model, we can directly use it downstream to generate the observation x. These two properties set our model apart from other work on VAEs with mixture priors [9, 10] aimed at unconditional density modelling. Another distinguishing feature of our approach is that we perform sampling-based variational inference on the mixing variable instead of integrating it out as is done in prior work, which is essential for scaling to a large number of memory addresses. Most existing memory-augmented generative models use soft attention with the weights dependent on the continuous latent variable to access the memory. This does not provide clean separation between inferring the address to access in memory and the latent factors of variation that account for the variability of the observation relative to the memory contents (see Figure 1). Or, alternatively, when the attention weights depend deterministically on the encoder, the retrieved memory content can not be directly used in the decoder. Our contributions in this paper are threefold: a) We interpret memory-read operations as conditional mixture distribution and use amortized variational inference for training; b) demonstrate that we can combine discrete memory addressing variables with continuous latent variables to build powerful models for generative few-shot learning that scale gracefully with the number of items in memory; and c) demonstrate that the KL divergence over the discrete variable a serves as a useful measure to monitor memory usage during inference and training. 2 Model and Training We will now describe the proposed model along with the variational inference procedure we use to train it. The generative model has the form Z X p(x|M) = p(a|M) p(z|ma ) p(x|z, ma ) dz (1) a z where x is the observation we wish to model, a is the addressing categorical latent variable, z the continuous latent vector, M the memory buffer and ma the memory contents at the ath address. The generative process proceeds by first sampling an address a from the categorical distribution p(a|M), retrieving the contents ma from the memory buffer M, and then sampling the observation x from a conditional variational auto-encoder with ma as the context conditioned on (Figure 1, B). 2 The intuition here is that if the memory buffer contains a set of templates, a trained model of this type should be able to produce observations by distorting a template retrieved from a randomly sampled memory location using the conditional variational autoencoder to account for the remaining variability. We can write the variational lower bound for the model in (1): log p(x|M) E a,z?q(?|M,x) [log p(x, z, a|M) log q(a, z|M, x)] where q(a, z|M, x) = q(a|M, x)q(z|ma , x). (2) (3) In the rest of the paper, we omit the dependence on M for brevity. We will now describe the components of the model and the variational posterior (3) in detail. The first component of the model is the memory buffer M. We here do not implement an explicit write operation but consider two possible sources for the memory content: Learned memory: In generative experiments aimed at better understanding the model?s behaviour we treat M as model parameters. That is we initialize M randomly and update its values using the gradient of the objective. Few-shot learning: In the generative few-shot learning experiments, before processing each minibatch, we sample |M| entries from the training data and store them in their raw (pixel) form in M. We ensure that the training minibatch {x1 , ..., x|B| } contains disjoint samples from the same character classes, so that the model can use M to find suitable templates for each target x. The second component is the addressing variable a 2 {1, ..., |M|} which selects a memory entry ma from the memory buffer M. The varitional posterior distribution q(a|x) is parameterized as a softmax over a similarity measure between x and each of the memory entries ma : q q (a|x) / exp Sq (ma , x), (4) where S (x, y) is a learned similarity function described in more detail below. Given a sample a from the posterior q (a|x), retreiving ma from M is a purely deterministic operation. Sampling from q(a|x) is easy as it amounts to computing its value for each slot in memory and sampling from the resulting categorical distribution. Given a, we can compute the probability of drawing that address under the prior p(a). We here use a learned prior p(a) that shares some parameters with q(a|x). Similarity functions: To obtain an efficient implementation for mini-batch training we use the same memory content M for the all training examples in a mini-batch and choose a specific form for the similarity function. We parameterize Sq (m, x) with two MLPs: h that embeds the memory content into the matching space and hq that does the same to the query x. The similarity is then computed as the inner product of the embeddings, normalized by the norm of the memory content embedding: hea , eq i ||ea ||2 where ea = h (ma ) , eq = hq (x). Sq (ma , x) = (5) (6) This form allows us to compute the similarities between the embeddings of a mini-batch of |B| observations and |M| memory entries at the computational cost of O(|M||B||e|), where |e| is the dimensionality of the embedding. We also experimented with several alternative similarity functions q such as the plain inner product (hea , eq i) and the cosine similarity (hea , e i/||ea || ? ||eq ||) and found that they did not outperform the above similarity function. For the unconditioneal prior p(a), we learn a query point ep 2 R|e| to use in similarity function (5) in place of eq . We share h between p(a) and q(a|x). Using a trainable p(a) allows the model to learn that some memory entries are more useful for generating new targets than others. Control experiments showed that there is only a very small degradation in performance when we assume a flat prior p(a) = 1/|M|. 2.1 Gradients and Training For the continuous variable z we use the methods developed in the context of variational autoencoders [1]. We use a conditional Gaussian prior p(z|ma ) and an approximate conditional posterior q(z|x, ma ). However, since we have a discrete latent variable a in the model we can not simply backpropagate gradients through it. Here we show how to use VIMCO [11] to estimate the gradients for 3 this model. With VIMCO, we essentially optimize the multi-sample variational bound [12, 13, 11]: " # K 1 X p(x, ma , z) log p(x) E log =L (7) K q(a, z|x) a(k) ?q(a|x) k=1 z(k) ?q(z|ma ,x) Multiple samples from the posterior enable VIMCO to estimate low-variance gradients for those parameters of the model which influence the non-differentiable discrete variable a. The corresponding gradient estimates are: ? ? X r? L ' ! (k) r? log p? (x, a(k) , z(k) ) r? log q? (z|a, x) (8) a(k) ,z (k) ? q(?|x) r L' X a(k) ,z (k) ! ? with ! (k) = P (k) (k) ? q(?|x) (k) r log q (a(k) |x) p(x, a(k) , z(k) ) q(a(k) , z(k) |x) X 0 1 log ! ? (k ) K 1 0 , ! ? (k) = ! ? (k) 1 X (k0 ) = log ! ? K 0 k and ! ! k ! (k) k 6=k For z-related gradients this is equivalent to IWAE [13]. Alternative gradient estimators for discrete latent variable models (e.g. NVIL [14], RWS [12] or Gumbel-max relaxation-based approaches [15, 16]) might work here too, but we have not investigated their effectiveness. Notice how the gradients r log p(x|z, a) provide updates for the memory contents ma (if necessary), while the gradients r log p(a) and r log q(a|x) provide updates for the embedding MLPs. The former update the mixture components while the latter update their relative weights. The log-likelihood bound (2) suggests that we can decompose the overall loss into three terms: the expected reconstruction error Ea,z?q [log p(x|a, z)] and the two KL terms which measure the information flow from the approximate posterior to the generative model for our latent variables: KL(q(a|x)||p(a)), and Ea?q [KL(q(z|a, x)||p(z|a))]. 3 Related work Attention and external memory are two closely related techniques that have recently become important building blocks for neural models. Attention has been widely used for supervised learning tasks such as translation, image classification and image captioning. External memory can be seen as an input or an internal state and attention mechanisms can either be used for selective reading or incremental updating. While most work involving memory and attention has been done in the context supervised learning, here we are interested in using them effectively in the generative setting. In [17] the authors use soft-attention with learned memory contents to augment models to have more parameters in the generative model. External memory as a way of implementing one-shot generalization was introduced in [18]. This was achieved by treating the exemplars conditioned on as memory entries accessed through a soft attention mechanism at each step of the incremental generative process similar to the one in DRAW [19]. Generative Matching Networks [20] are a similar architecture which uses a single-step VAE generative process instead of an iterative DRAW-like one. In both cases, soft attention is used to access the exemplar memory, with the address weights computed based on a learned similarity function between an observation at the address and a function of the latent state of the generative model. In contrast to this kind of deterministic soft addressing, we use hard attention, which stochastically picks a single memory entry and thus might be more appropriate in the few-shot setting. As the memory location is stochastic in our model, we perform variational inference over it, which has not been done for memory addressing in a generative model before. A similar approach has however been used for training stochastic attention for image captioning [21]. In the context of memory, hard attention has been used in RLNTM ? a version of the Neural Turing Machine modified to use stochastic hard addressing [22]. However, RLNTM has been trained using REINFORCE rather than variational inference. A number of architectures for VAEs augmented with mixture priors have 4 Figure 2: A: Typical learning curve when training a model to recall MNIST digits (M ? training data (each step); x ? M; |M| = 256): In the beginning the continuous latent variables model most of the variability of the data; after ? 100k update steps the stochastic memory component takes over and both the NLL bound and the KL(q(a|x)||p(a)) estimate approach log(256), the NLL of an optimal probabilistic lookup-table. B: Randomly selected samples from the MNIST model with learned memory: Samples within the same row use a common ma . been proposed, but they do not use the mixture component indicator variable to index memory and integrate out the variable instead [9, 10], which prevents them from scaling to a large number of mixing components. An alternative approach to generative few-shot learning proposed in [23] uses a hierarchical VAE to model a large number of small related datasets jointly. The statistical structure common to observations in the same dataset are modelled by a continuous latent vector shared among all such observations. Unlike our model, this model is not memory-based and does not use any form of attention. Generative models with memory have also been proposed for sequence modelling in [24], using differentiable soft addressing. Our approach to stochastic addressing is sufficiently general to be applicable in this setting as well and it would be interesting how it would perform as a plug-in replacement for soft addressing. 4 Experiments We optimize the parameters with Adam [25] and report experiments with the best results from learning rates in {1e-4, 3e-4}. We use minibatches of size 32 and K=4 samples from the approximate posterior q(?|x) to compute the gradients, the KL estimates, and the log-likelihood bounds. We keep the architectures deliberately simple and do not use autoregressive connections or IAF [5] in our models as we are primarily interested in the quantitative and qualitative behaviour of the memory component. 4.1 MNIST with fully connected MLPs We first perform a series of experiments on the binarized MNIST dataset [26]. We use 2 layered enand decoders with 256 and 128 hidden units with ReLU nonlinearities and a 32 dimensional Gaussian latent variable z. Train to recall: To investigate the model?s capability to use its memory to its full extent, we consider the case where it is trained to maximize the likelihood for random data points x which are present in M. During inference, an optimal model would pick the template ma that is equivalent to x with probability q(a|x)=1. The corresponding prior probability would be p(a) ? 1/|M|. Because there are no further variations that need to be modeled by z, its posterior q(z|x, m) can match the prior p(z|m), yielding a KL cost of zero. The model expected log likelihood would be -log |M|, equal to the log-likelihood of an optimal probabilistic lookup table. Figure 2A illustrates that our model converges to the optimal solution. We observed that the time to convergence depends on the size of the memory and with |M| > 512 the model sometimes fails to find the optimal solution. It is noteworthy that the trained model from Figure 2A can handle much larger memory sizes at test time, e.g. achieving NLL ? log(2048) given 2048 test set images in memory. This indicates that the matching MLPs for q(a|x) are sufficiently discriminative. 5 Figure 3: Approximate inference with q(a|x): Histogram and corresponding top-5 entries ma for two randomly selected targets. M contains 10 examples from 8 unseen test-set character classes. Figure 4: A: Generative one-shot sampling: Left most column is the testset example provided in M; remaining columns show randomly selected samples from p(x|M). The model was trained with 4 examples from 8 classes each per gradient step. B: Breakdown of the KL cost for different models trained with varying number of examples per class in memory. KL(q(a|x)||p(a)) increases from 2.0 to 4.5 nats as KL(q(z|ma , x)||p(z|ma )) decreases from 28.2 to 21.8 nats. As the number of examples per class increases, the model shifts the responsibility for modeling the data from the continuous variable z to the discrete a. The overall testset NLL for the different models improves from 75.1 to 69.1 nats. Learned memory: We train models with |M| 2 {64, 128, 256, 512, 1024} randomly initialized mixture components (ma 2 R256 ). After training, all models converged to an average KL(q(a|x)||p(a)) ? 2.5 ? 0.3 nats over both the training and the test set, suggesting that the model identified between e2.2 ? 9 and e2.8 ? 16 clusters in the data that are represented by a. The entropy of p(a) is significantly higher, indicating that multiple ma are used to represent the same data clusters. A manual inspection of the q(a|x) histograms confirms this interpretation. Although our model overfits slightly more to the training set, we do generally not observe a big difference between our model and the corresponding baseline VAE (a VAE with the same architecture, but without the top level mixture distribution) in terms of the final NLL. This is probably not surprising, because MNIST provides many training examples describing a relatively simple data manifold. Figure 2B shows samples from the model. 4.2 Omniglot with convolutional MLPs To apply the model to a more challenging dataset and to use it for generative few-shot learning, we train it on various versions of the Omniglot [27] dataset. For these experiments we use convolutional en- and decoders: The approximate posterior q(z|m, x) takes the concatenation of x and m as input and predicts the mean and variance for the 64 dimensional z. It consists of 6 convolutional layers with 3 ? 3 kernels and 48 or 64 feature maps each. Every second layer uses a stride of 2 to get an overall downsampling of 8 ? 8. The convolutional pyramid is followed by a fully-connected MLP with 1 hidden layer and 2|z| output units. The architecture of p(x|m, z) uses the same downscaling pyramid to map m to a |z|-dimensional vector, which is concatenated with z and upscaled with transposed convolutions to the full image size again. We use skip connections from the downscaling layers of m to the corresponding upscaling layers to preserve a high bandwidth path from m to x. To reduce overfitting, given the relatively small size of the Omniglot dataset, we tie the parameters of the convolutional downscaling layers in q(z|m) and p(x|m, z). The embedding MLPs for p(a) and q(a|x) use the same convolutional architecture and map images x and memory content ma into 6 Figure 5: Robustness to increasing memory size at test-time: A: Varying the number of confounding memory entries: At test-time we vary the number of classes in M. For an optimal model of disjoint data from C classes we expect L = average L per class + log C (dashed lines). The model was trained with 4 examples from 8 character classes in memory per gradient step. We also show our best soft-attenttion baseline model which was trained with 16 examples from two classes each gradient step. B: Memory contains examples from all 144 test-set character classes and we vary the number of examples per class. At C=0 we show the LL of our best unconditioned baseline VAE. The models were trained with 8 character classes and {1, 4, 8} examples per class in memory. a 128-dimensional matching space for the similarity calculations. We left their parameters untied because we did not observe any improvement nor degradation of performance when tying them. With learned memory: We run experiments on the 28 ? 28 pixel sized version of Omniglot which was introduced in [13]. The dataset contains 24,345 unlabeled examples in the training, and 8,070 examples in the test set from 1623 different character classes. The goal of this experiment is to show that our architecture can learn to use the top-level memory to model highly multi-modal input data. We run experiments with up to 2048 randomly initialized mixture components and observe that the model makes substantial use of them: The average KL(q(a|x)||p(a)) typically approaches log |M|, while KL(q(z|?)||p(z|?)) and the overall training-set NLL are significantly lower compared to the corresponding baseline VAE. However big models without regularization tend to overfit heavily (e.g. training-set NLL < 80 nats; testset NLL > 150 nats when using |M|=2048). By constraining the model size (|M|=256, convolutions with 32 feature maps) and adding 3e-4 L2 weight decay to all parameters with the exception of M, we obtain a model with a testset NLL of 103.6 nats (evaluated with K=5000 samples from the posterior), which is about the same as a two-layer IWAE and slightly worse than the best RBMs (103.4 and ?100 respectively, [13]). Few-shot learning: The 28 ? 28 pixel version [13] of Omniglot does not contain any alphabet or character-class labels. For few-shot learning we therefore start from the original dataset [27] and scale the 104 ? 104 pixel sized examples with 4 ? 4 max-pooling to 26 ? 26 pixels. We here use the 45/5 split introduced in [18] because we are mostly interested in the quantitative behaviour of the memory component, and not so much in finding optimal regularization hyperparameters to maximize performance on small datasets. For each gradient step, we sample 8 random character-classes from random alphabets. From each character-class we sample 4 examples and use them as targets x to form a minibatch of size 32. Depending on the experiment, we select a certain number of the remaining examples from the same character classes to populate M. We chose 8 character-classes and 4 examples per class for computational convenience (to obtain reasonable minibatch and memory sizes). In control experiments with 32 character classes per minibatch we obtain almost indistinguishable learning dynamics and results. To establish meaningful baselines, we train additional models with identical encoder and decoder architectures: 1) A simple, unconditioned VAE. 2) A memory-augmented generative model with soft-attention. Because the soft-attention weights have to depend solely on the variables in the generative model and may not take input directly from the encoder, we have to use z as the top-level latent variable: p(z), p(x|z, m(z)) and q(z|x). The overall structure of this model resembles the structure of prior work on memory-augmented generative models (see section 3 and Figure 1A), and is very similar to the one used in [20], for example. For the unconditioned baseline VAE we obtain a NLL of 90.8, while our memory augmented model reaches up to 68.8 nats. Figure 5 shows the scaling properties of our model when varying the number of conditioning examples at test-time. We observe only minimal degradation compared 7 Model Generative Matching Nets Generative Matching Nets Generative Matching Nets Variational Memory Addressing Variational Memory Addressing Variational Memory Addressing Variational Memory Addressing Ctest 1 2 4 1 2 4 16 1 83.3 86.4 88.3 86.5 87.2 87.5 89.6 2 78.9 84.9 87.3 83.0 83.3 83.3 85.1 3 75.7 82.4 86.7 79.6 80.9 81.2 81.5 4 72.9 81.0 85.4 79.0 79.3 80.7 81.9 5 70.1 78.8 84.0 76.5 79.1 79.5 81.3 10 59.9 71.4 80.2 76.2 77.0 78.6 79.8 19 45.8 61.2 73.7 73.9 75.0 76.7 77.0 Table 1: Our model compared to Generative Matching Networks [20]: GMNs have an extra stage that computes joint statistics over the memory context which gives the model a clear advantage when multiple conditiong examples per class are available. But with increasing number of classes C it quickly degrades. LL bounds were evaluated with K=1000 posterior samples. to a theoretically optimal model when we increase the number of concurrent character classes in memory up to 144, indicating that memory readout works reliably with |M| 2500 items in memory. The soft-attention baseline model reaches up to 73.4 nats when M contains 16 examples from 1 or 2 character-classes, but degrades rapidly with increasing number of confounding classes (see Figure 5A). Figure 3 shows histograms and samples from q(a|x), visually confirming that our model performs reliable approximate inference over the memory locations. We also train a model on the Omniglot dataset used in [20]. This split provides a relatively small training set. We reduce the number of feature channels and hidden layers in our MLPs and add 3e-4 L2 weight decay to all parameters to reduce overfitting. The model in [20] has a clear advantage when many examples from very few character classes are in memory because it was specifically designed to extract joint statistics from memory before applying the soft-attention readout. But like our own soft-attention baseline, it quickly degrades as the number of concurrent classes in memory is increased to 4 (table 1). Few-shot classification: Although this is not the main aim of this paper, we can use the trained model to perform discriminative few-shot classification: We can estimate p(c|x) ? P E [p(x, z, ma )/p(x)] or by using the feed forward approximation p(c|x) ? m has P a label c z?q(z|a,x) q(a|x). Without any further retraining or finetuneing we obtain classification accuracies ma has label c of 91%, 97%, 77% and 90% for 5-way 1-shot, 5-way 5-shot, 20-way 1-shot and 20-way 5-shot respectively with q(a|x). 5 Conclusions In our experiments we generally observe that the proposed model is very well behaved: we never used temperature annealing for the categorical softmax or other tricks to encourage the model to use memory. The interplay between p(a) and q(a|x) maintains exploration (high entropy) during the early phase of training and decreases naturally as the sampled ma become more informative. The KL divergences for the continuous and discrete latent variables show intuitively interpretable results for all our experiments: On the densely sampled MNIST dataset only a few distinctive mixture components are identified, while on the more disjoint and sparsely sampled Omniglot dataset the model chooses to use many more memory entries and uses the continuous latent variables less. By interpreting memory addressing as a stochastic operation, we gain the ability to apply a variational approximation which helps the model to perform precise memory lookups during inference and training. Compared to soft-attention approaches, we loose the ability to naively backprop through read-operations and we have to use approximations like VIMCO. However, our experiments strongly suggest that this can be a worthwhile trade-off. Our experiments also show that the proposed variational approximation is robust to increasing memory sizes: A model trained with 32 items in memory performed nearly optimally with more than 2500 items in memory at test-time. Beginning with M 48 our hard-attention implementation becomes noticeably faster in terms of wall-clock time per parameter update than the corresponding soft-attention baseline. Even though we use K=4 posterior samples during training and the soft-attention baseline only requires a single one. 8 Acknowledgments We thank our colleagues at DeepMind and especially Oriol Vinyals and Sergey Bartunov for insightful discussions. References [1] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013. [2] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of The 31st International Conference on Machine Learning, pages 1278?1286, 2014. [3] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672?2680, 2014. [4] Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015. [5] Diederik P Kingma, Tim Salimans, and Max Welling. Improving variational inference with inverse autoregressive flow. arXiv preprint arXiv:1606.04934, 2016. [6] Sreerupa Das, C. Lee Giles, and Guo zheng Sun. Learning context-free grammars: Capabilities and limitations of a recurrent neural network with an external stack memory. In In Proceedings of the Fourteenth Annual Conference of the Cognitive Science Society, pages 791?795. Morgan Kaufmann Publishers, 1992. [7] Sainbayar Sukhbaatar, arthur szlam, Jason Weston, and Rob Fergus. End-to-end memory networks. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems 28, pages 2440?2448. Curran Associates, Inc., 2015. [8] Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwi?nska, Sergio G?mez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626):471?476, 2016. [9] Nat Dilokthanakul, Pedro AM Mediano, Marta Garnelo, Matthew CH Lee, Hugh Salimbeni, Kai Arulkumaran, and Murray Shanahan. Deep unsupervised clustering with gaussian mixture variational autoencoders. arXiv preprint arXiv:1611.02648, 2016. [10] Eric Nalisnick, Lars Hertel, and Padhraic Smyth. Approximate inference for deep latent gaussian mixtures. In NIPS Workshop on Bayesian Deep Learning, 2016. [11] Andriy Mnih and Danilo J Rezende. Variational inference for monte carlo objectives. arXiv preprint arXiv:1602.06725, 2016. [12] J?rg Bornschein and Yoshua Bengio. Reweighted wake-sleep. arXiv preprint arXiv:1406.2751, 2014. [13] Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. arXiv preprint arXiv:1509.00519, 2015. [14] Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. arXiv preprint arXiv:1402.0030, 2014. [15] Chris J Maddison, Andriy Mnih, and Yee Whye Teh. The concrete distribution: A continuous relaxation of discrete random variables. arXiv preprint arXiv:1611.00712, 2016. [16] Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. stat, 1050:1, 2017. 9 [17] Chongxuan Li, Jun Zhu, and Bo Zhang. Learning to generate with memory. In International Conference on Machine Learning, pages 1177?1186, 2016. [18] Danilo Jimenez Rezende, Shakir Mohamed, Ivo Danihelka, Karol Gregor, and Daan Wierstra. One-shot generalization in deep generative models. arXiv preprint arXiv:1603.05106, 2016. [19] DRAW: A Recurrent Neural Network For Image Generation, 2015. [20] Sergey Bartunov and Dmitry P Vetrov. Fast adaptation in generative models with generative matching networks. arXiv preprint arXiv:1612.02192, 2016. [21] Jimmy Ba, Ruslan R Salakhutdinov, Roger B Grosse, and Brendan J Frey. Learning wake-sleep recurrent attention models. In Advances in Neural Information Processing Systems, pages 2593?2601, 2015. [22] Wojciech Zaremba and Ilya Sutskever. Reinforcement learning neural turing machines. arXiv preprint arXiv:1505.00521, 362, 2015. [23] Harrison Edwards and Amos Storkey. Towards a Neural Statistician. 2 2017. [24] Mevlana Gemici, Chia-Chun Hung, Adam Santoro, Greg Wayne, Shakir Mohamed, Danilo J Rezende, David Amos, and Timothy Lillicrap. Generative temporal models with memory. arXiv preprint arXiv:1702.04649, 2017. [25] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [26] Hugo Larochelle. Binarized mnist dataset http://www.cs.toronto.edu/~larocheh/public/ datasets/binarized_mnist/binarized_mnist_[train|valid|test].amat, 2011. [27] Brenden M Lake, Ruslan Salakhutdinov, and Joshua B Tenenbaum. Human-level concept learning through probabilistic program induction. Science, 350(6266):1332?1338, 2015. 10
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On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm Masaaki Imaizumi Institute of Statistical Mathematics RIKEN Center for Advanced Intelligence Project [email protected] Takanori Maehara RIKEN Center for Advanced Intelligence Project [email protected] Kohei Hayashi National Institute of Advanced Industrial Science and Technology RIKEN Center for Advanced Intelligence Project [email protected] Abstract Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop a randomized optimization method, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor. 1 Introduction Tensor decomposition is an essential tool for dealing with data represented as multidimensional arrays, or simply, tensors. Through tensor decomposition, we can determine latent factors of an input tensor in a low-dimensional multilinear space, which saves the storage cost and enables predicting missing elements. Note that, a different multilinear interaction among latent factors defines a different tensor decomposition model, which yields several variations of tensor decomposition. For general purposes, however, either Tucker decomposition [29] or CANDECOMP/PARAFAC (CP) decomposition [8] model is commonly used. In the past three years, an alternative tensor decomposition model, called tensor train (TT) decomposition [21] has actively been studied in machine learning communities for such as approximating the inference on a Markov random field [18], modeling supervised learning [19, 24], analyzing restricted Boltzmann machine [4], and compressing deep neural networks [17]. A key property is that, for higher-order tensors, TT decomposition provides more space-saving representation called TT format while preserving the representation power. Given an order-K tensor (i.e., a K-dimensional tensor), the space complexity of Tucker decomposition is exponential in K, whereas that of TT decomposition 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. is linear in K. Further, on TT format, several mathematical operations including the basic linear algebra operations can be performed efficiently [21]. Despite its potential importance, we face two crucial limitations when applying this decomposition to a much wider class of machine learning problems. First, its statistical performance is unknown. In Tucker decomposition and its variants, many authors addressed the generalization error and derived statistical bounds (e.g. [28, 27]). For example, Tomioka et al.[28] clarify the way in which using the convex relaxation of Tucker decomposition, the generalization error is affected by the rank (i.e., the dimensionalities of latent factors), dimension of an input, and number of observed elements. In contrast, such a relationship has not been studied for TT decomposition yet. Second, standard TT decomposition algorithms, such as alternating least squares (ALS) [6, 30] , require a huge computational cost. The main bottleneck arises from the singular value decomposition (SVD) operation to an ?unfolding? matrix, which is reshaped from the input tensor. The size of the unfolding matrix is huge and the computational cost grows exponentially in K. In this paper, we tackle the above issues and present a scalable yet statistically-guaranteed TT decomposition method. We first introduce a convex relaxation of the TT decomposition problem and its optimization algorithm via the alternating direction method of multipliers (ADMM). Based on this, a statistical error bound for tensor completion is derived, which achieves the same statistical efficiency as the convex version of Tucker decomposition does. Next, because the ADMM algorithm is not sufficiently scalable, we develop an alternative method by using a randomization technique. At the expense of losing the global convergence property, the dependency of K on the time complexity is reduced from exponential to quadratic. In addition, we show that a similar error bound is still guaranteed. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method using a real higher-order tensor. 2 2.1 Preliminaries Notation Let X ? RI1 ?????IK be the space of order-K tensors, where IkQdenotes the dimensionality of the k-th mode for k = 1, . . . , K. For brevity, we define I<k := k0 <k Ik0 ; similarly, I?k , Ik< and Ik? are defined. For a vector Y 2 Rd , [Y ]i denotes the i-th element of Y . Similarly, [X]i1 ,...,iK denotes the (i1 , . . . , iK ) elements of a tensor X 2 X . Let [X]i1 ,...,ik 1 ,:,ik+1 ,...,iK denote an Ik k dimensional vector (Xi1 ,...,ik 1 ,j,ik+1 ,...,iK )Ij=1 called the mode-k fiber. For a vector Y 2 Rd , kY k = (Y T Y )1/2 denotes the `2 -norm and kY k1 = maxi |[Y ]i | denotes the max norm. For tensors PI ...I X, X 0 2 X , an inner product is defined as hX, X 0 i := i11,...,iKK =1 X(i1 , . . . , iK )X 0 (i1 , . . . , iK ) P and kXkF = hX, Xi1/2 denotes the Frobenius norm. For a matrix Z, kZks := j j (Z) denotes the Schatten-1 norm, where j (?) is a j-th singular value of Z. 2.2 Tensor Train Decomposition Let us define a tuple of positive integers (R1 , . . . , RK 1 ) and an order-3 tensor Gk 2 RIk ?Rk 1 ?Rk for each k = 1, . . . , K. Here, we set R0 = RK = 1. Then, TT decomposition represents each element of X as follows: Xi1 ,...,iK = [G1 ]i1 ,:,: [G2 ]i2 ,:,: ? ? ? [GK ]iK ,:,: . (1) Note that [Gk ]ik ,:,: is an Rk 1 ? Rk matrix. We define G := {Gk }K k=1 as a set of the tensors, and let X(G) be a tensor whose elements are represented by G as (1). The tuple (R1 , . . . , RK 1 ) controls the complexity of TT decomposition, and it is called a Tensor Train (TT) rank. Note that TT decomposition is universal, i.e., any tensor can be represented by TT decomposition with sufficiently large TT rank [20]. When we evaluate the computational complexity, we assume the shape of G is roughly symmetric. That is, we assume there exist I, R 2 N such that Ik = O(I) for k = 1, . . . , K and Rk = O(R) for k = 1, . . . , K 1. 2 2.3 Tensor Completion Problem Suppose there exists a true tensor X ? 2 X that is unknown, and a part of the elements of X ? is ,...,IK observed with some noise. Let S ? {(j1 , j2 , . . . , jK )}Ij11,...,j be a set of indexes of the observed K =1 QK elements and n := |S| ? k=1 Ik be the number of observations. Let j(i) be an i-th element of S for i = 1, . . . , n, and yi denote i-th observation from X ? with noise. We consider the following observation model: yi = [X ? ]j(i) + ?i , (2) where ?i is i.i.d. noise with zero mean and variance 2 . For simplicity, we introduce observation vector Y := (y1 , . . . , yn ), noise vector E := (?1 , . . . , ?n ), and rearranging operator X : X ! Rn that randomly picks the elements of X. Then, the model (2) is rewritten as follows: Y = X(X ? ) + E. The goal of tensor completion is to estimate the true tensor X ? from the observation vector Y . Because the estimation problem is ill-posed, we need to restrict the degree of freedom of X ? , such as rank. Because the direct optimization of rank is difficult, its convex surrogation is alternatively used [2, 3, 11, 31, 22]. For tensor completion, the convex surrogation yields the following optimization problem [5, 14, 23, 26]: ? 1 min kY X(X)k2 + n kXks? , (3) X2? 2n where ? ? X is a convex subset of X , n 0 is a regularization coefficient, and k ? ks? is the PK 1 e e overlapped Schatten norm defined as kXks? := K k=1 kX(k) ks . Here, X(k) is the k-unfolding matrix defined by concatenating the mode-k fibers of X. The overlapped Schatten norm regularizes the rank of X in terms of Tucker decomposition [16, 28]. Although the Tucker rank of X ? is unknown in general, the convex optimization adjusts the rank depending on n . To solve the convex problem (3), the ADMM algorithm is often employed [1, 26, 28]. Since the overlapped Schatten norm is not differentiable, the ADMM algorithm avoids the differentiation of the regularization term by alternatively minimizing the augmented Lagrangian function iteratively. 3 Convex Formulation of TT Rank Minimization To adopt TT decomposition to the convex optimization problem as (3), we need the convex surrogation of TT rank. For that purpose, we introduce the Schatten TT norm [22] as follows: kXks,T := 1 K 1 K X1 k=1 kQk (X)ks := 1 K 1 K X1 X k=1 j (Qk (X)), (4) j where Qk : X ! RI?k ?Ik< is a reshaping operator that converts a tensor to a large matrix where the first k modes are combined into the rows and the rest K k modes are combined into the columns. Oseledets et al.[21] shows that the matrix rank of Qk (X) can bound the k-th TT rank of X, implying that the Schatten TT norm surrogates the sum of the TT rank. Putting the Schatten TT norm into (3), we obtain the following optimization problem: ? 1 min kY X(X)k2 + n kXks,T . (5) X2X 2n 3.1 ADMM Algorithm 1 K 1 To solve (5), we consider the augmented Lagrangian function L(x, {Zk }K k=1 , {?k }k=1 ), where Q Q I k x 2 R k is the vectorization of X, Zk is a reshaped matrices with size I?k ?Ik< , and ?k 2 R k Ik (0) (0) is a coefficient for constraints. Given initial points (x(0) , {Zk }k , {?k }k ), the `-th step of ADMM 3 is written as follows: x (`+1) (`+1) Zk (`+1) ?k = e T Y + n? ? = prox n /? 1 K 1 K X1 (`) (Vk (Zk ) (`) ?k ) k=1 ! /(1 + n?K), (`) (Vk 1 (x(`+1) + ?k )), k = 1, . . . , K, (`) = ?k + (x(`+1) (`+1) Vk (Zk )), k = 1, . . . , K. e is an n ? QIk matrix that works as the inversion mapping of X; Vk is a vectorizing Here, ? k=1 operator of an I?k ? Ik< matrix; prox(?) is the shrinkage operation of the singular values as proxb (W ) = U max{S bI, 0}V T , where U SV T is the singular value decomposition of W ; ? > 0 is a hyperparameter for a step size. We stop the iteration when the convergence criterion is satisfied (e.g. as suggested by Tomioka et al.[28]). Since the Schatten TT norm (4) is convex, the sequence of the variables of ADMM is guaranteed to converge to the optimal solution ([5, Theorem 5.1]). We refer to this algorithm as TT-ADMM. TT-ADMM requires huge resources in terms of both time and space. For the time complexity, the proximal operation of the Schatten TT norm, namely the SVD thresholding of Vk 1 , yields the dominant complexity, which is O(I 3K/2 ) time. For the space complexity, we have O(K) variables of size O(I K ), which requires O(KI K ) space. 4 Alternating Minimization with Randomization In this section, we consider reducing the space complexity for handling higher order tensors. The idea is simple: we only maintain the TT format of the input tensor rather than the input tensor itself. This leads the following optimization problem: ? 1 min kY X(X(G))k2 + n kX(G)ks,T . (6) G 2n Remember that G = {Gk }k is the set of TT components and X(G) is the tensor given by the TT format with G. Now we only need to store the TT components G, which drastically improves the space efficiency. 4.1 Randomized Schatten TT norm We approximate the optimization of the Schatten TT norm. To avoid the computation of exponentially large-scale SVDs in the Schatten TT norm, we employ a technique called the ?very sparse random projection? [12]. The main idea is that, if the size of a matrix is sufficiently larger than its rank, then its singular values (and vectors) are well preserved even after the projection by a sparse random matrix. This motivates us to use the Schatten TT norm over the random projection. Preliminary, we introduce tensors for the random projection. Let D1 , D2 2 N be the size of the matrix after projection. For each k = 1, . . . , K 1 and parameters, let ?k,1 2 RD1 ?I1 ?????Ik be a tensor whose elements are independently and identically distributed as follows: 8 p with probability 1/2s, <+ s/d1 [?k,1 ]d1 ,i1 ,...,ik = 0 p (7) with probability 1 1/s, : s/d1 with probability 1/2s, for i1 , . . . , ik and d1 = 1, . . . , D1 . Here, s > 0 is a hyperparameter controlling sparsity. Similarly, we introduce a tensor ?k,2 2 RD2 ?Ik+1 ?????IK 1 that is defined in the same way as ?k,1 . With ?k,1 and ?k,2 , let Pk : X ! RD1 ?D2 be a random projection operator whose element is defined as follows: [Pk (X)]d1 ,d2 = I1 X j1 =1 ??? IK X [?k,1 ]d1 ,j1 ,...,jk [X]j1 ,...,jK [?k,2 ]d2 ,jk+1 ,...,jK . jK =1 4 (8) Note that we can compute the above projection by using the facts that X has the TT format and the (k) projection matrices are sparse. Let ?j be a set of indexes of non-zero elements of ?k,j . Then, using the TT representation of X, (8) is rewritten as X [Pk (X(G))]d1 ,d2 = [?k,1 ]d1 ,j1 ,...,jk [G1 ]j1 ? ? ? [Gk ]jk (k) (j1 ,...,jk )2?1 X (k) (jk+1 ,...,jK )2?2 [Gk ]jk+1 ? ? ? [GK ]jK [?k,2 ]d2 ,jk+1 ,...,jK , (1) (2) If the projection matrices have only S nonzero elements (i.e., S = |?j | = |?j |), the computational cost of the above equation is O(D1 D2 SKR3 ). The next theorem guarantees that the Schatten-1 norm of Pk (X) approximates the original one. Theorem 1. Suppose X 2 X has TT rank (R1 , . . . , Rk ). Consider the reshaping operator Qk in (4), and the random operator Pk as (8) with tensors ?k,1 and ?k,2 defined as (7). If D1 , D2 max{R ) + log(1/?))/?2 }, and all the singular vectors u of Q(X)k are well-spread as k , 4(log(6Rkp P 3 j |uj | ? ?/(1.6k s), we have 1 ? kQk (X)ks ? kPk (X)ks ? (1 + ?)kQk (X)ks , Rk with probability at least 1 ?. Note that the well-spread condition can be seen as a stronger version of the incoherence assumption which will be discussed later. 4.2 Alternating Minimization Note that the new problem (6) is non-convex because X(G) does not form a convex set on X . However, if we fix G except for Gk , it becomes convex with respect to Gk . Combining with the random projection, we obtain the following minimization problem: " # K X1 1 n 2 min kY X(X(G))k + kPk0 (X(G))ks . (9) Gk 2n K 1 0 k =1 We solve this by the ADMM method for each k = 1, . . . , K. Let gk 2 RIk Rk 1 Rk be the vectorization of Gk , and Wk0 2 RD1 ?D2 be a matrix for the randomly projected matrix. The augmented 1 K 1 D1 D2 K 1 0 0 Lagrangian function is then given by Lk (gk , {Wk0 }K }k0 =1 k0 =1 , { k }k0 =1 ), where { k 2 R (0) (0) K 1 (0) K 1 are the Lagrange multipliers. Starting from initial points (gk , {Wk0 }k0 =1 , { k0 }k0 =1 ), the `-th ADMM step is written as follows: ! 1 ! K K X1 X1 1 (`+1) (`) (`) T T T T e gk = ? ?/n + ? ? Y /n + k0 k0 k0 (? Vk (Wk0 ) k0 ) , K 1 0 k0 =1 k =1 ? ? (`+1) (`+1) (`) Wk 0 = prox n /? Vek 1 ( k0 gk + k0 ) , k 0 = 1, . . . , K 1, (`+1) k0 = (`) k0 +( (`+1) k 0 gk (`+1) Vek (Wk0 )), k 0 = 1, . . . , K 1. Here, (k) 2 RD1 D2 ?Ik Rk 1 Rk is the matrix imitating the mapping Gk 7! Pk (X(Gk ; G\{Gk })), Vek is a vectorizing operator of D1 ? D2 matrix, and ? is an n ? Ik Rk 1 Rk matrix of the operator X X(?; G\{Gk }) with respect to gk . Similarly to the convex approach, we iterate the ADMM steps until convergence. We refer to this algorithm as TT-RAM, where RAM stands for randomized least square. The time complexity of TT-RAM at the `-th iteration is O((n + KD2 )KI 2 R4 ); the details are deferred to Supplementary material. The space complexity is O(n + KI 2 R4 ), where O(n) is for Y and O(KI 2 R4 ) is for the parameters. 5 5 Theoretical Analysis In this section, we investigate how the TT rank and the number of observations affect to the estimation error. Note that all the proofs of this section are deferred to Supplementary material. 5.1 Convex Solution To analyze the statistical error of the convex problem (5), we assume the incoherence of the reshaped version of X ? . Assumption 2. (Incoherence Assumption) There exists k 2 {1, . . . , K} such that a matrix Qk (X ? ) k has orthogonal singular vectors {ur 2 RI?k , vr 2 RIk< }R r=1 satisfying 1 max kPU ei k ? (?Rk /I?k ) 2 1?i?I<k 1 and max kPV ei k ? (?Rk /Ik< ) 2 1?i?I<k with some 0 ? ? < 1. Here, PU and PV are linear projections onto spaces spanned by {ur }r and {vr }r ; {ei }i is the natural basis. Intuitively, the incoherence assumption requires that the singular vectors for the matrix Qk (X ? ) are well separated. This type of assumption is commonly used in the matrix and tensor completion studies [2, 3, 31]. Under the incoherence assumption, the error rate of the solution of (5) is derived. b 2 X be Theorem 3. Let X ? 2 X be a true tensor with TT rank (R1 , . . . , RK 1 ), and let X ? the minimizer of (3). Suppose that n kX (E)k1 /n and that Assumption 2 for some k 0 2 {1, 2, . . . , K} is satisfied. If n Cm0 ?2k0 max{I?k0 , Ik0 < }Rk0 log3 max{I?k0 , Ik0 < } with a constant Cm0 , then with probability at least 1 CX , b kX X ? kF ? CX (max{I?k0 , Ik0 < }) n K K X1 p 3 and with a constant Rk . k=1 Theorem 3 states that the bound for the statistical error gets larger as the TT rank increases. In other words, completing a tensor is relatively easy as long as the tensor has small TT rank. Also, when we set n ! 0 as n increases, we can state the consistency of the minimizer. The result of Theorem 3 is similar to that obtained from the studies on matrix completion [3, 16] and tensor completion with the Tucker decomposition or SVD [28, 31]. Note that, although Theorem 3 is for tensor completion, the result can easily be generalized to other settings such as the tensor recovery or the compressed sensing problems. 5.2 TT-RAM Solution Prior to the analysis, let G ? be the true TT components such that X ? = X(G ? ). For simplification, we assume that the elements of G ? are normalized, i.e., kGk k = 1, 8k, and an Rk ? Ik 1 Ik matrix reshaped from G?k has a Rk row rank. In addition to the incoherence property (Assumption 2), we introduce an additional assumption on the initial point of the ALS iteration. K Assumption 4. (Initial Point Assumption) Let G init := {Ginit k }k=1 be the initial point of the ALS iteration procedure. Then, there exists a finite constant C that satisfies max k2{1,...,K} kGinit k G?k kF ? C . Assumption 4 requires that the initial point is sufficiently close to the true solutions G ? . Although the ALS method is not guaranteed to converge to the global optimum in general, Assumption 4 guarantees the convergence to the true solutions [25]. Now we can evaluate the error rate of the solution obtained by TT-RAM. 6 Theorem 5. Let X(G ? ) be the true tensor generated by G ? with TT rank (R1 , . . . , RK 1 ), and Gb = G t be the solution of TT-RAM at the t-th iteration. Further, suppose that Assumption 2 for some k 0 2 {1, 2, . . . , K} and Assumption 4 are satisfied, and suppose that Theorem (1) holds with ? > 0 for k = 1, . . . , K. Let Cm , CA , CB > 0 be 0 < < 1 be some constants. If Cm ?2k0 Rk0 max{I?k0 , Ik0 < } log3 max{I?k0 , Ik0 < }, P p and the number of iterations t satisfies t (log ) 1 {log(CB n K 1 (1 + ?) k Rk ) then with probability at least 1 ?(max{I?k0 , Ik0 < }) 3 and for n kX? (E)k1 /n, n b kX(G) X(G ? )kF ? CA (1 + ?) n log C }, K X1 p (10) Rk . k=1 Again, we can obtain the consistency of TT-RAM by setting n ! 0 as n increases. Since the setting of n corresponds to that of Theorem 3, the speed of convergence of TT-RAM in terms of n is equivalent to the speed of TT-ADMM. By comparing with the convex approach (Theorem 3), the error rate becomes slightly worse. Here, PK 1 p the term n k=1 Rk in (10) comes from the estimation by the alternating minimization, which linearly increases by K. This is because there are K optimization problems and their errors are accumulated to the final solution. The term (1 + ?) in (10) comes from the random projection. The size of the error ? can be arbitrary small by controlling the parameters of the random projection D1 , D2 and s. 6 Related Work To solve the tensor completion problem with TT decomposition, Wang et al.[30] and Grasedyck et al.[6] developed algorithms that iteratively solve minimization problems with respect to Gk for each k = 1, . . . , K. Unfortunately, the adaptivity of the TT rank is not well discussed. [30] assumed that the TT rank is given. Grasedyck et al.[6] proposed a grid search method. However, the TT rank is determined by a single parameter (i.e., R1 = ? ? ? = RK 1 ) and the search method lacks its generality. Furthermore, the scalability problem remains in both methods?they require more than O(I K ) space. Phien et al. [22] proposed a convex optimization method using the Schatten TT norm. However, because they employed an alternating-type optimization method, the global convergence of their method is not guaranteed. Moreover, since they maintain X directly and perform the reshape of X several times, their method requires O(I K ) time. Table 1 highlights the difference between the existing and our methods. We emphasize that our study is the first attempt to analyze the statistical performance of TT decomposition. In addition, TT-RAM is only the method that both time and space complexities do not grow exponentially in K. Method TCAM-TT[30] ADF for TT[6] SiLRTC-TT[22] TT-ADMM TT-RAM Global Convergence X Rank Adaptivity (search) X X X Time Complexity O(nIKR4 ) O(KIR3 + nKR2 ) O(I 3K/2 ) O(KI 3K/2 ) O((n + KD2 )KI 2 R4 ) Space Complexity O(I K ) O(I K ) O(KI K ) O(I K ) O(n + KI 2 R4 ) Statistical Bounds X X Table 1: Comparison of TT completion algorithms, with R is a parameter for the TT rank such that R = R1 = ? ? ? = RK 1 , I = I1 = ? ? ? = IK is dimension, K is the number of modes, n is the number of observed elements, and D is the dimension of random projection. 7 7.1 Experiments Validation of Statistical Efficiency Using synthetic data, we verify the theoretical bounds derived in Theorems 3 and 5. We first generate TT components G ? ; each component G?k is generated as G?k = G?k /kG?k kF where each 7 p b X ? kF against SRR P Rk with the order-4 Figure 1: Synthetic data: the estimation error kX k tensor (K = 4) and the order-5 tensor (K = 5). For each rank and n , we measure the error by 10 trials with different random seeds, which affect both the missing pattern and the initial points. Table 2: Electricity data: the prediction error and the runtime (in seconds). Method Tucker TCAM-TT ADF for TT SiLRTC-TT TT-ADMM TT-RAM K=5 Error Time 0.219 7.125 0.219 2.174 0.998 1.221 0.339 1.478 0.221 0.289 0.219 4.644 K Error 0.371 0.928 1.160 0.928 1.019 0.928 =7 Time 610.61 27.497 23.211 206.708 154.991 4.726 K Error N/A 0.928 1.180 N/A 1.061 0.928 =8 Time N/A 146.651 278.712 N/A 2418.00 7.654 K = 10 Error Time N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 1.173 7.968 element of G?k is sampled from the i.i.d. standard normal distribution. Then we generate Y by Q following the generative model (2) with the observation ratio n/ k Ik = 0.5 and the noise variance 0.01. We prepare two tensors of different size: an order-4 tensor of size 8 ? 8 ? 10 ? 10 and an order-5 tensor of size 5 ? 5 ? 7 ? 7 ? 7. At the order-4 tensor, the TT rank is set as (R1 , R2 , R3 ) where R1 , R2 , R3 2 {3, 5, 7}. At the order-5 tensor, the TT rank is set as (R1 , R2 , R3 , R4 ) where R1 , R2 , R3 , R4 2 {2, 4}. For estimation, we set the size of Gk and ?k as 10, which is larger than the true TT rank. The regularization coefficient n is selected from {1, 3, 5}. The parameters for random projection are set as s = 20 and D1 = D2 = 10. P p Figure 1 shows the relation between the estimation error and the sum of root rank (SRR) k Rk . The result of TT-ADMM shows that the empirical errors are linearly related to SSR which is shown by the theoretical result. The result of TT-RAM roughly replicates the theoretical relationship. 7.2 Higher-Order Markov Chain for Electricity Data We apply the proposed tensor completion methods for analyzing the electricity consumption data [13]. The dataset contains time series measurements of household electric power consumption for every minutes from December 2006 to November 2010 and it contains over 200, 000 observations. The higher-order Markov chain is a suitable method to represent long-term dependency, and it is a common tool of time-series analysis [7] and natural language processing [9]. Let {Wt }t be discrete-time random variables take values in a finite set B, and the order-K Markov chain describes the conditional distribution of Wt with given {W? }? <t as P (Wt |{W? }? <t ) = P (Wt |Wt 1 , . . . , Wt K ). As K increases, the conditional distribution of Wt can include more information from the past observations. We complete the missing values of K-th Markov transition of the electricity dataset. We discretize the value of the dataset into 10 values and set K 2 {5, 7, 8, 10}. Next, we empirically estimate the conditional distribution of size 10K using 200, 000 observations. Then, we create X by randomly selecting n = 10, 000 elements from the the conditional distribution and regarding the other elements as missing. After completion, the prediction error is measured. We select hyperparameters using a grid search with cross-validation. 8 Figure 2 compares the prediction error and the runtime by the related studies with TT decomposition. For reference, we also report those values by Tucker decomposition without TT. When K = 5, the rank adaptive methods achieve low estimation errors. As K increases, however, all the methods except for TT-RAM suffers from the scalability issue. Indeed, at K = 10, only TT-RAM works and the others does not due to exhausting memory. 8 Conclusion In this paper, we investigated TT decomposition from the statistical and computational viewpoints. To analyze its statistical performance, we formulated the convex tensor completion problem via the low-rank TT decomposition using the TT Schatten norm. In addition, because the optimization of the convex problem is infeasible, we developed an alternative algorithm called TT-RAM by combining with the ideas of random projection and alternating minimization. Based on this, we derived the error bounds of estimation for both the convex minimizer and the solution obtained by TT-RAM. The experiments supported our theoretical results and demonstrated the scalability of TT-RAM. Acknowledgement We thank Prof. Taiji Suzuki for comments that greatly improved the manuscript. M. Imaizumi is supported by Grant-in-Aid for JSPS Research Fellow (15J10206) from the JSPS. K. Hayashi is supported by ONR N62909-17-1-2138. References [1] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. 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Scalable L?evy Process Priors for Spectral Kernel Learning Phillip A. Jang Andrew E. Loeb Matthew B. Davidow Cornell University Andrew Gordon Wilson Abstract Gaussian processes are rich distributions over functions, with generalization properties determined by a kernel function. When used for long-range extrapolation, predictions are particularly sensitive to the choice of kernel parameters. It is therefore critical to account for kernel uncertainty in our predictive distributions. We propose a distribution over kernels formed by modelling a spectral mixture density with a L?evy process. The resulting distribution has support for all stationary covariances?including the popular RBF, periodic, and Mat?ern kernels? combined with inductive biases which enable automatic and data efficient learning, long-range extrapolation, and state of the art predictive performance. The proposed model also presents an approach to spectral regularization, as the L?evy process introduces a sparsity-inducing prior over mixture components, allowing automatic selection over model order and pruning of extraneous components. We exploit the algebraic structure of the proposed process for O(n) training and O(1) predictions. We perform extrapolations having reasonable uncertainty estimates on several benchmarks, show that the proposed model can recover flexible ground truth covariances and that it is robust to errors in initialization. 1 Introduction Gaussian processes (GPs) naturally give rise to a function space view of modelling, whereby we place a prior distribution over functions, and reason about the properties of likely functions under this prior (Rasmussen & Williams, 2006). Given data, we then infer a posterior distribution over functions to make predictions. The generalisation behavior of the Gaussian process is determined by its prior support (which functions are a priori possible) and its inductive biases (which functions are a priori likely), which are in turn encoded by a kernel function. However, popular kernels, and even multiple kernel learning procedures, typically cannot extract highly expressive hidden representations, as was envisaged for neural networks (MacKay, 1998; Wilson, 2014). To discover such representations, recent approaches have advocated building more expressive kernel functions. For instance, spectral mixture kernels (Wilson & Adams, 2013) were introduced for flexible kernel learning and extrapolation, by modelling a spectral density with a scale-location mixture of Gaussians, with promising results. However, Wilson & Adams (2013) specify the number of mixture components by hand, and do not characterize uncertainty over the mixture hyperparameters. As kernel functions become increasingly expressive and parametrized, it becomes natural to also adopt a function space view of kernel learning?to represent uncertainty over the values of the kernel function, and to reflect the belief that the kernel does not have a simple form. Just as we use Gaussian processes over functions to model data, we can apply the function space view a step further in a hierarchical model?with a prior distribution over kernels. In this paper, we introduce a scalable distribution over kernels by modelling a spectral density, the Fourier transform of a kernel, with a L?evy process. We consider both scale-location mixtures of Gaussians and Laplacians as basis functions for the L?evy process, to induce a prior over kernels that 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. gives rise to the sharply peaked spectral densities that often occur in practice?providing a powerful inductive bias for kernel learning. Moreover, this choice of basis functions allows our kernel function, conditioned on the L?evy process, to be expressed in closed form. This prior distribution over kernels also has support for all stationary covariances?containing, for instance, any composition of the popular RBF, Mat?ern, rational quadratic, gamma-exponential, or spectral mixture kernels. And unlike the spectral mixture representation in Wilson & Adams (2013), this proposed process prior allows for natural automatic inference over the number of mixture components in the spectral density model. Moreover, the priors implied by popular L?evy processes such as the gamma process and symmetric ?-stable process result in even stronger complexity penalties than `1 regularization, yielding sparse representations and removing mixture components which fit to noise. Conditioned on this distribution over kernels, we model data with a Gaussian process. To form a predictive distribution, we take a Bayesian model average of GP predictive distributions over a large set of possible kernel functions, represented by the support of our prior over kernels, weighted by the posterior probabilities of each of these kernels. This procedure leads to a non-Gaussian heavytailed predictive distribution for modelling data. We develop a reversible jump MCMC (RJ-MCMC) scheme (Green, 1995) to infer the posterior distribution over kernels, including inference over the number of components in the L?evy process expansion. For scalability, we pursue a structured kernel interpolation (Wilson & Nickisch, 2015) approach, in our case exploiting algebraic structure in the L?evy process expansion, for O(n) inference and O(1) predictions, compared to the standard O(n3 ) and O(n2 ) computations for inference and predictions with Gaussian processes. Flexible distributions over kernels will be especially valuable on large datasets, which often contain additional structure to learn rich statistical representations. The key contributions of this paper are summarized as follows: 1. The first fully probabilistic approach to inference with spectral mixture kernels ? to incorporate kernel uncertainty into our predictive distributions, yielding a more realistic coverage of extrapolations. This feature is demonstrated in Section 5.3. 2. Spectral regularization in spectral kernel learning. The L?evy process prior acts as a sparsityinducing prior on mixture components, automatically pruning extraneous components. This feature allows for automatic inference over model order, a key hyperparameter which must be hand tuned in the original spectral mixture kernel paper. 3. Reduced dependence on a good initialization, a key practical improvement over the original spectral mixture kernel paper. 4. A conceptually natural and interpretable function space view of kernel learning. 2 Background We provide a review of Gaussian and L?evy processes as models for prior distributions over functions. 2.1 Gaussian Processes A stochastic process f (x) is a Gaussian process (GP) if for any finite collection of inputs X = T {x1 , ? ? ? , xn } ? RD , the vector of function values [f (x1 ), ? ? ? , f (xn )] is jointly Gaussian. The distribution of a GP is completely determined by its mean function m(x), and covariance kernel k(x, x0 ). A GP used to specify a distribution over functions is denoted as f (x) ? GP(m(x), k(x, x0 )), where E[f (xi )] = m(xi ) and cov(f (x), f (x0 )) = k(x, x0 ). The generalization properties of the GP are encoded by the covariance kernel and its hyperparameters. By exploiting properties of joint Gaussian variables, we can obtain closed form expressions for conditional mean and covariance functions of unobserved function values given observed function T values. Given that f (x) is observed at n training inputs X with values f = [f (x1 ), ? ? ? , f (xn )] , the predictive distribution of the unobserved function values f? at n? testing inputs X? is given by f? |X? , X, ? ? N (? f? , cov(f? )), ? f? = mX? + KX? ,X K ?1 (f ? mX ), cov(f? ) = KX? ,X? ? X,X ?1 KX? ,X KX,X KX,X? . where KX? ,X for example denotes the n? ? n matrix of covariances evaluated at X? and X. 2 (1) (2) (3) The popular radial basis function (RBF) kernel has the following form: 2 kRBF (x, x0 ) = exp(?0.5 kx ? x0 k /`2 ). (4) GPs with RBF kernels are limited in their expressiveness and act primarily as smoothing interpolators, because the only covariance structure they can learn from data is the length scale `, which determines how quickly covariance decays with distance. Wilson & Adams (2013) introduce the more expressive spectral mixture (SM) kernel capable of extracting more complex covariance structures than the RBF kernel, formed by placing a scale-location mixture of Gaussians in the spectrum of the covariance kernel. The RBF kernel in comparison can only model a single Gaussian centered at the origin in frequency (spectral) space. 2.2 L?evy Processes A stochastic process {L(?)}??R+ is a L?evy process if it has stationary, independent increments and it is continuous in probability. In other words, L must satisfy 1. L(0) = 0, 2. L(?0 ), L(?1 ) ? L(?0 ), ? ? ? , L(?n ) ? L(?n?1 ) are independent ??0 ? ?1 ? ? ? ? ? ?n , d 3. L(?2 ) ? L(?1 ) = L(?2 ? ?1 ) ??2 ? ?1 , 4. lim P(|L(? + h) ? L(?)| ? ?) = 0 ?? > 0 ?? ? 0. h?0 15 10 f(x) By the L?evy-Khintchine representation, the distribution of a (pure jump) L?evy process is completely determined by its L?evy measure. That is, the characteristic function of L(?) is given by: log E[eiuL(?) ] = Z  ? eiu?? ? 1 ? iu ? ?1|?|?1 ?(d?). ?2 ?1 5 0 ?3 -5 Rd \{0} ?1 0 ?2 2 ?3 4 6 8 10 where the L?evy measure ?(d?) is any ?-finite x measure which satisfies the following integraFigure 1: Annotated realization of a compound bility condition Z Poisson process, a special case of a L?evy process. The ?j represent jump locations, and ?j represent (1 ? ? 2 )?(d?) < ?. Rd \{0} jump magnitudes. A L?evy process can be viewed as a combination of a Brownian motion with drift and a superposition of independent Poisson processes with differing jump sizes ?. The L?evy measure ?(d?) determines the expected number of Poisson events per unit of time for any particular jump size ?. The Brownian component of a L?evy process will not be considered for this model. For higher dimension input spaces ? ? ?, one defines the more general notion of L?evy random measure, which is also characterized by its L?evy measure ?(d?d?) (Wolpert et al., 2011) . We will show that the sample realizations of L?evy processes can be used to draw sample parameters for adaptive basis expansions. 2.3 L?evy Process Priors over Adaptive Expansions Suppose we wish to specify a o prior over the class of adaptive expansions: n PJ f : X ? R f (x) = j=1 ?j ?(x, ?j ) . Through a simple manipulation, we can rewrite f (x) into the form of a stochastic integral: Z Z J J J X X X f (x) = ?j ?(x, ?j ) = ?j ?(x, ?)??j (?)d? = ?(x, ?) ?j ??j (?)d? . j=1 j=1 ? ? j=1 | {z =dL(?) } Hence, by specifying a prior for the measure L(?), we can simultaneously specify a prior for all of the parameters {J, (?1 , ?1 ), ..., (?J , ?J )} of the expansion. L?evy random measures provide a 3 family of priors naturally suited for this purpose, as there is a one-to-one correspondence between the jump behavior of the L?evy prior and the components of the expansion. To illustrate this point, suppose the basis function parameters ?j are one-dimensional and consider the integral of dL(?) from 0 to ?. Z ?X Z ? J J X dL(?) = ?j ??j (?)d? = L(?) = ?j 1[0,?] (?j ). 0 0 j=1 j=1 PJ We see in Figure 1 that j=1 ?j 1[0,?] (?j ) resembles the sample path of a compound Poisson process, with the number of jumps J, jump sizes ?j , and jump locations ?j corresponding to the number of basis functions, basis function weights, and basis function parameters respectively. We can use a compound Poisson process to define a prior over all such piecewise constant paths. More generally, we can use a L?evy process to define a prior for L(?). Through the L?evy-Khintchine representation, the jump behavior of the prior is characterized by a L?evy measure ?(d?d?) which controls the mean number of Poisson events in every region of the parameter space, encoding the inductive biases of the model. As the number of parameters in this framework is random, we use a form of trans-dimensional reversible jump Markov chain Monte Carlo (RJ-MCMC) to sample the parameter space (Green, 2003). Popular L?evy processes such as the gamma process, symmetric gamma process, and the symmetric ?-stable process each possess desirable properties for different situations. The gamma process is able to produce strictly positive gamma distributed ?j without transforming the output space. The symmetric gamma process can produce both positive and negative ?j , and according to Wolpert et al. (2011) can achieve nearly all the commonly used isotropic geostatistical covariance functions. The symmetric ?-stable process can produce heavy-tailed distributions for ?j and is appropriate when one might expect the basis expansion to be dominated by a few heavily weighted functions. While one could dispense with L?evy processes and place Gaussian or Laplace priors on ?j to obtain `2 or `1 regularization on the expansions, respectively, a key benefit particular to these L?evy process priors are that the implied priors on the coefficients yield even stronger complexity penalties than `1 regularization. This property encourages sparsity in the expansions and permits scalability of our MCMC algorithm. Refer to the supplementary material for an illustration of the joint priors on coefficients, which exhibit concave contours in contrast to the convex elliptical and diamond contours of `2 and `1 regularization. Furthermore, in the log posterior for the L?evy process there is a log(J!) complexity penalty term which further encourages sparsity in the expansions. Refer to Clyde & Wolpert (2007) for further details. 3 L?evy Distributions over Kernels In this section, we motivate our choice of prior over kernel functions and describe how to generate samples from this prior distribution in practice. 3.1 L?evy Kernel Processes By Bochner?s Theorem (1959), a continuous stationary kernel can be represented as the Fourier dual of a spectral density: Z Z > > k(? ) = S(s)e2?is ? ds, S(s) = k(? )e?2?is ? d?. (5) RD RD Hence, the spectral density entirely characterizes a stationary kernel. Therefore, it can be desirable to model the spectrum rather than the kernel, since we can then view kernel estimation through the lens of density estimation. In order to emulate the sharp peaks that characterize frequency spectra of natural phenomena, we model the spectral density with a location-scale mixture of Laplacian components: ?j ??j |s??j | ?L (s, ?j ) = e , ?j ? (?j , ?j ) ? [0, fmax ] ? R+ . (6) 2 Then the full specification of the symmetric spectral mixture is J i X 1 h? ? ? S(s) = S(s) + S(?s) , S(s) = ?j ?L (s, ?j ). (7) 2 j=1 4 As Laplacian spikes have a closed form inverse Fourier transform, the spectral density S(s) represents the following kernel function: k(? ) = J X ?j j=1 ?2j cos(2??j ? ). ?2j + 4? 2 ? 2 (8) The parameters J, ?j , ?j , ?j can be interpreted through Eq. (8). The total number of terms to the mixture is J, while ?j is the scale of the j th frequency contribution, ?j is its central frequency, and ?j governs how rapidly the term decays (a high ? results in confident, long-term periodic extrapolation). Other basis functions can be used in place of ?L to model the spectrum as well. For example, if a Gaussian mixture is chosen, along with maximum likelihood estimation for the learning procedure, then we obtain the spectral mixture kernel (Wilson & Adams, 2013). As the spectral density S(s) takes the form of an adaptive expansion, we can define a L?evy prior over all such densities and hence all corresponding kernels of the above form. For a chosen basis function ?(s, ?) and L?evy measure ?(d?d?) we say that k(? ) is drawn from a L?evy kernel process (LKP), denoted as k(? ) ? LKP(?, ?). Wolpert et al. (2011) discuss the necessary regularity conditions for ? and ?. In summary, we propose the following hierarchical model over functions ? = x ? x0 , We now discuss how to generate samples from the L?evy kernel process in practice. In short, the kernel parameters are drawn according to {J, {(?j , ?j )}Jj=1 } ? L?evy(?(d?d?)), and then Eq. (8) is used to evaluate k ? LKP(?L , ?) at values of ? . Recall from Section 2.3 that the choice of L?evy measure ? is completely determined by the choice of the corresponding L?evy process and vice versa. Though the processes mentioned there produce sample paths with infinitely many jumps (and cannot be sampled directly), almost all jumps are infinitesimally small, and therefore these processes can be approximated in L2 by a compound Poisson process with a jump size distribution truncated by ?. Power 2 0.5 0 0 0 0.05 0.1 0.15 0.2 1 0.5 0 0 0.05 0.1 Frequency 0.15 0.2 0 Frequency 0.05 0.1 0.15 0.2 Frequency 0.2 K(?) Sampling L?evy Priors 1.5 1 4 0.1 0 -0.1 0 0.2 0.4 0.6 0.8 1 ? 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 0.5 f(X) 3.2 (9) Power Figure 2 shows three samples from the L?evy process specified through Eq. (7) and their corresponding covariance kernels. We also show one GP realization for each of the kernel functions. By placing a L?evy process prior over spectral densities, we induce a L?evy kernel process prior over stationary covariance functions. k(? ) ? LKP(?, ?). Power f (x)|k(? ) ? GP(0, k(? )), 0 -0.5 X Figure 2: Samples from a L?evy kernel mixture prior distribution. (top) Three spectra with Laplace components drawn from a L?evy process prior. (middle) The corresponding stationary covariance kernel functions and the prior mean with two standard deviations of the model, as determined by 10,000 samples. (bottom) GP samples with the respective covariance kernel functions. Once the desired L?evy process is chosen and the truncation bound is set, the basis expansion parameters are generated by drawing J ? Poisson(??+ ), and then drawing J i.i.d. samples ?1 , ? ? ? , ?J ? ?? (d?), and J i.i.d. samples ?1 , ? ? ? , ?J ? ?? (d?). Refer to the supplementary material for L2 error bounds and formulas for ??+ = ?? (R ? ?) for the gamma, symmetric gamma, and symmetric ?-stable processes. The form of ?? (?j ) also depends on the choice of L?evy process and can be found in the supplementary material, with further details in Wolpert et al. (2011). We choose to draw ? from an uninformed uniform prior over a reasonable range in the frequency domain, and ? from a gamma distribution, ? ? Gamma(a? , b? ). The choices for a? , b? , and the frequency limits are left as hyperparameters, which can have their own hyperprior distributions. After drawing the 3J values that specify 5 a L?evy process realization, the corresponding covariance function can be evaluated through the analytical expression for the inverse Fourier transform (e.g. Eq. (8) for Laplacian frequency mixture components). 4 Scalable Inference Given observed data D = {xi , yi }N i=1 , we wish to infer p(y(x? )|D, x? ) over some test set of inputs x? for interpolation and extrapolation. We model observations y(x) with a hierarchical model: y(x)|f (x) = f (x) + ?(x), f (x)|k(? ) ? GP(0, k(? )), k(? ) ? LKP(?, ?). iid ?(x) ? N (0, ? 2 ), 0 ? =x?x, (10) (11) (12) Computing the posterior distributions by marginalizing over the LKP will yield a heavy-tailed nonGaussian process for y(x? ) = y? given by an infinite Gaussian mixture model: Z H 1 X p(y? |D) = p(y? |k, D)p(k|D)dk ? p(y? |kh ), kh ? p(k|D). (13) H h=1 We compute this approximating sum using H RJ-MCMC samples (Green, 2003). Each sample draws a kernel from the posterior kh ? p(k|D) distribution. Each sample of kh enables us to draw a sample from the posterior predictive distribution p(y? |D), from which we can estimate the predictive mean and variance. Although we have chosen a Gaussian observation model in Eq. (10) (conditioned on f (x)), all of the inference procedures we have introduced here would also apply to non-Gaussian likelihoods, such as for Poisson processes with Gaussian process intensity functions, or classification. The sum in Eq. (13) requires drawing kernels from the distribution p(k|D). This is a difficult distribution to approximate, particularly because there is not a fixed number of parameters as J varies. We employ RJ-MCMC, which extends the capability of conventional MCMC to allow sequential samples of different dimensions to be drawn (Green, 2003). Thus, a posterior distribution is not limited to coefficients and other parameters of a fixed basis expansion, but can represent a changing number of basis functions, as required by the description of L?evy processes described in the previous section. Indeed, RJ-MCMC can be used to automatically learn the appropriate number of basis functions in an expansion. In the case of spectral kernel learning, inferring the number of basis functions corresponds to automatically learning the important frequency contributions to a GP kernel, which can lead to new interpretable insights into our data. 4.1 Initialization Considerations The choice of an initialization procedure is often an important practical consideration for machine learning tasks due to severe multimodality in a likelihood surface (Neal, 1996). In many cases, however, we find that spectral kernel learning with RJ-MCMC can automatically learn salient frequency contributions with a simple initialization, such as a uniform covering over a broad range of frequencies with many sharp peaks. The frequencies which are not important in describing the data are quickly attenuated or removed within RJ-MCMC learning. Typically only a few hundred RJ-MCMC iterations are needed to discover the salient frequencies in this way. Wilson (2014) proposes an alternative structured approach to initialization in previous spectral kernel modelling work. First, pass the (squared) data through a Fourier transform to obtain an empirical spectral density, which can be treated as observed. Next, fit the empirical spectral density using a standard Gaussian mixture density estimation procedure, assuming a fixed number of mixture components. Then, use the learned parameters of the Gaussian mixture as an initialization of the spectral mixture kernel hyperparameters, for Gaussian process marginal likelihood optimization. We observe successful adaptation of this procedure to our L?evy process method, replacing the approximation with Laplacian mixture terms and using the result to initialize RJ-MCMC. 4.2 Scalability As with other GP based kernel methods, the computational bottleneck lies in the evaluation of the log marginal likelihood during MCMC, which requires computing (KX,X + ? 2 I)?1 y and 6 log |KX,X + ? 2 I| for an n ? n kernel matrix KX,X evaluated at the n training points X. A direct approach through computing the Cholesky decomposition of the kernel matrix requires O(n3 ) computations and O(n2 ) storage, restricting the size of training sets to O(104 ). Furthermore, this computation must be performed at every iteration of RJ-MCMC, compounding standard computational constraints. However, this bottleneck can be readily overcome through the Structured Kernel Interpolation approach introduced in Wilson & Nickisch (2015), which approximates the kernel matrix as ? X,X 0 = MX KZ,Z M >0 for an exact kernel matrix KZ,Z evaluated on a much smaller set of K X m  n inducing points, and a sparse interpolation matrix MX which facilitates fast computations. The calculation reduces to O(n + g(m)) computations and O(n + g(m)) storage. As described in Wilson & Nickisch (2015), we can impose Toeplitz structure on KZ,Z for g(m) = m log m, allowing our RJ-MCMC procedure to train on massive datasets. 5 Experiments We conduct four experiments in total. In order to motivate our model for kernel learning in later experiments, we first demonstrate the ability of a L?evy process to recover?through direct regression?an observed noise-contaminated spectrum that is characteristic of sharply peaked naturally occurring spectra. In the second experiment we demonstrate the robustness of our RJMCMC sampler by automatically recovering the generative frequencies of a known kernel, even in presence of significant noise contamination and poor initializations. In the third experiment we demonstrate the ability of our method to infer the spectrum of airline passenger data, to perform long-range extrapolations on real data, and to demonstrate the utility of accounting for uncertainty in the kernel. In the final experiment we demonstrate the scalability of our method through training the model on a 100,000 data point sound waveform. Code is available at https: //github.com/pjang23/levy-spectral-kernel-learning. 5.1 50 Explicit Spectrum Modelling 40 30 f(x) We begin by applying a L?evy process directly for function modelling (known as LARK regression), with inference as described in Wolpert et al. (2011), and Laplacian basis functions. We choose an out of class test function proposed by Donoho & Johnstone (1993) that is standard in wavelet literature. The spatially inhomogeneous function is defined to represent spectral densities that arise in scientific and engineering applications. Gaussian i.i.d. noise is added to give a signal-to-noise ratio of 7, to be consistent with previous studies of the test function Wolpert et al. (2011). 20 10 0 0 1 2 3 4 5 6 7 8 9 10 x Figure 3: L?evy process regression on a noisy test function (black). The fit (red) captures the locations and scales of each spike while ignoring noise, but falls slightly short at its modes since the black spikes are parameterized as (1 + |x|)?4 rather than Laplacian. The noisy test function and LARK regression fit are shown in Figure 3. The synthetic spectrum is well characterized by the L?evy process, with no ?false positive? basis function terms fitting the noise owing to the strong regularization properties of the L?evy prior. By contrast, GP regression with an RBF kernel learns a length scale of 0.07 through maximum marginal likelihood training: the Gaussian process posterior can fit the sharp peaks in the test function only if it also overfits to the additive noise. The point of this experiment is to show that the L?evy process with Laplacian basis functions forms a natural prior over spectral densities. In other words, samples from this prior will typically look like the types of spectra that occur in practice. Thus, this process will have a powerful inductive bias when used for kernel learning, which we explore in the next experiments. 7 Based on these observed training data (depicted as black dots in Figure 4, right), we estimate the kernel of the Gaussian process by inferring its spectral density (Figure 4, left) using 1000 RJ-MCMC iterations. The empirical spectrum initialization described in section 4.1 results in the discovery of the two generative frequencies. Critically, we can also recover these salient frequencies even with a very poor initialization, as shown in Figure 4 (left). 400 5 300 f(X) Power 5.2 Ground Truth Recovery We next demonstrate the ability of our method to recover the generative frequencies of a known kernel and its robustness to noise and poor initializations. Data are generated from a GP with a kernel having two spectral Laplacian peaks, and partitioned into training and testing sets containing 256 points each. Moreover, the training data are contaminated with i.i.d. Gaussian noise (signal-to-noise ratio of 85%). 0 200 -5 100 -10 0 0.2 Frequency 0.4 0 10 20 30 40 50 X Figure 4: Ground truth recovery of known frequency components. (left) The spectrum of the Gaussian process that was used to generate the noisy training data is shown in black. From these noisy data and the erroneous spectral initialization shown in dashed blue, the maximum a posteriori estimate of the spectral density (over 1000 RJMCMC steps) is shown in red. A SM kernel also identifies the salient frequencies, but with broader support, shown in magenta. (right) Noisy training data are shown with a scatterplot, with withheld testing data shown in green. The learned posterior predictive distribution (mean in black, with 95% credible set in grey) captures the test data. For comparison, we also train a Gaussian SM kernel, initializing based on the empirical spectrum. The resulting kernel spectrum (Figure 4, magenta curve) does recover the salient frequencies, though with less confidence and higher overhead than even a poor initialization and spectral kernel learning with RJ-MCMC. 5.3 Spectral Kernel Learning for Long-Range Extrapolation We next demonstrate the ability of our method to perform long-range extrapolation on real data. Figure 5 shows a time series of monthly airline passenger data from 1949 to 1961 (Hyndman, 2005). The data show a long-term rising trend as well as a short term seasonal waveform, and an absence of white noise artifacts. As with Wilson & Adams (2013), the first 96 monthly data points are used to train the model and the last 48 months (4 years) are withheld as testing data, indicated in green. With an initialization from the empirical spectrum and 2500 RJ-MCMC steps, the model is able to automatically learn the necessary frequencies and the shape of the spectral density to capture both the rising trend and the seasonal waveform, allowing for accurate long-range extrapolations without pre-specifying the number of model components in advance. Figure 5: Learning of Airline passenger data. Training data is scatter plotted, with withheld testing data shown in green. The learned posterior distribution with the proposed approach (mean in black, with 95% credible set in grey) captures the periodicity and the rising trend in the test data. The analogous 95% interval using a GP with a SM kernel is illustrated in magenta. This experiment also demonstrates the impact of accounting for uncertainty in the kernel, as the withheld data often appears near or crosses the upper bound of the 95% predictive bands of the SM fit, whereas our model yields wider and more conservative predictive bands that wholly capture the test data. As the SM extrapolations are highly sensitive to the choice of parameter values, fixing the parameters of the kernel will yield overconfident predictions. The L?evy process prior allows us to account for a range of possible kernel parameters so we can achieve a more realistically broad coverage of possible extrapolations. Note that the L?evy process over spectral densities induces a prior over kernel functions. Figure 6 shows a side-by-side comparison of covariance function draws from the prior and posterior distributions over kernels. We see that sample covariance functions from the prior vary quite significantly, but are concentrated in the posterior, with movement towards the empirical covariance function. 8 Figure 6: Covariance function draws from the kernel prior (left) and posterior (right) distributions, with the empirical covariance function shown in black. After RJ-MCMC, the covariance distribution centers upon the correct frequencies and order of magnitude. We consider a 100,000 data point waveform, taken from the field of natural sound modelling (Turner, 2010). A L?evy kernel process is trained on a sound texture sample of howling wind with the middle 10% removed. Training involved initialization from the signal empirical covariance and 500 RJ-MCMC samples, and took less than one hour using an Intel i7 3.4 GHz CPU and 8 GB of memory. Four distinct mixture components in the model were automatically identified through the RJ-MCMC procedure. The learned kernel is then used for GP infilling with 900 training points, taken by down-sampling the training data, which is then applied to the original 44,100 Hz natural sound file for infilling. 0.4 0.2 f(X) 5.4 Scalability Demonstration A flexible and fully Bayesian approach to kernel learning can come with some additional computational overhead. Here we demonstrate the scalability that is achieved through the integration of SKI (Wilson & Nickisch, 2015) with our L?evy process model. 0 -0.2 -0.4 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 X (Seconds) Figure 7: Learning of a natural sound texture. A close-up of the training interval is displayed with the true waveform data scatter plotted. The learned posterior distribution (mean in black, with 95% credible set in grey) retains the periodicity of the signal within the corrupted interval. Three samples are drawn from the posterior distribution. The GP posterior distribution over the region of interest is shown in Figure 7, along with sample realizations, which appear to capture the qualitative behavior of the waveform. This experiment demonstrates the applicability of our proposed kernel learning method to large datasets, and show promise for extensions to higher dimensional data. 6 Discussion We introduced a distribution over covariance kernel functions that is well suited for modelling quasiperiodic data. We have shown how to place a L?evy process prior over the spectral density of a stationary kernel, and the resulting hierarchical model allows the incorporation of kernel uncertainty into the predictive distribution. Through the spectral regularization properties of L?evy process priors, we found that our trans-dimensional sampling procedure is suitable for automatically performing inference over model order, and is robust over initialization strategies. Finally, we incorporated structured kernel interpolation into our training and inference procedures for linear time scalability, enabling experiments on large datasets. The key advances over conventional spectral mixture kernels are in being able to interpretably and automatically discover the number of mixture components, and in representing uncertainty over the kernel. Here, we considered one dimensional inputs and stationary processes to most clearly elucidate the key properties of L?evy kernel processes. However, one could generalize this process to multidimensional non-stationary kernel learning by jointly inferring properties of transformations over inputs alongside the kernel hyperparameters. Alternatively, one could consider neural networks as basis functions in the L?evy process, inferring distributions over the parameters of the network and the numbers of basis functions as a step towards automating neural network architecture construction. 9 Acknowledgements. This work is supported in part by the Natural Sciences and Engineering Research Council of Canada (PGS-D 502888) and the National Science Foundation DGE 1144153 and IIS-1563887 awards. References Bochner, S. Lectures on Fourier Integrals.(AM-42), volume 42. Princeton University Press, 1959. Clyde, Merlise A and Wolpert, Robert L. Nonparametric function estimation using overcomplete dictionaries. Bayesian Statistics, 8:91?114, 2007. Donoho, D. and Johnstone, J.M. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3): 425?455, 1993. Green, P.J. Reversible jump monte carlo computation and bayesian model determination. Biometrika, 89(4):711?732, 1995. Green, P.J. Trans-dimensional Markov chain Monte Carlo, chapter 6. Oxford University Press, 2003. Hyndman, R.J. Time series data library. 2005. http://www-personal.buseco.monash. edu.au/?hyndman/TSDL/. MacKay, David J.C. Introduction to Gaussian processes. In Bishop, Christopher M. (ed.), Neural Networks and Machine Learning, chapter 11, pp. 133?165. Springer-Verlag, 1998. Neal, R.M. Bayesian Learning for Neural Networks. Springer Verlag, 1996. ISBN 0387947248. Rasmussen, C. E. and Williams, C. K. I. Gaussian processes for Machine Learning. The MIT Press, 2006. Turner, R. Statistical models for natural sounds. PhD thesis, University College London, 2010. Wilson, Andrew Gordon. Covariance kernels for fast automatic pattern discovery and extrapolation with Gaussian processes. PhD thesis, University of Cambridge, 2014. Wilson, Andrew Gordon and Adams, Ryan Prescott. Gaussian process kernels for pattern discovery and extrapolation. International Conference on Machine Learning (ICML), 2013. Wilson, Andrew Gordon and Nickisch, Hannes. Kernel interpolation for scalable structured Gaussian processes (KISS-GP). International Conference on Machine Learning (ICML), 2015. Wolpert, R.L., Clyde, M.A., and Tu, C. Stochastic expansions using continuous dictionaries: L?evy adaptive regression kernels. The Annals of Statistics, 39(4):1916?1962, 2011. 10
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Deep Hyperspherical Learning Weiyang Liu1 , Yan-Ming Zhang2 , Xingguo Li3,1 , Zhiding Yu4 , Bo Dai1 , Tuo Zhao1 , Le Song1 1 Georgia Institute of Technology 2 Institute of Automation, Chinese Academy of Sciences 3 University of Minnesota 4 Carnegie Mellon University {wyliu,tourzhao}@gatech.edu, [email protected], [email protected] Abstract Convolution as inner product has been the founding basis of convolutional neural networks (CNNs) and the key to end-to-end visual representation learning. Benefiting from deeper architectures, recent CNNs have demonstrated increasingly strong representation abilities. Despite such improvement, the increased depth and larger parameter space have also led to challenges in properly training a network. In light of such challenges, we propose hyperspherical convolution (SphereConv), a novel learning framework that gives angular representations on hyperspheres. We introduce SphereNet, deep hyperspherical convolution networks that are distinct from conventional inner product based convolutional networks. In particular, SphereNet adopts SphereConv as its basic convolution operator and is supervised by generalized angular softmax loss - a natural loss formulation under SphereConv. We show that SphereNet can effectively encode discriminative representation and alleviate training difficulty, leading to easier optimization, faster convergence and comparable (even better) classification accuracy over convolutional counterparts. We also provide some theoretical insights for the advantages of learning on hyperspheres. In addition, we introduce the learnable SphereConv, i.e., a natural improvement over prefixed SphereConv, and SphereNorm, i.e., hyperspherical learning as a normalization method. Experiments have verified our conclusions. 1 Introduction Recently, deep convolutional neural networks have led to significant breakthroughs on many vision problems such as image classification [9, 18, 19, 6], segmentation [3, 13, 1], object detection [3, 16], etc. While showing stronger representation power over many conventional hand-crafted features, CNNs often require a large amount of training data and face certain training difficulties such as overfitting, vanishing/exploding gradient, covariate shift, etc. The increasing depth of recently proposed CNN architectures have further aggravated the problems. To address the challenges, regularization techniques such as dropout [9] and orthogonality parameter constraints [21] have been proposed. Batch normalization [8] can also be viewed as an implicit regularization to the network, by normalizing each layer?s output distribution. Recently, deep residual learning [6] emerged as a promising way to overcome vanishing gradients in deep networks. However, [20] pointed out that residual networks (ResNets) are essentially an exponential ensembles of shallow networks where they avoid the vanishing/exploding gradient problem but do not provide direct solutions. As a result, training an ultra-deep network still remains an open problem. Besides vanishing/exploding gradient, network optimization is also very sensitive to initialization. Finding better initializations is thus widely studied [5, 14, 4]. In general, having a large parameter space is double-edged considering the benefit of representation power and the associated training difficulties. Therefore, proposing better learning frameworks to overcome such challenges remains important. In this paper, we introduce a novel convolutional learning framework that can effectively alleviate training difficulties, while giving better performance over dot product based convolution. Our idea 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. w ?(w,x) x g(?(w,x) ) SphereConv SphereConv Operator Operator ... w3 ... x x ... SphereConv Operator x ... w1 x SphereConv Operator x ... ... x w2 Cross-entropy w4 Softmax x x x Hyperspherical Convolutions Generalized Angular Softmax Loss Figure 1: Deep hyperspherical convolutional network architecture. is to project parameter learning onto unit hyperspheres, where layer activations only depend on the geodesic distance between kernels and input signals1 instead of their inner products. To this end, we propose the SphereConv operator as the basic module for our network layers. We also propose softmax losses accordingly under such representation framework. Specifically, the proposed softmax losses supervise network learning by also taking the SphereConv activations from the last layer instead of inner products. Note that the geodesic distances on a unit hypersphere is the angles between inputs and kernels. Therefore, the learning objective is essentially a function of the input angles and we call it generalized angular softmax loss in this paper. The resulting architecture is the hyperspherical convolutional network (SphereNet), which is shown in Fig. 1. Our key motivation to propose SphereNet is that angular information matters in convolutional representation learning. We argue this motivation from several aspects: training stability, training efficiency, and generalization power. SphereNet can also be viewed as an implicit regularization to the network by normalizing the activation distributions. The weight norm is no longer important since the entire network operates only on angles. And as a result, the `2 weight decay is also no longer needed in SphereNet. SphereConv to some extent also alleviates the covariate shift problem [8]. The output of SphereConv operators are bounded from ?1 to 1 (0 to 1 if considering ReLU), which makes the variance of each output also bounded. Our second intuition is that angles preserve the most abundant discriminative information in convolutional learning. We gain such intuition from 2D Fourier transform, where an image is decomposed by the combination of a set of templates with magnitude and phase information in 2D frequency domain. If one reconstructs an image with original magnitudes and random phases, the resulting images are generally not recognizable. However, if one reconstructs the image with random magnitudes and original phases. The resulting images are still recognizable. It shows that the most important structural information in an image for visual recognition is encoded by phases. This fact inspires us to project the network learning into angular space. In terms of low-level information, SphereConv is able to preserve the shape, edge, texture and relative color. SphereConv can learn to selectively drop the color depth but preserve the RGB ratio. Thus the semantic information of an image is preserved. SphereNet can also be viewed as a non-trivial generalization of [12, 11]. By proposing a loss that discriminatively supervises the network on a hypersphere, [11] achieves state-of-the-art performance on face recognition. However, the rest of the network remains a conventional convolution network. In contrast, SphereNet not only generalizes the hyperspherical constraint to every layer, but also to different nonlinearity functions of input angles. Specifically, we propose three instances of SphereConv operators: linear, cosine and sigmoid. The sigmoid SphereConv is the most flexible one with a parameter controlling the shape of the angular function. As a simple extension to the sigmoid SphereConv, we also present a learnable SphereConv operator. Moreover, the proposed generalized angular softmax (GA-Softmax) loss naturaly generalizes the angular supervision in [11] using the SphereConv operators. Additionally, the SphereConv can serve as a normalization method that is comparable to batch normalization, leading to an extension to spherical normalization (SphereNorm). SphereNet can be easily applied to other network architectures such as GoogLeNet [19], VGG [18] and ResNet [6]. One simply needs to replace the convolutional operators and the loss functions with the proposed SphereConv operators and hyperspherical loss functions. In summary, SphereConv can be viewed as an alternative to the original convolution operators, and serves as a new measure of correlation. SphereNet may open up an interesting direction to explore the neural networks. We ask the question whether inner product based convolution operator is an optimal correlation measure for all tasks? Our answer to this question is likely to be ?no?. 1 Without loss of generality, we study CNNs here, but our method is generalizable to any other neural nets. 2 2 2.1 Hyperspherical Convolutional Operator Definition The convolutional operator in CNNs is simply a linear matrix multiplication, written as F(w, x) = w> x + bF where w is a convolutional filter, x denotes a local patch from the bottom feature map and bF is the bias. The matrix multiplication here essentially computes the similarity between the local patch and the filter. Thus the standard convolution layer can be viewed as patch-wise matrix multiplication. Different from the standard convolutional operator, the hyperspherical convolutional (SphereConv) operator computes the similarity on a hypersphere and is defined as: Fs (w, x) = g(?(w,x) ) + bFs , (1) where ?(w,x) is the angle between the kernel parameter w and the local patch x. g(?(w,x) ) indicates a function of ?(w,x) (usually a monotonically decreasing function), and bFs is the bias. To simplify analysis and discussion, the bias terms are usually left out. The angle ?(w,x) can be interpreted as the geodesic distance (arc length) between w and x on a unit hypersphere. In contrast to the convolutional operator that works in the entire space, SphereConv only focuses on the angles between local patches and the filters, and therefore operates on the hypersphere space. In this paper, we present three specific instances of the SphereConv Operator. To facilitate the computation, we constrain the output of SphereConv operators to [?1, 1] (although it is not a necessary requirement). Linear SphereConv. In linear SphereConv operator, g is a linear function of ?(w,x) , with the form: g(?(w,x) ) = a?(w,x) + b, (2) where a and b are parameters for the linear SphereConv operator. In order to constrain the output range to [0, 1] while ?(w,x) ? [0, ?], we use a = ? ?2 and b = 1 (not necessarily optimal design). Cosine SphereConv. The cosine SphereConv operator is a nonlinear function of ?(w,x) , with its g being the form of 1 (3) 0.5 w x which can be reformulated as kwk . Therefore, it can be 2 kxk2 viewed as a doubly normalized convolutional operator, which bridges the SphereConv operator and convolutional operator. 0 g(?(w,x) ) = cos(?(w,x) ), T Sigmoid SphereConv. The Sigmoid SphereConv operator is derived from the Sigmoid function and its g can be written as g(?(w,x) ) = 1+ 1? ? exp(? 2k ) ? exp(? 2k ) ? 1 ? exp 1 + exp ?(w,x) k ?(w,x) k Cosine Linear Sigmoid (k=0.1) Sigmoid (k=0.3) Sigmoid (k=0.7) -0.5 -1 0 0.5 1 1.5 2 2.5 3 Figure 2: SphereConv operators. ? ? 2k ? ? 2k  , (4) where k > 0 is the parameter that controls the curvature of the function. While k is close to 0, g(?(w,x) ) will approximate the step function. While k becomes larger, g(?(w,x) ) is more like a linear function, i.e., the linear SphereConv operator. Sigmoid SphereConv is one instance of the parametric SphereConv family. With more parameters being introduced, the parametric SphereConv can have richer representation power. To increase the flexibility of the parametric SphereConv, we will discuss the case where these parameters can be jointly learned via back-prop later in the paper. 2.2 Optimization The optimization of the SphereConv operators is nearly the same as the convolutional operator and also follows the standard back-propagation. Using the chain rule, we have the gradient of the SphereConv with respect to the weights and the feature input: ?g(?(w,x) ) ?g(?(w,x) ) ??(w,x) = ? , ?w ??(w,x) ?w For different SphereConv operators, both lies in the ?g(?(w,x) ) ??(w,x) part. For ??(w,x) ?w , ?g(?(w,x) ) ?g(?(w,x) ) ??(w,x) = ? . ?x ??(w,x) ?x ??(w,x) ?w we have  wT x and ??(w,x) ?x (5) are the same, so the only difference T  w x ? arccos kwk ??(w,x) 2 kxk2 = , ?x ?x ? arccos kwk2 kxk2 ??(w,x) = , ?w ?w ?g(? (6) ) (w,x) which are straightforward to compute and therefore neglected here. Because ??(w,x) for the linear SphereConv, the cosine SphereConv and the Sigmoid SphereConv are a, ? sin(?(w,x) ) and ?2 exp(?(w,x) /k??/2k) k(1+exp(?(w,x) /k??/2k))2 respectively, all these partial gradients can be easily computed. 3 2.3 Theoretical Insights We provide a fundamental analysis for the cosine SphereConv operator in the case of linear neural network to justify that the SphereConv operator can improve the conditioning of the problem. In specific, we consider one layer of linear neural network, where the observation is F = U ? V ?> (ignore the bias), U ? ? Rn?k is the weight, and V ? ? Rm?k is the input that embeds weights from previous layers. Without loss of generality, we assume the columns satisfying kUi,: k2 = kVj,: k2 = 1 for all i = 1, . . . , n and j = 1, . . . , m, and consider min U ?Rn?k ,V ?Rm?k G(U , V ) = 21 kF ? U V > k2F . (7) This is closely related with the matrix factorization and (7) can be also viewed as the expected version for the matrix sensing problem [10]. The following lemma demonstrates a critical scaling issue of (7) for U and V that significantly deteriorate the conditioning without changing the objective of (7). Lemma 1. Consider a pair of global optimal points U , V satisfying F = U V > and Tr(V > V ? e = cU and Ve = V /c, then we have In ) ? Tr(U > U ? Im ). For any real c > 1, let U ? 2 2 2 max e , Ve )) = ?(c ?(? G(U , V ))), where ? = ?(? G(U ?min is the restricted condition number with ?max being the largest eigenvalue and ?min being the smallest nonzero eigenvalue. Lemma 1 implies that the conditioning of the problem (7) at a unbalanced global optimum scaled by a constant c is ?(c2 ) times larger than the conditioning of the problem at a balanced global optimum. Note that ?min = 0 may happen, thus we consider the restricted condition here. Similar results hold beyond global optima. This is an undesired geometric structure, which further leads to slow and unstable optimization procedures, e.g., using stochastic gradient descent (SGD). This motivates us to consider the SphereConv operator discussed above, which is equivalent to projecting data onto the hypersphere and leads to a better conditioned problem. Next, we consider our proposed cosine SphereConv operator for one-layer of the linear neural network. Based on our previous discussion on SphereConv, we consider an equivalent problem: min U ?Rn?k ,V ?Rm?k 1 1 GS (U , V ) = 21 kF ? DU U V > DV k2F ,  n?n (8) 1 1  where DU = diag kU1,: k2 , . . . , kUn,: k2 ? R and DV = diag kV1,: k2 , . . . , kVm,: k2 ? m?m R are diagonal matrices. We provide an analogous result to Lemma 1 for (8) . e , Ve )) = e = cU and Ve = V /c, then we have ?i (?2 GS (U Lemma 2. For any real c > 1, let U 2 2 e ?i (? GS (U , V )) for all i ? [(n + m)k] = {1, 2, . . . , (n + m)k} and ?(? G(U , Ve )) = ?(?2 G(U , V )), where ? is defined as in Lemma 1. We have from Lemma 2 that the issue of increasing condition caused by the scaling is eliminated by the SphereConv operator in the entire parameter space. This enhances the geometric structure over (7), which further results in improved convergence of optimization procedures. If we extend the result from one layer to multiple layers, the scaling issue propagates. Roughly speaking, when we train N layers, in the worst case, the conditioning of the problem can be cN times worse with a scaling factor c > 1. The analysis is similar to the one layer case, but the computation of the Hessian matrix and associated eigenvalues are much more complicated. Though our analysis is elementary, we provide an important insight and a straightforward illustration of the advantage for using the SphereConv operator. The extension to more general cases, e..g, using nonlinear activation function (e.g., ReLU), requires much more sophisticated analysis to bound the eigenvalues of Hessian for objectives, which is deferred to future investigation. 2.4 Discussion Comparison to convolutional operators. Convolutional operators compute the inner product between the kernels and the local patches, while the SphereConv operators compute a function of the angle between the kernels and local patches. If we normalize the convolutional operator in terms of both w and x, then the normalized convolutional operator is equivalent to the cosine SphereConv operator. Essentially, they use different metric spaces. Interestingly, SphereConv operators can also be interpreted as a function of the Geodesic distance on a unit hypersphere. Extension to fully connected layers. Because the fully connected layers can be viewed as a special convolution layer with the kernel size equal to the input feature map, the SphereConv operators could be easily generalized to the fully connected layers. It also indicates that SphereConv operators could be used not only to deep CNNs, but also to linear models like logistic regression, SVM, etc. 4 Network Regularization. Because the norm of weights is no longer crucial, we stop using the `2 weight decay to regularize the network. SphereNets are learned on hyperspheres, so we regularize the network based on angles instead of norms. To avoid redundant kernels, we want the kernels uniformly spaced around the hypersphere, but it is difficult to formulate such constraints. As a tradeoff, we encourage the orthogonality. Given a set of kernels W where the i-th column Wi is the weights of the i-th kernel, the network will also minimize kW > W ? Ik2F where I is an identity matrix. Determining the optimal SphereConv. In practice, we could treat different types of SphereConv as a hyperparameter and use the cross validation to determine which SphereConv is the most suitable one. For sigmoid SphereConv, we could also use the cross validation to determine its hyperparameter k. In general, we need to specify a SphereConv operator before using it, but prefixing a SphereConv may not be an optimal choice (even using cross validation). What if we treat the hyperparameter k in sigmoid SphereConv as a learnable parameter and use the back-prop to learn it? Following this idea, we further extend sigmoid SphereConv to a learnable SphereConv in the next subsection. SphereConv as normalization. Because SphereConv could partially address the covariate shift, it could also serve as a normalization method similar to batch normalization. Differently, SphereConv normalizes the network in terms of feature map and kernel weights, while batch normalization is for the mini-batches. Thus they do not contradict with each other and can be used simultaneously. 2.5 Extension: Learnable SphereConv and SphereNorm Learnable SphereConv. It is a natrual idea to replace the current prefixed SphereConv with a learnable one. There will be plenty of parametrization choices for the SphereConv to be learnable, and we present a very simple learnable SphereConv operator based on the sigmoid SphereConv. Because the sigmoid SphereConv has a hyperparameter k, we could treat it as a learnable parameter that can be updated by back-prop. In back-prop, k is updated using k t+1 = k t + ? ?L ?k where t denotes ?L the current iteration index and ?k can be easily computed by the chain rule. Usually, we also require k to be positive. The learning of k is in fact similar to the parameter learning in PReLU [5]. SphereNorm: hyperspherical learning as a normalization method. Similar to batch normalization (BatchNorm), we note that the hyperspherical learning can also be viewed as a way of normalization, because SphereConv constrain the output value in [?1, 1] ([0, 1] after ReLU). Different from BatchNorm, SphereNorm normalizes the network based on spatial information and the weights, so it has nothing to do with the mini-batch statistic. Because SphereNorm normalize both the input and weights, it could avoid covariate shift due to large weights and large inputs while BatchNorm could only prevent covariate shift caused by the inputs. In such sense, it will work better than BatchNorm when the batch size is small. Besides, SphereConv is more flexible in terms of design choices (e.g. linear, cosine, and sigmoid) and each may lead to different advantages. Similar to BatchNorm, we could use a rescaling strategy for the SphereNorm. Specifically, we rescale the output of SphereConv via ?Fs (w, x) + ? where ? and ? are learned by back-prop (similar to BatchNorm, the rescaling parameters can be either learned or prefixed). In fact, SphereNorm does not contradict with the BatchNorm at all and can be used simultaneously with BatchNorm. Interestingly, we find using both is empirically better than using either one alone. 3 Learning Objective on Hyperspheres For learning on hyperspheres, we can either use the conventional loss function such as softmax loss, or use some loss functions that are tailored for the SphereConv operators. We present some possible choices for these tailored loss functions. Weight-normalized Softmax Loss. The input feature and its label are denoted as xi and yi , respec P P fy tively. The original softmax loss can be written as L = N1 i Li = N1 i ? log Pe eifj where N j is the number of training samples and fj is the score of the j-th class (j ? [1, K], K is the number of classes). The class score vector f is usually the output of a fully connected layer W , so we have fj = Wj> xi + bj and fyi = Wy>i xi + byi in which xi , Wj , and Wyi are the i-th training sample, the j-th and yi -th column of W respectively. We can rewrite Li as  Li = ? log Wy> xi +byi e P j i e Wj> xi +bj   = ? log ekWyi kkxi k cos(?yi ,i )+byi P kW kkx k cos(? )+b j i j,i j je  , (9) where ?j,i (0 ? ?j,i ? ?) is the angle between vector Wj and xi . The decision boundary of the original softmax loss is determined by the vector f . Specifically in the binary-class case, the 5 decision boundary of the softmax loss is W1> x + b1 = W2> x + b2 . Considering the intuition of the SphereConv operators, we want to make the decision boundary only depend on the angles. To this end, we normalize the weights (kWj k = 1) and zero out the biases (bj = 0), following the intuition in [11] (sometimes we could keep the biases while data is imbalanced). The decision boundary becomes kxk cos(?1 ) = kxk cos(?2 ). Similar to SphereConv, we could generalize the decision boundary to kxkg(?1 ) = kxkg(?2 ), so the weight-normalized softmax (W-Softmax) loss can be written as  Li = ? log  ekxi kg(?yi ,i ) P kx kg(? ) , i j,i je (10) where g(?) can take the form of linear SphereConv, cosine SphereConv, or sigmoid SphereConv. Thus we also term these three difference weight-normalized loss functions as linear W-Softmax loss, cosine W-Softmax loss, and sigmoid W-Softmax loss, respectively. Generalized Angular Softmax Loss. Inspired by [11], we use a multiplicative parameter m to impose margins on hyperspheres. We propose a generalized angular softmax (GA-Softmax) loss which extends the W-Softmax loss to a loss function that favors large angular margin feature distribution. In general, the GA-Softmax loss is formulated as  Li = ? log  ekxi kg(m?yi ,i ) , P ekxi kg(m?yi ,i ) + j6=yi ekxi kg(?j,i ) (11) where g(?) could also have the linear, cosine and sigmoid form, similar to the W-Softmax loss. We can see A-Softmax loss [11] is exactly the cosine GA-Softmax loss and W-Softmax loss is the special case ? ], because cos(?j,i ) is only (m = 1) of GA-Sofmtax loss. Note that we usually require ?j,i ? [0, m monotonically decreasing in [0, ?]. To address this, [12, 11] construct a monotonically decreasing ? function recursively using the [0, m ] part of cos(m?j,i ). Although it indeed partially addressed the issue, it may introduce a number of saddle points (w.r.t. W ) in the loss surfaces. Originally, ?g ?? will be close to 0 only when ? is close to 0 and ?. However, in L-Softmax [12] or A-Softmax (cosine k? GA-Softmax), it is not the case. ?g ?? will be 0 when ? = m , k = 0, ? ? ? , m. It will possibly cause instability in training. The sigmoid GA-Softmax loss also has similar issues. However, if we use the linear GA-Softmax loss, this problem will be automatically solved and the training will possibly become more stable in practice. There will also be a lot of choices of g(?) to design a specific GA-Sofmtax loss, and each one has different optimization dynamics. The optimal one may depend on the task itself (e.g. cosine GA-Softmax has been shown effective in deep face recognition [11]). Discussion of Sphere-normalized Softmax Loss. We have also considered the sphere-normalized softmax loss (S-Softmax), which simultaneously normalizes the weights (Wj ) and the feature x. It seems to be a more natural choice than W-Softmax for the proposed SphereConv and makes the entire framework more unified. In fact, we have tried this and the empirical results are not that good, because the optimization seems to become very difficult. If we use the S-Softmax loss to train a network from scratch, we can not get reasonable results without using extra tricks, which is the reason we do not use it in this paper. For completeness, we give some discussions here. Normally, it is very difficult to make the S-Softmax loss value to be small enough, because we normalize the features to unit hypersphere. To make this loss work, we need to either normalize the feature to a value much larger than 1 (hypersphere with large radius) and then tune the learning rate or first train the network with the softmax loss from scratch and then use the S-Softmax loss for finetuning. 4 4.1 Experiments and Results Experimental Settings We will first perform comprehensive ablation study and exploratory experiments for the proposed SphereNets, and then evaluate the SphereNets on image classification. For the image classification task, we perform experiments on CIFAR10 (only with random left-right flipping), CIFAR10+ (with full data augmentation), CIFAR100 and large-scale Imagenet 2012 datasets [17]. General Settings. For CIFAR10, CIFAR10+ and CIFAR100, we follow the same settings from [7, 12]. For Imagenet 2012 dataset, we mostly follow the settings in [9]. We attach more details in Appendix B. For fairness, batch normalization and ReLU are used in all methods if not specified. All the comparisons are made to be fair. Compared CNNs have the same architecture with SphereNets. Training. Appendix A gives the network details. For CIFAR-10 and CIFAR-100, we use the ADAM, starting with the learning rate 0.001. The batch size is 128 if not specified. The learning rate is divided by 10 at 34K, 54K iterations and the training stops at 64K. For both A-Softmax and GA-Softmax 6 loss, we use m = 4. For Imagenet-2012, we use the SGD with momentum 0.9. The learning rate starts with 0.1, and is divided by 10 at 200K and 375K iterations. The training stops at 55K iteration. 4.2 Ablation Study and Exploratory Experiments We perform comprehensive Ablation and exploratory study on the SphereNet and evaluate every component individually in order to analyze its advantages. We use the 9-layer CNN as default (if not specified) and perform the image classification on CIFAR-10 without any data augmentation. SphereConv Operator / Loss Original Softmax Sigmoid (0.1) W-Softmax Sigmoid (0.3) W-Softmax Sigmoid (0.7) W-Softmax Linear W-Softmax Cosine W-Softmax A-Softmax (m=4) GA-Softmax (m=4) Sigmoid (0.1) Sigmoid (0.3) Sigmoid (0.7) Linear Cosine Original Conv 90.97 91.08 91.05 91.10 90.89 90.58 90.91 91.44 91.16 90.93 90.88 90.58 90.89 91.37 91.47 91.42 91.08 90.73 90.88 91.21 91.07 90.96 91.22 90.78 91.07 91.34 90.99 90.95 91.17 91.08 91.13 91.28 91.18 91.24 90.99 90.68 91.87 92.13 92.22 92.21 91.94 91.78 91.99 92.38 92.36 92.32 92.19 91.80 Table 1: Classification accuracy (%) with different loss functions. Comparison of different loss functions. We first evaluate all the SphereConv operators with different loss functions. All the compared SphereConv operators use the 9-layer CNN architecture in the experiment. From the results in Table 1, one can observe that the SphereConv operators consistently outperforms the original convolutional operator. For the compared loss functions except A-Softmax and GA-Softmax, the effect on accuracy seems to less crucial than the SphereConv operators, but sigmoid W-Softmax is more flexible and thus works slightly better than the others. The sigmoid SphereConv operators with a suitably chosen parameter also works better than the others. Note that, W-Softmax loss is in fact comparable to the original softmax loss, because our SphereNet optimizes angles and the W-Softmax is derived from the original softmax loss. Therefore, it is fair to compare the SphereNet with W-Softmax and CNN with softmax loss. From Table 1, we can see SphereConv operators are consistently better than the covolutional operators. While we use a large-margin loss function like the A-Softmax [11] and the proposed GA-Softmax, the accuracy can be further boosted. One may notice that A-Softmax is actually cosine GA-Softmax. The superior performance of A-Softmax with SphereNet shows that our architecture is more suitable for the learning of angular loss. Moreover, our proposed large-margin loss (linear GA-Softmax) performs the best among all these compared loss functions. Comparison of different network architectures. We are also interested in how our SphereConv operators work in different architectures. We evaluate all the proposed SphereConv operators with the same architecture of different layers and a totally different architecture (ResNet). Our baseline CNN architecture follows the design of VGG network [18] only with different convolutional layers. For fair comparison, we use cosine W-Softmax for all SphereConv operators and original softmax for original convolution operators. From the results in Table 2, one can see that SphereNets greatly outperforms the CNN baselines, usually with more than 1% improvement. While applied to ResNet, our SphereConv operators also work better than the baseline. Note that, we use the similar ResNet architecture from the CIFAR-10 experiment in [6]. We do not use data augmentation for CIFAR-10 in this experiment, so the ResNet accuracy is much lower than the reported one in [6]. Our results on different network architectures show consistent and significant improvement over CNNs. SphereConv Operator CNN-3 CNN-9 CNN-18 CNN-45 CNN-60 ResNet-32 SphereConv Operator Acc. (%) Sigmoid (0.1) Sigmoid (0.3) Sigmoid (0.7) Linear Cosine Original Conv 82.08 81.92 82.4 82.31 82.23 81.19 91.13 91.28 91.18 91.15 90.99 90.68 91.43 91.55 91.69 91.24 91.23 90.62 89.34 89.73 89.85 90.15 90.05 88.23 87.67 87.85 88.42 89.91 89.28 88.15 90.94 91.7 91.19 91.25 91.38 90.40 Sigmoid (0.1) Sigmoid (0.3) Sigmoid (0.7) Linear Cosine CNN w/o ReLU 86.29 85.67 85.51 85.34 85.25 80.73 Table 2: Classification accuracy (%) with different network architectures. Table 3: Acc. w/o ReLU. Comparison of different width (number of filters). We evaluate the SphereNet with different number of filters. Fig. 3(c) shows the convergence of different width of SphereNets. 16/32/48 means conv1.x, conv2.x and conv3.x have 16, 32 and 48 filters, respectively. One could observe that while the number of filters are small, SphereNet performs similarly to CNNs (slightly worse). However, while we increase the number of filters, the final accuracy will surpass the CNN baseline even faster and more stable convergence performance. With large width, we find that SphereNets perform consistently better than CNN baselines, showing that SphereNets can make better use of the width. Learning without ReLU. We notice that SphereConv operators are no longer a matrix multiplication, so it is essentially a non-linear function. Because the SphereConv operators already introduce certain 7 1 0.9 0.6 0.5 0.4 0.3 ResNet baseline on CIFAR10 ResNet baseline on CIFAR10+ SphereResNet (Sigmoid 0.3) on CIFAR10 SphereResNet (Sigmoid 0.3) on CIFAR10+ 0.2 0.1 0 0 1 2 3 4 Iteration 5 6 (a) ResNet vs. SphereResNet on CIFAR-10/10+ 7 x104 0.7 Testing Accuracy 0.7 0.6 0.5 CNN Baseline SphereNet (cosine) w/o orth. SphereNet (cosine) w/ orth. SphereNet (linear) w/ orth. SphereNet (Sigmoid 0.3) w/ orth. 0.4 0.3 0.2 0.1 0 1 2 3 4 Iteration 5 6 (b) CNN vs. SphereNet (orth.) on CIFAR-10 0.915 0.7 0.91 0.9 0.5 5.5 6 CNN 16/32/48 SphereNet 16/32/48 CNN 64/96/128 SphereNet 64/96/128 CNN 128/192/256 SphereNet 128/192/256 CNN 256/384/512 SphereNet 256/384/512 0.4 0.3 0.1 0.7 0.905 0.6 0.2 7 x10 4 69-layer CNN 69-layer SphereNet 0.8 0.8 0.8 Testing Accuracy Testing Accuracy 0.8 0.9 0.9 0 1 2 3 Iteration 4 5 6.5 4 x10 Testing Accuracy 1 0.9 0.6 0.5 0.4 0.3 0.2 0.1 6 0 x10 4 (c) Different width of SphereNet on CIFAR-10 0 0.5 1 1.5 2 Iteration 2.5 3 3.5 (d) Deep CNN vs. SphereNet on CIFAR-10 4 x10 4 Figure 3: Testing accuracy over iterations. (a) ResNet vs. SphereResNet. (b) Plain CNN vs. plain SphereNet. (c) Different width of SphereNet. (d) Ultra-deep plain CNN vs. ultra-deep plain SphereNet. non-linearity to the network, we evaluate how much gain will such non-linearity bring. Therefore, we remove the ReLU activation and compare our SphereNet with the CNNs without ReLU. The results are given in Table 3. All the compared methods use 18-layer CNNs (with BatchNorm). Although removing ReLU greatly reduces the classification accuracy, our SphereNet still outperforms the CNN without ReLU by a significant margin, showing its rich non-linearity and representation power. Convergence. One of the most significant advantages of SphereNet is its training stability and convergence speed. We evaluate the convergence with two different architectures: CNN-9 and ResNet-32. For fair comparison, we use the original softmax loss for all compared methods (including SphereNets). ADAM is used for the stochastic optimization and the learning rate is the same for all networks. From Fig. 3(a), the SphereResNet converges significantly faster than the original ResNet baseline in both CIFAR-10 and CIFAR-10+ and the final accuracy are also higher than the baselines. In Fig. 3(b), we evaluate the SphereNet with and without orthogonality constraints on kernel weights. With the same network architecture, SphereNet also converges much faster and performs better than the baselines. The orthogonality constraints also can bring performance gains in some cases. Generally from Fig. 3, one could also observe that the SphereNet converges fast and very stably in every case while the CNN baseline fluctuates in a relative wide range. Optimizing ultra-deep networks. Partially because of the alleviation of the covariate shift problem and the improvement of conditioning, our SphereNet is able to optimize ultra-deep neural networks without using residual units or any form of shortcuts. For SphereNets, we use the cosine SphereConv operator with the cosine W-Softmax loss. We directly optimize a very deep plain network with 69 stacked convolutional layers. From Fig. 3(d), one can see that the convergence of SphereNet is much easier than the CNN baseline and the SphereNet is able to achieve nearly 90% final accuracy. Frequency 4.3 Preliminary Study towards Learnable SphereConv Although the learnable SphereConv is not a main theme of this paper, we still run some preliminary evaluations on it. For the 0.3 proposed learnable sigmoid SphereConv, we learn the parameter conv1.1 conv2.1 k independently for each filter. It is also trivial to learn it in a conv3.1 0.2 layer-shared or network-shared fashsion. With the same 9-layer architecture used in Section 4.2, the learnable SphereConv (with 0.1 cosine W-Softmax loss) achieves 91.64% on CIFAR-10 (without 0 full data augmentation), while the best sigmoid SphereConv (with 0 0.2 0.4 0.6 0.8 1 The value of k cosine W-Softmax loss) achieves 91.22%. In Fig. 4, we also plot the frequency histogram of k in Conv1.1 (64 filters), Conv2.1 (96 Figure 4: Frequency histogram of k. filters) and Conv3.1 (128 filters) of the final learned SphereNet. From Fig. 4, we observe that each layer learns different distribution of k. The first convolutional layer (Conv1.1) tends to uniformly distribute k into a large range of values from 0 to 1, potentially extracting information from all levels of angular similarity. The fourth convolutional layer (Conv2.1) tends to learn more concentrated distribution of k than Conv1.1, while the seventh convolutional layer (Conv3.1) learns highly concentrated distribution of k which is centered around 0.8. Note that, we initialize all k with a constant 0.5 and learn them with the back-prop. 4.4 Evaluation of SphereNorm From Section 4.2, we could clearly see the convergence advantage of SphereNets. In general, we can view the SphereConv as a normalization method (comparable to batch normalization) that can be applied to all kinds of networks. This section evaluates the challenging scenarios where the minibatch size is small (results under 128 batch size could be found in Section 4.2) and we use the same 8 0.8 0.7 BatchNorm SphereNorm Rescaled SphereNorm SphereNorm w/ Orth. SphereNorm+BatchNorm 0.4 BatchNorm SphereNorm SphereNorm+BatchNorm 0.3 0.2 0 1 2 3 Iteration 4 5 (a) Mini-Batch Size = 4 6 x10 4 0.3 0.2 0.1 0.8 0.7 0.7 0.6 0.5 0.4 0.9 0.8 Testing Accuracy Testing Accuracy 0.6 0.5 0.1 Testing Accuracy 0.7 0.6 0.9 Testing Accuracy 0.9 0.8 0 1 2 3 Iteration 4 5 (b) Mini-Batch Size = 8 0.6 0.5 0.4 0.3 0.2 0.1 6 x10 4 0.5 BatchNorm SphereNorm Rescaled SphereNorm SphereNorm w/ Orth. SphereNorm+BatchNorm 0 1 2 3 Iteration 4 5 (c) Mini-Batch Size = 16 BatchNorm SphereNorm Rescaled SphereNorm SphereNorm w/ Orth. SphereNorm+BatchNorm 0.4 0.3 0.2 0.1 6 x10 4 0 1 2 3 Iteration 4 5 (d) Mini-Batch Size = 32 6 x10 4 Figure 5: Convergence under different mini-batch size on CIFAR-10 dataset (Same setting as Section 4.2). 9-layer CNN as in Section 4.2. To be simple, we use the cosine SphereConv as SphereNorm. The softmax loss is used in both CNNs and SphereNets. From Fig. 5, we could observe that SphereNorm achieves the final accuracy similar to BatchNorm, but SphereNorm converges faster and more stably. SphereNorm plus the orthogonal constraint helps convergence a little bit and rescaled SphereNorm does not seem to work well. While BatchNorm and SphereNorm are used together, we obtain the fastest convergence and the highest final accuracy, showing excellent compatibility of SphereNorm. 4.5 Image Classification on CIFAR-10+ and CIFAR-100 We first evaluate the SphereNet in a classic image classification task. We use the CIFAR-10+ and CIMethod CIFAR-10+ CIFAR-100 ELU [2] 94.16 72.34 FAR100 datasets and perform random flip (both horiFitResNet (LSUV) [14] 93.45 65.72 zontal and vertical) and random crop as data augmentaResNet-1001 [7] 95.38 77.29 tion (CIFAR-10 with full data augmentation is denoted Baseline ResNet-32 (softmax) 93.26 72.85 SphereResNet-32 (S-SW) 94.47 76.02 as CIFAR-10+). We use the ResNet-32 as a baseline arSphereResNet-32 (L-LW) 94.33 75.62 SphereResNet-32 (C-CW) 94.64 74.92 chitecture. For the SphereNet of the same architecture, SphereResNet-32 (S-G) 95.01 76.39 we evaluate sigmoid SphereConv operator (k = 0.3) with sigmoid W-Softmax (k = 0.3) loss (S-SW), lin- Table 4: Acc. (%) on CIFAR-10+ & CIFAR-100. ear SphereConv operator with linear W-Softmax loss (L-LW), cosine SphereConv operator with cosine W-Softmax loss (C-CW) and sigmoid SphereConv operator (k = 0.3) with GA-Softmax loss (S-G). In Table 4, we could see the SphereNet outperforms a lot of current state-of-the-art methods and is even comparable to the ResNet-1001 which is far deeper than ours. This experiment further validates our idea that learning on a hyperspheres constrains the parameter space to a more semantic and label-related one. Top5 Error Rate Top1 Error Rate 4.6 Large-scale Image Classification on Imagenet-2012 We evaluate SphereNets on large-scale Imagenet0.9 0.7 2012 dataset. We only use the minimum data ResNet-18 ResNet-18 SphereResNet-18-v1 SphereResNet-18-v1 0.8 0.6 augmentation strategy in the experiment (details SphereResNet-18-v2 SphereResNet-18-v2 0.7 0.5 are in Appendix B). For the ResNet-18 base0.6 0.4 line and SphereResNet-18, we use the same filter numbers in each layer. We develop two types of 0.5 0.3 SphereResNet-18, termed as v1 and v2 respec0.4 0.2 tively. In SphereResNet-18-v2, we do not use 0.3 0.1 0 1 2 3 4 5 0 1 2 3 4 5 SphereConv in the 1 ? 1 shortcut convolutions Iteration x10 Iteration x10 which are used to match the number of channels. Figure 6: Validation error (%) on ImageNet. In SphereResNet-18-v1, we use SphereConv in the 1 ? 1 shortcut convolutions. Fig. 6 shows the single crop validation error over iterations. One could observe that both SphereResNets converge much faster than the ResNet baseline, while SphereResNet18-v1 converges the fastest but yields a slightly worse yet comparable accuracy. SphereResNet-18-v2 not only converges faster than ResNet-18, but it also shows slightly better accuracy. 5 5 5 Limitations and Future Work Our work still has some limitations: (1) SphereNets have large performance gain while the network is wide enough. If the network is not wide enough, SphereNets still converge much faster but yield slightly worse (still comparable) recognition accuracy. (2) The computation complexity of each neuron is slightly higher than the CNNs. (3) SphereConvs are still mostly prefixed. Possible future work includes designing/learning a better SphereConv, efficiently computing the angles to reduce computation complexity, applications to the tasks that require fast convergence (e.g. reinforcement learning and recurrent neural networks), better angular regularization to replace orthogonality, etc. 9 Acknowledgements We thank Zhen Liu (Georgia Tech) for helping with the experiments and providing suggestions. This project was supported in part by NSF IIS-1218749, NIH BIGDATA 1R01GM108341, NSF CAREER IIS-1350983, NSF IIS-1639792 EAGER, NSF CNS-1704701, ONR N00014-15-1-2340, Intel ISTC, NVIDIA and Amazon AWS. Xingguo Li is supported by doctoral dissertation fellowship from University of Minnesota. Yan-Ming Zhang is supported by the National Natural Science Foundation of China under Grant 61773376. 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Learning Deep Structured Multi-Scale Features using Attention-Gated CRFs for Contour Prediction Dan Xu1 Wanli Ouyang2 Xavier Alameda-Pineda3 Elisa Ricci4 Xiaogang Wang5 Nicu Sebe1 1 The University of Trento, 2 The University of Sydney, 3 Perception Group, INRIA 4 University of Perugia, 5 The Chinese University of Hong Kong [email protected], [email protected], [email protected] [email protected], [email protected], [email protected] Abstract Recent works have shown that exploiting multi-scale representations deeply learned via convolutional neural networks (CNN) is of tremendous importance for accurate contour detection. This paper presents a novel approach for predicting contours which advances the state of the art in two fundamental aspects, i.e. multi-scale feature generation and fusion. Different from previous works directly considering multi-scale feature maps obtained from the inner layers of a primary CNN architecture, we introduce a hierarchical deep model which produces more rich and complementary representations. Furthermore, to refine and robustly fuse the representations learned at different scales, the novel Attention-Gated Conditional Random Fields (AG-CRFs) are proposed. The experiments ran on two publicly available datasets (BSDS500 and NYUDv2) demonstrate the effectiveness of the latent AG-CRF model and of the overall hierarchical framework. 1 Introduction Considered as one of the fundamental tasks in low-level vision, contour detection has been deeply studied in the past decades. While early works mostly focused on low-level cues (e.g. colors, gradients, textures) and hand-crafted features [3, 25, 22], more recent methods benefit from the representational power of deep learning models [31, 2, 38, 19, 24]. The ability to effectively exploit multi-scale feature representations is considered a crucial factor for achieving accurate predictions of contours in both traditional [29] and CNN-based [38, 19, 24] approaches. Restricting the attention on deep learning-based solutions, existing methods [38, 24] typically derive multi-scale representations by adopting standard CNN architectures and considering directly the feature maps associated to different inner layers. These maps are highly complementary: while the features from the first layers are responsible for predicting fine details, the ones from the higher layers are devoted to encode the basic structure of the objects. Traditionally, concatenation and weighted averaging are very popular strategies to combine multi-scale representations (see Fig. 1.a). While these strategies typically lead to an increased detection accuracy with respect to single-scale models, they severly simplify the complex relationship between multi-scale feature maps. The motivational cornerstone of this study is the following research question: is it worth modeling and exploiting complex relationships between multiple scales of a deep representation for contour detection? In order to provide an answer and inspired by recent works exploiting graphical models within deep learning architectures [5, 39], we introduce Attention-Gated Conditional Random Fields (AG-CRFs), which allow to learn robust feature map representations at each scale by exploiting the information available from other scales. This is achieved by incorporating an attention mechanism [27] seamlessly integrated into the multi-scale learning process under the form of gates [26]. Intuitively, 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. the attention mechanism will further enhance the quality of the learned multi-scale representation, thus improving the overall performance of the model. We integrated the proposed AG-CRFs into a two-level hierarchical CNN model, defining a novel Attention-guided Multi-scale Hierarchical deepNet (AMH-Net) for contour detection. The hierarchical network is able to learn richer multi-scale features than conventional CNNs, the representational power of which is further enhanced by the proposed AG-CRF model. We evaluate the effectiveness of the overall model on two publicly available datasets for the contour detection task, i.e. BSDS500 [1] and NYU Depth v2 [33]. The results demonstrate that our approach is able to learn rich and complementary features, thus outperforming state-of-the-art contour detection methods. Related work. In the last few years several deep learning models have been proposed for detecting contours [31, 2, 41, 38, 24, 23]. Among these, some works explicitly focused on devising multi-scale CNN models in order to boost performance. For instance, the Holistically-Nested Edge Detection method [38] employed multiple side outputs derived from the inner layers of a primary CNN and combine them for the final prediction. Liu et al. [23] introduced a framework to learn rich deep representations by concatenating features derived from all convolutional layers of VGG16. Bertasius et al. [2] considered skip-layer CNNs to jointly combine feature maps from multiple layers. Maninis et al. [24] proposed Convolutional Oriented Boundaries (COB), where features from different layers are fused to compute oriented contours and region hierarchies. However, these works combine the multi-scale representations from different layers adopting concatenation and weighted averaging schemes while not considering the dependency between the features. Furthermore, these works do not focus on generating more rich and diverse representations at each CNN layer. The combination of multi-scale representations has been also widely investigated for other pixel-level prediction tasks, such as semantic segmentation [43], visual saliency detection [21] and monocular depth estimation [39], and different deep architectures have been designed. For instance, to effectively aggregate the multi-scale information, Yu et al. [43] introduced dilated convolutions. Yang et al. [42] proposed DAG-CNNs where multi-scale feature outputs from different ReLU layers are combined through element-wise addition operator. However, none of these works incorporate an attention mechanism into a multi-scale structured feature learning framework. Attention models have been successfully exploited in deep learning for various tasks such as image classification [37], speech recognition [4] and image caption generation [40]. However, to our knowledge, this work is the first to introduce an attention model for estimating contours. Furthermore, we are not aware of previous studies integrating the attention mechanism into a probabilistic (CRF) framework to control the message passing between hidden variables. We model the attention as gates [26], which have been used in previous deep models such as restricted Boltzman machine for unsupervised feature learning [35], LSTM for sequence learning [12, 6] and CNN for image classification [44]. However, none of these works explore the possibility of jointly learning multi-scale deep representations and an attention model within a unified probabilistic graphical model. 2 2.1 Attention-Gated CRFs for Deep Structured Multi-Scale Feature Learning Problem Definition and Notation Given an input image I and a generic front-end CNN model with parameters Wc , we consider a set of S multi-scale feature maps F = {fs }Ss=1 . Being a generic framework, these feature maps can be the output of S intermediate CNN layers or of another representation, thus s is a virtual scale. The feature map at scale s, fs can be interpreted as a set of feature vectors, fs = {fsi }N i=1 , where N is the number of pixels. Opposite to previous works adopting simple concatenation or weighted averaging schemes [16, 38], we propose to combine the multi-scale feature maps by learning a set of latent feature maps hs = {his }N i=1 with a novel Attention-Gated CRF model sketched in Fig.1. Intuitively, this allows a joint refinement of the features by flowing information between different scales. Moreover, since the information from one scale may or may not be relevant for the pixels at another scale, we utilise the concept of gate, previously introduced in the literature in the case of graphical models [36], in our CRF formulation. These gates are binary random hidden variables that permit or block the flow of information between scales at every pixel. Formally, gsi e ,sr ? {0, 1} is the gate at pixel i of scale sr (receiver) from scale se (emitter), and we also write gse ,sr = {gsi e ,sr }N i=1 . i i Precisely, when gse ,sr = 1 then the hidden variable hsr is updated taking (also) into account the 2 I I fs hs 1 1 fs hs fs+1 hs+1 ??? fs ??? hs I fs 1 hs 1 (a) Multi-Scale Neural Network fs+1 ? ? ? fs hs+1 ? ? ? hs fs 1 gs gs,s+1 1,s hs 1 fs+1 ? ? ? hs+1 ? ? ? (c) Attention-Gated CRFs (b) Multi-Scale CRFs Figure 1: An illustration of different schemes for multi-scale deep feature learning and fusion. (a) the traditional approach (e.g. concatenation, weighted average), (b) CRF implementing multi-scale feature fusion (c) the proposed AG-CRF-based approach. information from the se -th layer, i.e. hse . As shown in the following, the joint inference of the hidden features and the gates leads to estimating the optimal features as well as the corresponding attention model, hence the name Attention-Gated CRFs. 2.2 Attention-Gated CRFs Given the observed multi-scale feature maps F of image I, the objective is to estimate the hidden multiscale representation H = {hs }Ss=1 and, accessorily the attention gate variables G = {gse ,sr }Sse ,sr =1 . To do that, we formalize the problem within a conditional random field framework and write the Gibbs distribution as P (H, G|I, ?) = exp (?E(H, G, I, ?)) /Z (I, ?), where ? is the set of parameters and E is the energy function. As usual, we exploit both unary and binary potentials to couple the hidden variables between them and to the observations. Importantly, the proposed binary potential is gated, and thus only active when the gate is open. More formally the general form1 of the energy function writes: XX XX E(H, G, I, ?) = ?h (his , fsi ) + gsi e ,sr ?h (hisr , hjse ) . (1) s | se ,sr i,j i {z Unary potential } | {z Gated pairwise potential } The first term of the energy function is a classical unary term that relates the hidden features to the observed multi-scale CNN representations. The second term synthesizes the theoretical contribution of the present study because it conditions the effect of the pair-wise potential ?h (hise , hjsr ) upon the gate hidden variable gsi e ,sr . Fig. 1c depicts the model formulated in Equ.(1). If we remove the attention gate variables, it becomes a general multi-scale CRFs as shown in Fig. 1b. Given that formulation, and as it is typically the case in conditional random fields, we exploit the mean-field approximation in order to derive a tractable inference procedure. Under this generic form, the mean-field inference procedure writes:   XX q(his ) ? exp ?h (his , fsi ) + Eq(gi 0 ) {gsi 0 ,s }Eq(hj 0 ) {?h (his , hjs0 )} , (2) s0 6=s  q(gsi 0 ,s ) ? exp gsi 0 ,s Eq(his ) nX j s ,s j s n o o Eq(hj 0 ) ?h (his , hjs0 ) , (3) s where Eq stands for the expectation with respect to the distribution q. Before deriving these formulae for our precise choice of potentials, we remark that, since the gate is a binary variable, the expectation of its value is the same as q(gsi 0 ,s = 1). By defining: nP n oo j i j Mis0 ,s = Eq(his ) E ? (h , h ) , the expected value of the gate writes: 0 h s s j q(h 0 ) s i ?s,s 0 = Eq(g i 0 s i ) {gs0 ,s } ,s = q(gsi 0 ,s  q(gsi 0 ,s = 1) = ? ?Mis0 ,s , = 0) + q(gsi 0 ,s = 1) (4) where ?() denotes the sigmoid function. This finding is specially relevant in the framework of CNN since many of the attention models are typically obtained after applying the sigmoid function to the 1 One could certainly include a unary potential for the gate variables as well. However this would imply that there is a way to set/learn the a priori distribution of opening/closing a gate. In practice we did not observe any notable difference between using or skipping the unary potential on g. 3 features derived from a feed-forward network. Importantly, since the quantity Mis0 ,s depends on the expected values of the hidden features his , the AG-CRF framework extends the unidirectional connection from the features to the attention model, to a bidirectional connection in which the expected value of the gate allows to refine the distribution of the hidden features as well. 2.3 AG-CRF Inference In order to construct an operative model we need to define the unary and gated potentials ?h and ?h . In our case, the unary potential corresponds to an isotropic Gaussian: ?h (his , fsi ) = ? ais i kh ? fsi k2 , 2 s (5) where ais > 0 is a weighting factor. The gated binary potential is specifically designed for a two-fold objective. On the one hand, we would like to learn and further exploit the relationships between hidden vectors at the same, as well as at different scales. On the other hand, we would like to exploit previous knowledge on attention models and include linear terms in the potential. Indeed, this would implicitly shape the gate variable to include a linear operator on the features. Therefore, we chose a bilinear potential: ? i Ki,j 0 h ? j0 , ?h (hi , hj 0 ) = h (6) s s s s,s s ? i = (hi> , 1)> and Ki,j 0 ? R(Cs +1)?(Cs0 +1) being Cs the size, i.e. the number of channels, where h s s s,s i,j i,j j,i> i,j of the representation at scale s. If we write this matrix as Ki,j s,s0 = (Ls,s0 , ls,s0 ; ls0 ,s , 1), then Ls,s0 i,j j,i exploits the relationships between hidden variables, while ls,s0 and ls0 ,s implement the classically used linear relationships of the attention models. In order words, ?h models the pair-wise relationships between features with the upper-left block of the matrix. Furthemore, ?h takes into account the linear relationships by completing the hidden vectors with the unity. In all, the energy function writes: X X ai XX s ? i Ki,j h ?j khis ? fsi k2 + gsi e ,sr h (7) E(H, G, I, ?) = ? sr sr ,se se . 2 s s ,s i,j i e r Under these potentials, we can consequently update the mean-field inference equations to:   ai X X i,j j i i i> ? 0 + li,j 0 ) , h f ) + ? h (L q(his ) ? exp ? s (khis k ? 2hi> 0 0 s s s,s s s,s s,s s 2 0 j ? j 0 is the expected a posteriori value of hj 0 . where h s s (8) s 6=s The previous expression implies that the a posteriori distribution for his is a Gaussian. The mean vector of the Gaussian and the function M write:   X X i,j j i,j  X i ? j> j,i ? i> i,j ? 0 +l 0 ) ?j ? i Li,j 0 h ? i = 1 ai f i + ?s,s (Ls,s0 h Mis0 ,s = h h 0 s s s s s,s s0 + hs ls,s0 + hs0 ls0 ,s s s,s i as 0 j j s 6=s which concludes the inference procedure. Furthermore, the proposed framework can be simplified to obtain the traditional attention models. In most of the previous studies, the attention variables are computed directly from the multi-scale features instead of computing them from the hidden variables. Indeed, since many of these studies do not propose a probabilistic formulation, there are no hidden variables and the attention is computed sequentially through the scales. We can emulate the same behavior within the AG-CRF framework by modifying the gated potential as follows: j,i ??h (hi , hj 0 , f i , f j0 ) = hi Li,j 0 hj 0 + f i> li,j 0 + f j> (9) 0 l 0 . s s s s s s,s s s s,s s s ,s This means that we keep the pair-wise relationships between hidden variables (as in any CRF) and let the attention model be generated by a linear combination of the observed features from the CNN, as it is traditionally done. The changes in the inference procedure are straightforward and reported in the supplementary material due to space constraints. We refer to this model as partially-latent AG-CRFs (PLAG-CRFs), whereas the more general one is denoted as fully-latent AG-CRFs (FLAG-CRFs). 2.4 Implementation with neural network for joint learning In order to infer the hidden variables and learn the parameters of the AG-CRFs together with those of the front-end CNN, we implement the AG-CRFs updates in neural network with several steps: 4 C ... C ... Front-End CNN C C C M D D D ... flD HIERARCHY 1 AG-CRF ... C M flC flM D D flC ... C C M D D D flM ... 0 AG-CRF D L D C D 0 AG-CRF D ... C fl D L D ... ... L D ... HIERARCHY 2 C Convolution D Deconvolution M Max-pooling L Loss AG-CRF C D L Figure 2: An overview of the proposed AMH-Net for contour detection. (i) message passing from the se -th scale to the current sr -th scale is performed with hse ?sr ? Lse ?sr ? hse , where ? denotes the convolutional operation and Lse ?sr denotes the corresponding convolution kernel, (ii) attention map estimation q(gse ,sr = 1) ? ?(hsr (Lse ?sr ? hse ) + lse ?sr ? hse + lsr ?se ? hsr ), where Lse ?sr , lse ?sr and lsr ?se are convolution kernels and represents element-wise product operation,Pand (iii) attention-gated message passing from other scales ? s = fs ? as and adding unary term: h r r r se 6=sr (q(gse ,sr = 1) hse ?sr ), where asr encodes the i effect of the asr for weighting the message and can be implemented as a 1 ? 1 convolution. The symbol ? denotes element-wise addition. In order to simplify the overall inference procedure, and because the magnitude of the linear term of ?h is in practice negligible compared to the quadratic term, we discard the message associated to the linear term. When the inference is complete, the final estimate is obtained by convolving all the scales. 3 Exploiting AG-CRFs with a Multi-scale Hierarchical Network AMH-Net Architecture. The proposed Attention-guided Multi-scale Hierarchical Network (AMHNet), as sketched in Figure 2, consists of a multi-scale hierarchical network (MH-Net) together with the AG-CRF model described above. The MH-Net is constructed from a front-end CNN architecture such as the widely used AlexNet [20], VGG [34] and ResNet [17]. One prominent feature of MH-Net is its ability to generate richer multi-scale representations. In order to do that, we perform distinct non-linear mappings (deconvolution D, convolution C and max-pooling M) upon fl , the CNN feature representation from an intermediate layer l of the front-end CNN. This leads to a three-way representation: flD , flC and flM . Remarkably, while D upsamples the feature map, C maintains its original size and M reduces it, and different kernel size is utilized for them to have different receptive fields, then naturally obtaining complementary inter- and multi-scale representations. The flC and flM are further aligned to the dimensions of the feature map flD by the deconvolutional operation. The hierarchy is implemented in two levels. The first level uses an AG-CRF model to fuse the three representations of each layer l, thus refining the CNN features within the same scale. The second level of the hierarchy uses an AG-CRF model to fuse the information coming from multiple CNN layers. The proposed hierarchical multi-scale structure is general purpose and able to involve an arbitrary number of layers and of diverse intra-layer representations. End-to-End Network Optimization. The parameters of the model consist of the front-end CNN parameters, Wc , the parameters to produce the richer decomposition from each layer l, Wl , the parameters of the AG-CRFs of the first level of the hierarchy, {WlI }L l=1 , and the parameters of the AG-CRFs of the second level of the hierarchy, WII . L is the number of intermediate layers used from the front-end CNN. In order to jointly optimize all these parameters we adopt deep supervision [38] and we add an optimization loss associated to each AG-CRF module. In addition, since the contour detection problem is highly unbalanced, i.e. contour pixels are significantly less than non-contour pixels, we employ the modified cross-entropy loss function of [38]. Given a training data 5 set D = {(Ip , Ep )}P p=1 consisting of P RGB-contour groundtruth pairs, the loss function ` writes:  X X   X  ` W = ? log P ekp = 1|Ip ; W + 1 ? ? log P ekp = 0|Ip ; W , (10) p ? ek p ?Ep + ek p ?Ep + ? + where ? = |E+ p |/(|Ep | + |Ep |), Ep is the set of contour pixels of image p and W is the set of all parameters. The optimization is performed via the back-propagation algorithm with standard stochastic gradient descent. AMH-Net for contour detection. After training of the whole AMH-Net, the optimized network parameters W are used for the contour detection task. Given a new test image I, the L + 1 classifiers ? l }L+1 = AMH-Net(I; W). The E ? l are obtained produce a set of contour prediction maps {E l=1 from the AG-CRFs with elementary operations as detailed in the supplementary material. We ? = inspire from [38] to fuse the multiple scale predictions thus obtaining an average prediction E P ? l El /(L + 1). 4 Experiments 4.1 Experimental Setup Datasets. To evaluate the proposed approach we employ two different benchmarks: the BSDS500 and the NYUDv2 datasets. The BSDS500 dataset is an extended dataset based on BSDS300 [1]. It consists of 200 training, 100 validation and 200 testing images. The groundtruth pixel-level labels for each sample are derived considering multiple annotators. Following [38, 41], we use all the training and validation images for learning the proposed model and perform data augmentation as described in [38]. The NYUDv2 [33] contains 1449 RGB-D images and it is split into three subsets, comprising 381 training, 414 validation and 654 testing images. Following [38] in our experiments we employ images at full resolution (i.e. 560 ? 425 pixels) both in the training and in the testing phases. Evaluation Metrics. During the test phase standard non-maximum suppression (NMS) [9] is first applied to produce thinned contour maps. We then evaluate the detection performance of our approach according to different metrics, including the F-measure at Optimal Dataset Scale (ODS) and Optimal Image Scale (OIS) and the Average Precision (AP). The maximum tolerance allowed for correct matches of edge predictions to the ground truth is set to 0.0075 for the BSDS500 dataset, and to .011 for the NYUDv2 dataset as in previous works [9, 14, 38]. Implementation Details. The proposed AMH-Net is implemented under the deep learning framework Caffe [18]. The implementation code is available on Github2 . The training and testing phase are carried out on an Nvidia Titan X GPU with 12GB memory. The ResNet50 network pretrained on ImageNet [8] is used to initialize the front-end CNN of AMH-Net. Due to memory constraints, our implementation only considers three scales, i.e. we generate multi-scale features from three different layers of the front-end CNN (i.e. res3d, res4f, res5c). In our CRF model we consider dependencies between all scales. Within the AG-CRFs, the kernel size for all convolutional operations is set to 3 ? 3 with stride 1 and padding 1. To simplify the model optimization, the parameters aisr are set as 0.1 for all scales during training. We choose this value as it corresponds to the best performance after cross-validation in the range [0, 1]. The initial learning rate is set to 1e-7 in all our experiments, and decreases 10 times after every 10k iterations. The total number of iterations for BSDS500 and NYUD v2 is 40k and 30k, respectively. The momentum and weight decay parameters are set to 0.9 and 0.0002, as in [38]. As the training images have different resolution, we need to set the batch size to 1, and for the sake of smooth convergence we updated the parameters only every 10 iterations. 4.2 Experimental Results In this section, we present the results of our evaluation, comparing our approach with several state of the art methods. We further conduct an in-depth analysis of our method, to show the impact of different components on the detection performance. Comparison with state of the art methods. We first consider the BSDS500 dataset and compare the performance of our approach with several traditional contour detection methods, including Felz-Hut [11], MeanShift [7], Normalized Cuts [32], ISCRA [30], gPb-ucm [1], SketchTokens [22], 2 https://github.com/danxuhk/AttentionGatedMulti-ScaleFeatureLearning 6 Figure 3: Qualitative results on the BSDS500 (left) and the NYUDv2 (right) test samples. The 2nd (4th) and 3rd (6th) columns are the ground-truth and estimated contour maps respectively. Table 1: BSDS500 dataset: quantitative results. Table 2: NYUDv2 dataset: quantitative results. Method ODS OIS AP Human .800 .800 - Felz-Hutt[11] Mean Shift[7] Normalized Cuts[32] ISCRA[30] gPb-ucm[1] Sketch Tokens[22] MCG[28] .610 .640 .641 .724 .726 .727 .747 .640 .680 .674 .752 .760 .746 .779 .560 .560 .447 .783 .727 .780 .759 DeepEdge[2] DeepContour[31] LEP[46] HED[38] CEDN[41] COB [24] RCF [23] (not comp.) .753 .756 .757 .788 .788 .793 .811 .772 .773 .793 .808 .804 .820 .830 .807 .797 .828 .840 .834 .859 ? AMH-Net (fusion) .798 .829 .869 Method ODS OIS AP gPb-ucm [1] OEF [15] Silberman et al. [33] SemiContour [45] SE [10] gPb+NG [13] SE+NG+ [14] .632 .651 .658 .680 .685 .687 .710 .661 .667 .661 .700 .699 .716 .723 .562 ? ? .690 .679 .629 .738 HED (RGB) [38] HED (HHA) [38] HED (RGB + HHA) [38] RCF (RGB) + HHA) [23] .720 .682 .746 .757 .734 .695 .761 .771 .734 .702 .786 ? AMH-Net (RGB) AMH-Net (HHA) AMH-Net (RGB+HHA) .744 .716 .771 .758 .729 .786 .765 .734 .802 MCG [28], LEP [46], and more recent CNN-based methods, including DeepEdge [2], DeepContour [31], HED [38], CEDN [41], COB [24]. We also report results of the RCF method [23], although they are not comparable because in [23] an extra dataset (Pascal Context) was used during RCF training to improve the results on BSDS500. In this series of experiments we consider AMH-Net with FLAG-CRFs. The results of this comparison are shown in Table 1 and Fig. 4a. AMH-Net obtains an F-measure (ODS) of 0.798, thus outperforms all previous methods. The improvement over the second and third best approaches, i.e. COB and HED, is 0.5% and 1.0%, respectively, which is not trivial to achieve on this challenging dataset. Furthermore, when considering the OIS and AP metrics, our approach is also better, with a clear performance gap. To perform experiments on NYUDv2, following previous works [38] we consider three different types of input representations, i.e. RGB, HHA [14] and RGB-HHA data. The results corresponding to the use of both RGB and HHA data (i.e. RGB+HHA) are obtained by performing a weighted average of the estimates obtained from two AMH-Net models trained separately on RGB and HHA representations. As baselines we consider gPb-ucm [1], OEF [15], the method in [33], SemiContour [45], SE [10], gPb+NG [13], SE+NG+ [14], HED [38] and RCF [23]. In this case the results are comparable to the RCF [23] since the experimental protocol is exactly the same. All of them are reported in Table 2 and Fig. 4b. Again, our approach outperforms all previous methods. In particular, the increased performance with respect to HED [38] and RCF [23] confirms the benefit of the proposed multi-scale feature learning and fusion scheme. Examples of qualitative results on the BSDS500 and the NYUDv2 datasets are shown in Fig. 3. Ablation Study. To further demonstrate the effectiveness of the proposed model and analyze the impact of the different components of AMH-Net on the countour detection task, we conduct an 7 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 [F=0.800] Human [F=0.798] AMH-Net [F=0.793] COB [F=0.788] CEDN [F=0.788] HED [F=0.757] LEP [F=0.756] DeepContour [F=0.753] DeepEdge [F=0.747] MCG [F=0.727] SketchTokens [F=0.726] UCM [F=0.724] ISCRA [F=0.641] Normalized Cuts [F=0.640] MeanShift [F=0.610] Felz-Hut 0.5 0.4 0.3 0.2 0.1 0 Precision Precision 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.6 0.5 [F=0.800] Human [F=0.771] AMH-Net [F=0.746] HED [F=0.706] SE+NG+ [F=0.695] SE [F=0.685] gPb+NG [F=0.680] SemiContour [F=0.658] Silberman [F=0.651] OEF [F=0.632] gPb-ucm 0.4 0.3 0.2 0.1 0.7 0.8 0.9 0 1 Recall 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Recall (a) BSDS500 (b) NYUDv2 Figure 4: Precision-Recall Curves on the BSDS500 and NYUDv2 test sets. ablation study considering the NYUDv2 dataset (RGB data). We tested the following models: (i) AMH-Net (baseline), which removes the first-level hierarchy and directly concatenates the feature maps for prediction, (ii) AMH-Net (w/o AG-CRFs), which employs the proposed multi-scale hierarchical structure but discards the AG-CRFs, (iii) AMH-Net (w/ CRFs), obtained by replacing our AG-CRFs with a multi-scale CRF model without attention gating, (iv) AMH-Net (w/o deep supervision) obtained removing intermediate loss functions in AMH-Net and (v) AMH-Net with the proposed two versions of the AG-CRFs model, i.e. PLAG-CRFs and FLAG-CRFs. The results of our comparison are shown in Table 3, where we also consider as reference traditional multi-scale deep learning models employing multi-scale representations, i.e. Hypercolumn [16] and HED [38]. These results clearly show the advantages of Table 3: Performance analysis on NYUDv2 RGB data. our contributions. The ODS F-measure of Method ODS OIS AP AMH-Net (w/o AG-CRFs) is 1.1% higher Hypercolumn [16] .718 .729 .731 than AMH-Net (baseline), clearly demonHED [38] .720 .734 .734 strating the effectiveness of the proposed hiAMH-Net (baseline) .711 .720 .724 erarchical network and confirming our intuAMH-Net (w/o AG-CRFs) .722 .732 .739 ition that exploiting more richer and diverse AMH-Net (w/ CRFs) .732 .742 .750 multi-scale representations is beneficial. Ta.725 .738 .747 AMH-Net (w/o deep supervision) ble 3 also shows that our AG-CRFs plays AMH-Net (w/ PLAG-CRFs) .737 .749 .746 a fundamental role for accurate detection, AMH-Net (w/ FLAG-CRFs) .744 .758 .765 as AMH-Net (w/ FLAG-CRFs) leads to an improvement of 1.9% over AMH-Net (w/o AG-CRFs) in terms of OSD. Finally, AMH-Net (w/ FLAG-CRFs) is 1.2% and 1.5% better than AMH-Net (w/ CRFs) in ODS and AP metrics respectively, confirming the effectiveness of embedding an attention mechanism in the multi-scale CRF model. AMH-Net (w/o deep supervision) decreases the overall performance of our method by 1.9% in ODS, showing the crucial importance of deep supervision for better optimization of the whole AMH-Net. Comparing the performance of the proposed two versions of the AG-CRF model, i.e. PLAG-CRFs and FLAG-CRFs, we can see that AMH-Net (FLAG-CRFs) slightly outperforms AMH-Net (PLAG-CRFs) in both ODS and OIS, while bringing a significant improvement (around 2%) in AP. Finally, considering HED [38] and Hypercolumn [16], it is clear that our AMH-Net (FLAG-CRFs) is significantly better than these methods. Importantly, our approach utilizes only three scales while for HED [38] and Hypercolumn [16] we consider five scales. We believe that our accuracy could be further boosted by involving more scales. 5 Conclusions We presented a novel multi-scale convolutional neural network for contour detection. 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On-the-fly Operation Batching in Dynamic Computation Graphs Graham Neubig? Language Technologies Institute Carnegie Mellon University [email protected] Yoav Goldberg? Computer Science Department Bar-Ilan University [email protected] Chris Dyer DeepMind [email protected] Abstract Dynamic neural network toolkits such as PyTorch, DyNet, and Chainer offer more flexibility for implementing models that cope with data of varying dimensions and structure, relative to toolkits that operate on statically declared computations (e.g., TensorFlow, CNTK, and Theano). However, existing toolkits?both static and dynamic?require that the developer organize the computations into the batches necessary for exploiting high-performance algorithms and hardware. This batching task is generally difficult, but it becomes a major hurdle as architectures become complex. In this paper, we present an algorithm, and its implementation in the DyNet toolkit, for automatically batching operations. Developers simply write minibatch computations as aggregations of single instance computations, and the batching algorithm seamlessly executes them, on the fly, using computationally efficient batched operations. On a variety of tasks, we obtain throughput similar to that obtained with manual batches, as well as comparable speedups over singleinstance learning on architectures that are impractical to batch manually.2 1 Introduction Modern CPUs and GPUs evaluate batches of arithmetic operations significantly faster than the sequential evaluation of the same operations. For example, performing elementwise operations takes nearly the same amount of time on the GPU whether operating on tens or on thousands of elements, and multiplying a few hundred different vectors by the same matrix is significantly slower than executing a single (equivalent) matrix?matrix product using an optimized GEMM implementation on either a GPU or a CPU. Thus, careful grouping of operations into batches that can execute efficiently in parallel is crucial for making the most of available hardware resources. Today, developers who write code to train neural networks are responsible for crafting most of this batch handling by hand. In some cases this is easy: when inputs and outputs are naturally represented as fixed sized tensors (e.g., images of a fixed size such those in the MNIST and CIFAR datasets, or regression problems on fixed sized vector inputs), and the computations required to process each instance are instance-invariant and expressible as standard operations on tensors (e.g., a series of matrix multiplications, convolutions, and elementwise nonlinearities), a suitably flexible tensor library ? Authors contributed equally. The proposed algorithm is implemented in DyNet (http://dynet.io/), and can be activated by using the ?--dynet-autobatch 1? command line flag. 2 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. RNN RNN RNN RNN masks L(1) m1 Y (1) x1 RNN (1) (1) x2 x3 RNN L(2) (1) x4 m3 m2 Y Y m4 Y y(1) L L1 L2 L3 L4 RNN RNN RNN RNN X1 X2 X3 X4 batches x1 (2) x2 (2) y(2) RNN RNN RNN (3) x1 (3) x2 (3) x3 L L (3) padding y(3) Figure 1: Two computation graphs for computing the loss on a minibatch of three training instances consisting of a sequence of input vectors paired with a fixed sized output vector. On the left is a ?conceptual? computation graph which shows the operations associated with computing the losses individually for each sequence and then aggregating them. The same computation is executed by the right-hand (?batched?) computation graph: it aggregates the inputs in order to make better use of modern processors. This comes with a price in complexity?the variable length of the sequences requires padding and masking operations. Our aim is for the user to specify the conceptual computation on the left, and let the framework take care of its efficient execution. that provides efficient implementations of higher-order generalizations of low-order operations makes manual batching straightforward. For example, by adding a leading or trailing dimension to the tensors representing inputs and outputs, multiple instances can be straightforwardly represented in a single data structure. In other words: in this scenario, the developer conceives of and writes code for the computation on an individual instance, packs several instances into a tensor as a ?minibatch?, and the library handles executing these efficiently in parallel. Unfortunately, this idealized scenario breaks when working with more complex architectures. Deep learning is increasingly being applied to problems whose inputs, outputs and intermediate representations do not fit easily into fixed sized tensors. For example, images vary in size and sequences in length; data may be structured as trees [29] or graphs [4, 17, 27], or the model may select its own computation conditional on the input [16, 28, 33]. In all these cases, while the desired computation is easy enough to write for a single instance, organizing the computational operations so that they make optimally efficient use of the hardware is nontrivial. Indeed, many papers that operate on data structures more complicated than sequences have avoided batching entirely [8, 15, 25]. In fact, until last year [7, 20], all published work on recursive (i.e., tree-structured) neural networks appears to have used single instance training. The premise of this work is that operation batching should not be the responsibility of the user, but instead should be a service provided by the framework. The user should only be responsible for specifying a large enough computation so that batching is possible (i.e, summing the losses of several instances, such as one sees in the left side of Figure 1), and the framework should take care of the lower-level details of operation batching, much like optimizing compilers or JIT optimizers in interpreted languages do.3 We take a large step towards this goal by introducing an efficient algorithm?and a corresponding implementation?for automatic batching in dynamically declared computation graphs.4 Our method relies on separating the graph construction from its execution, using operator overloading and lazy 3 This is in contrast to other existing options for automatic batching such as TensorFlow Fold, which require the user to learn an additional domain-specific language to turn computation into a format conducive to automatic batching [19]. 4 Computation graphs (often represented in a form called a Wengert list) are the data structures used to structure the evaluation of expressions and use reverse mode automatic differentiation to compute their derivatives [3]. Broadly, learning frameworks use two strategies to construct these: static and dynamic. In static toolkits (e.g., Theano [6], Tensorflow [1]) the computation graph is defined once and compiled, and then examples are fed into the same graph. In contrast, dynamic toolkits (e.g., DyNet [21], Chainer [32], PyTorch [http://pytorch.org]) construct the computation graph for each training instance (or minibatch) as the forward computation is executed. While dynamic declaration means that each minibatch can have its own computational architecture, the user is still responsible for batching operations themselves. 2 evaluation (?2). Once this separation is in place, we propose a fast batching heuristic that can be performed in real time, for each training instance (or minibatch), between the graph construction and its execution (?3). We extend the DyNet toolkit [21] with this capability. From the end-user?s perspective, the result is a simple mechanism for exploiting efficient data-parallel algorithms in networks that would be cumbersome to batch by hand. The user simply defines the computation independently for each instance in the batch (using standard Python or C++ language constructs), and the framework takes care of the rest. Experiments show that our algorithm compares favorably to manually batched code, that significant speed improvements are possible on architectures with no straightforward manual batching design, and that we obtain better performance than TensorFlow Fold [19], an alternative framework built to simulate dynamic graph definition and automatic batching on top of TensorFlow (?4). 2 Batching: Conception vs. Efficient Implementation To illustrate the challenges with batching, consider the problem of predicting a real-valued vector conditional on a sequence of input vectors (this example is chosen for its simplicity; experiments are conducted on more standard tasks). We assume that an input sequence of vectors is read sequentially by an RNN, and then the final state is used to make a prediction; the training loss is the Euclidean distance between the prediction and target. We compare two algorithms for computing this code: a na?ve, but developer-friendly one (whose computation graph is shown in the left part of Figure 1), which reflects how one conceives of what a batch loss computation is; and a computationally efficient? but more conceptually complex?version that batches up the computations so they are executed in parallel across the sequences (the right part of Figure 1). Na?ve (developer-friendly) batched implementation The left part of Figure 1 shows the computations that must be executed to compute the losses associated with three (b = 3) training instances, implemented na?vely. Pseudo-code for constructing the graph for each of the RNNs on the left using a dynamic declaration framework is as follows: function RNN-R EGRESSION -L OSS(x1:n , y; (W, U, b, c) = ?) h0 = 0 . Initial state of the RNN; ht 2 Rd . for t 2 1, 2, . . . , n do ht = tanh(W[ht 1 ; xt ] + b) ? = Uhn + c y L = ||? y y||22 return L Note that the code does not compute any value, but constructs a symbolic graph describing the computation. This can then be integrated into a batched training procedure: (i) function T RAIN -BATCH -NAIVE(T = {(x1:n(i) , y(i) )}bi=1 ; ?) N EW-G RAPH() for i 2 1, 2, . . . , b do . Na?vely loop over elements of batch. (i) (i) (i) L = RNN-R EGRESSION -L OSS(x1:n(i) , y ; ?) . Single instance loss. P (i) L= iL . Aggregate losses for all elements in batch. F ORWARD(L) @L @? = BACKWARD (L) ? = ? ? @L @? This code is simple to understand, uses basic flow control present in any programming language and simple mathematical operations. Unfortunately, executing it will generally be quite inefficient, since in the resulting computation graph each operation is performed sequentially without exploiting the fact that similar operations are being performed across the training instances. Efficient manually batched implementation To make good use of efficient data-parallel algorithms and hardware, it is necessary to batch up the operations so that the sequences are processed in parallel. The standard way to achieve this is by aggregating the inputs and outputs, altering the code as follows: 3 function RNN-R EGRESSION -BATCH -L OSS(X1:nmax , Y, n(1:b) ; (W, U, b, c) = ?) M=0 . Build loss mask; M 2 Rb?nmax . for i 2 1, 2, . . . , b do M[i,n(i) ] = 1 . Position where the final symbol in sequence i occurs. H0 = 0 . Initial states of the RNN (one per instance); Ht 2 Rd?b . for t 2 1, 2, . . . , nmax do Ht = tanh(W[Ht 1 ; Xt ] + b) . Addition broadcasts b over columns. ? t = UHt + c Y . Addition broadcasts c over columns. ? t Y)(mt 1> )||2 Lt = ||(Y . Compute masked losses (mt is the tth column of M). F P L = t Lt return L (i) function T RAIN -BATCH -M ANUAL(T = {(x1:n(i) , y(i) )}bi=1 ; ?) nmax = maxi n(i) for t 2 1, 2, . . . , nmax do . Build sequence of batch input matrices. Xt = 0 2 Rd?b for i 2 1, 2, . . . , b do (i) Xt,[?,i] = xt if t ? n(i) otherwise 0 . The ith column of Xt . (1) (2) (b) Y = [y y ??? y ] . Build batch of output targets. N EW-G RAPH() . Now that inputs are constructed, create graph, evaluate loss and gradient. L = RNN-R EGRESSION -BATCH -L OSS(X1:nmax , Y, n(1:b) ; ?) F ORWARD(L) @L @? = BACKWARD (L) ? = ? ? @L @? This code computes the same value as the na?ve implementation, it does so more efficiently, and it is significantly more complicated. Because the sequences processed by RNNs will generally be of different lengths (which is precisely why RNNs are useful!), it is necessary to pad the input representation with dummy values, and also to mask out the resulting losses at the right times. While these techniques are part of the inventory of skills that a good ML engineer has, they increase the difficulty of implementation and probability that bugs will be present in the code. Implementation comparison The na?ve algorithm has two advantages over manual batching. First, it is easy to implement: the way we conceive of a model is the way it is implemented, and errors with padding, masking, and batching are avoided. Second, the na?ve algorithm aggregates any single instance loss, whereas manual batching efforts are generally problem specific. For these reasons, one should strongly prefer the first algorithm; however, for efficiency reasons, batching matters. In the next section we turn to the problem of how to efficiently execute na?ve computation graphs so that they can take advantage of efficient batched implementations of operations. This provides the best of both worlds to developers: code is easy to write, but execution is fast. 3 An Algorithm for On-the-fly Batching Manual batching, discussed in the previous section, mostly operates by aggregating input instances and feeding them through a network. In RNNs, this means aggregating inputs that share a time step. This often require padding and masking, as input sizes may differ. It also restricts the kinds of operations that can be batched. In contrast, our method identifies and aggregates computation graph nodes that can be executed in a batched fashion for a given graph. This reduces the need for workarounds such as padding and masking, allows for seamless efficient execution also in architectures which are hard to conceptualize in the input-centric paradigm, and allows for the identification of batching opportunities that may not be apparent from an input-centric view. Our batching procedure operates in three steps (1) graph definition, (2) operation batching, and (3) computation. Here, steps (1) and (3) are shared with standard execution of computation graphs, while (2) corresponds to our proposed method. 3.1 Graph Definition First, we define the graph that represents the computation that we want to perform. From the user?s perspective, this is done by simply performing computation that they are interested in performing, such as that defined in the R NN -R EGRESSION -L OSS function from the previous example. While it is common for dynamic graph frameworks to interleave the graph definition and its forward execution, 4 we separate these parts by using lazy evaluation: we only perform forward evaluation when a resulting value is requested by the user through the calling of the F ORWARD function. The graph can be further extended after a call to F ORWARD, and further calls will lazily evaluate the delta of the computation. This allows the accumulation of large graph chunks before executing forward computations, providing ample opportunities for operation batching. 3.2 Operation Batching Next, given a computation graph, such as the one on the left side of Figure 1, our proposed algorithm converts it into a graph where operations that can be executed together are batched together. This is done in the two step process described below. Computing compatibility groups We first partition the nodes into compatibility groups, where nodes in the same group have the potential for batching. This is done by associating each node with a signature such that nodes that share the same signature are guaranteed to be able to be executed in a single operation if their inputs are ready. Signatures vary depending on the operation the node represents. For example, in nodes representing element-wise operations, all nodes with the same operation can be batched together, so the signature is simply the operation name (tanh, log, ...). In nodes where dimensions or other information is also relevant to whether the operations can be batched, this information is also included in the signature. For example, a node that picks a slice of the input matrix will also be dependent on the matrix size and range to slice, so the signature will look something like slice-400x500-100:200. In some other cases (e.g. a parameterized matrix multiply) we may remember the specific node ID of one of the inputs (e.g. node123 representing the matrix multiply parameters) while generalizing across other inputs (e.g. data or hidden state vectors on the right-hand side), resulting in a signature that would look something like matmul-node123-400x1. A more thorough discussion is given in Appendix A. Determining execution order A computation graph is essentially a job dependency graph where each node depends on its input (and by proxy the input of other preceding nodes on the path to its inputs). Our goal is to select an execution order in which (1) each node is executed after its dependencies; and (2) nodes that have the same signature and do not depend on each other are scheduled for execution on the same step (and will be executed in a single batched operation). Finding an optimal execution order that maximizes the amount of batching in the general case is NP hard [24]. We discuss two heuristic strategies for identifying execution orders that satisfy these requirements. Depth-based batching is used as a method for automatic batching in TensorFlow Fold [19]. This is done by calculating the depth of each node in the original computation graph, defined as the maximum length from a leaf node to the node itself, and batching together nodes that have an identical depth and signature. By construction, nodes of the same depth are not dependent on each-other, as all nodes will have a higher depth than their input, and thus this batching strategy is guaranteed to satisfy condition (1) above. However, this strategy will also miss some good batching opportunities. For example, the loss function calculations in Figure 1 are of different depths due to the different-lengthed sequences, and similar problems will occur in recurrent neural network language models, tree-structured neural networks, and a myriad of other situations. Agenda-based batching is a method we propose that does not depend solely on depth. The core of this method is an agenda that tracks ?available? nodes that have no unresolved dependencies. For each node, a count of its unresolved dependencies is maintained; this is initialized to be the number of inputs to the node. The agenda is initialized by adding nodes that have no incoming inputs (and thus no unresolved dependencies). At each iteration, we select a node from the agenda together with all of the available nodes in the same signature, and group them into a single batch operation. These nodes are then removed from the agenda, and the dependency counter of all of their successors are decremented. Any new zero-dependency nodes are added to the agenda. This process is repeated until all nodes have been processed. How do we prioritize between multiple available nodes in the agenda? Intuitively, we want to avoid prematurely executing nodes if there is a potential for more nodes of the same signature to be added to the agenda at a later point, resulting in better batching. A good example of this from our running example in Figure 1 is the loss-calculating nodes, which will be added to the agenda at different points due to becoming calculable after different numbers of RNN time steps. To capture this intuition, we introduce a heuristic method for prioritizing nodes based on the average depth of all nodes with their 5 signature, such that nodes with a lower average depth will be executed earlier. In general (with some exceptions), this tends to prioritize nodes that occur in earlier parts of the graph, which will result in the nodes in the later parts of the graph, such as these loss calculations, being executed later and hopefully batched together.5 Finally, this non-trivial batching procedure must be executed quickly so that overhead due to batch scheduling calculations doesn?t cancel out the efficiency gains from operation batching. To ensure this, we perform a number of optimizations in the implementation, which we detail in Appendix B. 3.3 Forward-backward Graph Execution and Update Once we have determined an execution order (including batching decisions), we perform calculations of the values themselves. In standard computation graphs, forward computation is done in topological order to calculate the function itself, and backward calculation is done in reverse topological order to calculate gradients. In our automatically batched evaluation, the calculation is largely similar with two exceptions: Single!batch node conversion First, it is necessary to convert single nodes into a batched node, which also requires modification of the underlying operations such as converting multiple matrixvector operations Whi to a single matrix-matrix operation WH. This is done internally in the library, while the user-facing API maintains the original unbatched computation graph structure, making this process invisible to the user. Ensuring contiguous memory To ensure that operations can be executed as a batch, the inputs to (i) the operations (e.g. the various vectors ht ) must be arranged in contiguous memory (e.g. a matrix Ht ). In some cases, it is necessary to perform a memory copy to arrange these inputs into contiguous memory, but in other cases the inputs are already contiguous and in the correct order, and in these cases we can omit the memory copy and use the inputs as-is.6 4 Experiments In this section we describe our experiments, designed to answer three main questions: (1) in situations where manual batching is easy, how close can the proposed method approach the efficiency of a program that uses hand-crafted manual batching, and how do the depth-based and agenda-based approaches compare (?4.1)? (2) in situations where manual batching is less easy, is the proposed method capable of obtaining significant improvements in efficiency (?4.2)? (3) how does the proposed method compare to TensorFlow Fold, an existing method for batching variably structured networks within a static declaration framework (?4.3)? 4.1 Synthetic Experiments Our first experiments stress-test our proposed algorithm in an ideal case for manual batching. Specifically, we train a model on a bi-directional LSTM sequence labeler [12, 23], on synthetic data where every sequence to be labeled is the same length (40). Because of this, manual batching is easy because we don?t have to do any padding or adjustment for sentences of different lengths. The network takes as input a size 200 embedding vector from a vocabulary of size 1000, has 2 layers of 256 hidden node LSTMs in either direction, then predicts a label from one of 300 classes. The batch size is 64.7 Within this setting we test various batching settings: Without or with manual mini-batching where we explicitly batch the word vector lookup, LSTM update, and loss calculation for each time step. 5 Even given this prioritization method it is still possible to have ties, in which case we break ties by calculating ?cheap? operations (e.g. tanh and other elementwise ops) before ?heavy? ones (e.g. matrix multiplies). 6 The implication of this is that batched computation will take up to twice as much memory as unbatched computation, but in practice the memory usage is much less than this. Like manually batched computation, memory usage can be controlled by adjusting the batch size appropriately so it fits in memory. 7 Experiments were run on a single Tesla K80 GPU or Intel Xeon 2.30GHz E5-2686v4 CPU. To control for variance in execution time, we perform three runs and report the fastest. We do not report accuracy numbers, as the functions calculated and thus accuracies are the same regardless of batching strategy. 6 CPU ms/ sent for graph 20 40 back graph 60 80 back calc update for graph 100 120 140 160 180 200 w/o Manual No Auto 0 By Depth By Agenda No Auto w/ Manual w/ Manual w/o Manual 0 GPU ms/ sent for calc By Depth By Agenda 0 2 4 6 8 20 for calc 40 80 back calc update 100 120 140 160 180 200 No Auto By Depth By Agenda No Auto By Depth By Agenda 0 10 12 14 16 18 20 back graph 60 2 4 6 8 10 12 14 16 18 20 Figure 2: Computation time for forward/backward graph construction or computation, as well as parameter update for a BiLSTM tagger without or with manual batching, and without, with depth-based, or with agenda-based automatic batching. Without on-the-fly batching (N OAUTO), with depth-based autobatching (B Y D EPTH), or with agendabased autobatching (B YAGENDA). We measure the speed of each method by ms/sec and also break down the percentage of computation time spent in (1) forward graph creation/on-the-fly batching, (2) forward computation, (3) backward graph creation, (4) backward computation, (5) parameter update. The results can be found in Figure 2. First, comparing the first row with the second two, we can see that the proposed on-the-fly batching strategy drastically reduces computation time per sentence, with B YAGENDA reducing per-sentence computation time from 193ms to 16.9ms on CPU and 54.6ms to 5.03ms on GPU, resulting in an approximately 11-fold increase in sentences processed per second (5.17!59.3 on CPU and 18.3!198 on GPU). B YAGENDA is faster than B Y D EPTH by about 15?30%, demonstrating that our more sophisticated agenda-based strategy is indeed more effective at batching together operations. Next, compared to manual batching without automatic batching (the fourth row), we can see that fully automatic batching with no manual batching is competitive, but slightly slower. The speed decrease is attributed to the increased overhead for computation graph construction and batch scheduling. However, even in this extremely idealized scenario where manual batching will be most competitive, the difference is relatively small (1.27? on CPU and 1.76? on GPU) compared to the extreme difference between the case of using no batching at all. Given that automatic batching has other major advantages such as ease of implementation, it may be an attractive alternative even in situations where manual batching is relatively easy. Finally, if we compare the fourth and fifth/sixth rows, we can see that on GPU, even with manual batching, automatic batching still provides gains in computational efficiency, processing sentences up to 1.1 times faster than without automatic batching. The reason for this can be attributed to the fact that our BiLSTM implementation performs manual batching across sentences, but not across time steps within the sentence. In contrast, the auto-batching procedure was able to batch the word embedding lookup and softmax operations across time-steps as well, reducing the number of GPU calls and increasing speed. This was not the case for CPU, as there is less to be gained from batching these less expensive operations. 4.2 Experiments on Difficult-to-batch Tasks Next, we extend our experiments to cases that are increasingly more difficult to manually batch. We use realistic dimension sizes for the corresponding tasks, and batches of size b = 64. Exact dimensions and further details on training settings are in Appendix C. BiLSTM: This is similar to the ideal case in the previous section, but trained on actual variable length sequences. BiLSTM w/char: This is the same as the BiLSTM tagger above, except that we use an additional BiLSTM over characters to calculate the embeddings over rare words. These sorts of 7 Table 1: Sentences/second on various training tasks for increasingly challenging batching scenarios. Task BiLSTM BiLSTM w/ char TreeLSTM Transition-Parsing CPU GPU N OAUTO B Y D EPTH B YAGENDA N OAUTO B Y D EPTH B YAGENDA 16.8 15.7 50.2 16.8 139 93.8 348 61.0 156 132 357 61.2 56.2 43.2 76.5 33.0 337 183 672 89.5 367 275 661 90.1 character-based embeddings have been shown to allow the model to generalize better [18], but also makes batching operations more difficult, as we now have a variably-lengthed encoding step that may or may not occur for each of the words in the input. Tree-structured LSTMs: This is the Tree-LSTM model of [31]. Here, each instance is a tree rather than a sequence, and the network structure follows the tree structures. As discussed in the introduction, this architecture is notoriously hard to manually batch. Transition-based Dependency Parsing: The most challenging case we evaluate is that of a transition-based system, such as a transition based parser with LSTM-based featureextraction [8, 9, 13] and exploration-based training [2, 5, 10]. Here, a sequence is encoded using an LSTM (or a bi-LSTM), followed by a series of predictions. Each prediction based on a subset of the encoded vectors, and the vectors that participate in each prediction, as well as the loss, are determined by the outcomes of the previous predictions. Here, batching is harder yet as the nature of the computation interleaves sampling from the model and training, and requires calling F ORWARD at each step, leaving the automatic-batcher very little room to play with. However, with only a small change to the computation, we can run b different parsers ?in parallel?, and potentially share the computation across the different systems in a given time-step. Concretely, we use a modified version of the B IST parser [14]. From the results in Table 1, we can see that in all cases automatic batching gives healthy improvements in computation time, 3.6x?9.2? on the CPU, and 2.7?8.6? on GPU. Furthermore, the agenda-based heuristic is generally more effective than the depth-based one. 4.3 Comparison to TensorFlow Fold We compare the TensorFlow Fold reference implementation of the Stanford Sentiment Treebank regression task [30], using the same TreeLSTM architecture [31].Figure 3 shows how many trees are processed per second by TF (excluding both evaluation of the dev set and static graph construction/optimization) on GPU and CPU relative to the performance of the B YAGENDA algorithm in DyNet (including graph construction time). The DyNet performance is better across the board stratified by hardware type. Furthermore, DyNet has greater throughput on CPU than TensorFlow Fold on GPU until batch sizes exceed 64. Additionally, we find that with single instance training, DyNet?s se- Figure 3: Comparison of runtime performance bequential evaluation processes 46.7 trees/second tween TensorFlow Fold and DyNet with autobatchon CPU, whereas autobatching processes 93.6 ing on TreeLSTMs (trees/sec). trees/second. This demonstrates that in complex architectures like TreeLSTMs, there are opportunities to batch up operations inside a single training instance, which are exploited by our batching algorithm. In addition, it should be noted that the DyNet implementation has the advantage that it is much more straightforward, relying on simple Python data structures and flow control to represent and traverse the trees, while the Fold implementation requires implementing the traversal and composition logic in a domain specific functional programming language (described in Section 3 of Looks et al. [19]). 8 5 Related Work Optimization of static algorithms is widely studied, and plays an important role in numerical libraries used in machine learning. Our work is rather different since the code/workload (as represented by the computation graph) is dynamically specified and must be executed rapidly, which precludes sophisticated statistic analysis. However, we review some of the important related work here. Automatic graph optimization and selection of kernels for static computation graphs is used in a variety of toolkits, including TensorFlow [1] and Theano [6]. Dynamic creation of optimally sized minibatches (similar to our strategy, except the computation graph is assumed to be static) that make good use of hardware resources has also been proposed for optimizing convolutional architectures [11]. The static nature of the computation makes this tools closer to optimizing compilers rather than efficient interpreters which are required to cope with the dynamic workloads encountered when dealing with dynamically structured computations. Related to this is the general technique of automatic vectorization, which is a mainstay of optimizing compilers. Recent work has begun to explore vectorization in the context of interpreted code which may cannot be compiled [26]. Our autobatching variant of DyNet similarly provides vectorized primitives that can be selected dynamically. Further afield, the problem of scheduling with batching decisions has been widely studied in operations research since at least the 1950s (for a recent survey, see [24]). Although the OR work deals with similar problems (e.g., scheduling work on machines that can process a ?family? of related item with minimal marginal cost over a single item), the standard algorithms from this field (which are often based on polynomial-time dynamic programs or approximations to NP-hard search problems) are too computationally demanding to execute in the inner loop of a learning algorithm. 6 Conclusion Deep learning research relies on empirical exploration of architectures. The rapid pace of innovation we have seen in the last several years has been enabled largely by tools that have automated the error-prone aspects of engineering, such as writing code that computes gradients. However, our contention is that operation batching is increasingly becoming another aspect of model coding that is error prone and amenable to automation. Our solution is a framework that lets programmers express computations naturally and relies on a smart yet lightweight interpreter to figure out how to execute the operations efficiently. Our hope is that this will facilitate the creation of new classes of models that better cope with the complexities of real-world data. Acknowledgements: The work of YG is supported by the Israeli Science Foundation (grant number 1555/15) and by the Intel Collaborative Research Institute for Computational Intelligence (ICRI-CI). References [1] Mart?n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. [2] Miguel Ballesteros, Yoav Goldberg, Chris Dyer, and Noah A. Smith. Training with exploration improves a greedy stack LSTM parser. 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Nonlinear Acceleration of Stochastic Algorithms Damien Scieur INRIA, ENS, PSL Research University, Paris France [email protected] Francis Bach INRIA, ENS, PSL Research University, Paris France [email protected] Alexandre d?Aspremont CNRS, ENS, PSL Research University, Paris France [email protected] Abstract Extrapolation methods use the last few iterates of an optimization algorithm to produce a better estimate of the optimum. They were shown to achieve optimal convergence rates in a deterministic setting using simple gradient iterates. Here, we study extrapolation methods in a stochastic setting, where the iterates are produced by either a simple or an accelerated stochastic gradient algorithm. We first derive convergence bounds for arbitrary, potentially biased perturbations, then produce asymptotic bounds using the ratio between the variance of the noise and the accuracy of the current point. Finally, we apply this acceleration technique to stochastic algorithms such as SGD, SAGA, SVRG and Katyusha in different settings, and show significant performance gains. 1 Introduction We focus on the problem min f (x) x?Rd (1) where f is a L-smooth and ?-strongly convex function with respect to the Euclidean norm, i.e., L ? ky ? xk2 ? f (y) ? f (x) ? ?f (x)T (y ? x) ? ky ? xk2 . 2 2 We consider a stochastic first-order oracle, which gives a noisy estimate of the gradient of f (x), with ?? f (x) = ?f (x) + ?, (2) where ? is a noise term with bounded variance. This is the case for example when f is a sum of strongly convex functions, and we only have access to the gradient of one randomly selected function. Stochastic optimization (2) is typically challenging as classical algorithms are not convergent (for example, gradient descent or Nesterov?s accelerated gradient). Even the averaged version of stochastic gradient descent with constant step size does not converge to the solution of (1), but to another point whose proximity to the real minimizer depends of the step size [Nedi?c and Bertsekas, 2001; Moulines and Bach, 2011]. When f is a finite sum of N functions, then algorithms such as SAG [Schmidt et al., 2013], SAGA [Defazio et al., 2014], SDCA [Shalev-Shwartz and Zhang, 2013] and SVRG [Johnson and Zhang, 2013] accelerate convergence using a variance reduction technique akin to control variate in MonteCarlo methods. Their rate of convergence depends on 1p? ?/L and thus does not exhibit an accelerated rate on par with the deterministic setting (in 1 ? ?/L). Recently a generic acceleration algorithm called Catalyst [Lin et al., 2015], based on the proximal point method improved this rate of convergence, but the practical performances highly depends on the input parameters. On the other hand, recent papers, for example [Shalev-Shwartz and Zhang, 2014] (Accelerated SDCA) and 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. [Allen-Zhu, 2016] (Katyusha), propose algorithms with accelerated convergence rates, if the strong convexity parameter is given. When f is a quadratic function then averaged SGD converges, but the rate of decay of initial conditions is very slow. Recently, some results have focused on accelerated versions of SGD for quadratic optimization, showing that with a two step recursion it is possible to enjoy both the optimal rate for the bias and variance terms [Flammarion and Bach, 2015], given an estimate of the ratio between the distance to the solution and the variance of ?. A novel generic acceleration technique was recently proposed by Scieur et al. [2016] in the deterministic setting. This uses iterates from a slow algorithm to extrapolate estimates of the solution with asymptotically optimal convergence rate. Moreover, this rate is reached without prior knowledge of the strong convexity constant, whose online estimation is still a challenge (even in the deterministic case [Fercoq and Qu, 2016]) but required if one wants to obtain optimal rates of convergence. Convergence bounds are derived by Scieur et al. [2016], tracking the difference between the deterministic first-order oracle of (1) and iterates from a linearized model. The main contribution of this paper is to extend the analysis to arbitrary perturbations, including stochastic ones, and to present numerical results when this acceleration method is used to speed up stochastic optimization algorithms. In Section 2 we recall the extrapolation algorithm, and quickly summarize its main convergence bounds in Section 3. In Section 4, we consider a stochastic oracle and analyze its asymptotic convergence in Section 5. Finally, in Section 6 we describe numerical experiments which confirm the theoretical bounds and show the practical efficiency of this acceleration. 2 Regularized Nonlinear Acceleration Consider the optimization problem min f (x) x?Rd where f is a L?smooth and ??strongly convex function [Nesterov, 2013]. Applying the fixed-step gradient method to this problem yields the following iterates 1 ?f (? xt ). L Let x? be the unique optimal point, this algorithm is proved to converge with x ?t+1 = x ?t ? k? xt ? x? k ? (1 ? ?)t k? x0 ? x? k (3) (4) where k ? k stands for the `2 norm and ? = ?/L ? [0, 1[ is the (inverse of the) condition number of f [Nesterov, 2013]. Using a two-step recurrence, the accelerated gradient descent by Nesterov [2013] achieves the improved convergence rate   ? x0 ? x? k . (5) k? xt ? x? k ? O (1 ? ?)t k? Indeed, (5) converges faster than (4) but the accelerated algorithm requires the knowledge of ? and L. Extrapolation techniques however obtain a similar convergence rate, but do not need estimates of the parameters ? and L. The idea is based on the comparison between the process followed by x ?i with a linearized model around the optimum (obtained by the first-order approximation of ?f (x)), written  1 xt+1 = xt ? ?f (x? ) +?2 f (x? )(xt ? x? ) , x0 = x ?0 . L | {z } =0 which can be rewritten as xt+1 ? x? = (I ? ?2 f (x? )/L)(xt ? x? ), x0 = x ?0 . (6) A better estimate of the optimum in (6) can be obtained by forming a linear combination of the iterates (see [Anderson, 1965; Cabay and Jackson, 1976; Me?ina, 1977]), with t X ci xi ? x?  kxt ? x? k, i=0 2 for some specific ci (either data driven, or derived from Chebyshev polynomials). These procedures were limited to quadratic functions only, i.e. when x ?i = xi but this was recently extended to generic convex problems by Scieur et al. [2016] and we briefly recall these results below. To simplify the notations, we write x ?t+1 = g(? xt ) (7) to be one step of algorithm g. We have that g is differentiable, Lipchitz-continuous with constant (1 ? ?) < 1, g(x? ) = x? and g 0 (x? ) is symmetric. For example, the gradient method (3) matches exactly this definition with g(x) = x ? ?f (x)/L. Running k steps of (7) produces a sequence {? x0 , ..., x ?k }, which we extrapolate using Algorithm 1 from Scieur et al. [2016]. Algorithm 1 Regularized Nonlinear Acceleration (RNA) Input: Iterates x ?0 , x ?1 , ..., x ?k+1 ? Rd produced by (7), and a regularization parameter ? > 0. ? = [? 1: Compute R r0 , ..., r?k ], where r?i = x ?i+1 ? x ?i is the ith residue. 2: Solve ? 2 + ?kck2 , c?? = argmin kRck cT 1=1 ? + ?I)z = 1 then set c?? = z/1T z. ?T R or equivalently solve (R Pk ? Output: Approximation of x computed as i=0 c??i x ?i For a good choice of ?, the output of Algorithm (1) is a much better estimate of the optimum than x ?k+1 (or any other points of the sequence). Using a simple grid search on a few values of ? is usually sufficient to improve convergence (see [Scieur et al., 2016] for more details). 3 Convergence of Regularized Nonlinear Acceleration We quickly summarize the argument behind the convergence of Algorithm (1). The theoretical bound compares x ?i to the iterates produced by the linearized model xt+1 = x? + ?g(x? )(xt ? x? ), x0 = x ?0 . (8) This sequence is useful to extend the convergence results to the nonlinear case, using sensivity analysis. We write c? the coefficients computed by Algorithm (1) from the ?linearized? sequence {x0 , ..., xk+1 } and the error term can be decomposed into three parts, k k k  k X X X X    ? ? ? ? ? ? ? ? c?i x ?i ? x ? ci xi ? x + c?i ? ci (xi ? x ) + c?i x ?i ? xi . (9) i=0 i=0 i=0 i=0 | {z } | {z } | {z } Acceleration Stability Nonlinearity Scieur et al. [2016] show that convergence is guaranteed as long as the errors (? xi ? x? ) and (xi ? x ?i ) converge to zero fast enough, which ensures a good rate of decay for the regularization parameter ?, leading to an asymptotic rate equivalent to the accelerated rate in (5). In this section, we will use results from Scieur et al. [2016] to bound each individual term, but in this paper we improve the final convergence result. The stability term (in c?? ? c? ) is bounded using the perturbation matrix ? T R, ? P , RT R ? R (10) ? are the matrices of residuals, where R and R R , [r0 ...rk ] ? , [? R r0 ...? rk ] rt = xt+1 ? xt , (11) r?t = x ?t+1 ? x ?t . (12) The proofs of the following propositions were obtained by Scieur et al. [2016]. 3 Proposition 3.1 (Stability). Let ?c? = c?? ? c? be the gap between the coefficients computed by Algorithm (1) using the sequences {? xi } and {xi } with regularization parameter ?. Let P = ?T R ? be defined in (10), (11) and (12). Then RT R ? R k?c? k ? This implies that the stability term is bounded by Pk k i=0 ?c?i (xi ? x? )k ? kP k ? ? kc k. (13) kP k ? ? kc k O(kx0 ? x? k). (14) The term Nonlinearity is bounded by the norm of the coefficients c?? (controlled thanks to the regularization parameter) times the norm of the noise matrix E = [x0 ? x ? 0 , x1 ? x ?1 , ..., xk ? x ?k ]. (15) ? Proposition 3.2 (Nonlinearity). Let c? be computed by Algorithm 1 using the sequence ? be defined in (12). The norm of c?? is {? x0 , ..., x ?k+1 } with regularization parameter ? and R bounded by q q k? c? k ? ? 2 +? kRk (k+1)? ? ?1 k+1 This bounds the nonlinearity term because P q k ? xi ? xi ) ? 1 + i=0 c?i (? 1+ ? 2 kRk ? . (16) ? 2 kEk kRk ? , ? k+1 (17) where E is defined in (15). These two propositions show that the regularization in Algorithm 1 limits the impact of the noise: the higher ? is, the smaller these terms are. It remains to control the acceleration term. For small ?, this term decreases as fast as the accelerated rate (5), as shown in the following proposition. Proposition 3.3 (Acceleration). Let Pk be the subspace of real polynomials of degree at most k and S? (k, ?) be the solution of the Regularized Chebychev Polynomial problem, S? (k, ?) , min max p?Pk x?[0,1??] ?, Let ? ? kx0 ?x? k2 p2 (x) + ?kpk2 s.t. p(1) = 1. be the normalized value of ?. The acceleration term is bounded by P p k ? 2 ? 2 ? i=0 c?i xi ? x? ? ?1 S? (k, ?)kx 0 ? x k ? ?kc k . (18) (19) We also get the following corollary, which will be useful for the asymptotic analysis of the rate of convergence of Algorithm 1. Corollary 3.4. If ? ? 0, the bound (19) becomes P  ? k k 1? ? i=0 c?i xi ? x? ? ?1 1+?? kx0 ? x? k. p p Proof. When ? = 0, (19) becomes ?1 S? (k, 0)kx0 ?x? k. The exact value of S? (k, 0) is obtained by using the coefficients of a re-scaled?Chebyshev polynomial, derived by Golub and Varga [1961]; ?? . Scieur et al. [2016], and is equal to 1? 1+ ? These last results controlling stability, nonlinearity and acceleration are proved by Scieur et al. [2016]. We now refine the final step of Scieur et al. [2016] to produce a global bound on the error that will allow us to extend these results to the stochastic setting in the next sections. Theorem 3.5. If Algorithm 1 is applied to the sequence x ?i with regularization parameter ?, it converges with rate s k r X ? k2 )kP k2 ? 2 1 1 O(kx ? x kEk kRk ? ? c??i x ?i ? kx0 ? x? kS?2 (k, ?) + + 1 + . (20) ?2 ?3 ? k+1 i=0 4 Proof. The proof is inspired by Scieur et al. [2016] and is straightforward. We can bound (9) using (14) (Stability), (17) (Nonlinearity) and (19) (Acceleration). It remains to maximize over the value of kc? k using the result of Proposition A.2. ? This last bound is not very explicit, in particular because of the regularized Chebyshev term S? (k, ?). ? The solution is well known when ? = 0 since it corresponds exactly to the rescaled Chebyshev polynomial [Golub and Varga, 1961], but as far as we know there is no known result about its regularized version, thus making the "finite-step" version hard to analyze. However, an asymptotic analysis simplifies it considerably. The next new proposition shows that when x0 is close to x? , then extrapolation converges as fast as in (5) in some cases. ? = O(kx0 ? x? k), kEk = O(kx0 ? x? k2 ) and kP k = O(kx0 ? x? k3 ). Proposition 3.6. Assume kRk If we chose ? = O(kx0 ? x? ks ) with s ? [2, 38 ] then the bound (20) becomes lim ? k kx0 ?x k?0 Pk ??i x ?i k i=0 c kx0 ? x? k ? 1 ?  ? k 1? ? ? . 1+ ? Proof. (Sketch) The proof is based on the fact that ? decreases slowly enough to ensure that the ? ? = Stability and Nonlinearity terms vanish over time, but fast enough to have ? kx0 ?x? k2 ? 0. Then it remains to bound S? (k, 0) with Corollary 3.4. The complete proof can be found in the Supplementary materials. Note: The assumptions are satisfied if we apply the gradient method on a twice differentiable, smooth and strongly convex function with Lipchitz-continuous Hessian [Scieur et al., 2016]. The efficiency of Algorithm 1 is thus ensured by two conditions. First, we need to be able to bound ? kP k and kEk by decreasing quantities. Second, we have to find a proper rate of decay for ? kRk, ? such that the stability and nonlinearity terms go to zero when perturbations also go to zero. If and ? these two conditions are met, then the accelerated rate in Proposition 3.6 holds. 4 Nonlinear and Noisy Updates In (7) we defined g(x) to be non linear, which generates a sequence x ?i . We now consider noisy iterates x ?t+1 = g(? xt ) + ?t+1 , (21) where ?t is a stochastic noise. To simplify notations, we write (21) as x ?t+1 = x? + G(? xt ? x? ) + ?t+1 , (22) where ?t is a stochastic noise (potentially correlated with the iterates xi ) with bounded mean ?t , k?t k ? ? and bounded covariance ?t  (? 2 /d)I. We also assume 0I  G  (1 ? ?)I and G is symmetric. For example, (22) can be linked to (21) if we set ?t = ?t + O(k? xt ? x? k2 ), which corresponds to the combination of the noise ?t+1 with the Taylor remainder of g(x) around x? , i.e., x ?t+1 = g(? xt ) + ?t+1 = g(x? ) + ?g(x? )(? xt ? x? ) + O(k? xt ? x? k) + ?t+1 . | {z } | {z } | {z } =x? =t+1 =G The recursion (22) is also valid when we apply the stochastic gradient method with fixed step size h to the quadratic problem minx 12 kAx ? bk2 . This corresponds to (22) with G = I ? hAT A and mean ? = 0. For the theoretical results, we will compare x ?t with their noiseless counterpart to control convergence, xt+1 = x? + G(xt ? x? ), 5 x0 = x ?0 . (23) 5 Convergence Analysis when Accelerating Stochastic Algorithms We will control convergence in expectation. Bound (9) now becomes # " k k X X h i h i ? ? c? kkEk . c?i xi ? x? + O(kx0 ? x? k)E k?c? k + E k? c?i x E ?i ? x ? (24) i=0 i=0 We now need to enforce bounds (14), (17) and (19) in expectation. The proofs of the two next propositions are in the supplementary material. For simplicity, we will omit all constants in what follows. Proposition 5.1. Consider the sequences xi and x ?i generated by (21) and (23). Then, ? E[kRk] ? O(kx0 ? x? k) + O(? + ?), E[kEk] ? O(? + ?), (25) (26) E[kP k] ? O((? + ?)kx0 ? x? k) + O((? + ?)2 ). (27) We define the following stochastic condition number ?+? . ?, kx0 ? x? k The Proposition 5.2 gives the result when injecting these bounds in (24). Proposition 5.2. The accuracy of extrapolation Algorithm 1 applied to the sequence {? x0 , ..., x ?k } generated by (21) is bounded by h P i !! r r k E k i=0 c??i x ? i ? x? k 1 ? 2 (1 + ? 2 ) O(? 2 (1 + ? )2 ) 2 ? +O ? + . (28) ? S? (k, ?) + ?3 ? kx0 ? x? k ?2 ? ? Consider a situation where ? is small, e.g. when using stochastic gradient descent with fixed step-size, ? and ? ensuring the with x0 far from x? . The following proposition details the dependence between ? upper convergence bound remains stable when ? goes to zero. ? = ?(? s ) with s ?]0, 2 [, we have the accelerated rate Proposition 5.3. When ? ? 0, if ? 3  ? k  Pk  ?? kx0 ? x? k. (29) E k i=0 c??i x ?i ? x? k ? ?1 1? 1+ ? Moreover, if ? ? ?, we recover the averaged gradient, i h  Pk  1 Pk ? E k i=0 c??i x ?i ? x? k = E k+1 x ? ? x . i i=0 ? = ?(? s ), using (28) we have Proof. Let ? q i h P k E i=0 c??i x ?i ? x? ? kx0 ? x? kS? (k, ? s ) ?12 O(? 2?3s (1 + ? )2 ) p +kx0 ? x? kO( ? 2 + ? 2?3s (1 + ? 2 )). Because s ?]0, 32 [, means 2 ? 3s > 0, thus lim? ?0 ? 2?3s = 0. The limits when ? ? 0 is thus exactly (29). If ? ? ?, we have also ? + ?kck2 = argminc:1T c=1 kck2 = lim c?? = lim argminc:1T c=1 kRck ??? ??? 1 k+1 which yields the desired result. Proposition 5.3 shows that Algorithm 1 is thus asymptotically optimal provided ? is well chosen because it recovers the accelerated rate for smooth and strongly convex functions when the perturbations goes to zero. Moreover, we recover Proposition 3.6 when t is the Taylor remainder, i.e. with ? = O(kx0 ? x? k2 ) and ? = 0, which matches the deterministic results. Algorithm 1 is particularly efficient when combined with a restart scheme [Scieur et al., 2016]. From a theoretical point of view, the acceleration peak arises for small values of k. Empirically, the 6 improvement is usually more important at the beginning, i.e. when k is small. Finally, the algorithmic complexity is O(k 2 d), which is linear in the problem dimension when k remains bounded. The benefits of extrapolation are limited in a regime where the noise dominates. However, when ? is relatively small then we can expect a significant speedup. This condition is satisfied in many cases, for example at the initial phase of the stochastic gradient descent or when optimizing a sum of functions with variance reduction techniques, such as SAGA or SVRG. 6 6.1 Numerical Experiments Stochastic gradient descent We want to solve the least-squares problem min F (x) = x?Rd 1 kAx ? bk2 , 2 where AT A satisfies ?I  (AT A)  LI. To solve this problem, we have access to the stochastic first-order oracle ?? F (x) = ?F (x) + ?, where ? is a zero-mean noise of covariance matrix ?  ?2 d I. We will compare several methods. ? SGD. Fixed step-size, xt+1 = xt ? L1 ?? F (xt ). ? Averaged SGD. Iterate xk is the mean of the k first iterations of SGD. ? AccSGD. The optimal two-step algorithm in Flammarion and Bach [2015], with optimal parameters (this implies kx0 ? x? k and ? are known exactly). ? RNA+SGD. The regularized nonlinear acceleration Algorithm 1 applied to a sequence of k ?6 ? T Rk/10 ? iterates of SGD, with k = 10 and ? = kR . By Proposition 5.2, we know that RNA+SGD will not converge to arbitrary precision because the noise is additive with a non-vanishing variance. However, Proposition 5.3 predicts an improvement of the convergence at the beginning of the process. We illustrate this behavior in Figure 1. We clearly see that at the beginning, the performance of RNA+SGD is comparable to that of the optimal accelerated algorithm. However, because of the restart strategy, in the regime where the level of noise becomes more important the acceleration becomes less effective and finally the convergence stalls, as for SGD. Of course, for practical purposes, the first regime is the most important because it effectively minimizes the generalization error [D?fossez and Bach, 2015; Jain et al., 2016]. 6.2 Finite sums of functions PN We focus on the composite problem minx?Rd F (x) = i=1 N1 fi (x) + ?2 kxk2 , where fi are convex and L-smooth functions and ? is the regularization parameter. We will use classical methods for minimizing F (x) such as SGD (with fixed step size), SAGA [Defazio et al., 2014], SVRG [Johnson and Zhang, 2013], and also the accelerated algorithm Katyusha [Allen-Zhu, 2016]. We will compare their performance with and without the (potential) acceleration provided by Algorithm 1 with restart after k data passes. The parameter ? is found by a grid search of size k, the size of the input sequence, but it adds only one data pass at each extrapolation. Actually, the grid search can be faster if we approximate F (x) with fewer samples, but we choose to present Algorithm 1 in its simplest version. We set k = 10 for all the experiments. In order to balance the complexity of the extrapolation algorithm and the optimization method we wait several data queries before adding the current point (the ?snapshot?) of the method to the sequence. Indeed, the extrapolation algorithm has a complexity of O(k 2 d) + O(N ) (computing the coefficients c?? and the grid search over ?). If we wait at least O(N ) updates, then the extrapolation method is of the same order of complexity as the optimization algorithm. ? SGD. We add the current point after N data queries (i.e. one epoch) and k snapshots of SGD cost kN data queries. 7 PSfrag replacements SGD Ave. SGD Acc. SGD RNA + SGD 10 4 PSfrag replacements SGD Ave. SGD Acc. SGD RNA + SGD f (x) ? f (x? ) PSfrag replacements 10 2 PSfrag replacements 10 0 SGD Ave. SGD Acc. SGD RNA + SGD 10 10 0 ? f (x) ? f (x ) 10 4 10 2 10 0 10 2 10 1 10 2 10 4 10 0 Iteration Iteration 10 2 10 4 Iteration replacements Left: ? = 10, ? = 10?2 . Center: ? PSfrag = 1000, ? = 10?2 . Right: ? = 1000, ? = 10?6 . PSfrag replacements SGD PSfrag replacements SGD Ave. SGD Acc. SGD RNA + SGD 10 2 f (x) ? f (x? ) f (x) ? f (x? ) 10 3 f (x) ? f (x? ) -2 10 0 SGD Ave. SGD Acc. SGD RNA + SGD SGD Ave. SGD Acc. SGD RNA + SGD 10 2 10 Ave. SGD Acc. SGD RNA + SGD 2 10 3 10 2 ? f (x) ? f (x ) 10 0 f (x) ? f (x? ) 10 1 10 0 10 0 10 -2 f (x) ? f (x? ) 10 0 f (x) ? f (x? ) 10 4 Iteration 10 2 10 0 10 2 Iteration 10 4 10 0 10 2 10 4 Iteration Left: ? = 10, ? = 1/d. Center: ? = 100, ? = 1/d. Right: ? = 1000, ? = 1/d. Figure 1: Comparison of performance between SGD, averaged SGD, Accelerated SGD [Flammarion and Bach, 2015] and RNA+SGD. We tested the performance on a matrix AT A of size d = 500, with (top) random eigenvalues between ? and 1 and (bottom) decaying eigenvalues from 1 to 1/d. We start at kx0 ? x? k = 104 , where x0 and x? are generated randomly. ? SAGA. We compute the gradient table exactly, then we add a new point after N queries, and k snapshots of SAGA cost (k + 1)N queries. Since we optimize a sum of quadratic or logistic losses, we used the version of SAGA which stores O(N ) scalars. ? SVRG. We compute the gradient exactly, then perform N queries (the inner-loop of SVRG), and k snapshots of SVRG cost 2kN queries. ? Katyusha. We compute the gradient exactly, then perform 4N gradient calls (the inner-loop of Katyusha), and k snapshots of Katyusha cost 3kN queries. We compare these various methods for solving least-squares regression and logistic regression on several datasets (Table 1), with several condition numbers ?: well (? = 100/N ), moderately (? = 1/N ) and badly (? = 1/100N ) conditioned. In this section, we present the numerical results on Sid (Sido0 dataset, where N = 12678 and d = 4932) with bad conditioning, see Figure 2. The other experiments are highlighted in the supplementary material. In Figure 2, we clearly see that both SGD and RNA+SGD do not converge. This is mainly due to the fact that we do not average the points. In any case, except for quadratic problems, the averaged version of SGD does not converge to the minimum of F with arbitrary precision. We also notice that Algorithm 1 is unable to accelerate Katyusha. This issue was already raised by Scieur et al. [2016]: when the algorithm has a momentum term (like Nesterov?s method), the underlying dynamical system is harder to extrapolate, in particular because the matrix presents in the linearized version of such systems is not symmetric. Because the iterates of SAGA and SVRG have low variance, their accelerated version converges faster to the optimum, and their performance are then comparable to Katyusha. In our experiments, Katyusha was faster than RNA+SAGA only once, when solving a least square problem on Sido0 8 f (x) ? f (x? ) PSfrag replacements PSfrag replacements 10 -5 10 -5 PSfrag replacements f (x) ? f (x? ) Epoch 10 -10 0 f (x) ? f (x? ) 200 10 -10 400 0 50 100 150 Time (sec) Epoch 200 Epoch Time (sec) PSfrag replacements f (x) ? f (x? ) f (x) ? f (x? ) Epoch PSfrag replacements Time (sec) 10 f (x) ? f (x? ) -5 10 -5 Epoch f (x) ? f (x? ) Time (sec) Epoch f (x) ? f (x? ) Time (sec) Epoch 10 -10 0 200 400 10 -10 0 Epoch SAGA SGD SVRG Katyusha RNA+SAGA 100 200 Time (sec) RNA+SGD 300 RNA+SVRG RNA+Kat. Figure 2: Optimization of quadratic loss (Top) and logistic loss (Bottom) with several algorithms, using the Sid dataset with bad conditioning. The experiments are done in Matlab. Left: Error vs epoch number. Right: Error vs time. with a bad condition number. Recall however that the acceleration Algorithm 1 does not require the specification of the strong convexity parameter, unlike Katyusha. Acknowledgments The authors would like to acknowledge support from a starting grant from the European Research Council (ERC project SIPA), from the European Union?s Seventh Framework Programme (FP7PEOPLE-2013-ITN) under grant agreement number 607290 SpaRTaN, as well as support from the chaire ?conomie des nouvelles donn?es with the data science joint research initiative with the fonds AXA pour la recherche and a gift from Soci?t? G?n?rale Cross Asset Quantitative Research. 9 References Allen-Zhu, Z. [2016], ?Katyusha: The first direct acceleration of stochastic gradient methods?, arXiv preprint arXiv:1603.05953 . Anderson, D. G. [1965], ?Iterative procedures for nonlinear integral equations?, Journal of the ACM (JACM) 12(4), 547?560. Cabay, S. and Jackson, L. [1976], ?A polynomial extrapolation method for finding limits and antilimits of vector sequences?, SIAM Journal on Numerical Analysis 13(5), 734?752. Defazio, A., Bach, F. and Lacoste-Julien, S. [2014], Saga: A fast incremental gradient method with support for non-strongly convex composite objectives, in ?Advances in Neural Information Processing Systems?, pp. 1646?1654. D?fossez, A. and Bach, F. [2015], Averaged least-mean-squares: Bias-variance trade-offs and optimal sampling distributions, in ?Artificial Intelligence and Statistics?, pp. 205?213. Fercoq, O. and Qu, Z. [2016], ?Restarting accelerated gradient methods with a rough strong convexity estimate?, arXiv preprint arXiv:1609.07358 . Flammarion, N. and Bach, F. [2015], From averaging to acceleration, there is only a step-size, in ?Conference on Learning Theory?, pp. 658?695. Golub, G. H. and Varga, R. S. [1961], ?Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order richardson iterative methods?, Numerische Mathematik 3(1), 147?156. Jain, P., Kakade, S. M., Kidambi, R., Netrapalli, P. and Sidford, A. [2016], ?Parallelizing stochastic approximation through mini-batching and tail-averaging?, arXiv preprint arXiv:1610.03774 . Johnson, R. and Zhang, T. [2013], Accelerating stochastic gradient descent using predictive variance reduction, in ?Advances in Neural Information Processing Systems?, pp. 315?323. Lin, H., Mairal, J. and Harchaoui, Z. [2015], A universal catalyst for first-order optimization, in ?Advances in Neural Information Processing Systems?, pp. 3384?3392. Me?ina, M. [1977], ?Convergence acceleration for the iterative solution of the equations x= ax+ f?, Computer Methods in Applied Mechanics and Engineering 10(2), 165?173. Moulines, E. and Bach, F. R. [2011], Non-asymptotic analysis of stochastic approximation algorithms for machine learning, in ?Advances in Neural Information Processing Systems?, pp. 451?459. Nedi?c, A. and Bertsekas, D. [2001], Convergence rate of incremental subgradient algorithms, in ?Stochastic optimization: algorithms and applications?, Springer, pp. 223?264. Nesterov, Y. [2013], Introductory lectures on convex optimization: A basic course, Vol. 87, Springer Science & Business Media. Schmidt, M., Le Roux, N. and Bach, F. [2013], ?Minimizing finite sums with the stochastic average gradient?, Mathematical Programming pp. 1?30. Scieur, D., d?Aspremont, A. and Bach, F. [2016], Regularized nonlinear acceleration, in ?Advances In Neural Information Processing Systems?, pp. 712?720. Shalev-Shwartz, S. and Zhang, T. [2013], ?Stochastic dual coordinate ascent methods for regularized loss minimization?, Journal of Machine Learning Research 14(Feb), 567?599. Shalev-Shwartz, S. and Zhang, T. [2014], Accelerated proximal stochastic dual coordinate ascent for regularized loss minimization., in ?ICML?, pp. 64?72. 10
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Optimized Pre-Processing for Discrimination Prevention Flavio P. Calmon Harvard University [email protected] Dennis Wei IBM Research AI [email protected] Karthikeyan Natesan Ramamurthy IBM Research AI [email protected] Bhanukiran Vinzamuri IBM Research AI [email protected] Kush R. Varshney IBM Research AI [email protected] Abstract Non-discrimination is a recognized objective in algorithmic decision making. In this paper, we introduce a novel probabilistic formulation of data pre-processing for reducing discrimination. We propose a convex optimization for learning a data transformation with three goals: controlling discrimination, limiting distortion in individual data samples, and preserving utility. We characterize the impact of limited sample size in accomplishing this objective. Two instances of the proposed optimization are applied to datasets, including one on real-world criminal recidivism. Results show that discrimination can be greatly reduced at a small cost in classification accuracy. 1 Introduction Discrimination is the prejudicial treatment of an individual based on membership in a legally protected group such as a race or gender. Direct discrimination occurs when protected attributes are used explicitly in making decisions, also known as disparate treatment. More pervasive nowadays is indirect discrimination, in which protected attributes are not used but reliance on variables correlated with them leads to significantly different outcomes for different groups. The latter phenomenon is termed disparate impact. Indirect discrimination may be intentional, as in the historical practice of ?redlining? in the U.S. in which home mortgages were denied in zip codes populated primarily by minorities. However, the doctrine of disparate impact applies regardless of actual intent. Supervised learning algorithms, increasingly used for decision making in applications of consequence, may at first be presumed to be fair and devoid of inherent bias, but in fact, inherit any bias or discrimination present in the data on which they are trained [Calders and ?liobait?e, 2013]. Furthermore, simply removing protected variables from the data is not enough since it does nothing to address indirect discrimination and may in fact conceal it. The need for more sophisticated tools has made discrimination discovery and prevention an important research area [Pedreschi et al., 2008]. Algorithmic discrimination prevention involves modifying one or more of the following to ensure that decisions made by supervised learning methods are less biased: (a) the training data, (b) the learning algorithm, and (c) the ensuing decisions themselves. These are respectively classified as pre-processing [Hajian, 2013], in-processing [Fish et al., 2016, Zafar et al., 2016, Kamishima et al., 2011] and post-processing approaches [Hardt et al., 2016]. In this paper, we focus on pre-processing since it is the most flexible in terms of the data science pipeline: it is independent of the modeling algorithm and can be integrated with data release and publishing mechanisms. Researchers have also studied several notions of discrimination and fairness. Disparate impact is addressed by the principles of statistical parity and group fairness [Feldman et al., 2015], which seek similar outcomes for all groups. In contrast, individual fairness [Dwork et al., 2012] mandates that similar individuals be treated similarly irrespective of group membership. For classifiers and other 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. predictive models, equal error rates for different groups are a desirable property [Hardt et al., 2016], as is calibration or lack of predictive bias in the predictions [Zhang and Neill, 2016]. The tension between the last two notions is described by Kleinberg et al. [2017] and Chouldechova [2016]; the work of Friedler et al. [2016] is in a similar vein. Corbett-Davies et al. [2017] discuss the trade-offs in satisfying prevailing notions of algorithmic fairness from a public safety standpoint. Since the present work pertains to pre-processing and not modeling, balanced error rates and predictive bias are less relevant criteria. Instead we focus primarily on achieving group fairness while also accounting for individual fairness through a distortion constraint. Existing pre-processing approaches include sampling or re-weighting the data to neutralize discriminatory effects [Kamiran and Calders, 2012], changing the individual data records [Hajian and Domingo-Ferrer, 2013], and using t-closeness [Li et al., 2007] for discrimination control [Ruggieri, 2014]. A common theme is the importance of balancing discrimination control against utility of the processed data. However, this prior work neither presents general and principled optimization frameworks for trading off these two criteria, nor allows connections to be made to the broader statistical learning and information theory literature via probabilistic descriptions. Another shortcoming is that individual distortion or fairness is not made explicit. In this work, we (i) introduce a probabilistic framework for discrimination-preventing preLearn/Apply Original data Transformed data Learn/Apply predictive ? , Y? )} {(X , Y )} processing in supervised learning, (ii) formu{(D , X Transformation ? D) model (Y? |X, late an optimization problem for producing preUtility: p p Individual distortion: (x , y ) (? x , y? ) processing transformations that trade off dis- Discriminatory Discrimination control: Y? D variable {D } crimination control, data utility, and individual distortion, (iii) characterize theoretical prop- Figure 1: The proposed pipeline for predictive learning erties of the optimization approach (e.g. con- with discrimination prevention. Learn mode applies vexity, robustness to limited samples), and (iv) with training data and apply mode with novel test data. benchmark the ensuing pre-processing transfor- Note that test data also requires transformation before mations on real-word datasets. Our aim in part is predictions can be obtained. to work toward a more unified view of existing pre-processing concepts and methods, which may help to suggest refinements. While discrimination and utility are defined at the level of probability distributions, distortion is controlled on a per-sample basis, thereby limiting the effect of the transformation on individuals and ensuring a degree of individual fairness. Figure 1 illustrates the supervised learning pipeline that includes our proposed discrimination-preventing pre-processing. i i i X,Y i i ? Y? X, i i i i i i i The work of Zemel et al. [2013] is closest to ours in also presenting a framework with three criteria related to discrimination control (group fairness), individual fairness, and utility. However, the criteria are manifested less directly than in our proposal. Discrimination control is posed in terms of intermediate features rather than outcomes, individual distortion does not take outcomes into account (being an `2 -norm between original and transformed features), and utility is specific to a particular classifier. Our formulation more naturally and generally encodes these fairness and utility desiderata. Given the novelty of our formulation, we devote more effort than usual to discussing its motivations and potential variations. We state conditions under which the proposed optimization problem is convex. The optimization assumes as input an estimate of the distribution of the data which, in practice, can be imprecise due to limited sample size. Accordingly, we characterize the possible degradation in discrimination and utility guarantees at test time in terms of the training sample size. To demonstrate our framework, we apply specific instances of it to a prison recidivism dataset [ProPublica, 2017] and the UCI Adult dataset [Lichman, 2013]. We show that discrimination, distortion, and utility loss can be controlled simultaneously with real data. We also show that the preprocessed data reduces discrimination when training standard classifiers, particularly when compared to the original data with and without removing protected variables. In the Supplementary Material (SM), we describe in more detail the resulting transformations and the demographic patterns that they reveal. 2 General Formulation We are given a dataset consisting of n i.i.d. samples {(Di , Xi , Yi )}ni=1 from a joint distribution pD,X,Y with domain D ? X ? Y. Here D denotes one or more protected (discriminatory) variables such as gender and race, X denotes other non-protected variables used for decision making, and Y is an outcome random variable. We use the term ?discriminatory? interchangeably with ?protected,? 2 and not in the usual statistical sense. For instance, Yi could represent a loan approval decision for individual i based on demographic information Di and credit score Xi . We focus in this paper on discrete (or discretized) and finite domains D and X and binary outcomes, i.e. Y = {0, 1}. There is no restriction on the dimensions of D and X. Our goal is to determine a randomized mapping pX, ? Y? |X,Y,D that (i) transforms the given dataset into n ? ? a new dataset {(Di , Xi , Yi )}i=1 which may be used to train a model, and (ii) similarly transforms ? i , Y?i ) is drawn independently from the same data to which the model is applied, i.e. test data. Each (X domain X ? Y as X, Y by applying pX, ? Y? |X,Y,D to the corresponding triplet (Di , Xi , Yi ). Since Di is retained as-is, we do not include it in the mapping to be determined. Motivation for retaining D is discussed later in Section 3. For test samples, Yi is not available at the input while Y?i may not be needed at the output. In this case, a reduced mapping pX|X,D is used as given later in (9). ? It is assumed that pD,X,Y is known along with its marginals and conditionals. This assumption is often satisfied using the empirical distribution of {(Di , Xi , Yi )}ni=1 . In Section 3, we state a result ensuring that discrimination and utility loss continue to be controlled if the distribution used to determine pX, ? Y? |X,Y,D differs from the distribution of test samples. We propose that the mapping pX, ? Y? |X,Y,D satisfy the three following properties. I. Discrimination Control. The first objective is to limit the dependence of the transformed outcome Y? on the protected variables D. We propose two alternative formulations. The first requires the conditional distribution pY? |D to be close to a target distribution pYT for all values of D,   J pY? |D (y|d), pYT (y) ? y,d ? d ? D, y ? {0, 1}, (1) where J(?, ?) denotes some distance function. In the second formulation, we constrain the conditional probability pY? |D to be similar for any two values of D:   J pY? |D (y|d1 ), pY? |D (y|d2 ) ? y,d1 ,d2 ? d1 , d2 ? D, y ? {0, 1}. (2) Note that the number of such constraints is O(|D|2 ) as opposed to O(|D|) constraints in (1). The choice of pYT in (1), and J and  in (1) and (2) should be informed by societal aspects, consultations with domain experts and stakeholders, and legal considerations such as the ?80% rule? [EEOC, 1979]. For this work, we choose J to be the following probability ratio measure: p J(p, q) = ? 1 . q (3) This metric is motivated by the ?80% rule.? The combination of (3) and (1) generalizes the extended lift criterion proposed in the literature [Pedreschi et al., 2012], while the combination of (3) and (2) generalizes selective and contrastive lift. The latter combination (2), (3) is used in the numerical results in Section 4. We note that the selection of a ?fair? target distribution pYT in (1) is not straightforward; see ?liobait?e et al. [2011] for one such proposal. Despite its practical motivation, we alert the reader that (3) may be unnecessarily restrictive when q is low. In (1) and (2), discrimination control is imposed jointly with respect to all protected variables, e.g. all combinations of gender and race if D consists of those two variables. An alternative is to take the protected variables one at a time, and impose univariate discrimination control. In this work, we opt for the more stringent joint discrimination control, although legal formulations tend to be of the univariate type. Formulations (1) and (2) control discrimination at the level of the overall population in the dataset. It is also possible to control discrimination within segments of the population by conditioning on additional variables B, where  B is a subset of X and Xis a collection of features. Constraint (1) would then generalize to J pY? |D,B (y|d, b), pYT |B (y|b) ? y,d,b for all d ? D, y ? {0, 1}, and b ? B. Similar conditioning or ?context? for discrimination has been explored before in Hajian and Domingo-Ferrer [2013] in the setting of association rule mining. For example, B could represent the fraction of a pool of applicants that applied to a certain department, which enables the metric to avoid statistical traps such as the Simpson?s paradox [Pearl, 2014]. One may wish to control for such 3 variables in determining the presence of discrimination, while ensuring that population segments created by conditioning are large enough to derive statistically valid inferences. Moreover, we note that there may exist inaccessible latent variables that drive discrimination, and the metrics used here are inherently limited by the available data. Recent definitions of fairness that seek to mitigate this issue include [Johnson et al., 2016] and [Kusner et al., 2017]. We defer further investigation of causality and conditional discrimination to future work. II. Distortion Control. The mapping pX, ? Y? |X,Y,D should satisfy distortion constraints with respect to the domain X ? Y. These constraints restrict the mapping to reduce or avoid altogether certain large changes (e.g. a very low credit score being mapped to a very high credit score). Given a distortion metric ? : (X ? Y)2 ? R+ , we constrain the conditional expectation of the distortion as, h i ? Y? )) | D = d, X = x, Y = y ? cd,x,y ? (d, x, y) ? D ? X ? Y. E ?((x, y), (X, (4) We assume that ?(x, y, x, y) = 0 for all (x, y) ? X ? Y. Constraint (4) is formulated with pointwise conditioning on (D, X, Y ) = (d, x, y) in order to promote individual fairness. It ensures that distortion is controlled for every combination of (d, x, y), i.e. every individual in the original dataset, and more importantly, every individual to which a model is later applied. By way of contrast, an average-case measure in which an expectation is also taken over D, X, Y may result in high distortion for certain (d, x, y), likely those with low probability. Equation (4) also allows the level of control cd,x,y to depend on (d, x, y) if desired. We also note that (4) is a property of the mapping pX, ? Y? |D,X,Y , and does not depend on the assumed distribution pD,X,Y . ? Y? in (4) encompasses several cases depending on the choices of the metric The expectation over X, ? and thresholds cd,x,y . If cd,x,y = 0, then no mappings with nonzero distortion are allowed for individuals with original values (d, x, y). If cd,x,y > 0, then certain mappings may still be disallowed by assigning them infinite distortion. Mappings with finite distortion are permissible subject to the budget cd,x,y . Lastly, if ? is binary-valued (perhaps achieved by thresholding a multi-valued distortion function), it can be seen as classifying mappings into desirable (? = 0) and undesirable ones (? = 1). Here, (4) reduces to a bound on the conditional probability of an undesirable mapping, i.e.,   ? Y? )) = 1 | D = d, X = x, Y = y ? cd,x,y . Pr ?((x, y), (X, (5) III. Utility Preservation. In addition to constraints on individual distortions, we also require that ? Y? ) be statistically close to the distribution of (X, Y ). This is to ensure that a the distribution of (X, model learned from the transformed dataset (when averaged over the protected variables D) is not too different from one learned from the original dataset, e.g. a bank?s existing policy for approving loans. For a givendissimilaritymeasure ? between probability distributions (e.g. KL-divergence), we require that ? pX, be small. ? Y? , pX,Y Optimization Formulation. Putting together the considerations from the three previous subsections, we arrive at the optimization problem below for determining a randomized transformation pX, ? Y? |X,Y,D ? ? mapping each sample (Di , Xi , Yi ) to (Xi , Yi ):   min ? pX, ? Y? , pX,Y pX, ? Y ? |X,Y,D   s.t. J pY? |D (y|d), pYT (y) ? y,d and h i ? Y? )) | D = d, X = x, Y = y ? cd,x,y ? (d, x, y) ? D ? X ? Y, E ?((x, y), (X, pX, ? Y? |X,Y,D is a valid distribution. (6) We choose to minimize the utility loss ? subject to constraints on individual distortion (4) and discrimination (we use (1) for concreteness, but (2) can be used instead), since it is more natural to place bounds on the latter two. The distortion constraints (4) are an essential component of the problem formulation (6). Without (4) and assuming that pYT = pY , it is possible to achieve perfect utility and non-discrimination ? i , Y?i ) from the original distribution pX,Y independently of any inputs, i.e. simply by sampling (X 4 pX, x, y?|x, y, d) = pX, x, y?) = pX,Y (? x, y?). Then ?(pX, ? Y? |X,Y,D (? ? Y? (? ? Y? , pX,Y ) = 0, and pY? |D (y|d) = pY? (y) = pY (y) = pYT (y) for all d ? D. Clearly, this solution is objectionable from the viewpoint of individual fairness, especially for individuals to whom a subsequent model is applied since it amounts to discarding an individual?s data and replacing it with a random sample from the population pX,Y . Constraint (4) seeks to prevent such gross deviations from occurring. The distortion constraints may, however, render the optimization infeasible, as illustrated in the SM. 3 Theoretical Properties I. Convexity. We show conditions under which (6) is a convex or quasiconvex optimization problem, and can thus be solved to optimality. The proof is presented in the SM. Proposition 1. Problem (6) is a (quasi)convex optimization if ?(?, ?) is (quasi)convex and J(?, ?) is quasiconvex in their respective first arguments (with the second arguments fixed). If discrimination constraint (2) is used in place of (1), then the condition on J is that it be jointly quasiconvex in both arguments. II. Generalizability of Discrimination Control. We now discuss the generalizability of discrimination guarantees (1) and (2) to unseen individuals, i.e. those to whom a model is applied. Recall from Section 2 that the proposed transformation retains the protected variables D. We first consider the case where models trained on the transformed data to predict Y? are allowed to depend on D. While such models may qualify as disparate treatment, the intent and effect is to better mitigate disparate impact resulting from the model. In this respect our proposal shares the same spirit with ?fair? affirmative action in Dwork et al. [2012] (fairer on account of distortion constraint (4)). Assuming that predictive models for Y? can depend on D, let Ye be the output of such a model ? To remove the separate issue of model accuracy, suppose for simplicity that the based on D and X. model provides a good approximation to the conditional distribution of Y? , i.e. pYe |X,D (e y |? x, d) ? ? p ? ? (e y |? x, d). Then for individuals in a protected group D = d, the conditional distribution of Ye Y |X,D is given by pYe |D (e y |d) = X x ? pYe |X,D (e y |? x, d)pX|D (? x|d) ? ? ? X x ? pY? |X,D (e y |? x, d)pX|D (? x|d) = pY? |D (e y |d). (7) ? ? Hence the model output pYe |D can also be controlled by (1) or (2). On the other hand, if D must be suppressed from the transformed data, perhaps to comply with legal ? and approximate requirements regarding its non-use, then a predictive model can depend only on X pY? |X? , i.e. pYe |X,D (e y |? x, d) = pYe |X? (e y |? x) ? pY? |X? (e y |? x). In this case we have ? pYe |D (e y |d) ? X x ? (8) pY? |X? (e y |? x)pX|D (? x|d), ? which in general is not equal to pY? |D (e y |d) in (7). The quantity on the right-hand side of (8) is less straightforward to control. We address this question in the SM. III. Training and Application Considerations. The proposed optimization framework has two modes of operation (Fig. 1): train and apply. In train mode, the optimization problem (6) is solved in order to determine a mapping pX, ? Y? |X,Y,D for randomizing the training set. The randomized training ? D) that approximates p ? ? , where ? are the set, in turn, is used to fit a classification model f? (X, Y |X,D parameters of the model. At apply time, a new data point (X, D) is received and transformed into ? D) through a randomized mapping p ? (X, is given by marginalizing ? X|X,D . The mapping pX|D,X ? over Y, Y : X pX|D,X (? x|d, x) = pX, x, y?|x, y, d)pY |X,D (y|x, d). (9) ? ? Y? |X,Y,D (? y,? y Assuming that the variable D is not suppressed, and that the marginals are known, then the utility and discrimination guarantees set during train time still hold during apply time, as discussed above. 5 However, the distortion control will inevitably change, since the mapping has been marginalized over Y . More specifically, the bound on the expected distortion for each sample becomes h h i i X ? Y? )) | D = d, X = x, Y | D = d, X = x ? E E ?((x, Y ), (X, pY |X,D (y|x, d)cx,y,d , cx,d . y?Y (10) If the distortion control values cx,y,d are independent of y, then the upper-bound on distortion set during training time still holds during apply time. Otherwise, (10) provides a bound on individual distortion at apply time. The same guarantee holds for the case when D is suppressed. IV. Robustness to Mismatched Prior Distribution Estimation. We may also consider the case where the distribution pD,X,Y used to determine the transformation differs from the distribution qD,X,Y of test samples. This occurs, for example, when pD,X,Y is the empirical distribution computed from n i.i.d. samples from an unknown distribution qD,X,Y . In this situation, discrimination control and utility are still guaranteed for samples drawn from qD,X,Y that are transformed using pY? ,X|X,Y,D , ? where the latter is obtained by solving (6) with pD,X,Y . In particular, denoting by qY? |D and qX, ? Y? ? and D when qD,X,Y is transformed using p ? ? the corresponding distributions for Y? , X , we Y       ,X|X,Y,D  have J pY? |D (y|d), pYT (y) ? J qY? |D (y|d), pYT (y) and ? pX,Y , pX, ? Y? ? ? qX,Y , qX, ? Y? for n sufficiently large (the distortion control constraints (4) only depend on pY? ,X|X,Y,D ). The next ? proposition provides an estimate of the rate of this convergence in terms of n and assuming pY,D (y, d) is fixed and bounded away from zero. Its proof can be found in the SM. Proposition 2. Let pD,X,Y be the empirical distribution obtained from n i.i.d. samples that is used to determine the mapping pY? ,X|X,Y,D , and qD,X,Y be the true distribution of the data, with support size ? m , |X ? Y ? D|. In addition, denote by qD,X, to ? Y? the joint distribution after applying pY? ,X|X,Y,D ?   samples from qD,X,Y . If for all y ? Y, d ? D we have pY,D (y, d) > 0, J pY? |D (y|d), pYT (y) ? , where J is given in (3), and   X ? pX,Y , pX, pX,Y (x, y) ? pX, ? Y? = ? Y? (x, y) ? ?, x,y with probability 1 ? ?,  n     o rm n  log ? log 1 + . (11) ? max J qY? |D (y|d), pYT (y) ? , ? qX,Y , qX, ? Y? ? ? . n m n Proposition 2 guarantees that, as long as n is sufficiently large, the utility and discrimination control guarantees will approximately hold when pX, ? Y? |Y,X,D is applied to fresh samples drawn from qD,X,Y . In particular, the utility and discrimination guarantees will converge to the ones used as parameters in q the optimization at a rate that is at least n1 log n. The distortion control guarantees (4) are a property of the mapping pX, ? Y? |Y,X,D , and do not depend on the distribution of the data. The convergence rate is tied to the support size, and for large m a dimensionality reduction step may be required to assuage generalization issues. The same upper bound on convergence rate holds for discrimination constraints of the form (2). 4 Experimental Results This section provides a numerical demonstration of running the data processing pipeline in Fig. 1. Our focus here is on the discrimination-accuracy trade-off obtained when the pre-processed data is used to train standard prediction algorithms. The SM presents additional results on the trade-off between discrimination control  and utility ? as well as an analysis of the optimized data transformations. We apply the pipeline to ProPublica?s COMPAS recidivism data [ProPublica, 2017] and the UCI Adult dataset [Lichman, 2013]. From the COMPAS dataset (7214 instances), we select severity of charge, number of prior crimes, and age category to be the decision variables (X). The outcome variable (Y ) is a binary indicator of whether the individual recidivated (re-offended), and race is set to be the protected variable (D). The encoding of categorical variables is described in the SM. For the Adult dataset (32561 instances), the features were categorized as protected variables (D): 6 gender (male, female); decision variables (X): age (quantized to decades) and education (quantized to years); and response variable (Y ): income (binary). Our proposed approach is benchmarked against two baselines, leaving the dataset as-is and suppressing the protected variable D during training and testing. We also compare against the learning fair representations (LFR) algorithm from Zemel et al. [2013]. As discussed in the introduction, LFR has fundamental differences from the proposed framework. In particular, LFR only considers binary-valued D, and consequently, we restrict D to be binary in the experiments presented here. However, our method is not restricted to D being binary or univariate. Illustrations of our method on non-binary D are provided in the SM. The details of applying our method to the datasets are as follows. For each train/test split, we approximate pD,X,Y using the empirical distribution of (D, X, Y ) in the training set and solve (6) using a standard convex solver [Diamond datasets the utility metric  and Boyd,2016].PFor both 1 ? is the total variation distance, i.e. ? pX,Y , pX, = ? Y? (x, y) , the ? Y? x,y pX,Y (x, y) ? pX, 2 distortion constraint is the combination of (2) and (3), and two levels of discrimination control are used,  = {0.05, 0.1}. The distortion function ? is chosen differently for the two datasets as described below, based on the differing semantics of the variables in the two applications. The specific values were chosen for demonstration purposes to be reasonable to our judgment and can easily be tuned according to the desires of a practitioner. We emphasize that the distortion values were not selected to optimize the results presented here. All experiments run in minutes on a standard laptop. Distortion function for COMPAS: We use the expected distortion constraint in (4) with cd,x,y = 0.4, 0.3 for d being respectively African-American and Caucasian. The distortion function ? has the following behavior. Jumps of more than one category in age and prior counts are heavily discouraged by a high distortion penalty (104 ) for such transformations. We impose the same penalty on increases in recidivism (change of Y from 0 to 1). Both these choices are made in the interest of individual fairness. Furthermore, for every jump to an adjacent category for age and prior counts, a penalty of 1 is assessed, and a similar jump in charge degree incurs a penalty of 2. Reduction in recidivism (1 to 0) has a penalty of 2. The total distortion for each individual is the sum of squares of distortions for each attribute of X. Distortion function for Adult: We use three conditional probability constraints of the form in (5). In constraint i, the distortion function returns 1 in case (i) and 0 otherwise: (1) if income is decreased, age is not changed and education is increased by at most 1 year, (2) if age is changed by a decade and education is increased by at most 1 year regardless of the change of income, (3) if age is changed by more than a decade or education is lowered by any amount or increased by more than 1 year. The corresponding probability bounds cd,x,y are 0.1, 0.05, 0 (no dependence on d, x, y). As a consequence, and in the same broad spirit as for COMPAS, decreases in income, small changes in age, and small increases in education (events (1), (2)) are permitted with small probabilities, while larger changes in age and education (event (3)) are not allowed at all. Once the optimized randomized mapping pX, ? Y? |D,X,Y is determined, we apply it to the training set to obtain a new perturbed training set, which is then used to fit two classifiers: logistic regression (LR) and random forest (RF). For the test set, we first compute the test-time mapping pX|D,X in (9) using ? pX, ? Y? |D,X,Y and pD,X,Y estimated from the training set. We then independently randomize each pX|D,X ? test sample (di , xi ) using pX|D,X , preserving the protected variable D, i.e. (di , xi ) ?????? (di , x ?i ). ? Each trained classifier f is applied to the transformed test samples, obtaining an estimate yei = f (di , x ?i ) which is evaluated P against yi . These estimates induce an empirical posterior distribution given by pYe |D (1|d) = n1d {?xi ,di }:di =d f (di , x ?i ), where nd is the number of samples with di = d. For the two baselines, the above procedure is repeated without data transformation except for dropping D throughout for the second baseline (D is still used to compute the discrimination of the resulting classifier). Due to the lack of available code, we implemented LFR ourselves in Python and solved the associated optimization problem using the SciPy package. The parameters for LFR were set as recommended in Zemel et al. [2013]: Az = 50 (group fairness), Ax = 0.01 (individual fairness), and Ay = 1 (prediction accuracy). The results did not significantly change within a reasonable variation of these three parameters. 7 0.710 0.705 0.70 0.69 0.700 AUC AUC 0.71 LR LR?+?Dropping?D LFR LR?+?Our?approach(.05) LR?+?Our?approach(.1) 0.715 0.695 0.690 0.67 0.685 0.66 0.680 0.675 RF RF?+?Dropping?D LFR RF?+?Our?approach(.05) RF?+?Our?approach(.1) 0.65 0.00 0.82 0.05 0.10 0.15 0.20 0.25 Discrimination 0.30 0.35 0.40 0.0 0.82 LR LR?+?Dropping?D LFR LR?+?Our?approach(.05) LR?+?Our?approach(.1) 0.81 0.80 0.1 0.2 Discrimination 0.3 0.4 RF RF?+?Dropping?D LFR RF?+?Our?approach(.05) RF?+?Our?approach(.1) 0.81 0.80 0.79 AUC AUC 0.68 0.79 0.78 0.77 0.78 0.76 0.77 0.75 0.76 0.00 0.25 0.50 0.75 1.00 Discrimination 1.25 1.50 0.74 1.75 0.00 0.25 0.50 0.75 1.00 Discrimination 1.25 1.50 1.75 Figure 2: Discrimination-AUC plots for two different classifiers. Top row is for COMPAS dataset, and bottom row for UCI Adult dataset. First column is logistic regression (LR), and second column is random forests (RF). Results. We report the trade-off between two metrics: (i) the empirical discrimination of the classifier on the test set, given by maxd,d0 ?D J(pYe |D (1|d), pYe |D (1|d0 )), and (ii) the empirical accuracy, measured by the Area under ROC (AUC) of yei = f (di , x ?i ) compared to yi , using 5-fold cross validation. Fig. 2 presents the operating points achieved by each procedure in the discrimination-accuracy space as measured by these metrics. For the COMPAS dataset, there is significant discrimination in the original dataset, which is reflected by both LR and RF when the data is not transformed. Dropping the D variable reduces discrimination with a negligible impact on classification. However discrimination is far from removed since the features X are correlated with D, i.e. there is indirect discrimination. LFR with the recommended parameters is successful in further reducing discrimination while still achieving high prediction performance for the task. Our proposed optimized pre-processing approach successfully decreases the empirical discrimination close to the target  values (x-axis). Deviations are expected due to the approximation of Y? , the output of the transformation, by Ye , the output of each classifier, and also due to the randomized nature of the method. The decreased discrimination comes at an accuracy cost, which is greater in this case than for LFR. A possible explanation is that LFR is free to search across different representations whereas our method is restricted by the chosen distortion metric and having to preserve the domain of the original variables. For example, for COMPAS we heavily penalize increases in recidivism from 0 to 1 as well as large changes in prior counts and age. When combined with the other constraints in the optimization, this may alter the joint distribution after perturbation and by extension the classifier output. Increased accuracy could be obtained by relaxing the distortion constraint, as long as this is acceptable to the practitioner. We highlight again that our distortion metric was not chosen to explicitly optimize performance on this task, and should be guided by the practitioner. Nevertheless, we do successfully obtain a controlled reduction of discrimination while avoiding unwanted deviations in the randomized mapping. For the Adult dataset, dropping the protected variable does significantly reduce discrimination, in contrast with COMPAS. Our method further reduces discrimination towards the target  values. The loss of prediction performance is again due to satisfying the distortion and discrimination constraints. On the other hand, LFR with the recommended parameters provides only a small reduction in discrimination. We note that this does not contradict the results in Zemel et al. [2013], since here we have adopted a multiplicative discrimination metric (3) whereas Zemel et al. [2013] used an additive metric. Moreover, we reduced the Adult dataset to 31 binary features which is different from Zemel et al. [2013] where they additionally considered the test dataset for Adult (12661 instances) also and created 103 binary features. By varying the LFR parameters, it is possible to attain low empirical discrimination but with a large loss in prediction performance (below the plotted range). Thus, we do not claim that our method outperforms LFR since different operating points can be achieved by 8 adjusting parameters in either approach. In our approach however, individual fairness is explicitly maintained through the design of the distortion metric and discrimination is controlled directly by a single parameter , whereas the relationship is less clear with LFR. 5 Conclusions We proposed a flexible, data-driven optimization framework for probabilistically transforming data in order to reduce algorithmic discrimination, and applied it to two datasets. When used to train standard classifiers, the transformed dataset led to a fairer classification when compared to the original dataset. The reduction in discrimination comes at an accuracy penalty due to the restrictions imposed on the randomized mapping. Moreover, our method is competitive with others in the literature, with the added benefit of enabling an explicit control of individual fairness and the possibility of multivariate, non-binary protected variables. 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YASS: Yet Another Spike Sorter JinHyung Lee1 , David Carlson2 , Hooshmand Shokri1 , Weichi Yao1 , Georges Goetz3 , Espen Hagen4 , Eleanor Batty1 , EJ Chichilnisky3 , Gaute Einevoll5 , and Liam Paninski1 1 Columbia University, 2 Duke University, 3 Stanford University, 4 University of Oslo, 5 Norwegian University of Life Sciences Abstract Spike sorting is a critical first step in extracting neural signals from large-scale electrophysiological data. This manuscript describes an efficient, reliable pipeline for spike sorting on dense multi-electrode arrays (MEAs), where neural signals appear across many electrodes and spike sorting currently represents a major computational bottleneck. We present several new techniques that make dense MEA spike sorting more robust and scalable. Our pipeline is based on an efficient multistage ?triage-then-cluster-then-pursuit? approach that initially extracts only clean, high-quality waveforms from the electrophysiological time series by temporarily skipping noisy or ?collided? events (representing two neurons firing synchronously). This is accomplished by developing a neural network detection method followed by efficient outlier triaging. The clean waveforms are then used to infer the set of neural spike waveform templates through nonparametric Bayesian clustering. Our clustering approach adapts a ?coreset? approach for data reduction and uses efficient inference methods in a Dirichlet process mixture model framework to dramatically improve the scalability and reliability of the entire pipeline. The ?triaged? waveforms are then finally recovered with matching-pursuit deconvolution techniques. The proposed methods improve on the state-of-the-art in terms of accuracy and stability on both real and biophysically-realistic simulated MEA data. Furthermore, the proposed pipeline is efficient, learning templates and clustering faster than real-time for a ' 500-electrode dataset, largely on a single CPU core. 1 Introduction The analysis of large-scale multineuronal spike train data is crucial for current and future neuroscience research. These analyses are predicated on the existence of reliable and reproducible methods that feasibly scale to the increasing rate of data acquisition. A standard approach for collecting these data is to use dense multi-electrode array (MEA) recordings followed by ?spike sorting? algorithms to turn the obtained raw electrical signals into spike trains. A crucial consideration going forward is the ability to scale to massive datasets: MEAs currently scale up to the order of 104 electrodes, but efforts are underway to increase this number to 106 electrodes1 . At this scale any manual processing of the obtained data is infeasible. Therefore, automatic spike sorting for dense MEAs has enjoyed significant recent attention [15, 9, 51, 24, 36, 20, 33, 12]. Despite these efforts, spike sorting remains the major computational bottleneck in the scientific pipeline when using dense MEAs, due both to the high computational cost of the algorithms and the human time spent on manual postprocessing. To accelerate progress on this critical scientific problem, our proposed methodology is guided by several main principles. First, robustness is critical, since hand-tuning and post-processing is not 1 DARPA Neural Engineering System Design program BAA-16-09 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Algorithm 1 Pseudocode for the complete proposed pipeline. Input: time-series of electrophysiological data V 2 RT ?C , locations 2 R3 [waveforms, timestamps] Detection(V) % (Section 2.2) % ?Triage? noisy waveforms and collisions (Section 2.4): [cleanWaveforms, cleanTimestamps] Triage(waveforms, timestamps) % Build a set of representative waveforms and summary statistics (Section 2.5) [representativeWaveforms, sufficientStatistics] coresetConstruction(cleanWaveforms) % DP-GMM clustering via divide-and-conquer (Sections 2.6 and 2.7) [{representativeWaveformsi , sufficientStatisticsi }i=1,... ] splitIntoSpatialGroups(representativeWaveforms, sufficientStatistics, locations) for i=1,. . . do % Run efficient inference for the DP-GMM [clusterAssignmentsi ] SplitMergeDPMM(representativeWaveformsi , sufficientStatisticsi ) end for % Merge spatial neighborhoods and similar templates [allClusterAssignments, templates] mergeTemplates({clusterAssignmentsi }i=1,... , {representativeWaveformsi }i=1,... , locations) % Pursuit stage to recover collision and noisy waveforms [finalTimestamps, finalClusterAssignments] deconvolution(templates) return [finalTimestamps, finalClusterAssignments] feasible at scale. Second, scalability is key. To feasibly process the oncoming data deluge, we use efficient data summarizations wherever possible and focus computational power on the ?hard cases,? using cheap fast methods to handle easy cases. Next, the pipeline should be modular. Each stage in the pipeline has many potential feasible solutions, and the pipeline is improved by rapidly iterating and updating each stage as methodology develops further. Finally, prior information is leveraged as much as possible; we share information across neurons, electrodes, and experiments in order to extract information from the MEA datastream as efficiently as possible. We will first outline the methodology that forms the core of our pipeline in Section 2.1, and then we demonstrate the improvements in performance on simulated data and a 512-electrode recording in Section 3. Further supporting results appear in the appendix. 2 2.1 Methods Overview The inputs to the pipeline are the band-pass filtered voltage recordings from all C electrodes and their spatial layout, and the end result of the pipeline is the set of K (where K is determined by the algorithm) binary neural spike trains, where a ?1? in the time series reflects a neural action potential from the kth neuron at the corresponding time point. The voltage signals are spatially whitened prior to processing and are modeled as the superposition of action potentials and background Gaussian noise [12]. Spatial whitening is performed by removing potential spikes using amplitude thresholding and estimating the whitening filter under a Gaussianity assumption. Succinctly, the pipeline is a multistage procedure as follows: (i) detecting waveforms and extracting features, (ii) screening outliers and collided waveforms, (iii) clustering, and (iv) inferring missed and collided spikes. Pseudocode for the flow of the pipeline can be found in Algorithm 1. A brief overview is below, followed by additional details. Our overall strategy can be considered a hybrid of a matching pursuit approach (similar to that employed by [36]) and a classical clustering approach, generalized and adapted to the large dense MEA setting. Our guiding philosophy is that it is essential to properly handle ?collisions? between simultaneous spikes [37, 12], since collisions distort the extracted feature space and hinder clustering. A typical approach to this issue utilizes matching pursuit methods (or other sparse deconvolution strategies), but these methods are relatively computationally expensive compared to clustering primitives. This led us to a ?triage-then-cluster-then-pursuit? approach: we ?triage? collided or overly noisy waveforms, putting them aside during the feature extraction and clustering stages, and later recover these spikes during a final ?pursuit? or deconvolution stage. The triaging begins during the detection stage in Section 2.2, where we develop a neural network based detection method that 2 significantly improves sensitivity and selectivity. For example, on a simulated 30 electrode dataset with low SNR, the new approach reduces false positives and collisions by 90% for the same rate of true positives. Furthermore, the neural network is significantly better at the alignment of signals, which improves the feature space and signal-to-noise power. The detected waveforms then are projected to a feature space and restricted to a local spatial subset of electrodes as in [24] in Section 2.3. Next, in Section 2.4 an outlier detection method further ?triages? the detected waveforms and reduces false positives and collisions by an additional 70% while removing only a small percentage of real detections. All of these steps are achievable in nearly linear time. Simulations demonstrate that this large reduction in false positives and collisions dramatically improves accuracy and stability. Following the above steps, the remaining waveforms are partitioned into distinct neurons via clustering. Our clustering framework is based on the Dirichlet Process Gaussian Mixture Model (DP-GMM) approach [48, 9], and we modify existing inference techniques to improve scalability and performance. Succinctly, each neuron is represented by a distinct Gaussian distribution in the feature space. Directly calculating the clustering on all of the channels and all of the waveforms is computationally infeasible. Instead, the inference first utilizes the spatial locality via masking [24] from Section 2.3. Second, the inference procedure operates on a coreset of representative points [13] and the resulting approximate sufficient statistics are used in lieu of the full dataset (Section 2.5). Remarkably, we can reduce a dataset with 100k points to a coreset of ' 10k points with trivial accuracy loss. Next, split and merge methods are adapted to efficiently explore the clustering space [21, 24] in Section 2.6. Using these modern scalable inference techniques is crucial for robustness because they empirically find much more sensible and accurate optima and permit Bayesian characterization of posterior uncertainty. For very large arrays, instead of operating on all channels simultaneously, each distinct spatial neighborhood is processed by a separate clustering algorithm that may be run in parallel. This parallelization is crucial for processing very large arrays because it allows greater utilization of computer resources (or multiple machines). It also helps improve the efficacy of the split-merge inference by limiting the search space. This divide-and-conquer approach and the post-processing to stitch the results together is discussed in Section 2.7. The computational time required for the clustering algorithm scales nearly linearly with the number of electrodes C and the experiment time. After the clustering stage is completed, the means of clusters are used as templates and collided and missed spikes are inferred using the deconvolution (or ?pursuit? [37]) algorithm from Kilosort [36], which recovers the final set of binary spike trains. We limit this computationally expensive approach only to sections of the data that are not well handled by the rest of the pipeline, and use the faster clustering approach to fill in the well-explained (i.e. easy) sections. We note finally that when memory is limited compared to the size of the dataset, the preprocessing, spike detection, and final deconvolution steps are performed on temporal minibatches of data; the other stages operate on significantly reduced data representations, so memory management issues typically do not arise here. See Section B.4 for further details on memory management. 2.2 Detection The detection stage extracts temporal and spatial windows around action potentials from the noisy raw electrophysiological signal V to use as inputs in the following clustering stage. The number of clean waveform detections (true positives) should be maximized for a given level of detected collision and noise events (false positives). Because collisions corrupt feature spaces [37, 12] and will simply be recovered during pursuit stage, they are not included as true positives at this stage. In contrast to the plethora of prior work on hand-designed detection rules (detailed in Section C.1), we use a data-driven approach with neural networks to dramatically improve both detection efficacy and alignment quality. The neural network is trained to return only clean waveforms on real data, not collisions, so it de facto performs a preliminary triage prior to the main triage stage in Section 2.4. The crux of the data-driven approach is the availability of prior training data. We are targeting the typical case that an experimental lab performs repeated experiments using the same recording setup from day to day. In this setting hand-curated or otherwise validated prior sorts are saved, resulting in abundant training data for a given experimental preparation. In the supplemental material, we discuss the construction of a training set (including data augmentation approaches) in Section C.2, the architecture and training of the network in Section C.3, the detection using the network in Section C.4, empirical performance in Section C.5, and scalability in Section C.5. This strategy is effective when 3 this training data exists; however, many research groups are currently using single electrodes and do not have dense MEA training data. Thus it is worth emphasizing that here we train the detector only on a single electrode. We have also experimented with training and evaluating on multiple electrodes with good success; however, these results are more specialized and are not shown here. A key result is that our neural network dramatically improves both the temporal and spatial alignment of detected waveforms. This improved alignment improves the fidelity of the feature space and the signal-to-noise power, and the result of the improved feature space can clearly be seen by comparing the detected waveform features from one standard detection approach (SpikeDetekt [24]) in Figure 1 (left) to the detected waveform features from our neural network in Figure 1 (middle). Note that the output of the neural net detection is remarkably more Gaussian compared to SpikeDetekt. 2.3 Feature Extraction and Mask Creation Following detection we have a collection of N events defined as Xn 2 RR?C for n = 1, . . . , N , each with an associated detection time tn . Recall that C is the total number of electrodes, and R is the number of time samples, in our case chosen to correspond to 1.5ms. Next features are extracted by using uncentered Principal Components Analysis (PCA) on each channel separately with P features per channel. Each waveform Xn is transformed to the feature space Yn . To handle duplicate spikes, Yn is kept only if cn = arg max{||ync ||c2Ncn }, where Ncn contains all electrodes in the local neighborhood of electrode cn . To address the increasing dimensionality, spikes are localized by using the sparse masking vector {mn } 2 [0, 1]C method of [24], where the mask should be set to 1 only where the signal exists. The sparse vector reduces the dimensionality and facilitates sparse updates to improve computational efficiency. We give additional mathematical details in Supplemental Section D. We have also experimented with an autoencoder framework to standardize the feature extraction across channels and facilitate online inference. This approach performed similarly to PCA and is not shown here, but will be addressed in depth in future work. 2.4 Collision Screening and Outlier Triaging Many collisions and outliers remain even after our improved detection algorithm. Because these events destabilize the clustering algorithms, the pipeline benefits from a ?triage? stage to further screen collisions and noise events. Note that triaging out a small fraction of true positives is a minor concern at this stage because they will be recovered in the final deconvolution step. We use a two-fold approach to perform this triaging. First, obvious collisions with nearly overlapping spike times and spatial locations are removed. Second, k-Nearest Neighbors (k-NN) is used to detect outliers in the masked feature space based on [27]. To develop a computationally efficient and effective approach, waveforms are grouped based on their primary (highest-energy) channel, and then k-NN is run for each channel. Empirically, these approximations do not suffer in efficacy compared to using the full spatial area. When the dimensionality of P , the number of features per channel, is low, a kd-tree can find neighbors in O(N log N ) average time. We demonstrate that this method is effective for triaging false positives and collisions in Figure 1 (middle). 2.5 Coreset Construction ?Big data? improves density estimates for clustering, but the cost per iteration naively scales with the amount of data. However, often data has some redundant features, and we can take advantage of these redundancies to create more efficient summarizations of the data. Then running the clustering algorithm on the summarized data should scale only with the number of summary points. By choosing representative points (or a ?coreset") carefully we can potentially describe huge datasets accurately with a relatively small number of points [19, 13, 2]. There is a sizable literature on the construction of coresets for clustering problems; however, the number of required representative points to satisfy the theoretical guarantees is infeasible in this problem domain. Instead, we propose a simple approach to build coresets that empirically outperforms existing approaches in our experiments by forcing adequate coverage of the complete dataset. We demonstrate in Supplemental Figure S6 that this approach can cover clusters completely missed by existing approaches, and show the chosen representative points on data in Figure 1 (right). This algorithm is based on recursively performing k-means; we provide pseudocode and additional details 4 NN-triaged NN-kept coreset PC 2 SpikeDetekt PC 1 Figure 1: Illustration of Neural Network Detection, Triage, and Coreset from a primate retinal ganglion cell recording. The first column shows spike waveforms from SpikeDetekt in their PCA space. Due to poor alignment, clusters have a non-Gaussian shape with many outliers. The second column shows spike waveforms from our proposed neural network detection in the PCA space. After triaging outliers, the clusters have cleaner Gaussian shapes in the recomputed feature space. The last column illustrates the coreset. The size of each coreset diamond represents its weight. For visibility, only 10% of data are plotted. in in Supplemental Section E. The worst case time complexity is nearly linear with respect to the number of representative points, the number of detected spikes, and the number of channels. The algorithm ends by returning G representative points, their sufficient statistics, and masks. 2.6 Efficient Inference for the Dirichlet Process Gaussian Mixture Model For the clustering step we use a Dirichlet Process Gaussian Mixture Model (DP-GMM) formulation, which has been previously used in spike sorting [48, 9], to adaptively choose the number of mixture components (visible neurons). In contrast to these prior approaches, here we adopt a Variational Bayesian split-merge approach to explore the clustering space [21] and to find a more robust and higher-likelihood optimum. We address the high computational cost of this approach with several key innovations. First, following [24], we fit a mixture model on the virtual masked data to exploit the localized nature of the data. Second, following [9, 24], the covariance structure is approximated as a block-diagonal to reduce the parameter space and computation. Finally, we adapted the methodology to work with the representative points (coreset) rather than the raw data, resulting in a highly scalable algorithm. A more complete description of this stage can be found in Supplemental Section F, with pseudocode in Supplemental Algorithm S2. In terms of computational costs, the dominant cost per iteration in the DPMM algorithm is the calculation of data to cluster assignments, which in our framework will scale at O(GmP ? 2 K), where m ? is the average number of channels maintained in the mask for each of the representative points, G is the number of representative points, and P is the number of features per channel. This is in stark contrast to a scaling of O(N C 2 P 2 K + P 3 ) without our above modifications. Both K and G are expected to scale linearly with the number of electrodes and sublinearly with the length of the recording. Without further modification, the time complexity in the above clustering algorithm would depend on the square of the number of electrodes for each iteration; fortunately, this can be reduced to a linear dependency based on a divide-and-conquer approach discussed below in Section 2.7. 5 % of x(%) Stable Clusters 80 60 40 20 0 100 90 80 70 60 Stability % Threshold 50 Accuracy (High Collision ViSAPy) # of x(%) Accurate Clusters % of x(%) Stable Clusters # of x(%) Accurate Clusters Stability (High Collision ViSAPy) 15 10 5 0 100 YASS KiloSort Mountain SpyKING 90 80 70 60 True Positive % Threshold 50 Stability (Low SNR ViSAPy) 60 40 YASS Kilosort Mountain SpyKing 20 0 100 90 80 70 60 Stability % Threshold 50 Accuracy (Low SNR ViSAPy) 15 10 5 0 100 90 80 70 60 True Positive % Threshold 50 Figure 2: Simulation results on 30-channel ViSAPy datasets. Left panels show the result on ViSAPy with high collision rate and Right panels show the result on ViSAPy with low SNR setting. (Top) stability metric (following [5]) and percentage of total discovered clusters above a certain stability measure. The noticeable gap between stability of YASS and the other methods results from a combination of high number of stable clusters and lower number of total clusters. (Bottom) These results show the number of clusters (out of a ground truth of 16 units) above a varying quality threshold for each pipeline. For each level of accuracy, the number of clusters that pass that threshold is calculated to demonstrate the relative quality of the competing algorithms on this dataset. Empirically, our pipeline (YASS) outperforms other methods. 2.7 Divide and Conquer and Template Merging Neural action potentials have a finite spatial extent [6]. Therefore, the spikes can be divided into distinct groups based on the geometry of the MEA and the local position of each neuron, and each group is then processed independently. Thus, each group can be processed in parallel, allowing for high data throughput. This is crucial for exploiting parallel computer resources and limited memory structures. Second, the split-and-merge approach in a DP-GMM is greatly hindered when the numbers of clusters is very high [21]. The proposed divide and conquer approach addresses this problem by greatly reducing the number of clusters within each subproblem, allowing the split and merge algorithm to be targeted and effective. To divide the data based on the spatial location of each spike, the primary channel cn is determined for every point in the coreset based on the channel with maximum energy, and clustering is applied on each channel. Because neurons may now end up on multiple channels, it is necessary to merge templates from nearby channels as a post-clustering step. When the clustering is completed, the mean of each cluster is taken as a template. Because waveforms are limited to their primary channel, some neurons may have ?overclustered? and have a distinct mixture component on distinct channels. Also, overclustering can occur from model mismatch (non-Gaussianity). Therefore, it is necessary to merge waveforms. Template merging is performed based on two criteria, the angle and the amplitude of templates, using the best alignment on all temporal shifts between two templates. The pseudocode to perform this merging is shown in Supplemental Algorithm S3. Additional details can be found in Supplemental Section G. 6 40 YASS Kilosort Mountain SpyKing 20 0 100 90 80 70 60 Stability % Threshold 50 # of x(%) Accurate Clusters % of x(%) Stable Clusters Stability 60 Accuracy 30 20 10 0 100 90 80 70 60 True Positive % Threshold 50 Figure 3: Performance comparison of spike sorting pipelines on primate retina data. (Left) The same type of plot as in the top panels of Figure 2. (Right) The same type of plot as in the bottom panels of Figure 2 compared to the ?gold standard? sort. YASS demonstrates both improved stability and also per-cluster accuracy. 2.8 Recovering Triaged Waveforms and Collisions After the previous steps, we apply matching pursuit [36] to recover triaged waveforms and collisions. We detail the available choices for this stage in Supplemental Section I. 3 Performance Comparison We evaluate performance to compare several algorithms (detailed in Section 3.1) to our proposed methodology on both synthetic (Section 3.2) and real (Section 3.3) dense MEA recordings. For each synthetic dataset we evaluate the ability to capture ground truth in addition to the per-cluster stability metrics. For the ground truth, inferred clusters are matched with ground truth clusters via the Hungarian algorithm, and then the per-cluster accuracy is calculated as the number of assignments shared between the inferred cluster and the ground truth cluster over the total number of waveforms in the inferred cluster. For the per-cluster stability metric, we use the method from Section 3.3 of [5] with the rate scaling parameter of the Poisson processes set to 0.25. This method evaluates how robust individual clusters are to perturbations of the dataset. In addition, we provide runtime information to empirically evaluate the computational scaling of each approach. The CPU runtime was calculated on a single core of a six-core i7 machine with 32GB of RAM. GPU runtime is given from a Nvidia Titan X within the same machine. 3.1 Competing Algorithms We compare our proposed pipeline to three recently proposed approaches for dense MEA spike sorting: KiloSort [36], Spyking Circus [51], and MountainSort [31]. Kilosort, Spyking Cricus, and MountainSort were downloaded on January 30, 2017, May 26th, 2017, and June 7th, 2017, respectively. We dub our algorithm Yet Another Spike Sorter (YASS). We discuss additional details on the relationships between these approaches and our pipeline in Supplemental Section I. All results are shown with no manual post-processing. 3.2 Synthetic Datasets First, we used the biophysics-based spike activity generator ViSAPy [18] to generate multiple 30channel datasets with different noise levels and collision rates. The detection network was trained on the ground truth from a low signal-to-noise level recording. Then, the trained neural network is applied to all signal-to-noise levels. The neural network dramatically outperforms existing detection methodologies on these datasets. For a given level of true positives, the number of false positives can be reduced by an order of magnitude. The properties of the learned network are shown in Supplemental Figures S4 and S5. Performance is evaluated on the known ground truth. For each level of accuracy, the number of clusters that pass that threshold is calculated to demonstrate the relative quality of the competing 7 Detection (GPU) Data Ext. Triage Coreset Clustering Template Ext. Total 1m7s 42s 11s 34s 3m12s 54s 6m40s Table 1: Running times of the main processes on 512-channel primate retinal recording of 30 minutes duration. Results shown using a single CPU core, except for the detection step (2.2), which was run on GPU. We found that full accuracy was achieved after processing just one-fifth of this dataset, leading to significant speed gains. Data Extraction refers to waveform extraction and Performing PCA (2.3). Triage, Coreset, and Clustering refer to 2.4, 2.5, and 2.6, respectively. Template Extraction describes revisiting the recording to estimate templates and merging them (2.7). Each step scales approximately linearly (Section B.3). algorithms on this dataset. Empirically, our pipeline (YASS) outperforms other methods. This is especially true in low SNR settings, as shown in Figure 2. The per-cluster stability metric is also shown in Figure 2. The stability result demonstrates that YASS has significantly fewer low-quality clusters than competing methods. 3.3 Real Datasets To examine real data, we focused on 30 minutes of extracellular recordings of the peripheral primate retina, obtained ex-vivo using a high-density 512-channel recording array [30]. The half-hour recording was taken while the retina was stimulated with spatiotemporal white noise. A ?gold standard" sort was constructed for this dataset by extensive hand validation of automated techniques, as detailed in Supplemental Section H. Nonstationarity effects (time-evolution of waveform shapes) were found to be minimal in this recording (data not shown). We evaluate the performance of YASS and competing algorithms using 4 distinct sets of 49 spatially contiguous electrodes. Note that the gold standard sort here uses the information from the full 512-electrode array, while we examine the more difficult problem of sorting the 49-electrode data; we have less information about the cells near the edges of this 49-electrode subset, allowing us to quantify the performance of the algorithms over a range of effective SNR levels. By comparing the inferred results to the gold standard, the cluster-specific true positives are determined in addition to the stability metric. The results are shown in Figure 3 for one of the four sets of electrodes, and the remaining three sets are shown in Supplemental Section B.1. As in the simulated data, compared to KiloSort, which had the second-best overall performance on this dataset, YASS has dramatically fewer low-stability clusters. Finally, we evaluate the time required for each step in the YASS pipeline (Table 1). Importantly, we found that YASS is highly robust to data limitations: as shown in Supplemental Figure S3 and Section B.3, using only a fraction of the 30 minute dataset has only a minor impact on performance. We exploit this to speed up the pipeline. Remarkably, running primarily on a single CPU core (only the detect step utilizes a GPU here), YASS achieves a several-fold speedup in template and cluster estimation compared to the next fastest competitor2 , Kilosort, which was run in full GPU mode and spent about 30 minutes on this dataset. We plan to further parallelize and GPU-ize the remaining steps in our pipeline next, and expect to achieve significant further speedups. 4 Conclusion YASS has demonstrated state-of-the-art performance in accuracy, stability, and computational efficiency; we believe the tools presented here will have a major practical and scientific impact in large-scale neuroscience. In our future work, we plan to continue iteratively updating our modular pipeline to better handle template drift, refractory violations, and dense collisions. Lastly, YASS is available online at https://github.com/paninski-lab/yass 2 Spyking Circus took over a day to process this dataset. Assuming linear scaling based on smaller-scale experiments, Mountainsort is expected to take approximately 10 hours. 8 Acknowledgements This work was partially supported by NSF grants IIS-1546296 and IIS-1430239, and DARPA Contract No. N66001-17-C-4002. References [1] D. Arthur and S. Vassilvitskii. k-means++: The advantages of careful seeding. In ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2007. [2] O. Bachem, M. Lucic, and A. Krause. Coresets for nonparametric estimation-the case of dp-means. In ICML, 2015. [3] B. Bahmani, B. Moseley, A. Vattani, R. Kumar, and S. Vassilvitskii. Scalable k-means++. Proceedings of the VLDB Endowment, 2012. [4] I. N. Bankman, K. O. Johnson, and W. Schneider. Optimal detection, classification, and superposition resolution in neural waveform recordings. IEEE Trans. Biomed. Eng. 1993. [5] A. H. Barnett, J. F. Magland, and L. F. Greengard. Validation of neural spike sorting algorithms without ground-truth information. J. Neuro. Methods, 2016. [6] G. Buzs?ki. Large-scale recording of neuronal ensembles. Nature neuroscience, 2004. [7] T. Campbell, J. Straub, J. W. F. III, and J. P. How. Streaming, Distributed Variational Inference for Bayesian Nonparametrics. In NIPS, 2015. [8] D. Carlson, V. 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Fast bayesian inference in dirichlet process mixture models. J. Comp. and Graphical Stat., 2011. [47] A. B. Wiltschko, G. J. Gage, and J. D. Berke. Wavelet filtering before spike detection preserves waveform shape and enhances single-unit discrimination. J. Neuro. Methods, 2008. 10 [48] F. Wood and M. J. Black. A nonparametric bayesian alternative to spike sorting. J. Neuro. Methods, 2008. [49] F. Wood, M. J. Black, C. Vargas-Irwin, M. Fellows, and J. P. Donoghue. On the variability of manual spike sorting. IEEE TBME 2004. [50] X. Yang and S. A. Shamma. A totally automated system for the detection and classification of neural spikes. IEEE Trans. Biomed. Eng. 1988. [51] P. Yger, G. L. Spampinato, E. Esposito, B. Lefebvre, S. Deny, C. Gardella, M. Stimberg, F. Jetter, G. Zeck, S. Picaud, et al. Fast and accurate spike sorting in vitro and in vivo for up to thousands of electrodes. bioRxiv, 2016. [52] L. Zelnik-Manor and P. Perona. Self-tuning spectral clustering. 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A Practice Strategy for Robot Learning Control Terence D. Sanger Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology, room E25-534 Cambridge, MA 02139 [email protected] Abstract "Trajectory Extension Learning" is a new technique for Learning Control in Robots which assumes that there exists some parameter of the desired trajectory that can be smoothly varied from a region of easy solvability of the dynamics to a region of desired behavior which may have more difficult dynamics. By gradually varying the parameter, practice movements remain near the desired path while a Neural Network learns to approximate the inverse dynamics. For example, the average speed of motion might be varied, and the inverse dynamics can be "bootstrapped" from slow movements with simpler dynamics to fast movements. This provides an example of the more general concept of a "Practice Strategy" in which a sequence of intermediate tasks is used to simplify learning a complex task. I show an example of the application of this idea to a real 2-joint direct drive robot arm. 1 INTRODUCTION The most general definition of Adaptive Control is one which includes any controller whose behavior changes in response to the controlled system's behavior. In practice, this definition is usually restricted to modifying a small number of controller parameters in order to maintain system stability or global asymptotic stability of the errors during execution of a single trajectory (Sastry and Bodson 1989, for review). Learning Control represents a second level of operation, since it uses Adaptive Con335 336 Sanger trol to modify parameters during repeated performance trials of a desired trajectory so that future trials result in greater accuracy (Arimoto et al. 1984). In this paper I present a third level called a "Practice Strategy", in which Learning Control is applied to a sequence of intermediate trajectories leading ultimately to the true desired trajectory. I claim that this can significantly increase learning speed and make learning possible for systems which would otherwise become unstable. 1.1 LEARNING CONTROL During repeated practice of a single desired trajectory, the actual trajectory followed by the robot may be significantly different. Many Learning Control algorithms modify the commands stored in a sequence memory to minimize this difference (Atkeson 1989, for review). However, the performance errors are usually measured in a sensory coordinate system, while command corrections must be made in the motor coordinate system. If the relationship between these two coordinate systems is not known, then command corrections might be in the wrong direction and inadvertently worsen performance. However, if the practice trajectory is close to the desired trajectory, then the errors will be small and the relationship between command and sensory errors can be approximated by the system Jacobian. An alternative to a stored command sequence is to use a Neural Network to learn an approximation to the inverse dynamics in the region of interest (Sanner and Slotine 1992, Yabuta and Yamada 1991, Atkeson 1989). In this case, the commands and results from the actual movement are used as training data for the network, and smoothness properties are assumed such that the error on the desired trajectory will decrease. However, a significant problem with this method is that if the actual practice trajectory is far from the desired trajectory, then its inverse dynamics information will be of little use in training the inverse dynamics for the desired trajectory. In fact, the network may achieve perfect approximation on the actual trajectory while still making significant errors on the desired trajectory. In this case, learning will stop (since the training error is zero) leading to the phenomenon of "learning lock-up" (An et al. 1988). So whether Learning Control uses a sequence memory or a Neural Network, learning may proceed poorly if large errors are made during the initial practice movements. 1.2 PRACTICE STRATEGIES I define a "practice strategy" as a sequence of trajectories such that the first element in the sequence is any previously learned trajectory, and the last element in the sequence is the ultimate desired trajectory. A well designed practice strategy will result in a seqence for which learning control of the trajectory for any particular step is simplified if prior steps have already been learned. This will occur if learning of prior trajectories reduces the initial performance error for subsequent trajectories, so that a network will be less likely to experience learning lock-up. One example of a practice strategy is a three-step sequence in which the intermediate step is a set of independently executable subtasks which partition the desired trajectory into discrete pieces. Another example is a multi-step sequence in which intermediate steps are a set of trajectories which are somehow related to the desired trajectory. In this paper I present a multi-step sequence which gradually A Practice Strategy for Robot Learning Control ---~-------, I I " A u N P A y " a. Figure 1: Training signals for network learning. transforms some known trajectory into the desired trajectory by varying a single parameter. This method has the advantage of not requiring detailed knowledge of the task structure in order to break it up into meaningful subtasks, and conditions for convergence can be stated explicitly. It has a close relationship to Continuation Methods for solving differential equations, and can be considered to be a particular application of the Banach Extension Theorem. 2 METHODS As in (Sanger 1992), we need to specify 4 aspects of the use of a neural network within a control system: 1. the networks' function in the control system, 2. the network learning algorithm which modifies the connection weights, 3. the training signals used for network learning, and 4. the practice strategy used to generate sample movements. The network's function is to learn the inverse dynamics of an equilibrium-point controlled plant (Shadmehr 1990). The LMS-tree learning algorithm trains the network (Sanger 1991b, Sanger 1991a). The training signals are determined from the actual practice data using either "Actual Trajectory Training" or "Desired Trajectory Training", as defined below. And the practice strategy is "Trajectory Extension Learning", in which a parameter of the movement is gradually modified during training. 337 338 Sanger 2.1 TRAINING SIGNALS Figure 1 shows the general structure of the network and training signals. A desired trajectory y is fed into the network N to yield an estimated command U. This command is then applied to the plant Pcx where the subscript indicates that the plant is parameterized by the variable a. Although the true command u which achieves y is unknown, we do know that the estimated command u produces y, so these signals are used for training by comparing the network response to y given by ~ = Ny to the known value and subtracting these to yield the training error 6,. u Normally, network training would use this error signal to modify the network output for inputs near y, and I refer to this as "Actual Trajectory Training". However, if y is far from y then no change in response may occur at y and this may lead even more quickly to learning lock-up. Therefore an alternative is to use the error 6fJ to train the network output for inputs near y. I refer to this as "Desired Trajectory Training", and in the figure it is represented by the dotted arrow. The following discussion will summarize the convergence conditions and theorems presented in (Sanger 1992). Define Ru . (1 - N P(x))u = u - U to be an operator which maps commands into command errors for states x on the desired trajectory. Similarly, let Ru = (1 - = u- ~ map commands into command errors for states x on the actual trajectory. N P( x))u Convergence depends upon the following assumptions: A1: The plant P is smooth and invertible with respect to both the state x and the input u with Lipschitz constants k'z; and ku, and it has stable zero-dynamics. A2: The network N is smooth with Lipschitz constant kN. A3: Network learning reduces the error in response to a pair (y, 6y ). A4: The change in network output in response to training is smooth with Lipschitz constant kL. A5: There exists a smoothly controllable parameter a such that an inverse dynamics solution is available at a = ao, and the desired performance occurs when a = ad. A6: The change in command required to produce a desired output after any change in a is bounded by the change in a multiplied by a constant kcx ? A 7: The change in plant response for any fixed input is bounded by the change in a multiplied by a constant kp ? Under assumptions A1-A3 we can prove convergence of Desired Trajectory Training: Theorem 1: If there exists a k Rn such that II R nu - Rnull < kRn lI u - ull A Practice Strategy for Robot Learning Control then if the learning rate 0 < 'Y :::; 1, If k Rn < 1 and 'Y :::; 1, then the network output u approaches the correct command u. Under assumptions A1-A4, we can prove convergence of Actual Trajectory Training: Theorem 2: If there exists a kRn such that IIRn u - Rnull < kRn lIu - illl then if the learning rate 0 < 'Y :::; 1, 2.2 TRAJECTORY EXTENSION LEARNING Let a be some modifiable parameter of the plant such that for a = ao there exists a simple inverse dynamics solution, and we seek a solution when a = ad. For example, if the plant uses Equilibrium Point Control (Shadmehr 1990), then at low speeds the inverse dynamics behave like a perfect servo controller yielding desired trajectories without the need to solve the dynamics. We can continue to train a learning controller as the average speed of movement (a) is gradually increased. The inverse dynamics learned at one speed provide an approximation to the inverse dynamics for a slightly faster speed, and thus the performance errors remain small during practice. This leads to significantly faster learning rates and greater likelihood that the conditions for convergence at any given speed will be satisfied. Note that unlike traditional learning schemes, the error does not decrease monotonically with practice, but instead maintains a steady magnitude as the speed increases, until the network is no longer able to approximate the inverse dynamics. The following is a summary of a result from (Sanger 1992). Let a change from al to a2, and let P = Pal and P' = Pa2 . Then under assumptions AI-A7 we can prove convergence of Trajectory Extension Learning: Theorem 3: If there exists a kR such that for a = then for a al = a2 IIR'u' - R'illl < kRllu' - ull + (2k a + kNkp)la2 - all This shows that given the smoothness assumptions and a small enough change in a, the error will continue to decrease. 339 340 Sanger 3 EXAMPLE Figure 2 shows the result of 15 learning trials performed by a real direct-drive twojoint robot arm on a sampled desired trajectory. The initial trial required 11.5 seconds to execute, and the speed was gradually increased until the final trial required only 4.5 seconds. Simulated equilibrium point control was used (Bizzi et al. 1984) with stiffness and damping coefficients of 15 nm/rad and 1.5 nm/rad/sec, respectively. The grey line in figure 2 shows the equilibrium point control signal which generated the actual movement represented by the solid line. The difference between these two indicates the nontrivial nature of the dynamics calculations required to derive the control signal from the desired trajectory. Note that without Trajectory Extension Learning, the network does not converge and the arm becomes unstable. The neural network was an LMS tree (Sanger 1991b, Sanger 1991a) with 10 Gaussian basis functions for each of the 6 input dimensions, and a total of 15 subtrees were grown per joint (see (Sanger 1992) for further explanation). 4 CONCLUSION Trajectory Extension Learning is one example of the way in which a practice strategy can be used to improve convergence for Learning Control. This or other types of practice strategies might be able to increase the performance of many different types of learning algorithms both within and outside the Control domain. Such strategies may also provide a theoretical model for the practice strategies used by humans to learn complex tasks, and the theoretical analysis and convergence conditions could potentially lead to a deeper understanding of human motor learning and successful techniques for optimizing performance. Acknowledgements Thanks are due to Simon Giszter, Reza Shadmehr, Sandro Mussa-Ivaldi, Emilio Bizzi, and many people at the NIPS conference for their comments and criticisms. This report describes research done within the laboratory of Dr. Emilio Bizzi in the department of Brain and Cognitive Sciences at MIT. The author was supported during this work by a National Defense Science and Engineering Graduate Fellowship, and by NIH grants 5R37 AR26710 and 5ROINS09343 to Dr. Bizzi. References An C. H., Atkeson C. G., Hollerbach J. M., 1988, Model-Based Control of a Robot Manipulator, MIT Press, Cambridge, MA. Arimoto S., Kawamura S., Miyazaki F., 1984, Bettering operation of robots by learning, Journal of Robotic Systems, 1(2):123-140. Atkeson C. G., 1989, Learning arm kinematics and dynamics, Ann. Rev. Neurosci., 12:157-183. Bizzi E., Accornero N., Chapple W., Hogan N., 1984, Posture control and trajectory formation during arm movement, J. Neurosci, 4:2738-2744. Sanger T. D., 1991a, A tree-structured adaptive network for function approximation in high dimensional spaces, IEEE Trans. Neural Networks, 2(2):285-293. A Practice Strategy for Robot Learning Control Sanger T. D., 1991b, A tree-structured algorithm for reducing computation in networks with separable basis functions, Neural Computation, 3(1):67-78. Sanger T. D., 1992, Neural network learning control of robot manipulators using gradually increasing task difficulty, submitted to IEEE Trans. Robotics and Automation. Sanner R. M., Slotine J.-J. E., 1992, Gaussian networks for direct adaptive control, IEEE Trans. Neural Networks, in press. Also MIT NSL Report 910303, 910503, March 1991 and Proc. American Control Conference, Boston pages 2153-2159, June 1991. Sastry S., Bodson M., 1989, Adaptive Control: Stability, Convergence, and Robustness, Prentice Hall, New Jersey. Shadmehr R., 1990, Learning virtual equilibrium trajectories for control of a robot arm, Neural Computation, 2:436-446. Yabuta T., Yamada T., 1991, Learning control using neural networks, Proc. IEEE Int'l ConJ. on Robotics and Automation, Sacramento, pages 740-745. Figure 2: Dotted line is the desired trajectory, solid line is the actual trajectory, and the grey line is the equilibrium point control trajectory. 341
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Independence clustering (without a matrix) Daniil Ryabko INRIA Lillle, 40 avenue de Halley, Villeneuve d?Ascq, France [email protected] Abstract The independence clustering problem is considered in the following formulation: given a set S of random variables, it is required to find the finest partitioning {U1 , . . . , Uk } of S into clusters such that the clusters U1 , . . . , Uk are mutually independent. Since mutual independence is the target, pairwise similarity measurements are of no use, and thus traditional clustering algorithms are inapplicable. The distribution of the random variables in S is, in general, unknown, but a sample is available. Thus, the problem is cast in terms of time series. Two forms of sampling are considered: i.i.d. and stationary time series, with the main emphasis being on the latter, more general, case. A consistent, computationally tractable algorithm for each of the settings is proposed, and a number of fascinating open directions for further research are outlined. 1 Introduction Many applications face the situation where a set S = {x1 , . . . , xN } of samples has to be divided into clusters in such a way that inside each cluster the samples are dependent, but the clusters between themselves are as independent as possible. Here each xi may itself be a sample or a time series xi = X1i , . . . , Xni . For example, in financial applications, xi can be a series of recordings of prices of a stock i over time. The goal is to find the segments of the market such that different segments evolve independently, but within each segment the prices are mutually informative [15, 17]. In biological applications, each xi may be a DNA sequence, or may represent gene expression data [28, 20], or, in other applications, an fMRI record [4, 13]. The staple approach to this problem in applications is to construct a matrix of (pairwise) correlations between the elements, and use traditional clustering methods, e.g., linkage-based methods or k means and its variants, with this matrix [15, 17, 16]. If mutual information is used, it is used as a (pairwise) proximity measure between individual inputs, e.g. [14]. We remark that pairwise independence is but a surrogate for (mutual) independence, and, in addition, correlation is a surrogate for pairwise independence. There is, however, no need to resort to surrogates unless forced to do so by statistical or computational hardness results. We therefore propose to reformulate the problem from the first principles, and then show that it is indeed solvable both statistically and computationally ? but calls for completely different algorithms. The formulation proposed is as follows. Given a set S = (x1 , . . . , xN ) of random variables, it is required to find the finest partitioning {U1 , . . . , Uk } of S into clusters such that the clusters U1 , . . . , Uk are mutually independent. To our knowledge, this problem in its full generality has not been addressed before. A similar informal formulation appears in the work [1] that is devoted to optimizing a generalization of the ICA objective. However, the actual problem considered only concerns the case of tree-structured dependence, which allows for a solution based on pairwise measurements of mutual information. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Note that in the fully general case pairwise measurements are useless, as are, furthermore, bottom-up (e.g., linkage-based) approaches. Thus, in particular, a proximity matrix cannot be used for the analysis. Indeed, it is easy to construct examples in which any pair or any small group of elements are independent, but are dependent when the same group is considered jointly with more elements. For instance, consider a group of Bernoulli 1/2-distributed random variables x1 , . . . , xN +1 , where PN x1 , . . . , xN are i.i.d. and xN +1 = i=1 xi mod 2. Note that any N out of these N + 1 random variables are i.i.d., but together the N + 1 are dependent. Add then two more groups like this, say, y1 , . . . , yN +1 and z1 , . . . , zN +1 that have the exact same distribution, with the groups of x, y and z mutually independent. Naturally, these are the three clusters we would want to recover. However, if we try to cluster the union of the three, then any algorithm based on pairwise correlations will return an essentially arbitrary result. What is more, if we try to find clusters that are pairwise independent, then, for example, the clustering {(xi , yi , zi )i=1..N } of the input set into N + 1 clusters appears correct, but, in fact, the resulting clusters are dependent. Of course, real-world data does not come in the form of summed up Bernoulli variables, but this simple example shows that considering independence of small subsets may be very misleading. The considered problem is split into two parts considered separately: the computational and the statistical part. This is done by first considering the problem assuming the joint distribution of all the random variables is known, and is accessible via an oracle. Thus, the problem becomes computational. A simple, computationally efficient algorithm is proposed for this case. We then proceed to the time-series formulations: the distribution of (x1 , . . . , xN ) is unknown, but a sample (X11 , . . . , X1N ), . . . , (Xn1 , . . . , XnN ) is provided, so that xi can be identified with the time series X1i , . . . , Xni . The sample may be either independent and identically distributed (i.i.d.), or, in a more general formulation, stationary. As one might expect, relying on the existing statistical machinery, the case of known distributions can be directly extended to the case of i.i.d. samples. Thus, we show that it is possible to replace the oracle access with statistical tests and estimators, and then use the same algorithm as in the case of known distributions. The general case of stationary samples turns out to be much more difficult, in particular because of a number of strong impossibility results. In fact, it is challenging already to determine what is possible and what is not from the statistical point of view. In this case, it is not possible to replicate the oracle access to the distribution, but only its weak version that we call fickle oracle. We find that, in this case, it is only possible to have a consistent algorithm for the case of known k. An algorithm that has this property is constructed. This algorithm is computationally feasible when the number of clusters k is small, as its complexity is O(N 2k ). Besides, a measure of information divergence is proposed for time-series distributions that may be of independent interest, since it can be estimated consistently without any assumptions at all on the distributions or their densities (the latter may not exist). The main results of this work are theoretical. The goal is to determine, as a first step, what is possible and what is not from both statistical and computational points of view. The main emphasis is placed on highly dependent time series, as warranted by the applications cited above, leaving experimental investigations for future work. The contribution can be summarized as follows: ? a consistent, computationally feasible algorithm for known distributions, unknown number of clusters, and an extension to the case of unknown distributions and i.i.d. samples; ? an algorithm that is consistent under stationary ergodic sampling with arbitrary, unknown distributions, but with a known number k of clusters; ? an impossibility result for clustering stationary ergodic samples with k unknown; ? an information divergence measure for stationary ergodic time-series distributions along with its estimator that is consistent without any extra assumptions; In addition, an array of open problems and exciting directions for future work is proposed. Related work. Apart from the work on independence clustering mentioned above, it is worth pointing out the relation to some other problems. First, the proposed problem formulation can be viewed as a Bayesian-network learning problem: given an unknown network, it is required to split it into independent clusters. In general, learning a Bayesian network is NP-hard [5], even for rather restricted classes of networks (e.g., [18]). Here the problem we consider is much less general, which is why it admits a polynomial-time solution. A related clustering problem, proposed in [23] (see also [12]) is clustering time series with respect to distribution. Here, it is required to put two time series samples x1 , x2 into the same cluster if and only if their distribution is the same. Similar to the independence clustering introduced here, this problem admits a consistent algorithm if the samples are i.i.d. (or 2 mixing) and the number of distributions (clusters) is unknown, and in the case of stationary ergodic samples if and only if k is known. 2 Set-up and preliminaries A set S := {x1 , . . . , xN } is given, where we will either assume that the joint distribution of xi is known, or else that the distribution is unknown but a sample (X11 , . . . , Xn1 ), . . . , (X1N , . . . , XnN ) is i given. In the latter case, we identify each xi with the sequence (sample) X1i , . . . , Xni , or X1..n for short, of length n. The lengths of the samples are the same only for the sake of notational convenience; it is easy to generalize all algorithms to the case of different sample lengths ni , but the asymptotic would then be with respect to n := mini=1..N ni . It is assumed that Xji ? X := R are real-valued, but extensions to more general cases are straightforward. For random variables A, B, C we write (A ? B)|C to say that A is conditionally independent of B given C, and A ? B ? C to say that A, B and C are mutually independent. The (unique up to a permutation) partitioning U := {U1 , . . . , Uk } of the set S is called the groundtruth clustering if U1 , . . . , Uk are mutually independent (U1 ? ? ? ? ? Uk ) and no refinement of U has this property. A clustering algorithm is consistent if it outputs the ground-truth clustering, and it is asymptotically consistent if w.p. 1 it outputs the ground-truth clustering from some n on. P For a discrete A-valued r.v. X its Shannon entropy is defined as H(X) := a?A ?P (X = a) log P (X = a), lettingR 0 log 0 = 0. For a distribution with a density f its (differential) entropy is defined as H(X) =: ? f (x) log f (x). For two random variables X, Y their mutual information I(X, Y ) is defined as I(X, Y ) = H(X) + H(Y ) ? H(X, Y ). For discrete random variables, as well as for continuous ones with aP density, X ? Y if and only if I(X, Y ) = 0; see, e.g., [6]. Likewise, I(X1 , . . . , Xm ) is defined as i=1..m H(Xi ) ? H(X1 , . . . , Xm ). For the sake of convenience, in the next two sections we make the assumption stated below. However, we will show (Sections 5,6) that this assumption can be gotten rid of as well. Assumption 1. All distributions in question have densities bounded away from zero on their support. 3 Known distributions As with any statistical problem, it is instructive to start with the case where the (joint) distribution of all the random variables in question is known. Finding out what can be done and how to do it in this case helps us to set the goals for the (more realistic) case of unknown distributions. Thus, in this section, x1 , . . . , xN are not time series, but random variables whose joint distribution is known to the statistician. The access to this distribution is via an oracle; specifically, our oracle will provide answers to the following questions about mutual information (where, for convenience, we assume that the mutual information with the empty set is 0): Oracle TEST. Given sets of random variables A, B, C, D ? {x1 , . . . , xN } answer whether I(A, B) > I(C, D). Remark 1 ( Conditional independence oracle). Equivalently, one can consider an oracle that answers conditional independence queries of the form (A ? B)|C. The definition above is chosen for the sake of continuity with the next section, and it also makes the algorithm below more intuitive. However, in order to test conditional independence statistically one does not have to use mutual information, but may resort to any other divergence measure instead. The proposed algorithm (see the pseudocode listing below) works as follows. It attempts to split the input set recursively into two independent clusters, until it is no longer possible. To split a set in two, it starts with putting one element x from the input set S into a candidate cluster C := {x}, and measures its mutual information I(C, R) with the rest of the set, R := S \ C. If I(C, R) is already 0 then we have split the set into two independent clusters and can stop. Otherwise, the algorithm then takes the elements out of R one by one without replacement and each time looks whether I(C, R) has decreased. Once such an element is found, it is moved from R to C and the process is restarted from the beginning with C thus updated. Note that, if we have started with I(C, R) > 0, then taking elements out of R without replacement we eventually should find a one that decreases I(C, R), since I(C, ?) = 0 and I(C, R) cannot increase in the process. 3 Theorem 1. The algorithm CLIN outputs the correct clustering using at most 2kN 2 oracle calls. Proof. We shall first show that the procedure for splitting a set into two indeed splits the input set into two independent sets, if and only if such two sets exist. First, note that if I(C, S \ C) = 0 then C ? R and the function terminates. In the opposite case, when I(C, S \ C) > 0, by removing an element from R := S \ C, I(C, R) can only decrease (indeed, h(C|R) ? h(C|R \ {x}) by information processing inequality). Eventually when all elements are removed, I(C, R) = I(C, {}) = 0, so there must be an element x removing which decreases I(C, R). When such an element x found it is moved to C. Note that, in this case, indeed x? \C. However, it is possible that removing an element x from R does not reduce I(C, R), yet x? \C. This is why the while loop is needed, that is, the whole process has to be repeated until no elements can be moved to C. By the end of each for loop, we have either found at least one element to move to C, or we have assured that C ? S \ C and the loop terminates. Since there are only finitely many elements in S \ C, the while loop eventually terminates. Moreover, each of the two loops (while and for) terminates in at most N iterations. Finally, notice that if (C1 , C2 ) ? C3 and C1 ? C2 then also C1 ? C2 ? C3 , which means that by repeating the Split function recursively we find the correct clustering. From the above, the bound on the number of oracle calls is easily obtained by direct calculation. 4 I.I.D. sampling In this section we assume that the distribution of (x1 , . . . , xN ) is not known, but an i.i.d. sample Figure 1: CLIN: cluster with k unknown, (X11 , . . . , X1N ), . . . , (Xn1 , . . . , XnN ) is provided. We ideni given an oracle for MI tify xi with the (i.i.d.) time series X1..n . Formally, N X INPUT: The set S. valued processes is just a single X N -valued process. The (C1 , C2 ) := Split(S) latter can be seen as a matrix (Xji )i=1..N,j=1..? , where i if C2 6= ? then each row i is the sample xi = X1..n.. and each column j Output:CLIN (C1 ), CLIN (C2 ) is what is observed at time j: Xj1 ..XjN . else The case of i.i.d. samples is not much different from the Output: C1 case of a known distribution. What we need is to replace end if the oracle test with (nonparametric) statistical tests. First, Function Split(Set S of samples) a test for independence is needed to replace the oracle call Initialize: C := {x1 }, R := S \ C; TEST(I(C, R) > 0) in the while loop. Such tests are while TEST(I(C; R) > 0) do indeed available, see, for example, [8]. Second, we need for each x ? R do if TEST(I(C; R)>I(C; R \ {x})) an estimator of mutual information I(X, Y ), or, which is sufficient, for entropies, but with a rate of convergence. then If the rate of convergence is known to be asymptotically move x from R to C bounded by, say, t(n), then, in order to construct an asympbreak the for loop totically consistent test, we can take any t0 (n) ? 0 such else that t(n) = o(t0 (n)) and decide our inequality as folmove x from R to M ? ? lows: if I(C; R \ {x}) < I(C; R) ? t0 (n) then say that end if I(C; R \ {x}) < I(C; R). The required rates of converend for ? gence, which are of order n under Assumption 1, can be M := {}, R := S \ C; found in [3]. end while Return(C,R) Given the result of the previous section, it is clear that if END function the oracle is replaced by the tests described, then CLIN is a.s. consistent. Thus, we have demonstrated the following. Theorem 2. Under Assumption 1, there is an asymptotically consistent algorithm for independence clustering with i.i.d. sampling. Remark 2 (Necessity of the assumption). The independence test of [8] does not need Assumption 1, as it is distribution-free. Since the mutual information is defined in terms of densities, if we want to completely get rid of Assumption 1, we would need to use some other measure of dependence for the test. One such measure is defined in the next section already for the general case of process distributions. However, the rates of convergence of its empirical estimates under i.i.d. sampling remain to be studied. 4 Remark 3 (Estimators vs. tests). As noted in Remark 1, the tests we are using are, in fact, tests for (conditional) independence: testing I(C; R) > I(C; R \ {x}) is testing for (C ? {x}|R \ {x}). Conditional independence can be tested directly, without estimating I (see, for example 27), potentially allowing for tighter performance guarantees under more general conditions. 5 Stationary sampling We now enter the hard mode. The general case of stationary sampling presents numerous obstacles, often in the form of theoretical impossibility results: there are (provably) no rates of convergence, no independence test, and 0 mutual information rate does not guarantee independence. Besides, some simple-looking questions regarding the existence of consistent tests, which indeed have simple answers in the i.i.d. case, remain open in the stationary ergodic case. Despite all this, a computationally feasible asymptotically consistent independence clustering algorithm can be obtained, although only for the case of a known number of clusters. This parallels the situation of clustering according to the distribution [23, 12]. In this section we assume that the distribution of (x1 , . . . , xN ) is not known, but a jointly stationary ergodic sample (X11 , . . . , X1N ), . . . , (Xn1 , . . . , XnN ) is provided. Thus, xi is a stationary ergodic time i series X1..n . Here is also where we drop Assumption 1; in particular, densities do not have to exist. This new relaxed set of assumptions can be interpreted as using a weaker oracle, as explained in Remark 5 below. We start with preliminaries about stationary processes, followed by impossibility results, and concluding with an algorithm for the case of known k. 5.1 Preliminaries: stationary ergodic processes Stationary, ergodicity, information rate. (Time-series) distributions, or processes, are measures on the space (X ? , FX ? ), where FX ? is the Borel sigma-algebra of X ? . Recall that N X -valued process is just a single X N -valued process. So the distributions are on the space ((X N )? , F(AN )? ); this will be often left implicit. For a sequence x ? An and a set B ? B denote ?(x, B) the frequency with which the sequence x falls in the set B. A process ? is stationary if ?(X1..|B| = B) = ?(Xt..t+|B|?1 = B) for any measurable B ? X ? and t ? N. We further abbreviate ?(B) := ?(X1..|B| = B). A stationary process ? is called (stationary) ergodic if the frequency of occurrence of each measurable B ? X ? in a sequence X1 , X2 , . . . generated by ? tends to its a priori (or limiting) probability a.s.: ?(limn?? ?(X1..n , B) = ?(B)) = 1. By virtue of the ergodic theorem, this definition can be shown to be equivalent to the more standard definition of stationary ergodic processes given in terms of shift-invariant sets [26]. Denote S and E the sets of all stationary and stationary ergodic processes correspondingly. The ergodic decomposition theorem for stationary processes (see, e.g., 7) states that any stationary process can be expressed as a mixture of stationary ergodic processes. That is, a stationary process ? can be envisaged as first selecting a stationary ergodic distribution according to some measure W? over the set of all such distributions, and then using this ergodic distribution to generate the sequence. More R formally, for any ? ? S there is a measure W? on (S, FS ), such that W? (E) = 1, and ?(B) = dW? (?)?(B), for any B ? FX ? . For a stationary time series x, its m-order entropy hm (x) is defined as EX1..m?1 h(Xm |X1..m?1 ) (so the usual Shannon entropy is the entropy of order 0). By stationarity, the limit limm?? hm exists 1 and equals limm?? m h(X1..m ) (see, for example, [6] for more details). This limit is called entropy rate and is denoted h? . For l stationary processes xi = (X1i , . . . , Xni , . . . ), i = 1..l, the m-order Pl mutual information is defined as Im (x1 , . . . , xl ) := i=1 hm (xi ) ? hm (x1 , . . . , xl ) and the mutual information rate is defined as the limit I? (x1 , . . . , xl ) := lim Im (x1 , . . . , xl ). m?? (1) Discretisations and a metric. For each m, l ? N, let B m,l be a partitioning of X m into 2l sets such that jointly they generate Fm of X m , i.e. ?(?l?N B m,l ) = Fm . The distributional distance between a pair of process distributions ?1 , ?2 is defined as follows [7]: d(?1 , ?2 ) = ? X m,l=1 X wm wl B?B m,l 5 |?1 (B) ? ?2 (B)|, (2) where we set wj := 1/j(j + 1), but any summable sequence of positive weights may be used. As shown in [22], empirical estimates of this distance are asymptotically consistent for arbitrary stationary ergodic processes. These estimates are used in [23, 12] to construct time-series clustering algorithms for clustering with respect to distribution. Here we will only use this distance in the impossibility results. Basing on these ideas,PGy?rfi [9] suggested to use a similar construction for P ? studying independence, namely d(?1 , ?2 ) = m,l=1 wm wl A,B?B m,l |?1 (A)?2 (B) ? ?(A ? B)|, where ?1 and ?2 are the two marginals of a process ? on pairs, and noted that its empirical estimates are asymptotically consistent. The distance we will use is similar, but is based on mutual information. 5.2 Impossibility results First of all, while the definition of ergodic processes guarantees convergence of frequencies to the corresponding probabilities, this convergence can be arbitrary slow [26]: there are no meaningful bounds on |?(X1..n , 0) ? ?(X1 = 0)| in terms of n for ergodic ?. This means that we cannot use tests for (conditional) independence of the kind employed in the i.i.d. case (Section 4). Thus, the first question we have to pose is whether it is possible to test independence, that is, to say 1 2 whether x1 ? x2 based on a stationary ergodic samples X1..n , X1..n . Here we show that the answer in a certain sense is negative, but some important questions remain open. 1 2 An (independence) test ? is a function that takes two samples X1..n , X1..n and a parameter ? ? (0, 1), called the confidence level, and outputs a binary answer: independent or not. A test ? is ?-level 1 2 consistent if, for every stationary ergodic distribution ? over a pair of samples (X1..n.. , X1..n.. ), for 1 2 every confidence level ?, ?(?? (X1..n , X1..n ) = 1) < ? if the marginal distributions of the samples 1 2 are independent, and ?? (X1..n , X1..n ) converges to 1 as n ? ? with ?-probability 1 otherwise. The next proposition can be established using the criterion of [25]. Recall that, for ? ? S, the measure W? over E is its ergodic decomposition. The criterion states that there is an ?-level consistent test for H0 against E \ H0 if an only if W? (H0 ) = 1 for every ? ? cl H0 . Proposition 1. There is no ?-level consistent independence test (jointly stationary ergodic samples). Proof. The example is based on the so-called translation process, constructed as follows. Fix some irrational ? ? (0, 1) and select r0 ? [0, 1] uniformly at random. For each i = 1..n.. let ri = (ri?1 + ?) mod 1 (the previous element is shifted by ? to the right, considering the [0,1] interval looped). The samples Xi are obtained from ri by thresholding at 1/2, i.e. Xi := I{ri > 0.5} (here ri can be considered hidden states). This process is stationary and ergodic; besides, it has 0 entropy rate [26], and this is not the last of its peculiarities. Take now two independent copies of this process to obtain a pair (x1 , x2 ) = (X11 , X12 . . . , Xn1 , Xn2 , . . . ). The resulting process on pairs, which we denote ?, is stationary, but it is not ergodic. To see the latter, observe that the difference between the corresponding hidden states remains constant. In fact, each initial state (r1 , r2 ) corresponds to an ergodic component of our process on pairs. By the same argument, these ergodic components are not independent. Thus, we have taken two independent copies of a stationary ergodic process, and obtained a stationary process which is not ergodic and whose ergodic components are pairs of processes that are not independent! To apply the criterion cited above, it remains to show that the process ? we constructed can be obtained as a limit of stationary ergodic processes on pairs. To see this, consider, for each ?, a process ?? , whose construction is identical to ? except that instead of shifting the hidden states by ? we shift them by ? + u?i where u?i are i.i.d. uniformly random on [??, ?]. It is easy to see that lim??0 ?? = ? in distributional distance, and all ?? are stationary ergodic. Thus, if H0 is the set of all stationary ergodic distributions on pairs, we have found a distribution ? ? cl H0 such that W? (H0 ) = 0. Thus, there is no consistent test that could provide a given level of confidence under H0 , even if only asymptotic consistency is required under H1 . However, a yet weaker notion of consistency might suffice to construct asymptotically consistent clustering algorithms. Namely, we could ask for a test whose answer converges to either 0 or 1 according to whether the distributions generating the samples are independent or not. Unfortunately, it is not known whether a test consistent in this weaker sense exists or not. I conjecture that it does not. The conjecture is based not only on the result above, but also on the result of [24] that shows that there is no such test for the related problem of homogeneity testing, that is, for testing whether two given samples have the same or different distributions. This negative result holds even if the distributions are independent, binary-valued, the 6 difference is restricted to P (X0 = 0), and, finally, for a smaller family of processes (B-processes). Thus, for now what we can say is that there is no test for independence available that would be consistent under ergodic sampling. Therefore, we cannot distinguish even between the cases of 1 and 2 clusters. Thus, in the following it is assumed that the number of clusters k is given. The last problem we have to address is mutual information for processes. The analogue of mutual information for stationary processes is the mutual information rate (1). Unfortunately, 0 mutual information rate does not imply independence. This is manifest on processes with 0 entropy rate, for example those of the example in the proof of Proposition 1. What happens is that, if two processes are dependent, then indeed at least one of the m-order entropy rates Im is non-zero, but the limit may still be zero. Since we do not know in advance which Im to take, we will have to consider all of them, as is explained in the next subsection. 5.3 Clustering with the number of clusters known The quantity introduced below, which we call sum-information, will serve as an analogue of mutual information in the i.i.d. case, allowing us to get around the problem that the mutual information rate may be 0 for a pair of dependent stationary ergodic processes. Defined in the same vein as the distributional distance (2), this new quantity is a weighted sum over all the mutual informations up to time n; in addition, all the individual mutual informations are computed for quantized versions of random variables in question, with decreasing cell size of quantization, keeping all the mutual information resulting from different quantizations. The latter allows us not to require the existence of densities. Weighting is needed in order to be able to obtain consistent empirical estimates of the theoretical quantity under study. First, for a process x = (X1 , . . . , Xn , . . . ) and for each m, l ? N define the l?th quantized version [X1..m ]l of X1..m as the index of the cell of B m,l to which X1..m belongs. Recall that each of the partitions B m,l has cell size 2l , and that wl := 1/l(l + 1). Definition 1 (sum-information). For stationary x1 ..xN define the sum-information ! ? ? N X X X 1 1 s i l 1 N I(x1 , . . . , xN ) := wm wl h([X1..m ] ) ? h([X1..m ]l , . . . , [X1..m ]l ) (3) m l m=1 i=1 l=1 The next lemma follows from the fact that ?l?N B m,l generates Fm and ?m?N Fm generates F? . Lemma 1. sI(x1 , . . . , xN ) = 0 if and only if x1 , . . . , xN are mutually independent. ? n ([X i ]l ) of entropy are defined by replacing unknown probabilities by The empirical estimates h 1..m ? frequencies; the estimate sbI n (x1 , . . . , xN ) of is obtained by replacing h in (3) with h. Remark 4 (Computing sbI n ). The expression (3) might appear to hint at a computational disaster, as it involves two infinite sums, and, in addition, the number of elements in the sum inside h([]l ) grows exponentially in l. However, it is easy to see that, when we replace the probabilities with frequencies, all but a finite number of summands are either zero or can be collapsed (because they are constant). Moreover, the sums can be further truncated so that the total computation becomes quasilinear in n. This can be done exactly the same way as for distributional distance, as described in [12, Section 5]. The following lemma can be proven analogously to the corresponding statement about consistency of empirical estimates of the distributional distance, given in [22, Lemma 1]. Lemma 2. Let the distribution ? of x1 , . . . , xN be jointly stationary ergodic. Then sb I n (x1 , . . . , xk ) ? sI(x1 , . . . , xN ) ?-a.s. This lemma alone is enough to establish the existence of a consistent clustering algorithm. To see this, first consider the following problem, which is the ?independence? version of the classical statistical three-sample problem. The 3-sample-independence problem. Three samples x1 , x2 , x3 , are given, and it is known that either (x1 , x2 ) ? x3 or x1 ? (x2 , x3 ) but not both. It is required to find out which one is the case. Proposition 2. There exists an algorithm for solving the 3-sample-independence problem that is asymptotically consistent under ergodic sampling. 7 Indeed, it is enough to consider an algorithm that compares sbI n ((x1 , x2 ), x3 ) and sbI n (x1 , (x2 , x3 )) and answers according to whichever is smaller. The independence clustering problem which we are after is a generalisation of the 3-sampleindependence problem to N samples. We can also have a consistent algorithm for the clustering problem, simply comparing all possible clusterings U1 , . . . , Uk of the N samples given and selecting whichever minimizes sbI n (U1 , . . . , Uk ). Such an algorithm is of course not practical, since the number of computations it makes must be exponential in N and k. We will show that the number of candidate clustering can be reduced dramatically, making the problem amenable to computation. The proposed algorithm CLINk (Algorithm 2 below) works similarly to CLIN, but with some important difFigure 2: CLINk: cluster given k and an ferences. Like before, the main procedure is to attempt estimator of mutual sum-information to split the given set of samples into two clusters. This Consider all the clusterings obtained splitting procedure starts with a single element x1 and by applying recursively the function estimates its sum-information sbI(x1 , R) with the rest of Split to each of the sets in each of the elements, R. It then takes the elements out of R one the candidate partitions, starting with by one without replacement, each time measuring how the input set S, until k clusters are this changes sbI(x , R). As before, once and if we find an 1 obtained. Output the clustering U that element that is not independent of x , this change will 1 minimizes sbI(U ) be positive. However, unlike in the i.i.d. case, here we Function Split(Set S of samples) cannot test whether this change is 0. Yet, we can say that Initialize: C := {x1 }, R := S \ C, if, among the tested elements, there is one that gives a P := {} non-zero change in sI, then one of such elements will be while R 6= ? do the one that gives the maximal change in sbI (provided, of Initialize:M := {}, d := 0; sb I to xmax:= index of any x in R course, that we have enough data for the estimates s be close enough to the theoretical values I). Thus, we keep each split that arises from such a maximal-change elAdd (C, R) to P ement, resulting in O(N 2 ) candidate splits for the case of for each x ? R do 2 clusters. For k clusters, we have to consider all the comr := s?I(C, R) binations of the splits, resulting in O(N 2k?2 ) candidate move x from R to M clusterings. Then select the one that minimizes sbI. r0 := s?I(C, R); d0 := r ? r0 0 if d > d then Theorem 3. CLINk is asymptotically consistent under d := d0 , xmax:=index of(x) ergodic sampling. This algorithm makes at most N 2k?2 end if calls to the estimator of mutual sum-information. end for Proof. The consistency of sbI (Lemma 2) implies that, for Move xxmax from M to C; R := every ? > 0, from some n on w.p. 1, all the estimates of S\C s I the algorithm uses will be within ? of their sI values. end while Since I(U1 , . . . , Uk ) = 0 if and only if U1 , . . . , Uk is Return(List of candidate splits P) the correct clustering (Lemma 1), it is enough to show END function that, assuming all the sbI estimates are close enough to the sI values, the clustering that minimizes sbI(U1 , . . . , Uk ) is among those the algorithm searchers through, that is, among the clusterings obtained by applying recursively the function Split to each of the sets in each of the candidate partitions, starting with the input set S, until k clusters are obtained. To see the latter, on each iteration of the while loop, we either already have a correct candidate split in P, that is, a split (U1 , U2 ) such that sI(U1 , U2 ) = 0, or we find (executing the for loop) an element x0 to add to the set C such that C? \x0 . Indeed, if at least one such element x0 exists, then among all such elements there is one that maximizes the difference d0 . Since the set C is initialized as a singleton, a correct split is eventually found if it exists. Applying the same procedure exhaustively to each of the elements of each of the candidate splits producing all the combinations of k candidate clusterings, under the assumption that all the estimates sbI are sufficiently close the corresponding values, we are guaranteed to have the one that minimizes I(U1 , . . . , Uk ) among the output. Remark 5 (Fickle oracle). Another way to look at the difference between the stationary and the i.i.d. cases is to consider the following ?fickle? version of the oracle test of Section 3. Consider the oracle that, as before, given sets of random variables A, B, C, D ? {x1 , . . . , xN } answers whether sI(A, B) > sI(C, D). However, the answer is only guaranteed to be correct in the case 8 s I(A, B) 6= sI(C, D). If sI(A, B) = sI(C, D) then the answer is arbitrary (and can be considered adversarial). One can see that Lemma 2 guarantees the existence of the oracle that has the requisite fickle correctness property asymptotically, that is, w.p. 1 from some n on. It is also easy to see that Algorithm 2 can be rewritten in terms of calls to such an oracle. 6 Generalizations, future work A general formulation of the independence clustering problem has been presented, and attempt has been made to trace out broadly the limits of what is possible and what is not possible in this formulation. In doing so, clear-cut formulations have been favoured over utmost generality, and over, on the other end of the spectrum, precise performance guarantees. Thus, many interesting questions have been left out; some of these are outlined in this section. Beyond time series. For the case when the distribution of the random variables xi is unknown, we i have assumed that a sample X1..n is available for each i = 1..N . Thus, each xi is represented by a time series. A time series is but one form the data may come in. Other ways include functional data, mutli-dimensional- or continuous-time processes, or graphs. Generalizations to some of these models, such as, for example, space-time stationary processes, are relatively straightforward, while others require more care. Some generalizations to infinite stationary graphs may be possible along the lines of [21]. In any case, the generalization problem is statistical (rather than algorithmic). If the number of clusters is unknown, we need to be able to replace the emulate the oracle test of section 3 with statistical tests. As explained in Section 4, it is sufficient to find a test for conditional independence, or an estimator of entropy along with guarantees on its convergence rates. If these are not available, as is the case of stationary ergodic samples, we can still have a consistent algorithm for k known, as long as we have an asymptotically consistent estimator of mutual information (without rates), or, more generally, if we can emulate the fickle oracle (Remark 5). Beyond independence. The problem formulation considered rests on the assumption that there exists a partition U1 , . . . , Uk of the input set S such that U1 , . . . , Uk are jointly independent, that is, such that I(U1 , . . . , Uk ) = 0. In reality, perhaps, nothing is really independent, and so some relaxations are in order. It is easy to introduce some thresholding in the algorithms (replacing 0 in each test by some threshold ?) and derive some basic consistency guarantees for the resulting algorithms. The general problem formulation is to find a finest clustering such that I(U1 , . . . , Uk ) > ?, for a given ? (note that, unlike in the independence case of ? = 0, such a clustering may not be unique). If one wants to get rid of ?, a tree of clusterings may be considered for all ? ? 0, which is a common way to treat unknown parameters in the clustering literature (e.g.,[2]). Another generalization can be obtained by considering the problem from the graphical model point of view. The random variables xi are vertices of a graph, and edges represent dependencies. In this representation, clusters are connected components of the graph. A generalization then is to clusters that are the smallest components that are connected (to each other) by at most l edges, where l is a parameter. Yet another generalization would be to decomposable distributions of [10]. Performance guarantees. Non-asymptotic results (finite-sample performance guarantees) can be obtained under additional assumptions, using the corresponding results on (conditional) independence tests and on estimators of divergence between distributions. Here it is worth noting that we are not restricted to using the mutual information I, but any measure of divergence can be used, for example, R?nyi divergence; a variety of relevant estimators and corresponding bounds, obtained under such assumptions as H?lder continuity, can be found in [19, 11]. From any such bounds, at least some performance guarantees for CLIN can be obtained simply using the union bound over all the invocations of the tests. Complexity. The algorithmic aspects of the problem have only been started upon in this work. Thus, it remains to find out what is the computational complexity of the studied problem. So far, we have presented only some upper bounds, by constructing algorithms and bounding their complexity (kN 2 for CLIN and N 2k for CLINk). Lower bounds (and better upper bounds) are left for future work. A subtlety worth noting is that, for the case of known distributions, the complexity may be affected by the choice of the oracle. In other words, some calculations may be ?pushed? inside the oracle. In this regard, it may be better to consider the oracle for testing conditional independence, rather than a comparison of mutual informations, as explained in Remarks 1, 3. The complexity of the stationary-sampling version of the problem can be studied using the fickle oracle of Remark 5. The consistency of the algorithm should then be established for every assignment of those answers of the oracle that are arbitrary (adversarial). 9 References [1] Francis R Bach and Michael I Jordan. Beyond independent components: trees and clusters. 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Fast amortized inference of neural activity from calcium imaging data with variational autoencoders Artur Speiser12 , Jinyao Yan3 , Evan Archer4?, Lars Buesing4?, Srinivas C. Turaga3? and Jakob H. Macke1?? 1 research center caesar, an associate of the Max Planck Society, Bonn, Germany 2 IMPRS Brain and Behavior Bonn/Florida 3 HHMI Janelia Research Campus 4 Columbia University [email protected], [email protected], [email protected] Abstract Calcium imaging permits optical measurement of neural activity. Since intracellular calcium concentration is an indirect measurement of neural activity, computational tools are necessary to infer the true underlying spiking activity from fluorescence measurements. Bayesian model inversion can be used to solve this problem, but typically requires either computationally expensive MCMC sampling, or faster but approximate maximum-a-posteriori optimization. Here, we introduce a flexible algorithmic framework for fast, efficient and accurate extraction of neural spikes from imaging data. Using the framework of variational autoencoders, we propose to amortize inference by training a deep neural network to perform model inversion efficiently. The recognition network is trained to produce samples from the posterior distribution over spike trains. Once trained, performing inference amounts to a fast single forward pass through the network, without the need for iterative optimization or sampling. We show that amortization can be applied flexibly to a wide range of nonlinear generative models and significantly improves upon the state of the art in computation time, while achieving competitive accuracy. Our framework is also able to represent posterior distributions over spike-trains. We demonstrate the generality of our method by proposing the first probabilistic approach for separating backpropagating action potentials from putative synaptic inputs in calcium imaging of dendritic spines. 1 Introduction Spiking activity in neurons leads to changes in intra-cellular calcium concentration which can be measured by fluorescence microscopy of synthetic calcium indicators such as Oregon Green BAPTA-1 [1] or genetically encoded calcium indictors such as GCaMP6 [2]. Such calcium imaging has become important since it enables the parallel measurement of large neural populations in a spatially resolved and minimally invasive manner [3, 4]. Calcium imaging can also be used to study neural activity at subcellular resolution, e.g. for measuring the tuning of dendritic spines [5, 6]. However, due to the indirect nature of calcium imaging, spike inference algorithms must be used to infer the underlying neural spiking activity leading to measured fluorescence dynamics. ? current affiliation: Cogitai.Inc current affiliation: DeepMind ? equal contribution ? current primary affiliation: Centre for Cognitive Science, Technical University Darmstadt ? 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Most commonly-used approaches to spike inference [7, 8, 9, 10, 11, 12, 13, 14] are based on carefully designed generative models that describe the process by which spiking activity leads to fluorescence measurements. Spikes are treated as latent variables, and spike-prediction is performed by inferring both the parameters of the model and the spike latent variables from fluorescence time series, or ?traces? [7, 8, 9, 10]. The advantage of this approach is that it does not require extensive ground truth data for training, since simultaneous electrophysiological and fluorescence recordings of neural activity are difficult to acquire, and that prior knowledge can be incorporated in the specification of the generative model. The accuracy of the predictions depends on the faithfulness of the generative model of the transformation of spike trains into fluorescence measurements [14, 12]. The disadvantage of this approach is that spike-inference requires either Markov-Chain Monte Carlo (MCMC) or Sequential Monte-Carlo techniques to sample from the posterior distribution over spike-trains or alternatively, iterative optimization to obtain an approximate maximum a-posteriori (MAP) prediction. Currently used approaches rely on bespoke, model-specific inference algorithms, which can limit the flexibility in designing suitable generative models. Most commonly used methods are based on simple phenomenological (and often linear) models [7, 8, 9, 10, 13]. Recently, a small number of cell-attached electrophysiological recordings of neural activity have become available, with simultaneous fluorescence calcium measurements in the same neurons. This has made it possible to train powerful and fast classifiers to perform spike-inference in a discriminative manner, precluding the need for accurate generative models of calcium dynamics [15]. The disadvantage of this approach is that it can require large labeled data-sets for every new combination of calcium indicator, cell-type and microscopy method, which can be expensive or impossible to acquire. Further, these discriminative methods do not easily allow the incorporation of prior knowledge about the generative process. Finally, current classification approaches yield only pointwise predictions of spike probability (i.e. firing rates), independent across time, and ignore temporal correlations in the posterior distribution of spikes. Sampled spikes Predicted probability Backward RNN Forward RNN 1D CNN Figure 1: Amortized inference for predicting spikes from imaging data. A) Our goal is to infer a spike train s from an observed time-series of fluorescence-measurements f . We assume that we have a generative model of fluorescence given spikes with (unknown) parameters ?, and we simultaneously learn ? as well as a ?recognition model? which approximates the posterior over spikes s given f and which can be used for decoding a spike train from imaging data. B) We parameterize the recognition-model by a multi-layer network architecture: Fluorescence-data is first filtered by a deep 1D convolutional network (CNN), providing input to a stochastic forward running recurrent neural network (RNN) which predicts spike-probabilities and takes previously sampled spikes as additional input. An additional deterministic RNN runs backward in time and provides further context. Here, we develop a new spike inference framework called DeepSpike (DS) based on the variational autoencoder technique which uses stochastic variational inference (SVI) to teach a classifier to predict spikes in an unsupervised manner using a generative model. This new strategy allows us to combine the advantages of generative [7] and discriminative approaches [15] into a single fast classifier-based method for spike inference. In the variational autoencoder framework, the classifier is called a recognition model and represents an approximate posterior distribution over spike trains from which samples can be drawn in an efficient manner. Once trained to perform spike inference on one dataset, the recognition model can be applied to perform inference on statistically similar datasets without any retraining: The computational cost of variational spike inference is amortized, dramatically speeding up inference at test-time by exploiting fast, classifier based recognition models. 2 We introduce two recognition models: The first is a temporal convolutional network which produces a posterior distribution which is factorized in time, similar to standard classifier-based methods [15]. The second is a recurrent neural network-based recognition model, similar to [16, 17] which can represent any correlated posterior distribution in the non-parametric limit. Once trained, both models perform spike inference with state-of-the-art accuracy, and enable simultaneous spike inference for populations as large as 104 in real time on a single GPU. We show the generality of this black-box amortized inference method by demonstrating its accuracy for inference with a classic linear generative model [7, 8], as well as two nonlinear generative models [12]. Finally, we show an extension of the spike inference method to simultaneous inference and demixing of synaptic inputs from backpropagating somatic action potentials from simultaneous somatic and dendritic calcium imaging. 2 2.1 Amortized inference using variational autoencoders Approach and training procedure We observe fluorescence traces fti , t = 1 . . . T i representing noisy measurements of the dynamics of somatic calcium concentration in neurons i = 1 . . . N . We assume a parametrised, probabilistic, differentiable generative model p?i (f |s) with (unknown) parameters ?i . The generative model predicts a fluorescence trace given an underlying binary spike train si , where sit = 1 indicates that the neuron i produced an action potential in the interval indexed by t. Our goal is to infer a latent spike-train s given only fluorescence observations f . We will solve this problem by training a deep neural network as a ?recognition model? [18, 19, 20] parametrized by weights ?. Use of a recognition model enables fast computation of an approximate posterior distribution over spike trains from a fluorescence trace q? (s|f ). We will share one recognition model across multiple cells, i.e. that q? (si |f i ) ? p?i (si |f i ) for each i. We describe an unsupervised training procedure which jointly optimizes parameters of the generative model ? and the recognition network ? in order to maximize a lower bound on the log likelihood of the observed data, log p(f ) [19, 18, 20]. We learn the parameters ? and ? simultaneously by jointly maximizing LK (?, ?), a multi-sample importance-weighting lower bound on the log likelihood log p(f ) given by [21] # " K k X p (s , f ) 1 ? LK (?, ?) = Es1 ,...,sK ?q? (s|f ) log ? log p(f ), (1) K q? (sk |f ) k=1 k where s are spike trains sampled from the recognition model q? (s|f ). This stochastic objective involves drawing K samples from the recognition model, and evaluating their likelihood by passing them through the generative model. When K = 1, the bound reduces to the evidence lower bound (ELBO). Increasing K yields a tighter lower bound (than the ELBO) on the marginal log likelihood, at the cost of additional training time. We found that increasing the number of samples leads to better fits of the generative model; in our experiments, we used K = 64. To train ? and ? by stochastic gradient ascent, we must estimate the gradient ??,? L(?, ?). As our recognition model produces an approximate posterior over binary spike trains, the gradients have to be estimated based on samples. Obtaining functional estimates of the gradients ?? L(?, ?) with respect to parameters of the recognition model is challenging and relies on constructing effective control variates to reduce variance [22]. We use the variational inference for monte carlo objectives (VIMCO) approach of [23] to produce low-variance unbiased estimates of the gradients ??,? LK (?, ?). The generative training procedure could be augmented with a supervised cost term [24, 25], resulting in a semi-supervised training method. Gradient optimization: We use ADAM [26], an adaptive gradient update scheme, to perform online stochastic gradient ascent. The training data is cut into short chunks of several hundred time-steps and arranged in batches containing samples from a single cell. As we train only one recognition model but multiple generative models in parallel, we load the respective generative model and ADAM parameters at each iteration. Finally, we use norm-clipping to scale the gradients acting on the recognition model: the norm of all gradients is calculated, and if it exceeds a fixed threshold the gradients are rescaled. While norm-clipping was introduced to prevent exploding gradients in RNNs 3 [27], we found it to be critical to achieve high performance both for RNN and CNN architectures in our learning problem. Very small threshold values (0.02) empirically yielded best results. 2.2 Generative models p? (f |s) To demonstrate that our computational strategy can be applied to a wide range of differentiable models in a black-box manner, we consider four generative models: a simple, but commonly used linear model of calcium dynamics [7, 8, 9, 10], two more sophisticated nonlinear models which additionally incorporate saturation and facilitation resulting from the dynamics of calcium binding to the calcium sensor, and finally a multi-dimensional model for dendritic imaging data. Linear auto-regressive generative model (SCF): We use the name SCF for the classic linear convolutional generative model used in [7, 8, 9, 10], since this generative process is described by the Spikes st , which linearly impact Calcium concentration ct , which in turn determines the observed Fluorescence intensity ft , ct = p X ?t0 ct?t0 + ?st , ft = ?ct + ? + et , (2) t0 =1 with linear auto-regressive dynamics of order p for the calcium concentration with parameters ?, spike-amplitude ?, gain ?, constant fluorescence baseline ?, and additive measurement noise et ? N (0, ? 2 ). Nonlinear auto-regressive and sensor dynamics generative models (SCDF & MLphys): As examples of nonlinear generative models [28], we consider two simple models of the discrete-time dynamics of the calcium sensor or dye. In the first (SCDF), the concentration of fluorescent dye molecules dt is a function of the somatic Calcium concentration ct , and has Dynamics dt ? dt?1 = ?on c?t ([D] ? dt?1 ) ? ?off dt?1 , ft = ?dt + ? + et , (3) where ?on and ?off are the rates at which the calcium sensor binds and unbinds calcium ions, and ? is a Hill coefficient. We constrained these parameters to be non-negative. [D] is the total concentration of the dye molecule in the soma, which sets the maximum possible value of dt . The richer dynamics of the SCDF model allow for facilitation of fluorescence at low firing rates, and saturation at high rates. The parameters of the SCDF model are ? = {?, ?, ?, ?on , ?off , ?, [D], ? 2 }. The second nonlinear model (MLphys) is a discrete-time version of the MLspike generative model [12], simplified by not including a model of the time-varying baseline. The dynamics for ft and ct are as above, with ? = 1. We replace the dynamics for dt by dt ? dt?1 = 1 ?on (1 + ?((c0 + ct )? ? c?0 ))( ((c0 + ct )? ? c?0 ) ? dt?1 ). (1 + ?((c0 + ct )? ? c?0 )) (4) Multi-dimensional soma + dendrite generative model (DS-F-DEN): The dendritic generative model is a multi-dimensional SCDF model that incorporates back-propagating action potentials (bAPs). The calcium concentration at the cell body (superscript c) is generated as for SCDF, whereas for the spine (superscript s), there are two components: synaptic inputs and bAPs from the soma, cct = p X ?tc0 cct?t0 + ? c sct , cst = t0 =1 p X ?ts0 cst?t0 + ? s sst + ? bs sct , (5) t0 =1 where ? bs are the amplitude coefficients of bAPs for different spine locations, and c ? {1, ..., Nc }, s ? {1, ..., Ns }. The spines and soma share the same dye dynamics as in (3). The parameters of the 2 dendritic integration model are ? = {?s,c , ?s,c , ?s,c , ?on , ?off , ?, [D], ?s,c }. We note that this simple generative model does not attempt to capture the full complexity of nonlinear processing in dendrites (e.g. it does not incorporate nonlinear phenomena such as dendritic plateau potentials). Its goal is to separate local influences (synaptic inputs) from global events (bAPs, or potentially regenerative dendritic events). 4 2.3 Recognition models: parametrization of the approximate posterior q? (s|f ) The goal of the recognition model is to provide a fast and efficient approximation q? (s|f ) to the true posterior p(s|f ) over discrete latent spike trains s. We will use both a factorized, localized approximation (parameterized as a convolutional neural network), and a more flexible, non-factorized and non-localized approximation (parameterized using additional recurrent neural networks). Convolutional neural network: Factorized posterior approximation (DS-F) In [15], it was reported that good spike-prediction performance can be achieved by making the spike probability q? (st |ft??...t+? ) depend on a local window of the fluorescence trace of length 2? + 1 centered at t when training such a model fully supervised. We implement a scaled up version of this idea, using a deep neural network which is convolutional in time as the recognition model. We use architectures with up to five hidden layers and ? 20 filters per layer with Leaky ReLUs units [29]. The output layer uses a sigmoid nonlinearity to compute the Bernoulli spike probabilities q? (st |f ). Recurrent neural network: Capturing temporal correlations in the posterior (DS-NF) The fully-factorized posterior approximation (DS-F) above ignores temporal correlations in the posterior over spike trains. Such correlations can be useful in modeling uncertainty in the precise timing of a spike, which induces negative correlations between nearby time bins. To model temporal correlations, we developed a RNN-based non-factorizing distribution which can approach the true posterior in the non-parametric limit (see figure 1B). Similar to [16], we use Q the temporal ordering over spikes and factorize the joint distribution over spikes as q? (s|f ) = t q? (st |f, s0 , ..., st?1 ), by conditioning spikes at t on all previously sampled spikes. Our RNN uses a CNN as described above to extract features from the input trace. Additional input is provided by a a backwards RNN which also receives input from the CNN features. The outputs of the forward RNN and CNN are transformed into Bernoulli spike probabilities q? (st |f ) through a dense sigmoid layer. This probability and the sample drawn from it are relayed to the forward RNN in the next time step. Forward and backward RNN have a single layer with 64 gated recurrent units each [30]. 2.4 Details of synthetic and real data and evaluation methodology We evaluated our method on simulated and experimental data. From our SCF and SCDF generative models for spike-inference, we simulated traces of length T = 104 assuming a recording frequency of 60 Hz. Initial parameters where obtained by fitting the models to real data (see below), and heterogeneity across neurons was achieved by randomly perturbing parameters. We used 50 neurons each for training and validation and 100 neurons in the test set. For each cell, we generated three traces with firing rates of 0.6, 0.9 and 1.1 Hz, assuming i.i.d. spikes. Finally, we compared methods on two-photon imaging data from 9 + 11 cells from [2], which is available at www.crcns.org. Layer 2/3 pyramidal neurons in mouse visual cortex were imaged at 60 Hz using the genetically encoded calcium-indicators GCaMP6s and GCaMP6f, while action-potentials were measured electrophysiologically using cell-attached recordings. Data was pre-processed by removing a slow moving baseline using the 5th percentile in a window of 6000 time steps. Furthermore we used this baseline estimate to calculate ?F/F . Cross-validated results where obtained using 4 folds, where we trained and validated on 3/4 of the cells in each dataset and tested on the remaining cells to highlight the potential for amortized inference. Early stopping was performed based on the the correlation achieved on the train/validation set, which was evaluated every 100 update steps. We report results using the cross-correlation between true and predicted spike-rates, at the sampling discretization of 16.6 ms for simulated data and 40 ms for real data. As the predictions of our DS-NF model are not deterministic, we sample 30 times from the model and average over the resulting probability distributions to obtain an estimate of the marginal probability before we calculate crosscorrelations. We used multiple generative models to show that our inference algorithm is not tied to a particular model: SCDF for the experiments depicted in Fig. 2, SCF for a comparison with established methods based on this linear model (Table 1, column 1), and MLphys on real data as it is used by the current state-of-the-art inference algorithm (Table 1, columns 2 & 3, Fig. 3). 5 True spikes Trace Reconstruction | DS-F Reconstruction | DS-NF D B 300 1.00 0.5 2 4 6 8 C 0 0 0 2 4 Time in seconds 6 8 0.8 0.6 2 3 Mean correlation: 0.80 1.0 0.0 1 Sampled spikes / True spike 0.40.4 500 400 300 200 100 0 0 Loglikelihood 100 DS-F DS-NF 100 Single cell inference Sampled spiketrains Marginal probability 200 (True spiketrain) 200 Correlated posterior A 300 400 500 1.0 Correlation (Marginal probability) 0.8 0.6 Mean correlation: 0.77 0.6 0.8 Amortized network 0.4 1.0 0.4 0.6 0.8 Factorized posterior 1.0 Figure 2: Model-inversion with variational autoencoders, simulated data A) Illustration of factorized (CNN, DS-F) and non-factorized posterior approximation (RNN, DS-NF) on simulated data (SCDF generative model). DS-NF yields more accurate reconstructions, but both methods lead to similar marginal predictions (i.e. predicted firing rates, bottom). B) Number of spikes sampled for every true spike for the factorized (red) and non-factorized (red) posterior. The correlated posterior consistently samples the correct number of spikes while still accounting for the uncertainty in the spike timing. C) Performance of amortized vs non-amortized inference on simulated data. D) Scatter plots of achieved log-likelihood of the true spike train under the posterior model (top) and achieved correlation coefficients between the marginalized spiking probabilities and true spike trains (bottom). 3 3.1 Results Stochastic variational spike inference of factorized and correlated posteriors We first illustrate our approach on synthetic data, and compare our two different architectures for recognition models. We simulated data from the SCDF nonlinear generative model and trained DeepSpike unsupervised using the same SCDF model. While only the more expressive recognition model (DS-NF) is able to achieve a close-to-perfect reconstructions of the fluorescence traces (Fig. 2 A, top row), both approaches yield similar marginal firing rate predictions (second row). However, as the factorized model does not model correlations in the posterior, it yields higher variance in the number of spikes reconstructed for each true spike (Fig. 2 B). This is because the factorized model can not capture that a fluorescence increase might be ?explained away? by a spike that has just been sampled, i.e. it can not capture the difference between uncertainty in spike-timing and uncertainty in (local) spike-counts. Therefore, while both approaches predict firing rates similarly well on simulated data (as quantified using correlation, Fig. 2 D), the DS-NF model assigns higher posterior probability to the true spike trains. 3.2 Amortizing inference leads to fast and accurate test-time inference In principle, our unsupervised learning procedure could be re-trained on every data-set of interest. However, it also allows for amortizing inference by sharing one recognition model across multiple cells, and applying the recognition model directly on new data without additional training for fast test-time performance. Amortized inference allows for the recognition model to be used for inference in the same way as a network that was trained fully supervised. Since there is no variational optimization at test time, inference with this network is just as fast as inference with a supervised network. Similarly to supervised learning, there will be limitations on the ability of this network to generalize to different imaging conditions or indicators that where not included in the training set. To test if our recognition model generalizes well enough for amortized inference to work across multiple cells, as well as on cells it did not see during training, we trained one DS-NF model on 50 6 cells (simulated data, SCDF) and evaluated its performance on a non-overlapping set of 30 cells. For comparison, we also trained 30 DS-NF models separately, on each of those cells? this amounts to standard variational inference using a neural network to parametrize the posterior approximation, but without amortizing inference. We found that amortizing inference only causes a small drop in performance (Fig. 2 C). However, this drop in performance is offset by the the large gain in computational efficiency as training a neural network takes several orders of magnitude more time then applying it at test time. Inference using the DS-F model only requires a single forward pass through a convolutional network to predict firing rates, and DS-NF requires running a stochastic RNN for each sampled spike train. While the exact running-time of each of these applications will depend on both implementation and hardware, we give rough indications of computational speed number estimated on an Intel(R) Xeon(R) CPU E5-2697 v3. On the CPU, our DS-F approach takes 0.05 s to process a single trace of 10K time steps, when using a network appropriate for 60 Hz data. This is on the same order as the 0.07 s (Intel Core i5 2.7 GHz CPU) reported by [31] for their OASIS algorithm, which is currently the fastest available implementation for constrained deconvolution (CDEC) of SCF, but restricted to this linear generative model. The DS-NF algorithm requires 4.6 s which still compares favourably to MLspike which takes 9.2 s (evaluated on the same CPU). As our algorithm is implemented in Theano [32] it can be easily accelerated and allows for massive parallelization on a single GPU. On a GTX Titan X, DS-F and DS-NF take 0.001 s and 1.5 s, respectively. When processing 500 traces in parallel, DS-NF becomes only 2.5 times slower. Extrapolating from these results, this implies that even when using the DS-NF algorithm, we would be able to perform spike-inference on 1 hour of recordings at 60 Hz for 500 cells in less then 90 s. Table 1: Performance comparison. Values are correlations between predicted marginal probabilities and ground truth spikes. Algorithm DS-F DS-NF CDEC [10] MCMC [9] MLSpike [12] DS-F-DEN Foopsi-RR [2] 3.3 Dataset SCF-Sim. 0.88 ? 0.01 0.89 ? 0.01 0.86 ? 0.01 0.87 ? 0.01 GCaMP6s 0.74 ? 0.02 0.72 ? 0.02 0.39 ? 0.03 * 0.47 ? 0.03 * 0.60 ? 0.02 * GCaMP6f 0.74 ? 0.02 0.73 ? 0.02 0.58 ? 0.02 * 0.53 ? 0.03 * 0.67 ? 0.01 * Dendritic dataset Soma Spine 0.84 ? 0.01 0.66 ? 0.02 0.78 ? 0.01 0.60 ? 0.01 DS achieves competitive results on simulated and publicly available imaging data The advantages of our framework (black-box inference for different generative models, fast testtime performance through amortization, correlated posteriors through RNNs) are only useful if the approach can also achieve competitive performance. To demonstrate that this is the case, we compare our approach to alternative generative-model based spike prediction methods on data sampled from the SCF model? as this is the generative model underlying commonly used methods [10, 9], it is difficult to beat their performance on this data. We find that both DS-F and DS-NF achieve competitive performance, as measured by correlation between predicted firing rates and true (simulated) spike trains (Table 1, left column. Values are means and standard error of the mean calculated over cells). To evaluate our performance on real data we compare to the current state-of-the-art method for spike inference based on generative models[12]. For these experiments we trained separate models on each of the GCaMP variants using the MLspike generative model. We achieve competitive accuracy to the results in [12] (see Table 1, values marked with an asterisk are taken from [12], Fig. 6d) and clearly outperform methods that are based on the linear SCF model. We note that, while our method performs inference in an unsupervised fashion and is trained using an un-supervised objective, we initialized our generative model with the mean values given in [12] (Fig. S6a), which were obtained using ground truth data. An example of inference and reconstruction using the DS-NF model is shown in Fig. 3. The reconstruction based on the true spikes (purple line) was obtained using the generative model parameters which had been acquired from unsupervised learning. This explains why the reconstruction using the inferred spikes is more accurate and suggests that there is a mismatch 7 GCaMP6s Corr. posterior Marginal probability Corr: 0.73 True spikes Spikes: 41.74 / 35.0 Trace Prediction | Infered spiketrain Prediction | True spiketrain 1.0 0.5 0.0 0 10 20 30 40 Time in seconds 50 Figure 3: Inference and reconstruction using the DS-NF algorithm on GECI data. The reconstruction based on the inferred spike trains (blue) shows that the algorithm converges to a good joint model while the reconstruction based on the true spikes (purple) shows a mismatch of the generative model for high activity which results in an overestimate of the overall firing rate. between the MLphys model and the true data-generating generating process. Developing more accurate generative models would therefore likely further increase the performance of the algorithm. Marginal probability Marginal probability True soma spikes Soma trace Inferred: DS-F-DEN Inferred: FOOPSI-RR 1.0 0.5 0.0 True synaptic inputs Spine trace 1.0 0.5 0.0 0 2 4 6 8 10 12 Time in seconds Cell cartoon Figure 4: Inference of somatic spikes and synaptic input spikes from simulated dendritic imaging data. We simulated imaging data from our generative model, and compared our approach (DS-F-DEN) to an analysis inspired by [2] (Foopsi-RR), and found that our method can extract synaptic inputs more accurately. Traces at the soma and spines are used to infer somatic spikes and synaptic inputs at spines. Top: somatic trace and predictions. DS-F-DEN produces better predictions at the soma since it uses all traces to infer global events. Bottom: spine trace and predictions. DS-F-DEN performs better in terms of extracting synaptic inputs. 3.4 Extracting putative synaptic inputs from calcium imaging in dendritic spines We generalized the DeepSpike variational-inference approach to perform simultaneous inference of backpropagating APs and synaptic inputs, imaged jointly across the entire neuronal dendritic arbor. We illustrate this idea on synthetic data based on the DS-F-DEN generative model (5). We simulated 15 cells each with 10 dendritic spines with a range of firing rates and noise levels. We then used a multi-input multi-output convolutional neural network (CNN, DS-F) in the non-amortized setting to infer a fully-factorized Bernoulli posterior distribution over global action potentials and local synaptic events. We compared our results to an analysis technique inspired by [2] which we call Foopsi-RR. We first apply constrained deconvolution [33] to somatic and dendritic calcium traces, and then use robust 8 linear regression to identify and subtract deconvolved components of the spine signal that correlated with global back-propagated action potential. Compared to the method suggested by [2], our model is significantly more accurate. The average correlation of our model is 0.84 for soma and 0.78 for spines, whereas for Foopsi-RR the average correlation is 0.66 for soma and 0.60 for spines (Table 1). 4 Discussion Spike inference is an important step in the analysis of fluorescence imaging. We here propose a strategy based on variational autoencoders that combines the advantages of generative [7] and discriminative approaches [15]. The generative model makes it possible to incorporate knowledge about underlying mechanisms and thus learn from unlabeled data. A simultaneously-learned recognition network allows fast test-time performance, without the need for expensive optimization or MCMC sampling. This opens up the possibility of scaling up spike inference to very large neural populations [34], and to real-time and closed-loop applications. Furthermore, our approach is able to estimate full posteriors rather than just marginal firing rates. It is likely that improvements in performance and interpretability will result from the design of better, biophysically accurate and possibly dye-, cell-type- and modality-specific models of the fluorescence measurement process, the dynamics of neurons [28] and indicators, as well as from taking spatial information into account. Our goal here is not to design such models or to improve accuracy per se, but rather to develop an inference strategy which can be applied to a large class of such potential generative models without model-specific modifications: A trained recognition model that can invert, and provide fast test-time performance, for any such model while preserving performance in spike-detection. Our recognition model is designed to serve as the common approximate posterior for multiple, possibly heterogeneous populations of cells, requiring an expressive model. These assumptions are supported by prior work [15] and our results on simulated and publicly available data, but might be suboptimal or not appropriate in other contexts, or for other performance measures. In particular, we emphasize that our comparisons are based on a specific data-set and performance measure which is commonly used for comparing spike-inference algorithms, but which can in itself not provide conclusive evidence for performance in other settings and measures. Our approach includes rich posterior approximations [35] based on RNNs to make predictions using longer context-windows and modelling posterior correlations. Possible extensions include causal recurrent recognition models for real-time spike inference, which would require combining them with fast algorithms for detecting regions of interest from imaging-movies [10, 36]. Another promising avenue is extending our variational inference approach so it can also learn from available labeled data to obtain a semisupervised algorithm [37]. As a statistical problem, spike inference has many similarities with other analysis problems in biological imaging? an underlying, sparse signal needs to be reconstructed from spatio-temporal imaging observations, and one has substantial prior knowledge about the image-formation process which can be encapsulated in generative models. As a concrete example of generalization, we proposed an extension to multi-dimensional inference of inputs from dendritic imaging data, and illustrated it on simulated data. We expect the approach pursued here to also be applicable in other inference tasks, such as the localization of particles from fluorescence microscopy [38]. 5 Acknowledgements We thank T. W. Chen, K. Svoboda and the GENIE project at Janelia Research Campus for sharing their published GCaMP6 data, available at http://crcns.org. We also thank T. Deneux for sharing his results for comparison and comments on the manuscript and D. Greenberg, L. Paninski and A. Mnih for discussions. 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Adaptive Active Hypothesis Testing under Limited Information Fabio Cecchi Eindhoven University of Technology, Eindhoven, The Netherlands [email protected] Nidhi Hegde Nokia Bell Labs, Paris-Saclay, France [email protected] Abstract We consider the problem of active sequential hypothesis testing where a Bayesian decision maker must infer the true hypothesis from a set of hypotheses. The decision maker may choose for a set of actions, where the outcome of an action is corrupted by independent noise. In this paper we consider a special case where the decision maker has limited knowledge about the distribution of observations for each action, in that only a binary value is observed. Our objective is to infer the true hypothesis with low error, while minimizing the number of action sampled. Our main results include the derivation of a lower bound on sample size for our system under limited knowledge and the design of an active learning policy that matches this lower bound and outperforms similar known algorithms. 1 Introduction We consider the problem of active sequential hypothesis testing with incomplete information. The original problem, first studied by Chernoff [1], is one where a Bayesian decision maker must infer the correct hypothesis from a set of J hypotheses. At each step the decision maker may choose from W actions where the outcome of an action is a random variable that depends on the action and the true (hidden) hypothesis. In prior work, the probability distribution functions on the outcomes are assumed to be known. In the present work we assume that these distributions are not known, and only some rough information about the outcomes of the actions is known, to be made more precise further on. Active hypothesis testing is an increasingly important problem these days, with applications that include the following. (a) Medical diagnostics ([2]) systems that include clinical trials for testing a new treatment, or diagnostics of a new disease. (b) Crowdsourcing: online platforms for task-worker matching such as Amazon?s Mechanical Turk or TaskRabbit, where, as new tasks arrive, they must be matched to workers capable of working on them. (c) Customer hotline centres or Q&A forums: online platforms such as StackExchange where questions are submitted, and users with varying capabilities are available for providing an answer. This includes customer service centres where customer tickets are submitted and the nature of the problem must be learned before its treatment (an example where supervised learning techniques are used is [3]). (d) Content search problems where an incoming image must be matched to known contents, as studied in [4]. We now informally describe our model. In the general instance of our problem, the true hypothesis, ?? is one in a set of J hypotheses, J = {?1 , . . . , ?J }, and a set of W actions is available, where the outcomes of the actions depend on the true hypothesis. When the true hypothesis is ?j and 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. action w is chosen, a noisy outcome Xw,j ? J is observed, whose distribution, pw,j (?) ? P(J ), is given. The objective then is to select an action at each step so as to infer the true hypothesis in a minimum number of steps, with a given accuracy. In our model, we assume that the decision maker has limited information about the outcome distributions. We define the principal set of an action w as Jw ? J . When action w is sampled, a noisy binary outcome y ? {?1, 1} is observed, which gives an indication on whether the action classifies the hypothesis in the set Jw . The quality of action w, ?w is related to the noise in the outcome. Rather than the distributions pw,j (?), we assume that the decision maker only has knowledge of the principal set Jw and quality ?w of each action. 1.1 Related work Since the seminal work by Chernoff [1], active hypothesis testing and variants of the problemhave been studied through various perspectives (see [5] for a brief survey). Chernoff derived a simple heuristic algorithm whose performance is shown to achieve asymptotic optimality in the regime where the probability of error vanishes. Specifically, it is shown that as the probability of error ? decreases the expected number of samples needed by Chernoff?s algorithm grows as ? log(?). Most of the past literature in active sequential hypothesis testing has dealt with extensions of Chernoff?s model, and has shown that Chernoff?s algorithm performs well in more general settings [6, 7]. A notable exception is [8], where the impact of the number of hypotheses is analyzed and an algorithm that performs better than Chernoff?s benchmark is provided for the case of large values of J. Our work differs from prior work in a few ways. First, the hypothesis need not be locally identifiable. While in [1] each action is able to distinguish each pair of hypotheses, we assume that each hypothesis is globally identifiable, i.e., each pair of hypotheses can be discerned by at least one action. This is a common assumption in the area of distributed hypothesis testing ([9, 10]) and a weaker assumption than that of Chernoff. Note that dropping this assumption is not novel in itself, and has been done in other work such as [8]. Second, a novel extension in our work, differing from [8] is that we do not assume full knowledge on the actions? statistical parameters. The responses of actions are noisy, and in past literature the probability distributions governing them was assumed to be known. In our model, we drop this assumption, and we only require to know a lower bound ?w > 1/2 on the probability that action w will provide a correct response, no matter the hypothesis we want to test. As far as we know, no previous work in active sequential learning has tackled the problem of incomplete statistical information and we believe that such an extension may provide a non-negligible impact in real-life applications. Active hypothesis testing is similar to the problem of Bayesian active learning. This latter perspective in considered in [11] where noisy Bayesian active learning setting is used on the hypothesis testing problem with asymmetric noise and a heuristic based on the extrinsic Jensen-Shannon (EJS) divergence [12] is proposed. As in [8], full knowledge of the probability distributions governing the noise is available. In contrast, in our work we consider a more restricted model where, only a binary outcome with noise is given by the actions on the large hypothesis space. Inference with binary responses is considered in work on generalized binary search (GBS) [13], which is special case where the label set (outcome of actions) is binary with the case of symmetric, non-peristent noise. Our work differs from this type of work in that we consider asymmetric label-dependent noise, that is, ?w varies with action w. We thus position our work between [11, 8] and [13]. While the former assumes full knowledge on the noise distributions, we assume that only a binary response is provided and only a lower bound on the value that governs the outcome is known, and while the latter considers symmetric noise, we extend to asymmetric label-dependent noise. Our contribution. Our main objective is to investigate the minimum sample query size of this system for a certain level of accuracy in the inference of the true hypothesis, and to design efficient policies for this inference. Our contributions in the present paper are as follows. First, we consider the system under limited knowledge of outcome distribution. This restricted scenario adds a significant constraint for the action selection policy, and the belief vector update policy. To the best of our knowledge, this restricted scenario has not been considered in past literature. Second, under the limited knowledge constraint, we propose the Incomplete-Bayesian Adaptive Gradient (IBAG) policy which includes a belief vector update rule that we call Incomplete-Bayesian, and an action selection rule, named Adaptive Gradient, that follows the drift of the (unknown) coordinate of interest in the 2 belief vector. Third, we derive a lower bound on the sample size for the system under incomplete information, and show that the performance of IBAG matches this bound. We also carry out numerical experiments to compare IBAG to prior work. 2 Model The classic model of the active sequential learning problem consists in sequentially selecting one of several available sensing actions, in order to collect enough information to identify the true hypothesis, as considered in [1]. We thus consider a system where a decision maker has at his disposal a finite set of actions W = {1, . . . , W }, and there are a set of J = |J | < ? possible hypothesis, J = {?1 , . . . , ?J }. (For the rest of the paper, we refer to a hypothesis only by its index, i.e., j for hypothesis ?j , for ease of notation.) When the true hypothesis is j and action w is sensed, the outcome Xw,j ? J is sampled from the distribution pw,j (?) ? P(J ), i.e., P{Xw,j = j 0 } = pw,j (j 0 ). In our model, we assume to have limited information about the actions and this affects the classic model in two ways. First, for every sampled action w, a binary outcome y ? {?1, 1} is observed, indicating whether the inference of hypothesis by this action is in Jw or not, i.e., the response observed is Yw,j ? {?1, 1} where  1, if Xw,j ? Jw , Yw,j = ?1, if Xw,j ? / Jw . The subset Jw ? J is assumed to be known, and it is described by the matrix g ? {?1, 1}W ?J where  1, if j ? Jw , gw,j = (1) ?1, if j ? / Jw . Observe that the probability an action w correctly identifies the subset to which the true hypothesis P j belongs is given by qw,j := P{Yw,j = gw,j } = j 0 :gw,j =gw,j0 pw,j (j 0 ). However, as a second restriction, instead of knowing qw,j , the capacity, or quality, of an action w is captured by ?w where we assume that qw,j ? ?w , ? j ? J , w ? W. (2) We thus characterize each action by its principal set, Jw , and its quality, ?w . Assumption 1. For every action w ? W, the principal sets Jw ? J and the quality ?w ? (1/2, 1) are known. Denote by ?w = 2?w ? 1 where ?w ? [?m , ?M ] and ?m , ?M ? (0, 1). Since each action can only indicate whether the hypothesis belongs to a subset or not, there must exist an action w ? W for which j1 and j2 belong to different subsets, for all pairs j1 , j2 ? J . Define the subset Wj1 ,j2 ? W as Wj1 ,j2 = {w ? W : gw,j1 gw,j2 = ?1}. Assumption 2. For every j1 , j2 ? J , the subset Wj1 ,j2 is nonempty, i.e., each hypothesis is globally identifiable. For every action w ? W and hypothesis j ? J we define the subsets Jw,+j and Jw,?j which are, respectively, given by the hypotheses that action w cannot and can distinguish from j, i.e., Jw,+j = {j 0 ? J : gw,j 0 gw,j = 1}, Jw,?j = {j 0 ? J : gw,j 0 gw,j = ?1}. Note that w ? Wj1 ,j2 if and only if j2 ? Jw,?j1 (or equivalently j1 ? Jw,?j2 ). We aim to design a simple algorithm to infer the correct hypothesis using as few actions as possible. The true hypothesis will be denoted by j ? ? J . The learning process is captured by the evolution of the belief vector ?(t) ? P(J ), where ?j (t) denotes the decision maker?s confidence at time t that the true hypothesis is j. At the initial step t = 1, the belief vector ?(1) ? P(J ) is initialized so that ?j (1) > 0, j ? J . Since we assume to initially lack any information on the true hypothesis, without loss of generality, we set ?j (1) = 1/J for every j ? J . At every step t ? 1, according to the belief vector ?(t), the decision maker determines the next action to sense FW (?(t)) = w(t) ? W according to some selection rule FW (?). The outcome y(t) ? {?1, 1} from the chosen action w(t) is used to update the belief vector according to an update rule F U ?(t), w(t), y(t) = ?(t + 1) ? P(J ). The algorithm ends at time T ? , and the 3 inferred hypothesis is given by ?j = arg maxj?J ?j (T ? ) . Sensing actions is stopped when one of the posteriors is larger than 1 ? ?, for some ? > 0: T ? = inf {max ?j (t) > 1 ? ?}. (3) t?0 j?J 3 The Incomplete-Bayesian update rule We now describe how the decision maker updates the belief vector after he observes the outcome of an action. Given a belief vector ? ? P(J ) and the observation y ? {?1, 1} obtained from action w ? W, define   q , y = gw,j , ?w , y = gw,j , f?(y, j, w) = w,j f (y, j, w) = 1 ? qw,j , y = ?gw,j , 1 ? ?w , y = ?gw,j . Note that f?(y, j, w) denotes the probability of having outcome y given that the action w is chosen and the true hypothesis is j. The standard Bayesian update rule is given by the map F B U (?, w, y), where B FU,j (?, w, y) = P f?(y,j,w)?j . f?(y,i,w)?i In our model, however, the values qw,j for w ? W are unknown to i?J the decision maker. Hence, we introduce the Incomplete Bayesian (IB) update rule, which mimics the Bayesian rule, but with limited knowledge on outcome probailities. The IB update rule is given by the map F U (?, w, y), where f (y, j, w)?j . i?J f (y, i, w)?i FU,j (?, w, y) = P (4) Observe that Bayesian and IB update rules are identical when qw,j = ?w . In practice, the ?j (t) evolves according to both the quality of the chosen action, ?w , and the relation between this action?s principal set Jw and the current state of the belief vector ?(t). This dependence is formalized in the following lemma whose proof is included in the supplementary material, Section B. Lemma 1. Given ?(t) ? P(J ) and w(t) ? W, then it holds that ? ? 1, w.p. ? ? ? ? ? indic1{w(t) ? / Wj ,j }, ?j ? (t) ?j ? (t + 1) = ? 1+?w(t) , w.p. 1{w(t) ? Wj ? ,j }qw(t),j ? , ? ?j (t + 1) ?j (t) 1??w(t) ? ? 1?? ? w(t) ? , w.p. 1{w(t) ? Wj ? ,j }(1 ? qw(t),j ? ). 1+?w(t) 3.1 A lower bound on the sample size Note that the IB update rule alone sets some constraints on the performance. In particular, if we require the error probability to be low, then the expected number of samples is necessarily larger than a certain quantity depending on the model parameters. We show that this quantity asymptotically grows as ? log ? in the asymptotic regime where ? ? 0. Theorem 1. Assume the IB update rule is applied to the belief vector and that lim P{?j ? (T ? ) ? ?} ? ?? < 1. ??0 Then, there exist functions K0l (?), K1l (?) such that E[T ? ] ? K1l (?) log 1 + K0l (?), ? lim Kil (?) ? Kil > 0, ??0 for i = 0, 1. The proof of this result is presented in the supplement, Section A.2. We sketch the proof here. We first define ?j (t) St (j1 , j2 ) = log 1 , S(j1 , j2 ) = ST ? (j1 , j2 ), ?j2 (t) P and show that, on the one hand, if P{?j 6= j ? } is small, then j6=j ? S(j ? , j) is large with high P probability, and on the other hand, if t is small, then j6=j ? St (j ? , j) is small with high probability. 4 We use these properties to derive a lower bound on the tail probability of T ? , and thus on its expected value. Further, we can control the belief vector evolution by deriving bounds on the ratio between coordinates of the belief vector under the IB policy. Specifically, in the supplementary material Section A.3, we bound the probability that ?j (t) > ?j ? (t) at a certain time, and investigate how this probability evolves with t. 4 4.1 Adaptive Gradient: the action selection policy A gradient-based selection policy We now present an action selection policy that, together with the IB update rule, defines our active learning algorithm, which we call the Incomplete-Bayesian Adaptive Gradient (IBAG) policy. We will then analyze the complete algorithm showing that its performance asymptotically matches the lower bound provided in Theorem 1 as ? ? 0. We focus on the j ? -th coordinate of the belief vector, and define the drift at time t as Dw (?(t)) = E[?j ? (t + 1)|?(t), w(t) = w] ? ?j ? (t). Simple algebra and (4) yield the following Lemma. Lemma 2. It holds that q ? ? ? + ? ?  w,j w w w,?j ? (t) Dw (?(t)) = 4?w ?j ? (t)?w,?j ? (t) 2 , 1 ? ?2w 1 ? 2?w,?j ? (t) where ?w,+j = X ?j , X ?w,?j = j?Jw,+j (5) ?j . j?Jw,?j Assume for a moment that we know the true hypothesis j ? and qw,j ? for every w ? W. Then, in order to let ?j ? (t) grow as much as possible, we would greedily select the action w which maximizes Dw (?(t)). Our worker selection policy will attempt to mimic as closely as possible this greedy policy, while operating without complete information. L Lemma 3. It holds that Dw (?(t)) ? Dw (?(t)), where L Dw (?(t)) = 4?j ? (t) and ??w (t) = min 2 ?2w ??w (t) 1 ? ?2w 1 ? 2??w (t) n X X ?j (t), j?Jw 2 , (6) o ?j (t) j ?J / w The proof follows from the fact that Dw (?(t)) is increasing both in qw,j ? and ?w,?j ? (t) for every w ? W, and the observation that that qw,j ? ? ?w and ?w,?j ? (t) ? ??w (t). L Note that Dw (?(t)) provides us a tight lower bound on the expected growth of the coordinate of the L true hypothesis if action w is chosen at step t. Indeed, Dw (?(t)) can be decomposed to a part that ? uses the j -th coordinate of the belief vector and a part than can be computed without knowing j ? . The Adaptive Gradient (AG) selection rule, then chooses at step t, the action wD (t) ? W such that wD (t) = FW (?(t)) = arg max G(??w , ?w ), w?W G(v, d) = d2 v 2 1 ? d2 1 ? 2v 2 , (7) i.e., we select the action maximizing the current lower bound on the expected growth of the j ? coordinate of the belief vector. Ties are broken uniformly. Remark: Assume the actions have different costs of sensing. The AG selection rule can then be generalized as follows: c wD (t) = FW (?(t)) = arg max w?W 5 G(??w , ?w ) . cw (8) 4.2 An upper bound We now present our main result. We show that the expected number of samples required by our algorithm IBAG asymptotically matches the lower bound obtained in Theorem 1. Theorem 2. Under the IBAG algorithm, there exist constants K0u , K1u > 0 independent of ? such that 1 E[T ? ] ? K1u log + K0u . ? The proof is provided in supplementary material, Section A.5. This result is based on the intuition that IBAG never selects an action that is too uninformative relative to the other actions. Specifically, the information provided by an action w at time t depends on its quality ?w and outcome over the subset Jw,?j ? . In other words, the value ?w,?j ? must decrease to 0, hence the higher this value is for a given action w, the more we can still learn from sensing this action. As a proxy for ?w,?j ? we use ??w which also must be as large as possible. The following lemma, whose proof is given in supplementary material, Section B, provides bounds on the relative quality of ??wD (t) compared to ??w . Lemma 4. For every w ? W, it holds that ??wD (t) ? 5 ?m ?M ??w . Numerical results We now present numerical results based on simulations. In order to gain practical insight, we will focus on a task labelling application. A task labelling problem might arise in a crowdsourcing scenrio such as Amazon?s Mechanical Turk or Content search problems where an incoming image must be matched to known contents. The mapping to the hypothesis testing problem is as follows. The set of hypotheses J corresponds to the set of task labels, with j ? the true hypothesis being the latent task label that must be inferred. The set of W actions corresponds to W workers who perform the labelling when sampled, where pw,j (j 0 ) is the probability that worker w assigns the task the label j 0 when the true label is j. For each worker w, we will call Jw the expertise of the worker (principal set of the actions), and ?w will be the quality of the worker. We will first investigate the impact of the lack of exact knowledge, i.e., the difference between ?w and qw,j , that we call slack. We then compare our algorithm to that in [1] and that of [13] for a few scenarios of interest. 5.1 The effect of the slack Here we present a simulated scenario with J = 100, W = 15, and fixed subsets {Jw }w?W satisfying Assumption 2. We set ? ? 0.001, and assume the incoming job-type to be j ? = 1. In Figure 1 we present the results of 1000 runs of the simulation for every instance of respectively the first and second scenario described below. Recalling that the simulation stops as soon as maxj ?j (t) > 1 ? ?, we specify that out of the entire set of simulations of these scenarios the algorithm never failed to infer the correct incoming job type j ? = 1. For both scenarios, in Figure 1(left) we display the averaged sample paths of the coordinate ?j ? (t) and in Figure 1(right) the average sample size required for the decision maker to make an inference. The performance upper bound is pessimistic. In the first set of simulations, scenario A, we fix the quality vector ? with ?w ? (0.55, 0.6) for every worker w ? W. We then let the parameter s vary in {0, .05, .1, .15, .2, .25, .3} and assume qw,j ? = ?w + s for every w ? W. In Theorem 2 we proved an upper bound for E[T ? ] when the IBAG algorithm is employed. It can be observed that the upper bound does not depend on qw,j ? , but only on ?w . In fact, the upper bound is obtained by looking at the worst case scenario, where qw,j ? = ?w for every w ? W and j ? J . As the slack s grows, the performance of the algorithm drastically improves even if it is not reflected in the upper bound term. Robustness to perturbations in estimate of worker skills. In the second set of simulations, scenario B we fix the quality vector qw,j ? ? (0.85, 0.9) for every worker w ? W. We then let the parameter s vary in {0, .05, .1, .15, .2, .25, .3} and set ?w = qw,j ? ? s for every w ? W. It is observed that the IBAG algorithm performs well even when the decision maker?s knowledge of the skills is not precise, and he decides to play safe by reducing the lower bound ?(w). 6 (a) Scenario A (b) Scenario B Figure 1: ((a), (b) left) Empirical average of the sample paths of the process ?j ? (t), ((a), (b) right) Empirical average of the sample size T ? . We therefore deduce that the learning process strongly depends on the true skills of the worker qw,j (Figure 1(a)), however their exact knowledge is not fundamental for IBAG to behave well (Figure 1(b)) - it is robust to small perturbations. 5.2 Comparison to existing algorithms Chernoff algorithm. As we mentioned, most of the existing sequential hypothesis testing algorithms are based on Chernoff?s algorithm presented in [1]. Such an algorithm, at step t identifies the job-types j1 , j2 ? J associated with the two highest values of ?(t) and selects the class of workers wC that best distinguishes j1 and j2 , i.e., wC = arg maxw?Wj1 ,j2 ?w . In the asymptotic regime with ? ? 0, the expected sample size required by the Chernoff?s algorithm is of order ? log ?, exactly as with IBAG. This has been proven ([1, 8]) in the case with full knowledge of the matrix pw,j (?). What we emphasize here is that by focusing only on the two highest components of ?(t), the decision maker loses information that might help him make a better selection of worker w(t). In particular, Chernoff?s algorithm bases its decision largely on the workers? skills and thus does not behave as well as it should when these are not informative enough. Soft-Decision GBS algorithm. The algorithm proposed in [13] generalizes the intuition behind optimal GBS algorithms in noiseless environments. This P algorithm, given a belief vector P ?(t) at step t picks the worker w ? such that w ? = arg minw j?J ?j gw,j = arg minw j?Jw ?j ? P j ?J / w ?j = arg maxw {??w }. Intuitively, the Soft-Decision GBS algorithm selects the worker that is the most "unsure", in the sense that the worker splits the belief vector as evenly as possible. Since the model in [13] does not allow for different qualities of the workers (noise is symmetric there), this feature does not play a role on the worker selection policy. Note that when the quality of all workers are identical, the Soft-Decision GBS and the IBAG algorithms are identical. In [13], an asymptotic performance analysis is presented, and under certain constraints on the problem geometry, it is shown that the sample size required is of order ? log ? + log J, and once again the performance in terms of the error probability matches with IBAG. We now compare our algorithm IBAG with the Chernoff algorithm under three scenarios and with Soft-Decision GBS only for the third scenario where the quality ?w or workers (noise in GBS) differ among the workers. In the first scenario, we set J = 32, j ? = 1, and ? = 0.003. We assume two kinds of worker classes. We have 5 ?generalist? workers, each of whom has |Jw | = J/2 = 16 and moreover for every pair of job types (j1 , j2 ) there exists a generalist belonging to Wj1 ,j2 . In addition, we have 32 ?specialist? workers who can distinguish exactly 1 job-type, i.e., |Jw | = 1. We assume that there is one specialist per job-type, and note that among them there is also w? such that Jw? = {j ? }. We consider two cases: in case A, the skills of the workers are identical, ?w = 0.8 for every w ? W, and in case B we drop the generalists? skill level to ?w = 0.75. We assume qw,j = ?w for every w ? W and j ? J . In the second scenario, we set J = 30 with only specialists present. We set ? = 0.003 and j ? = 1. In this scenario we consider two cases as well, in case A ?w = 0.7 for every worker, while in case B we drop the skill level of the specialist on job-type j ? to 0.65, representing a situation where the system is ill-prepared for an incoming job. We assume qw,j = ?w for every w ? W and j ? J . We display the results for both scenarios in Figure 2. In Figure 2(top) we display boxplots of the number of queries required and in Figure 2(bottom) we show the expectation of the number of queries per kind of worker. In both scenarios, the performance of Chernoff?s algorithm is drastically 7 (a) Scenario 1 (b) Scenario 2 (c) Scenario 3 Figure 2: (top) Boxplot of the sample size T ? . (bottom) Empirical expected number of times the different groups of workers are queried. weakened by only a tiny variation in ?w , yielding a very different behavior. In the first scenario, although it is very informative to query the generalists in an early explorative stage, under Chernoff?s algorithm the selection of the workers relies too much on the skill levels and therefore always queries the specialists. The IBAG algorithm, on the other hand, sensibly decides at each step on the trade-off between getting rough information on a larger set of job pairs, or getting more precise information on a smaller set, and seems to better grasp this quality vs quantity dilemma. Similarly, in case B of the second scenario, the low-quality workers (the specialist in j ? ) are never selected by Chernoff?s algorithm, even if their responses have a large impact on the growth of ?j ? (t). For both cases A and B we see that IBAG outperforms Chernoff. In the third scenario we set J = 32, W = 42, and ? = 0.03. We have five low-quality generalist workers with ?w = 0.55, five high-quality generalist workers with ?w = 0.75. The remaining 32 workers are specialists with ?w = 0.8. The plots comparing all three algorithms is shown in Figure 2(iii). We observe again that the Chernoff algorithm never queries generalists and performs the worst. IBAG outperforms Soft-GBS because it queries high-quality workers preferentially while Soft-GBS doesn?t consider quality. 6 Discussion and conclusion We have presented and analyzed the IBAG algorithm, an intuitive active sequential learning algorithm which requires only a rough knowledge of the quality and principal set of each available action. The algorithm is shown to be competitive and in many cases outperforms Chernoff?s algorithm, the benchmark in the area. As far as we know, this is the first attempt to analyze a scenario where the decision maker has limited knowledge of the system parameters. In Section 5 we studied through simulations, the effect of this lack of exact knowledge on the performances of the system, in order to quantify the tradeoff between caution, i.e., how close ?w is to qw,j , and the cost. The numerical analysis suggests that a moderate caution does not worsen drastically the performance. In the supplement Section C we analyze formally this tradeoff and show results on how cautious the decision maker can be while still ensuring good performance. A further element of incomplete knowledge would be to allow slight perturbations on the principal sets of the actions. In the present paper we have assumed to know with certainty, for every w ? W and j ? J , whether w has j in its principal set (j ? Jw ), or not. In future work we will investigate the impact of uncertainty in the expertise, for instance having j ? Jw with some probability pj,w . 8 As a last remark, it would be interesting to analyze the model when the different actions have heterogeneous costs. Note that the IBAG algorithm naturally extends to such case, as mentioned in equation (8). The IBAG algorithm in the framework of the task-worker system could give definitive answers on whether it is better to sample a response from a cheap worker with a general expertise and low skill or from more expensive workers with narrow expertise and higher skill. References [1] H. Chernoff, ?Sequential design of experiments,? The Annals of Mathematical Statistics, vol. 30, no. 3, pp. 755?770, 1959. [2] S. Berry, B. Carlin, J. Lee, and P. Muller, Bayesian Adaptive Methods for Clinical Trials. CRC press, 2010. [3] S. C. Hui and G. Jha, ?Data mining for customer service support,? Information & Management, vol. 38, no. 1, pp. 1?13, 2000. [4] N. Vaidhiyan, S. P. Arun, and R. Sundaresan, ?Active sequential hypothesis testing with application to a visual search problem,? in 2012 IEEE International Symposium on Information Theory Proceedings (ISIT), pp. 2201?2205, IEEE, 2012. [5] B. Ghosh, ?A brief history of sequential analysis,? Handbook of Sequential Analysis, vol. 1, 1991. [6] A. Albert, ?The sequential design of experiments for infinitely many states of nature,? The Annals of Mathematical Statistics, vol. 32, pp. 774?799, 1961. [7] J. Kiefer and J. Sacks, ?Asymptotically optimum sequential inference and design,? The Annals of Mathematical Statistics, vol. 34, pp. 705?750, 1963. [8] M. Naghshvar and T. Javidi, ?Active sequential hypothesis testing,? The Annals of Statistics, vol. 41, no. 6, pp. 2703?2738, 2013. [9] A. Lalitha, A. Sarwate, and T. Javidi, ?Social learning and distributed hypothesis testing,? in Information Theory (ISIT), 2014 IEEE International Symposium on, pp. 551?555, IEEE, 2014. [10] R. Olfati-Saber, J. Fax, and R. Murray, ?Consensus and cooperation in networked multi-agent systems,? Proceedings of the IEEE, vol. 95, no. 1, pp. 215?233, 2007. [11] M. Naghshvar, T. Javidi, and K. Chaudhuri, ?Noisy bayesian active learning,? in Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on, pp. 1626?1633, IEEE, 2012. [12] M. Naghshvar and T. Javidi, ?Extrinsic jensen-shannon divergence with application in active hypothesis testing,? in IEEE International Symposium on Information Theory (ISIT), 2012. [13] R. Nowak, ?Noisy generalized binary search,? in Advances in neural information processing systems, pp. 1366?1374, 2009. 9
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Streaming Weak Submodularity: Interpreting Neural Networks on the Fly Ethan R. Elenberg Department of Electrical and Computer Engineering The University of Texas at Austin [email protected] Moran Feldman Department of Mathematics and Computer Science Open University of Israel [email protected] Alexandros G. Dimakis Department of Electrical and Computer Engineering The University of Texas at Austin [email protected] Amin Karbasi Department of Electrical Engineering Department of Computer Science Yale University [email protected] Abstract In many machine learning applications, it is important to explain the predictions of a black-box classifier. For example, why does a deep neural network assign an image to a particular class? We cast interpretability of black-box classifiers as a combinatorial maximization problem and propose an efficient streaming algorithm to solve it subject to cardinality constraints. By extending ideas from Badanidiyuru et al. [2014], we provide a constant factor approximation guarantee for our algorithm in the case of random stream order and a weakly submodular objective function. This is the first such theoretical guarantee for this general class of functions, and we also show that no such algorithm exists for a worst case stream order. Our algorithm obtains similar explanations of Inception V3 predictions 10 times faster than the state-of-the-art LIME framework of Ribeiro et al. [2016]. 1 Introduction Consider the following combinatorial optimization problem. Given a ground set N of N elements and a set function f : 2N 7! R 0 , find the set S of size k which maximizes f (S). This formulation is at the heart of many machine learning applications such as sparse regression, data summarization, facility location, and graphical model inference. Although the problem is intractable in general, if f is assumed to be submodular then many approximation algorithms have been shown to perform provably within a constant factor from the best solution. Some disadvantages of the standard greedy algorithm of Nemhauser et al. [1978] for this problem are that it requires repeated access to each data element and a large total number of function evaluations. This is undesirable in many large-scale machine learning tasks where the entire dataset cannot fit in main memory, or when a single function evaluation is time consuming. In our main application, each function evaluation corresponds to inference on a large neural network and can take a few seconds. In contrast, streaming algorithms make a small number of passes (often only one) over the data and have sublinear space complexity, and thus, are ideal for tasks of the above kind. Recent ideas, algorithms, and techniques from submodular set function theory have been used to derive similar results in much more general settings. For example, Elenberg et al. [2016a] used the concept of weak submodularity to derive approximation and parameter recovery guarantees for 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. nonlinear sparse regression. Thus, a natural question is whether recent results on streaming algorithms for maximizing submodular functions [Badanidiyuru et al., 2014, Buchbinder et al., 2015, Chekuri et al., 2015] extend to the weakly submodular setting. This paper answers the above question by providing the first analysis of a streaming algorithm for any class of approximately submodular functions. We use key algorithmic components of S IEVE -S TREAMING [Badanidiyuru et al., 2014], namely greedy thresholding and binary search, combined with a novel analysis to prove a constant factor approximation for -weakly submodular functions (defined in Section 3). Specifically, our contributions are as follows. ? An impossibility result showing that, even for 0.5-weakly submodular objectives, no randomized streaming algorithm which uses o(N ) memory can have a constant approximation ratio when the ground set elements arrive in a worst case order. ? S TREAK: a greedy, deterministic streaming algorithm for maximizing -weakly submodular functions which uses O(" 1 k log k) memory and has an approximation ratio of (1 ") 2 ? p (3 e /2 2 2 e /2 ) when the ground set elements arrive in a random order. ? An experimental evaluation of our algorithm in two applications: nonlinear sparse regression using pairwise products of features and interpretability of black-box neural network classifiers. The above theoretical impossibility result is quite surprising since it stands in sharp contrast to known streaming algorithms for submodular objectives achieving a constant approximation ratio even for worst case stream order. One advantage of our approach is that, while our approximation guarantees are in terms of , our algorithm S TREAK runs without requiring prior knowledge about the value of . This is important since the weak submodularity parameter is hard to compute, especially in streaming applications, as a single element can alter drastically. We use our streaming algorithm for neural network interpretability on Inception V3 [Szegedy et al., 2016]. For that purpose, we define a new set function maximization problem similar to LIME [Ribeiro et al., 2016] and apply our framework to approximately maximize this function. Experimentally, we find that our interpretability method produces explanations of similar quality as LIME, but runs approximately 10 times faster. 2 Related Work Monotone submodular set function maximization has been well studied, starting with the classical analysis of greedy forward selection subject to a matroid constraint [Nemhauser et al., 1978, Fisher et al., 1978]. For the special case of a uniform matroid constraint, the greedy algorithm achieves an approximation ratio of 1 1/e [Fisher et al., 1978], and a more involved algorithm obtains this ratio also for general matroid constraints [C?alinescu et al., 2011]. In general, no polynomial-time algorithm can have a better approximation ratio even for a uniform matroid constraint [Nemhauser and Wolsey, 1978, Feige, 1998]. However, it is possible to improve upon this bound when the data obeys some additional guarantees [Conforti and Cornu?jols, 1984, Vondr?k, 2010, Sviridenko et al., 2015]. For maximizing nonnegative, not necessarily monotone, submodular functions subject to a general matroid constraint, the state-of-the-art randomized algorithm achieves an approximation ratio of 0.385 [Buchbinder and Feldman, 2016b]. Moreover, for uniform matroids there is also a deterministic algorithm achieving a slightly worse approximation ratio of 1/e [Buchbinder and Feldman, 2016a]. The reader is referred to Bach [2013] and Krause and Golovin [2014] for surveys on submodular function theory. A recent line of work aims to develop new algorithms for optimizing submodular functions suitable for large-scale machine learning applications. Algorithmic advances of this kind include S TOCHASTIC -G REEDY [Mirzasoleiman et al., 2015], S IEVE -S TREAMING [Badanidiyuru et al., 2014], and several distributed approaches [Mirzasoleiman et al., 2013, Barbosa et al., 2015, 2016, Pan et al., 2014, Khanna et al., 2017b]. Our algorithm extends ideas found in S IEVE -S TREAMING and uses a different analysis to handle more general functions. Additionally, submodular set functions have been used to prove guarantees for online and active learning problems [Hoi et al., 2006, Wei et al., 2015, Buchbinder et al., 2015]. Specifically, in the online setting corresponding to our setting 2 (i.e., maximizing a monotone function subject to a cardinality constraint), Chan et al. [2017] achieve a competitive ratio of about 0.3178 when the function is submodular. The concept of weak submodularity was introduced in Krause and Cevher [2010], Das and Kempe [2011], where it was applied to the specific problem of feature selection in linear regression. Their main results state that if the data covariance matrix is not too correlated (using either incoherence or restricted eigenvalue assumptions), then maximizing the goodness of fit f (S) = RS2 as a function of the feature set S is weakly submodular. This leads to constant factor approximation guarantees for several greedy algorithms. Weak submodularity was connected with Restricted Strong Convexity in Elenberg et al. [2016a,b]. This showed that the same assumptions which imply the success of regularization also lead to guarantees on greedy algorithms. This framework was later used for additional algorithms and applications [Khanna et al., 2017a,b]. Other approximate versions of submodularity were used for greedy selection problems in Horel and Singer [2016], Hassidim and Singer [2017], Altschuler et al. [2016], Bian et al. [2017]. To the best of our knowledge, this is the first analysis of streaming algorithms for approximately submodular set functions. Increased interest in interpretable machine learning models has led to extensive study of sparse feature selection methods. For example, Bahmani et al. [2013] consider greedy algorithms for logistic regression, and Yang et al. [2016] solve a more general problem using `1 regularization. Recently, Ribeiro et al. [2016] developed a framework called LIME for interpreting black-box neural networks, and Sundararajan et al. [2017] proposed a method that requires access to the network?s gradients with respect to its inputs. We compare our algorithm to variations of LIME in Section 6.2. 3 Preliminaries First we establish some definitions and notation. Sets are denoted with capital letters, and all big O notation is assumed to be scaling with respect to N (the number of elements in the input stream). Given a set function f , we often use the discrete derivative f (B | A) , f (A [ B) f (A). f is monotone if f (B | A) 0, 8A, B and nonnegative if f (A) 0, 8A. Using this notation one can define weakly submodular functions based on the following ratio. Definition 3.1 (Weak Submodularity, adapted from Das and Kempe [2011]). A monotone nonnegative set function f : 2N 7! R 0 is called -weakly submodular for an integer r if P j2S\L f (j | L) ? r, min , L,S?N : f (S | L) |L|,|S\L|?r where the ratio is considered to be equal to 1 when its numerator and denominator are both 0. This generalizes submodular functions by relaxing the diminishing returns property of discrete derivatives. It is easy to show that f is submodular if and only if |N | = 1. Definition 3.2 (Approximation Ratio). A streaming maximization algorithm ALG which returns a set S has approximation ratio R 2 [0, 1] if E[f (S)] R ? f (OP T ), where OP T is the optimal solution and the expectation is over the random decisions of the algorithm and the randomness of the input stream order (when it is random). Formally our problem is as follows. Assume that elements from a ground set N arrive in a stream at either random or worst case order. The goal is then to design a one pass streaming algorithm that given oracle access to a nonnegative set function f : 2N 7! R 0 maintains at most o(N ) elements in memory and returns a set S of size at most k approximating max f (T ) , |T |?k up to an approximation ratio R( k ). Ideally, this approximation ratio should be as large as possible, and we also want it to be a function of k and nothing else. In particular, we want it to be independent of k and N . To simplify notation, we use in place of k in the rest of the paper. Additionally, proofs for all our theoretical results are deferred to the Supplementary Material. 3 4 Impossibility Result To prove our negative result showing that no streaming algorithm for our problem has a constant approximation ratio against a worst case stream order, we first need to construct a weakly submodular set function fk . Later we use it to construct a bad instance for any given streaming algorithm. Fix some k 1, and consider the ground set Nk = {ui , vi }ki=1 . For ease of notation, let us define for every subset S ? Nk u(S) = |S \ {ui }ki=1 | , v(S) = |S \ {vi }ki=1 | . Now we define the following set function: fk (S) = min{2 ? u(S) + 1, 2 ? v(S)} 8 S ? Nk . Lemma 4.1. fk is nonnegative, monotone and 0.5-weakly submodular for the integer |Nk |. Since |Nk | = 2k, the maximum value of fk is fk (Nk ) = 2 ? v(Nk ) = 2k. We now extend the ground set of fk by adding to it an arbitrary large number d of dummy elements which do not affect fk at all. Clearly, this does not affect the properties of fk proved in Lemma 4.1. However, the introduction of dummy elements allows us to assume that k is an arbitrary small value compared to N , which is necessary for the proof of the next theorem. In a nutshell, this proof is based on the observation that the elements of {ui }ki=1 are indistinguishable from the dummy elements as long as no element of {vi }ki=1 has arrived yet. Theorem 4.2. For every constant c 2 (0, 1] there is a large enough k such that no randomized streaming algorithm that uses o(N ) memory to solve max|S|?2k fk (S) has an approximation ratio of c for a worst case stream order. We note that fk has strong properties. In particular, Lemma 4.1 implies that it is 0.5-weakly submodular for every 0 ? r ? |N |. In contrast, the algorithm we show later assumes weak submodularity only for the cardinality constraint k. Thus, the above theorem implies that worst case stream order precludes a constant approximation ratio even for functions with much stronger properties compared to what is necessary for getting a constant approximation ratio when the order is random. The proof of Theorem 4.2 relies critically on the fact that each element is seen exactly once. In other words, once the algorithm decides to discard an element from its memory, this element is gone forever, which is a standard assumption for streaming algorithms. Thus, the theorem does not apply to algorithms that use multiple passes over N , or non-streaming algorithms that use o(N ) writable memory, and their analysis remains an interesting open problem. 5 Streaming Algorithms In this section we give a deterministic streaming algorithm for our problem which works in a model in which the stream contains the elements of N in a random order. We first describe in Section 5.1 such a streaming algorithm p assuming access to a value ? which approximates a ? f (OP T ), where a is a shorthand for a = ( 2 e /2 1)/2. Then, in Section 5.2 we explain how this assumption can be removed to obtain S TREAK and bound its approximation ratio, space complexity, and running time. 5.1 Algorithm with access to ? Consider Algorithm 1. In addition to the input instance, this algorithm gets a parameter ? 2 [0, a ? f (OP T )]. One should think of ? as close to a ? f (OP T ), although the following analysis of the algorithm does not rely on it. We provide an outline of the proof, but defer the technical details to the Supplementary Material. Theorem 5.1. The expected value of the set produced by Algorithm 1 is at least p p ? 3 e /2 2 2 e /2 ? = ? ? ( 2 e /2 1) . a 2 4 Algorithm 1 T HRESHOLD G REEDY (f, k, ? ) Let S ?. while there are more elements do Let u be the next element. if |S| < k and f (u | S) ? /k then Update S S [ {u}. end if end while return: S Algorithm 2 S TREAK(f, k, ") Let m 0, and let I be an (originally empty) collection of instances of Algorithm 1. while there are more elements do Let u be the next element. if f (u) m then Update m f (u) and um u. end if Update I so that it contains an instance of Algorithm 1 with ? = x for every x 2 {(1 ")i | i 2 Z and (1 ")m/(9k 2 ) ? (1 ")i ? mk}, as explained in Section 5.2. Pass u to all instances of Algorithm 1 in I. end while return: the best set among all the outputs of the instances of Algorithm 1 in I and the singleton set {um }. Proof (Sketch). Let E be the event that f (S) < ? , where S is the output produced by Algorithm 1. Clearly f (S) ? whenever E does not occur, and thus, it is possible to lower bound the expected value of f (S) using E as follows. Observation 5.2. Let S denote the output of Algorithm 1, then E[f (S)] (1 Pr[E]) ? ? . The lower bound given by Observation 5.2 is decreasing in Pr[E]. Proposition 5.4 provides another lower bound for E[f (S)] which increases with Pr[E]. An important ingredient of the proof of this proposition is the next observation, which implies that the solution produced by Algorithm 1 is always of size smaller than k when E happens. Observation 5.3. If at some point Algorithm 1 has a set S of size k, then f (S) ?. The proof of Proposition 5.4 is based on the above observation and on the observation that the random arrival order implies that every time that an element of OP T arrives in the stream we may assume it is a random element out of all the OP T elements that did not arrive yet. Proposition 5.4. For the set S produced by Algorithm 1, 1 ? E[f (S)] ? ? [Pr[E] e /2 ] ? f (OP T ) 2 2? ? . The theorem now follows by showing that for every possible value of Pr[E] the guarantee of the theorem is impliedp by either Observation 5.2 or Proposition 5.4.pSpecifically, the former happens when Pr[E] ? 2 2 e /2 and the later when Pr[E] 2 2 e /2 . 5.2 Algorithm without access to ? In this section we explain how to get an algorithm which does not depend on ? . Instead, S TREAK (Algorithm 2) receives an accuracy parameter " 2 (0, 1). Then, it uses " to run several instances of Algorithm 1 stored in a collection denoted by I. The algorithm maintains two variables throughout its execution: m is the maximum value of a singleton set corresponding to an element that the algorithm already observed, and um references an arbitrary element satisfying f (um ) = m. 5 The collection I is updated as follows after each element arrival. If previously I contained an instance of Algorithm 1 with a given value for ? , and it no longer should contain such an instance, then the instance is simply removed. In contrast, if I did not contain an instance of Algorithm 1 with a given value for ? , and it should now contain such an instance, then a new instance with this value for ? is created. Finally, if I contained an instance of Algorithm 1 with a given value for ? , and it should continue to contain such an instance, then this instance remains in I as is. Theorem 5.5. The approximation ratio of S TREAK is at least p 3 e /2 2 2 e (1 ") ? 2 /2 . The proof of Theorem 5.5 shows that in the final collection I there is an instance of Algorithm 1 whose ? provides a good approximation for a ? f (OP T ), and thus, this instance of Algorithm 1 should (up to some technical details) produce a good output set in accordance with Theorem 5.1. It remains to analyze the space complexity and running time of S TREAK. We concentrate on bounding the number of elements S TREAK keeps in its memory at any given time, as this amount dominates the space complexity as long as we assume that the space necessary to keep an element is at least as large as the space necessary to keep each one of the numbers used by the algorithm. Theorem 5.6. The space complexity of S TREAK is O(" 1 k log k) elements. The running time of Algorithm 1 is O(N f ) where, abusing notation, f is the running time of a single oracle evaluation of f . Therefore, the running time of S TREAK is O(N f " 1 log k) since it uses at every given time only O(" 1 log k) instances of the former algorithm. Given multiple threads, this can be improved to O(N f + " 1 log k) by running the O(" 1 log k) instances of Algorithm 1 in parallel. 6 Experiments Running Time (s) 600 400 200 0 Random Streak(0.75) Streak(0.1) Local Search 1.00 Oracle Evaluations Generalization Accuracy Log Likelihood We evaluate the performance of our streaming algorithm on two sparse feature selection applications.1 Features are passed to all algorithms in a random order to match the setting of Section 5. 0.95 0.90 0.85 0.80 0.75 0.70 Random Streak(0.75) k=20 Streak(0.1) Local Search k=40 k=80 15000 10000 5000 0 Random Streak(0.75) Streak(0.1) Local Search Random Streak(0.75) Streak(0.1) Local Search 400000 300000 200000 100000 0 k=20 (a) Performance k=40 k=80 (b) Cost Figure 1: Logistic Regression, Phishing dataset with pairwise feature products. Our algorithm is comparable to L OCAL S EARCH in both log likelihood and generalization accuracy, with much lower running time and number of model fits in most cases. Results averaged over 40 iterations, error bars show 1 standard deviation. 6.1 Sparse Regression with Pairwise Features In this experiment, a sparse logistic regression is fit on 2000 training and 2000 test observations from the Phishing dataset [Lichman, 2013]. This setup is known to be weakly submodular under mild data assumptions [Elenberg et al., 2016a]. First, the categorical features are one-hot encoded, increasing 1 Code for these experiments is available at https://github.com/eelenberg/streak. 6 700 2500 2000 Running Time (s) Log Likelihood 650 600 Random Streak(0.75) Streak(0.5) Streak(0.2) Streak(0.1) Streak(0.05) Local Search 550 500 0 10 101 102 103 104 1500 1000 500 0 105 Running Time (s) (a) Sparse Regression LIME + Max Wts LIME + FS LIME + Lasso Streak (b) Interpretability Figure 2: 2(a): Logistic Regression, Phishing dataset with pairwise feature products, k = 80 features. By varying the parameter ", our algorithm captures a time-accuracy tradeoff between R ANDOM S UBSET and L OCAL S EARCH. Results averaged over 40 iterations, standard deviation shown with error bars. 2(b): Running times of interpretability algorithms on the Inception V3 network, N = 30, k = 5. Streaming maximization runs 10 times faster than the LIME framework. Results averaged over 40 total iterations using 8 example explanations, error bars show 1 standard deviation. the feature dimension to 68. Then, all pairwise products are added for a total of N = 4692 features. To reduce computational cost, feature products are generated and added to the stream on-the-fly as needed. We compare with 2 other algorithms. R ANDOM S UBSET selects the first k features from the random stream. L OCAL S EARCH first fills a buffer with the first k features, and then swaps each incoming feature with the feature from the buffer which yields the largest nonnegative improvement. Figure 1(a) shows both the final log likelihood and the generalization accuracy for R ANDOM S UBSET, L OCAL S EARCH, and our S TREAK algorithm for " = {0.75, 0.1} and k = {20, 40, 80}. As expected, the R ANDOM S UBSET algorithm has much larger variation since its performance depends highly on the random stream order. It also performs significantly worse than L OCAL S EARCH for both metrics, whereas S TREAK is comparable for most parameter choices. Figure 1(b) shows two measures of computational cost: running time and the number of oracle evaluations (regression fits). We note S TREAK scales better as k increases; for example, S TREAK with k = 80 and " = 0.1 (" = 0.75) runs in about 70% (5%) of the time it takes to run L OCAL S EARCH with k = 40. Interestingly, our speedups are more substantial with respect to running time. In some cases S TREAK actually fits more regressions than L OCAL S EARCH, but still manages to be faster. We attribute this to the fact that nearly all of L OCAL S EARCH?s regressions involve k features, which are slower than many of the small regressions called by S TREAK. Figure 2(a) shows the final log likelihood versus running time for k = 80 and " 2 [0.05, 0.75]. By varying the precision ", we achieve a gradual tradeoff between speed and performance. This shows that S TREAK can reduce the running time by over an order of magnitude with minimal impact on the final log likelihood. 6.2 Black-Box Interpretability Our next application is interpreting the predictions of black-box machine learning models. Specifically, we begin with the Inception V3 deep neural network [Szegedy et al., 2016] trained on ImageNet. We use this network for the task of classifying 5 types of flowers via transfer learning. This is done by adding a final softmax layer and retraining the network. We compare our approach to the LIME framework [Ribeiro et al., 2016] for developing sparse, interpretable explanations. The final step of LIME is to fit a k-sparse linear regression in the space of interpretable features. Here, the features are superpixels determined by the SLIC image segmentation algorithm [Achanta et al., 2012] (regions from any other segmentation would also suffice). The number of superpixels is bounded by N = 30. After a feature selection step, a final regression is performed on only the selected features. The following feature selection methods are supplied by 7 LIME: 1. Highest Weights: fits a full regression and keep the k features with largest coefficients. 2. Forward Selection: standard greedy forward selection. 3. Lasso: `1 regularization. We introduce a novel method for black-box interpretability that is similar to but simpler than LIME. As before, we segment an image into N superpixels. Then, for a subset S of those regions we can create a new image that contains only these regions and feed this into the black-box classifier. For a given model M , an input image I, and a label L1 we ask for an explanation: why did model M label image I with label L1 . We propose the following solution to this problem. Consider the set function f (S) giving the likelihood that image I(S) has label L1 . We approximately solve max f (S) , |S|?k using S TREAK. Intuitively, we are limiting the number of superpixels to k so that the output will include only the most important superpixels, and thus, will represent an interpretable explanation. In our experiments we set k = 5. Note that the set function f (S) depends on the black-box classifier and is neither monotone nor submodular in general. Still, we find that the greedy maximization algorithm produces very good explanations for the flower classifier as shown in Figure 3 and the additional experiments in the Supplementary Material. Figure 2(b) shows that our algorithm is much faster than the LIME approach. This is primarily because LIME relies on generating and classifying a large set of randomly perturbed example images. 7 Conclusions We propose S TREAK, the first streaming algorithm for maximizing weakly submodular functions, and prove that it achieves a constant factor approximation assuming a random stream order. This is useful when the set function is not submodular and, additionally, takes a long time to evaluate or has a very large ground set. Conversely, we show that under a worst case stream order no algorithm with memory sublinear in the ground set size has a constant factor approximation. We formulate interpretability of black-box neural networks as set function maximization, and show that S TREAK provides interpretable explanations faster than previous approaches. We also show experimentally that S TREAK trades off accuracy and running time in nonlinear sparse regression. One interesting direction for future work is to tighten the bounds of Theorems 5.1 and 5.5, which are nontrivial but somewhat loose. For example, there is a gap between the theoretical guarantee of the state-of-the-art algorithm for submodular functions and our bound for = 1. However, as our algorithm performs the same computation as that state-of-the-art algorithm when the function is submodular, this gap is solely an analysis issue. Hence, the real theoretical performance of our algorithm is better than what we have been able to prove in Section 5. 8 Acknowledgments This research has been supported by NSF Grants CCF 1344364, 1407278, 1422549, 1618689, ARO YIP W911NF-14-1-0258, ISF Grant 1357/16, Google Faculty Research Award, and DARPA Young Faculty Award (D16AP00046). 8 (a) (b) (c) (d) Figure 3: Comparison of interpretability algorithms for the Inception V3 deep neural network. We have used transfer learning to extract features from Inception and train a flower classifier. In these four input images the flower types were correctly classified (from (a) to (d): rose, sunflower, daisy, and daisy). We ask the question of interpretability: why did this model classify this image as rose. We are using our framework (and the recent prior work LIME [Ribeiro et al., 2016]) to see which parts of the image the neural network is looking at for these classification tasks. As can be seen S TREAK correctly identifies the flower parts of the images while some LIME variations do not. More importantly, S TREAK is creating subsampled images on-the-fly, and hence, runs approximately 10 times faster. Since interpretability tasks perform multiple calls to the black-box model, the running times can be quite significant. 9 References Radhakrishna Achanta, Appu Shaji, Kevin Smith, Aurelien Lucchi, Pascal Fua, and Sabine S?sstrunk. SLIC Superpixels Compared to State-of-the-art Superpixel Methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(11):2274?2282, 2012. Jason Altschuler, Aditya Bhaskara, Gang (Thomas) Fu, Vahab Mirrokni, Afshin Rostamizadeh, and Morteza Zadimoghaddam. Greedy Column Subset Selection: New Bounds and Distributed Algorithms. In ICML, pages 2539?2548, 2016. Francis R. Bach. Learning with Submodular Functions: A Convex Optimization Perspective. Foundations and Trends in Machine Learning, 6, 2013. Ashwinkumar Badanidiyuru, Baharan Mirzasoleiman, Amin Karbasi, and Andreas Krause. Streaming Submodular Maximization: Massive Data Summarization on the Fly. In KDD, pages 671?680, 2014. Sohail Bahmani, Bhiksha Raj, and Petros T. Boufounos. Greedy Sparsity-Constrained Optimization. Journal of Machine Learning Research, 14:807?841, 2013. Rafael da Ponte Barbosa, Alina Ene, Huy L. Nguyen, and Justin Ward. The Power of Randomization: Distributed Submodular Maximization on Massive Datasets. In ICML, pages 1236?1244, 2015. Rafael da Ponte Barbosa, Alina Ene, Huy L. Nguyen, and Justin Ward. A New Framework for Distributed Submodular Maximization. In FOCS, pages 645?654, 2016. Andrew An Bian, Baharan Mirzasoleiman, Joachim M. Buhmann, and Andreas Krause. Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains. In AISTATS, pages 111?120, 2017. Niv Buchbinder and Moran Feldman. Deterministic Algorithms for Submodular Maximization Problems. In SODA, pages 392?403, 2016a. Niv Buchbinder and Moran Feldman. Constrained Submodular Maximization via a Non-symmetric Technique. CoRR, abs/1611.03253, 2016b. URL http://arxiv.org/abs/1611.03253. Niv Buchbinder, Moran Feldman, and Roy Schwartz. Online Submodular Maximization with Preemption. In SODA, pages 1202?1216, 2015. Gruia C?alinescu, Chandra Chekuri, Martin P?l, and Jan Vondr?k. Maximizing a Monotone Submodular Function Subject to a Matroid Constraint. SIAM J. Comput., 40(6):1740?1766, 2011. T-H. Hubert Chan, Zhiyi Huang, Shaofeng H.-C. Jiang, Ning Kang, and Zhihao Gavin Tang. Online Submodular Maximization with Free Disposal: Randomization Beats 1/4 for Partition Matroids. In SODA, pages 1204?1223, 2017. Chandra Chekuri, Shalmoli Gupta, and Kent Quanrud. Streaming Algorithms for Submodular Function Maximization. In ICALP, pages 318?330, 2015. Michele Conforti and G?rard Cornu?jols. Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discrete Applied Mathematics, 7(3):251?274, March 1984. Abhimanyu Das and David Kempe. Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection. In ICML, pages 1057?1064, 2011. Ethan R. Elenberg, Rajiv Khanna, Alexandros G. Dimakis, and Sahand Negahban. Restricted Strong Convexity Implies Weak Submodularity. CoRR, abs/1612.00804, 2016a. URL http: //arxiv.org/abs/1612.00804. Ethan R. Elenberg, Rajiv Khanna, Alexandros G. Dimakis, and Sahand Negahban. Restricted Strong Convexity Implies Weak Submodularity. In NIPS Workshop on Learning in High Dimensions with Structure, 2016b. Uriel Feige. A Threshold of ln n for Approximating Set Cover. Journal of the ACM (JACM), 45(4): 634?652, 1998. 10 Marshall L. Fisher, George L. Nemhauser, and Laurence A. Wolsey. An analysis of approximations for maximizing submodular set functions?II. In M. L. Balinski and A. J. Hoffman, editors, Polyhedral Combinatorics: Dedicated to the memory of D.R. Fulkerson, pages 73?87. Springer Berlin Heidelberg, Berlin, Heidelberg, 1978. Avinatan Hassidim and Yaron Singer. Submodular Optimization Under Noise. In COLT, pages 1069?1122, 2017. Steven C. H. Hoi, Rong Jin, Jianke Zhu, and Michael R. Lyu. Batch Mode Active Learning and its Application to Medical Image Classification. In ICML, pages 417?424, 2006. Thibaut Horel and Yaron Singer. Maximization of Approximately Submodular Functions. In NIPS, 2016. Rajiv Khanna, Ethan R. Elenberg, Alexandros G. Dimakis, Joydeep Ghosh, and Sahand Negahban. On Approximation Guarantees for Greedy Low Rank Optimization. In ICML, pages 1837?1846, 2017a. Rajiv Khanna, Ethan R. Elenberg, Alexandros G. Dimakis, Sahand Negahban, and Joydeep Ghosh. Scalable Greedy Support Selection via Weak Submodularity. In AISTATS, pages 1560?1568, 2017b. Andreas Krause and Volkan Cevher. Submodular Dictionary Selection for Sparse Representation. In ICML, pages 567?574, 2010. Andreas Krause and Daniel Golovin. Submodular Function Maximization. Tractability: Practical Approaches to Hard Problems, 3:71?104, 2014. Moshe Lichman. UCI machine learning repository, 2013. URL http://archive.ics.uci.edu/ ml. Baharan Mirzasoleiman, Amin Karbasi, Rik Sarkar, and Andreas Krause. Distributed Submodular Maximization: Identifying Representative Elements in Massive Data. NIPS, pages 2049?2057, 2013. Baharan Mirzasoleiman, Ashwinkumar Badanidiyuru, Amin Karbasi, Jan Vondr?k, and Andreas Krause. Lazier Than Lazy Greedy. In AAAI, pages 1812?1818, 2015. George L. Nemhauser and Laurence A. Wolsey. Best Algorithms for Approximating the Maximum of a Submodular Set Function. Math. Oper. Res., 3(3):177?188, August 1978. George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions?I. Mathematical Programming, 14(1):265?294, 1978. Xinghao Pan, Stefanie Jegelka, Joseph E. Gonzalez, Joseph K. Bradley, and Michael I. Jordan. Parallel Double Greedy Submodular Maximization. In NIPS, pages 118?126, 2014. Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. ?Why Should I Trust You?? Explaining the Predictions of Any Classifier. In KDD, pages 1135?1144, 2016. Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic Attribution for Deep Networks. In ICML, pages 3319?3328, 2017. Maxim Sviridenko, Jan Vondr?k, and Justin Ward. Optimal approximation for submodular and supermodular optimization with bounded curvature. In SODA, pages 1134?1148, 2015. Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the Inception Architecture for Computer Vision. In CVPR, pages 2818?2826, 2016. Jan Vondr?k. Submodularity and curvature: the optimal algorithm. RIMS K?ky?roku Bessatsu B23, pages 253?266, 2010. Kai Wei, Iyer Rishabh, and Jeff Bilmes. Submodularity in Data Subset Selection and Active Learning. ICML, pages 1954?1963, 2015. Zhuoran Yang, Zhaoran Wang, Han Liu, Yonina C. Eldar, and Tong Zhang. Sparse Nonlinear Regression: Parameter Estimation and Asymptotic Inference. ICML, pages 2472?2481, 2016. 11
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Successor Features for Transfer in Reinforcement Learning Andr? Barreto, Will Dabney, R?mi Munos, Jonathan J. Hunt, Tom Schaul, David Silver, Hado van Hasselt {andrebarreto,wdabney,munos,jjhunt,schaul,davidsilver,hado}@google.com DeepMind Abstract Transfer in reinforcement learning refers to the notion that generalization should occur not only within a task but also across tasks. We propose a transfer framework for the scenario where the reward function changes between tasks but the environment?s dynamics remain the same. Our approach rests on two key ideas: successor features, a value function representation that decouples the dynamics of the environment from the rewards, and generalized policy improvement, a generalization of dynamic programming?s policy improvement operation that considers a set of policies rather than a single one. Put together, the two ideas lead to an approach that integrates seamlessly within the reinforcement learning framework and allows the free exchange of information across tasks. The proposed method also provides performance guarantees for the transferred policy even before any learning has taken place. We derive two theorems that set our approach in firm theoretical ground and present experiments that show that it successfully promotes transfer in practice, significantly outperforming alternative methods in a sequence of navigation tasks and in the control of a simulated robotic arm. 1 Introduction Reinforcement learning (RL) provides a framework for the development of situated agents that learn how to behave while interacting with the environment [21]. The basic RL loop is defined in an abstract way so as to capture only the essential aspects of this interaction: an agent receives observations and selects actions to maximize a reward signal. This setup is generic enough to describe tasks of different levels of complexity that may unroll at distinct time scales. For example, in the task of driving a car, an action can be to turn the wheel, make a right turn, or drive to a given location. Clearly, from the point of view of the designer, it is desirable to describe a task at the highest level of abstraction possible. However, by doing so one may overlook behavioral patterns and inadvertently make the task more difficult than it really is. The task of driving to a location clearly encompasses the subtask of making a right turn, which in turn encompasses the action of turning the wheel. In learning how to drive an agent should be able to identify and exploit such interdependencies. More generally, the agent should be able to break a task into smaller subtasks and use knowledge accumulated in any subset of those to speed up learning in related tasks. This process of leveraging knowledge acquired in one task to improve performance on other tasks is called transfer [25, 11]. In this paper we look at one specific type of transfer, namely, when subtasks correspond to different reward functions defined in the same environment. This setup is flexible enough to allow transfer to happen at different levels. In particular, by appropriately defining the rewards one can induce different task decompositions. For instance, the type of hierarchical decomposition involved in the driving example above can be induced by changing the frequency at which rewards are delivered: 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. a positive reinforcement can be given after each maneuver that is well executed or only at the final destination. Obviously, one can also decompose a task into subtasks that are independent of each other or whose dependency is strictly temporal (that is, when tasks must be executed in a certain order but no single task is clearly ?contained? within another). The types of task decomposition discussed above potentially allow the agent to tackle more complex problems than would be possible were the tasks modeled as a single monolithic challenge. However, in order to apply this divide-and-conquer strategy to its full extent the agent should have an explicit mechanism to promote transfer between tasks. Ideally, we want a transfer approach to have two important properties. First, the flow of information between tasks should not be dictated by a rigid diagram that reflects the relationship between the tasks themselves, such as hierarchical or temporal dependencies. On the contrary, information should be exchanged across tasks whenever useful. Second, rather than being posed as a separate problem, transfer should be integrated into the RL framework as much as possible, preferably in a way that is almost transparent to the agent. In this paper we propose an approach for transfer that has the two properties above. Our method builds on two conceptual pillars that complement each other. The first is a generalization of Dayan?s [7] successor representation. As the name suggests, in this representation scheme each state is described by a prediction about the future occurrence of all states under a fixed policy. We present a generalization of Dayan?s idea which extends the original scheme to continuous spaces and also facilitates the use of approximation. We call the resulting scheme successor features. As will be shown, successor features lead to a representation of the value function that naturally decouples the dynamics of the environment from the rewards, which makes them particularly suitable for transfer. The second pillar of our framework is a generalization of Bellman?s [3] classic policy improvement theorem that extends the original result from one to multiple decision policies. This novel result shows how knowledge about a set of tasks can be transferred to a new task in a way that is completely integrated within RL. It also provides performance guarantees on the new task before any learning has taken place, which opens up the possibility of constructing a library of ?skills? that can be reused to solve previously unseen tasks. In addition, we present a theorem that formalizes the notion that an agent should be able to perform well on a task if it has seen a similar task before?something clearly desirable in the context of transfer. Combined, the two results above not only set our approach in firm ground but also outline the mechanics of how to actually implement transfer. We build on this knowledge to propose a concrete method and evaluate it in two environments, one encompassing a sequence of navigation tasks and the other involving the control of a simulated two-joint robotic arm. 2 Background and problem formulation As usual, we assume that the interaction between agent and environment can be modeled as a Markov decision process (MDP, Puterman, [19]). An MDP is defined as a tuple M ? (S, A, p, R, ?). The sets S and A are the state and action spaces, respectively; here we assume that S and A are finite whenever such an assumption facilitates the presentation, but most of the ideas readily extend to continuous spaces. For each s ? S and a ? A the function p(?|s, a) gives the next-state distribution upon taking action a in state s. We will often refer to p(?|s, a) as the dynamics of the MDP. The reward received at a transition s ? ? s0 is given by the random variable R(s, a, s0 ); usually one is interested in the expected value of this variable, which we will denote by r(s, a, s0 ) or by r(s, a) = ES 0 ?p(?|s,a) [r(s, a, S 0 )]. The discount factor ? ? [0, 1) gives smaller weights to future rewards. The objective of the agent in RL is to find a policy ??a mapping from states P to actions?that ? maximizes the expected discounted sum of rewards, also called the return Gt = i=0 ? i Rt+i+1 , where Rt = R(St , At , St+1 ). One way to address this problem is to use methods derived from dynamic programming (DP), which heavily rely on the concept of a value function [19]. The action-value function of a policy ? is defined as Q? (s, a) ? E? [Gt | St = s, At = a] , (1) ? where E [?] denotes expected value when following policy ?. Once the action-value function of a particular policy ? is known, we can derive a new policy ? 0 which is greedy with respect to Q? (s, a), that is, ? 0 (s) ? argmaxa Q? (s, a). Policy ? 0 is guaranteed to be at least as good as (if not better than) policy ?. The computation of Q? (s, a) and ? 0 , called policy evaluation and policy improvement, define the basic mechanics of RL algorithms based on DP; under certain conditions their successive application leads to an optimal policy ? ? that maximizes the expected return from every s ? S [21]. 2 In this paper we are interested in the problem of transfer, which we define as follows. Let T, T 0 be two sets of tasks such that T 0 ? T , and let t be any task. Then there is transfer if, after training on T , the agent always performs as well or better on task t than if only trained on T 0 . Note that T 0 can be the empty set. In this paper a task will be defined as a specific instantiation of the reward function R(s, a, s0 ) for a given MDP. In Section 4 we will revisit this definition and make it more formal. 3 Successor features In this section we present the concept that will serve as a cornerstone for the rest of the paper. We start by presenting a simple reward model and then show how it naturally leads to a generalization of Dayan?s [7] successor representation (SR). Suppose that the expected one-step reward associated with transition (s, a, s0 ) can be computed as r(s, a, s0 ) = ?(s, a, s0 )> w, 0 d 0 (2) d where ?(s, a, s ) ? R are features of (s, a, s ) and w ? R are weights. This assumption is not restrictive because we are not making any assumptions about ?(s, a, s0 ): if we have ?i (s, a, s0 ) = r(s, a, s0 ) for some i, for example, we can clearly recover any reward function exactly. To simplify the notation, let ?t = ?(st , at , st+1 ). Then, by simply rewriting the definition of the action-value function in (1) we have Q? (s, a) = E? [rt+1 + ?rt+2 + ... | St = s, At = a] h i > = E? ?> w + ?? w + ... | S = s, A = a t t t+1 t+2  > ? P? i?t =E ?i+1 | St = s, At = a w = ? ? (s, a)> w. i=t ? (3) The decomposition (3) has appeared before in the literature under different names and interpretations, as discussed in Section 6. Since here we propose to look at (3) as an extension of Dayan?s [7] SR, we call ? ? (s, a) the successor features (SFs) of (s, a) under policy ?. The ith component of ? ? (s, a) gives the expected discounted sum of ?i when following policy ? starting from (s, a). In the particular case where S and A are finite and ? is a tabular representation 2 of S ? A ? S?that is, ?(s, a, s0 ) is a one-hot vector in R|S| |A| ?? ? (s, a) is the discounted sum of occurrences, under ?, of each possible transition. This is essentially the concept of SR extended from the space S to the set S ? A ? S [7]. One of the contributions of this paper is precisely to generalize SR to be used with function approximation, but the exercise of deriving the concept as above provides insights already in the tabular 2 case. To see this, note that in the tabular case the entries of w ? R|S| |A| are the function r(s, a, s0 ) and suppose that r(s, a, s0 ) 6= 0 in only a small subset W ? S ? A ? S. From (2) and (3), it is clear that the cardinality of W, and not of S ? A ? S, is what effectively defines the dimension of the representation ? ? , since there is no point in having d > |W|. Although this fact is hinted at by Dayan [7], it becomes more apparent when we look at SR as a particular case of SFs. SFs extend SR in two other ways. First, the concept readily applies to continuous state and action spaces without any modification. Second, by explicitly casting (2) and (3) as inner products involving feature vectors, SFs make it evident how to incorporate function approximation: as will be shown, these vectors can be learned from data. The representation in (3) requires two components to be learned, w and ? ? . Since the latter is the expected discounted sum of ? under ?, we must either be given ? or learn it as well. Note ? is a supervised learning problem, so we can use that approximating r(s, a, s0 ) ? ?(s, a, s0 )> w ? too) [9]. As for ? ? , we note ? (and potentially ?, well-understood techniques from the field to learn w that ? ? (s, a) = ?t+1 + ?E ? [? ? (St+1 , ?(St+1 )) | St = s, At = a], (4) that is, SFs satisfy a Bellman equation in which ?i play the role of rewards?something also noted by Dayan [7] regarding SR. Therefore, in principle any RL method can be used to compute ? ? [24]. The SFs ? ? summarize the dynamics induced by ? in a given environment. As shown in (3), this allows for a modular representation of Q? in which the MDP?s dynamics are decoupled from its 3 rewards, which are captured by the weights w. One potential benefit of having such a decoupled representation is that only the relevant module must be relearned when either the dynamics or the reward changes, which may serve as an argument in favor of adopting SFs as a general approximation scheme for RL. However, in this paper we focus on a scenario where the decoupled value-function approximation provided by SFs is exploited to its full extent, as we discuss next. 4 Transfer via successor features We now return to the discussion about transfer in RL. As described, we are interested in the scenario where all components of an MDP are fixed, except for the reward function. One way of formalizing this model is through (2): if we suppose that ? ? Rd is fixed, any w ? Rd gives rise to a new MDP. Based on this observation, we define M? (S, A, p, ?)? {M (S, A, p, r, ?)|r(s, a, s0 )= ?(s, a, s0 )> w}, (5) that is, M? is the set of MDPs induced by ? through all possible instantiations of w. Since what differentiates the MDPs in M? is essentially the agent?s goal, we will refer to Mi ? M? as a task. The assumption is that we are interested in solving (a subset of) the tasks in the environment M? . Definition (5) is a natural way of modeling some scenarios of interest. Think, for example, how the desirability of water or food changes depending on whether an animal is thirsty or hungry. One way to model this type of preference shifting is to suppose that the vector w appearing in (2) reflects the taste of the agent at any given point in time [17]. Further in the paper we will present experiments that reflect this scenario. For another illustrative example, imagine that the agent?s goal is to produce and sell a combination of goods whose production line is relatively stable but whose prices vary considerably over time. In this case updating the price of the products corresponds to picking a new w. A slightly different way of motivating (5) is to suppose that the environment itself is changing, that is, the element wi indicates not only desirability, but also availability, of feature ?i . In the examples above it is desirable for the agent to build on previous experience to improve its performance on a new setup. More concretely, if the agent knows good policies for the set of tasks M ? {M1 , M2 , ..., Mn }, with Mi ? M? , it should be able to leverage this knowledge to improve its behavior on a new task Mn+1 ?that is, it should perform better than it would had it been exposed to only a subset of the original tasks, M0 ? M. We can assess the performance of an agent on task Mn+1 based on the value function of the policy followed after wn+1 has become available but before any policy improvement has taken place in Mn+1 .1 More precisely, suppose that an agent has been exposed to each one of the tasks Mi ? M0 . Based on this experience, and on the new wn+1 , the agent computes a policy ? 0 that will define its initial behavior in Mn+1 . Now, if we repeat the 0 experience replacing M0 with M, the resulting policy ? should be such that Q? (s, a) ? Q? (s, a) for all (s, a) ? S ? A. Now that our setup is clear we can start to describe our solution for the transfer problem discussed above. We do so in two stages. First, we present a generalization of DP?s notion of policy improvement whose interest may go beyond the current work. We then show how SFs can be used to implement this generalized form of policy improvement in an efficient and elegant way. 4.1 Generalized policy improvement One of the fundamental results in RL is Bellman?s [3] policy improvement theorem. In essence, the theorem states that acting greedily with respect to a policy?s value function gives rise to another policy whose performance is no worse than the former?s. This is the driving force behind DP, and most RL algorithms that compute a value function are exploiting Bellman?s result in one way or another. In this section we extend the policy improvement theorem to the scenario where the new policy is to be computed based on the value functions of a set of policies. We show that this extension can be done in a natural way, by acting greedily with respect to the maximum over the value functions available. Our result is summarized in the theorem below. 1 Of course wn+1 can, and will be, learned, as discussed in Section 4.2 and illustrated in Section 5. Here we assume that wn+1 is given to make our performance criterion clear. 4 Theorem 1. (Generalized Policy Improvement) Let ?1 , ?2 , ..., ?n be n decision policies and let ? ?1 , Q ? ?2 , ..., Q ? ?n be approximations of their respective action-value functions such that Q ? ?i (s, a)| ?  for all s ? S, a ? A, and i ? {1, 2, ..., n}. |Q?i (s, a) ? Q (6) Define ? ?i (s, a). ?(s) ? argmax max Q i a (7) Then, 2  i 1?? for any s ? S and a ? A, where Q? is the action-value function of ?. Q? (s, a) ? max Q?i (s, a) ? (8) The proofs of our theoretical results are in the supplementary material. As one can see, our theorem covers the case where the policies? value functions are not computed exactly, either because function approximation is used or because some exact algorithm has not be run to completion. This error is captured by  in (6), which re-appears as a penalty term in the lower bound (8). Such a penalty is inherent to the presence of approximation in RL, and in fact it is identical to the penalty incurred in the single-policy case (see e.g. Bertsekas and Tsitsiklis?s Proposition 6.1 [5]). In order to contextualize generalized policy improvement (GPI) within the broader scenario of DP, suppose for a moment that  = 0. In this case Theorem 1 states that ? will perform no worse than all of the policies ?1 , ?2 , ..., ?n . This is interesting because in general maxi Q?i ?the function used to induce ??is not the value function of any particular policy. It is not difficult to see that ? will be strictly better than all previous policies if no single policy dominates all other policies, that is, ? ?i (s, a) ? argmaxi maxa Q ? ?i (s0 , a) = ? for some s, s0 ? S. If one policy does if argmaxi maxa Q dominate all others, GPI reduces to the original policy improvement theorem. If we consider the usual DP loop, in which policies of increasing performance are computed in sequence, our result is not of much use because the most recent policy will always dominate all others. Another way of putting it is to say that after Theorem 1 is applied once adding the resulting ? to the set {?1 , ?2 , ..., ?n } will reduce the next improvement step to standard policy improvement, and thus the policies ?1 , ?2 , ..., ?n can be simply discarded. There are however two situations in which our result may be of interest. One is when we have many policies ?i being evaluated in parallel. In this case GPI provides a principled strategy for combining these policies. The other situation in which our result may be useful is when the underlying MDP changes, as we discuss next. 4.2 Generalized policy improvement with successor features We start this section by extending our notation slightly to make it easier to refer to the quantities involved in transfer learning. Let Mi be a task in M? defined by wi ? Rd . We will use ?i? to refer ?? to an optimal policy of MDP Mi and use Qi i to refer to its value function. The value function of ?i? ?? when executed in Mj ? M? will be denoted by Qj i . Suppose now that an agent has computed optimal policies for the tasks M1 , M2 , ..., Mn ? M? . Sup? ?1? ?2? ?n pose further that when presented with a new task Mn+1 the agent computes {Qn+1 , Qn+1 , ..., Qn+1 }, the evaluation of each ?i? under the new reward function induced by wn+1 . In this case, applying the GPI theorem to the newly-computed set of value functions will give rise to a policy that performs at least as well as a policy based on any subset of these, including the empty set. Thus, this strategy satisfies our definition of successful transfer. There is a caveat, though. Why would one waste time computing the value functions of ?1? , ?2? , ..., ?n? , whose performance in Mn+1 may be mediocre, if the same amount of resources can be allocated to compute a sequence of n policies with increasing performance? This is where SFs come into play. ?? Suppose that we have learned the functions Qi i using the representation scheme shown in (3). Now, if the reward changes to rn+1 (s, a, s0 ) = ?(s, a, s0 )> wn+1 , as long as we have wn+1 we can compute ?? ? i the new value function of ?i? by simply making Qn+1 (s, a) = ? ?i (s, a)> wn+1 . This reduces the ?? i computation of all Qn+1 to the much simpler supervised problem of approximating wn+1 . ?? i Once the functions Qn+1 have been computed, we can apply GPI to derive a policy ? whose performance on Mn+1 is no worse than the performance of ?1? , ?2? , ..., ?n? on the same task. A 5 question that arises in this case is whether we can provide stronger guarantees on the performance of ? by exploiting the structure shared by the tasks in M? . The following theorem answers this question in the affirmative. ?? Theorem 2. Let Mi ? M? and let Qi j be the action-value function of an optimal policy of ? ? ? ? ?1 , Q ? ?2 , ..., Q ? ?n } such that Mj ? M? when executed in Mi . Given approximations {Q i i i ?? ? j ? ?j (s, a) ?  (9) Qi (s, a) ? Q i ? ? ?j (s, a). Finally, let for all s ? S, a ? A, and j ? {1, 2, ..., n}, let ?(s) ? argmaxa maxj Q i ?max = maxs,a ||?(s, a)||, where || ? || is the norm induced by the inner product adopted. Then, ?? Qi i (s, a) ? Q?i (s, a) ? 2 (? minj ||wi ? wj || + ) . 1 ? ? max (10) Note that we used Mi rather than Mn+1 in the theorem?s statement to remove any suggestion of order among the tasks. Theorem 2 is a specialization of Theorem 1 for the case where the set of value functions used to compute ? are associated with tasks in the form of (5). As such, it provides stronger guarantees: instead of comparing the performance of ? with that of the previously-computed policies ?j , Theorem 2 quantifies the loss incurred by following ? as opposed to one of Mi ?s optimal policies. ?? As shown in (10), the loss Qi i (s, a) ? Q?i (s, a) is upper-bounded by two terms. The term 2?max minj ||wi ? wj ||/(1 ? ?) is of more interest here because it reflects the structure of M? . This term is a multiple of the distance between wi , the vector describing the task we are currently interested in, and the closest wj for which we have computed a policy. This formalizes the intuition that the agent should perform well in task wi if it has solved a similar task before. More generally, the term in question relates the concept of distance in Rd with difference in performance in M? . Note that this correspondence depends on the specific set of features ? used, which raises the interesting question of how to define ? such that tasks that are close in Rd induce policies that are also similar in some sense. Regardless of how exactly ? is defined, the bound (10) allows for powerful extrapolations. For example, by covering the relevant subspace of Rd with balls of appropriate radii centered at wj we can provide performance guarantees for any task w [14]. This corresponds to building a library of options (or ?skills?) that can be used to solve any task in a (possibly infinite) set [22]. In Section 5 we illustrate this concept with experiments. Although Theorem 2 is inexorably related to the characterization of M? in (5), it does not depend on the definition of SFs in any way. Here SFs are the mechanism used to efficiently apply the protocol suggested by Theorem 2. When SFs are used the value function approximations are given by ? ? ?j? (s, a)> w ? ?j? are computed and stored when the agent is learning ? ?j (s, a) = ? ? i . The modules ? Q i the tasks Mj ; when faced with a new task Mi the agent computes an approximation of wi , which is a ? ?i? . Note supervised learning problem, and then uses the policy ? defined in Theorem 2 to learn ? ? ? ?j? and w ?i that we do not assume that either ? ?j or wi is computed exactly: the effect of errors in ? are accounted for by the term  appearing in (9). As shown in (10), if  is small and the agent has seen enough tasks the performance of ? on Mi should already be good, which suggests it may also ? ?i? . speed up the process of learning ? Interestingly, Theorem 2 also provides guidance for some practical algorithmic choices. Since in an ? ?j? stored in memory, the corresponding actual implementation one wants to limit the number of SFs ? ? ?i? only ? j can be used to decide which ones to keep. For example, one can create a new ? vectors w ?i ?w ? j || is above a given threshold; alternatively, once the maximum number of SFs when minj ||w ? ?k? , where k = argmin ||w ? j || (here wi is the current task). has been reached, one can replace ? j ?i ?w 5 Experiments In this section we present our main experimental results. Additional details, along with further results and analysis, can be found in Appendix B of the supplementary material. The first environment we consider involves navigation tasks defined over a two-dimensional continuous space composed of four rooms (Figure 1). The agent starts in one of the rooms and must reach a 6 goal region located in the farthest room. The environment has objects that can be picked up by the agent by passing over them. Each object belongs to one of three classes determining the associated reward. The objective of the agent is to pick up the ?good? objects and navigate to the goal while avoiding ?bad? objects. The rewards associated with object classes change at every 20 000 transitions, giving rise to very different tasks (Figure 1). The goal is to maximize the sum of rewards accumulated over a sequence of 250 tasks, with each task?s rewards sampled uniformly from [?1, 1]3 . We defined a straightforward instantia? tion of our approach in which both w ? ? and ? are computed incrementally in order to minimize losses induced by (2) and (4). Every time the task changes the ? ?i is stored and a new ? ? ?i+1 current ? is created. We call this method SFQL as a reference to the fact that SFs are learned through an algorithm analogous to Q-learning (QL)?which is used as a baseline in our comparisons [27] . As a Figure 1: Environment layout and some examples of optimore challenging reference point we re- mal trajectories associated with specific tasks. The shapes port results for a transfer method called of the objects represent their classes; ?S? is the start state probabilistic policy reuse [8]. We adopt and ?G? is the goal. a version of the algorithm that builds on QL and reuses all policies learned. The resulting method, PRQL, is thus directly comparable to SFQL. The details of QL, PRQL, and SFQL, including their pseudo-codes, are given in Appendix B. We compared two versions of SFQL. In the first one, called SFQL-?, we assume the agent has access to features ? that perfectly predict the rewards, as in (2). The second version of our agent had to ? ? Rh directly from data collected by QL in the first 20 tasks. Note that learn an approximation ? h may not coincide with the true dimension of ?, which in this case is 4; we refer to the different ? followed the multi-task learning instances of our algorithm as SFQL-h. The process of learning ? protocol proposed by Caruana [6] and Baxter [2], and described in detail in Appendix B. The results of our experiments can be seen in Figure 2. As shown, all versions of SFQL significantly outperform the other two methods, with an improvement on the average return of more than 100% when compared to PRQL, which itself improves on QL by around 100%. Interestingly, SFQL-h seems to achieve good overall performance faster than SFQL-?, even though the latter uses features that allow for an exact representation of the rewards. One possible explanation is that, unlike their counterparts ?i , the features ??i are activated over most of the space S ? A ? S, which results in a dense pseudo-reward signal that facilitates learning. The second environment we consider is a set of control tasks defined in the MuJoCo physics engine [26]. Each task consists in moving a two-joint torque-controlled simulated robotic arm to a SFQL-8 SFQL-? / SFQL-4 PRQL Q-Learning Figure 2: Average and cumulative return per task in the four-room domain. SFQL-h receives no ? Error-bands show one standard error over 30 runs. reward during the first 20 tasks while learning ?. 7 Normalized Return Normalized Return SFDQN DQN Tasks Trained (b) Average performance on test tasks. Task 1 Task 2 Task 3 Task 4 Tasks Trained (a) Performance on training tasks (faded dotted lines in the background are DQN?s results). (c) Colored and gray circles depict training and test targets, respectively. Figure 3: Normalized return on the reacher domain: ?1? corresponds to the average result achieved by DQN after learning each task separately and ?0? corresponds to the average performance of a randomly-initialized agent (see Appendix B for details). SFDQN?s results were obtained using the GPI policies ?i (s) defined in the text. Shading shows one standard error over 30 runs. specific target location; thus, we refer to this environment as ?the reacher domain.? We defined 12 tasks, but only allowed the agents to train in 4 of them (Figure 3c). This means that the agent must be able to perform well on tasks that it has never experienced during training. In order to solve this problem, we adopted essentially the same algorithm as above, but we replaced QL with Mnih et al.?s DQN?both as a baseline and as the basic engine underlying the SF agent [15]. The resulting method, which we call SFDQN, is an illustration of how our method can be naturally combined with complex nonlinear approximators such as neural networks. The features ?i used by SFDQN are the negation of the distances to the center of the 12 target regions. As usual in experiments of this type, we give the agents a description of the current task: for DQN the target coordinates are given as inputs, while for SFDQN this is provided as an one-hot vector wt ? R12 [12]. Unlike in the ? ?i through losses previous experiment, in the current setup each transition was used to train all four ? ? (s, a)> wi . derived from (4). Here ?i is the GPI policy on the ith task: ?i (s) ? argmaxa maxj ? j Results are shown in Figures 3a and 3b. Looking at the training curves, we see that whenever a task is selected for training SFDQN?s return on that task quickly improves and saturates at nearoptimal performance. The interesting point to be noted is that, when learning a given task, SFDQN?s performance also improves in all other tasks, including the test ones, for which it does not have specialized policies. This illustrates how the combination of SFs and GPI can give rise to flexible agents able to perform well in any task of a set of tasks with shared dynamics?which in turn can be seen as both a form of temporal abstraction and a step towards more general hierarchical RL [22, 1]. 6 Related work Mehta et al.?s [14] approach for transfer learning is probably the closest work to ours in the literature. There are important differences, though. First, Mehta et al. [14] assume that both ? and w are always observable quantities provided by the environment. They also focus on average reward RL, in which the quality of a decision policy can be characterized by a single scalar. This reduces the process of selecting a policy for a task to one decision made at the outset, which is in clear contrast with GPI. 8 The literature on transfer learning has other methods that relate to ours [25, 11]. Among the algorithms designed for the scenario considered here, two approaches are particularly relevant because they also reuse old policies. One is Fern?ndez et al.?s [8] probabilistic policy reuse, adopted in our experiments and described in Appendix B. The other approach, by Bernstein [4], corresponds to using our method ? ?i from scratch at each new task. but relearning all ? When we look at SFs strictly as a representation scheme, there are clear similarities with Littman et al.?s [13] predictive state representation (PSR). Unlike SFs, though, PSR tries to summarize the dynamics of the entire environment rather than of a single policy ?. A scheme that is perhaps closer to SFs is the value function representation sometimes adopted in inverse RL [18]. SFs are also related to Sutton et al.?s [23] general value functions (GVFs), which extends the notion of value function to also include ?pseudo-rewards.? If we see ?i as a pseudo-reward, ?i? (s, a) becomes a particular case of GVF. Beyond the technical similarities, the connection between SFs and GVFs uncovers some principles underlying both lines of work that, when contrasted, may benefit both. On one hand, Sutton et al.?s [23] and Modayil et al.?s [16] hypothesis that relevant knowledge about the world can be expressed as many predictions naturally translates to SFs: if ? is expressive enough, the agent should be able to represent any relevant reward function. Conversely, SFs not only provide a concrete way of using this knowledge, they also suggest a possible criterion to select the pseudo-rewards ?i , since ultimately we are only interested in features that help in the approximation ? ? r(s, a, s0 ). ?(s, a, s0 )> w Another generalization of value functions that is related to SFs is Schaul et al.?s [20] universal value function approximators (UVFAs). UVFAs extend the notion of value function to also include as an argument an abstract representation of a ?goal,? which make them particularly suitable for transfer. ? ?j? (s, a)> w ? ? used in our framework can be seen as a function of s, a, and w?the The function maxj ? latter a generic way of representing a goal?, and thus in some sense this representation is a UVFA. ? is simply the description This connection raises an interesting point: since under this interpretation w of a task, it can in principle be a direct function of the observations, which opens up the possibility of ? even before seeing any rewards. the agent determining w As discussed, our approach is also related to temporal abstraction and hierarchical RL: if we look at ? ? as instances of Sutton et al.?s [22] options, acting greedily with respect to the maximum over their value functions corresponds in some sense to planning at a higher level of temporal abstraction (that is, each ? ? (s, a) is associated with an option that terminates after a single step). This is the view adopted by Yao et al. [28], whose universal option model closely resembles our approach in some aspects (the main difference being that they do not do GPI). Finally, there have been previous attempts to combine SR and neural networks. Kulkarni et al. ? ? (s, a), ?(s, ? a, s0 ) and w. ? [10] and Zhang et al. [29] propose similar architectures to jointly learn ? Although neither work exploits SFs for GPI, they both discuss other uses of SFs for transfer. In principle the proposed (or similar) architectures can also be used within our framework. 7 Conclusion This paper builds on two concepts, both of which are generalizations of previous ideas. The first one is SFs, a generalization of Dayan?s [7] SR that extends the original definition from discrete to continuous spaces and also facilitates the use of function approximation. The second concept is GPI, formalized in Theorem 1. As the name suggests, this result extends Bellman?s [3] classic policy improvement theorem from a single to multiple policies. Although SFs and GPI are of interest on their own, in this paper we focus on their combination to induce transfer. The resulting framework is an elegant extension of DP?s basic setting that provides a solid foundation for transfer in RL. As a complement to the proposed transfer approach, we derived a theoretical result, Theorem 2, that formalizes the intuition that an agent should perform well on a novel task if it has seen a similar task before. We also illustrated with a comprehensive set of experiments how the combination of SFs and GPI promotes transfer in practice. We believe the proposed ideas lay out a general framework for transfer in RL. By specializing the basic components presented one can build on our results to derive agents able to perform well across a wide variety of tasks, and thus extend the range of environments that can be successfully tackled. 9 Acknowledgments The authors would like to thank Joseph Modayil for the invaluable discussions during the development of the ideas described in this paper. We also thank Peter Dayan, Matt Botvinick, Marc Bellemare, and Guy Lever for the excellent comments, and Dan Horgan and Alexander Pritzel for their help with the experiments. Finally, we thank the anonymous reviewers for their comments and suggestions to improve the paper. References [1] Andrew G. Barto and Sridhar Mahadevan. Recent advances in hierarchical reinforcement learning. Discrete Event Dynamic Systems, 13(4):341?379, 2003. [2] Jonathan Baxter. A model of inductive bias learning. Journal of Artificial Intelligence Research, 12:149?198, 2000. [3] Richard E. Bellman. Dynamic Programming. Princeton University Press, 1957. [4] Daniel S. Bernstein. Reusing old policies to accelerate learning on new MDPs. Technical report, Amherst, MA, USA, 1999. [5] Dimitri P. Bertsekas and John N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996. [6] Rich Caruana. Multitask learning. Machine Learning, 28(1):41?75, 1997. [7] Peter Dayan. Improving generalization for temporal difference learning: The successor representation. Neural Computation, 5(4):613?624, 1993. [8] Fernando Fern?ndez, Javier Garc?a, and Manuela Veloso. Probabilistic policy reuse for inter-task transfer learning. Robotics and Autonomous Systems, 58(7):866?871, 2010. [9] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, 2002. [10] Tejas D. Kulkarni, Ardavan Saeedi, Simanta Gautam, and Samuel J Gershman. Deep successor reinforcement learning. arXiv preprint arXiv:1606.02396, 2016. [11] Alessandro Lazaric. Transfer in Reinforcement Learning: A Framework and a Survey. Reinforcement Learning: State-of-the-Art, pages 143?173, 2012. [12] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015. [13] Michael L. Littman, Richard S. Sutton, and Satinder Singh. Predictive representations of state. In Advances in Neural Information Processing Systems (NIPS), pages 1555?1561, 2001. [14] Neville Mehta, Sriraam Natarajan, Prasad Tadepalli, and Alan Fern. Transfer in variable-reward hierarchical reinforcement learning. Machine Learning, 73(3), 2008. [15] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness, Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland, Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and Demis Hassabis. Human-level control through deep reinforcement learning. Nature, 518(7540):529?533, 2015. [16] Joseph Modayil, Adam White, and Richard S. Sutton. Multi-timescale nexting in a reinforcement learning robot. Adaptive Behavior, 22(2):146?160, 2014. [17] Sriraam Natarajan and Prasad Tadepalli. Dynamic preferences in multi-criteria reinforcement learning. In Proceedings of the International Conference on Machine Learning (ICML), pages 601?608, 2005. 10 [18] Andrew Ng and Stuart Russell. Algorithms for inverse reinforcement learning. In Proceedings of the International Conference on Machine Learning (ICML), pages 663?670, 2000. [19] Martin L. Puterman. Markov Decision Processes?Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., 1994. [20] Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. Universal Value Function Approximators. In International Conference on Machine Learning (ICML), pages 1312?1320, 2015. [21] Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. MIT Press, 1998. [22] Richard S. Sutton, Doina Precup, and Satinder Singh. Between MDPs and semi-MDPs: a framework for temporal abstraction in reinforcement learning. Artificial Intelligence, 112: 181?211, 1999. [23] Richard S. Sutton, Joseph Modayil, Michael Delp, Thomas Degris, Patrick M. Pilarski, Adam White, and Doina Precup. Horde: A scalable real-time architecture for learning knowledge from unsupervised sensorimotor interaction. In International Conference on Autonomous Agents and Multiagent Systems, pages 761?768, 2011. [24] Csaba Szepesv?ri. Algorithms for Reinforcement Learning. Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers, 2010. [25] Matthew E. Taylor and Peter Stone. Transfer learning for reinforcement learning domains: A survey. Journal of Machine Learning Research, 10(1):1633?1685, 2009. [26] Emanuel Todorov, Tom Erez, and Yuval Tassa. MuJoCo: A physics engine for model-based control. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 5026?5033, 2012. [27] Christopher Watkins and Peter Dayan. Q-learning. Machine Learning, 8:279?292, 1992. [28] Hengshuai Yao, Csaba Szepesv?ri, Richard S Sutton, Joseph Modayil, and Shalabh Bhatnagar. Universal option models. In Advances in Neural Information Processing Systems (NIPS), pages 990?998. 2014. [29] Jingwei Zhang, Jost Tobias Springenberg, Joschka Boedecker, and Wolfram Burgard. Deep reinforcement learning with successor features for navigation across similar environments. CoRR, abs/1612.05533, 2016. 11
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Counterfactual Fairness Matt Kusner ? The Alan Turing Institute and University of Warwick [email protected] Joshua Loftus ? New York University [email protected] Chris Russell ? The Alan Turing Institute and University of Surrey [email protected] Ricardo Silva The Alan Turing Institute and University College London [email protected] Abstract Machine learning can impact people with legal or ethical consequences when it is used to automate decisions in areas such as insurance, lending, hiring, and predictive policing. In many of these scenarios, previous decisions have been made that are unfairly biased against certain subpopulations, for example those of a particular race, gender, or sexual orientation. Since this past data may be biased, machine learning predictors must account for this to avoid perpetuating or creating discriminatory practices. In this paper, we develop a framework for modeling fairness using tools from causal inference. Our definition of counterfactual fairness captures the intuition that a decision is fair towards an individual if it the same in (a) the actual world and (b) a counterfactual world where the individual belonged to a different demographic group. We demonstrate our framework on a real-world problem of fair prediction of success in law school. 1 Contribution Machine learning has spread to fields as diverse as credit scoring [20], crime prediction [5], and loan assessment [25]. Decisions in these areas may have ethical or legal implications, so it is necessary for the modeler to think beyond the objective of maximizing prediction accuracy and consider the societal impact of their work. For many of these applications, it is crucial to ask if the predictions of a model are fair. Training data can contain unfairness for reasons having to do with historical prejudices or other factors outside an individual?s control. In 2016, the Obama administration released a report2 which urged data scientists to analyze ?how technologies can deliberately or inadvertently perpetuate, exacerbate, or mask discrimination." There has been much recent interest in designing algorithms that make fair predictions [4, 6, 10, 12, 14, 16?19, 22, 24, 36?39]. In large part, the literature has focused on formalizing fairness into quantitative definitions and using them to solve a discrimination problem in a certain dataset. Unfortunately, for a practitioner, law-maker, judge, or anyone else who is interested in implementing algorithms that control for discrimination, it can be difficult to decide which definition of fairness to choose for the task at hand. Indeed, we demonstrate that depending on the relationship between a protected attribute and the data, certain definitions of fairness can actually increase discrimination. ? Equal contribution. This work was done while JL was a Research Fellow at the Alan Turing Institute. https://obamawhitehouse.archives.gov/blog/2016/05/04/big-risks-big-opportunities-intersection-big-dataand-civil-rights 2 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. In this paper, we introduce the first explicitly causal approach to address fairness. Specifically, we leverage the causal framework of Pearl [30] to model the relationship between protected attributes and data. We describe how techniques from causal inference can be effective tools for designing fair algorithms and argue, as in DeDeo [9], that it is essential to properly address causality in fairness. In perhaps the most closely related prior work, Johnson et al. [15] make similar arguments but from a non-causal perspective. An alternative use of causal modeling in the context of fairness is introduced independently by [21]. In Section 2, we provide a summary of basic concepts in fairness and causal modeling. In Section 3, we provide the formal definition of counterfactual fairness, which enforces that a distribution over possible predictions for an individual should remain unchanged in a world where an individual?s protected attributes had been different in a causal sense. In Section 4, we describe an algorithm to implement this definition, while distinguishing it from existing approaches. In Section 5, we illustrate the algorithm with a case of fair assessment of law school success. 2 Background This section provides a basic account of two separate areas of research in machine learning, which are formally unified in this paper. We suggest Berk et al. [1] and Pearl et al. [29] as references. Throughout this paper, we will use the following notation. Let A denote the set of protected attributes of an individual, variables that must not be discriminated against in a formal sense defined differently by each notion of fairness discussed. The decision of whether an attribute is protected or not is taken as a primitive in any given problem, regardless of the definition of fairness adopted. Moreover, let X denote the other observable attributes of any particular individual, U the set of relevant latent attributes which are not observed, and let Y denote the outcome to be predicted, which itself might be contaminated with historical biases. Finally, Y? is the predictor, a random variable that depends on A, X and U , and which is produced by a machine learning algorithm as a prediction of Y . 2.1 Fairness There has been much recent work on fair algorithms. These include fairness through unawareness [12], individual fairness [10, 16, 24, 38], demographic parity/disparate impact [36], and equality of opportunity [14, 37]. For simplicity we often assume A is encoded as a binary attribute, but this can be generalized. Definition 1 (Fairness Through Unawareness (FTU)). An algorithm is fair so long as any protected attributes A are not explicitly used in the decision-making process. Any mapping Y? : X ? Y that excludes A satisfies this. Initially proposed as a baseline, the approach has found favor recently with more general approaches such as Grgic-Hlaca et al. [12]. Despite its compelling simplicity, FTU has a clear shortcoming as elements of X can contain discriminatory information analogous to A that may not be obvious at first. The need for expert knowledge in assessing the relationship between A and X was highlighted in the work on individual fairness: Definition 2 (Individual Fairness (IF)). An algorithm is fair if it gives similar predictions to similar individuals. Formally, given a metric d(?, ?), if individuals i and j are similar under this metric (i.e., d(i, j) is small) then their predictions should be similar: Y? (X (i) , A(i) ) ? Y? (X (j) , A(j) ). As described in [10], the metric d(?, ?) must be carefully chosen, requiring an understanding of the domain at hand beyond black-box statistical modeling. This can also be contrasted against population level criteria such as Definition 3 (Demographic Parity (DP)). A predictor Y? satisfies demographic parity if P (Y? |A = 0) = P (Y? |A = 1). Definition 4 (Equality of Opportunity (EO)). A predictor Y? satisfies equality of opportunity if P (Y? = 1|A = 0, Y = 1) = P (Y? = 1|A = 1, Y = 1). These criteria can be incompatible in general, as discussed in [1, 7, 22]. Following the motivation of IF and [15], we propose that knowledge about relationships between all attributes should be taken into consideration, even if strong assumptions are necessary. Moreover, it is not immediately clear 2 for any of these approaches in which ways historical biases can be tackled. We approach such issues from an explicit causal modeling perspective. 2.2 Causal Models and Counterfactuals We follow Pearl [28], and define a causal model as a triple (U, V, F ) of sets such that ? U is a set of latent background variables,which are factors not caused by any variable in the set V of observable variables; ? F is a set of functions {f1 , . . . , fn }, one for each Vi ? V , such that Vi = fi (pai , Upai ), pai ? V \{Vi } and Upai ? U . Such equations are also known as structural equations [2]. The notation ?pai ? refers to the ?parents? of Vi and is motivated by the assumption that the model factorizes as a directed graph, here assumed to be a directed acyclic graph (DAG). The model is causal in that, given a distribution P (U ) over the background variables U , we can derive the distribution of a subset Z ? V following an intervention on V \ Z. An intervention on variable Vi is the substitution of equation Vi = fi (pai , Upai ) with the equation Vi = v for some v. This captures the idea of an agent, external to the system, modifying it by forcefully assigning value v to Vi , for example as in a randomized experiment. The specification of F is a strong assumption but allows for the calculation of counterfactual quantities. In brief, consider the following counterfactual statement, ?the value of Y if Z had taken value z?, for two observable variables Z and Y . By assumption, the state of any observable variable is fully determined by the background variables and structural equations. The counterfactual is modeled as the solution for Y for a given U = u where the equations for Z are replaced with Z = z. We denote it by YZ?z (u) [28], and sometimes as Yz if the context of the notation is clear. Counterfactual inference, as specified by a causal model (U, V, F ) given evidence W , is the computation of probabilities P (YZ?z (U ) | W = w), where W , Z and Y are subsets of V . Inference proceeds in three steps, as explained in more detail in Chapter 4 of Pearl et al. [29]: 1. Abduction: for a given prior on U , compute the posterior distribution of U given the evidence W = w; 2. Action: substitute the equations for Z with the interventional values z, resulting in the modified set of equations Fz ; 3. Prediction: compute the implied distribution on the remaining elements of V using Fz and the posterior P (U |W = w). 3 Counterfactual Fairness Given a predictive problem with fairness considerations, where A, X and Y represent the protected attributes, remaining attributes, and output of interest respectively, let us assume that we are given a causal model (U, V, F ), where V ? A ? X. We postulate the following criterion for predictors of Y . Definition 5 (Counterfactual fairness). Predictor Y? is counterfactually fair if under any context X = x and A = a, P (Y?A?a (U ) = y | X = x, A = a) = P (Y?A?a0 (U ) = y | X = x, A = a), (1) for all y and for any value a0 attainable by A. This notion is closely related to actual causes [13], or token causality in the sense that, to be fair, A should not be a cause of Y? in any individual instance. In other words, changing A while holding things which are not causally dependent on A constant will not change the distribution of Y? . We also emphasize that counterfactual fairness is an individual-level definition. This is substantially different from comparing different individuals that happen to share the same ?treatment? A = a and coincide on the values of X, as discussed in Section 4.3.1 of [29] and the Supplementary Material. Differences between Xa and Xa0 must be caused by variations on A only. Notice also that this definition is agnostic with respect to how good a predictor Y? is, which we discuss in Section 4. Relation to individual fairness. IF is agnostic with respect to its notion of similarity metric, which is both a strength (generality) and a weakness (no unified way of defining similarity). Counterfactuals and similarities are related, as in the classical notion of distances between ?worlds? corresponding to different counterfactuals [23]. If Y? is a deterministic function of W ? A ? X ? U , as in several of 3 U A Prejudiced U A UA Y A UY (c) X Y (a) X Y (b) UA A UA Qualifications Prejudiced Employed (d) Y UY Qualifications A Employed Y a Employed a Ya a0 Employed a0 Ya UY 0 (e) Figure 1: (a), (b) Two causal models for different real-world fair prediction scenarios. See Section 3.1 for discussion. (c) The graph corresponding to a causal model with A being the protected attribute and Y some outcome of interest, with background variables assumed to be independent. (d) Expanding the model to include an intermediate variable indicating whether the individual is employed with two (latent) background variables Prejudiced (if the person offering the job is prejudiced) and Qualifications (a measure of the individual?s qualifications). (e) A twin network representation of this system [28] under two different counterfactual levels for A. This is created by copying nodes descending from A, which inherit unaffected parents from the factual world. our examples to follow, then IF can be defined by treating equally two individuals with the same W in a way that is also counterfactually fair. Relation to Pearl et al. [29]. In Example 4.4.4 of [29], the authors condition instead on X, A, and the observed realization of Y? , and calculate the probability of the counterfactual realization Y?A?a0 differing from the factual. This example conflates the predictor Y? with the outcome Y , of which we remain agnostic in our definition but which is used in the construction of Y? as in Section 4. Our framing makes the connection to machine learning more explicit. 3.1 Examples To provide an intuition for counterfactual fairness, we will consider two real-world fair prediction scenarios: insurance pricing and crime prediction. Each of these correspond to one of the two causal graphs in Figure 1(a),(b). The Supplementary Material provides a more mathematical discussion of these examples with more detailed insights. Scenario 1: The Red Car. A car insurance company wishes to price insurance for car owners by predicting their accident rate Y . They assume there is an unobserved factor corresponding to aggressive driving U , that (a) causes drivers to be more likely have an accident, and (b) causes individuals to prefer red cars (the observed variable X). Moreover, individuals belonging to a certain race A are more likely to drive red cars. However, these individuals are no more likely to be aggressive or to get in accidents than any one else. We show this in Figure 1(a). Thus, using the red car feature X to predict accident rate Y would seem to be an unfair prediction because it may charge individuals of a certain race more than others, even though no race is more likely to have an accident. Counterfactual fairness agrees with this notion: changing A while holding U fixed will also change X and, consequently, Y? . Interestingly, we can show (Supplementary Material) that in a linear model, regressing Y on A and X is equivalent to regressing on U , so off-the-shelf regression here is counterfactually fair. Regressing Y on X alone obeys the FTU criterion but is not counterfactually fair, so omitting A (FTU) may introduce unfairness into an otherwise fair world. Scenario 2: High Crime Regions. A city government wants to estimate crime rates by neighborhood to allocate policing resources. Its analyst constructed training data by merging (1) a registry of residents containing their neighborhood X and race A, with (2) police records of arrests, giving each resident a binary label with Y = 1 indicating a criminal arrest record. Due to historically segregated housing, the location X depends on A. Locations X with more police resources have larger numbers of arrests Y . And finally, U represents the totality of socioeconomic factors and policing practices that both influence where an individual may live and how likely they are to be arrested and charged. This can all be seen in Figure 1(b). In this example, higher observed arrest rates in some neighborhoods are due to greater policing there, not because people of different races are any more or less likely to break the law. The label Y = 0 4 does not mean someone has never committed a crime, but rather that they have not been caught. If individuals in the training data have not already had equal opportunity, algorithms enforcing EO will not remedy such unfairness. In contrast, a counterfactually fair approach would model differential enforcement rates using U and base predictions on this information rather than on X directly. In general, we need a multistage procedure in which we first derive latent variables U , and then based on them we minimize some loss with respect to Y . This is the core of the algorithm discussed next. 3.2 Implications One simple but important implication of the definition of counterfactual fairness is the following: Lemma 1. Let G be the causal graph of the given model (U, V, F ). Then Y? will be counterfactually fair if it is a function of the non-descendants of A. Proof. Let W be any non-descendant of A in G. Then WA?a (U ) and WA?a0 (U ) have the same distribution by the three inferential steps in Section 2.2. Hence, the distribution of any function Y? of the non-descendants of A is invariant with respect to the counterfactual values of A. This does not exclude using a descendant W of A as a possible input to Y? . However, this will only be possible in the case where the overall dependence of Y? on A disappears, which will not happen in general. Hence, Lemma 1 provides the most straightforward way to achieve counterfactual fairness. In some scenarios, it is desirable to define path-specific variations of counterfactual fairness that allow for the inclusion of some descendants of A, as discussed by [21, 27] and the Supplementary Material. Ancestral closure of protected attributes. Suppose that a parent of a member of A is not in A. Counterfactual fairness allows for the use of it in the definition of Y? . If this seems counterintuitive, then we argue that the fault should be at the postulated set of protected attributes rather than with the definition of counterfactual fairness, and that typically we should expect set A to be closed under ancestral relationships given by the causal graph. For instance, if Race is a protected attribute, and Mother?s race is a parent of Race, then it should also be in A. Dealing with historical biases and an existing fairness paradox. The explicit difference between Y? and Y allows us to tackle historical biases. For instance, let Y be an indicator of whether a client defaults on a loan, while Y? is the actual decision of giving the loan. Consider the DAG A ? Y , shown in Figure 1(c) with the explicit inclusion of set U of independent background variables. Y is the objectively ideal measure for decision making, the binary indicator of the event that the individual defaults on a loan. If A is postulated to be a protected attribute, then the predictor Y? = Y = fY (A, U ) is not counterfactually fair, with the arrow A ? Y being (for instance) the result of a world that punishes individuals in a way that is out of their control. Figure 1(d) shows a finer-grained model, where the path is mediated by a measure of whether the person is employed, which is itself caused by two background factors: one representing whether the person hiring is prejudiced, and the other the employee?s qualifications. In this world, A is a cause of defaulting, even if mediated by other variables3 . The counterfactual fairness principle however forbids us from using Y : using the twin network 4 of Pearl [28], we see in Figure 1(e) that Ya and Ya0 need not be identically distributed given the background variables. In contrast, any function of variables not descendants of A can be used a basis for fair decision making. This means that any variable Y? defined by Y? = g(U ) will be counterfactually fair for any function g(?). Hence, given a causal model, the functional defined by the function g(?) minimizing some predictive error for Y will satisfy the criterion, as proposed in Section 4.1. We are essentially learning a projection of Y into the space of fair decisions, removing historical biases as a by-product. Counterfactual fairness also provides an answer to some problems on the incompatibility of fairness criteria. In particular, consider the following problem raised independently by different authors (e.g., 3 For example, if the function determining employment fE (A, P, Q) ? I(Q>0,P =0 or A6=a) then an individual with sufficient qualifications and prejudiced potential employer may have a different counterfactual employment value for A = a compared to A = a0 , and a different chance of default. 4 In a nutshell, this is a graph that simultaneously depicts ?multiple worlds? parallel to the factual realizations. In this graph, all multiple worlds share the same background variables, but with different consequences in the remaining variables depending on which counterfactual assignments are provided. 5 [7, 22]), illustrated below for the binary case: ideally, we would like our predictors to obey both Equality of Opportunity and the predictive parity criterion defined by satisfying P (Y = 1 | Y? = 1, A = 1) = P (Y = 1 | Y? = 1, A = 0), as well as the corresponding equation for Y? = 0. It has been shown that if Y and A are marginally associated (e.g., recidivism and race are associated) and Y is not a deterministic function of Y? , then the two criteria cannot be reconciled. Counterfactual fairness throws a light in this scenario, suggesting that both EO and predictive parity may be insufficient if Y and A are associated: assuming that A and Y are unconfounded (as expected for demographic attributes), this is the result of A being a cause of Y . By counterfactual fairness, we should not want to use Y as a basis for our decisions, instead aiming at some function Y?A of variables which are not caused by A but are predictive of Y . Y? is defined in such a way that is an estimate of the ?closest? Y?A to Y according to some preferred risk function. This makes the incompatibility between EO and predictive parity irrelevant, as A and Y?A will be independent by construction given the model assumptions. 4 Implementing Counterfactual Fairness As discussed in the previous Section, we need to relate Y? to Y if the predictor is to be useful, and we restrict Y? to be a (parameterized) function of the non-descendants of A in the causal graph following Lemma 1. We next introduce an algorithm, then discuss assumptions that can be used to express counterfactuals. 4.1 Algorithm Let Y? ? g? (U, XA ) be a predictor parameterized by ?, such as a logistic regression or a neural network, and where XA ? X are non-descendants of A. Given a loss function l(?, ?) such as squared loss or log-likelihood, and training data D ? {(A(i) , X (i) , Y (i) )} for i = 1, 2, . . . , n, we Pn (i) define L(?) ? i=1 E[l(y (i) , g? (U (i) , xA )) | x(i) , a(i) ]/n as the empirical loss to be minimized with respect to ?. Each expectation is with respect to random variable U (i) ? PM (U | x(i) , a(i) ) where PM (U | x, a) is the conditional distribution of the background variables as given by a causal model M that is available by assumption. If this expectation cannot be calculated analytically, Markov chain Monte Carlo (MCMC) can be used to approximate it as in the following algorithm. 1: procedure FAIR L EARNING(D, M) . Learned parameters ?? (i) (i) For each data point i ? D, sample m MCMC samples U1 , . . . , Um ? PM (U | x(i) , a(i) ). Let D0 be the augmented dataset where each point (a(i) , x(i) , y (i) ) in D is replaced with the (i) corresponding m points {(a(i) , x(i) , y (i) , uj )}. P 0 0 (i0 ) 4: ?? ? argmin? i0 ?D0 l(y (i ) , g? (U (i ) , xA )). 5: end procedure 2: 3: At prediction time, we report Y? ? E[Y? (U ? , x?A ) | x? , a? ] for a new data point (a? , x? ). Deconvolution perspective. The algorithm can be understood as a deconvolution approach that, given observables A ? X, extracts its latent sources and pipelines them into a predictive model. We advocate that counterfactual assumptions must underlie all approaches that claim to extract the sources of variation of the data as ?fair? latent components. As an example, Louizos et al. [24] start from the DAG A ? X ? U to extract P (U | X, A). As U and A are not independent given X in this representation, a type of penalization is enforced to create a posterior Pf air (U |A, X) that is close to the model posterior P (U | A, X) while satisfying Pf air (U |A = a, X) ? Pf air (U |A = a0 , X). But this is neither necessary nor sufficient for counterfactual fairness. The model for X given A and U must be justified by a causal mechanism, and that being the case, P (U | A, X) requires no postprocessing. As a matter of fact, model M can be learned by penalizing empirical dependence measures between U and pai for a given Vi (e.g. Mooij et al. [26]), but this concerns M and not Y? , and is motivated by explicit assumptions about structural equations, as described next. 6 4.2 Designing the Input Causal Model Model M must be provided to algorithm FAIR L EARNING. Although this is well understood, it is worthwhile remembering that causal models always require strong assumptions, even more so when making counterfactual claims [8]. Counterfactuals assumptions such as structural equations are in general unfalsifiable even if interventional data for all variables is available. This is because there are infinitely many structural equations compatible with the same observable distribution [28], be it observational or interventional. Having passed testable implications, the remaining components of a counterfactual model should be understood as conjectures formulated according to the best of our knowledge. Such models should be deemed provisional and prone to modifications if, for example, new data containing measurement of variables previously hidden contradict the current model. We point out that we do not need to specify a fully deterministic model, and structural equations can be relaxed as conditional distributions. In particular, the concept of counterfactual fairness holds under three levels of assumptions of increasing strength: Level 1. Build Y? using only the observable non-descendants of A. This only requires partial causal ordering and no further causal assumptions, but in many problems there will be few, if any, observables which are not descendants of protected demographic factors. Level 2. Postulate background latent variables that act as non-deterministic causes of observable variables, based on explicit domain knowledge and learning algorithms5 . Information about X is passed to Y? via P (U | x, a). Level 3. Postulate a fully deterministic model with latent variables. For instance, the distribution P (Vi | pai ) can be treated as an additive error model, Vi = fi (pai )+ei [31]. The error term ei then becomes an input to Y? as calculated from the observed variables. This maximizes the information extracted by the fair predictor Y? . 4.3 Further Considerations on Designing the Input Causal Model One might ask what we can lose by defining causal fairness measures involving only noncounterfactual causal quantities, such as enforcing P (Y? = 1 | do(A = a)) = P (Y? = 1 | do(A = a0 )) instead of our counterfactual criterion. The reason is that the above equation is only a constraint on an average effect. Obeying this criterion provides no guarantees against, for example, having half of the individuals being strongly ?negatively? discriminated and half of the individuals strongly ?positively? discriminated. We advocate that, for fairness, society should not be satisfied in pursuing only counterfactually-free guarantees. While one may be willing to claim posthoc that the equation above masks no balancing effect so that individuals receive approximately the same distribution of outcomes, that itself is just a counterfactual claim in disguise. Our approach is to make counterfactual assumptions explicit. When unfairness is judged to follow only some ?pathways? in the causal graph (in a sense that can be made formal, see [21, 27]), nonparametric assumptions about the independence of counterfactuals may suffice, as discussed by [27]. In general, nonparametric assumptions may not provide identifiable adjustments even in this case, as also discussed in our Supplementary Material. If competing models with different untestable assumptions are available, there are ways of simultaneously enforcing a notion of approximate counterfactual fairness in all of them, as introduced by us in [32]. Other alternatives include exploiting bounds on the contribution of hidden variables [29, 33]. Another issue is the interpretation of causal claims involving demographic variables such as race and sex. Our view is that such constructs are the result of translating complex events into random variables and, despite some controversy, we consider counterproductive to claim that e.g. race and sex cannot be causes. An idealized intervention on some A at a particular time can be seen as a notational shortcut to express a conjunction of more specific interventions, which may be individually doable but jointly impossible in practice. It is the plausibility of complex, even if impossible to practically manipulate, causal chains from A to Y that allows us to claim that unfairness is real [11]. Experiments for constructs exist, such as randomizing names in job applications to make them race-blind. They do not contradict the notion of race as a cause, and can be interpreted as an intervention on a particular aspect of the construct ?race,? such as ?race perception? (e.g. Section 4.4.4 of [29]). 5 In some domains, it is actually common to build a model entirely around latent constructs with few or no observable parents nor connections among observed variables [2]. 7 5 Illustration: Law School Success We illustrate our approach on a practical problem that requires fairness, the prediction of success in law school. A second problem, understanding the contribution of race to police stops, is described in the Supplementary Material. Following closely the usual framework for assessing causal models in the machine learning literature, the goal of this experiment is to quantify how our algorithm behaves with finite sample sizes while assuming ground truth compatible with a synthetic model. Problem definition: Law school success The Law School Admission Council conducted a survey across 163 law schools in the United States [35]. It contains information on 21,790 law students such as their entrance exam scores (LSAT), their grade-point average (GPA) collected prior to law school, and their first year average grade (FYA). Given this data, a school may wish to predict if an applicant will have a high FYA. The school would also like to make sure these predictions are not biased by an individual?s race and sex. However, the LSAT, GPA, and FYA scores, may be biased due to social factors. We compare our framework with two unfair baselines: 1. Full: the standard technique of using all features, including sensitive features such as race and sex to make predictions; 2. Unaware: fairness through unawareness, where we do not use race and sex as features. For comparison, we generate predictors Y? for all models using logistic regression. Fair prediction. As described in Section 4.2, there are three ways in which we can model a counterfactually fair predictor of FYA. Level 1 uses any features which are not descendants of race and sex for prediction. Level 2 models latent ?fair? variables which are parents of observed variables. These variables are independent of both race and sex. Level 3 models the data using an additive error model, and uses the independent error terms to make predictions. These models make increasingly strong assumptions corresponding to increased predictive power. We split the dataset 80/20 into a train/test set, preserving label balance, to evaluate the models. As we believe LSAT, GPA, and FYA are all biased by race and sex, we cannot use any observed features to construct a counterfactually fair predictor as described in Level 1. In Level 2, we postulate that a latent variable: a student?s knowledge (K), affects GPA, LSAT, and FYA scores. The causal graph corresponding to this model is shown in Figure 2, (Level 2). This is a short-hand for the distributions: K R S GPA ? N (bG + wG K + wG R + wG S, ?G ), K R S LSAT ? Poisson(exp(bL + wL K + wL R + wL S)), FYA ? N (wFK K + wFR R + wFS S, 1), K ? N (0, 1) We perform inference on this model using an observed training set to estimate the posterior distribution of K. We use the probabilistic programming language Stan [34] to learn K. We call the predictor constructed using K, Fair K. black $ white asian $ white Level 2 Level 3 0 ?1.0 ?0.5 0.0 0.5 ?0.5 swapped 0.0 0.5 type original original swapped swapped 1 1.5 0.5 ?1.0 0.0 V V pred_zfya FYA FYA 0.4 0.8 0.0 counterfactual 0.5 pred_zfya 1.5 1.5 type original swapped 1.0 type original original swapped 1.0 swapped 0.5 0.0 ?0.4 ?0.5 2.0 0.5 0.0 0.0 pred_zfya FYA 0.5 type original swapped 1.0 0.5 0.0 0.0 pred_zfya density density density density 0.5 V ?0.5 2.0 type 1.0 ?0.5 0 ?1.0 pred_zfya 2.0 1.5 density density Sex type original 1 0 ?1.0 pred_zfya 2.0 Unaware Sex Know density density FYA Race LSAT swapped original data 0 Race 2 2 type original 1 density density FYA ?F density density LSAT ?L 2 1 female $ male 3 3 type density density GPA ?G Full 2 GPA mexican $ white 3 density density 3 0.0 ?0.4 0.0 V V pred_zfya FYA FYA 0.4 0.8 ?0.4 0.0 V V 0.4 0.8 pred_zfya FYA FYA Figure 2: Left: A causal model for the problem of predicting law school success fairly. Right: Density plots of predicted FYAa and FYAa0 . In Level 3, we model GPA, LSAT, and FYA as continuous variables with additive error terms independent of race and sex (that may in turn be correlated with one-another). This model is shown 8 Table 1: Prediction results using logistic regression. Note that we must sacrifice a small amount of accuracy to ensuring counterfactually fair prediction (Fair K, Fair Add), versus the models that use unfair features: GPA, LSAT, race, sex (Full, Unaware). Full Unaware Fair K Fair Add RMSE 0.873 0.894 0.929 0.918 in Figure 2, (Level 3), and is expressed by: R S GPA = bG + wG R + wG S + G , G ? p(G ) R S LSAT = bL + wL R + wL S + L , L ? p(L ) FYA = bF + wFR R + wFS S + F , F ? p(F ) We estimate the error terms G , L by first fitting two models that each use race and sex to individually predict GPA and LSAT. We then compute the residuals of each model (e.g., G = GPA? Y?GPA (R, S)). We use these residual estimates of G , L to predict FYA. We call this Fair Add. Accuracy. We compare the RMSE achieved by logistic regression for each of the models on the test set in Table 1. The Full model achieves the lowest RMSE as it uses race and sex to more accurately reconstruct FYA. Note that in this case, this model is not fair even if the data was generated by one of the models shown in Figure 2 as it corresponds to Scenario 3. The (also unfair) Unaware model still uses the unfair variables GPA and LSAT, but because it does not use race and sex it cannot match the RMSE of the Full model. As our models satisfy counterfactual fairness, they trade off some accuracy. Our first model Fair K uses weaker assumptions and thus the RMSE is highest. Using the Level 3 assumptions, as in Fair Add we produce a counterfactually fair model that trades slightly stronger assumptions for lower RMSE. Counterfactual fairness. We would like to empirically test whether the baseline methods are counterfactually fair. To do so we will assume the true model of the world is given by Figure 2, (Level 2). We can fit the parameters of this model using the observed data and evaluate counterfactual fairness by sampling from it. Specifically, we will generate samples from the model given either the observed race and sex, or counterfactual race and sex variables. We will fit models to both the original and counterfactual sampled data and plot how the distribution of predicted FYA changes for both baseline models. Figure 2 shows this, where each row corresponds to a baseline predictor and each column corresponds to the counterfactual change. In each plot, the blue distribution is density of predicted FYA for the original data and the red distribution is this density for the counterfactual data. If a model is counterfactually fair we would expect these distributions to lie exactly on top of each other. Instead, we note that the Full model exhibits counterfactual unfairness for all counterfactuals except sex. We see a similar trend for the Unaware model, although it is closer to being counterfactually fair. To see why these models seem to be fair w.r.t. to sex we can look at weights of the DAG which generates the counterfactual data. Specifically the DAG weights from (male,female) to GPA are (0.93,1.06) and from (male,female) to LSAT are (1.1,1.1). Thus, these models are fair w.r.t. to sex simply because of a very weak causal link between sex and GPA/LSAT. 6 Conclusion We have presented a new model of fairness we refer to as counterfactual fairness. It allows us to propose algorithms that, rather than simply ignoring protected attributes, are able to take into account the different social biases that may arise towards individuals based on ethically sensitive attributes and compensate for these biases effectively. We experimentally contrasted our approach with previous fairness approaches and show that our explicit causal models capture these social biases and make clear the implicit trade-off between prediction accuracy and fairness in an unfair world. We propose that fairness should be regulated by explicitly modeling the causal structure of the world. Criteria based purely on probabilistic independence cannot satisfy this and are unable to address how unfairness is occurring in the task at hand. By providing such causal tools for addressing fairness questions we hope we can provide practitioners with customized techniques for solving a wide array of fairness modeling problems. 9 Acknowledgments This work was supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1. CR acknowledges additional support under the EPSRC Platform Grant EP/P022529/1. We thank Adrian Weller for insightful feedback, and the anonymous reviewers for helpful comments. 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Prototypical Networks for Few-shot Learning Jake Snell University of Toronto? Vector Institute Kevin Swersky Twitter Richard Zemel University of Toronto Vector Institute Canadian Institute for Advanced Research Abstract We propose Prototypical Networks for the problem of few-shot classification, where a classifier must generalize to new classes not seen in the training set, given only a small number of examples of each new class. Prototypical Networks learn a metric space in which classification can be performed by computing distances to prototype representations of each class. Compared to recent approaches for few-shot learning, they reflect a simpler inductive bias that is beneficial in this limited-data regime, and achieve excellent results. We provide an analysis showing that some simple design decisions can yield substantial improvements over recent approaches involving complicated architectural choices and meta-learning. We further extend Prototypical Networks to zero-shot learning and achieve state-ofthe-art results on the CU-Birds dataset. 1 Introduction Few-shot classification [22, 18, 15] is a task in which a classifier must be adapted to accommodate new classes not seen in training, given only a few examples of each of these classes. A naive approach, such as re-training the model on the new data, would severely overfit. While the problem is quite difficult, it has been demonstrated that humans have the ability to perform even one-shot classification, where only a single example of each new class is given, with a high degree of accuracy [18]. Two recent approaches have made significant progress in few-shot learning. Vinyals et al. [32] proposed Matching Networks, which uses an attention mechanism over a learned embedding of the labeled set of examples (the support set) to predict classes for the unlabeled points (the query set). Matching Networks can be interpreted as a weighted nearest-neighbor classifier applied within an embedding space. Notably, this model utilizes sampled mini-batches called episodes during training, where each episode is designed to mimic the few-shot task by subsampling classes as well as data points. The use of episodes makes the training problem more faithful to the test environment and thereby improves generalization. Ravi and Larochelle [24] take the episodic training idea further and propose a meta-learning approach to few-shot learning. Their approach involves training an LSTM [11] to produce the updates to a classifier, given an episode, such that it will generalize well to a test-set. Here, rather than training a single model over multiple episodes, the LSTM meta-learner learns to train a custom model for each episode. We attack the problem of few-shot learning by addressing the key issue of overfitting. Since data is severely limited, we work under the assumption that a classifier should have a very simple inductive bias. Our approach, Prototypical Networks, is based on the idea that there exists an embedding in which points cluster around a single prototype representation for each class. In order to do this, we learn a non-linear mapping of the input into an embedding space using a neural network and take a class?s prototype to be the mean of its support set in the embedding space. Classification is then performed for an embedded query point by simply finding the nearest class prototype. We ? Initial work done while at Twitter. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. c2 c2 x c1 v1 x c1 c3 c3 (a) Few-shot v2 v3 (b) Zero-shot Figure 1: Prototypical Networks in the few-shot and zero-shot scenarios. Left: Few-shot prototypes ck are computed as the mean of embedded support examples for each class. Right: Zero-shot prototypes ck are produced by embedding class meta-data vk . In either case, embedded query points are classified via a softmax over distances to class prototypes: p? (y = k|x) ? exp(?d(f? (x), ck )). follow the same approach to tackle zero-shot learning; here each class comes with meta-data giving a high-level description of the class rather than a small number of labeled examples. We therefore learn an embedding of the meta-data into a shared space to serve as the prototype for each class. Classification is performed, as in the few-shot scenario, by finding the nearest class prototype for an embedded query point. In this paper, we formulate Prototypical Networks for both the few-shot and zero-shot settings. We draw connections to Matching Networks in the one-shot setting, and analyze the underlying distance function used in the model. In particular, we relate Prototypical Networks to clustering [4] in order to justify the use of class means as prototypes when distances are computed with a Bregman divergence, such as squared Euclidean distance. We find empirically that the choice of distance is vital, as Euclidean distance greatly outperforms the more commonly used cosine similarity. On several benchmark tasks, we achieve state-of-the-art performance. Prototypical Networks are simpler and more efficient than recent meta-learning algorithms, making them an appealing approach to few-shot and zero-shot learning. 2 2.1 Prototypical Networks Notation In few-shot classification we are given a small support set of N labeled examples S = {(x1 , y1 ), . . . , (xN , yN )} where each xi ? RD is the D-dimensional feature vector of an example and yi ? {1, . . . , K} is the corresponding label. Sk denotes the set of examples labeled with class k. 2.2 Model Prototypical Networks compute an M -dimensional representation ck ? RM , or prototype, of each class through an embedding function f? : RD ? RM with learnable parameters ?. Each prototype is the mean vector of the embedded support points belonging to its class: X 1 ck = f? (xi ) (1) |Sk | (xi ,yi )?Sk Given a distance function d : RM ? RM ? [0, +?), Prototypical Networks produce a distribution over classes for a query point x based on a softmax over distances to the prototypes in the embedding space: exp(?d(f? (x), ck )) p? (y = k | x) = P (2) 0 k0 exp(?d(f? (x), ck )) Learning proceeds by minimizing the negative log-probability J(?) = ? log p? (y = k | x) of the true class k via SGD. Training episodes are formed by randomly selecting a subset of classes from the training set, then choosing a subset of examples within each class to act as the support set and a 2 Algorithm 1 Training episode loss computation for Prototypical Networks. N is the number of examples in the training set, K is the number of classes in the training set, NC ? K is the number of classes per episode, NS is the number of support examples per class, NQ is the number of query examples per class. R ANDOM S AMPLE(S, N ) denotes a set of N elements chosen uniformly at random from set S, without replacement. Input: Training set D = {(x1 , y1 ), . . . , (xN , yN )}, where each yi ? {1, . . . , K}. Dk denotes the subset of D containing all elements (xi , yi ) such that yi = k. Output: The loss J for a randomly generated training episode. V ? R ANDOM S AMPLE({1, . . . , K}, NC ) . Select class indices for episode for k in {1, . . . , NC } do Sk ? R ANDOM S AMPLE(DVk , NS ) . Select support examples Qk ? R ANDOM S AMPLE(DVk \ Sk , NQ ) . Select query examples X 1 ck ? f? (xi ) . Compute prototype from support examples NC (xi ,yi )?Sk end for J ?0 for k in {1, . . . , NC } do for (x, y) in Qk do " # X 1 J ?J+ d(f? (x), ck )) + log exp(?d(f? (x), ck0 )) NC NQ 0 k end for end for . Initialize loss . Update loss subset of the remainder to serve as query points. Pseudocode to compute the loss J(?) for a training episode is provided in Algorithm 1. 2.3 Prototypical Networks as Mixture Density Estimation For a particular class of distance functions, known as regular Bregman divergences [4], the Prototypical Networks algorithm is equivalent to performing mixture density estimation on the support set with an exponential family density. A regular Bregman divergence d? is defined as: d? (z, z0 ) = ?(z) ? ?(z0 ) ? (z ? z0 )T ??(z0 ), (3) where ? is a differentiable, strictly convex function of the Legendre type. Examples of Bregman divergences include squared Euclidean distance kz ? z0 k2 and Mahalanobis distance. Prototype computation can be viewed in terms of hard clustering on the support set, with one cluster per class and each support point assigned to its corresponding class cluster. It has been shown [4] for Bregman divergences that the cluster representative achieving minimal distance to its assigned points is the cluster mean. Thus the prototype computation in Equation (1) yields optimal cluster representatives given the support set labels when a Bregman divergence is used. Moreover, any regular exponential family distribution p? (z|?) with parameters ? and cumulant function ? can be written in terms of a uniquely determined regular Bregman divergence [4]: p? (z|?) = exp{zT ? ? ?(?) ? g? (z)} = exp{?d? (z, ?(?)) ? g? (z)} Consider now a regular exponential family mixture model with parameters ? = p(z|?) = K X ?k p? (z|?k ) = k=1 K X (4) {?k , ?k }K k=1 : ?k exp(?d? (z, ?(?k )) ? g? (z)) (5) k=1 Given ?, inference of the cluster assignment y for an unlabeled point z becomes: ?k exp(?d? (z, ?(?k ))) 0 k0 ?k exp(?d? (z, ?(?k ))) p(y = k|z) = P (6) For an equally-weighted mixture model with one cluster per class, cluster assignment inference (6) is equivalent to query class prediction (2) with f? (x) = z and ck = ?(?k ). In this case, 3 Prototypical Networks are effectively performing mixture density estimation with an exponential family distribution determined by d? . The choice of distance therefore specifies modeling assumptions about the class-conditional data distribution in the embedding space. 2.4 Reinterpretation as a Linear Model A simple analysis is useful in gaining insight into the nature of the learned classifier. When we use Euclidean distance d(z, z0 ) = kz ? z0 k2 , then the model in Equation (2) is equivalent to a linear model with a particular parameterization [21]. To see this, expand the term in the exponent: > ?kf? (x) ? ck k2 = ?f? (x)> f? (x) + 2c> k f? (x) ? ck ck (7) The first term in Equation (7) is constant with respect to the class k, so it does not affect the softmax probabilities. We can write the remaining terms as a linear model as follows: > > > 2c> k f? (x) ? ck ck = wk f? (x) + bk , where wk = 2ck and bk = ?ck ck (8) We focus primarily on squared Euclidean distance (corresponding to spherical Gaussian densities) in this work. Our results indicate that Euclidean distance is an effective choice despite the equivalence to a linear model. We hypothesize this is because all of the required non-linearity can be learned within the embedding function. Indeed, this is the approach that modern neural network classification systems currently use, e.g., [16, 31]. 2.5 Comparison to Matching Networks Prototypical Networks differ from Matching Networks in the few-shot case with equivalence in the one-shot scenario. Matching Networks [32] produce a weighted nearest neighbor classifier given the support set, while Prototypical Networks produce a linear classifier when squared Euclidean distance is used. In the case of one-shot learning, ck = xk since there is only one support point per class, and Matching Networks and Prototypical Networks become equivalent. A natural question is whether it makes sense to use multiple prototypes per class instead of just one. If the number of prototypes per class is fixed and greater than 1, then this would require a partitioning scheme to further cluster the support points within a class. This has been proposed in Mensink et al. [21] and Rippel et al. [27]; however both methods require a separate partitioning phase that is decoupled from the weight updates, while our approach is simple to learn with ordinary gradient descent methods. Vinyals et al. [32] propose a number of extensions, including decoupling the embedding functions of the support and query points, and using a second-level, fully-conditional embedding (FCE) that takes into account specific points in each episode. These could likewise be incorporated into Prototypical Networks, however they increase the number of learnable parameters, and FCE imposes an arbitrary ordering on the support set using a bi-directional LSTM. Instead, we show that it is possible to achieve the same level of performance using simple design choices, which we outline next. 2.6 Design Choices Distance metric Vinyals et al. [32] and Ravi and Larochelle [24] apply Matching Networks using cosine distance. However for both Prototypical Networks and Matching Networks any distance is permissible, and we found that using squared Euclidean distance can greatly improve results for both. For Protypical Networks, we conjecture this is primarily due to cosine distance not being a Bregman divergence, and thus the equivalence to mixture density estimation discussed in Section 2.3 does not hold. Episode composition A straightforward way to construct episodes, used in Vinyals et al. [32] and Ravi and Larochelle [24], is to choose Nc classes and NS support points per class in order to match the expected situation at test-time. That is, if we expect at test-time to perform 5-way classification and 1-shot learning, then training episodes could be comprised of Nc = 5, NS = 1. We have found, however, that it can be extremely beneficial to train with a higher Nc , or ?way?, than will be used at test-time. In our experiments, we tune the training Nc on a held-out validation set. Another consideration is whether to match NS , or ?shot?, at train and test-time. For Prototypical Networks, we found that it is usually best to train and test with the same ?shot? number. 4 2.7 Zero-Shot Learning Zero-shot learning differs from few-shot learning in that instead of being given a support set of training points, we are given a class meta-data vector vk for each class. These could be determined in advance, or they could be learned from e.g., raw text [8]. Modifying Prototypical Networks to deal with the zero-shot case is straightforward: we simply define ck = g? (vk ) to be a separate embedding of the meta-data vector. An illustration of the zero-shot procedure for Prototypical Networks as it relates to the few-shot procedure is shown in Figure 1. Since the meta-data vector and query point come from different input domains, we found it was helpful empirically to fix the prototype embedding g to have unit length, however we do not constrain the query embedding f . 3 Experiments For few-shot learning, we performed experiments on Omniglot [18] and the miniImageNet version of ILSVRC-2012 [28] with the splits proposed by Ravi and Larochelle [24]. We perform zero-shot experiments on the 2011 version of the Caltech UCSD bird dataset (CUB-200 2011) [34]. 3.1 Omniglot Few-shot Classification Omniglot [18] is a dataset of 1623 handwritten characters collected from 50 alphabets. There are 20 examples associated with each character, where each example is drawn by a different human subject. We follow the procedure of Vinyals et al. [32] by resizing the grayscale images to 28 ? 28 and augmenting the character classes with rotations in multiples of 90 degrees. We use 1200 characters plus rotations for training (4,800 classes in total) and the remaining classes, including rotations, for test. Our embedding architecture mirrors that used by Vinyals et al. [32] and is composed of four convolutional blocks. Each block comprises a 64-filter 3 ? 3 convolution, batch normalization layer [12], a ReLU nonlinearity and a 2 ? 2 max-pooling layer. When applied to the 28 ? 28 Omniglot images this architecture results in a 64-dimensional output space. We use the same encoder for embedding both support and query points. All of our models were trained via SGD with Adam [13]. We used an initial learning rate of 10?3 and cut the learning rate in half every 2000 episodes. No regularization was used other than batch normalization. We trained Prototypical Networks using Euclidean distance in the 1-shot and 5-shot scenarios with training episodes containing 60 classes and 5 query points per class. We found that it is advantageous to match the training-shot with the test-shot, and to use more classes (higher ?way?) per training episode rather than fewer. We compare against various baselines, including the Neural Statistician [7], Meta-Learner LSTM [24], MAML [9], and both the fine-tuned and non-fine-tuned versions of Matching Networks [32]. We computed classification accuracy for our models averaged over 1,000 randomly generated episodes from the test set. The results are shown in Table 1 and to our knowledge are competitive with state-of-the-art on this dataset. Figure 2 shows a sample t-SNE visualization [20] of the embeddings learned by Prototypical Networks. We visualize a subset of test characters from the same alphabet in order to gain better insight, despite the fact that classes in actual test episodes are likely to come from different alphabets. Even though the visualized characters are minor variations of each other, the network is able to cluster the hand-drawn characters closely around the class prototypes. Figure 2: A t-SNE visualization of the embeddings learned by Prototypical networks on the Omniglot dataset. A subset of the Tengwar script is shown (an alphabet in the test set). Class prototypes are indicated in black. Several misclassified characters are highlighted in red along with arrows pointing to the correct prototype. 5 Table 1: Few-shot classification accuracies on Omniglot. ? Uses non-standard train/test splits. Model M ATCHING N ETWORKS [32] M ATCHING N ETWORKS [32] N EURAL S TATISTICIAN [7] MAML [9]? P ROTOTYPICAL N ETWORKS (O URS ) Dist. Fine Tune Cosine Cosine Euclid. N Y N N N 5-way Acc. 1-shot 5-shot 20-way Acc. 1-shot 5-shot 98.1% 97.9% 98.1% 98.7% 98.8% 93.8% 93.5% 93.2% 95.8% 96.0% 98.9% 98.7% 99.5% 99.9% 99.7% 98.5% 98.7% 98.1% 98.9% 98.9% Table 2: Few-shot classification accuracies on miniImageNet. All accuracy results are averaged over 600 test episodes and are reported with 95% confidence intervals. ? Results reported by [24]. 5-way Acc. Model BASELINE N EAREST NEIGHBORS? M ATCHING N ETWORKS [32]? M ATCHING N ETWORKS FCE [32]? M ETA -L EARNER LSTM [24]? MAML [9] P ROTOTYPICAL N ETWORKS (O URS ) 3.2 Dist. Fine Tune 1-shot 5-shot Cosine Cosine Cosine Euclid. N N N N N N 28.86 ? 0.54% 43.40 ? 0.78% 43.56 ? 0.84% 43.44 ? 0.77% 48.70 ? 1.84% 49.42 ? 0.78% 49.79 ? 0.79% 51.09 ? 0.71% 55.31 ? 0.73% 60.60 ? 0.71% 63.15 ? 0.91% 68.20 ? 0.66% miniImageNet Few-shot Classification The miniImageNet dataset, originally proposed by Vinyals et al. [32], is derived from the larger ILSVRC-12 dataset [28]. The splits used by Vinyals et al. [32] consist of 60,000 color images of size 84 ? 84 divided into 100 classes with 600 examples each. For our experiments, we use the splits introduced by Ravi and Larochelle [24] in order to directly compare with state-of-the-art algorithms for few-shot learning. Their splits use a different set of 100 classes, divided into 64 training, 16 validation, and 20 test classes. We follow their procedure by training on the 64 training classes and using the 16 validation classes for monitoring generalization performance only. We use the same four-block embedding architecture as in our Omniglot experiments, though here it results in a 1,600-dimensional output space due to the increased size of the images. We also use the same learning rate schedule as in our Omniglot experiments and train until validation loss stops improving. We train using 30-way episodes for 1-shot classification and 20-way episodes for 5-shot classification. We match train shot to test shot and each class contains 15 query points per episode. We compare to the baselines as reported by Ravi and Larochelle [24], which include a simple nearest neighbor approach on top of features learned by a classification network on the 64 training classes. The other baselines are two non-fine-tuned variants of Matching Networks (both ordinary and FCE) and the Meta-Learner LSTM. in the non-fine-tuned setting because the fine-tuning procedure as proposed by Vinyals et al. [32] is not fully described. As can be seen in Table 2, Prototypical Networks achieves state-of-the-art by a wide margin on 5-shot accuracy. We conducted further analysis, to determine the effect of distance metric and the number of training classes per episode on the performance of Prototypical Networks and Matching Networks. To make the methods comparable, we use our own implementation of Matching Networks that utilizes the same embedding architecture as our Prototypical Networks. In Figure 3 we compare cosine vs. Euclidean distance and 5-way vs. 20-way training episodes in the 1-shot and 5-shot scenarios, with 15 query points per class per episode. We note that 20-way achieves higher accuracy than 5-way and conjecture that the increased difficulty of 20-way classification helps the network to generalize better, because it forces the model to make more fine-grained decisions in the embedding space. Also, using Euclidean distance improves performance substantially over cosine distance. This effect is even more pronounced for Prototypical Networks, in which computing the class prototype as the mean of embedded support points is more naturally suited to Euclidean distances since cosine distance is not a Bregman divergence. 6 70% 80% Matching / Proto. Nets 5-shot Accuracy (5-way) 1-shot Accuracy (5-way) 80% 60% 50% 40% 30% 20% 70% Matching Nets Proto. Nets 60% 50% 40% 30% 20% 5-way Cosine 5-way Euclid. 20-way Cosine 1-shot 5-way Cosine 20-way Euclid. 5-way Euclid. 20-way Cosine 5-shot 20-way Euclid. Figure 3: Comparison showing the effect of distance metric and number of classes per training episode on 5-way classification accuracy for both Matching Networks and Prototypical Networks on miniImageNet. The x-axis indicates configuration of the training episodes (way, distance, and shot), and the y-axis indicates 5-way test accuracy for the corresponding shot. Error bars indicate 95% confidence intervals as computed over 600 test episodes. Note that Matching Networks and Prototypical Networks are identical in the 1-shot case. Table 3: Zero-shot classification accuracies on CUB-200. Model ALE [1] SJE [2] S AMPLE C LUSTERING [19] SJE [2] DS-SJE [25] DA-SJE [25] S YNTHESIZED C LASSIFIERS [6] P ROTOTYPICAL N ETWORKS (O URS ) Z HANG AND S ALIGRAMA [36] 3.3 Image Features 50-way Acc. 0-shot Fisher AlexNet AlexNet GoogLeNet GoogLeNet GoogLeNet GoogLeNet GoogLeNet VGG-19 26.9% 40.3% 44.3% 50.1% 50.4% 50.9% 54.7% 54.8% 55.3% ? 0.8 CUB Zero-shot Classification In order to assess the suitability of our approach for zero-shot learning, we also run experiments on the Caltech-UCSD Birds (CUB) 200-2011 dataset [34]. The CUB dataset contains 11,788 images of 200 bird species. We closely follow the procedure of Reed et al. [25] in preparing the data. We use their splits to divide the classes into 100 training, 50 validation, and 50 test. For images we use 1,024dimensional features extracted by applying GoogLeNet [31] to middle, upper left, upper right, lower left, and lower right crops of the original and horizontally-flipped image2 . At test time we use only the middle crop of the original image. For class meta-data we use the 312-dimensional continuous attribute vectors provided with the CUB dataset. These attributes encode various characteristics of the bird species such as their color, shape, and feather patterns. We learned a simple linear mapping on top of both the 1024-dimensional image features and the 312-dimensional attribute vectors to produce a 1,024-dimensional output space. For this dataset we found it helpful to normalize the class prototypes (embedded attribute vectors) to be of unit length, since the attribute vectors come from a different domain than the images. Training episodes were constructed with 50 classes and 10 query images per class. The embeddings were optimized via SGD with Adam at a fixed learning rate of 10?4 and weight decay of 10?5 . Early stopping on validation loss was used to determine the optimal number of epochs for retraining on the training plus validation set. Table 3 shows that we achieve state-of-the-art results when compared to methods utilizing attributes as class meta-data. We compare our method to variety of zero-shot learning methods, including other embedding approaches such as ALE [1], SJE [2], and DS-SJE/DA-SJE [25]. We also compare to a recent clustering approach [19] which trains an SVM on a learned feature space obtained by fine2 Features downloaded from https://github.com/reedscot/cvpr2016. 7 tuning AlexNet [16]. The Synthesized Classifiers approach of [6] is a manifold learning technique that aligns the class meta-data space with the visual model space, and the method of Zhang and Saligrama [36] is a structured prediction approach trained on top of VGG-19 features [30]. Since Zhang and Saligrama [36] is a randomized method, we include their reported error bars in Table 3. Our Protypical Networks outperform Synthesized Classifiers and are within error bars of Zhang and Saligrama [36], while being a much simpler approach than either. We also ran an additional set of zero-shot experiments with stronger class meta-data. We extracted 1,024-dimensional meta-data vectors for each CUB-200 class using the pretrained Char CNN-RNN model of [25], then trained zero-shot Prototypical Networks using the same procedure described above except we used a 512-dimensional output embedding, as chosen via validation accuracy. We obtained test accuracy of 58.3%, compared to the 54.0% accuracy obtained by DS-SJE [25] with a Char CNN-RNN model. Moreover, our result exceeds the 56.8% accuracy attained by DS-SJE with even stronger Word CNN-RNN class-metadata representations. Taken together, these zero-shot classification results demonstrate that our approach is general enough to be applied even when the data points (images) are from a different domain relative to the classes (attributes). 4 Related Work The literature on metric learning is vast [17, 5]; we summarize here the work most relevant to our proposed method. Neighborhood Components Analysis (NCA) [10] learns a Mahalanobis distance to maximize K-nearest-neighbor?s (KNN) leave-one-out accuracy in the transformed space. Salakhutdinov and Hinton [29] extend NCA by using a neural network to perform the transformation. Large margin nearest neighbor (LMNN) classification [33] also attempts to optimize KNN accuracy but does so using a hinge loss that encourages the local neighborhood of a point to contain other points with the same label. The DNet-KNN [23] is another margin-based method that improves upon LMNN by utilizing a neural network to perform the embedding instead of a simple linear transformation. Of these, our method is most similar to the non-linear extension of NCA [29] because we use a neural network to perform the embedding and we optimize a softmax based on Euclidean distances in the transformed space, as opposed to a margin loss. A key distinction between our approach and non-linear NCA is that we form a softmax directly over classes, rather than individual points, computed from distances to each class?s prototype representation. This allows each class to have a concise representation independent of the number of data points and obviates the need to store the entire support set to make predictions. Our approach is also similar to the nearest class mean approach [21], where each class is represented by the mean of its examples. This approach was developed to rapidly incorporate new classes into a classifier without retraining, however it relies on a linear embedding and was designed to handle the case where the novel classes come with a large number of examples. In contrast, our approach utilizes neural networks to non-linearly embed points and we couple this with episodic training in order to handle the few-shot scenario. Mensink et al. [21] attempt to extend their approach to also perform non-linear classification, but they do so by allowing classes to have multiple prototypes. They find these prototypes in a pre-processing step by using k-means on the input space and then perform a multi-modal variant of their linear embedding. Prototypical Networks, on the other hand, learn a non-linear embedding in an end-to-end manner with no such pre-processing, producing a non-linear classifier that still only requires one prototype per class. In addition, our approach naturally generalizes to other distance functions, particularly Bregman divergences. The center loss proposed by Wen et al. [35] for face recognition is similar to ours but has two main differences. First, they learn the centers for each class as parameters of the model whereas we compute protoypes as a function of the labeled examples within each episode. Second, they combine the center loss with a softmax loss in order to prevent representations collapsing to zero, whereas we construct a softmax loss from our prototypes which naturally prevents such collapse. Moreover, our approach is designed for the few-shot scenario rather than face recognition. A relevant few-shot learning method is the meta-learning approach proposed in Ravi and Larochelle [24]. The key insight here is that LSTM dynamics and gradient descent can be written in effectively the same way. An LSTM can then be trained to itself train a model from a given episode, with the performance goal of generalizing well on the query points. MAML [9] is another meta-learning approach to few-shot learning. It seeks to learn a representation that is easily fit to new data with few 8 steps of gradient descent. Matching Networks and Prototypical Networks can also be seen as forms of meta-learning, in the sense that they produce simple classifiers dynamically from new training episodes; however the core embeddings they rely on are fixed after training. The FCE extension to Matching Networks involves a secondary embedding that depends on the support set. However, in the few-shot scenario the amount of data is so small that a simple inductive bias seems to work well, without the need to learn a custom embedding for each episode. Prototypical Networks are also related to the Neural Statistician [7] from the generative modeling literature, which extends the variational autoencoder [14, 26] to learn generative models of datasets rather than individual points. One component of the Neural Statistician is the ?statistic network? which summarizes a set of data points into a statistic vector. It does this by encoding each point within a dataset, taking a sample mean, and applying a post-processing network to obtain an approximate posterior over the statistic vector. Edwards and Storkey [7] test their model for one-shot classification on the Omniglot dataset by considering each character to be a separate dataset and making predictions based on the class whose approximate posterior over the statistic vector has minimal KL-divergence from the posterior inferred by the test point. Like the Neural Statistician, we also produce a summary statistic for each class. However, ours is a discriminative model, as befits our discriminative task of few-shot classification. With respect to zero-shot learning, the use of embedded meta-data in Prototypical Networks resembles the method of [3] in that both predict the weights of a linear classifier. The DS-SJE and DA-SJE approach of [25] also learns deep multimodal embedding functions for images and class meta-data. Unlike ours, they learn using an empirical risk loss. Neither [3] nor [25] uses episodic training, which allows us to help speed up training and regularize the model. 5 Conclusion We have proposed a simple method called Prototypical Networks for few-shot learning based on the idea that we can represent each class by the mean of its examples in a representation space learned by a neural network. We train these networks to specifically perform well in the few-shot setting by using episodic training. The approach is far simpler and more efficient than recent meta-learning approaches, and produces state-of-the-art results even without sophisticated extensions developed for Matching Networks (although these can be applied to Prototypical Networks as well). We show how performance can be greatly improved by carefully considering the chosen distance metric, and by modifying the episodic learning procedure. We further demonstrate how to generalize Prototypical Networks to the zero-shot setting, and achieve state-of-the-art results on the CUB-200 dataset. A natural direction for future work is to utilize Bregman divergences other than squared Euclidean distance, corresponding to class-conditional distributions beyond spherical Gaussians. We conducted preliminary explorations of this, including learning a variance per dimension for each class. This did not lead to any empirical gains, suggesting that the embedding network has enough flexibility on its own without requiring additional fitted parameters per class. Overall, the simplicity and effectiveness of Prototypical Networks makes it a promising approach for few-shot learning. Acknowledgements We would like to thank Marc Law, Sachin Ravi, Hugo Larochelle, Renjie Liao, and Oriol Vinyals for helpful discussions. This work was supported by the Samsung GRP project and the Canadian Institute for Advanced Research. References [1] Zeynep Akata, Florent Perronnin, Zaid Harchaoui, and Cordelia Schmid. Label-embedding for attributebased classification. In IEEE Computer Vision and Pattern Recognition, pages 819?826, 2013. [2] Zeynep Akata, Scott Reed, Daniel Walter, Honglak Lee, and Bernt Schiele. Evaluation of output embeddings for fine-grained image classification. In IEEE Computer Vision and Pattern Recognition, 2015. 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Triple Generative Adversarial Nets Chongxuan Li, Kun Xu, Jun Zhu?, Bo Zhang Dept. of Comp. Sci. & Tech., TNList Lab, State Key Lab of Intell. Tech. & Sys., Center for Bio-Inspired Computing Research, Tsinghua University, Beijing, 100084, China {licx14, xu-k16}@mails.tsinghua.edu.cn, {dcszj, dcszb}@mail.tsinghua.edu.cn Abstract Generative Adversarial Nets (GANs) have shown promise in image generation and semi-supervised learning (SSL). However, existing GANs in SSL have two problems: (1) the generator and the discriminator (i.e. the classifier) may not be optimal at the same time; and (2) the generator cannot control the semantics of the generated samples. The problems essentially arise from the two-player formulation, where a single discriminator shares incompatible roles of identifying fake samples and predicting labels and it only estimates the data without considering the labels. To address the problems, we present triple generative adversarial net (Triple-GAN), which consists of three players?a generator, a discriminator and a classifier. The generator and the classifier characterize the conditional distributions between images and labels, and the discriminator solely focuses on identifying fake image-label pairs. We design compatible utilities to ensure that the distributions characterized by the classifier and the generator both converge to the data distribution. Our results on various datasets demonstrate that Triple-GAN as a unified model can simultaneously (1) achieve the state-of-the-art classification results among deep generative models, and (2) disentangle the classes and styles of the input and transfer smoothly in the data space via interpolation in the latent space class-conditionally. 1 Introduction Deep generative models (DGMs) can capture the underlying distributions of the data and synthesize new samples. Recently, significant progress has been made on generating realistic images based on Generative Adversarial Nets (GANs) [7, 3, 22]. GAN is formulated as a two-player game, where the generator G takes a random noise z as input and produces a sample G(z) in the data space while the discriminator D identifies whether a certain sample comes from the true data distribution p(x) or the generator. Both G and D are parameterized as deep neural networks and the training procedure is to solve a minimax problem: min max U (D, G) = Ex?p(x) [log(D(x))] + Ez?pz (z) [log(1 ? D(G(z)))], G D where pz (z) is a simple distribution (e.g., uniform or normal) and U (?) denotes the utilities. Given a generator and the defined distribution pg , the optimal discriminator is D(x) = p(x)/(pg (x) + p(x)) in the nonparametric setting, and the global equilibrium of this game is achieved if and only if pg (x) = p(x) [7], which is desired in terms of image generation. GANs and DGMs in general have also proven effective in semi-supervised learning (SSL) [11], while retaining the generative capability. Under the same two-player game framework, Cat-GAN [26] generalizes GANs with a categorical discriminative network and an objective function that minimizes the conditional entropy of the predictions given the real data while maximizes the conditional entropy ? J. Zhu is the corresponding author. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. CE ?? , ?? ? ?(?, ?) ?? ? ?(?) C ?? ? ?? (?) ?? , ?? ? ?(?, ?) ?? , ?? ~?? (?, ?) CE ?? , ?? ~?? (?, ?) G ?? ? ?(?) D A/R A A/R Figure 1: An illustration of Triple-GAN (best view in color). The utilities of D, C and G are colored in blue, green and yellow respectively, with ?R? denoting rejection, ?A? denoting acceptance and ?CE? denoting the cross entropy loss for supervised learning. ?A?s and ?R?s are the adversarial losses and ?CE?s are unbiased regularizations that ensure the consistency between pg , pc and p, which are the distributions defined by the generator, classifier and true data generating process, respectively. of the predictions given the generated samples. Odena [20] and Salimans et al. [25] augment the categorical discriminator with one more class, corresponding to the fake data generated by the generator. There are two main problems in existing GANs for SSL: (1) the generator and the discriminator (i.e. the classifier) may not be optimal at the same time [25]; and (2) the generator cannot control the semantics of the generated samples. For the first problem, as an instance, Salimans et al. [25] propose two alternative training objectives that work well for either classification or image generation in SSL, but not both. The objective of feature matching works well in classification but fails to generate indistinguishable samples (See Sec.5.2 for examples), while the other objective of minibatch discrimination is good at realistic image generation but cannot predict labels accurately. The phenomena are not analyzed deeply in [25] and here we argue that they essentially arise from the two-player formulation, where a single discriminator has to play two incompatible roles?identifying fake samples and predicting labels. Specifically, assume that G is optimal, i.e p(x) = pg (x), and consider a sample x ? pg (x). On one hand, as a discriminator, the optimal D should identify x as a fake sample with non-zero probability (See [7] for the proof). On the other hand, as a classifier, the optimal D should always predict the correct class of x confidently since x ? p(x). It conflicts as D has two incompatible convergence points, which indicates that G and D may not be optimal at the same time. Moreover, the issue remains even given imperfect G, as long as pg (x) and p(x) overlaps as in most of the real cases. Given a sample form the overlapped area, the two roles of D still compete by treating the sample differently, leading to a poor classifier2 . Namely, the learning capacity of existing two-player models is restricted, which should be addressed to advance current SSL results. For the second problem, disentangling meaningful physical factors like the object category from the latent representations with limited supervision is of general interest [30, 2]. However, to our best knowledge, none of the existing GANs can learn the disentangled representations in SSL, though some work [22, 5, 21] can learn such representations given full labels. Again, we believe that the problem is caused by their two-player formulation. Specifically, the discriminators in [26, 25] take a single data instead of a data-label pair as input and the label information is totally ignored when justifying whether a sample is real or fake. Therefore, the generators will not receive any learning signal regarding the label information from the discriminators and hence such models cannot control the semantics of the generated samples, which is not satisfactory. To address these problems, we present Triple-GAN, a flexible game-theoretical framework for both classification and class-conditional image generation in SSL, where we have a partially labeled dataset. We introduce two conditional networks?a classifier and a generator to generate pseudo labels given real data and pseudo data given real labels, respectively. To jointly justify the quality of the samples from the conditional networks, we define a single discriminator network which has the sole role of distinguishing whether a data-label pair is from the real labeled dataset or not. The resulting model is called Triple-GAN because not only are there three networks, but we consider three joint distributions, i.e. the true data-label distribution and the distributions defined by the conditional networks (See Figure 1 for the illustration of Triple-GAN). Directly motivated by the desirable equilibrium that both the classifier and the conditional generator are optimal, we carefully design 2 The results of minibatch discrimination approach in [25] well support our analysis. 2 compatible utilities including adversarial losses and unbiased regularizations (See Sec. 3), which lead to an effective solution to the challenging SSL task, justified both in theory and practice. In particular, theoretically, instead of competing as stated in the first problem, a good classifier will result in a good generator and vice versa in Triple-GAN (See Sec. 3.2 for the proof). Furthermore, the discriminator can access the label information of the unlabeled data from the classifier and then force the generator to generate correct image-label pairs, which addresses the second problem. Empirically, we evaluate our model on the widely adopted MNIST [14], SVHN [19] and CIFAR10 [12] datasets. The results (See Sec. 5) demonstrate that Triple-GAN can simultaneously learn a good classifier and a conditional generator, which agrees with our motivation and theoretical results. Overall, our main contributions are two folded: (1) we analyze the problems in existing SSL GANs [26, 25] and propose a novel game-theoretical Triple-GAN framework to address them with carefully designed compatible objectives; and (2) we show that on the three datasets with incomplete labels, Triple-GAN can advance the state-of-the-art classification results of DGMs substantially and, at the same time, disentangle classes and styles and perform class-conditional interpolation. 2 Related Work Recently, various approaches have been developed to learn directed DGMs, including Variational Autoencoders (VAEs) [10, 24], Generative Moment Matching Networks (GMMNs) [16, 6] and Generative Adversarial Nets (GANs) [7]. These criteria are systematically compared in [28]. One primal goal of DGMs is to generate realistic samples, for which GANs have proven effective. Specifically, LAP-GAN [3] leverages a series of GANs to upscale the generated samples to high resolution images through the Laplacian pyramid framework [1]. DCGAN [22] adopts (fractionally) strided convolution layers and batch normalization [8] in GANs and generates realistic natural images. Recent work has introduced inference networks in GANs. For instance, InfoGAN [2] learns explainable latent codes from unlabeled data by regularizing the original GANs via variational mutual information maximization. In ALI [5, 4], the inference network approximates the posterior distribution of latent variables given true data in unsupervised manner. Triple-GAN also has an inference network (classifier) as in ALI but there exist two important differences in the global equilibria and utilities between them: (1) Triple-GAN matches both the distributions defined by the generator and classifier to true data distribution while ALI only ensures that the distributions defined by the generator and inference network to be the same; (2) the discriminator will reject the samples from the classifier in Triple-GAN while the discriminator will accept the samples from the inference network in ALI, which leads to different update rules for the discriminator and inference network. These differences naturally arise because Triple-GAN is proposed to solve the existing problems in SSL GANs as stated in the introduction. Indeed, ALI [5] uses the same approach as [25] to deal with partially labeled data and hence it still suffers from the problems. In addition, Triple-GAN outperforms ALI significantly in the semi-supervised classification task (See comparison in Table. 1). To handle partially labeled data, the conditional VAE [11] treats the missing labels as latent variables and infer them for unlabeled data. ADGM [17] introduces auxiliary variables to build a more expressive variational distribution and improve the predictive performance. The Ladder Network [23] employs lateral connections between a variation of denoising autoencoders and obtains excellent SSL results. Cat-GAN [26] generalizes GANs with a categorical discriminator and an objective function. Salimans et al. [25] propose empirical techniques to stabilize the training of GANs and improve the performance on SSL and image generation under incompatible learning criteria. Triple-GAN differs significantly from these methods, as stated in the introduction. 3 Method We consider learning DGMs in the semi-supervised setting,3 where we have a partially labeled dataset with x denoting the input data and y denoting the output label. The goal is to predict the labels y for unlabeled data as well as to generate new samples x conditioned on y. This is different from the unsupervised setting for pure generation, where the only goal is to sample data x from a generator to fool a discriminator; thus a two-player game is sufficient to describe the process as in GANs. 3 Supervised learning is an extreme case, where the training set is fully labeled. 3 In our setting, as the label information y is incomplete (thus uncertain), our density model should characterize the uncertainty of both x and y, therefore a joint distribution p(x, y) of input-label pairs. A straightforward application of the two-player GAN is infeasible because of the missing values on y. Unlike the previous work [26, 25], which is restricted to the two-player framework and can lead to incompatible objectives, we build our game-theoretic objective based on the insight that the joint distribution can be factorized in two ways, namely, p(x, y) = p(x)p(y|x) and p(x, y) = p(y)p(x|y), and that the conditional distributions p(y|x) and p(x|y) are of interest for classification and classconditional generation, respectively. To jointly estimate these conditional distributions, which are characterized by a classifier network and a class-conditional generator network, we define a single discriminator network which has the sole role of distinguishing whether a sample is from the true data distribution or the models. Hence, we naturally extend GANs to Triple-GAN, a three-player game to characterize the process of classification and class-conditional generation in SSL, as detailed below. 3.1 A Game with Three Players Triple-GAN consists of three components: (1) a classifier C that (approximately) characterizes the conditional distribution pc (y|x) ? p(y|x); (2) a class-conditional generator G that (approximately) characterizes the conditional distribution in the other direction pg (x|y) ? p(x|y); and (3) a discriminator D that distinguishes whether a pair of data (x, y) comes from the true distribution p(x, y). All the components are parameterized as neural networks. Our desired equilibrium is that the joint distributions defined by the classifier and the generator both converge to the true data distribution. To this end, we design a game with compatible utilities for the three players as follows. We make the mild assumption that the samples from both p(x) and p(y) can be easily obtained.4 In the game, after a sample x is drawn from p(x), C produces a pseudo label y given x following the conditional distribution pc (y|x). Hence, the pseudo input-label pair is a sample from the joint distribution pc (x, y) = p(x)pc (y|x). Similarly, a pseudo input-label pair can be sampled from G by first drawing y ? p(y) and then drawing x|y ? pg (x|y); hence from the joint distribution pg (x, y) = p(y)pg (x|y). For pg (x|y), we assume that x is transformed by the latent style variables z given the label y, namely, x = G(y, z), z ? pz (z), where pz (z) is a simple distribution (e.g., uniform or standard normal). Then, the pseudo input-label pairs (x, y) generated by both C and G are sent to the single discriminator D for judgement. D can also access the input-label pairs from the true data distribution as positive samples. We refer the utilities in the process as adversarial losses, which can be formulated as a minimax game: min max U (C, G, D) =E (x,y)?p(x,y) [log D(x, y)] + ?E(x,y)?pc (x,y) [log(1 ? D(x, y))] C,G D +(1 ? ?)E(x,y)?pg (x,y) [log(1 ? D(G(y, z), y))], (1) where ? ? (0, 1) is a constant that controls the relative importance of generation and classification and we focus on the balance case by fixing it as 1/2 throughout the paper. The game defined in Eqn. (1) achieves its equilibrium if and only if p(x, y) = (1 ? ?)pg (x, y) + ?pc (x, y) (See details in Sec. 3.2). The equilibrium indicates that if one of C and G tends to the data distribution, the other will also go towards the data distribution, which addresses the competing problem. However, unfortunately, it cannot guarantee that p(x, y) = pg (x, y) = pc (x, y) is the unique global optimum, which is not desirable. To address this problem, we introduce the standard supervised loss (i.e., cross-entropy loss) to C, RL = E(x,y)?p(x,y) [? log pc (y|x)], which is equivalent to the KL-divergence between pc (x, y) and p(x, y). Consequently, we define the game as: ? (C, G, D) =E (x,y)?p(x,y) [log D(x, y)] + ?E(x,y)?p (x,y) [log(1 ? D(x, y))] min max U c C,G D +(1 ? ?)E(x,y)?pg (x,y) [log(1 ? D(G(y, z), y))] + RL . (2) ? has the unique global optimum for C and G. It will be proven that the game with utilities U 3.2 Theoretical Analysis and Pseudo Discriminative Loss 4 In semi-supervised learning, p(x) is the empirical distribution of inputs and p(y) is assumed same to the distribution of labels on labeled data, which is uniform in our experiment. 4 Algorithm 1 Minibatch stochastic gradient descent training of Triple-GAN in SSL. for number of training iterations do ? Sample a batch of pairs (xg , yg ) ? pg (x, y) of size mg , a batch of pairs (xc , yc ) ? pc (x, y) of size mc and a batch of labeled data (xd , yd ) ? p(x, y) of size md . ? Update D by ascending along its stochastic gradient: ? ? X X X 1 ? 1 ? ? ??d? ( log D(xd , yd ))+ log(1?D(xc , yc ))+ log(1?D(xg , yg ))? . md mc mg (xd ,yd ) (xc ,yc ) (xg ,yg ) ? L and R ? P of RL and RP respectively. ? Compute the unbiased estimators R ? Update C by descending along its stochastic gradient: ? ? X ? ? L + ?P R ?P? . ?? c ? pc (yc |xc ) log(1 ? D(xc , yc )) + R mc (xc ,yc ) ? Update G by descending along ? ? its stochastic gradient: X 1 ? ? log(1 ? D(xg , yg ))? . ?? g ? mg (xg ,yg ) end for We now provide a formal theoretical analysis of Triple-GAN under nonparametric assumptions and introduce the pseudo discriminative loss, which is an unbiased regularization motivated by the global equilibrium. For clarity of the main text, we defer the proof details to Appendix A. First, we can show that the optimal D balances between the true data distribution and the mixture distribution defined by C and G, as summarized in Lemma 3.1. Lemma 3.1 For any fixed C and G, the optimal D of the game defined by the utility function U (C, G, D) is: p(x, y) ? , (3) DC,G (x, y) = p(x, y) + p? (x, y) where p? (x, y) := (1 ? ?)pg (x, y) + ?pc (x, y) is a mixture distribution for ? ? (0, 1). ? Given DC,G , we can omit D and reformulate the minimax game with value function U as: V (C, G) = maxD U (C, G, D), whose optimal point is summarized as in Lemma 3.2. Lemma 3.2 The global minimum of V (C, G) is achieved if and only if p(x, y) = p? (x, y). We can further show that C and G can at least capture the marginal distributions of data, especially for pg (x), even there may exist multiple global equilibria, as summarized in Corollary 3.2.1. Corollary 3.2.1 Given p(x, y) = p? (x, y), the marginal distributions are the same for p, pc and pg , i.e. p(x) = pg (x) = pc (x) and p(y) = pg (y) = pc (y). Given the above result that p(x, y) = p? (x, y), C and G do not compete as in the two-player based formulation and it is easy to verify that p(x, y) = pc (x, y) = pg (x, y) is a global equilibrium point. However, it may not be unique and we should minimize an additional objective to ensure the ? (C, G, D) in problem (2), as stated below. uniqueness. In fact, this is true for the utility function U ? (C, G, D) is achieved if and only if p(x, y) = pg (x, y) = Theorem 3.3 The equilibrium of U pc (x, y). The conclusion essentially motivates our design of Triple-GAN, as we can ensure that both C and G will converge to the true data distribution if the model has been trained to achieve the optimum. ? , which allows us to regularize our model for stable We can further show another nice property of U and better convergence in practice without bias, as summarized below. 5 Corollary 3.3.1 Adding any divergence (e.g. the KL divergence) between any two of the joint ? as the additional distributions or the conditional distributions or the marginal distributions, to U ? regularization to be minimized, will not change the global equilibrium of U . Because label information is extremely insufficient in SSL, we propose pseudo discriminative loss RP = Epg [? log pc (y|x)], which optimizes C on the samples generated by G in the supervised manner. Intuitively, a good G can provide meaningful labeled data beyond the training set as extra side information for C, which will boost the predictive performance (See Sec. 5.1 for the empirical evidence). Indeed, minimizing pseudo discriminative loss with respect to C is equivalent to minimizing DKL (pg (x, y)||pc (x, y)) (See Appendix A for proof) and hence the global equilibrium remains following Corollary 3.3.1. Also note that directly minimizing DKL (pg (x, y)||pc (x, y)) is infeasible since its computation involves the unknown likelihood ratio pg (x, y)/pc (x, y). The pseudo discriminative loss is weighted by a hyperparameter ?P . See Algorithm 1 for the whole training procedure, where ?c , ?d and ?g are trainable parameters in C, D and G respectively. 4 Practical Techniques In this section we introduce several practical techniques used in the implementation of Triple-GAN, which may lead to a biased solution theoretically but work well for challenging SSL tasks empirically. One crucial problem of SSL is the small size of the labeled data. In Triple-GAN, D may memorize the empirical distribution of the labeled data, and reject other types of samples from the true data distribution. Consequently, G may collapse to these modes. To this end, we generate pseudo labels through C for some unlabeled data and use these pairs as positive samples of D. The cost is on introducing some bias to the target distribution of D, which is a mixture of pc and p instead of the pure p. However, this is acceptable as C converges quickly and pc and p are close (See results in Sec.5). Since properly leveraging the unlabeled data is key to success in SSL, it is necessary to regularize C heuristically as in many existing methods [23, 26, 13, 15] to make more accurate predictions. We consider two alternative losses on the unlabeled data. The confidence loss [26] minimizes the conditional entropy of pc (y|x) and the cross entropy between   p(y) and pc (y), weighted by a hyperparameter ?B , as RU = Hpc (y|x) + ?B Ep ? log pc (y) , which encourages C to make predictions confidently and be balanced on the unlabeled data. The consistency loss [13] penalizes the network if it predicts the same unlabeled data inconsistently given different noise , e.g., dropout masks, as RU = Ex?p(x) ||pc (y|x, ) ? pc (y|x, 0 )||2 , where || ? ||2 is the square of the l2 -norm. We use the confidence loss by default except on the CIFAR10 dataset (See details in Sec. 5). Another consideration is to compute the gradients of Ex?p(x),y?pc (y|x) [log(1 ? D(x, y))] with respect to the parameters ?c in C, which involves summation over the discrete random variable y, i.e. the class label. On one hand, integrating out the class label is time consuming. On the other hand, directly sampling one label to approximate the expectation via the Monte Carlo method makes the feedback of the discriminator not differentiable with respect to ?c . As the REINFORCE algorithm [29] can deal with such cases with discrete variables, we use a variant of it for the endto-end training of our classifier. The gradients in the original REINFORCE algorithm should be Ex?p(x) Ey?pc (y|x) [??c log pc (y|x) log(1 ? D(x, y))]. In our experiment, we find the best strategy is to use most probable y instead of sampling one to approximate the expectation over y. The bias is small as the prediction of C is rather confident typically. 5 Experiments We now present results on the widely adopted MNIST [14], SVHN [19], and CIFAR10 [12] datasets. MNIST consists of 50,000 training samples, 10,000 validation samples and 10,000 testing samples of handwritten digits of size 28 ? 28. SVHN consists of 73,257 training samples and 26,032 testing samples and each is a colored image of size 32 ? 32, containing a sequence of digits with various backgrounds. CIFAR10 consists of colored images distributed across 10 general classes?airplane, automobile, bird, cat, deer, dog, frog, horse, ship and truck. There are 50,000 training samples and 10,000 testing samples of size 32 ? 32 in CIFAR10. We split 5,000 training data of SVHN and 6 Table 1: Error rates (%) on partially labeled MNIST, SHVN and CIFAR10 datasets, averaged by 10 runs. The results with ? are trained with more than 500,000 extra unlabeled data on SVHN. MNIST n = 100 SVHN n = 1000 M1+M2 [11] VAT [18] Ladder [23] Conv-Ladder [23] ADGM [17] SDGM [17] MMCVA [15] 3.33 (?0.14) 2.33 1.06 (?0.37) 0.89 (?0.50) 0.96 (?0.02) 1.32 (?0.07) 1.24 (?0.54) 36.02 (?0.10) CatGAN [26] Improved-GAN [25] ALI [5] Triple-GAN (ours) 1.39 (?0.28) 0.93 (?0.07) Algorithm CIFAR10 n = 4000 24.63 20.40 (?0.47) 22.86 ? 16.61(?0.24)? 4.95 (?0.18) ? 8.11 (?1.3) 7.3 5.77(?0.17) 0.91 (?0.58) 19.58 (?0.58) 18.63 (?2.32) 18.3 16.99 (?0.36) Table 2: Error rates (%) on MNIST with different number of labels, averaged by 10 runs. Algorithm Improved-GAN [25] Triple-GAN (ours) n = 20 n = 50 n = 200 16.77 (?4.52) 4.81 (?4.95) 2.21 (?1.36) 1.56 (?0.72) 0.90 (?0.04) 0.67 (?0.16) CIFAR10 for validation if needed. On CIFAR10, we follow [13] to perform ZCA for the input of C but still generate and estimate the raw images using G and D. We implement our method based on Theano [27] and here we briefly summarize our experimental settings.5 Though we have an additional network, the generator and classifier of Triple-GAN have comparable architectures to those of the baselines [26, 25] (See details in Appendix F). The pseudo discriminative loss is not applied until the number of epochs reach a threshold that the generator could generate meaningful data. We only search the threshold in {200, 300}, ?P in {0.1, 0.03} and the global learning rate in {0.0003, 0.001} based on the validation performance on each dataset. All of the other hyperparameters including relative weights and parameters in Adam [9] are fixed according to [25, 15] across all of the experiments. Further, in our experiments, we find that the training techniques for the original two-player GANs [3, 25] are sufficient to stabilize the optimization of Triple-GAN. 5.1 Classification For fair comparison, all the results of the baselines are from the corresponding papers and we average Triple-GAN over 10 runs with different random initialization and splits of the training data and report the mean error rates with the standard deviations following [25]. Firstly, we compare our method with a large body of approaches in the widely used settings on MNIST, SVHN and CIFAR10 datasets given 100, 1,000 and 4,000 labels6 , respectively. Table 1 summarizes the quantitative results. On all of the three datasets, Triple-GAN achieves the state-of-the-art results consistently and it substantially outperforms the strongest competitors (e.g., Improved-GAN) on more challenging SVHN and CIFAR10 datasets, which demonstrate the benefit of compatible learning objectives proposed in Triple-GAN. Note that for a fair comparison with previous GANs, we do not leverage the extra unlabeled data on SVHN, while some baselines [17, 15] do. Secondly, we evaluate our method with 20, 50 and 200 labeled samples on MNIST for a systematical comparison with our main baseline Improved-GAN [25], as shown in Table 2. Triple-GAN consistently outperforms Improved-GAN with a substantial margin, which again demonstrates the benefit of Triple-GAN. Besides, we can see that Triple-GAN achieves more significant improvement as the number of labeled data decreases, suggesting the effectiveness of the pseudo discriminative loss. Finally, we investigate the reasons for the outstanding performance of Triple-GAN. We train a single C without G and D on SVHN as the baseline and get more than 10% error rate, which shows that G is important for SSL even though C can leverage unlabeled data directly. On CIFAR10, the baseline 5 6 Our source code is available at https://github.com/zhenxuan00/triple-gan We use these amounts of labels as default settings throughout the paper if not specified. 7 (a) Feature Matching (b) Triple-GAN (c) Automobile (d) Horse Figure 2: (a-b) Comparison between samples from Improved-GAN trained with feature matching and Triple-GAN on SVHN. (c-d) Samples of Triple-GAN in specific classes on CIFAR10. (a) SVHN data (b) SVHN samples (c) CIFAR10 data (d) CIFAR10 samples Figure 3: (a) and (c) are randomly selected labeled data. (b) and (d) are samples from Triple-GAN, where each row shares the same label and each column shares the same latent variables. (a) SVHN (b) CIFAR10 Figure 4: Class-conditional latent space interpolation. We first sample two random vectors in the latent space and interpolate linearly from one to another. Then, we map these vectors to the data level given a fixed label for each class. Totally, 20 images are shown for each class. We select two endpoints with clear semantics on CIFAR10 for better illustration. (a simple version of ? model [13]) achieves 17.7% error rate. The smaller improvement is reasonable as CIFAR10 is more complex and hence G is not as good as in SVHN. In addition, we evaluate Triple-GAN without the pseudo discriminative loss on SVHN and it achieves about 7.8% error rate, which shows the advantages of compatible objectives (better than the 8.11% error rate of ImprovedGAN) and the importance of the pseudo discriminative loss (worse than the complete Triple-GAN by 2%). Furthermore, Triple-GAN has a comparable convergence speed with Improved-GAN [25], as shown in Appendix E. 5.2 Generation We demonstrate that Triple-GAN can learn good G and C simultaneously by generating samples in various ways with the exact models used in Sec. 5.1. For fair comparison, the generative model and the number of labels are the same to the previous method [25]. In Fig. 2 (a-b), we first compare the quality of images generated by Triple-GAN on SVHN and the Improved-GAN with feature matching [25],7 which works well for semi-supervised classification. We can see that Triple-GAN outperforms the baseline by generating fewer meaningless samples and 7 Though the Improved-GAN trained with minibatch discrimination [25] can generate good samples, it fails to predict labels accurately. 8 clearer digits. Further, the baseline generates the same strange sample four times, labeled with red rectangles in Fig. 2 . The comparison on MNIST and CIFAR10 is presented in Appendix B. We also evaluate the samples on CIFAR10 quantitatively via the inception score following [25]. The value of Triple-GAN is 5.08 ? 0.09 while that of the Improved-GAN trained without minibatch discrimination [25] is 3.87 ? 0.03, which agrees with the visual comparison. We then illustrate images generated from two specific classes on CIFAR10 in Fig. 2 (c-d) and see more in Appendix C. In most cases, Triple-GAN is able to generate meaningful images with correct semantics. Further, we show the ability of Triple-GAN to disentangle classes and styles in Fig. 3. It can be seen that Triple-GAN can generate realistic data in a specific class and the latent factors encode meaningful physical factors like: scale, intensity, orientation, color and so on. Some GANs [22, 5, 21] can generate data class-conditionally given full labels, while Triple-GAN can do similar thing given much less label information. Finally, we demonstrate the generalization capability of our Triple-GAN on class-conditional latent space interpolation as in Fig. 4. Triple-GAN can transit smoothly from one sample to another with totally different visual factors without losing label semantics, which proves that Triple-GANs can learn meaningful latent spaces class-conditionally instead of overfitting to the training data, especially labeled data. See these results on MNIST in Appendix D. Overall, these results confirm that Triple-GAN avoid the competition between C and G and can lead to a situation where both the generation and classification are good in semi-supervised learning. 6 Conclusions We present triple generative adversarial networks (Triple-GAN), a unified game-theoretical framework with three players?a generator, a discriminator and a classifier, to do semi-supervised learning with compatible utilities. With such utilities, Triple-GAN addresses two main problems of existing methods [26, 25]. Specifically, Triple-GAN ensures that both the classifier and the generator can achieve their own optima respectively in the perspective of game theory and enable the generator to sample data in a specific class. Our empirical results on MNIST, SVHN and CIFAR10 datasets demonstrate that as a unified model, Triple-GAN can simultaneously achieve the state-of-the-art classification results among deep generative models and disentangle styles and classes and transfer smoothly on the data level via interpolation in the latent space. Acknowledgments The work is supported by the National NSF of China (Nos. 61620106010, 61621136008, 61332007), the MIIT Grant of Int. Man. Comp. Stan (No. 2016ZXFB00001), the Youth Top-notch Talent Support Program, Tsinghua Tiangong Institute for Intelligent Computing, the NVIDIA NVAIL Program and a Project from Siemens. References [1] Peter Burt and Edward Adelson. The Laplacian pyramid as a compact image code. IEEE Transactions on communications, 1983. [2] Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. InfoGAN: Interpretable representation learning by information maximizing generative adversarial nets. In NIPS, 2016. 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Conditional image synthesis with auxiliary classifier gans. arXiv preprint arXiv:1610.09585, 2016. [22] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015. [23] Antti Rasmus, Mathias Berglund, Mikko Honkala, Harri Valpola, and Tapani Raiko. Semisupervised learning with ladder networks. In NIPS, 2015. [24] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. arXiv preprint arXiv:1401.4082, 2014. [25] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In NIPS, 2016. [26] Jost Tobias Springenberg. Unsupervised and semi-supervised learning with categorical generative adversarial networks. arXiv preprint arXiv:1511.06390, 2015. 10 [27] Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016. URL http://arxiv.org/abs/1605. 02688. [28] Lucas Theis, A?ron van den Oord, and Matthias Bethge. A note on the evaluation of generative models. arXiv preprint arXiv:1511.01844, 2015. [29] Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229?256, 1992. [30] Jimei Yang, Scott E Reed, Ming-Hsuan Yang, and Honglak Lee. Weakly-supervised disentangling with recurrent transformations for 3d view synthesis. In NIPS, 2015. 11
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Efficient Sublinear-Regret Algorithms for Online Sparse Linear Regression with Limited Observation Shinji Ito NEC Corporation [email protected] Hanna Sumita National Institute of Informatics [email protected] Daisuke Hatano National Institute of Informatics [email protected] Akihiro Yabe NEC Corporation [email protected] Naonori Kakimura Keio University [email protected] Takuro Fukunaga JST, PRESTO [email protected] Ken-ichi Kawarabayashi National Institute of Informatics [email protected] Abstract Online sparse linear regression is the task of applying linear regression analysis to examples arriving sequentially subject to a resource constraint that a limited number of features of examples can be observed. Despite its importance in many practical applications, it has been recently shown that there is no polynomialtime sublinear-regret algorithm unless NP?BPP, and only an exponential-time sublinear-regret algorithm has been found. In this paper, we introduce mild assumptions to solve the problem. Under these assumptions, we present polynomialtime sublinear-regret algorithms for the online sparse linear regression. In addition, thorough experiments with publicly available data demonstrate that our algorithms outperform other known algorithms. 1 Introduction In online regression, a learner receives examples one by one, and aims to make a good prediction from the features of arriving examples, learning a model in the process. Online regression has attracted attention recently in the research community in managing massive learning data.In realworld scenarios, however, with resource constraints, it is desired to make a prediction with only a limited number of features per example. Such scenarios arise in the context of medical diagnosis of a disease [3] and in generating a ranking of web pages in a search engine, in which it costs to obtain features or only partial features are available in each round. In both these examples, predictions need to be made sequentially because a patient or a search query arrives online. To resolve the above issue of limited access to features, Kale [7] proposed online sparse regression. In this problem, a learner makes a prediction for the labels of examples arriving sequentially over a number of rounds. Each example has d features that can be potentially accessed by the learner. However, in each round, the learner can acquire the values of at most k 0 features out of the d features, where k 0 is a parameter set in advance. The learner then makes a prediction for the label of the example. After the prediction, the true label is revealed to the learner, and the learner suffers a loss for making an incorrect prediction. The performance of the prediction is measured here by the standard notion of regret, which is the difference between the total loss of the learner and the total 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Table 1: Computational complexity of online sparse linear regression. Assumptions (1) (2) (a) X X X X X X X X X Time complexity (b) X Hard [5] Hard (Theorem 1) Polynomial time (Algorithms 1, 2) Polynomial time (Algorithm 3) loss of the best predictor. In [7], the best predictor is defined as the best k-sparse linear predictor, i.e., the label is defined as a linear combination of at most k features. Online sparse regression is a natural online variant of sparse regression; however, its computational complexity was not well known until recently, as Kale [7] raised a question of whether it is possible to achieve sublinear regret in polynomial time for online sparse linear regression. Foster et al. [5] answered the question by proving that no polynomial-time algorithm achieves sublinear regret unless NP?BPP. Indeed, this hardness result holds even when observing ?(k log d) features per example. On the positive side, they also proposed an exponential-time algorithm with sublinear regret, when we can observe at least k + 2 features in each round. However, their algorithm is not expected to  work efficiently in practice. In fact, the algorithm enumerates all the kd0 possibilities to determine k 0 features in each round, which requires exponential time for any instance. Our contributions. In this paper, we show that online sparse linear regression admits a polynomial-time algorithm with sublinear regret, under mild practical assumptions. First, we assume that the features of examples arriving online are determined by a hidden distribution (Assumption (1)), and the labels of the examples are determined by a weighted average of k features, where the weights are fixed through all rounds (Assumption (2)). These are natural assumptions in the online linear regression. However, Foster et al. [5] showed that no polynomial-time algorithm can achieve sublinear regret unless NP?BPP even under these two assumptions.1 Owing to this hardness, we introduce two types of conditions on the distribution of features, both of which are closely related to the restricted isometry property (RIP) that has been studied in the literature of sparse recovery. The first condition, which we call linear independence of features (Assumption (a)), is stronger than RIP. This condition roughly says that all the features are linearly independent. The second condition, which we call compatibility (Assumption (b)), is weaker than RIP. Thus, an instance having RIP always satisfies the compatibility condition. Under these assumptions, we propose the following three algorithms. Here, T is the number of rounds. ? ? Algorithm 1: A polynomial-time algorithm that achieves O( k0d?k T ) regret, under Assumptions (1), (2), and (a), which requires at least k + 2 features to be observed per example. ? 16 ? Algorithm 2: A polynomial-time algorithm that achieves O( dT + kd016 ) regret, under Assumptions (1), (2), and (a), which requires at least k features to be observed per example. ? 16 ? Algorithm 3: A polynomial-time algorithm that achieves O( dT + kd016 ) regret, under Assumptions (1), (2), and (b), which requires at least k features to be observed per example. ? We can also construct an algorithm achieving O( k0d?k T ) regret under Assumption (b) for the case where k 0 ? k + 2, analogous to Algorithm 1, but we omit it due to space limitations. Assumptions (1)+(2)+(a) or (1)+(2)+(b) seem to be minimal assumptions needed to achieve sublinear regret in polynomial time. Indeed, as listed in Table 1, the problem is hard if any one of the assumptions is violated, where hard means that no polynomial-time algorithm can achieve sublinear regret unless NP?BPP. Note that Assumption (a) is stronger than (b). In addition to proving theoretical regret bounds of our algorithms, we perform thorough experiments to evaluate the algorithms. We verified that our algorithms outperform the exponential-time algorithm [5] in terms of computational complexity as well as performance of the prediction. Our algorithms also outperform (baseline) heuristic-based algorithms and algorithms proposed in [2, 6] 1 Although the statement in [5] does not mention the assumptions, its proof indicates that the hardness holds even with these assumptions. 2 for online learning based on limited observation. Moreover, we observe that our algorithms perform well even for a real dataset, which may not satisfy our assumptions (deciding whether the model satisfies our assumptions is difficult; for example, the RIP parameter cannot be approximated within any constant factor under a reasonable complexity assumption [9]). Thus, we can conclude that our algorithm is applicable in practice. Overview of our techniques. One naive strategy for choosing a limited number of features is to choose ?large-weight? features in terms of estimated ground-truth regression weights. This strategy, however, does not achieve sublinear regret, as it ignores small-weight features. When we have Assumption (a), we show that if we observe two more features chosen uniformly at random, together with the largest k features, we can make a good prediction. More precisely, using the observed features, we output the label that minimizes the least-square loss function, based on the technique using an unbiased estimator of the gradient [2, 6] and the regularized dual averaging (RDA) method (see, e.g., [11, 4]). This idea gives Algorithm 1, and the details are given in Section 4. The reason why we use RDA is that it is efficient in terms of computational time and memory space as pointed out in [11] and, more importantly, we will combine this with the `1 regularization later. However, this requires at least k + 2 features to be observed in each round. To avoid the requirement of two extra observations, the main idea is to employ Algorithm 1 with a partial dataset. As a by-product of Algorithm 1, we can estimate the ground-truth regression weight vector with high probability, even without observing extra features in each round. We use the ground-truth weight vector estimated by Algorithm 1 to choose k features. Combining this idea with RDA adapted for the sparse regression gives Algorithm 2 (Section 5.1) under Assumption (a). The compatibility condition (Assumption (b)) is often used in LASSO (Least Absolute Shrinkage and Selection Operator), and it is known that minimization with an `1 regularizer converges to the sparse solution under the compatibility condition [1]. We introduce `1 regularization into Algorithm 1 to estimate the ground-truth regression weight vector when we have Assumption (b) instead of Assumption (a). This gives Algorithm 3 (Section 5.2). Related work. In the online learning problem, a learner aims to predict a model based on the arriving examples. Specifically, in the linear function case, a learner predicts the coefficient wt of a linear function wt> xt whenever an example with features xt arrives in round t. The learner then PT suffers a loss `t (wt ) = (yt ? wt> xt )2 . The aim is to minimize the total loss t=1 (`t (wt ) ? `t (w)) for an arbitrary w. It is ? known that both the gradient descent method [12] and the dual averaging method [11] attain an O( T ) regret even for the more general convex function case. However, these methods require access to all features of the examples. In linear regression with limited observation, the limited access to features in regression has been considered [2, 6]. In this problem, a learner can acquire only the values of at most k 0 features among d features. The purpose here is to estimate a good weight vector, e.g., minimize the loss function `(w) or the loss function with `1 regularizer `(w) + kwk1 . Let us note that, even if we obtain a good weight vector w with small `(w), we cannot always compute w> xt from limited observation of xt and, hence, in our setting the prediction error might not be as small as `(w). Thus, our setting uses a different loss function, defined in Section 2, to minimize the prediction error. Another problem incorporating the limited access is proposed by Zolghadr et al. [13]. Here, instead of observing k 0 features, one considers the situation where obtaining a feature has an associated cost. In each round, one chooses a set of features to pay some amount of money, and the purpose is to minimize the sum of the regret and the total cost. They designed an exponential-time algorithm for the problem. Online sparse linear regression has been studied in [5, 7], but only an exponential-time algorithm has been proposed so far. In fact, Foster et al. [5] suggested designing an efficient algorithm for a special class of the problem as future work. The present paper aims to follow this suggestion. Recently, Kale et al. [8]2 presented computationally efficient algorithms to achieve sublinear regret under the assumption that input features satisfy RIP. Though this study includes similar results to ours, we can realize some differences. Our paper considers the assumption of the compatibility condition without extra observation (i.e., the case of k 0 = k), whereas Kale et al. [8] studies a 2 The paper [8] was published after our manuscript was submitted. 3 stronger assumption with extra observation (k 0 ? k + 2) that yields a smaller regret bound than ours. They also studies the agnostic (adversarial) setting. 2 Problem setting Online sparse linear regression. We suppose that there are T rounds, and an example arrives online in each round. Each example is represented by d features and is associated with a label, where features and labels are all real numbers. We denote the features of the example arriving in round t by xt = (xt1 , . . . , xtd )> ? {x ? Rd | kxk ? 1}, where the norm k ? k without subscripts denotes the `2 norm. The label of each example is denoted by yt ? [?1, 1]. The purpose of the online sparse regression is to predict the label yt ? R from a partial observation of xt in each round t = 1, . . . , T . The prediction is made through the following four steps: (i) we choose a set St ? [d] := {1, . . . , d} of features to observe, where |St | is restricted to be at most k 0 ; (ii) observe the selected features {xti }i?St ; (iii) on the basis of observation {xti }i?St , estimate a predictor y?t of yt ; and (iv) observe the true value of yt . From St , we define Dt ? Rd?d to be the diagonal matrix such that its (i, i)th entries are 1 for i ? St and the other entries are 0. Then, observing the selected features {xti }i?St in (ii) is equivalent to observing Dt xt . The predictor y?t is computed by y?t = wt> Dt xt in (iii). Throughout the paper, we assume the following conditions, corresponding to Assumptions (1) and (2) in Section 1, respectively. Assumption (1) There exists a weight vector w? ? Rd such that kwk ? 1 and yt = w?> xt + t for all t = 1, . . . , T , where t ? D , independent and identically distributed (i.i.d.), and E[t ] = 0, E[t 2 ] = ? 2 . There exists a distribution Dx on Rd such that xt ? Dx , i.i.d. and independent of {t }. Assumption (2) The true weight vector w? is k-sparse, i.e., S ? = supp(w? ) = {i ? [d] | wi? 6= 0} satisfies |S ? | ? k. Regret. The performance of the prediction is evaluated based on the regret RT (w) defined by RT (w) = T X (? yt ? yt ) 2 ? t=1 T X (w> xt ? yt )2 . (1) t=1 Our goal is to achieve smaller regret RT (w) for an arbitrary w ? Rd such that kwk ? 1 and kwk0 ? k. For random inputs and randomized algorithms, we consider the expected regret maxw:kwk0 ?k,kwk?1 E[RT (w)]. Define the loss function `t (w) = (w> xt ? yt )2 . If we compute a predictor y?t = wt> Dt xt using a weight vector wt = (wt1 , . . . , wtd )> ? Rd in each step, we can rewrite the regret RT (w) in (1) using Dt and wt as T X RT (w) = (`t (Dt wt ) ? `t (w)) (2) t=1 because (? yt ? yt )2 = (wt> Dt xt ? yt )2 = `t (Dt wt ). It is worth noting that if our goal is only to construct wt that minimizes the loss function `t (wt ), then the definition of the regret should be RT0 (w) = T X (`t (wt ) ? `t (w)). (3) t=1 However, the goal of online sparse regression involves predicting yt from the limited observation. Hence, we use (2) to evaluate the performance. In terms of the regret defined by (3), several algorithms based on limited observation have been developed. For ? example, the algorithms proposed by Cesa-Bianchi et al. [3] and Hazan and Koren [6] achieve O( T ) regret of (3). 4 3 Extra assumptions on features of examples Foster et al. [5] showed that Assumptions (1) and (2) are not sufficient to achieve sublinear regret. Owing to this observation, we impose extra assumptions. d?d Let V := E[x> and let L be the Cholesky decomposition of V (i.e., V = L> L). Denote t xt ] ? R the largest and the smallest singular values of L by ?1 and ?d , respectively. Under Assumption (1) in Section 2, we have ?1 ? 1 because, for arbitrary unit vector u ? Rd , it holds that u> V u = E[(u> x)2 ] ? 1. For a vector w ? R[d] and S ? [d], we let wS denote the restriction of w onto S. For S ? [d], S c denotes [d] \ S. We assume either one of the following conditions holds. (a) Linear independence of features: ?d > 0. (b) Compatibility: There exists a constant ?0 > 0 that satisfies ?20 kwS ? k21 ? kw> V w for all w ? Rd with kw(S ? )c k1 ? 2kwS ? k1 . We assume the linear independence of features in Sections 4 and 5.1, and the compatibility in Section 5.2 to develop efficient algorithms. Note that condition (a) means that L is non-singular, and so is V . In other words, condition (a) indicates that the features in xt are linearly independent. This is the reason why we call condition (a) the ?linear independence of features? assumption. Note that the linear independence of features does not imply the stochastic independence of features. Conditions (a) and (b) are closely related to RIP. Indeed, condition (b) is a weaker assumption than RIP, and RIP is weaker than condition (a), i.e., (a) linear independence of features =? RIP =? (b) compatibility (see, e.g., [1]). We now clarify how the above two assumptions are connected to the regret. The expectation of the loss function `t (w) is equal to Ext ,yt [`t (w)] = Ext ?Dx ,t ?D [(w> xt ? w?> xt ? t )2 ] ? > ? 2 = Ext ?Dx [((w ? w? )> xt )2 ] + Et ?D [> t t ] = (w ? w ) V (w ? w ) + ? for all t, where the second equality comes from E[t ] = 0 and that xt and t are independent. Denote this function by `(w), and then `(w) is minimized when w = w? . If Dt and wt are determined independently of xt and yt , the expectation of the regret RT (w) satisfies E[RT (w)] = E[ T T X X (`(Dt wt ) ? `(w))] ? E[ (`(Dt wt ) ? `(w? ))] t=1 t=1 = E[ T X T X (Dt wt ? w? )> V (Dt wt ? w? )] = E[ kL(Dt wt ? w? )k2 ]. t=1 (4) t=1 We bound (4) in the analysis. Hardness result. Similarly to [5], we can show that it remains hard under Assumptions (1), (2), and (a). Refer to Appendix A for the proof. Theorem 1. Let D be any positive constant, and let cD ? (0, 1) be a constant dependent on D. Suppose that Assumptions (1) and (2) hold with k = O(dcD ) and k 0 = bkD ln dc. If an algorithm for the online sparse regression problem runs in poly(d, T ) time per iteration and achieves a regret at most poly(d, 1/?d )T 1?? in expectation for some constant ? > 0, then NP?BPP. 4 Algorithm with extra observations and linear independence of features In this section, we present Algorithm 1. Here we assume k 0 ? k + 2, in addition to the linear independence of features (Assumption (a)). The additional assumption will be removed in Section 5. As noted in Section 2, our algorithm first computes a weight vector wt , chooses a set St of k 0 features to be observed, and computes a label y?t by y?t = wt> Dt xt in each round t. In addition, ?t of the gradient gt of the loss function `t (w) at our algorithm constructs an unbiased estimator g w = wt , i.e., gt = ?w `t (wt ) = 2xt (x> w ? yt ) at the end of the round. In the following, we t t ?t in round t, respectively, assuming that wt0 , St0 , and g ?t0 are describe how to compute wt , St , and g computed in the previous rounds t0 = 1, . . . , t ? 1. The entire algorithm is described in Algorithm 1. 5 Algorithm 1 Input: {xt , yt } ? Rd ? R, {?t } ? R>0 , k 0 ? 2 and k1 ? 0 such that k1 ? k 0 ? 2. ? 0 = 0. 1: Set h 2: for t = 1, . . . , T do 3: Define wt by (5) and define St by Observe(wt , k 0 , k1 ). 4: Observe Dt xt and output y?t := wt> Dt xt . ?t = h ? t?1 + g ?t by (6) and set h ?t 5: Observe yt and define g 6: end for ?1 , . . . , g ?t?1 to estimate wt by the dual averaging method as follows. Computing wt . We use g ? t?1 = Pt?1 g ?j , which is the average of all estimators of gradients computed in the preDefine h j=1 vious rounds. Moreover, let (?1 , . . . , ?T ) be a monotonically non-decreasing sequence of positive numbers. From these, we define wt by   1 ?t 2 > ? , ? h (5) wt = arg min ht?1 w + kwk = ? ? t?1 k} t?1 2 d max{?t , kh w?R ,kwk?1 Computing St . Let k1 be an integer such that k1 ? k 0 ? 2. We define Ut ? [d] as the set of the k1 largest features with respect to wt , i.e., choose Ut so that |Ut | = k1 and all i ? Ut and j ? [d] \ Ut satisfy |wti | ? |wtj |. Let Vt be the set of (k 0 ? k1 ) elements chosen from [d] \ Ut uniformly at random. Then our algorithm observes the set St = Ut ? Vt of the k 0 features. We call this procedure to obtain St Observe(wt , k 0 , k1 ). Observation 1. We observe that Ut ? St and Prob[i, j ? St ] ? Thus, Prob[i, j ? St ] > 0 for all i, j ? [d] if k 0 ? k1 + 2. (t) (k0 ?k1 )(k0 ?k1 ?1) d(d?1) =: Cd,k0 ,k1 . (t) For simplicity, we use the notation pi = Prob[i ? St ] and pij = Prob[i, j ? St ] for i, j ? [d]. d?d ? t = (? ? t = Dt x> ?t . Define X be a matrix Computing g xtij ) ? Rd?d by X t xt Dt and let Xt ? R (t) . Similarly, whose (i, j)-th entry is x ?tij /pij . It follows that Xt is an unbiased estimator of xt x> t (t) defining zt = (zti ) ? Rd by zti = xti /pi for i ? St and zti = 0 for i ? / St , we see that zt is an ?t to be unbiased estimator of xt . Using Xt and zt , we define g ?t = 2Xt wt ? 2yt zt . g (6) Regret ? bound of Algorithm 1. Let us show that the regret achieved by Algorithm 1 is O( k0d?k T ) in expectation. Theorem 2. Suppose that the linear independence of features is satisfied and k ? k 0 ? 2. Let k1 be an arbitrary integer such that k ?  k1 ? k 0 ? 2. Then, for arbitrary w ? Rd with kwk ? 1,  ? P T ? 1 T +1 Algorithm 1 achieves E[RT (w)] ? ?32 C 160 . By setting ?t = 8 t/Cd,k0 ,k1 t=1 ?t + 2 d,k ,k1 d for each t = 1, . . . , T , we obtain s ? 24 d(d ? 1) E[RT (w)] ? 2 ? T + 1. (7) ?d (k 0 ? k1 )(k 0 ? k1 ? 1) The rest of this section is devoted to proving Theorem 2. By (4), it suffices to evaluate PT E[ t=1 kL(Dt wt ? w? )k2 ] instead of E[RT (w)]. The following lemma asserts that each term of (4) can be bounded, assuming the linear independence of features. Proofs of all lemmas are given in the supplementary material. Lemma 3. Suppose that the linear independence of features is satisfied. If St ? Ut , kL(Dt wt ? w? )k2 ? 6 3 kL(wt ? w? )k2 . ?d2 (8) Proof. We have ? ? kL(Dt wt ? w? )k2 ? ?12 kDt wt ? w? k2 = ?12 ? X (wti ? wi? )2 + i?S ? ?St X wi?2 + i?S ? \St ? X 2? wti i?St \S ? ? X ? ?12 ?kwt ? w? k2 + wi?2 ? , (9) i?S ? \St where the second inequality holds since wi? = 0 for i ? [d] \ S ? . It holds that X X X  2 wi?2 ? wi?2 ? 2wti + 2(wti ? wi? )2 i?S ? \St i?S ? \Ut X ?2 2 wti +2 i?Ut \S ? X i?S ? \Ut (wti ? wi? )2 ? 2kwt ? w? k2 . (10) i?S ? \Ut The first and third inequalities come from Ut ? St and the definition of Ut . Putting (10) into (9), we have 3? 2 kL(Dt wt ? w? )k2 ? 3?12 kwt ? w? k2 ? 21 kL(wt ? w? )k2 . ?d It follows from the above lemma that, if wt converges to w? , we have Dt wt = w? , and hence St PT PT includes the support of w? . Moreover, it holds that t=1 E[kL(wt ? w? )k2 ] = E[ t=1 (`t (wt ) ? PT `t (w? ))] = E[RT0 (w? )], since wt is independent of xt and yt . Thus, to bound t=1 E[kL(wt ? w? )k2 ], we shall evaluate E[RT0 (w? )]. Lemma 4 ([11]). Suppose that wt is defined by (5) for each t = 1, . . . , T , and w ? Rd satisfies kwk ? 1. Let Gt = E[k? gt k2 ] for t = 1, . . . , T . Then, E[RT0 (w)] ? T X 1 ?T +1 Gt + . ? 2 t=1 t (11) ? ? If Gt = O(1) and ?t = ?( t), the right-hand side of (11) is O( T ). The following lemma shows (t) that this is true if pij = ?(1). Lemma 5. Suppose that the linear independence of features is satisfied. Let t ? [T ], and let q be a (t) (t) positive number such that q ? min{pi , pij }. Then we have Gt ? 16/q. We are now ready to prove Theorem 2. Proof of Theorem 2. The expectation E[RT (w)] of the regret is bounded as E[RT (w)] ? PT PT 3 3 ? 2 ? 2 0 ? t=1 E[kL(wt ? w )k ] = ?d2 E[RT (w )], where the first t=1 E[kL(Dt wt ? w )k ] ? ?d2 inequality comes from (4) and the second comes from Lemma 3. From Lemma 4, E[RT0 (w? )] PT 1 is bounded by E[RT0 (w? )] ? HT := Gt + ?T2+1 . Lemma 5 and Observation 1 yield p t=1 ?t PT Gt ? 16/Cd,k0 ,k1 . Hence, for ?t = 8 Cd,k0 ,k1 t, HT satisfies HT ? t=1 C 16 + ?T2+1 = d,k0 ,k1 ?t ? ? PT ? 2 + ?C 4 T + 1 ? 8 ?C 1 T + 1. Combining the above three inequalit=1 C t d,k0 ,k1 d,k0 ,k1 d,k0 ,k1 ties, we obtain (7). 5 5.1 Algorithms without extra observations Algorithm 2: Assuming (a) the linear independence of features 3? 2 In Section 4, Lemma 3 showed a connection between RT and RT0 : E[RT (w)] ? ?d 12 E[RT0 (w? )] ? under Ut ? St . Then, Lemmas 4 and 5 gave an upper bound of E[RT0 (w? )]: E[RT0 (w? )] = O( T ) 7 (t) (t) under pij = ?(1). In the case of k 0 = k, however, the conditions Ut ? St and pij = ?(1) may not be satisfied simultaneously, since, if Ut ? St and |St | = k 0 = k ? k1 = |Ut |, then we have (t) Ut = St , which means pij = 0 for i ? / Ut or j ? / Ut . Thus, we cannot use both relationships for the analysis. In Algorithm 2, we bound RT (w) without bounding RT0 (w). Let us describe an idea of Algorithm 2. To achieve the claimed regret, ? we first define a subset J of {1, 2, . . . , T } by the set of squares, i.e., J = {s2 | s = 1, . . . , b T c}. Let ts denote the s-th smallest number in J for each s = 1, . . . , |J|. In each round t, the algorithm computes St , a weight ? t , and a vector Dt g ?t , where g ?t is the gradient of `t (w)P ? t . In addition, if t = ts , vector w at w = Dt w ? s := 1s sj=1 wj , and an unbiased estimator the algorithm computes other weight vectors ws and w ?s of the gradient of the loss function `t (w) at ws . g ? s is defined as the At the beginning of round t, if t = ts , the algorithm first computes ws , and w average of w1 , . . . , ws . Roughly speaking, ws is the weight vector computed with Algorithm 1 applied to the examples (xt1 , yt1 ), . . . , (xts , yts ), setting k1 to be at most k ? 2. Then, we can ? s is a consistent estimator of w? . This step is only performed if t ? J. Then St is show that w ? s , where s is the largest number such that ts ? t. Thus, St does not change for any defined from w ? t from D1 g ?1 , . . . , Dt?1 g ?t?1 , and predicts t ? [ts , ts+1 ? 1]. After this, the algorithm computes w ? t> Dt xt . At the end of the round, the true label yt is observed, and Dt g ?t the label of xt as y?t := w ?s is computed as in Algorithm 1. We is computed from wt and (Dt xt , yt ). In addition, if t = ts , g ?s for computing ws0 with s0 > s in the subsequent rounds ts0 . need g The following theorem bounds the regret of Algorithm 2. See the supplementary material for details of the algorithm and the proof of the theorem. Theorem 6. Suppose that (a), the linear independence of features, is satisfied and k ? k 0 . Then, there exists a polynomial-time algorithm such that E[RT (w)] is at most 1 2 ? 2 2 X X ? ? Cd,k 0 ,0 (T 4 ? 1)|wi | ?d 4096 2 ? 8(1+ d) T + 1+12T )+4 |wi? |( 2 |wi | exp(? ?4 4 +1) , 18432 C d,k0 ,0 wi ?d ? ? i?S i?S for arbitrary w ? Rd with kwk ? 1, where Cd,k0 ,0 = 5.2 0 0 k (k ?1) d(d?1) 02 = O( kd2 ).2 Algorithm 3: Assuming (b) the compatibility condition Algorithm 3 adopts the same strategy as Algorithm 2 except for the procedure for determining ws ? s . In the analysis of Algorithm 2, we show that, and w ?to achieve the claimed regret, it suffices to PT generate {St } that satisfies t=1 Prob[i ? was satisfied / St ] = O( T ) for i ? S ? . The condition P ? s = sj=1 wj /s. by defining St as the set of k largest features with respect to a weight vector w ? s computed in Algorithm 2 converges to w? , The linear independence of features guarantees that w and hence {St } defined as above possesses the required property. Unfortunately, if the assumption ? s does of the independence of features is not satisfied, e.g., if we have almost same features, then w not converge to w? . However, if we introduce an `1 -regularization to the minimization problem in ? s to a weighted average of the modified vectors the definition of ws and change the definition of w w1 , . . . , ws , then we can generate a required set {St } under the compatibility assumption. See the supplementary material for details and the proof of the following theorem. Theorem 7. Suppose that (b), the compatibility assumption, is satisfied and k ? k 0 . Then, there exists a polynomial-time algorithm such that E[RT (w)] is at most p 1 X X ? ? 0 ,0 C T 4 ?1|wi? |2 ?20 64 ? 364 k 2 d,k |wi? | exp(? )+4 |wi? |( 2 +1)2 , 8(1+ d) T +1 + 12T 5832k Cd,k0 ,0 wi?4 ?40 ? ? i?S i?S for arbitrary w ? Rd with kwk ? 1, where Cd,k0 ,0 = 3 0 0 k (k ?1) d(d?1) 02 = O( kd2 ).3,4 The asymptotic regret bound mentioned in Section 1, can be yielded by bounding the second term with ?1 the aid of the following: maxT ?0 T exp(??T ? ) = (??) ? exp(?1/?) for arbitrary ? > 0, ? > 0. 4 Note that ?0 is the constant appearing in Assumption (b) in Section 3. 8 6 Experiments In this section, we compare our algorithms with the following four baseline algorithms: (i) a greedy method that chooses the k 0 largest features with respect to wt computed as in Algorithm 1; (ii) a uniform-random method that chooses k 0 features uniformly at random; (iii) the algorithm of [6] (called AELR); and (iv) the algorithm of [5] (called FKK). Owing to space limitations, we only present typical results here. Other results and the detailed descriptions on experiment settings are provided in the supplementary material. Synthetic data. First we show results on two kinds of synthetic datasets: instances with (d, k, k 0 ) and instances with (d, k1 , k). We set k1 = k in the setting of (d, k, k 0 ) and k 0 = k in the setting of (d, k1 , k). The instances with (d, k, k 0 ) assume that Algorithm 1 can use the ground truth k, while Algorithm 1 cannot use k in the instances with (d, k1 , k). For each (d, k, k 0 ) and (d, k1 , k), we executed all algorithms on five instances with T = 5000 and computed the averages of regrets and run time, respectively. When (d, k, k 0 ) = (20, 5, 7), FKK spent 1176 s on average, while AELR spent 6 s, and the others spent at most 1 s. Figures 1 and 2 plot the regrets given by (1) over the number of rounds on a typical instance with (d, k, k 0 ) = (20, 5, 7). Tables 2 and 3 summarize the average regrets at T = 5000, where A1, A2, A3, G, and U denote Algorithm 1, 2, 3, greedy, and uniform random, respectively. We observe that Algorithm 1 achieves smallest regrets in the setting of (d, k, k 0 ), whereas Algorithms 2 and 3 are better than Algorithm 1 in the setting of (d, k1 , k). The results match our theoretical results. 4000 3000 2000 2000 1000 1000 0 1000 2000 T 3000 4000 5000 Figure 1: Plot of regrets with (d, k, k 0 ) = (20, 5, 7) 1.00 0.75 0.50 0.25 0 0 Algorithm 1 Algorithm 2 Algorithm 3 greedy uniform random AELR 1.25 T 3000 5000 1.50 ? (yt? ? yt)2 4000 1e8 Algorithm 1 Algorithm 2 Algorithm 3 greedy uniform random AELR FKK 6000 RT 5000 RT 7000 Algorithm 1 Algorithm 2 Algorithm 3 greedy uniform random AELR FKK 6000 t=0 7000 0 1000 2000 T 3000 4000 5000 0.00 0 Figure 2: Plot of regrets with (d, k1 , k) = (20, 5, 7) 10000 20000 30000 T 40000 50000 Figure 3: CT-slice datasets Table 2: Values of RT /102 when changing Table 3: Values of RT /102 when changing (d, k, k 0 ). (d, k1 , k). (d, k1 , k) (10,2,4) A1 1.53 A2 2.38 A3 3.60 G 33.28 U 25.73 AELR 60.76 FKK 24.05 (d, k1 , k) (10,2,4) A1 26.88 A2 20.59 A3 17.19 G 43.03 U 60.02 AELR 64.75 FKK 58.71 Real data. We next conducted experiments using a CT-slice dataset, which is available online [10]. Each data consists of 384 features retrieved from 53500 CT images associated with a label that denotes the relative position of an image on the axial axis. We executed all algorithms except FKK, which does not work due to its expensive run time. Since we do not know the ground-truth regression weights, we measure the performance by the first term of (1), i.e., square loss of predictions. Figure 3 plots the losses over the number of rounds. The parameters are k1 = 60 and k 0 = 70. For this instance, the run times of Algorithms 1 and 2, greedy, uniform random, and AELR were 195, 35, 147, 382, and 477 s, respectively. We observe that Algorithms 2 and 3 are superior to the others, which implies that Algorithm 2 and 3 are suitable for instances where the ground truth k is not known, such as real data-based instances. Acknowledgement This work was supported by JST ERATO Grant Number JPMJER1201, Japan. References [1] P. B?uhlmann and S. van de Geer. Statistics for high-dimensional data. 2011. 9 [2] N. Cesa-Bianchi, S. Shalev-Shwartz, and O. Shamir. Some impossibility results for budgeted learning. In Joint ICML-COLT workshop on Budgeted Learning, 2010. [3] N. Cesa-Bianchi, S. Shalev-Shwartz, and O. Shamir. Efficient learning with partially observed attributes. Journal of Machine Learning Research, 12:2857?2878, 2011. [4] X. Chen, Q. Lin, and J. Pena. Optimal regularized dual averaging methods for stochastic optimization. In Advances in Neural Information Processing Systems, pages 395?403, 2012. [5] D. Foster, S. Kale, and H. Karloff. Online sparse linear regression. In 29th Annual Conference on Learning Theory, pages 960?970, 2016. [6] E. Hazan and T. Koren. Linear regression with limited observation. In Proceedings of the 29th International Conference on Machine Learning (ICML-12), pages 807?814, 2012. [7] S. Kale. Open problem: Efficient online sparse regression. In Proceedings of The 27th Conference on Learning Theory, pages 1299?1301, 2014. [8] S. Kale, Z. Karnin, T. Liang, and D. P?al. Adaptive feature selection: Computationally efficient online sparse linear regression under rip. In Proceedings of the 34th International Conference on Machine Learning (ICML-17), pages 1780?1788, 2017. [9] P. Koiran and A. Zouzias. Hidden cliques and the certification of the restricted isometry property. IEEE Trans. Information Theory, 60(8):4999?5006, 2014. [10] M. Lichman. UCI machine learning repository, 2013. [11] L. Xiao. Dual averaging methods for regularized stochastic learning and online optimization. Journal of Machine Learning Research, 11:2543?2596, 2010. [12] M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on Machine Learning (ICML-03), pages 928?936, 2003. [13] N. Zolghadr, G. Bart?ok, R. Greiner, A. Gy?orgy, and C. Szepesv?ari. Online learning with costly features and labels. In Advances in Neural Information Processing Systems, pages 1241?1249, 2013. 10
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Mapping distinct timescales of functional interactions among brain networks Mali Sundaresan Centre for Neuroscience Indian Institute of Science Bangalore, India 560 012 [email protected] Arshed Nabeel Centre for Neuroscience Indian Institute of Science Bangalore, India 560 012 [email protected] Devarajan Sridharan? Centre for Neuroscience Indian Institute of Science Bangalore, India 560 012 [email protected] Abstract Brain processes occur at various timescales, ranging from milliseconds (neurons) to minutes and hours (behavior). Characterizing functional coupling among brain regions at these diverse timescales is key to understanding how the brain produces behavior. Here, we apply instantaneous and lag-based measures of conditional linear dependence, based on Granger-Geweke causality (GC), to infer network connections at distinct timescales from functional magnetic resonance imaging (fMRI) data. Due to the slow sampling rate of fMRI, it is widely held that GC produces spurious and unreliable estimates of functional connectivity when applied to fMRI data. We challenge this claim with simulations and a novel machine learning approach. First, we show, with simulated fMRI data, that instantaneous and lag-based GC identify distinct timescales and complementary patterns of functional connectivity. Next, we analyze fMRI scans from 500 subjects and show that a linear classifier trained on either instantaneous or lag-based GC connectivity reliably distinguishes task versus rest brain states, with ?80-85% cross-validation accuracy. Importantly, instantaneous and lag-based GC exploit markedly different spatial and temporal patterns of connectivity to achieve robust classification. Our approach enables identifying functionally connected networks that operate at distinct timescales in the brain. 1 Introduction Processes in the brain occur at various timescales. These range from the timescales of milliseconds for extremely rapid processes (e.g. neuron spikes), to timescales of tens to hundreds of milliseconds for processes coordinated across local populations of neurons (e.g. synchronized neural oscillations), to timescales of seconds for processes that are coordinated across diverse brain networks (e.g. language) and even up to minutes, hours or days for processes that involve large-scale neuroplastic changes (e.g. learning a new skill). Coordinated activity among brain regions that mediate each of these cognitive processes would manifest in the form of functional connections among these regions at the corresponding timescales. Characterizing patterns of functional connectivity that occur at these different timescales is, hence, essential for understanding how the brain produces behavior. Measures of linear dependence and feedback, based on Granger-Geweke causality (GC) [10][11]), have been used to estimate instantaneous and lagged functional connectivity in recordings of brain activity made with electroencephalography (EEG, [6]), and electrocorticography (ECoG, [3]). However, the application of GC measures to brain recordings made with functional magnetic resonance imaging (fMRI) remains controversial [22][20][2]. Because the hemodynamic response is produced and sampled at a timescale (seconds) several orders of magnitude slower than the underlying neural ? Corresponding author 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. processes (milliseconds), previous studies have argued that GC measures, particularly lag-based GC, produce spurious and unreliable estimates of functional connectivity from fMRI data [22][20]. Three primary confounds have been reported with applying lag-based GC to fMRI data. First, systematic hemodynamic lags: a slower hemodynamic response in one region, as compared to another could produce a spurious directed GC connection from the second to the first [22] [4]. Second, in simulations, measurement noise added to the signal during fMRI acquisition was shown to produce significant degradation in GC functional connectivity estimates [20]. Finally, downsampling recordings to the typical fMRI sampling rate (seconds), three orders of magnitude slower than the timescale of neural spiking (milliseconds), was shown to effectively eliminate all traces of functional connectivity inferred by GC [20]. Hence, a previous, widely cited study argued that same-time correlation based measures of functional connectivity, such as partial correlations, fare much better than GC for estimating functional connectivity from fMRI data [22]. The controversy over the application of GC measures to fMRI data remains unresolved to date, primarily because of the lack of access to ?ground truth?. On the one hand, claims regarding the efficacy of GC estimates based on simulations, are only as valid as the underlying model of hemodynamic responses. Because the precise mechanism by which neural responses generate hemodynamic responses is an active area of research [7], strong conclusions cannot be drawn based on simulated fMRI data alone. On the other hand, establishing ?ground truth? validity for connections estimated by GC on fMRI data require concurrent, brain-wide invasive neurophysiological recordings during fMRI scans, a prohibitive enterprise. Here, we seek to resolve this controversy by introducing a novel application of machine learning that works around these criticisms. We estimate instantaneous and lag-based GC connectivity, first, with simulated fMRI time series under different model network configurations and, next, from real fMRI time series (from 500 human subjects) recorded under different task conditions. Based on the GC connectivity matrices, we train a linear classifier to discriminate model network configurations or subject task conditions, and assess classifier accuracy with cross validation. Our results show that instantaneous and lag-based GC connectivity estimated from empirical fMRI data can distinguish task conditions with over 80% cross-validation accuracies. To permit such accurate classification, GC estimates of functional connectivity must be robustly consistent within each model configuration (or task condition) and reliably different across configurations (or task conditions). In addition, drawing inspiration from simulations, we show that GC estimated on real fMRI data downsampled to 3x-7x the original sampling rate provides novel insights into functional brain networks that operate at distinct timescales. 2 2.1 Simulations and Theory Instantaneous and lag-based measures of conditional linear dependence The linear relationship among two multivariate signals x and y conditioned on a third multivariate signal z can be measured as the sum of linear feedback from x to y (Fx?y ), linear feedback from y to x (Fy?x ), and instantaneous linear feedback (Fx?y ) [11][16]. To quantify these linear relationships, we model the future of each time series in terms of their past values with a wellestablished multivariate autoregressive (MVAR) model (detailed in Supplementary Material, Section S1). Briefly, Fx?y is a measure of the improvement in the ability to predict the future values of y given the past values of x, over and above what can be predicted from the past values of z and y, itself (and vice versa for Fy?x ). Fx?y , on the other hand, measures the instantaneous influence between x and y conditioned on z (see Supplementary Material, Section S1). We refer to Fx?y , as instantaneous GC (iGC), and Fx?y Fy?x as lag-based GC or directed GC (dGC), with the direction of the influence (x to y or vice versa) being indicated by the arrow. The ?full? measure of linear dependence and feedback Fx,y is given by : Fx,y = Fx?y + Fy?x + Fx?y (1) Fx,y measures the complete conditional linear dependence between two time series. If, at a given instant, no aspect of one time series can be explained by a linear model containing all the values (past and present) of the other, Fx,y will evaluate to zero [16]. These measures are firmly grounded in information theory and statistical inferential frameworks [9]. 2 B Destination C -200ms-1 D Network H E F C F slow 0.02 2s B C Network J D E F 6s-1 -6s-1 D slow -0.02 0.02 Re fast 0 0.002 Source A B C D E F 50ms 5s Fast interaction (50ms) 0.002 fast Im Im E Network J 0.002 Re B A Node time-constant Network H -0.02 D 50ms 6s-1 Connection strength C A B Destination Source A 0 intermediate 0.002 600s Amplitude (a.u.) A Slow interaction (5s) 600s Time (s) Figure 1: Network simulations. (A) Network configuration H. (Left) Connectivity matrix. Red vs. blue: Excitatory vs. inhibitory connections. Deeper hues: Higher connection strengths. Non-zero value at (i, j) corresponds to a connection from node j to node i (column to row). Sub-network A-B-C operates at a fast timescale (50 ms) whereas D-E-F operates at a slow timescale (2 s). (Right) Network schematic showing the connectivity matrix as a graph. (B) Network configuration J. Conventions are the same as in A. (C) The eigenspectra of networks H (left) and J (right). (D) Simulated time series in network configuration J with fast (top panel) and slow (bottom panel) dynamics, corresponding to nodes A-B and E-F, respectively. Within each panel, the top plot is the simulated neural time series, and the bottom plot is the simulated fMRI time series. 2.2 Simulating functional interactions at different timescales To test the ability of GC measures to reliably recover functional interactions at different timescales, we simulated fMRI time series for model networks with two configurations of directed connectivity. Simulated fMRI time series were generated using a two-stage model (2): the first stage involved a latent variable model that described neural dynamics, and the second stage that convolved these dynamics with the hemodynamic response function (HRF) to obtain the simulated fMRI time series. y =H ?x x? = Ax + ? (2) where A is the neural (?ground truth?) connectivity matrix, x is the neural time series, x? is dx/dt, H is the canonical hemodynamic response function (HRF; simulated with spm_hrf in SPM8 software), ? is the convolution operation, y is the simulated BOLD time series, and ? is i.i.d Gaussian noise. Other than noise ?, other kinds of external input were not included in these simulations. Similar models have been employed widely for simulating fMRI time series data previously [22][2][20]. First, we sought to demonstrate the complementary nature of connections estimated by iGC and dGC. For this, we used network configuration H, shown in Fig. 1A. Note that this corresponds to two non-interacting sub-networks, each operating at distinctly different timescales (50 ms and 2000 ms node decay times, respectively) as revealed by the eigenspectrum of the connectivity matrix (Fig. 1C). For convenience, we term these two timescales as ?fast? and ?slow?. Moreover, each sub-network operated with a distinct pattern of connectivity, either purely feedforward, or with feedback (E-I). Dynamics were simulated with a 1 ms integration step (Euler scheme), convolved with the HRF and then downsampled to 0.5 Hz resolution (interval of 2 s) to match the sampling rate (repeat time, TR) of typical fMRI recordings. Second, we sought to demonstrate the ability of dGC to recover functional interactions at distinct timescales. For this, we simulated a different network configuration J, whose connectivity matrix 3 A iGC A D B E C F Ground Truth 0.06 dGC 0 B A B C D E F Ground Truth 0.02 0 50ms 500ms 5s Sampling Interval Figure 2: Connectivity estimated from simulated data. (A) iGC and dGC values estimated from simulated fMRI time series, network H. (Leftmost) Ground truth connectivity used in simulations. (Top) Estimated iGC connectivity matrix (left) and significant connections (right, p<0.05) estimated by a bootstrap procedure using 1000 phase scrambled surrogates[18]. (Bottom) Same as top panel, but for dGC. (B) dGC estimates from simulated fMRI time series, network J, sampled at three different sampling intervals: 50 ms (left), 500 ms (middle) and 5 s (right). In each case the estimated dGC matrix and significant connections are shown, with the same conventions as in panel (A). is shown in Fig. 1B. This network comprised three non-interacting sub-networks operating at three distinct timescales (50 ms, 0.5 s, and 5 s node decay times; eigenspectrum in Fig. 1C). As before, simulated dynamics were downsampled at various rates ? 20 Hz, 2 Hz, 0.2 Hz ? corresponding to sampling intervals of 50 ms, 0.5 s, and 5 s, respectively. The middle interval (0.5 s) is closest to the repeat time (TR=0.7 s) of the experimental fMRI data used in our analyses; the first and last intervals were chosen to be one order of magnitude faster and slower, respectively. Sufficiently long (3000 s) simulated fMRI timeseries were generated for each network configuration (H and J). Sample time series from a subset of these simulations before and after hemodynamic convolution and downsampling are shown in Fig. 1D. 2.3 Instantaneous and lag-based GC identify complementary connectivity patterns Our goal was to test if the ground truth neural connectivity matrix (A in equation 2) could be estimated by applying iGC and dGC to the fMRI time series y. dGC was estimated from the time series with the MVGC toolbox (GCCA mode) [1][19] and iGC was estimated from the MVAR residuals [16]. For simulations with network configuration H, iGC and dGC identified connectivity patterns that differed in two key respects (Fig. 2A). First, iGC identified feedforward interactions at both fast and slow timescales whereas dGC was able to estimate only the slow interactions, which occurred at a timescale comparable to the sampling rate of the measurement. Second, dGC was able to identify the presence of the E-I feedback connection at the slow timescale, whereas iGC entirely failed to estimate this connection. In the Supplementary Material (Section S2), we show theoretically why iGC can identify mutually excitatory or mutually inhibitory feedback connections, but fails to identify the presence of reciprocal excitatory-inhibitory (E-I) feedback connections, particularly when the connection strengths are balanced. For simulations with network configuration J, dGC identified distinct connections depending on the sampling rate. At the highest sampling rate (20 Hz), connections at the fastest timescales (50 ms) were estimated most effectively, whereas at the slowest sampling rates (0.2 Hz), only the slowest timescale connections (5 s) were estimated; intermediate sampling rates (2 Hz) estimated connections at intermediate timescales (0.5 s). Thus, dGC estimated robustly those connections whose process timescale was closest to the sampling rate of the data. The first finding ? that connections at fast timescales (50 ms) could not be estimated from data sampled at much lower rates (0.2 Hz) ? is expected, and in line with previous findings. However, the converse finding ? that the slowest timescale connections (5 s) could not be detected at the fastest sampling rates (20 Hz) ? was indeed surprising. To better understand these puzzling findings, we performed simulations over a wide range of sampling rates for each of these connection timescales; the results are shown in Supplementary Figure S1. dGC values (both with and without convolution with the hemodynamic response function) systematically increased from baseline, peaked at a sampling rate corresponding to the process timescale and decreased rapidly at higher sampling rates, matching 4 recent analytical findings[2]. Thus, dGC for connections at the at a particular timescale was highest when the data were sampled at a rate that closely matched that timescale. Two key conclusions emerged from these simulations. First, functional connections estimated by dGC can be distinct from and complementary to connections identified by iGC, both spatially and temporally. Second, connections that operate at distinct timescales can be detected by estimating dGC on data sampled at distinct rates that match the timescales of the underlying processes. 3 Experimental Validation We demonstrated the success of instantaneous and lag-based GC to accurately estimate functional connectivity with simulated fMRI data. Nevertheless, application of GC measures to real fMRI data is fraught with significant caveats, associated with hemodynamic confounds and measurement noise, as described above. We asked whether, despite these confounds, iGC and dGC would be able to produce reliable estimates of connectivity in real fMRI data. Moreover, as with simulated data, would iGC and dGC reveal complementary patterns of connectivity that varied reliably with different task conditions? 3.1 Machine learning, cross-validation and recursive feature elimination We analyzed minimally preprocessed brain scans of 500 subjects, drawn from the Human Connectome Project (HCP) database [12]. We analyzed data from resting state and seven other task conditions (total of 4000 scans; Supplementary Table S1). In the main text we present results for classifying the resting state from the language task; the other classifications are reported in the Supplementary Material. The language task involves subjects listening to short segments of stories and evaluating semantic content in the stories. This task is expected to robustly engage a network of language processing regions in the brain. The resting state scans served as a ?task-free? baseline, for comparison. Brain volumes were parcellated with a 14-network atlas [21] (see Supplementary Material Section S3; Supplementary Table S2). Network time series were computed by averaging time series across all voxels in a given network using Matlab and SPM8. These multivariate network time series were then fit with an MVAR model (Supplementary Material Section S1). Model order was determined with the Akaike Information Criterion for each subject, was typically 1, and did not change with further downsampling of the data (see next section). The MVAR model fit was then used to estimate both an instantaneous connectivity matrix using iGC (Fx?y ) and a lag-based connectivity matrix using dGC (Fx?y ). The connection strengths in these matrices were used as feature vectors in a linear classifier based on support vector machines (SVMs) for high dimensional predictor data. We used Matlab?s fitclinear function, optimizing hyperparameters using a 5-fold approach: by estimating hyperparameters with five sets of 100 subjects in turn, and measuring classification accuracies with the remaining 400 subjects; the only exception was for the classification analysis with averaging GC matrices (Fig. 3B) for which classification was run with default hyperparameters (regularization strength = 1/(cardinality of training-set), ridge penalty). The number of features for iGC-based classification was 91 (upper triangular portion of the symmetric 14?14 iGC matrix) and for dGC-based classification was 182 (all entries of the 14?14 dGC matrix, barring self-connections on the main diagonal). Based on these functional connectivity features, we asked if we could reliably predict the task condition (e.g. language versus resting). Classification performance was tested with leave-one-out and k-fold crossvalidation. We also assessed the significance of the classification accuracy with permutation testing [14] (Supplementary Material, Section S4). Finally, we wished to identify a key set of connections that permitted accurately classifying task from resting states. To accomplish this, we applied a two-stage recursive feature elimination (RFE) algorithm [5], which identified a minimal set of features that provided maximal cross validation accuracy (generalization performance). Details are provided in the Supplementary Material (Section S5, Supplementary Figs. S2-S3). 5 A 100 dGC iGC B 100 fGC 90 Accuracy Accuracy 90 80 70 60 50 80 70 Classi?cation using dGC Classi?cation using iGC 60 14-Network 90-Node 0 10 20 30 40 50 No. of Subjects Parcellation Scheme Figure 3: Classification based on GC connectivity estimates in real data. (A) Leave-one-out classification accuracies for different GC measures for the 14-network parcellation (left) and the 90-node parcellation (right). Within each group, the first two bars represent the classification accuracy with dGC and iGC respectively. The third bar is the classification accurcay with fGC (see equation 1). Chance: 50% (two-way classification). Error-bars: Clopper-Pearson binomial confidence intervals. (B) Classification accuracy when the classifier is tested on average GC matrices, as a function of number of subjects being averaged (see text for details). 3.2 Instantaneous and lag-based GC reliably distinguish task from rest Both iGC and dGC connectivity were able to distinguish task from resting state significantly above chance (Fig. 3A). Average leave-one-out cross validation accuracy was 80.0% with iGC and 83.4% with dGC (Fig. 3A, left). Both iGC and dGC classification exhibited high precision and recall at identifying language task (precision= 0.81, recall= 0.78 for iGC and precision= 0.85, recall= 0.81 for dGC). k-fold (k=10) cross-validation accuracy was also similar for both the GC measures (79.4% for iGC and 83.7% for dGC). dGC and iGC are complementary measures of linear dependence, by their definition. We asked if combining them would produce better classification performance. We combined dGC and iGC in two ways. First, we performed classification after pooling features (connectivity matrices) across both dGC and iGC (?iGC ? dGC?). Second, we estimated the full GC measure (Fx,y ), which is a direct sum of dGC and iGC estimates (see equation 1). Both of these approaches yielded marginally higher classification accuracies ? 88.2% for iGC ? dGC and 84.6% for fGC ? than dGC or iGC alone. Next, we asked if classification would be more accurate if we averaged the GC measures across a few subjects, to remove uncorrelated noise (e.g. measurement noise) in connectivity estimates. For this, the data were partitioned into two groups of 250 subjects: a training (T) group and a test (S) group. The classifier was trained on group T and the classifier prediction was tested by averaging GC matrices across several folds of S, each fold containing a few (m=2,4,5,10 or 25) subjects. Prediction accuracy for both dGC and iGC reached ?90% with averaging as few as two subjects? GC matrices, and reached ?100%, with averaging 10 subjects? matrices (Fig. 3B). We also tested if these classification accuracies were brain atlas or cognitive task specific. First, we tested an alternative atlas with 90 functional nodes based on a finer regional parcellation of the 14 functional networks [21]. Classification accuracies for iGC and fGC improved (87.9% and 90.8%, respectively), and for dGC remained comparable (81.4%), to the 14 network case (Fig. 3A, right). Second, we performed the same GC-based classification analysis for six other tasks drawn from the HCP database (Supplementary Table S1) . We discovered that all of the remaining six tasks could be classified from the resting state with accuracy comparable to the language versus resting classification (Supplementary Fig. S4). Finally, we asked how iGC and dGC classification accuracies would compare to those of other functional connectivity estimators. For example, partial correlations (PC) have been proposed as a robust measure of functional connectivity in previous studies [22]. Classification accuracies for PC varied between 81-96% across tasks (Supplementary Fig. S5B). PC?s better performance is expected: estimators based on same-time covariance are less susceptible to noise than those based on lagged covariance, a result we derive analytically in the Supplementary Material (Section S6). Also, when classifying language task versus rest, PC and iGC relied on largely overlapping connections (?60% overlap) whereas PC and dGC relied on largely non-overlapping connections (?25% overlap; Supplementary Fig. S5C). These results highlight the complementary nature of PC and dGC connectivity. Moreover, we demonstrate, both with simulations and with real-data, that 6 A B 0 0.4309 0 0.4001 0.5320 0 0.6161 0 0 0.7246 Accuracy 1 D-DMN LECN RECN A-SAL P-SAL LANG AUD SENMOT BG PREC V-DMN VISPA PR-VIS HI-VIS 1 31 61 # Features 0.5 1 31 61 91 121 151 181 1 Sampling rate 1x (0.72s) 31 61 91 121 151 181 1 3x (2.16s) 31 61 91 121 151 181 1 5x (3.60s) 31 61 91 121 151 181 7x (5.04s) Figure 4: Maximally discriminative connections identified with RFE (A) (Top) iGC connections that were maximally discriminative between the language task and resting state, identified using recursive feature elimination (RFE). Darker gray shades denote more discriminative connections (higher beta weights) (Bottom) RFE curves, with classification accuracy plotted as a function of the number of remaining features. The dots mark the elbow-points of the RFE curves, corresponding to the optimal number of discriminative connections. (B) Same as in (A), except that RFE was performed on dGC connectivity matrices with data sampled at 1x, 3x, 5x, and 7x of the original sampling interval (TR=0.72 s). Non-zero value at (i, j) corresponds to a connection from node j to node i (column to row). classification accuracy with GC typically increased with more scan timepoints, consistent with GC being an information theoretic measure (Supplementary Fig. S6). These superior classification accuracies show that, despite conventional caveats for estimating GC with fMRI data, both iGC and dGC yield functional connectivity estimates that are reliable across subjects. Moreover, dGC?s lag-based functional connectivity provides a robust feature space for classifying brain states into task or rest. In addition, we found that dGC connectivity can be used to predict task versus rest brain states with near-perfect (>95-97%) accuracy, by averaging connectivity estimates across as few as 10 subjects, further confirming the robustness of these estimates. 3.3 Characterizing brain functional networks at distinct timescales Recent studies have shown that brain regions, across a range of species, operate at diverse timescales. For example, a recent calcium imaging study demonstrated the occurrence of fast (?100 ms) and slow (?1 s) functional interactions in mouse cortex [17]. In non-human primates, cortical brain regions operate at a hierarchy of intrinsic timescales, with the sensory cortex operating at faster timescales compared to prefrontal cortex [13]. In the resting human brain, cortical regions organize into a hierarchy of functionally-coupled networks characterized by distinct timescales [24]. It is likely that these characteristic timescales of brain networks are also modulated by task demands. We asked if the framework presented in our study could characterize brain networks operating at distinct timescales across different tasks (and rest) from fMRI data. We had already observed, in simulations, that instantaneous and lag-based GC measures identified functional connections that operate at different timescales (Fig. 2A). We asked if these measures could identify connections at fast versus slow timescales (compared to TR=0.72s) that were specific to task verus rest, from fMRI recordings. To identify these task-specific connections, we performed recursive feature elimination (described in Supplementary Material, Section S5) with the language task and resting state scans, separately with iGC and dGC features (connections). Prior to analysis of real data, we validated RFE by applying it to estimate key differences in two simulated networks (Supplementary Material Fig. S2 and Fig. S3). RFE accurately identified connections that differed in simulation ?ground truth?: specifically, differences in fast timescale connections were identified by iGC, and in slow timescale connections by dGC. When applied to the language task versus resting state fMRI data, RFE identified a small subset of 18(/91) connections based on iGC (Fig. 4A), and an overlapping but non-identical set of 17(/182) connections based on dGC (Fig. 4B); these connections were key to distinguishing task (language) 7 from resting brain states. Specifically, the highest iGC beta weights, corresponding to the most discriminative iGC connections, occurred among various cognitive control networks, including the anterior and posterior salience networks, the precuneus and the visuospatial network (Fig. 5A). Some of these connections were also detected by dGC. Nevertheless, the highest dGC beta weights occurred for connections to and from the language network, for example from the language network to dorsal default mode network and from the precuneus to the language network (Fig. 5B). Notably, these latter connections were important for classification based on dGC, but not based on iGC. Moreover, iGC identified a connection between the language network and the basal ganglia whereas dGC, in addition, identified the directionality of the connection, as being from the language network to the basal ganglia. In summary, dGC and iGC identified several complementary connections, but dGC alone identified many connections with the language network, indicating that slow processes in this network significantly distinguished language from resting states. Next, we tested whether estimating dGC after systematically downsampling the fMRI time series would permit identifying maximally discriminative connections at progressively slower timescales. To avoid degradation of GC estimates because of fewer numbers of samples with downsampling (by decimation), we concatenated the different downsampled time series to maintain an identical total number of samples. RFE was applied to GC estimates based on data sampled at different rates: 1.4 Hz, 0.5 Hz, 0.3 Hz and 0.2 Hz corresponding to 1x, 3x, 5x, and 7x of TR (sampling period of 0.72 s, 2.16 s, 3.6 s and 5.04 s), respectively. RFE with dGC identified 17(/182) key connections at each of these timescales (Fig. 4B). Interestingly, some connections manifested in dGC estimates across all sampling rates. For instance, the connection from the precuneus to the language network was important for classification across all sampling rates (Fig. 5C). On the other hand, connections between the language network and various other networks manifested at specific sampling rates only. For instance an outgoing connection from the language network to the basal ganglia manifested only at the 1.4 Hz sampling rate, to the visuospatial network and default mode networks only at 0.5 Hz, to the higher-visual network only at 0.2-0.3 Hz, and an incoming connection from the anterior salience only at 0.2 Hz. None of these connections were identified by the iGC classifier (compare Fig. 5A and 5C). Similar timescale generic and timescale specific connections were observed in other tasks as well (Supplementary Fig. S7). Despite downsampling, RFE accuracies were significantly above chance, although accuracies decreased at lower sampling rates (Fig. 4 lower panels) [20]. Thus, dGC identified distinct connectivity profiles for data sampled at different timescales, without significantly compromising classification performance. Finally, we sought to provide independent evidence to confirm whether these network connections operated at different timescales. For this, we estimated the average cross coherence (Supplementary Material, Section S7) between the fMRI time series of two connections from the language network that were identified by RFE exclusively at 0.2-0.3 Hz (language to higher visual) and 0.5 Hz (language to visuospatial) sampling rates, respectively (Fig. 5C). Each connection exhibited an extremum in the coherence plot at a frequency which closely matched the respective connection?s timescale (Fig. 5D). These findings, from experimental data, provide empirical validation to our simulation results, which indicate that estimating dGC on downsampled data is a tenable approach for identifying functional connections that operate at specific timescales. 4 Conclusions These results contain three novel insights. First, we show that two measures of conditional linear dependence ? instantaneous and directed Granger-Geweke causality ? provide robust measures of functional connectivity in the brain, resolving over a decade of controversy in the field [23][22]. Second, functional connections identified by iGC and dGC carry complementary information, both in simulated and in real fMRI recordings. In particular, dGC is a powerful approach for identifying reciprocal excitatory-inhibitory connections, which are easily missed by iGC and other same-time correlation based metrics like partial correlations [22]. Third, when processes at multiple timescales exist in the data, our results show that downsampling the time series to different extents provides an effective method for recovering connections at these distinct timescales. Our simulations highlight the importance of capturing emergent timescales in simulations of neural data. For instance, a widely-cited study [22] employed purely feedforward connectivity matrices with a 50 ms neural timescale in their simulations, and argued that functional connections are not reliably inferred with GC on fMRI data. However, such connectivity matrices preclude the occurrence of 8 A D-DMN D-DMN RECN A-SAL PREC V-DMN VISPA LANG PREC V-DMN LANG BG PREC PREC VISPA V-DMN VISPA HI-VIS D Both HI-VIS D-DMN RECN A-SAL 0.01 P-SAL 0.01 0.32Hz LANG BG PREC VISPA RECN LANG Coherence dGC only A-SAL All HI-VIS iGC only RECN 0.19 Hz P-SAL VISPA HI-VIS 0.28 Hz A-SAL P-SAL LANG AUD BG PREC V-DMN AUD BG 0.46 Hz D-DMN A-SAL P-SAL B 1.39 Hz VISPA 0.46Hz Freq 0.7Hz -0.01 HI-VIS LANG?VISPA (dGC 3x) 0.14Hz Coherence D-DMN LECN RECN A-SAL P-SAL LANG AUD SENMOT BG PREC V-DMN VISPA PR-VIS HI-VIS C Freq 0.28Hz 0.7Hz -0.01 LANG?HI-VIS (dGC 5x) Figure 5: Connectivity at different timescales. (A-B) Discriminative connections identified exclusively by iGC (teal), exclusively by dGC (blue), or by both (yellow). Each connection is represented as a band going from a source node on the left to a destination node on the right. (C) (Top) Discriminative connections identified by dGC, exclusively at different sampling intervals (1x, 3x, 5x, 7x TR). (D) (Left) Directed connection between language network and visuospatial network identified by dGC with fMRI data sampled at 0.5 Hz (sampling interval, 3x TR). (Right) Directed connection between language network and higher visual network identified by dGC with fMRI data sampled at 0.3 Hz (sampling interval, 5x TR). (Lower plots) Cross coherence between respective network time series. Shaded area: Frequencies from Fs /2 to Fs , where Fs is the sampling rate of the fMRI timeseries from which dGC was estimated. slower, behaviorally relevant timescales of seconds, which readily emerge in the presence of feedback connections, both in simulations [8][15] and in the brain [17][24]. Our simulations explicitly incorporated these slow timescales to show that connections at these timescales could be robustly estimated with GC on simulated fMRI data. Moreover, we show that such slow interactions also occur in human brain networks. Our approach is particularly relevant for studies that seek to investigate dynamic functional connectivity with slow sampling techniques, such as fMRI or calcium imaging. Our empirical validation of the robustness of GC measures, by applying machine learning to fMRI data from 500 subjects (and 4000 functional scans), is widely relevant for studies that seek to apply GC to estimate directed functional networks from fMRI data. Although, scanner noise or hemodynamic confounds can influence GC estimates in fMRI data [20][4], our results demonstrate that dGC contains enough directed connectivity information for robust prediction, reaching over 95% validation accuracy with averaging even as few as 10 subjects? connectivity matrices (Fig. 3B). These results strongly indicate the existence of slow information flow networks in the brain that can be meaningfully inferred from fMRI data. Future work will test if these functional networks influence behavior at distinct timescales. Acknowledgments. This research was supported by a Wellcome Trust DBT-India Alliance Intermediate Fellowship, a SERB Early Career Research award, a Pratiksha Trust Young Investigator award, a DBT-IISc Partnership program grant, and a Tata Trusts grant (all to DS). We would like to thank Hritik Jain for help with data analysis. References [1] L. Barnett and A. K. Seth. 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Rangarajan, and M. Ding. Analyzing information flow in brain networks with nonparametric granger causality. NeuroImage, 41(2):354 ? 362, 2008. [7] K. J. Friston, A. Mechelli, R. Turner, and C. J. Price. Nonlinear responses in fmri: the balloon model, volterra kernels, and other hemodynamics. NeuroImage, 12(4):466?477, 2000. [8] S. Ganguli, J. W. Bisley, J. D. Roitman, M. N. Shadlen, M. E. Goldberg, and K. D. Miller. One-dimensional dynamics of attention and decision making in lip. Neuron, 58(1):15?25, 2008. [9] I. M. Gel?fand and A. M. Yaglom. Calculation of the amount of information about a random function contained in another such function. American Mathematical Society Translations, 12(1):199?246, 1959. [10] J. Geweke. Measurement of linear dependence and feedback between multiple time series. Journal of the American statistical association, 77(378):304?313, 1982. [11] J. F. Geweke. Measures of conditional linear dependence and feedback between time series. Journal of the American Statistical Association, 79(388):907?915, 1984. [12] M. F. Glasser, S. N. Sotiropoulos, J. A. Wilson, T. S. Coalson, B. Fischl, J. L. Andersson, J. Xu, S. Jbabdi, M. Webster, J. R. Polimeni, et al. The minimal preprocessing pipelines for the human connectome project. Neuroimage, 80:105?124, 2013. [13] J. D. Murray, A. Bernacchia, D. J. Freedman, R. Romo, J. D. Wallis, X. Cai, C. Padoa-Schioppa, T. Pasternak, H. Seo, D. Lee, et al. A hierarchy of intrinsic timescales across primate cortex. Nature neuroscience, 17(12):1661?1663, 2014. [14] M. Ojala and G. C. Garriga. Permutation tests for studying classifier performance. Journal of Machine Learning Research, 11(Jun):1833?1863, 2010. [15] K. Rajan and L. Abbott. Eigenvalue spectra of random matrices for neural networks. Physical review letters, 97(18):188104, 2006. [16] A. Roebroeck, E. Formisano, and R. Goebel. Mapping directed influence over the brain using granger causality and fmri. 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377 EXPERIMENTAL DEMONSTRATIONS OF OPTICAL NEURAL COMPUTERS Ken Hsu, David Brady, and Demetri Psaltis Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 ABSTRACT We describe two expriments in optical neural computing. In the first a closed optical feedback loop is used to implement auto-associative image recall. In the second a perceptron-Iike learning algorithm is implemented with photorefractive holography. INTRODUCTION The hardware needs of many neural computing systems are well matched with the capabilities of optical systems l ,2,3. The high interconnectivity required by neural computers can be simply implemented in optics because channels for optical signals may be superimposed in three dimensions with little or no cross coupling. Since these channels may be formed holographically, optical neural systems can be designed to create and maintain interconnections very simply. Thus the optical system designer can to a large extent avoid the analytical and topological problems of determining individual interconnections for a given neural system and constructing physical paths for these interconnections. An archetypical design for a single layer of an optical neural computer is shown in Fig. 1. Nonlinear thresholding elements, neurons, are arranged on two dimensional planes which are interconnected via the third dimension by holographic elements. The key concerns in implementing this design involve the need for suitable nonlinearities for the neural planes and high capacity, easily modifiable holographic elements. While it is possible to implement the neural function using entirely optical nonlinearities, for example using etalon arrays\ optoelectronic two dimensional spatial light modulators (2D SLMs) suitable for this purpose are more readily available. and their properties, i.e. speed and resolution, are well matched with the requirements of neural computation and the limitations imposed on the system by the holographic interconnections 5 ,6. Just as the main advantage of optics in connectionist machines is the fact that an optical system is generally linear and thus allows the superposition of connections, the main disadvantage of optics is that good optical nonlinearities are hard to obtain. Thus most SLMs are optoelectronic with a non-linearity mediated by electronic effects. The need for optical nonlinearities arises again when we consider the formation of modifiable optical interconnections, which must be an all optical process. In selecting @ American Institute of Physics 1988 378 a holographic material for a neural computing application we would like to have the capability of real-time recording and slow erasure. Materials such as photographic film can provide this only with an impractical fixing process. Photorefractive crystals are nonlinear optical materials that promise to have a relatively fast recording response and long term memory4,5,6,7,B. '. '. . " ..... .. '. .. .' - ~ :-w:-=7 -~--- " . '. ...... . '. Fourier lens hologro.phlc I"IealuI"I Fourier lens Figure 1. Optical neural computer architecture. In this paper we describe two experimental implementations of optical neural computers which demonstrate how currently available optical devices may be used in this application. The first experiment we describe involves an optical associative loop which uses feedback through a neural plane in the form of a pinhole array and a separate thresholding plane to implement associate regeneration of stored patterns from correlated inputs. This experiment demonstrates the input-output dynamics of an optical neural computer similar to that shown in Fig. 1, implemented using the Hughes Liquid Crystal Light Valve. The second experiment we describe is a single neuron optical perceptron implemented with a photorefractive crystal. This experiment demonstrates how the learning dynamics of long term memory may be controlled optically. By combining these two experiments we should eventually be able to construct high capacity adaptive optical neural computers. OPTICAL ASSOCIATIVE LOOP A schematic diagram of the optical associative memory loop is shown in Fig. 2. It is comprised of two cascaded Vander Lugt correlators9. The input section of the system from the threshold device P1 through the first hologram P2 to the pinhole array P3 forms the first correlator. The feedback section from P3 through the second hologram P4 back to the threshold device P1 forms the second correlator. An array of pinholes sits on the back focal plane of L2, which coincides with the front focal plane of L3. The purpose of the pinholes is to link the first and the second (reversed) correlator to form a closed optical feedback loop 10. There are two phases in operating this optical loop, the learning phase and the recal phase. In the learning phase, the images to be stored are spatially multiplexed and entered simultaneously on the threshold device. The 379 thresholded images are Fourier transformed by the lens Ll. The Fourier spectrum and a plane wave reference beam interfere at the plane P2 and record a Fourier transform hologram. This hologram is moved to plane P4 as our stored memory. We then reconstruct the images from the memory to form a new input to make a second Fourier transform hologram that will stay at plane P2. This completes the learning phase. In the recalling phase an input is imaged on the threshold Input ~~~*+++~~~~ device. This image is correlated with the reference images in the hologram at P2. If the correlation between the input and one of the stored images is high a bright peak appears at one of the pinholes. This peak is sampled by ~ -,.....,.- Second Pinhole Hologram Array - -.... L z the pinhole to reconstruct the stored I I image from the hologram at P4. The reconstructed beam is then imaged back to the threshold device to form a closed loop. If the overall optical gain Figure. 2. All-optical associative in the loop exceeds the loss the loop loop. The threshold device is a LCLV, signal will grow until the threshold and the holograms are thermoplastic device is saturated. In this case, we plates. can cutoff the external input image and the optical loop will be latched at the stable memory. The key elements in this optical loop are the holograms, the pinhole array, and the threshold device. If we put a mirror 10 or a phase conjugate mirror 7 ,11 at the pinhole plane P3 to reflect the correlation signal back through the system then we only need one hologram to form a closed loop. The use of two holograms, however, improves system performance. We make the hologram at P2 with a high pass characteristic so that the input section of the loop has high spectral discrimination. On the other hand we want the images to be reconstructed with high fidelity to the original images. Thus the hologram at plane P4 must have broadband characteristics. We use a diffuser to achieve this when making this hologram. Fig. 3a shows the original images. Fig. 3b and Fig. 3c are the images reconstructed from first and second holograms, respectively. As desired, Fig. 3b is a high pass version of the stored image while Fig. 3c is broadband . Each of the pinholes at the correlation plane P3 has a diameter of 60 j.lm. The separations between the pinholes correspond to the separations of the input images at plane P 1. If one of the stored images appears at P 1 there will be a bright spot at the corresponding pinhole on plane P3. If the input image shifts to the position of another image the correlation peak will also 380 ,. ~ ?.. ? ' .a:..J ~ a. . , .'" . ~ ~. ( i \~ .~ -y::' . .. ?Il,... .' .r I K~?';t L ? ? ? .# b. c. Figure 3. (a) The original images. (b)The reconstructed images from the highpass hologram P2. (c) The reconstructed images from the band-pass hologram P4. shift to another pinhole. But if the shift is not an exact image spacing the correlation peak can not pass the pinhole and we lose the feedback signal. Therefore this is a loop with "discrete" shift invariance. Without the pinholes the cross-correlation noise and the auto-correlation peak will be fed back to the loop together and the reconstructed images won't be recognizable. There is a compromise between the pinhole size and the loop performance. Small pinholes allow good memory discrimination and sharp reconstructed images, but can cut the signal to below the level that can be detected by the threshold device and reduce the tolerance of the system to shifts in the input. The function of the pinhole array in this system might also be met by a nonlinear spatial light modulator, in which case we can achieve full shift invariance 12 ? The threshold device at plane PI is a Hughes Liquid Crystal Light Valve. The device has a resolution of 16 Ip/mm and uniform aperture of 1 inch diameter. This gives us about 160,000 neurons at PI. In order to compensate for the optical loss in the loop, which is on the order of 10- 5 , we need the neurons to provide gain on the order of 105. In our system this is achieved by placing a Hamamatsu image intensifier at the write side of the LCLV. Since the microchannel plate of the image intensifier can give gains of 104 , the combination of the LCLV and the image intensifier can give gains of 10 6 with sensitivity down to n W /cm 2 . The optical gain in the loop can be adjusted by changing the gain of the image intensifier. Since the activity of neurons and the dynamics of the memory loop is a continuously evolving phenomenon, we need to have a real time device to monitor and record this behavior. We do this by using a prism beam splitter to take part of the read out beam from the LCLV and image it onto a CCD camera. The output is displayed on a CRT monitor and also recorded on a video tape recorder. Unfortunately, in a paper we can only show static pictures taken from the screen. We put a window at the CCD plane so that each time we can pick up one of the stored images. Fig. 4a shows the read out image 381 a. b. c. Figure 4. (a) The external input to the optical loop. (b) The feedback image superimposed with the input image. (c) The latched loop image. from the LCLV which comes from the external input shifted away from its stored position. This shift moves its correlation peak so that it does not match the position of the pinhole. Thus there is no feedback signal going through the loop. If we cut off the input image the read out image will die out with a characteristic time on the order of 50 to 100 ms, corresponding to the response time of the LCLV. Now we shift the input image around trying to search for the correct position. Once the input image comes close enough to the correct position the correlation peak passes through the right pinhole, giving a strong feedback signal superimposed with the external input on the neurons. The total signal then goes through the feedback loop and is amplified continuously until the neurons are saturated. Depending on the optical gain of the neurons the time required for the loop to reach a stable state is between 100 ms and several seconds. Fig. 4b shows the superimposed images of the external input and the loop images. While the feedback signal is shifted somewhat with respect to the input, there is sufficient correlation to induce recall. If the neurons have enough gain then we can cut off the input and the loop stays in its stable state. Otherwise we have to increase the neuron gain until the loop can sustain itself. Fig. 4c shows the image in the loop with the input removed and the memory latched. If we enter another image into the system, again we have to shift the input within the window to search the memory until we are close enough to the correct position. Then the loop will evolve to another stable state and give a correct output. The input images do not need to match exactly with the memory. Since the neurons can sense and amplify the feedback signal produced by a partial match between the input and a stored image, the stored memory can grow in the loop. Thus the loop has the capability to recall the complete memory from a partial input. Fig. 5a shows the image of a half face input into the system. Fig. 5b shows the overlap of the input with the complete face from the memory. Fig. 5c shows the stable state of the loop after we cut off the external input. In order to have this associative behavior the input must have enough correlation with the stored memory to yield a strong feedback signal. For instance, the loop does not respond to the the presentation of a picture of 382 a. c. Figure 5. (a) Partial face used as the external input. (b) The superimposed images of the partial input with the complete face recalled by the loop. (c) The complete face latched in the loop. a. b. c. Figure 6. (a) Rotated image used as the external input. (b) The superimposed images of the input with the recalled image from the loop. (c) The image latched in the optical loop. a person not stored in memory. Another way to demonstrate the associative behavior of the loop is to use a rotated image as the input. Experiments show that for a small rotation the loop can recognize the image very quickly. As the input is rotated more, it takes longer for the loop to reach a stable state. If it is rotated too much, depending on the neuron gain, the input won't be recognizable. Fig. 6a shows the rotated input. Fig. 6b shows the overlap of loop image with input after we turn on the loop for several seconds. Fig. 6c shows the correct memory recalled from the loop after we cut the input. There is a trade-off between the degree of distortion at the input that the system can tolerate and its ability to discriminate against patterns it has not seen before. In this system the feedback gain (which can be adjusted through the image intensifier) controls this trade-off. PHOTOREFRACTIVE PERCEPTRON Holograms are recorded in photorefractive crystals via the electrooptic modulation of the index of refraction by space charge fields created by the migration of photogenerated charge 13 ,14. Photorefractive crystals are attractive for optical neural applications because they may be used to store 383 long term interactions between a very large number of neurons. While photorefractive recording does not require a development step, the fact that the response is not instantaneous allows the crystal to store long term traces of the learning process. Since the photorefractive effect arises from the reversible redistribution of a fixed pool of charge among a fixed set of optically addressable trapping sites, the photorefractive response of a crystal does not deteriorate with exposure. Finally, the fact that photorefractive holograms may extend over the entire volume of the crystal has previously been shown to imply that as many as 10 10 interconnections may be stored in a single crystal with the independence of each interconnection guaranteed by an appropriate spatial arrangement of the interconnected neurons 6 ,5. In this section we consider a rudimentary optical neural system which uses the dynamics of photorefractive crystals to implement perceptron-like learning. The architecture of this system is shown schematically in Fig. 7. The input to the system, x, corresponds to a two dimensional pattern recorded from a video monitor onto a liquid crystal light valve. The light valve transfers this pattern on a laser beam. This beam is split into two paths which cross in a photorefractive crystal. The light propagating along each path is focused such that an image of the input pattern is formed on the crystal. The images along both paths are of the same size and are superposed on the crystal, which is assumed to be thinner than the depth of focus of the images. The intensity diffracted from one of the two paths onto the other by a hologram stored in the crystal is isolated by a polarizer and spatially integrated by a single output detector. The thresholded output of this detector corresponds to the output of a neuron in a perceptron. laser ~---,t+ - PB LCL V TV --f4HJ ucl BS$- - COl"lputer Xtal PM Figure 7. Photorefractive perceptron. PB is a polarizing beam splitter. Ll and L2 are imaging lenses. WP is a quarter waveplate. PM is a piezoelectric mirror. P is a polarizer. D is a detector. Solid lines show electronic control. Dashed lines show the optical path. The ith component of the input to this system corresponds to the intensity in the ith pixel of the input pattern. The interconnection strength, Wi, between the ith input and the output neuron corresponds to the diffraction efficiency of the hologram taking one path into the other at the ith pixel of the image plane. While the dynamics of Wi can be quite complex in some geometries 384 and crystals, it is possible to show from the band transport model for the photorefractive effect that under certain circumstances the time development of Wi may be modeled by (1) where m(s) and 4>(s) are the modulation depth and phase, respectively, of the interference pattern formed in the crystal between the light in the two paths 15 ? T is a characteristic time constant for crystal. T is inversely proportional to the intensity incident on the ith pixel of the crystal. Using Eqn. 1 it is possible to make Wi(t) take any value between 0 and W m l1Z by properly exposing the ith pixel of the crystal to an appropriate modulation depth and intensity. The modulation depth between two optical beams can be adjusted by a variety of simple mechanisms. In Fig. 7 we choose to control met) using a mirror mounted on a piezoelectric crystal. By varying the frequency and the amplitude of oscillations in the piezoelectric crystal we can electronically set both met) and 4>(t) over a continuous range without changing the intensity in the optical beams or interrupting readout of the system. With this control over met) it is possible via the dynamics described in Eqn. (1) to implement any learning algorithm for which Wi can be limited to the range (0, w maz ). The architecture of Fig. 7 classifies input patterns into two classes according to the thresholded output of the detector. The goal of a learning algorithm for this system is to correctly classify a set of training patterns. The perceptron learning algorithm involves simply testing each training vector and adding training vectors which yield too Iowan output to the weight vector and subtracting training vectors which yield too high an output from the weight vector until all training vectors are correctly classified 16. This training algorithm is described by the equation L\wi = aXj where alpha is positive (negative) if the output for x is too low (high). An optical analog of this method is implemented by testing each training pattern and exposing the crystal with each incorrectly classified pattern. Training vectors that yield a high output when a low output is desired are exposed at zero modulation depth . Training vectors that yield a low output when high output is desired are exposed at a modulation depth of one. The weight vector for the k + 1th iteration when erasure occurs in the kth iteration is given by (2) where we assume that the exposure time, L\t, is much less than T. Note that since T is inversely proportional to the intensity in the ith pixel, the change in 385 Wi is proportional to the ith input. The weight vector at the k + 1th iteration when recording occurs in the kth iteration is given by -2~t -~t _ / -~t -~t wi(k+ 1) = e-r-Wi(k) +2y Wi(k)Wmcue-r- (l-e-r- ) +wmaz(l-e-r-) To lowest order in 6.t .,. 2 (3) and ~, Eqn. (3) yields w m .... _/ ~t ~t 2 wi(k + 1) = wi(k) + 2y wi(k)Wmaz(-) + Wmaz(-) T T (4) Once again the change in Wi is proportional to the ith input. We have implemented the architecture of Fig. 7 using a SBN60:Ce crystal provided by the Rockwell International Science Center. We used the 488 nm line of an argon ion laser to record holograms in this crystal. Most of the patterns we considered were laid out on 10 x 10 grids of pixels, thus allowing 100 input channels. Ultimately, the number of channels which may be achieved using this architecture is limited by the number of pixels which may be imaged onto the crystal with a depth of focus sufficient to isolate each pixel along the length of the crystal. - ?? +.+ Y ....... ? ? ? 1 3 2 ..... ? ? 4 Figure 8. Training patterns. ... 1'1 j Ia. 8. ! l t I , 0 0 aCOftClS ~ W CIII) Figure 9. Output in the second training cycle. Using the variation on the perceptron learning algorithm described above with a fixed exposure times ~tr and ~te for recording and erasing, we have been able to correctly classify various sets of input patterns. One particular set which we used is shown in Fig. 8. In one training sequence, we grouped patterns 1 and 2 together with a high output and patterns 3 and 4 together with a low output. After all four patterns had been presented four times, the system gave the correct output for all patterns. The weights stored in the crystal were corrected seven times, four times by recording and three by erasing. Fig . 9a shows the output of the detector as pattern 1 is recorded in the second learning cycle. The dashed line in this figure corresponds to the threshold level. Fig. 9b shows the output of the detector as pattern 3 is erased in the second learning cycle. 386 CONCLUSION The experiments described in this paper demonstrate how neural network architectures can be implemented using currently available optical devices. By combining the recall dynamics of the first system with the learning capability of the second, we can construct sophisticated optical neural computers. ACKNOWLEDGEMENTS The authors thank Ratnakar Neurgaonkar and Rockwell International for supplying the SBN crystal used in our experiments and Hamamatsu Photonics K.K. for assistance with image intesifiers. We also thank Eung Gi Paek and Kelvin Wagner for their contributions to this research. This research is supported by the Defense Advanced Research Projects Agency, the Army Research Office, and the Air Force Office of Scientific Research. REFERENCES 1. Y. S. Abu-Mostafa and D. Psaltis, Scientific American, pp.88-95, March, 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 1987. D. Psaltis and N. H. Farhat, Opt. Lett., 10,(2),98(1985). A. D. Fisher, R. C. Fukuda, and J. N. Lee, Proc. SPIE 625, 196(1986). K. Wagner and D. Psaltis, Appl. opt., 26(23), pp.5061-5076(1987). D. Psaltis, D. Brady, and K. Wagner, Applied optics, March 1988. D. Psaltis, J. Yu, X. G. Gu, and H. Lee, Second Topical Meeting on Optical Computing, Incline Village, Nevada, March 16-18,1987. A. Yariv, S.-K. Kwong, and K. Kyuma, SPIE proc. 613-01,(1986). D. Z. Anderson, Proceedings of the International Conference on Neural Networks, San Diego, June 1987. A. B. Vander Lugt, IEEE Trans. Inform. Theory, IT-I0(2), pp.139145(1964). E. G. Paek and D. Psaltis, Opt. Eng., 26(5), pp.428-433(1987). Y. Owechko, G. J. Dunning, E. Marom, and B. H. Soffer, Appl. Opt. 26,(10) ,1900(1987). D. Psaltis and J. Hong, Opt. Eng. 26,10(1987). N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics, 22,949(1979). J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51,1297(1980). T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, Prog. Quan. Electr. 10,77(1985). F. Rosenblatt, ' Principles of Neurodynamics: Perceptron and the Theory of Brain Mechanisms, Spartan Books, Washington,(1961).
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137 On the Power of Neural Networks for Solving Hard Problems J ehoshua Bruck Joseph W. Goodman Information Systems Laboratory Departmen t of Electrical Engineering Stanford University Stanford, CA 94305 Abstract This paper deals with a neural network model in which each neuron performs a threshold logic function. An important property of the model is that it always converges to a stable state when operating in a serial mode [2,5]. This property is the basis of the potential applications of the model such as associative memory devices and combinatorial optimization [3,6]. One of the motivations for use of the model for solving hard combinatorial problems is the fact that it can be implemented by optical devices and thus operate at a higher speed than conventional electronics. The main theme in this work is to investigate the power of the model for solving NP-hard problems [4,8], and to understand the relation between speed of operation and the size of a neural network. In particular, it will be shown that for any NP-hard problem the existence of a polynomial size network that solves it implies that NP=co-NP. Also, for Traveling Salesman Problem (TSP), even a polynomial size network that gets an ?-approximate solution does not exist unless P=NP. The above results are of great practical interest, because right now it is possible to build neural networks which will operate fast but are limited in the number of neurons. 1 Background The neural network model is a discrete time system that can be represented by a weighted and undirected graph. There is a weight attached to each edge of the graph and a threshold value attached to each node (neuron) of the graph. ? American Institute of Physics 1988 138 The order of the network is the number of nodes in the corresponding graph. Let N be a neural network of order n; then N is uniquely defined by (W, T) where: ? W is an n X n symmetric matrix, Wii is equal to the weight attached to edge (i, j) . ? T is a vector of dimension n, Ti denotes the threshold attached to node i. Every node (neuron) can be in one of two possible states, either 1 or -1. The state of node i at time t is denoted by Vi(t). The state of the neural network at time t is the vector V(t). The next state of a node is computed by: Vi(t + 1) = sgn(H,(t)) = { where ~1 ~t~;2i~ 0 (1) n Hi(t) = L WiiVj(t) - Ti i=l The next state of the network, i.e. V(t + 1), is computed from the current state by performing the evaluation (1) at a subset of the nodes of the network, to be denoted by S. The modes of operation are determined by the method by which the set S is selected in each time interval. If the computation is performed at a single node in any time interval, i.e. 1S 1= 1, then we will say that the network is operating in a serial mode; if 1S 1= n then we will say that that the network is operating in a fully parallel mode. All the other cases, i.e. 1 <I S 1< n will be called parallel modes of operation. The set S can be chosen at random or according to some deterministic rule. A state V(t) is called stable iff V(t) = sgn(WV(t) - T), i.e. there is no change in the state of the network no matter what the mode of operation is. One of the most important properties of the model is the fact that it always converges to a stable state while operating in a serial mode. The main idea in the proof of the convergence property is to define a so called energy function and to show that this energy function is nondecreasing when the state of the network changes. The energy function is: (2) An important note is that originally the energy function was defined such that it is nonincreasing [5]; we changed it such that it will comply with some known graph problems (e.g. Min Cut). A neural network will always get to a stable state which corresponds to a local maximum in the energy function. This suggests the use of the network as a 139 device for performing a local search algorithm for finding a maximal value of the energy function [6]. Thus, the network will perform a local search by operating in a random and serial mode. It is also known [2,9] that maximization of E associated with a given network N in which T = 0 is equivalent to finding the Minimum Cut in N. Actually, many hard problems can be formulated as maximization of a quadratic form (e.g. TSP [6)) and thus can be mapped to a neural network. . 2 The Main Results The set of stable states is the set of possible final solutions that one will get using the above approach. These final solutions correspond to local maxima of the energy function but do not necessarily correspond to global optima of the corresponding problem. The main question is: suppose we allow the network to operate for a very long time until it converges; can we do better than just getting some local optimum? i.e., is it possible to design a network which will always find the exact solution (or some guaranteed approximation) of the problem? Definition: Let X be an instance of problem. Then 1 X 1 denotes the size of X, that is, the number of bits required to represent X. For example, for X being an instance of TSP, 1 X I is the number of bits needed to represent the matrix of the distances between cities. Definition: Let N be a neural network. Then 1 N 1 denotes the size of the network N. Namely, the number of bits needed to represent Wand T. Let us start by defining the desired setup for using the neural network as a model for solving hard problems. Consider an optimization problem L, we would like to have for every instance X of L a neural network N x with the following properties: ? Every local maximum of the energy function associated with N x corresponds to a global optimum of X . ? The network N x is small, that is, in 1X I. I Nx 1 is bounded by some polynomial Moreover, we would like to have an algorithm, to be denoted by A L , which given an instance X E L, generates the description for N x in polynomial (in I X I) time. Now, we will define the desired setup for using the neural network as a model for finding approximate solutions for hard problems. Definition: Let Eglo be the global maximum of the energy function. Let Eloc 140 be a local maximum of the energy function. We will say that a local maximum is an f-approximate of the global iff: Eglo - Eloc --:;.--< Eglo f - The setup for finding approximate solutions is similar to the one for finding exact solutions. For fo> 0 being some fixed number. We would like to have a network N x~ in which every local maximum is an f-approximate of the global and that the global corresponds to an optimum of X. The network N x? should be small, namely, 1 N x~ 1 should be bounded by a polynomial in 1 X I. Also, we would like to have an algorithm AL~, such that, given an instance X E L, it generates the description for N x? in polynomial (in 1 X I) time. Note that in both the exact case and the approximate case we do not put any restriction on the time it takes the network to converge to a solution (it can be exponential) . A t this point the reader should convince himself that the above description is what he imagined as the setup for using the neural network model for solving hard problems, because that is what the following definition is about. Definition: We will say that a neural network for solving (or finding an fapproximation of) a problem L exists if the algorithm AL (or ALJ which generates the description of N x (or Nx~) exists. The main results in the paper are summarized by the following two propositions. The first one deals with exact solutions of NP-hard problems while the second deals with approximate solutions to TSP. Proposition 1 Let L be an NP-hard problem. Then the existence of a neural network for solving L implies that NP = co-NP. Proposition 2 Let f > 0 be some fixed number. The existence of a neural network for finding an f-approximate solution to TSP implies that P=NP. Both (P=NP) and (NP=co-NP) are believed to be false statements, hence, we can not use the model in the way we imagine. The key observation for proving the above propositions is the fact that a single iteration in a neural network takes time which is bounded by a polynomial in the size of the instance of the corresponding problem. The proofs of the above two propositions follow directly from known results in complexity theory and should not be considered as new results in complexity theory. 141 3 The Proofs Proof of Proposition 1: The proof follows from the definition of the classes NP and co-NP, and Lemma 1. The definitions and the lemma appear in Chapters 15 and 16 in [8] and also in Chapters 2 and 7 in [4]. Lemma 1 If the complement of an NP-complete problem is in NP, then NP=co-NP. Let L be an NP-hard problem. Suppose there exists a neural network that solves L. Let 1 be an NP-complete problem. By definition, 1 can be polynomialy reduced to L. Thus, for every instance X E 1, we have a neural network such that from any of its global maxima we can efficiently recognize whether X is a 'yes' or a 'no' instance of 1. We claim that we have a nondeterministic polynomial time algorithm to decide that a given instance X E 1 is a 'no' instance. Here is how we do it: for X E 1 we construct the neural network that solves it by using the reduction to L. We then check every state of the network to see if it is a local maximum (that is done in polynomial time). In case it is a local maximum, we check if the instance is a 'yes' or a 'no' instance (this is also done in polynomial time). Thus, we have a nondeterministic polynomial time algorithm to recognize any 'no' instance of 1. Thus, the complement of the problem 1 is in NP. But 1 is an NP-complete problem, hence, from Lemma 1 it follows that NP=co-NP. 0 Proof of Proposition 2: The result is a corollary of the results in [7], the reader can refer to it for a more complete presentation. The proof uses the fact that the Restricted Hamiltonian Circuit (RHC) is an NP-complete problem. Definiton of RHC: Given a graph G = (V, E) and a Hamiltonian path in G. The question is whether there is a Hamiltonian circuit in G? It is proven in [7] that RHC is NP-complete. Suppose there exists a polynomial size neural network for finding an f-approximate solution to TSP. Then it can be shown that an instance X E RHC can be reduced to an instance X E TSP, such that in the network N x the following holds: if the Hamiltonian path that is given in X corresponds to a local maximum in N x? then X is a 'no' instance; else, if it does not correspond to a local maximum in N x? then X is a 'yes' instance. Note that we can check for locality in polynomial time. Hence, the existence of N xe for all X E TSP implies that we have a polynomial time algorithm for RHC. 0 ? 142 4 Concluding Remarks 1. In Proposition 1 we let I W I and I T I be arbitrary but bounded by a polynomial in the size of a given instance of a problem. If we assume that I W I and I T I are fixed for all instances then a similar result to Proposition 1 can be proved without using complexity theory; this result appears in [1]. 2. The network which corresponds to TSP, as suggested in [6], can not solve the TSP with guaranteed quality. However, one should note that all the analysis in this paper is a worst case type of analysis. So, it might be that there exist networks that have good behavior on the average. 3. Proposition 1 is general to all NP-hard problems while Proposition 2 is specific to TSP. Both propositions hold for any type of networks in which an iteration takes polynomial time. 4. Clearly, every network has an algorithm which is equivalent to it, but an algorithm does not necessarily have a corresponding network. Thus, if we do not know of an algorithmic solution to a problem we also will not be able to find a network which solves the problem. If one believes that the neural network model is a good model (e.g. it is amenable to implementation with optics), one should develop techniques to program the network to perform an algorithm that is known to have some guaranteed good behavior. Acknowledgement: Support of the U.S. Air Force Office of Scientific Research is gratefully acknowledged. References [1] Y. Abu Mostafa, Neural Networks for Computing? in Neural Networks for Computing, edited by J. Denker (AlP Conference Proceedings no. 151, 1986). [2] J. Bruck and J. Sanz, A Study on Neural Networks, IBM Tech Rep, RJ 5403, 1986. To appear in International Journal of Intelligent Systems, 1988. [3] J. Bruck and J. W. Goodman, A Generalized Convergence Theorem for Neural Networks and its Applications in Combinatorial Optimization, IEEE First ICNN, San-Diego, June 1987. [4] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, 1979. 143 [5] J. J. Hopfield, Neural Networks and Physical Systems with Emergent Collective Computational Abilities, Proc. Nat. Acad. Sci .. USA, Vol. 79, pp. 2554-2558, 1982. [6] J. J. Hopfield and D. W. Tank, Neural Computations of Decisions in Optimization Problems, BioI. Cybern. 52, pp. 141-152, 1985. [7] C. H. Papadimitriou and K. Steiglitz, On the Complexity of Local Search for the Traveling Salesman Problem, SIAM J. on Comp., Vol. 6, No.1, pp. 76-83, 1977. [8] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algo:rithms and Complexity, Prentice-Hall, Inc., 1982. [9] J. C. Picard and H. D. Ratliff, Minimum Cuts and Related Problems, Networks, Vol 5, pp. 357-370, 1974.
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Neural Network Model Selection Using Asymptotic Jackknife Estimator and Cross-Validation Method Yong Liu Department of Physics and Institute for Brain and Neural Systems Box 1843, Brown University Providence, RI, 02912 Abstract Two theorems and a lemma are presented about the use of jackknife estimator and the cross-validation method for model selection. Theorem 1 gives the asymptotic form for the jackknife estimator. Combined with the model selection criterion, this asymptotic form can be used to obtain the fit of a model. The model selection criterion we used is the negative of the average predictive likehood, the choice of which is based on the idea of the cross-validation method. Lemma 1 provides a formula for further exploration of the asymptotics of the model selection criterion. Theorem 2 gives an asymptotic form of the model selection criterion for the regression case, when the parameters optimization criterion has a penalty term. Theorem 2 also proves the asymptotic equivalence of Moody's model selection criterion (Moody, 1992) and the cross-validation method, when the distance measure between response y and regression function takes the form of a squared difference. 1 INTRODUCTION Selecting a model for a specified problem is the key to generalization based on the training data set. In the context of neural network, this corresponds to selecting an architecture. There has been a substantial amount of work in model selection (Lindley, 1968; Mallows, 1973j Akaike, 1973; Stone, 1977; Atkinson, 1978j Schwartz, 599 600 Liu 1978; Zellner, 1984; MacKay, 1991; Moody, 1992; etc.). In Moody's paper (Moody, 1992), the author generalized Akaike Information Criterion (AIC) (Akaike, 1973) in the regression case and introduced the term effective number of parameters. It is thus of great interest to see what the link between this criterion and the crossvalidation method (Stone, 1974) is and what we can gain from it, given the fact that AIC is asymptotically equivalent to the cross-validation method (Stone, 1977). In the method of cross-validation (Stone, 1974), a data set, which has a data point deleted from the original training data set, is used to estimate the parameters of a model by optimizing a parameters optimization criterion. The optimal parameters thus obtained are called the jackknife estimator (Miller, 1974). Then the predictive likelihood of the deleted data point is calculated, based on the estimated parameters. This is repeated for each data point in the original training data set. The fit of the model, or the model selection criterion, is chosen as the negative of the average of these predictive likelihoods. However, the computational cost of estimating parameters for different data point deletion is expensive. In section 2, we obtained an asymptotic formula (theorem 1) for the jackknife estimator based on optimizing a parameters optimization criterion with one data point deleted from the training data set. This somewhat relieves the computational cost mentioned above. This asymptotic formula can be used to obtain the model selection criterion by plugging it into the criterion. Furthermore, in section 3, we obtained the asymptotic form of the model selection criterion for the general case (Lemma 1) and for the special case when the parameters optimization criterion has a penalty term (theorem 2). We also proved the equivalence of Moody's model selection criterion (Moody, 1992) and the cross-validation method (theorem 2). Only sketchy proofs are given when these theorems and lemma are introduced. The detail of the proofs are given in section 4. 2 APPROXIMATE JACKKNIFE ESTIMATOR Let the parameters optimization criterion, with data set w = {(:Vi, yd, i = 1, ... , n} and parameters 9, be Cw (9), and let W-i denote the data set with ith data point deleted from w. If we denote 8 and 8_ i as the optimal parameters for criterion Cw (9) and Cw _.(9), respectively, \79 as the derivative with respect to 9 and superscript t ~s transpose, we have the following theorem about the relationship between 8 and 9_ i ? Theorem 1 If the criterion function C w (9) is an infinite- order differentiable function and its derivatives are bounded around 8. The estimator 8-i (also called jackknife estimator (Miller, 1974)) can be approzimated as 8_ i - 8~ -(\79\7~Cw(8) - \79\7~Ci(8?-1\79Ci(8) (1) in which Ci(9) = Cw(9) - Cw_.(9). Proof. Use the Taylor expansion of equation \7 9Cw_.(8_d terms higher than the second order. o around 9. Ignore Model Selection Using Asymptotic Jackknife Estimator & Cross-Validation Method Example 1: Using the generalized maximum likelihood method from Bayesian analysis 1 (Berger, 1985), if 7r( 0) is the prior on the parameters and the observations are mutually independent, for which the distribution is modeled as ylx ,..... f(Ylx, 0), the parameters optimization criterion is Thus Ci(O) = logf(Yilxi, 0). If we ignore the influence of the deleted data point in the dt nominator of equation 1, we have (3) Example 2: In the special case of example I, with noninformative prior 7r( 0) = 1, the criterion is the ordinary log-likelihood function, thus L 9_i-9~- [ VeV~logf(Yj lxj,O) j-lVelogf(Yilxi,O). (4) (xi,Y.:)Ew 3 CROSS-VALIDATION METHOD AND MODEL SELECTION CRITERION Hereafter we use the negative of the average predictive likelihood, or, 1 L Tm(w) = -n logf(Yi lXi, O-i) (5) (x"y.:)Ew as the model selection criterion, in which n is the size of the training data set w, mE ..Vi denotes parametric probability models f(Ylx, 0) and .tVi is the set of all the models in consideration. It is well known that Tm(w) is an unbiased estimator of r(00,9(.)), the risk of using the model m and estimator 0, when the true parameters are 00 and the training data set is w (Stone, 1974; Efron and Gong, 1983; etc.), i.e., r(Oo, 0(.)) E{Tm(w)} E{ -logf(Ylx, 9(w))} E{ -~ L logf(Yj IXj, O(w)) } (6) (x] ,Y] )Ew ... in which Wn = {( Xj , Yj ), j = I, ... k} is the test data set, 9(.) is an implicit function of the training data set wand it is the estimator we decide to use after we have observed the training data set w. The expectation above is taken over the randomness of w, x, Y and W n . The optimal model will be the one that minimizes this criterion. This procedure of using 9_ t and Tm(w) to obtain an estimation of risk is often called the cross-validation method (Stone, 1974; Efron and Gong, 1983) . Remark: After we have obtained 9 for a model, we can use equation 1 to calculate 9_ i for each i, and put the resulting 9_ i into equation 5 to get the fit of the model, thus we will be able to compare different models m E .tVi. 1 Strictly speaking, it is a method to find the posterior mode. 601 602 Liu Lemma 1 If the probability model f(ylx, 8), as a function,. of 8, is differentiable up to infinite order and its derivatives are bounded around 8. The approximation to the model selection criterion, equation 5, can be written as Tm(w) ~ -~n L logf(Yi lXi, 8) - (Xi,Yi)Ew L ~n V'~logf(Yi lXi, 8)(8_ i - 8) (7) (Xi,Yi)Ew Proof. Igoring the terms higher than the second order of the Taylor expansion of logf(Yj IXj, 8_ i ) around 8 will yield the result. Ezample 2 (continued): Using equation 4, we have, for the modei selection criterion, 1 n 1 n " L" logf(Yi lXi, 9) A (xi,y.)Ew 2: V'~logf(Yi lXi, 8)A -IV' /}logf(Yi lXi, 8). (8) (:Ci,y.)Ew in which A = E(X]'Y3)EW V'/}V'~logf(Yjlxj,9). If the model f(Ylx,8) is the true one, the second term is asymptotically equal to p, the number of parameters in the model. So the model selection criterion is - log-likelihood + number of parameters of the model. This is the well known Akaike's Information Criterion (AIC) (Akaike, 1973). Ezample 1( continued): Consider the probability model f(Ylx,8) = ,8exp( - 1 20'2 (9) E(y, 1}/}( X))) in which,8 is a normalization factor, E(y, 1}/}(x)) is a distance measure between Y and regression function 1}/} (x). E(?) as function of 9 is assumed differentiable. Denoting 2 U(8,~, w) E(Xi,Yi)EW E(Yi' 1}/}(xd) - 20'2log1T(81~), we have the following theorem, = Theorem 2 For the model specified in equation 9 and the parameters optimization criterion specified in equation 2 (ezample 1), under regular condition, the unbiased estimator of ~ 2: E(Yi,1}e(xd)} (10) V'~E(Yi,1}e(xd){V'/}V'~U(8,).,w)}-IV'/}E(Yi,1}9(xd)? (11) E{ (xi,y.)Ew .. asymptotically equals to 1"" L n 1 n E(Yi' 1}e(x~)) + (:Ci,y .. )Ew L (Xi,Yi)Ew 2For example, 1r(OIA) = Alp(O, (72 fA), this corresponds to U(O, A, w) = L (z"Yi)Ew ?(Yi,l1s(xi)) + A02 + const(A, (72). Model Selection Using Asymptotic Jackknife Estimator & Cross-Validation Method For the case when ?(Y, 179(Z)) = (y -179(Z))2, we get, for the asymptotic equivalency of the equation 11, (12) in wh.ich W = {(Zi,Yi), i = 1, ... , n} is the training data set, Wn = {(zi,yd, ~ = 1, ... , k} is the test data set, and ?(8,w) = ~ L(:z:"y,)EW E(Yi,179(Zi)). Proof. This result comes directly from theorem 1 and lemma 1. Some asymptotic technique has to be used. Remark: The result in equation 12 was first proposed by Moody (Moody, 1992). The effective number of parameters formulated in his paper corresponds to the summation in equation 12. Since the result in this theorem comes directly from the asymptotics of the cross-validation method and the jackknife estimator, it gives the equivalency proof between Moody's model selection criterion and the crossvalidation method. The detailed proof of this theorem, presented in section 4, is in spirit the same as the one presented in Stone's paper about the proof of the asymptotic equivalence of Ale and the cross-validation method (Stone, 1977). 4 DETAILED PROOF OF LEl\1MAS AND THEOREMS In order to prove theorem 1, lemma 1 and theorem 2, we will present three auxiliary lemmas first. Lemma 2 For random variable sequence Zn and Yn, if limn-+co Zn liffin-+co Yn = z, then Zn and Yn are asymptotically equivalent. Z and Proof. This comes from the definition of asymptotic equivalence. Because asymptotically the two random variable will behave the same as random variable z. Lemma 3 Consider the summation Li h(Zi' Ydg(Zi' z). If E(h(z, y)lz, z) is a constant c independent of z, y, z, then the summation is asymptotically equivalent to cLig(Zi'Z). Proof. According to the theorem of large number, lim ~ ' " h(Zi' Yi)g(Zt, z) n-+co n ~ E(h(z, y)g(z, z)) t E(E(h(z, y)lz, z)g(z, z)) = cE(g(z, z)) which is the same as the limit of ~ Li g(Zt' z). Using lemma 2, we get the result of this lemma. Lemma 4 If T}9 (.) and g( 8, .) are differentiable up to the second order, and the model Y = T}9 (z) + f with f ,...., ,/V (0, (]'2) is the true model, the second derivative with 603 604 Liu respect to 8 of n i=l evaluated at the minimum of U, i. e., iJ, is asymptotically independent of random variable {Yi, i = 1, ... , n}. Pro~of. Explicit calculation of the second derivative of U with respect to 8, evaluated at 8, gives n V9V~U(iJ,'\,W) = 2:LV977J(1;JV~179(:Z:t) i=l i=l + As n approaches infinite, the effect of the second term in U vanishes, iJ approach the mean squared error estimator with infinite amount of data points, or the true parameters 80 of the model (consistency of MSE estimator (Jennrich, 1969)), E(y779(z)) approaches E(Y-779 0(Z)) which is O. According to lemma 2 and lemma 3, the second term of this second derivative vanishes asymptotically. So as n approaches infinite, the second derivative of U with respect to 8, evaluated at iJ, approaches n V' 9V~U(80), '\, w) = 2 which is independent of {Yi, i lemma is readily obtained. L V' 97790 (zi)V~7790 (Zi) + V' 9V~g( 8 0 , ,\) = 1, ... , n}. According to lemma 2, the result of this Now we give the detailed proof of theorem 1, lemma 1 and theorem 2. Proof of Theorem 1. The jackknife estimator iJ- i satisfies, V 9Cw_ . (ILi) The Taylor expansion of the left side of this equation around 8 gives VeCW_i(iJ) + VeV~Cw_.(iJ)(iJ_i - iJ) + O(liJ- i 91 2 ) - O. =0 According to the definition of iJ and iJ_ i , their difference is thus a small quantity. Also because of the boundness of the derivatives, we can ignore higher order terms in the Taylor expansion and get the approximation iJ- i - iJ ~ -(V9V~CW_i(iJ))-1V'9Cw_.(iJ) Since 9 satisfies V' 9Cw(iJ) = 0, we can rewrite this equation and obtain equation 1. Proof of Lemma 1. The Taylor expansion of 10gf(Yi IZi' iJ-d around iJ is 10gf(Yi IZi, iJ-d = 10gf(Yi IZi, iJ) + V'~logf(Yi IZi, iJ)(iJ_ i - iJ) + O(liJ- i - 91 2 ) Putting this into equation 5 and ignoring higher order terms for the same argument as that presented in the proof of theorem 1, we readily get equation 7. Proof of Theorem 2. Up to an additive constant dependent only on ,\ and cr 2 , the optimization criterion, or equation 2, can be rewritten as (13) Model Selection Using Asymptotic Jackknife Estimator & Cross-Validation Method Now putting equation 9 and 13 into equation 3, we get, 0 ~ -{V' 9V'~U(8, .\, w)} -IV' 9?(Yi, 17e(:z:d) O-i - (14) Putting equation 14 into equation 7, we get, for the model selection criterion, Tm(w) = n1 '~ " 1 ?(Yi, 17e(:z:d) 2u2 + (:t:"Yi)Ew 1 r.. t ~ }-l V'9?(Yi,17e(:Z:i )) 2u1 2 V'9?(Yi, 17e(:z:d){V'9V'9t U (O,>.,w) '~ " (15) (:t:i,Yi)Ew Recall the discussion associated with equation 6 and now E{ -"k1 ~ 10gf(Y;I:Z:j,0)} '~ " = E{"k1 (:t:"y,)Ew" '~ " 2u1 2 ?(Yj,17e(:Z:;))} (16) (:t:"Yj)Ew" after some simple algebra, we can obtain the unbiased estimator of equation 10. The result is equation 15 multiplied by 2u 2 , or equation 11. Thus we prove the first part of the theorem. Now consider the case when ?(Y,179(:Z:)) = (y -179(:z:))2 (17) The second term of equation 11 now becomes ~ L -17e(:z:d)2V'~17e(:Z:i){V'9V'~U(8, >',w)}-1V'917e(:Z:i) 4(Yi (18) (:t:"Yi)Ew As n approaches infinite, 0 approach the true parameters 0o , V' 917e(:Z:') approaches V'9179 0 (:Z:.) and E((y -17e(:z:)))2 asymptotically equals to u 2 ? Using lemma 4 and lemma 3, we get, for the asymptotic equivalency of equation 18, .!..u2 n 2V'~17?(:z:d{V'9V'~U(0,>.,W)}-12V'917?(:z:d L (19) If we use notation ?(O,w) = ~ L(:t:"Yi)EW ?(Yi,179(:z:d), with ?(Y,179(:Z:)) of the form specified in equation 17, we can get, a -a V'9 n ?(0,w) = -2V'9179(:Z:i) Yi (20) Combining this with equation 19 and equation 11, we can readily obtain equation 12. 5 SUMMARY In this paper, we used asymptotics to obtain the jackknife estimator, which can be used to get the fit of a model by plugging it into the model selection criterion. Based on the idea of the cross-validation method, we used the negative of the average predicative likelihood as the model selection criterion. We also obtained the asymptotic form of the model selection criterion and proved that when the parameters optimization criterion is the mean squared error plus a penalty term, this asymptotic form is the same as the form presented by (Moody, 1992). This also served to prove the asymptotic equivalence of this criterion to the method of cross-validation. 605 606 Liu Acknowledgements The author thanks all the members of the Institute for Brain and Neural Systems, in particular, Professor Leon N Cooper for reading the draft of this paper, and Dr. Nathan Intrator, Michael P. Perrone and Harel Shouval for helpful comments. This research was supported by grants from NSF, ONR and ARO. References Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov and Czaki, editors, Proceedings of the 2nd International Symposium on Information Theory, pages 267-281. Atkinson, A. C. (1978). Posterior probabilities for choosing a regression model. Biometrika, 65:39-48. Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. SpringerVerlag. Efron, B. and Gong, G. (1983). A leisurely look at the bootstrap, the jackknife and cross-validation. Amer. Stat., 37:36-48. Jennrich, R. (1969). Asymptotic properties of nonlinear least squares estimators. Ann. Math. Stat., 40:633-643. Lindley, D. V. (1968). The choice of variables in multiple regression (with discussion). J. Roy. Stat. Soc., Ser. B, 30:31-66. MacKay, D. (1991). Bayesian methods for adaptive models. PhD thesis, California Institute of Technology. Mallows, C. L. (1973). Some comments on Cpo Technometrics, 15:661-675. Miller, R. G. (1974). The jackknife - a review. Biometrika, 61:1-15. Moody, J. E. (1992). The effective number of parameters, an analysis of generalization and regularization in nonlinear learning system. In Moody, J. E., Hanson, S. J., and Lippmann, R. P., editors, Advances in Neural Information Processing System 4. Morgan Kaufmann Publication. Schwartz, G. (1978). Estimating the dimension of a model. Ann. Stat, 6:461-464. Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions (with discussion). J. Roy. Stat. Soc., Ser. B. Stone, M. (1977). An asymptotic equivalence of choice of model by cross-validation and Akaike's criterion. J. Roy. Stat. Soc., Ser. B, 39(1):44-47. Zellner, A. (1984). Posterior odds ratios for regression hypotheses: General consideration and some specific results. In Zellner, A., editor, Basic Issues in Econometrics, pages 275-305. University of Chicago Press.
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Multi-Armed Bandits with Metric Movement Costs Tomer Koren Google Brain [email protected] Roi Livni Princeton University [email protected] Yishay Mansour Tel Aviv University and Google [email protected] Abstract We consider the non-stochastic Multi-Armed Bandit problem in a setting where there is a fixed and known metric on the action space that determines a cost for switching between any pair of actions. The loss of the online learner has two components: the first is the usual loss of the selected actions, and the second is an additional loss due to switching between actions. Our main contribution gives a tight characterization of the expected minimax regret in this setting, in terms of a complexity measure C of the underlying metric which depends on its covering numbers. In finite metric spaces with k actions, we ? give an efficient algorithm 1/3T 2/3, kT }), and show that this is e that achieves regret of the form O(max{C the best possible. Our regret bound generalizes previous known regret bounds ? 1/3T 2/3, kT }) e for some special cases: (i) the unit-switching cost regret ?(max{k ? 2/3, kT }) where e where C = ?(k), and (ii) the interval metric with regret ?(max{T C = ?(1). For infinite metrics spaces with Lipschitz loss functions, we derive a e d+1 d+2 ) where d ? 1 is the Minkowski dimension of the tight regret bound of ?(T space, which is known to be tight even when there are no switching costs. 1 Introduction Multi-Armed Bandit (MAB) is perhaps one of the most well studied model for learning that allows to incorporate settings with limited feedback. In its simplest form, MAB can be thought of as a game between a learner and an adversary: At first, the adversary chooses an arbitrary sequence of losses `1, . . . , `T (possibly adversarially). Then, at each round the learner chooses an action it from a finite set of actions K. At the end of each round, the learner gets to observe her loss `t (it ), and only the loss of her chosen action. The objective of the PTlearner is to minimize her (external) regret, defined as the expected difference between her loss, t=1 `t (it ), and the loss of the best action in hindsight, i.e., P mini ?K Tt=1 `t (i). One simplification of the MAB is that it assumes that the learner can switch between actions without any cost, this is in contrast to online algorithms that maintain a state and have a cost of switching between states. One simple intermediate solution is to add further costs to the learner that penalize movements between actions. (Since we compare the learner to the single best action, the adversary has no movement and hence no movement cost.) This approach has been studied in the MAB with unit switching costs [2, 12], where the learner is not only penalized for her loss but also pays a unit cost for any time she switches between actions. This simple penalty implicitly advocates the construction of algorithms that avoid frequent fluctuation in their decisions. Regulating switching has been successfully applied to many interesting instances such as buffering problems [16], limited-delay lossy coding [19] and dynamic pricing with patient buyers [15]. The unit switching cost assumes that any pair of actions have the same cost, which in many scenarios is far from true. For example, consider an ice-cream vendor on a beach, where his actions are to select a location and price. Clearly, changing location comes at a cost, while changing prices might come with no cost. In this case we can define a interval metric (the coast line) and the movement cost is the distance. A more involved case is a hot-dog vendor in Manhattan, which needs to select a location 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. and price. Again, it makes sense to charge a switching cost between locations according to their distance, and in this case the Manhattan-distance seems the most appropriate. Such settings are at the core of our model for MAB with movement cost. The authors of [24] considered a MAB problem equipped with an interval metric, i.e, the actions are [0, 1] and the movement cost is the distance between the actions. They proposed a new online algorithm, called the Slowly Moving Bandit (SMB) algorithm, that achieves optimal regret bound for this setting, and applied it to a dynamic pricing problem with patient buyers to achieve a new tight regret bound. The objective of this paper is to handle general metric spaces, both finite and infinite. We show how to generalize the SMB algorithm and its analysis to design optimal moving-cost algorithms for any metric space over finite decision space. Our main result identifies an intrinsic complexity measure of the metric space, which we call the covering/packing complexity, and give a tight characterization of the expected movement regret in terms of the complexity of the underlying metric. In particular, in finite metric spaces of complexity C with k actions, we give a regret bound of the ? 1/3T 2/3, kT }) and present an efficient algorithm that achieves it. We also give a e form O(max{C ? 1/3T 2/3, kT }) lower bound that applies to any metric with complexity C. e matching ?(max{C We extend out results to general continuous metric spaces. For such a settings we clearly have to make some assumption about the losses, and we make the rather standard assumption that the losses are Lipchitz with respect to the underlying metric. In this setting our results depend on a quite different complexity measures: the upper and lower Minkowski dimensions of the space, thus exhibiting a phase transition between the finite case (that corresponds to Minkowski dimension zero) and the e d+1 d+2 ) where d ? 1 is the upper infinite case. Specifically, we give an upper bound on the regret of O(T Minkowski dimension. When the upper and lower Minkowski dimensions coincide?which is the case in many natural spaces, such as normed vector spaces?the latter bound matches a lower bound of [10] that holds even when there are no switching costs. Thus, a surprising implication of our result is that in infinite actions spaces (of bounded Minkowski dimension), adding movement costs do not add to the complexity of the MAB problem! Our approach extends the techniques of [24] for the SMB algorithm, which was designed to optimize over an interval metric, which is equivalent to a complete binary Hierarchally well-Separated Tree (HST) metric space. By carefully balancing and regulating its sampling distributions, the SMB algorithm avoids switching between far-apart nodes in the tree and possibly incurring large movement costs with respect to the associated metric. We show that the SMB regret guarantees are much more general than just binary balanced trees, and give an analysis of the SMB algorithm when applied to general HSTs. As a second step, we show that a rich class of trees, on which the SMB algorithm can be applied, can be used to upper-bound any general metric. Finally, we reduce the case of an infinite metric space to the finite case via simple discretization, and show that this reduction gives rise to the Minkowski dimension as a natural complexity measure. All of these contractions turn out to be optimal (up to logarithmic factors), as demonstrated by our matching lower bounds. 1.1 Related Work Perhaps the most well known?classical algorithm for non-stochastic bandit is the Exp3 Algorithm [4] e kT) without movement costs. However, for general MAB algorithms that guarantee a regret of O( there are no guarantees for slow movement between actions. In fact, it is known that in a worst case e ?(T) switches between actions are expected (see [12]). A simple case of MAB with movement cost is the uniform metric, i.e., when the distance between any two actions is the same. This setting has seen intensive study, both in terms of analyzing optimal regret rates [2, 12], as well as applications [16, 19, 15]. Our main technical tools for achieving lower bounds is through the lower bound of Dekel et al. [12] that achieve such bound for this special case. The general problem of bandits with movement costs has been first introduced in [24], where the authors gave an efficient algorithm for a 2-HST binary balanced tree metric, as well as for evenly spaced points on the interval. The main contribution of this paper is a generalization of these results to general metric spaces. There is a vast and vigorous study of MAB in continuous spaces [23, 11, 5, 10, 32]. These works relate the change in the payoff to the change in the action. Specifically, there has been a vast research on Lipschitz MAB with stochastic payoffs [22, 29, 30, 21, 26], where, roughly, the expected reward is Lipschitz. For applying our results in continuous spaces we too need to assume Lipschitz losses, 2 however, our metric defines also the movement cost between actions and not only relates the losses of similar actions. Our general findings is that in Euclidean spaces, one can achieve the same regret bounds when movement cost is applied. Thus, the SMB algorithm can achieve the optimal regret rate. One can model our problem as a deterministic Markov Decision Process (MDP), where the states are the MAB actions and in every state there is an action to move the MDP to a given state (which correspond to switching actions). The payoff would be the payoff of the MAB action associated with the state plus the movement cost to the next state. The work of Ortner [28] studies deterministic MDP where the payoffs are stochastic, and also allows for a fixed uniform switching cost. The work of Even-Dar et al. [13] and it extensions [27, 33] studies a MDP where the payoffs are adversarial but there is full information of the payoffs. Latter this work was extended to the bandit model by Neu et al. [27]. This line of works imposes various assumptions regarding the MDP and the benchmark policies, specifically, that the MDP is ?mixing? and that the policies considered has full support stationary distributions, assumptions that clearly fail in our very specific setting. Bayesian MAB, such as in the Gittins index (see [17]), assume that the payoffs are from some stochastic process. It is known that when there are switching costs then the existence of an optimal index policy is not guaranteed [6]. There have been some works on special cases with a fixed uniform switching cost [1, 3]. The most relevant work is that of Guha and Munagala [18] which for a general metric over the actions gives a constant approximation off-line algorithm. For a survey of switching costs in this context see [20]. The MAB problem with movement costs is related to the literature on online algorithms and the competitive analysis framework [8]. A prototypical online problem is the Metrical Task System (MTS) presented by Borodin et al. [9]. In a metrical task system there are a collection of states and a metric over the states. Similar to MAB, the online algorithm at each time step moves to a state, incurs a movement cost according to the metric, and suffers a loss that corresponds to that state. However, unlike MAB, in an MTS the online algorithm is given the loss prior to selecting the new state. Furthermore, competitive analysis has a much more stringent benchmark: the best sequence of actions in retrospect. Like most of the regret minimization literature, we use the best single action in hindsight as a benchmark, aiming for a vanishing average regret. One of our main technical tools is an approximation from above of a metric via a Metric Tree (i.e., 2-HST). k-HST metrics have been vastly studied in the online algorithms starting with [7]. The main goal is to derive a simpler metric representation (using randomized trees) that will both upper and lower bound the given metric. The main result is to show a bound of O(log n) on the expected stretch of any edge, and this is also the best possible [14]. It is noteworthy that for bandit learning, and in contrast with these works, an upper bound over the metric suffices to achieve optimal regret rate. This is since in online learning we compete against the best static action in hindsight, which does not move at all and hence has zero movement cost. In contrast, in a MTS, where one compete against the best dynamic sequence of actions, one needs both an upper a lower bound on the metric. 2 Problem Setup and Background In this section we recall the setting of Multi-armed Bandit with Movement Costs introduced in [24], and review the necessary background required to state our main results. 2.1 Multi-armed Bandits with Movement Costs In the Multi-armed Bandits (MAB) with Movement Costs problem, we consider a game between an online learner and an adversary continuing for T rounds. There is a set K, possibly infinite, of actions (or ?arms?) that the learner can choose from. The set of actions is equipped with a fixed and known metric ? that determines a cost ?(i, j) ? [0, 1] for moving between any pair of actions i, j ? K. Before the game begins, an adversary fixes a sequence `1, . . . , `T : K 7? [0, 1] of loss functions assigning loss values in [0, 1] to actions in K (in particular, we assume an oblivious adversary). Then, on each round t = 1, . . . , T, the learner picks an action it ? K, possibly at random. At the end of each round t, the learner gets to observe her loss (namely, `t (it )) and nothing else. In contrast with the standard MAB setting, in addition to the loss `t (it ) the learner suffers an additional cost due to her movement between actions, which is determined by the metric and is equal to ?(it , it?1 ). Thus, the total cost at round t is given by `t (it ) + ?(it?1, it ). 3 The goal of the learner, over the course of T rounds of the game, is to minimize her expected movement-regret, which is defined as the difference between her (expected) total costs and the total costs of the best fixed action in hindsight (that incurs no movement costs); namely, the movement regret with respect to a sequence `1:T of loss vectors and a metric ? equals " # T T T X X X RegretMC (`1:T , ?) = E `t (it ) + ?(it , it?1 ) ? min `t (i) . t=1 t=2 i ?K t=1 Here, the expectation is taken with respect to the learner?s randomization in choosing the actions i1, . . . , iT ; notice that, as we assume an oblivious adversary, the loss functions `t are deterministic and cannot depend on the learner?s randomization. 2.2 Basic Definitions in Metric Spaces We recall basic notions in metric space that govern the regret in the MAB with movement costs setting. Throughout we assume a bounded metric space (K, ?), where for normalization we assume ?(i, j) ? [0, 1] for all i, j ? K. Given a point i ? K we will denote by B (i) = { j ? K : ?(i, j) ?  } the ball of radius  around i. The following definitions are standard. Definition 1 (Packing numbers). A subset P ? K in a metric space (K, ?) is an -packing if the sets p {B (i)}i ?P are disjoint sets. The -packing number of ?, denoted N (?), is the maximum cardinality of any -packing of K. Definition 2 (Covering numbers). A subset C ? K in a metric space (K, ?) is an -covering if K ? ?i ?C B (i). The -covering number of K, denoted Nc (?), is the minimum cardinality of any -covering of K. Tree metrics and HSTs. We recall the notion of a tree metric, and in particular, a metric induced by an Hierarchically well-Separated (HST) Tree; see [7] for more details. Any weighted tree defines a metric over the vertices, by considering the shortest path between each two nodes. An HST tree (2-HST tree, to be precise) is a rooted weighted tree such that: 1) the edge weight from any node to each of its children is the same and 2) the edge weight along any path from the root to a leaf are decreasing by a factor 2 per edge. We will also assume that all leaves are of the same depth in the tree (this does not imply that the tree is complete). Given a tree T we let depth(T ) denote its height, which is the maximal length of a path from any leaf to the root. Let level(v) be the level of a node v ? T , where the level of the leaves is 0 and the level of the root is depth(T ). Given nodes u, v ? T , let LCA(u, v) be their least common ancestor node in T . The metric which we next define is equivalent (up to a constant factor) to standard tree?metric induced over the leaves by an HST. By a slight abuse of terminology, we will call it HST metric: Definition 3 (HST metric). Let K be a finite set and let T be a tree whose leaves are at the same depth and are indexed by elements of K. Then the HST metric ? T over K induced by the tree T is defined as follows: ? T (i, j) = 2level(LCA(i, j)) 2depth(T) ? i, j ? K. For a HST metric ? T , observe that the packing number and covering number are simple to characterize: for all 0 ? h < depth(T ) we have that for  = 2h?H , p Nc (? T ) = N (? T ) = {v ? T : level(v) = h} . Complexity measures for finite metric spaces. We next define the two notions of complexity that, as we will later see, governs the complexity of MAB with metric movement costs. Definition 4 (covering complexity). The covering complexity of a metric space (K, ?) denoted Cc (?) is given by Cc (?) = sup  ? Nc (?). 0< <1 4 Definition 5 (packing complexity). The packing complexity of a metric space (K, ?) denoted Cp (?) is given by p Cp (?) = sup  ? N (?). 0< <1 For a HST metric, the two complexity measures coincide as its packing and covering numbers are the same. Therefore, for a HST metric ? T we will simply denote the complexity of (K, ? T ) as C(T ). In p fact, it is known that in any metric space N (?) ? Nc (?) ? Np/2 (?) for all  > 0. Thus, for a general metric space we obtain that Cp (?) ? Cc (?) ? 2Cp (?). (1) Complexity measures for infinite metric spaces. For infinite metric spaces, we require the following definition. Definition 6 (Minkowski dimensions). Let (K, ?) be a bounded metric space. The upper Minkowski dimension of (K, ?), denoted D(?), is defined as D(?) = lim sup  ?0 p log Nc (?) log N (?) = lim sup . log(1/) log(1/)  ?0 Similarly, the lower Minkowski dimension is denoted by D(?) and is defined as D(?) = lim inf  ?0 p log Nc (?) log N (?) = lim inf .  ?0 log(1/) log(1/) We refer to [31] for more background on the Minkowski dimensions and related notions in metric spaces theory. 3 Main Results We now state the main results of the paper, which give a complete characterization of the expected regret in the MAB with movement costs problem. 3.1 Finite Metric Spaces The following are the main results of the paper. Theorem 7 (Upper Bound). Let (K, ?) be a finite metric space over |K | = k elements with diameter ? 1 and covering complexity Cc = Cc (?). There exists an algorithm such that for any sequence of loss functions `1, . . . , `T guarantees that ?   e max Cc1/3T 2/3, kT . RegretMC (`1:T , ?) = O Theorem 8 (Lower Bound). Let (K, ?) be a finite metric space over |K | = k elements with diameter ? 1 and packing complexity Cp = Cp (?). For any algorithm there exists a sequence `1, . . . , `T of loss functions such that ?   e max Cp1/3T 2/3, kT . RegretMC (`1:T , ?) = ? For the detailed proofs, see the full version of the paper [25]. Recalling Eq. (1), we see that the regret bounds obtained in Theorems 7 and 8 are matching up to logarithmic factors. Notice that the tightness is achieved per instance; namely, for any given metric we are able to fully characterize the regret?s rate of growth as a function of the intrinsic properties of the metric. (In particular, this is substantially stronger than demonstrating a specific metric for which the upper bound cannot be improved.) Note that for the lower bound statement in Theorem 8 we require that the diameter of K is bounded away from zero, where for simplicity we assume a constant bound of 1. Such an assumption is necessary to avoid degenerate metrics. Indeed, when the diameter is very small, the problem?reduces to the standard MAB setting without any additional costs and we obtain a regret rate of ?( kT). Notice how the above results extend known instances of the problem from previous work: for uniform movement costs (i.e., unit switching costs) over K = {1, . . . , k} we have Cc = ?(k), so that the 5 ? 1/3T 2/3, kT }), which recovers the results in [2, 12]; and for a 2-HST binary e obtain bound is ?(max{k ? 2/3, kT }), which e balanced tree with k leaves, we have Cc = ?(1) and the resulting bound is ?(max{T is identical to the bound proved in [24]. The 2-HST regret bound in [24] was primarily used to obtain regret bounds for the action space K = [0, 1]. In the next section we show how this technique is extended for infinite metric space to obtain regret bounds that depend on the dimensionality of the action space. 3.2 Infinite Metric Spaces When (K, ?) is an infinite metric space, without additional constraints on the loss functions, the problem becomes ill-posed with a linear regret rate, even without movement costs. Therefore, one has to make additional assumptions on the loss functions in order to achieve sublinear regret. One natural assumption, which is common in previous work, is to assume that the loss functions `1, . . . , `T are all 1-Lipschitz with respect to the metric ?. Under this assumption, we have the following result. Theorem 9. Let (K, ?) be a metric space with diameter ? 1 and upper Minkowski dimension d = D(?), such that d ? 1. There exists a strategy such that for any sequence of loss functions `1, . . . , `T , which are all 1-Lipschitz with respect to ?, guarantees that d+1  e T d+2 RegretMC (`1:T , ?) = O . We refer the full version of the paper [25] for a proof of the theorem. Again, we observe that the above result extend the case of K = [0, 1] where d = 1. Indeed, for Lipschitz functions over the interval a e 2/3 ) was achieved in [24], which is exactly the bound we obtain above. tight regret bound of ?(T e d+1 d+2 ) is known for MAB in metric spaces with Lipschitz cost We mention that a lower bound of ?(T functions?even without movement costs?where d = D(?) is the lower Minkowski dimension. Theorem 10 (Bubeck et al. [10]). Let (K, ?) be a metric space with diameter ? 1 and lower Minkowski dimension d = D(?), such that d ? 1. Then for any learning algorithm, there exists a sequence of loss function `1, . . . , `d+1 are all 1-Lipschitz with respect to ?, such that the regret (without T , which  e T d+2 . movement costs) is ? In many natural metric spaces in which the upper and lower Minkowski dimensions coincide (e.g., normed spaces), the bound of Theorem 9 is tight up to logarithmic factors in T. In particular, and quite surprisingly, we see that the movement costs do not add to the regret of the problem! It is important to note that Theorem 9 holds only for metric spaces whose (upper) Minkowski dimension is at least 1. Indeed, finite metric spaces are of Minkowski dimension zero, and as we ? demonstrated in Section 3.1 above, a O( T) regret bound is not achievable. Finite matric spaces are associated with a complexity measure which is very different from the Minkowski dimension (i.e., the covering/packing complexity). In other words, we exhibit a phase transition between dimension d = 0 and d ? 1 in the rate of growth of the regret induced by the metric. 4 Algorithms In this section we turn to prove Theorem 7. Our strategy is much inspired by the approach in [24], and we employ a two-step approach: First, we consider the case that the metric is a HST metric; we then turn to deal with general metrics, and show how to upper-bound any metric with a HST metric. 4.1 Tree Metrics: The Slowly-Moving Bandit Algorithm In this section we analyze the simplest case of the problem, in which the metric ? = ? T is induced by a HST tree T (whose leaves are associated with actions in K). In this case, our main tool is the Slowly-Moving Bandit (SMB) algorithm [24]: we demonstrate how it can be applied to general tree metrics, and analyze its performance in terms of intrinsic properties of the metric. We begin by reviewing the SMB algorithm. In order to present the algorithm we require few additional notations. The algorithm receives as input a tree structure over the set of actions K, and its operation depends on the tree structure. We fix a HST tree T and let H = depth(T ). For any level 0 ? h ? H and action i ? K, let Ah (i) be the set of leaves of T that share a common ancestor with i at level h 6 (recall that level h = 0 is the bottom?most level corresponding to the singletons). In terms of the tree metric we have that Ah (i) = { j : ? T (i, j) ? 2?H+h }. The SMB algorithm is presented in Algorithm 1. The algorithm is based on the multiplicative update method, in the spirit of Exp3 algorithms [4]. Similarly to Exp3, the algorithm computes at each round t an estimator `et to the loss vector `t using the single loss value `t (it ) observed. In addition to being an (almost) unbiased estimate for the true loss vector, the estimator `et used by SMB has the additional property of inducing slowly-changing sampling distributions pt : This is done by choosing at random a level ht of the tree to be rebalanced (in terms of the weights maintained by the algorithm): As a result, the marginal probabilities pt+1 (Aht (i)) are not changed at round t. In turn, and in contrast with Exp3, the algorithm choice of action at round t + 1 is not purely sampled from pt , but rather conditioned on our last choice of level ht . This is informally justified by the fact that pt and pt+1 agree on the marginal distribution of Aht (it ), hence we can think of the level drawn at round t as if it were drawn subject to pt+1 (Aht ) = pt (Aht ). Input: A tree T with a set of finite leaves K, ? > 0. Initialize: H = depth(T ), Ah (i) = B2?H +h (i), ?i ? K, 0 ? h ? H Initialize p1 = unif(K), h0 = H and i0 ? p1 For t = 1, . . . , T: (1) Choose action it ? pt ( ? | Aht ?1 (it?1 )), observe loss `t (it ) (2) Choose ?t,0, . . . , ?t, H?1 ? {?1} uniformly at random; let ht = min{0 ? h ? H : ?t,h < 0} where ?t, H = ?1 (3) Compute vectors `?t,0, . . . , `?t, H?1 recursively via 1{it = i} `?t,0 (i) = `t (it ), pt (i) and for all h ? 1: 1 `?t,h (i) = ? ln ? X j ? Ah (i) pt ( j) ??(1+?t, h?1 )`?t, h?1 (j) e pt (Ah (i)) ! (4) Define Et = {i : pt (Ah (i)) < 2h ? for some 0 ? h < H} and set: ( 0 if it ? Et ; e `t = P H?1 ? ? `t,0 + h=0 ?t,h `t,h otherwise (5) Update: pt+1 (i) = P k pt (i) e?? `t (i) j=1 e pt ( j) e?? `et (j) ?i ? K Algorithm 1: The SMB algorithm. A key observation is that by directly applying SMB to the metric ? T , we can achieve the following regret bound: Theorem 11. Let (K, ? T ) be a metric space defined by a 2-HST T with depth(T ) = H and complexity C(T ) = C. Using SMB algorithm we can achieve the following regret bound:  p  RegretMC (`1:T , ? T ) = O H 2 H T Clog C + H2?H T . (2) To show Theorem 11, we adapt the analysis of [24] (that applies only to complete binary HSTs) to handle more general HSTs. We defer this part of our analysis to the full version of the paper [25], since it follows from a technical modification of the original proof. For a tree that is either too deep or too shallow, Eq. (2) may not necessarily lead to a sublinear regret bound, let alone optimal. The main idea behind achieving optimal regret bound for a general tree, is to modify it until one of two things happen: Either we have optimized the depth so that the two terms in the left-hand side of Eq. (2) are of same order: In that case, we will show that one can achieve 7 regret rate of order O(C(T )1/3T 2/3 ). If we fail to do?that, we show that the first term in the left-hand side is the dominant one, and it will be of order O( kT). For trees that are in some sense ?well behaved" we have the following Corollary of Theorem 11. Corollary 12. Let (K, ? T ) be a metric space defined by a tree T over |K | = k leaves with depth(T ) = H and complexity C(T ) = C. Assume that T satisfies the following: ? (1) 2?H HT ? 2 H HCT; (2) One of the following is true: (a) 2 H C ? k; p (b) 2?(H?1) (H ? 1)T ? 2 H?1 (H ? 1)CT. ?   e max C 1/3T 2/3, kT . Then, the SMB algorithm can be used to attain RegretMC (`1:T , ? T ) = O The following establishes Theorem 7 for the special case of tree metrics. Lemma 13. For any tree T and time horizon T, there exists a tree T 0 (over the same set K of k leaves) that satisfies the conditions of Corollary 12, such that ? T 0 ? ? T and C(T 0) = C(T ). Furthermore, T 0 can be constructed efficiently from T (i.e., in time polynomial in |K| and T). Hence, ? applying  e max C(T )1/3T 2/3, kT . SMB to the metric space (K, ? T 0 ) leads to RegretMC (`1:T , ? T ) = O We refer to [25] for the proofs of both results. 4.2 General Finite Metrics Finally, we obtain the general finite case as a corollary of the following. Lemma 14. Let (K, ?) be a finite metric space. There exists a tree metric ? T over K (with |K | = k) such that 4? T , dominates ? (i.e., such that 4? T (i, j) ? ?(i, j) for all i, j ? K) for which C(T ) = O(Cc (?) log k). Furthermore, T can be constructed efficiently. Proof. Let H be such that the minimal distance in ? is larger than 2?H . For each r = 2?1, 2?2, . . . , 2?H we let {Br (i {1,r } ), . . . , Br (i {mr ,r } )} = Br be a covering of K of size Nrc (T ) log k using balls of radius r. Note that finding a minimal set of balls of radius r that covers K is exactly the set cover problem. Hence, we can efficiently approximate it (to within a O(log k) factor) and construct the sets Br . We now construct a tree graph, whose nodes are associated with the cover balls: The leaves correspond to singleton balls, hence correspond to the action space. For each leaf i we find an action a1 (i) ? K such that: i ? B2?H +1 (a1 (i)) ? B2?H +1 . If there is more than one, we arbitrarily choose one, and we connect an edge between i and B2?H +1 (a1 (i)). We continue in this manner inductively to define ar (i) for every a and r < 1: given ar?1 (i) we find an action ar (i) such that ar?1 (i) ? B2?H +r (ar (i)) ? B2?H +r , and we connect an edge from B2?H +r ?1 (ar?1 (i)) and B2?H +r (ar (i)). We now claim that the metric induced by the tree graph dominates up to factor 4 the original metric. Let i, j ? K such that ? T (i, j) < 2?H+r then by construction there are i, a1 (i), a2 (i), . . . ar (i) and j, a1 ( j), a2 ( j), . . . ar ( j), such that ar (i) = ar ( j) and for which it holds that ?(as (i), as?1 (i)) ? 2?H+s and similarly ?(as ( j), as?1 ( j)) ? 2?H+s for every s ? r. Denoting a0 (i) = i and a0 ( j) = j, we have that r r X X ?(i, j) ? ?(as?1 (i), as (i)) + ?(as?1 ( j), as ( j)) s=1 r X ?2 s=1 2?H+s ? 2?2?H ?2r+1 ? 4? T (i, j).  s=1 4.3 Infinite Metric Spaces Finally, we address infinite spaces by discretizing the space K and reducing to the finite case. Recall that in this case we also assume that the loss functions are Lipschitz. Proof of Theorem 9. Given the definition of the covering dimension d = D(?) ? 1, it is straightforward that for some constant C > 0 (that might depend on the metric ?) it holds that Nrc (?) ? Cr ?d for 8 all r > 0. Fix some  > 0, and take a minimal 2-covering K 0 of K of size |K 0 | ? C(2)?d ? C ?d . Observe that by restricting the algorithm to pick actions from K 0, we might lose at most O(T) in the regret. Also, since K 0 is minimal, the distance between any two elements in K 0 is at least , thus the covering complexity of the space has Cc (?) = sup r ? Nrc (?) ? C sup r ?d+1 ? C ?d+1, r ? r ? as we assume that d ? 1. Hence, by Theorem 7 and the Lipschitz assumption, there exists an algorithm for which    2 d 1 e max  ? d?1 3 T 3 ,  ? 2 T 2 , T . RegretMC (`1:T , ?) = O 1 e d+1 d+2 ) A simple computation reveals that  = ?(T ? d+2 ) optimizes the above bound, and leads to O(T movement regret.  Acknowledgements RL is supported in funds by the Eric and Wendy Schmidt Foundation for strategic innovations. YM is supported in part by a grant from the Israel Science Foundation, a grant from the United States-Israel Binational Science Foundation (BSF), and the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). References [1] R. Agrawal, M. V. Hegde, and D. Teneketzis. Asymptotically efficient adaptive allocation rules for the multiarmed bandit problem with switching costs. IEEE Transactions on Optimal Control, 33(10):899?906, 1988. [2] R. Arora, O. Dekel, and A. Tewari. Online bandit learning against an adaptive adversary: from regret to policy regret. In Proceedings of the 29th International Conference on Machine Learning (ICML-12), pages 1503?1510, 2012. [3] M. Asawa and D. Teneketzis. Multi-armed bandits with switching penalties. IEEE Transactions on Automatic Control, 41(3):328?348, 1996. [4] P. Auer, N. Cesa-Bianchi, Y. Freund, and R. E. Schapire. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing, 32(1):48?77, 2002. [5] P. Auer, R. 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[18] S. Guha and K. Munagala. Multi-armed bandits with metric switching costs. In International Colloquium on Automata, Languages, and Programming, pages 496?507. Springer, 2009. [19] A. Gyorgy and G. Neu. Near-optimal rates for limited-delay universal lossy source coding. IEEE Transactions on Information Theory, 60(5):2823?2834, 2014. [20] T. Jun. A survey on the bandit problem with switching costs. De Economist, 152(4):513?541, 2004. [21] R. Kleinberg and A. Slivkins. Sharp dichotomies for regret minimization in metric spaces. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 827?846. Society for Industrial and Applied Mathematics, 2010. [22] R. Kleinberg, A. Slivkins, and E. Upfal. Multi-armed bandits in metric spaces. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 681?690. ACM, 2008. [23] R. D. Kleinberg. Nearly tight bounds for the continuum-armed bandit problem. In Advances in Neural Information Processing Systems, pages 697?704, 2004. [24] T. Koren, R. Livni, and Y. Mansour. Bandits with movement costs and adaptive pricing. In COLT, 2017. [25] T. Koren, R. Livni, and Y. Mansour. Multi-armed bandits with metric movement costs. arXiv preprint arXiv:1710.08997, 2017. [26] S. Magureanu, R. Combes, and A. Proutiere. Lipschitz bandits: Regret lower bound and optimal algorithms. In COLT, pages 975?999, 2014. [27] G. Neu, A. Gy?rgy, C. Szepesv?ri, and A. Antos. Online markov decision processes under bandit feedback. IEEE Trans. Automat. Contr., 59(3):676?691, 2014. [28] R. Ortner. Online regret bounds for markov decision processes with deterministic transitions. Theor. Comput. Sci., 411(29-30):2684?2695, 2010. [29] A. Slivkins. Multi-armed bandits on implicit metric spaces. In Advances in Neural Information Processing Systems, pages 1602?1610, 2011. [30] A. Slivkins, F. Radlinski, and S. Gollapudi. 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Learning A Structured Optimal Bipartite Graph for Co-Clustering 1 Feiping Nie1 , Xiaoqian Wang2 , Cheng Deng3 , Heng Huang2? School of Computer Science, Center for OPTIMAL, Northwestern Polytechnical University, China 2 Department of Electrical and Computer Engineering, University of Pittsburgh, USA 3 School of Electronic Engineering, Xidian University, China [email protected],[email protected] [email protected],[email protected] Abstract Co-clustering methods have been widely applied to document clustering and gene expression analysis. These methods make use of the duality between features and samples such that the co-occurring structure of sample and feature clusters can be extracted. In graph based co-clustering methods, a bipartite graph is constructed to depict the relation between features and samples. Most existing co-clustering methods conduct clustering on the graph achieved from the original data matrix, which doesn?t have explicit cluster structure, thus they require a post-processing step to obtain the clustering results. In this paper, we propose a novel co-clustering method to learn a bipartite graph with exactly k connected components, where k is the number of clusters. The new bipartite graph learned in our model approximates the original graph but maintains an explicit cluster structure, from which we can immediately get the clustering results without post-processing. Extensive empirical results are presented to verify the effectiveness and robustness of our model. 1 Introduction Clustering has long been a fundamental topic in unsupervised learning. The goal of clustering is to partition data into different groups. Clustering methods have been successfully applied to various areas, such as document clustering [3, 17], image segmentation [18, 7, 8] and bioinformatics [16, 14]. In clustering problems, the input data is usually formatted as a matrix, where one dimension represents samples and the other denotes features. Each sample can be seen as a data point characterized by a vector in the feature space. Alternatively, each feature can be regarded as a vector spanning in the sample space. Traditional clustering methods propose to cluster samples according to their distribution on features, or conversely, cluster features in terms of their distribution on samples. In several types of data, such as document data and gene expression data, duality exists between samples and features. For example, in document data, we can reasonably assume that documents can be clustered based on their relations with different word clusters, while word clusters are formed according to their associations with distinct document clusters. However, in the one-sided clustering mechanism, the duality between samples and features is not taken into consideration. To make full use of the duality information, co-clustering methods (also known as bi-clustering methods) are proposed. The co-clustering mechanism takes advantage of the co-occurring cluster structure among features and samples to strengthen the clustering performance and gain better interpretation of the pragmatic meaning of the clusters. ? This work was partially supported by U.S. NSF-IIS 1302675, NSF-IIS 1344152, NSF-DBI 1356628, NSF-IIS 1619308, NSF-IIS 1633753, NIH AG049371. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Several co-clustering methods have been put forward to depict the relations between samples and features. In the graph based methods, the co-occurring structure between samples and features is usually treated as a bipartite graph, where the weights of edges indicate the relations between sample-feature pairs. In the left part of Fig. 1 we show an illustration of such bipartite graph, where the blue nodes on the left represent features while red nodes on the right show samples. The affinity between the features and samples is denoted by the weight of the corresponding edge. For example, Bij denotes the affinity between the i-th feature and the j-sample. In [4], the authors propose to minimize the cut between samples and features, which is equivalent to conducting spectral clustering on the bipartite graph. However, in this method, since the original graph doesn?t display an explicit cluster structure, it still calls for the post-processing step like K-mean clustering to obtain the final clustering indicators, which may not be optimal. To address this problem, in this paper, we propose a novel graph based co-clustering model to learn a bipartite graph with exactly k connected components, where k is the number of clusters. The new bipartite graph learned in our model approximates the original graph but maintains an explicit cluster structure, from which we can directly get the clustering results without post-processing steps. To achieve such an ideal structure of the new bipartite graph, we impose constraints on the rank of its Laplacian or normalized Laplacian matrix and derive algorithms to optimize the objective. We conduct several experiments to evaluate the effectiveness and robustness of our model. On both synthetic and benchmark datasets we gain equivalent or even better clustering results than other related methods. Notations: Throughout the paper, all the matrices are written as uppercase. For matrix M , the ij-th element of M is denoted by mij . The trace of matrix M is denoted by T r(M ). The `2 -norm of vector v is denoted by kvk2 , the Frobenius norm of matrix M is denoted by kM kF . 2 Bipartite Spectral Graph Partitioning Revisited The classic Bipartite Spectral Graph Partitioning (BSGP) method [4] is very effective for co-clustering. In order to simultaneously partition the rows and columns of a data matrix B ? Rn1 ?n2 , we first view B as the weight matrix of a bipartite graph, where the left-side nodes are the n1 rows of B, the right-side nodes are the n2 columns of B, and the weight to connect the i-th left-side node and the j-th right-side node is bij (see Fig.1). The procedure of BSGP is as follows: 1 1 ? ? 1) Calculate A? = Du 2 BDv 2 , where the diagonal matrices Du and Dv are defined in Eq.(6). ? respectively. 2) Calculate U and V , which are the leading k left and right singular vectors of A, 3) Run the K-means on the rows of F defined in Eq. (6) to obtain the final clustering results. The bipartite graph can be viewed as an undirected weighted graph G = {V, A} with n = n1 + n2 nodes, where V is the node set and the affinity matrix A ? Rn?n is   0 B A= (1) BT 0 In the following, we will show that the BSGP method essentially performs spectral clustering with normalized cut on the graph G. Suppose the graph G is partitioned into k components V = {V1 , V2 , ..., Vk } . According to the spectral clustering, the normalized cut on the graph G = {V, A} is defined as Ncut = k X cut(Vi , V\Vi ) assoc(Vi , V) P assoc(Vi , V) = i?Vi ,j?V aij . (2) i=1 where cut(Vi , V\Vi ) = P i?Vi ,j?V\Vi aij ; Let Y ? Rn?k be the partition indicator matrix, i.e., yij = 1 indicates the i-th node is partitioned into the j-th component. Then minimizing the normalized cut defined in Eq.(2) can be rewritten as the following problem: k X yiT Lyi min (3) Y y T Dyi i=1 i 2 Figure 1: Illustration of the structured optimal bipartite graph. where yi is the i-th column of Y , L = D P ? A ? Rn?n is the Laplacian matrix, and D ? Rn?n is the diagonal degree matrix defined as dii = j aij . 1 Let Z = Y (Y T DY )? 2 , and denote the identity matrix by I, then problem (3) can be rewritten as min Z T DZ=I 1 1 T r(Z T LZ) (4) 1 Further, denotes F = D 2 Z = D 2 Y (Y T DY )? 2 , then the problem (4) can be rewritten as ? ) min T r(F T LF F T F =I 1 (5) 1 ? = I ? D? 2 AD? 2 is the normalized Laplacian matrix. where L We rewrite F and D as the following block matrices:    U Du F = ,D = V  Dv (6) where U ? Rn1 ?k , V ? Rn2 ?k , Du ? Rn1 ?n1 , Dv ? Rn2 ?n2 . Then according to the definition of A in Eq. (1), the problem (5) can be further rewritten as max ?1 U T U +V T V =I ?1 T r(U T Du 2 BDv 2 V ) (7) Note that in addition to the constraint U T U + V T V = I, the U, V should be constrained to be discrete values according to the definitions of U and V . This discrete constraint makes the problem very difficult to solve. To address it, we first remove the discrete constraint to make the problem (7) solvable with Lemma 1 , and then run K-means on U and V to get the discrete solution. Lemma 1 Suppose M ? Rn1 ?n2 , X ? Rn1 ?k , Y ? Rn2 ?k . The optimal solutions to the problem max X T X+Y T Y =I ? are X = 22 U1 , Y = respectively. T r(X T M Y ) (8) ? 2 2 V1 , where U1 , V1 are the leading k left and right singular vectors of M , Proof: Denote the Lagrangian function of the problem is L(X, Y, ?) = T r(X T AY )?T r(?(X T X + Y T Y ? I)) By setting the derivative of L(X, Y, ?) w.r.t. X to zero, we have AY = X?. By taking the derivative of L(X, Y, ?) w.r.t. Y to zero, we have AT X = Y ?. Thus AAT X = AY ? = X?2 . Therefore, the optimal solution X should be the eigenvectors of AAT , i.e, the left singular vectors of M . Similarly, the optimal solution Y should be the right singular vectors of M . Since it is a maximization problem, the optimal solution X, Y should be the leading k left and right singular vectors of M , respectively.  According to Lemma 1, if the discrete constraint on U and V is not considered, the optimal solution ?1 ?1 U and V to the problem (7) are the leading k left and right singular vectors of A? = Du 2 BDv 2 , respectively. Since the solution U and V are not discrete values, we need to run the K-means on the rows of F defined in Eq.(6) to obtain the final clustering results. 3 3 3.1 Learning Structured Optimal Bipartite Graph for Co-Clustering Motivation We can see from the previous section that the given B or A does not have a very clear clustering structure (i.e., A is not a block diagonal matrix with proper permutation) and the U and V are not discrete values, thus we need run the K-means to obtain the final clustering results. However, K-means is very sensitive to the initialization, which makes the clustering performance unstable and suboptimal. To address this challenging and fundamental problem, we target to learn a new graph similarity matrix S ? Rn?n or P ? Rn1 ?n2 as   0 P S= , (9) PT 0 such that the new graph is more suitable for clustering task. In our strategy, we learn an S that has exact k connected components, see Fig. 1. Obviously such a new graph can be considered as the ideal graph for clustering task with providing clear clustering structure. If S has exact k connected components, we can directly obtain the final clustering result based on S, without running K-means or other discretization procedures as traditional graph based clustering methods have to do. The learned structured optimal graph similarity matrix S should be as close as possible to the given graph affinity matrix A, so we propose to solve the following problem: 2 min P ?0,P 1=1,S?? kS ? AkF (10) where ? is the set of matrices S ? Rn?n which have exact k connected components. According to the special structure of A and S in Eq. (1) and Eq. (9), the problem (10) can be written as 2 min kP ? BkF (11) P ?0,P 1=1,S?? The problem (11) seems very difficult to solve since the constraint S ? ? is intractable to handle. In the next subsection, we will propose a novel and efficient algorithm to solve this problem. 3.2 Optimization If the similarity matrix S is nonnegative, then the Laplacian matrix LS = DS ? S associated with S has an important property as follows [13, 12, 11, 2]. Theorem 1 The multiplicity k of the eigenvalue 0 of the Laplacian matrix LS is equal to the number of connected components in the graph associated with S. Theorem 1 indicates that if rank(LS ) = n ? k, the constraint S ? ? will be held. Therefore, the problem (11) can be rewritten as: 2 min P ?0,P 1=1,rank(LS )=n?k kP ? BkF (12) Suppose ?i (LS ) is the i-th smallest eigenvalue of LS . Note that ?i (LS ) ? 0 because LS is positive semi-definite. The problem (12) is equivalent to the following problem for a large enough ?: min P ?0,P 1=1 2 kP ? BkF + ? k X ?i (LS ) (13) i=1 When ? is large enough (note that ?i (LS ) ? 0 for every i), the optimal solution S to the problem Pk (13) will make the second term i=1 ?i (LS ) to be zero, and thus the constraint rank(LS ) = n ? k in the problem (12) would be satisfied. According to the Ky Fan?s Theorem [6], we have: k X i=1 ?i (LS ) = min F ?Rn?k ,F T F =I 4 T r(F T LS F ) (14) Therefore, the problem (13) is further equivalent to the following problem 2 min kP ? BkF + ?T r(F T LS F ) (15) s.t. P ? 0, P 1 = 1, F ? Rn?k , F T F = I The problem (15) is much easier to solve compared with the rank constrained problem (12). We can apply the alternating optimization technique to solve this problem. P,F When P is fixed, the problem (15) becomes: min F ?Rn?k ,F T F =I T r(F T LS F ) (16) The optimal solution F is formed by the k eigenvectors of LS corresponding to the k smallest eigenvalues. When F is fixed, the problem (15) becomes min P ?0,P 1=1 2 kP ? BkF + ?T r(F T LS F ) According to the property of Laplacian matrix, we have the following relationship: n n 1 XX 2 T r(F T LS F ) = kfi ? fj k2 sij 2 i=1 j=1 (17) (18) where fi is the i-th row of F . Thus according to the structure of S defined in Eq.(9), Eq.(18) can be rewritten as n2 n1 X X 2 kfi ? fj k2 pij T r(F T LS F ) = (19) i=1 j=1 Based on Eq. (19), the problem (17) can be rewritten as n1 X n2 X 2 2 (pij ? bij ) + ?kfi ? fj k2 pij min P ?0,P 1=1 (20) i=1 j=1 Note that the problem (20) is independent between different i, so we can solve the following problem 2 individually for each i. Denote vij = kfi ? fj k2 , and denote vi as a vector with the j-th element as vij (same for pi and bi ), then for each i, the problem (20) can be written in the vector form as 2 pi ? (bi ? ? vi ) min (21) T 2 2 pi 1=1,pi ?0 This problem can be solved by an efficient iterative algorithm [9]. The detailed algorithm to solve the problem (15) is summarized in Algorithm 1. In the algorithm, we can only update the m nearest similarities for each data points in P and thus the complexity of updating P and updating F (only need to compute top k eigenvectors on very sparse matrix) can be reduced significantly. Nevertheless, Algorithm 1 needs to conduct eigen-decomposition on an n ? n(n = n1 + n2 ) matrix in each iteration, which is time consuming. In the next section, we will propose another optimization algorithm, which only needs to conduct SVD on an n1 ? n2 matrix in each iteration, and thus is much more efficient than Algorithm 1. Algorithm 1 Algorithm to solve the problem (15). input B ? Rn1 ?n2 , cluster number k, a large enough ?. output P ? Rn1 ?n2 and thus S ? Rn?n defined in Eq.(9) with exact k connected components. Initialize F ? Rn?k , which is formed by the k eigenvectors of L = D ? A corresponding to the k smallest eigenvalues, A is defined in Eq. (1). while not converge do 1. For each i, update the i-th row of P by solving the problem (21), where the j-th element of 2 vi is vij = kfi ? fj k2 . 2. Update F , which is formed by the k eigenvectors of LS = DS ? S corresponding to the k smallest eigenvalues. end while 5 4 Speed Up the Model 1 1 ? S = I ?D? 2 SD? 2 If the similarity matrix S is nonnegative, then the normalized Laplacian matrix L S S associated with S also has an important property as follows [11, 2]. ? S is equal to Theorem 2 The multiplicity k of the eigenvalue 0 of the normalized Laplacian matrix L the number of connected components in the graph associated with S. ? S ) = n ? k, the constraint S ? ? will be hold. Therefore, the Theorem 2 indicates that if rank(L problem (11) can also be rewritten as 2 min ? S )=n?k P ?0,P 1=1,rank(L kP ? BkF (22) Similarly, the problem (22) is equivalent to the following problem for a large enough value of ?: 2 ?S F ) min kP ? BkF + ?T r(F T L P,F (23) s.t. P ? 0, P 1 = 1, F ? Rn?k , F T F = I Again, we can apply the alternating optimization technique to solve problem (23). 1 1 ? S = I ? D? 2 SD? 2 , the problem (23) becomes When P is fixed, since L S S max ?1 F ?Rn?k ,F T F =I ?1 T r(F T DS 2 SDS 2 F ) We rewrite F and DS as the following block matrices:    U DSu F = , DS = V (24)  (25) DSv where U ? Rn1 ?k , V ? Rn2 ?k , DSu ? Rn1 ?n1 , DSv ? Rn2 ?n2 . Then according to the definition of S in Eq. (9), the problem (24) can be further rewritten as max U T U +V T V =I ?1 ?1 T r(U T DSu2 P DSv2 V ) (26) According to Lemma 1, the optimal solution U and V to the problem (26) are the leading k left and ?1 ?1 right singular vectors of S? = DSu2 P DSv2 , respectively. When F is fixed, the problem (23) becomes 2 ?S F ) min kP ? BkF + ?T r(F T L P s.t. (27) P ? 0, P 1 = 1 According to the property of normalized Laplacian matrix, we have the following relationship: 2 n n 1 XX fj fi T? T r(F LS F ) = (28) ? ? p sij 2 i=1 j=1 di dj 2 2 f fj i ,the problem ? ? Thus according to the structure of S defined in Eq.(9), and denote vij = d ? i dj 2 (27) can be rewritten as min P ?0,P 1=1 n1 X n2 X 2 (pij ? bij ) + ?vij pij , i=1 j=1 which has the same form as in Eq. (20) and thus can be solved efficiently. The detailed algorithm to solve the problem (23) is summarized in Algorithm 2. In the algorithm, we can also only update the m nearest similarities for each data points in P and thus the complexity of updating P and updating F can be reduced significantly. 6 Note that Algorithm 2 only needs to conduct SVD on an n1 ? n2 matrix in each iteration. In some cases, min(n1 , n2 )  (n1 + n2 ), thus Algorithm 2 is much more efficient than Algorithm 1. Therefore, in the next section, we use Algorithm 2 to conduct the experiments. Algorithm 2 Algorithm to solve the problem (23). input B ? Rn1 ?n2 , cluster number k, a large enough ?. output P ? Rn1 ?n2 and thus S ? Rn?n defined in Eq.(9) with exact k connected components. ? = I ?D? 12 AD? 21 corresponding Initialize F ? Rn?k , which is formed by the k eigenvectors of L to the k smallest eigenvalues, A is defined in Eq. (1). while not converge do 1. For each i, update the i-th row of P by solving the problem (21), where the j-th element of 2 f fj i . vi is vij = ?d ? ? i dj   2 U 2. Update F = , where U and V are the leading k left and right singular vectors of V   ?1 ?1 DSu . S? = DSu2 P DSv2 respectively and DS = DSv end while 5 Experimental Results In this section, we conduct multiple experiments to evaluate our model. We will first introduce the experimental settings throughout the section and then present evaluation results on both synthetic and benchmark datasets. 5.1 Experimental Settings We compared our method (denoted by SOBG) with two related co-clustering methods, including Bipartite Spectral Graph Partition (BSGP) [4] and Orthogonal Nonnegative Matrix Tri-Factorizations (ONMTF) [5]. Also, we introduced several one-sided clustering methods to the comparison, which are K-means clustering, Normalized Cut (NCut) and Nonnegative Matrix Factorization (NMF). For methods requiring a similarity graph as the input, i.e., NCut and NMF, we adopted the self-tuning Gaussian method [19] to construct the graph, where the number of neighbors was set to be 5 and the ? value was self-tuned. In the experiment, there are four methods involving K-means clustering, which are K-means, NCut, BSGP and ONMTF (the latter three methods need K-means as the post-processing step to get the clustering results). When running K-means we used 100 random initializations for all these four methods and recorded the average performance over these 100 runs as well as the best one with respect to the K-means objective function value. In our method, to accelerate the algorithmic procedure, we determined the parameter ? in an heuristic way: first specify the value of ? with an initial guess; next, we computed the number of zero ? S in each iteration, if it was larger than k, then divided ? by 2; if smaller then eigenvalues in L multiplied ? by 2; otherwise we stopped the iteration. The number of clusters was set to be the ground truth. The evaluation of different methods was based on the percentage of correctly clustered samples, i.e., clustering accuracy. 5.2 Results on Synthetic Data In this subsection, we first apply our method to the synthetic data as a sanity check. The synthetic data is constructed as a two-dimensional matrix, where rows and columns come from three clusters respectively. Row clusters and column clusters maintain mutual dependence, i.e., rows and columns from the first cluster form a block along the diagonal of the data matrix, and this also holds true for the second and third cluster. The number of rows for each cluster is 20, 30 and 40 respectively, while the number of columns is 30, 40 and 50. Each block is generated randomly with elements i.i.d. sampled from Gaussian distribution N (0, 1). Also, we add noise to the ?non-block" area of the data matrix, i.e., all entries in the matrix excluding elements in the three clusters. The noise can be denoted as r ? ?, where ? is Gaussian noise i.i.d. sampled from Gaussian distribution N (0, 1) and r 7 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 (a) Noise = 0.6 (b) Noise = 0.6 (c) Noise = 0.7 (d) Noise = 0.7 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 (e) Noise = 0.8 (f) Noise = 0.8 (g) Noise = 0.9 (h) Noise = 0.9 Figure 2: Illustration of the data matrix in different settings of noise. Different rows of figures come from different settings of noise. In each row, figures on the left column are the original data matrices generated in the experiment, while on the right column display the bipartite matrix B learned in our model which approximates the original data matrix and maintains the block structure. Clustering Accuracy(%) on Rows Clustering Accuracy(%) on Columns Methods K-means NCut NMF BSGP ONMTF SOBG K-means NCut NMF BSGP ONMTF SOBG Noise = 0.6 99.17 99.17 98.33 100.00 99.17 100.00 100.00 100.00 100.00 100.00 100.00 100.00 Noise = 0.7 97.50 95.00 95.00 93.33 97.50 100.00 95.56 91.11 90.00 93.33 95.56 100.00 Noise = 0.8 71.67 46.67 46.67 62.50 71.67 98.33 51.11 60.00 47.78 63.33 51.11 100.00 Noise = 0.9 39.17 38.33 37.50 40.00 39.17 84.17 46.67 38.89 37.78 46.67 46.67 87.78 Table 1: Clustering accuracy comparison on rows and columns of the synthetic data in different portion of noise. is the portion of noise. We set r to be {0.6, 0.7, 0.8, 0.9} respectively so as to evaluate the robustness of different methods under the circumstances of various disturbance. We apply all comparing methods to the synthetic data and assess their ability to cluster the rows and columns. One-sided clustering methods are applied to the data twice (once to cluster rows and the other time to cluster columns) such that clustering accuracy on these two dimensions can be achieved. Co-clustering methods can obtain clustering results on both dimensions simultaneously in one run. In Table 1 we summarize the clustering accuracy comparison on both rows and columns under different settings of noise. In Fig. 2 we display the corresponding original data matrix and the bipartite matrix B learned in our model. We can notice that when the portion of noise r is relatively low, i.e., 0.6 and 0.7, the block structure of the original data is clear, then all methods perform fairly well in clustering both rows and columns. However, as r increases, the block structure in the original data blurs thus brings obstacles to the clustering task. With high portion of noise, all other methods seem to be disturbed to a large extent while our method shows apparent robustness. Even when the portion of noise becomes as high as 0.9, such that the structure of clusters in the original data becomes hard to distinguish with eyes, our method still excavates a reasonable block arrangement with a clustering accuracy of over 80%. Also, we can find that co-clustering methods usually outperform one-sided clustering methods since they utilize the interrelations between rows and columns. The interpretation of the co-clustering structure strengthens the performance, which conforms to our theoretical analysis. 8 Methods Ave K-means Best Ave NCut Best NMF Ave BSGP Best Ave ONMTF Best SOBG Reuters21578 40.86?4.59 32.77 26.92?0.93 29.18 30.91 11.44?0.39 11.26 17.57?1.95 27.90 43.94 LUNG 61.91?6.00 71.43 69.67?14.26 79.80 75.86 64.95?5.06 70.94 61.31?10.34 71.43 78.82 Prostate-MS 46.47?3.26 45.34 46.86?1.19 47.20 47.83 46.27?0.00 46.27 45.46?3.18 45.34 62.73 prostateCancerPSA410 64.15?9.40 62.92 55.06?0.00 55.06 55.06 57.30?0.00 57.30 62.92?0.00 62.92 69.66 Table 2: Clustering accuracy comparison on four benchmark datasets. For the four methods involving K-means clustering, i.e., K-means, NCut, BSGP and ONMTF, their average performance (Ave) over 100 repetitions and the best one (Best) w.r.t. K-means objective function value were both reported. 5.3 Results on Benchmark Data In this subsection, we use four benchmark datasets for the evaluation. There are one document dataset and three gene expression datasets participating in the experiment, the property of which is introduced in details as below. Reuters21578 dataset is processed and downloaded from http://www.cad.zju.edu.cn/ home/dengcai/Data/TextData.html. It contains 8293 documents in 65 topics. Each document is depicted by its frequency on 18933 terms. LUNG dataset [1] provides a source for the study of lung cancer. It has 203 samples in five classes, among which there are 139 adenocarcinoma (AD), 17 normal lung (NL), 6 small cell lung cancer (SMCL), 21 squamous cell carcinoma (SQ) as well as 20 pulmonary carcinoid (COID) samples. Each sample has 3312 genes. Prostate-MS dataset [15] contains a total of 332 samples from three different classes, which are 69 samples diagnosed as prostate cancer, 190 samples of benign prostate hyperplasia, as well as 63 normal samples showing no evidence of disease. Each sample has 15154 genes. ProstateCancerPSA410 dataset [10] describes gene information of patients with prostate-specific antigen (PSA)-recurrent prostate cancer. It includes a total of 89 samples from two classes. Each sample has 15154 genes. Before the clustering process, feature scaling was performed on each dataset such that features are on the same scale of [0, 1]. Also, the `2 -norm of each feature was normalized to 1. Table 2 summarizes the clustering accuracy comparison on these benchmark datasets. Our method performs equally or even better than the alternatives on all these datasets. This verifies the effectiveness of our method in the practical situation. There is an interesting phenomenon that the advantage of our method tends to be more obvious for higher dimensional data. This is because high-dimensional features make the differences in the distance between samples to be smaller thus the cluster structure of the original data becomes vague. In this case, since our model is more robust compared with the alternative methods (verified in the synthetic experiments), we can get better clustering results. 6 Conclusions In this paper, we proposed a novel graph based co-clustering model. Different from existing methods which conduct clustering on the graph achieved from the original data, our model learned a new bipartite graph with explicit cluster structure. By imposing the rank constraint on the Laplacian matrix of the new bipartite graph, we guaranteed the learned graph to have exactly k connected components, where k is the number of clusters. From this ideal structure of the new bipartite graph learned in our model, the obvious clustering structure can be obtained without resorting to post-processing steps. We presented experimental results on both synthetic data and four benchmark datasets, which validated the effectiveness and robustness of our model. 9 References [1] A. Bhattacharjee, W. G. Richards, J. Staunton, C. 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Learning Low-Dimensional Metrics Lalit Jain ? University of Michigan Ann Arbor, MI 48109 [email protected] Blake Mason ? University of Wisconsin Madison, WI 53706 [email protected] Robert Nowak University of Wisconsin Madison, WI 53706 [email protected] Abstract This paper investigates the theoretical foundations of metric learning, focused on three key questions that are not fully addressed in prior work: 1) we consider learning general low-dimensional (low-rank) metrics as well as sparse metrics; 2) we develop upper and lower (minimax) bounds on the generalization error; 3) we quantify the sample complexity of metric learning in terms of the dimension of the feature space and the dimension/rank of the underlying metric; 4) we also bound the accuracy of the learned metric relative to the underlying true generative metric. All the results involve novel mathematical approaches to the metric learning problem, and also shed new light on the special case of ordinal embedding (aka non-metric multidimensional scaling). 1 Low-Dimensional Metric Learning This paper studies the problem of learning a low-dimensional Euclidean metric from comparative judgments. Specifically, consider a set of n items with high-dimensional features xi 2 Rp and suppose we are given a set of (possibly noisy) distance comparisons of the form sign(dist(xi , xj ) dist(xi , xk )), for a subset of all possible triplets of the items. Here we have in mind comparative judgments made by humans and the distance function implicitly defined according to human perceptions of similarities and differences. For example, the items could be images and the xi could be visual features automatically extracted by a machine. Accordingly, our goal is to learn a p ? p symmetric positive semi-definite (psd) matrix K such that the metric dK (xi , xj ) := (xi xj )T K(xi xj ), where dK (xi , xj ) denotes the squared distance between molecules i and j with respect to a matrix K, predicts the given distance comparisons as well as possible. Furthermore, it is often desired that the metric is low-dimensional relative to the original high-dimensional feature representation (i.e., rank(K) ? d < p). There are several motivations for this: ? Learning a high-dimensional metric may be infeasible from a limited number of comparative judgments, and encouraging a low-dimensional solution is a natural regularization. ? Cognitive scientists are often interested in visualizing human perceptual judgments (e.g., in a two-dimensional representation) and determining which features most strongly influence human perceptions. For example, educational psychologists in [1] collected comparisons between visual representations of chemical molecules in order to identify a small set of visual features that most significantly influence the judgments of beginning chemistry students. ? It is sometimes reasonable to hypothesize that a small subset of the high-dimensional features dominate the underlying metric (i.e., many irrelevant features). ? Downstream applications of the learned metric (e.g., for classification purposes) may benefit from robust, low-dimensional metrics. ? Authors contributed equally to this paper and are listed alphabetically. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. (b) A sparse and low rank psd matrix (a) A general low rank psd matrix Figure 1: Examples of K for p = 20 and d = 7. The sparse case depicts a situation in which only some of the features are relevant to the metric. With this in mind, several authors have proposed nuclear norm and `1,2 group lasso norm regularization to encourage low-dimensional and sparse metrics as in Fig. 1b (see [2] for a review). Relative to such prior work, the contributions of this paper are three-fold: 1. We develop novel upper bounds on the generalization error and sample complexity of learning lowdimensional metrics from triplet distance comparisons. Notably, unlike previous generalization bounds, our bounds allow one to easily quantify how the feature space dimension p and rank or sparsity d < p of the underlying metric impacts the sample complexity. 2. We establish minimax lower bounds for learning low-rank and sparse metrics that match the upper bounds up to polylogarithmic factors, demonstrating the optimality of learning algorithms for the first time. Moreover, the upper and lower bounds demonstrate that learning sparse (and low-rank) metrics is essentially as difficult as learning a general low-rank metric. This suggests that nuclear norm regularization may be preferable in practice, since it places less restrictive assumptions on the problem. 3. We use the generalization error bounds to obtain model identification error bounds that quantify the accuracy of the learned K matrix. This problem has received very little, if any, attention in the past and is crucial for interpreting the learned metrics (e.g., in cognitive science applications). This is a bit surprising, since the term ?metric learning? strongly suggests accurately determining a metric, not simply learning a predictor that is parameterized by a metric. 1.1 Comparison with Previous Work There is a fairly large body of work on metric learning which is nicely reviewed and summarized in the monograph [2], and we refer the reader to it for a comprehensive summary of the field. Here we discuss a few recent works most closely connected to this paper. Several authors have developed generalization error bounds for metric learning, as well as bounds for downstream applications, such as classification, based on learned metrics. To use the terminology of [2], most of the focus has been on must-link/cannot-link constraints and less on relative constraints (i.e., triplet constraints as considered in this paper). Generalization bounds based on algorithmic robustness are studied in [3], but the generality of this framework makes it difficult to quantify the sample complexity of specific cases, such as low-rank or sparse metric learning. Rademacher complexities are used to establish generalization error bounds in the must-link/cannot-link situation in [4, 5, 6], but do not consider the case of relative/triplet constraints. The sparse compositional metric learning framework of [7] does focus on relative/triplet constraints and provides generalization error bounds in terms of covering numbers. However, this work does not provide bounds on the covering numbers, making it difficult to quantify the sample complexity. To sum up, prior work does not quantify the sample complexity of metric learning based on relative/triplet constraints in terms of the intrinsic problem dimensions (i.e., dimension p of the high-dimensional feature space and the dimension of the underlying metric), there is no prior work on lower bounds, and no prior work quantifying the accuracy of learned metrics themselves (i.e., only bounds on prediction errors, not model identification errors). Finally we mention that Fazel et a.l [8] also consider the recovery of sparse and low rank matrices from linear observations. Our situation is very different, our matrices are low rank because they are sparse - not sparse and simultaneously low rank as in their case. 2 2 The Metric Learning Problem Consider n known points X := [x1 , x2 , . . . , xn ] 2 Rp?n . We are interested in learning a symmetric positive semidefinite matrix K that specifies a metric on Rp given ordinal constraints on distances between the known points. Let S denote a set of triplets, where each t = (i, j, k) 2 S is drawn uniformly at random from the full set of n n 2 1 triplets T := {(i, j, k) : 1 ? i 6= j 6= k ? n, j < k}. For each triplet, we observe a yt 2 {?1} which is a noisy indication of the triplet constraint dK (xi , xj ) < dK (xi , xk ). Specifically we assume that each t has an associated probability qt of yt = 1, and all yt are statistically independent. c from S that predicts triplets as well as possible. Objective 1: Compute an estimate K In many instances, our triplet measurements are noisy observations of triplets from a true positive semi-definite matrix K ? . In particular we assume qt > 1/2 () dK ? (xi , xj ) < dK ? (xi , xk ) . We can also assume an explicit known link function, f : R ! [0, 1], so that qt = f (dK ? (xi , xj ) dK ? (xi , xk )). Objective 2: Assuming an explicit known link function f estimate K ? from S. 2.1 Definitions and Notation Our triplet observations are nonlinear transformations of a linear function of the Gram matrix G := X T KX. Indeed for any triple t = (i, j, k), define M t (K) := dK (xi , xj ) = xTi Kxk + dK (xi , xk ) T xk Kxi xTi Kxj xTj Kxi + xTj Kxj xTk Kxk . So for every t 2 S, yt is a noisy measurement of sign(M t (K)). This linear operator may also be expressed as a matrix M t := xi xTk + xk xTi xi xTj xj xTi + xj xTj xk xTk , so that M t (K) = hM t , Ki = Trace(M Tt K). We will use M t to denote the operator and associated matrix interchangeably. Ordering the elements of T lexicographically, we let M denote the linear map, n 1 M(K) = (M t (K)| for t 2 T ) 2 Rn( 2 ) Given a PSD matrix K and a sample, t 2 S, we let `(yt hM t , Ki) denote the loss of K with respect to t; e.g., the 0-1 loss {sign(yt hM t ,Ki)6=1} , the hinge-loss max{0, 1 yt hM t , Ki}, or the logistic loss log(1 + exp( yt hM t , Ki)). Note that we insist that our losses be functions of our triplet differences hM t , Ki. Note that this makes our losses invariant to rigid motions of the points xi . Other models proposed for metric learning use scale-invariant loss functions [9]. For a given loss `, we then define the empirical risk with respect to our set of observations S to be X bS (K) := 1 R `(yt hM t , Ki). |S| t2S This is an unbiased estimator of the true risk R(K) := E[`(yt hM t , Ki)] where the expectation is taken with respect to a triplet t selected uniformly at random and the random value of yt . Finally, we let I n denote the identity matrix in Rn , 1n the n-dimensional vector of all ones and V := I n n1 1n 1Tn the centering matrix. In particular if X 2 Rp?n is a set of points, XV subtracts the mean of the columns of X from each column. We say that X is centered if XV = 0, or equivalently X1n = 0. If G is the Gram matrix of the set of points X, i.e. G = X T X, then we say that G is centered if X is centered or if equivalently, G1n = 0. Furthermore we use k ? k? to denote the nuclear norm, and k ? k1,2 to denote the mixed `1,2 norm of a matrix, the sum of the `2 norms of its rows. Unless otherwise specified, we take k ? k to be the standard operator norm when applied to matrices and the standard Euclidean norm when applied to vectors. Finally we define the K-norm of a vector as kxk2K := xT Kx. 3 2.2 Sample Complexity of Learning Metrics. In most applications, we are interested in learning a matrix K that is low-rank and positivesemidefinite. Furthermore as we will show in Theorem 2.1, such matrices can be learned using fewer samples than general psd matrices. As is common in machine learning applications, we relax the rank constraint to a nuclear norm constraint. In particular, let our constraint set be K , = {K 2 Rp?p |K positive-semidefinite, kKk? ? , maxhM t , Ki ? }. t2T Up to constants, a bound on hM t , Ki is a bound on xTi Kxi . This bound along with assuming our bS (K) from R(K) crucial loss function is Lipschitz, will lead to a tighter bound on the deviation of R in our upper bound theorem. c := Let K ? := minK2K , R(K) be the true risk minimizer in this class, and let K bS (K) be the empirical risk minimizer. We achieve the following prediction error minK2K , R bounds for the empirical risk minimzer. Theorem 2.1. Fix , , > 0. In addition assume that max1?i?n kxi k2 = 1. If the loss function ` is L-Lipschitz, then with probability 1 0s 1 s T 2 kXX k log p 140 2 log p A 2L2 2 log 2/ n c R(K) R(K ? ) ? 4L @ + + |S| |S| |S| Note that past generalization error bounds in the metric learning literature have failed to quantify the precise dependence on observation noise, dimension, rank, and our features X. Consider the fact that a p ? p matrix with rank d has O(dp) degrees of freedom. With that in mind, one expects the sample complexity to be also roughly O(dp). We next show that this intuition is correct if the original representation X is isotropic (i.e., has no preferred direction). The Isotropic Case. Suppose that x1 , ? ? ? , xn , n > p, are drawn independently from the isotropic Gaussian N (0, p1 I). Furthermore, suppose that K ? = ppd U U T with U 2 Rp?d is a generic (dense) orthogonal matrix with unit norm columns. The factor ppd is simply the scaling needed so that the average magnitude of the entries in K ? is a constant, independent of the dimensions p and d. In this case, rank(K ? ) = d and kK ? kF = trace(U T Up) = p. These two facts imply p that the tightest ? bound on the nuclear norm of Kq is kK ? k? ? p d. Thus, we take = p d for the nuclear pp U T xi ? N (0, I d ) and note that kxi k2 = kz i k2 ? 2 . norm constraint. Now let z i = K d d Therefore, Ekxi k2K = d and it follows from standard concentration bounds that with large probability maxi kxi k2K ? 5d log n =: see [10]. Also, because the xi ? N (0, p1 I) it follows that if n > p log p, say, then with large probability kXX T k ? 5n/p. We now plug these calculations into Theorem 2.1 to obtain the following corollary. p Corollary 2.1.1 (Sample complexity for isotropic points). Fix > 0, set = p d, and assume that kXX T k = O(n/p) and := maxi kxi k2K = O(d log n). Then for a generic K ? 2 K , , as constructed above, with probability at least 1 , 0s 1 2 dp(log p + log n) c A R(K) R(K ? ) = O @ |S| This bound agrees with the intuition that the sample complexity should grow roughly like dp, the degrees of freedom on K ? . Moreover, our minimax lower bound in Theorem 2.3 below shows that, ignoring logarithmic factors, the general upper bound in Theorem 2.1 is unimprovable in general. Beyond low rank metrics, in many applications it is reasonable to assume that only a few of the features are salient and should be given nonzero weight. Such a metric may be learned by insisting K to be row sparse in addition to being low rank. Whereas learning a low rank K assumes that distance is well represented in a low dimensional subspace, a row sparse (and hence low rank) K defines a metric using only a subset of the features. Figure 1 gives a comparison of a low rank versus a low rank and sparse matrix K. 4 Analogous to the convex relaxation of rank by the nuclear norm, it is common to relax row sparsity by using the mixed `1,2 norm. In fact, the geometry of the `1,2 and nuclear norm balls are tightly related as the following lemma shows. Lemma 2.2. For a symmetric positive semi-definite matrix K 2 Rp?p , kKk? ? kKk1,2 . Proof. kKk1,2 v p uX X u p t = K 2i,j i=1 j=1 p X K i,i = Trace(K) = i=1 p X i (K) i=1 = kKk? This implies that the `1,2 ball of a given radius is contained inside the nuclear norm ball of the same radius. In particular, it is reasonable to assume that it is easier to learn a K that is sparse in addition to being low rank. Surprisingly, however, the following minimax bound shows that this is not necessarily the case. To make this more precise, we will consider optimization over the set K0 , = {K 2 Rp?p |K positive-semidefinite, kKk1,2 ? , maxhM t , Ki ? }. t2T Furthermore, we must specify the way in which our data could be generated from noisy triplet observations of a fixed K ? . To this end, assume the existence of a link function f : R ! [0, 1] so that qt = P(yt = 1) = f (M t (K ? )) governs the observations. There is a natural associated logarithmic loss function `f corresponding to the log-likelihood, where the loss of an arbitrary K is `f (yt hM t , Ki) = {yt = 1} log 1 + f (hM t , Ki) {yt =1} log 1 1 f (hM t , Ki) Theorem 2.3. Choose a link function f and let `f be the associated logarithmic loss. For p sufficiently large, then there exists a choice of , , X, and |S| such that s kXX T k C13 ln 4 2 n c inf sup E[R(K)] R(K) C 2 |S| c K2K0 K , where C = Cf2 32 r inf |x|? f (x)(1 f (x)) sup|?|? f 0 (?)2 with Cf = inf |x|? f 0 (x), C1 is an absolute constant, and the c of K from |S| samples. infimum is taken over all estimators K Importantly, up to polylogarithmic factors and constants, our minimax lower over the `1,2 ball bound matches the upper bound over the nuclear norm ball given in Theorem 2.1. In particular, in the worst case, learning a sparse and low rank matrix K is no easier than learning a K that is simply low rank. However in many realistic cases, a slight performance gain is seen from optimizing over the `1,2 ball when K ? is row sparse, while optimizing over the nuclear norm ball does better when K ? is dense. We show examples of this in the Section 3. The proof is given in the supplementary materials. Note that if is in a bounded range, then the constant C has little effect. For the case that f is the 1 1 yt hM t ,Ki logistic function, Cf . Likewise, the term under the root will be also be 4e 4e bounded for in a constant range. The terms in the constant C arise when translating from risk and a KL-divergence to squared distance and reflects the noise in the problem. 2.3 Sample Complexity Bounds for Identification Under a general loss function and arbitrary K ? , we can not hope to convert our prediction error bounds into a recovery statement. However in this section we will show that as long as K ? is low rank, and if we choose the loss function to be the log loss `f of a given link function f as defined prior to the statement of Theorem 2.3, recovery is possible. Firstly note that under these assumptions we have an explicit formula for the risk, 1 X 1 1 R(K) = f (hM t , K ? i) log + (1 f (hM t , K ? i)) log |T | f (hM t , Ki) 1 f (hM t , Ki) t2T 5 and R(K) R(K ? ) = 1 X KL(f (hM t , K ? i)||f (hM t , Ki)). |T | t2T c approximates R(K ? ) well, The following theorem shows that if the excess risk is small, i.e. R(K) ? c approximates M(K ) well. The proof, given in the supplementary materials, uses then M(K) standard Taylor series arguments to show the KL-divergence is bounded below by squared-distance. Lemma 2.4. Let Cf = inf |x|? f 0 (x). Then for any K 2 K , , 2Cf2 kM(K) |T | M(K ? )k2 ? R(K) R(K ? ). The following may give us hope that recovering K ? from M(K ? ) is trivial, but the linear operator M is non-invertible in general, as we discuss next. To see why, we must consider a more general class of operators defined on Gram matrices. Given a symmetric matrix G, define the operator Lt by Lt (G) = 2Gik 2Gij + Gjj Gkk If G = X T KX then Lt (G) = M t (K), and more so M t = XLt X T . Analogous to M, we will combine the Lt operators into a single operator L, L(G) = (Lt (G)| for t 2 T ) 2 Rn( n 1 2 ). Lemma 2.5. The null space of L is one dimensional, spanned by V = I n 1 T n 1n 1n . The proof is contained in the supplementary materials. In particular we see that M is not invertible in general, adding a serious complication to our argument. However L is still invertible on the subset of centered symmetric matrices orthogonal to V , a fact that we will now exploit. We can decompose G into V and a component orthogonal to V denoted H, G=H+ GV hG,V i , kV k2F ? where G := and under the assumption that G is centered, G = kGk n 1 . Remarkably, the following lemma tells us that a non-linear function of H uniquely determines G. Lemma 2.6. If n > d + 1, and G is rank d and centered, then G is an eigenvalue of H with multiplicity n d 1. In addition, given another Gram matrix G0 of rank d0 , G0 G is an eigenvalue of H H 0 with multiplicity at least n d d0 1. Proof. Since G is centered, 1n 2 ker G, and in particular dim(1? d 1. If n \ ker G) = n x 2 1? n \ ker G, then Gx = Hx + G V x ) Hx = G x. 0 ? For the second statement, notice that dim(1n \ ker G G ) n d d0 1. A similar argument then applies. If n > 2d, then the multiplicity of the eigenvalue G is at least n/2. So we can trivially identify it from the spectrum of H. This gives us a non-linear way to recover G from H. c Indeed the above lemma implies that Now we can return to the task of recovering K ? from M(K). ? ? G (and hence K if X is full rank) can be recovered from H ? by computing an eigenvalue of H ? . c = M(K). c However H ? is recoverable from L(H ? ), which is itself well approximated by L(H) The proof of the following theorem makes this argument precise. c is rank d0 , n > d + d0 + 1, X is rank p and X T K ? X Theorem 2.7. Assume that K ? is rank d, K ? ? Tc n 1 and X KX are all centered. Let Cd,d0 = 1 + . Then 0 (n d d n min (XX |T | where T 2 ) c kK min (XX T 20s 2LCd,d0 4@ K ? k2F ? Cf2 140 1) T 2 kXX k n |S| ) is the smallest eigenvalue of XX T . 6 log p 1 s 3 2L2 2 log 2 2 log p A 5 + + |S| |S| The proof, given in the supplementary materials, relies on two key components, Lemma 2.6 and a type of restricted isometry property for M on V ? . Our proof technique is a streamlined and more general approach similar to that used in the special case of ordinal embedding. In fact, our new bound improves on the recovery bound given in [11] for ordinal embedding. We have several remarks about the bound in the theorem. If X is well conditioned, e.g. isotropic, then n min (XX T )2 T n ? p12 , so the left hand side is the average squared error min (XX ) ? p . In that case |T | c is approximately of the recovery. In most applications the rank of the empirical risk minimizer K equal to the rank of K ? , i.e. d ? d0 . Note that If d + d0 ? 12 (n 1) then Cd,d0 ? 3. Finally, the assumption that X T K ? X are centered can be guaranteed by centering X, which has no impact on the triplet differences hM t , K ? i, or insisting that K ? is centered. As mentioned above Cf will be have little effect assuming that our measurements hM t , Ki are bounded. 2.4 Applications to Ordinal Embedding In the ordinal embedding setting, there are a set of items with unknown locations, z 1 , ? ? ? , z n 2 Rd and a set of triplet observations S where as in the metric learning case observing yt = 1, for a triplet t = (i, j, k) is indicative of the kz i z j k2 ? kz i z k k2 , i.e. item i is closer to j than k. The goal is to recover the z i ?s, up to rigid motions, by recovering their Gram matrix G? from these comparisons. Ordinal embedding case reduces to metric learning through the following observation. Consider the case when n = p and X = I p , i.e. the xi are standard basis vectors. Letting K ? = G? , we see that kxi xj k2K = kz i z j k2 . So in particular, Lt = M t for each triple t, and observations are exactly comparative distance judgements. Our results then apply, and extend previous work on sample complexity in the ordinal embedding setting given in [11]. In particular, though Theorem 5 in b will converge to G? , they [11] provides a consistency guarantee that the empirical risk minimizer G do not provide a convergence rate. We resolve this issue now. p In their work, it is assumed that kz i k2 ? and kGk? ? dn . In particular, sample complexity results of the form O(dn log n) are obtained. However, these results are trivial in the following sense, if kz i k2 ? then kGk? ? n, and their results (as well as our upper bound) implies that true sample complexity is significantly smaller, namely O( n log n) which is independent of the ambient dimension d. As before, assume an explicit link function f with Lipschitz constant L, so the samples are noisy observations governed by G? , and take the loss to be the logarithmic loss associated to f . We obtain the following improved recovery bound in this case. The proof is immediate from Theorem 2.7. Corollary 2.7.1. Let G? be the Gram matrix of n centered points in d dimensions with kG? k2F = 2 2 n b b d . Let G = minkGk? ? n,kGk1 ? RS (G) and assume that G is rank d, with n > 2d + 1. Then, s ! b G ? k2 kG LCd,d n log n F =O n2 Cf2 |S| 3 Experiments To validate our complexity and recovery guarantees, we ran the following simulations. We generate iid x1 , ? ? ? , xn ? N (0, p1 I), with n = 200, and K ? = ppd U U T for a random orthogonal matrix U 2 Rp?d with unit norm columns. In Figure 2a, K ? has d nonzero rows/columns. In Figure 2b, K ? is a dense rank-d matrix. We compare the performance of nuclear norm and `1,2 regularization in each setting against an unconstrained baseline where we only enforce that K be psd. Given a fixed number of samples, each method is compared in terms of the relative excess risk, the relative squared recovery error, been trimmed for readability. c K ? k2 kK F kK ? k2F c R(K ? ) R(K) , R(K ? ) and , averaged over 20 trials. The y-axes of both plots have In the case that K ? is sparse, `1,2 regularization outperforms nuclear norm regularization. However, in the case of dense low rank matrices, nuclear norm reularization is superior. Notably, as expected from our upper and lower bounds, the performances of the two approaches seem to be within constant 7 factors of each other. Therefore, unless there is strong reason to believe that the underlying K ? is sparse, nuclear norm regularization achieves comparable performance with a less restrictive modeling assumption. Furthermore, in the two settings, both the nuclear norm and `1,2 constrained methods outperform the unconstrained baseline, especially in the case where K ? is low rank and sparse. To empirically validate our sample complexity results, we compute the number of samples averaged over 20 runs to achieve a relative excess risk of less than 0.1 in Figure 3. First, we fix p = 100 and increment d from 1 to 10. Then we fix d = 10 and increment p from 10 to 100 to clearly show the linear dependence of the sample complexity on d and p as demonstrated in Corollary 2.1.1. To our knowledge, these are the first results quantifying the sample complexity in terms of the number of features, p, and the embedding dimension, d. (a) Sparse low rank metric (b) Dense low rank metric Figure 2: `1,2 and nuclear norm regularization performance (a) d varying (b) p varying Figure 3: Number of samples to achieve relative excess risk < 0.1 Acknowledgments This work was partially supported by the NSF grants CCF-1218189 and IIS1623605 8 References [1] Martina A Rau, Blake Mason, and Robert D Nowak. How to model implicit knowledge? similarity learning methods to assess perceptions of visual representations. In Proceedings of the 9th International Conference on Educational Data Mining, pages 199?206, 2016. [2] Aur?lien Bellet, Amaury Habrard, and Marc Sebban. Metric learning. Synthesis Lectures on Artificial Intelligence and Machine Learning, 9(1):1?151, 2015. [3] Aur?lien Bellet and Amaury Habrard. Robustness and generalization for metric learning. Neurocomputing, 151:259?267, 2015. [4] Zheng-Chu Guo and Yiming Ying. Guaranteed classification via regularized similarity learning. Neural Computation, 26(3):497?522, 2014. [5] Yiming Ying, Kaizhu Huang, and Colin Campbell. Sparse metric learning via smooth optimization. In Advances in neural information processing systems, pages 2214?2222, 2009. [6] Wei Bian and Dacheng Tao. Constrained empirical risk minimization framework for distance metric learning. IEEE transactions on neural networks and learning systems, 23(8):1194?1205, 2012. [7] Yuan Shi, Aur?lien Bellet, and Fei Sha. Sparse compositional metric learning. arXiv preprint arXiv:1404.4105, 2014. [8] Samet Oymak, Amin Jalali, Maryam Fazel, Yonina C Eldar, and Babak Hassibi. Simultaneously structured models with application to sparse and low-rank matrices. IEEE Transactions on Information Theory, 61(5):2886?2908, 2015. [9] Eric Heim, Matthew Berger, Lee Seversky, and Milos Hauskrecht. Active perceptual similarity modeling with auxiliary information. arXiv preprint arXiv:1511.02254, 2015. [10] Kenneth R Davidson and Stanislaw J Szarek. Local operator theory, random matrices and banach spaces. Handbook of the geometry of Banach spaces, 1(317-366):131, 2001. [11] Lalit Jain, Kevin G Jamieson, and Rob Nowak. Finite sample prediction and recovery bounds for ordinal embedding. In Advances In Neural Information Processing Systems, pages 2703?2711, 2016. [12] Mark A Davenport, Yaniv Plan, Ewout Van Den Berg, and Mary Wootters. 1-bit matrix completion. Information and Inference: A Journal of the IMA, 3(3):189?223, 2014. [13] Joel A. Tropp. An introduction to matrix concentration inequalities, 2015. [14] Felix Abramovich and Vadim Grinshtein. Model selection and minimax estimation in generalized linear models. IEEE Transactions on Information Theory, 62(6):3721?3730, 2016. [15] Florentina Bunea, Alexandre B Tsybakov, Marten H Wegkamp, et al. Aggregation for gaussian regression. The Annals of Statistics, 35(4):1674?1697, 2007. [16] Philippe Rigollet and Alexandre Tsybakov. Exponential screening and optimal rates of sparse estimation. The Annals of Statistics, pages 731?771, 2011. [17] Jon Dattorro. Convex Optimization & Euclidean Distance Geometry. Meboo Publishing USA, 2011. 9
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The Marginal Value of Adaptive Gradient Methods in Machine Learning Ashia C. Wilson] , Rebecca Roelofs] , Mitchell Stern] , Nathan Srebro? , and Benjamin Recht] {ashia,roelofs,mitchell}@berkeley.edu, [email protected], [email protected] ? ] University of California, Berkeley Toyota Technological Institute at Chicago Abstract Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drastically different solutions than gradient descent (GD) or stochastic gradient descent (SGD). We construct an illustrative binary classification problem where the data is linearly separable, GD and SGD achieve zero test error, and AdaGrad, Adam, and RMSProp attain test errors arbitrarily close to half. We additionally study the empirical generalization capability of adaptive methods on several stateof-the-art deep learning models. We observe that the solutions found by adaptive methods generalize worse (often significantly worse) than SGD, even when these solutions have better training performance. These results suggest that practitioners should reconsider the use of adaptive methods to train neural networks. 1 Introduction An increasing share of deep learning researchers are training their models with adaptive gradient methods [3, 12] due to their rapid training time [6]. Adam [8] in particular has become the default algorithm used across many deep learning frameworks. However, the generalization and out-ofsample behavior of such adaptive gradient methods remains poorly understood. Given that many passes over the data are needed to minimize the training objective, typical regret guarantees do not necessarily ensure that the found solutions will generalize [17]. Notably, when the number of parameters exceeds the number of data points, it is possible that the choice of algorithm can dramatically influence which model is learned [15]. Given two different minimizers of some optimization problem, what can we say about their relative ability to generalize? In this paper, we show that adaptive and non-adaptive optimization methods indeed find very different solutions with very different generalization properties. We provide a simple generative model for binary classification where the population is linearly separable (i.e., there exists a solution with large margin), but AdaGrad [3], RMSProp [21], and Adam converge to a solution that incorrectly classifies new data with probability arbitrarily close to half. On this same example, SGD finds a solution with zero error on new data. Our construction suggests that adaptive methods tend to give undue influence to spurious features that have no effect on out-of-sample generalization. We additionally present numerical experiments demonstrating that adaptive methods generalize worse than their non-adaptive counterparts. Our experiments reveal three primary findings. First, with the same amount of hyperparameter tuning, SGD and SGD with momentum outperform adaptive methods on the development/test set across all evaluated models and tasks. This is true even when the adaptive methods achieve the same training loss or lower than non-adaptive methods. Second, adaptive methods often display faster initial progress on the training set, but their performance quickly 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. plateaus on the development/test set. Third, the same amount of tuning was required for all methods, including adaptive methods. This challenges the conventional wisdom that adaptive methods require less tuning. Moreover, as a useful guide to future practice, we propose a simple scheme for tuning learning rates and decays that performs well on all deep learning tasks we studied. 2 Background The canonical optimization algorithms used to minimize risk are either stochastic gradient methods or stochastic momentum methods. Stochastic gradient methods can generally be written ? (wk ), ?k rf wk+1 = wk (2.1) ? (wk ) := rf (wk ; xi ) is the gradient of some loss function f computed on a batch of data where rf k x ik . Stochastic momentum methods are a second family of techniques that have been used to accelerate training. These methods can generally be written as wk+1 = wk ? (wk + ?k rf k (wk wk 1 )) + k (wk wk The sequence of iterates (2.2) includes Polyak?s heavy-ball method (HB) with Accelerated Gradient method (NAG) [19] with k = k . (2.2) 1 ). = 0, and Nesterov?s k Notable exceptions to the general formulations (2.1) and (2.2) are adaptive gradient and adaptive momentum methods, which choose a local distance measure constructed using the entire sequence of iterates (w1 , ? ? ? , wk ). These methods (including AdaGrad [3], RMSProp [21], and Adam [8]) can generally be written as ? (wk + ?k Hk 1 rf wk+1 = wk k (wk wk 1 )) + k Hk 1 Hk 1 (wk wk (2.3) 1 ), where Hk := H(w1 , ? ? ? , wk ) is a positive definite matrix. Though not necessary, the matrix Hk is usually defined as 0( )1/2 1 k X A, Hk = diag @ ?i g i g i (2.4) i=1 ? (wk + k (wk wk 1 )), and ?k is where ? ? denotes the entry-wise or Hadamard product, gk = rf some set of coefficients specified for each algorithm. That is, Hk is a diagonal matrix whose entries are the square roots of a linear combination of squares of past gradient components. We will use the fact that Hk are defined in this fashion in the sequel. For the specific settings of the parameters for many of the algorithms used in deep learning, see Table 1. Adaptive methods attempt to adjust an algorithm to the geometry of the data. In contrast, stochastic gradient descent and related variants use the `2 geometry inherent to the parameter space, and are equivalent to setting Hk = I in the adaptive methods. Gk ?k k SGD HB NAG AdaGrad I I I Gk 1 + D k ? ? 0 0 0 ? 2 Gk RMSProp 1 + (1 2 )Dk ? ? 0 0 0 0 2 1 k Gk 2 Adam (1 1+ 1 ? 11 1 (1 1 0 2) k 2 Dk 1 k 1 k 1 ) 1 k 1 Table 1: Parameter settings of algorithms used in deep learning. Here, Dk = diag(gk gk ) and Gk := Hk Hk . We omit the additional ? added to the adaptive methods, which is only needed to ensure non-singularity of the matrices Hk . In this context, generalization refers to the performance of a solution w on a broader population. Performance is often defined in terms of a different loss function than the function f used in training. For example, in classification tasks, we typically define generalization in terms of classification error rather than cross-entropy. 2 2.1 Related Work Understanding how optimization relates to generalization is a very active area of current machine learning research. Most of the seminal work in this area has focused on understanding how early stopping can act as implicit regularization [22]. In a similar vein, Ma and Belkin [10] have shown that gradient methods may not be able to find complex solutions at all in any reasonable amount of time. Hardt et al. [17] show that SGD is uniformly stable, and therefore solutions with low training error found quickly will generalize well. Similarly, using a stability argument, Raginsky et al. [16] have shown that Langevin dynamics can find solutions than generalize better than ordinary SGD in non-convex settings. Neyshabur, Srebro, and Tomioka [15] discuss how algorithmic choices can act as implicit regularizer. In a similar vein, Neyshabur, Salakhutdinov, and Srebro [14] show that a different algorithm, one which performs descent using a metric that is invariant to re-scaling of the parameters, can lead to solutions which sometimes generalize better than SGD. Our work supports the work of [14] by drawing connections between the metric used to perform local optimization and the ability of the training algorithm to find solutions that generalize. However, we focus primarily on the different generalization properties of adaptive and non-adaptive methods. A similar line of inquiry has been pursued by Keskar et al. [7]. Hochreiter and Schmidhuber [4] showed that ?sharp? minimizers generalize poorly, whereas ?flat? minimizers generalize well. Keskar et al. empirically show that Adam converges to sharper minimizers when the batch size is increased. However, they observe that even with small batches, Adam does not find solutions whose performance matches state-of-the-art. In the current work, we aim to show that the choice of Adam as an optimizer itself strongly influences the set of minimizers that any batch size will ever see, and help explain why they were unable to find solutions that generalized particularly well. 3 The potential perils of adaptivity The goal of this section is to illustrate the following observation: when a problem has multiple global minima, different algorithms can find entirely different solutions when initialized from the same point. In addition, we construct an example where adaptive gradient methods find a solution which has worse out-of-sample error than SGD. To simplify the presentation, let us restrict our attention to the binary least-squares classification problem, where we can easily compute closed the closed form solution found by different methods. In least-squares classification, we aim to solve minimizew RS [w] := 12 kXw yk22 . (3.1) Here X is an n ? d matrix of features and y is an n-dimensional vector of labels in { 1, 1}. We aim to find the best linear classifier w. Note that when d > n, if there is a minimizer with loss 0 then there is an infinite number of global minimizers. The question remains: what solution does an algorithm find and how well does it perform on unseen data? 3.1 Non-adaptive methods Most common non-adaptive methods will find the same solution for the least squares objective (3.1). Any gradient or stochastic gradient of RS must lie in the span of the rows of X. Therefore, any method that is initialized in the row span of X (say, for instance at w = 0) and uses only linear combinations of gradients, stochastic gradients, and previous iterates must also lie in the row span of X. The unique solution that lies in the row span of X also happens to be the solution with minimum Euclidean norm. We thus denote wSGD = X T (XX T ) 1 y. Almost all non-adaptive methods like SGD, SGD with momentum, mini-batch SGD, gradient descent, Nesterov?s method, and the conjugate gradient method will converge to this minimum norm solution. The minimum norm solutions have the largest margin out of all solutions of the equation Xw = y. Maximizing margin has a long and fruitful history in machine learning, and thus it is a pleasant surprise that gradient descent naturally finds a max-margin solution. 3 3.2 Adaptive methods Next, we consider adaptive methods where Hk is diagonal. While it is difficult to derive the general form of the solution, we can analyze special cases. Indeed, we can construct a variety of instances where adaptive methods converge to solutions with low `1 norm rather than low `2 norm. For a vector x 2 Rq , let sign(x) denote the function that maps each component of x to its sign. Lemma 3.1 Suppose there exists a scalar c such that X sign(X T y) = cy. Then, when initialized at w0 = 0, AdaGrad, Adam, and RMSProp all converge to the unique solution w / sign(X T y). In other words, whenever there exists a solution of Xw = y that is proportional to sign(X T y), this is precisely the solution to which all of the adaptive gradient methods converge. Proof We prove this lemma by showing that the entire trajectory of the algorithm consists of iterates whose components have constant magnitude. In particular, we will show that wk = for some scalar k. k sign(X T y) , The initial point w0 = 0 satisfies the assertion with Now, assume the assertion holds for all k ? t. Observe that rRS (wk + k (wk wk 1 )) = X T (X(wk + = XT ( = {( k k + + k (wk k( k k( k wk 1 )) k 1 ))X 0 = 0. y) sign(X T y) y T k 1 ))c 1} X y T = ?k X y, where the last equation defines ?k . Hence, letting gk = rRS (wk + k (wk wk 1 )), we also have 0( 0( 1 )1/2 1 )1/2 k k X X A = diag @ Hk = diag @ ?s g s g s ?s ?2s |X T y|A = ?k diag |X T y| , s=1 s=1 where |u| denotes the component-wise absolute value of a vector and the last equation defines ?k . In sum, wk+1 = wk ?k Hk 1 rf (wk + k (wk ? ?k ?k k ?k 1 = + ( k k ?k ?k wk k 1) 1 )) + t Hk 1 Hk 1 (wk wk 1) sign(X T y), proving the claim.1 This solution is far simpler than the one obtained by gradient methods, and it would be surprising if such a simple solution would perform particularly well. We now turn to showing that such solutions can indeed generalize arbitrarily poorly. 3.3 Adaptivity can overfit Lemma 3.1 allows us to construct a particularly pernicious generative model where AdaGrad fails to find a solution that generalizes. This example uses infinite dimensions to simplify bookkeeping, but one could take the dimensionality to be 6n. Note that in deep learning, we often have a number of parameters equal to 25n or more [20], so this is not a particularly high dimensional example by contemporary standards. For i = 1, . . . , n, sample the label yi to be 1 with probability p and 1 with probability 1 p for some p > 1/2. Let xi be an infinite dimensional vector with entries 8 yi j = 1 > > < 1 j = 2, 3 xij = . > 1 j = 4 + 5(i 1), . . . , 4 + 5(i 1) + 2(1 yi ) > : 0 otherwise 1 In the event that X T y has a component equal to 0, we define 0/0 = 0 so that the update is well-defined. 4 In other words, the first feature of xi is the class label. The next 2 features are always equal to 1. After this, there is a set of features unique to xi that are equal to 1. If the class label is 1, then there is 1 such unique feature. If the class label is 1, then there are 5 such features. Note that the only discriminative feature useful for classifying data outside the training set is the first one! Indeed, one can perform perfect classification using only the first feature. The other features are all useless. Features 2 and 3 are constant, and each of the remaining features only appear for one example in the data set. However, as we will see, algorithms without such a priori knowledge may not be able to learn these distinctions. Take n samples and consider the AdaGrad solution for minimizing 12 ||Xw y||2 . First we show that Pn the conditions of Lemma 3.1 hold. Let b = i=1 yi and assume for the sake of simplicity that b > 0. This will happen with arbitrarily high probability for large enough n. Define u = X T y and observe that 8 8 > > n j=1 1 j=1 > > > > < < b j = 2, 3 1 j = 2, 3 uj = and sign(uj ) = j+1 y if j > 3 and x = 1 =1 > > j b 5 c,j > >yj if j > 3 and xb j+1 5 c,j > > : : 0 otherwise 0 otherwise Thus we have hsign(u), xi i = yi + 2 + yi (3 2yi ) = 4yi as desired. Hence, the AdaGrad solution wada / sign(u). In particular, wada has all of its components equal to ?? for some positive constant ? . Now since wada has the same sign pattern as u, the first three components of wada are equal to each other. But for a new data point, xtest , the only features that are nonzero in both xtest and wada are the first three. In particular, we have hwada , xtest i = ? (y (test) + 2) > 0 . Therefore, the AdaGrad solution will label all unseen data as a positive example! Now, we turn to the minimum 2-norm solution. Let P and N denote the set of positive and negative examples respectively. Let n+ = |P| and n = |N |. Assuming ?i = ?+ when yi = 1 and ?i = ? when form wSGD = X T ? = P yi = 1,Pwe have that the minimum norm solution will have the T i2P ?+ xi + j2N ? xj . These scalars can be found by solving XX ? = y. In closed form we have 4n + 3 4n+ + 1 ?+ = and ? = . (3.2) 9n+ + 3n + 8n+ n + 3 9n+ + 3n + 8n+ n + 3 The algebra required to compute these coefficients can be found in the Appendix. For a new data point, xtest , again the only features that are nonzero in both xtest and wSGD are the first three. Thus we have hwSGD , xtest i = y test (n+ ?+ n ? ) + 2(n+ ?+ + n ? ) . Using (3.2), we see that whenever n+ > n /3, the SGD solution makes no errors. A formal construction of this example using a data-generating distribution can be found in Appendix C. Though this generative model was chosen to illustrate extreme behavior, it shares salient features with many common machine learning instances. There are a few frequent features, where some predictor based on them is a good predictor, though these might not be easy to identify from first inspection. Additionally, there are many other features which are sparse. On finite training data it looks like such features are good for prediction, since each such feature is discriminatory for a particular training example, but this is over-fitting and an artifact of having fewer training examples than features. Moreover, we will see shortly that adaptive methods typically generalize worse than their non-adaptive counterparts on real datasets. 4 Deep Learning Experiments Having established that adaptive and non-adaptive methods can find different solutions in the convex setting, we now turn to an empirical study of deep neural networks to see whether we observe a similar discrepancy in generalization. We compare two non-adaptive methods ? SGD and the heavy ball method (HB) ? to three popular adaptive methods ? AdaGrad, RMSProp and Adam. We study performance on four deep learning problems: (C1) the CIFAR-10 image classification task, (L1) 5 Name Network type Architecture Dataset Framework C1 Deep Convolutional cifar.torch CIFAR-10 Torch L1 2-Layer LSTM torch-rnn War & Peace Torch L2 2-Layer LSTM + Feedforward span-parser Penn Treebank DyNet L3 3-Layer LSTM emnlp2016 Penn Treebank Tensorflow Table 2: Summaries of the models we use for our experiments.2 character-level language modeling on the novel War and Peace, and (L2) discriminative parsing and (L3) generative parsing on Penn Treebank. In the interest of reproducibility, we use a network architecture for each problem that is either easily found online (C1, L1, L2, and L3) or produces state-of-the-art results (L2 and L3). Table 2 summarizes the setup for each application. We take care to make minimal changes to the architectures and their data pre-processing pipelines in order to best isolate the effect of each optimization algorithm. We conduct each experiment 5 times from randomly initialized starting points, using the initialization scheme specified in each code repository. We allocate a pre-specified budget on the number of epochs used for training each model. When a development set was available, we chose the settings that achieved the best peak performance on the development set by the end of the fixed epoch budget. CIFAR-10 did not have an explicit development set, so we chose the settings that achieved the lowest training loss at the end of the fixed epoch budget. Our experiments show the following primary findings: (i) Adaptive methods find solutions that generalize worse than those found by non-adaptive methods. (ii) Even when the adaptive methods achieve the same training loss or lower than non-adaptive methods, the development or test performance is worse. (iii) Adaptive methods often display faster initial progress on the training set, but their performance quickly plateaus on the development set. (iv) Though conventional wisdom suggests that Adam does not require tuning, we find that tuning the initial learning rate and decay scheme for Adam yields significant improvements over its default settings in all cases. 4.1 Hyperparameter Tuning Optimization hyperparameters have a large influence on the quality of solutions found by optimization algorithms for deep neural networks. The algorithms under consideration have many hyperparameters: the initial step size ?0 , the step decay scheme, the momentum value 0 , the momentum schedule k , the smoothing term ?, the initialization scheme for the gradient accumulator, and the parameter controlling how to combine gradient outer products, to name a few. A grid search on a large space of hyperparameters is infeasible even with substantial industrial resources, and we found that the parameters that impacted performance the most were the initial step size and the step decay scheme. We left the remaining parameters with their default settings. We describe the differences between the default settings of Torch, DyNet, and Tensorflow in Appendix B for completeness. To tune the step sizes, we evaluated a logarithmically-spaced grid of five step sizes. If the best performance was ever at one of the extremes of the grid, we would try new grid points so that the best performance was contained in the middle of the parameters. For example, if we initially tried step sizes 2, 1, 0.5, 0.25, and 0.125 and found that 2 was the best performing, we would have tried the step size 4 to see if performance was improved. If performance improved, we would have tried 8 and so on. We list the initial step sizes we tried in Appendix D. For step size decay, we explored two separate schemes, a development-based decay scheme (devdecay) and a fixed frequency decay scheme (fixed-decay). For dev-decay, we keep track of the best validation performance so far, and at each epoch decay the learning rate by a constant factor if the model does not attain a new best value. For fixed-decay, we decay the learning rate by a constant factor every k epochs. We recommend the dev-decay scheme when a development set is available; 2 Architectures can be found at the following links: (1) cifar.torch: https://github. com/szagoruyko/cifar.torch; (2) torch-rnn: https://github.com/jcjohnson/torch-rnn; (3) span-parser: https://github.com/jhcross/span-parser; (4) emnlp2016: https://github.com/ cdg720/emnlp2016. 6 (a) CIFAR-10 (Train) (b) CIFAR-10 (Test) Figure 1: Training (left) and top-1 test error (right) on CIFAR-10. The annotations indicate where the best performance is attained for each method. The shading represents ? one standard deviation computed across five runs from random initial starting points. In all cases, adaptive methods are performing worse on both train and test than non-adaptive methods. not only does it have fewer hyperparameters than the fixed frequency scheme, but our experiments also show that it produces results comparable to, or better than, the fixed-decay scheme. 4.2 Convolutional Neural Network We used the VGG+BN+Dropout network for CIFAR-10 from the Torch blog [23], which in prior work achieves a baseline test error of 7.55%. Figure 1 shows the learning curve for each algorithm on both the training and test dataset. We observe that the solutions found by SGD and HB do indeed generalize better than those found by adaptive methods. The best overall test error found by a non-adaptive algorithm, SGD, was 7.65 ? 0.14%, whereas the best adaptive method, RMSProp, achieved a test error of 9.60 ? 0.19%. Early on in training, the adaptive methods appear to be performing better than the non-adaptive methods, but starting at epoch 50, even though the training error of the adaptive methods is still lower, SGD and HB begin to outperform adaptive methods on the test error. By epoch 100, the performance of SGD and HB surpass all adaptive methods on both train and test. Among all adaptive methods, AdaGrad?s rate of improvement flatlines the earliest. We also found that by increasing the step size, we could drive the performance of the adaptive methods down in the first 50 or so epochs, but the aggressive step size made the flatlining behavior worse, and no step decay scheme could fix the behavior. 4.3 Character-Level Language Modeling Using the torch-rnn library, we train a character-level language model on the text of the novel War and Peace, running for a fixed budget of 200 epochs. Our results are shown in Figures 2(a) and 2(b). Under the fixed-decay scheme, the best configuration for all algorithms except AdaGrad was to decay relatively late with regards to the total number of epochs, either 60 or 80% through the total number of epochs and by a large amount, dividing the step size by 10. The dev-decay scheme paralleled (within the same standard deviation) the results of the exhaustive search over the decay frequency and amount; we report the curves from the fixed policy. Overall, SGD achieved the lowest test loss at 1.212 ? 0.001. AdaGrad has fast initial progress, but flatlines. The adaptive methods appear more sensitive to the initialization scheme than non-adaptive methods, displaying a higher variance on both train and test. Surprisingly, RMSProp closely trails SGD on test loss, confirming that it is not impossible for adaptive methods to find solutions that generalize well. We note that there are step configurations for RMSProp that drive the training loss 7 below that of SGD, but these configurations cause erratic behavior on test, driving the test error of RMSProp above Adam. 4.4 Constituency Parsing A constituency parser is used to predict the hierarchical structure of a sentence, breaking it down into nested clause-level, phrase-level, and word-level units. We carry out experiments using two stateof-the-art parsers: the stand-alone discriminative parser of Cross and Huang [2], and the generative reranking parser of Choe and Charniak [1]. In both cases, we use the dev-decay scheme with = 0.9 for learning rate decay. Discriminative Model. Cross and Huang [2] develop a transition-based framework that reduces constituency parsing to a sequence prediction problem, giving a one-to-one correspondence between parse trees and sequences of structural and labeling actions. Using their code with the default settings, we trained for 50 epochs on the Penn Treebank [11], comparing labeled F1 scores on the training and development data over time. RMSProp was not implemented in the used version of DyNet, and we omit it from our experiments. Results are shown in Figures 2(c) and 2(d). We find that SGD obtained the best overall performance on the development set, followed closely by HB and Adam, with AdaGrad trailing far behind. The default configuration of Adam without learning rate decay actually achieved the best overall training performance by the end of the run, but was notably worse than tuned Adam on the development set. Interestingly, Adam achieved its best development F1 of 91.11 quite early, after just 6 epochs, whereas SGD took 18 epochs to reach this value and didn?t reach its best F1 of 91.24 until epoch 31. On the other hand, Adam continued to improve on the training set well after its best development performance was obtained, while the peaks for SGD were more closely aligned. Generative Model. Choe and Charniak [1] show that constituency parsing can be cast as a language modeling problem, with trees being represented by their depth-first traversals. This formulation requires a separate base system to produce candidate parse trees, which are then rescored by the generative model. Using an adapted version of their code base,3 we retrained their model for 100 epochs on the Penn Treebank. However, to reduce computational costs, we made two minor changes: (a) we used a smaller LSTM hidden dimension of 500 instead of 1500, finding that performance decreased only slightly; and (b) we accordingly lowered the dropout ratio from 0.7 to 0.5. Since they demonstrated a high correlation between perplexity (the exponential of the average loss) and labeled F1 on the development set, we explored the relation between training and development perplexity to avoid any conflation with the performance of a base parser. Our results are shown in Figures 2(e) and 2(f). On development set performance, SGD and HB obtained the best perplexities, with SGD slightly ahead. Despite having one of the best performance curves on the training dataset, Adam achieves the worst development perplexities. 5 Conclusion Despite the fact that our experimental evidence demonstrates that adaptive methods are not advantageous for machine learning, the Adam algorithm remains incredibly popular. We are not sure exactly as to why, but hope that our step-size tuning suggestions make it easier for practitioners to use standard stochastic gradient methods in their research. In our conversations with other researchers, we have surmised that adaptive gradient methods are particularly popular for training GANs [18, 5] and Q-learning with function approximation [13, 9]. Both of these applications stand out because they are not solving optimization problems. It is possible that the dynamics of Adam are accidentally well matched to these sorts of optimization-free iterative search procedures. It is also possible that carefully tuned stochastic gradient methods may work as well or better in both of these applications. 3 While the code of Choe and Charniak treats the entire corpus as a single long example, relying on the network to reset itself upon encountering an end-of-sentence token, we use the more conventional approach of resetting the network for each example. This reduces training efficiency slightly when batches contain examples of different lengths, but removes a potential confounding factor from our experiments. 8 It is an exciting direction of future work to determine which of these possibilities is true and to understand better as to why. Acknowledgements The authors would like to thank Pieter Abbeel, Moritz Hardt, Tomer Koren, Sergey Levine, Henry Milner, Yoram Singer, and Shivaram Venkataraman for many helpful comments and suggestions. RR is generously supported by DOE award AC02-05CH11231. MS and AW are supported by NSF Graduate Research Fellowships. NS is partially supported by NSF-IIS-13-02662 and NSF-IIS15-46500, an Inter ICRI-RI award and a Google Faculty Award. BR is generously supported by NSF award CCF-1359814, ONR awards N00014-14-1-0024 and N00014-17-1-2191, the DARPA Fundamental Limits of Learning (Fun LoL) Program, a Sloan Research Fellowship, and a Google Faculty Award. (a) War and Peace (Training Set) (b) War and Peace (Test Set) (c) Discriminative Parsing (Training Set) (d) Discriminative Parsing (Development Set) (e) Generative Parsing (Training Set) (f) Generative Parsing (Development Set) Figure 2: Performance curves on the training data (left) and the development/test data (right) for three experiments on natural language tasks. The annotations indicate where the best performance is attained for each method. The shading represents one standard deviation computed across five runs from random initial starting points. 9 References [1] Do Kook Choe and Eugene Charniak. Parsing as language modeling. In Jian Su, Xavier Carreras, and Kevin Duh, editors, Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, EMNLP 2016, Austin, Texas, USA, November 1-4, 2016, pages 2331?2336. The Association for Computational Linguistics, 2016. [2] James Cross and Liang Huang. Span-based constituency parsing with a structure-label system and provably optimal dynamic oracles. In Jian Su, Xavier Carreras, and Kevin Duh, editors, Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, Austin, Texas, pages 1?11. The Association for Computational Linguistics, 2016. [3] John C. Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12:2121?2159, 2011. [4] Sepp Hochreiter and J?rgen Schmidhuber. Flat minima. Neural Computation, 9(1):1?42, 1997. [5] Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. arXiv:1611.07004, 2016. [6] Andrej Karparthy. A peek at trends in machine learning. https://medium.com/@karpathy/ a-peek-at-trends-in-machine-learning-ab8a1085a106. Accessed: 2017-05-17. [7] Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. In The International Conference on Learning Representations (ICLR), 2017. [8] D.P. Kingma and J. Ba. Adam: A method for stochastic optimization. The International Conference on Learning Representations (ICLR), 2015. [9] Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. In International Conference on Learning Representations (ICLR), 2016. [10] Siyuan Ma and Mikhail Belkin. Diving into the shallows: a computational perspective on large-scale shallow learning. arXiv:1703.10622, 2017. [11] Mitchell P. Marcus, Mary Ann Marcinkiewicz, and Beatrice Santorini. Building a large annotated corpus of english: The penn treebank. COMPUTATIONAL LINGUISTICS, 19(2):313?330, 1993. [12] H. Brendan McMahan and Matthew Streeter. Adaptive bound optimization for online convex optimization. In Proceedings of the 23rd Annual Conference on Learning Theory (COLT), 2010. [13] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning (ICML), 2016. [14] Behnam Neyshabur, Ruslan Salakhutdinov, and Nathan Srebro. Path-SGD: Path-normalized optimization in deep neural networks. In Neural Information Processing Systems (NIPS), 2015. [15] Behnam Neyshabur, Ryota Tomioka, and Nathan Srebro. In search of the real inductive bias: On the role of implicit regularization in deep learning. In International Conference on Learning Representations (ICLR), 2015. [16] Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learning via stochastic gradient Langevin dynamics: a nonasymptotic analysis. arXiv:1702.03849, 2017. [17] Benjamin Recht, Moritz Hardt, and Yoram Singer. Train faster, generalize better: Stability of stochastic gradient descent. In Proceedings of the International Conference on Machine Learning (ICML), 2016. 10 [18] Scott Reed, Zeynep Akata, Xinchen Yan, Lajanugen Logeswaran, Bernt Schiele, and Honglak Lee. Generative adversarial text to image synthesis. In Proceedings of The International Conference on Machine Learning (ICML), 2016. [19] Ilya Sutskever, James Martens, George Dahl, and Geoffrey Hinton. On the importance of initialization and momentum in deep learning. In Proceedings of the International Conference on Machine Learning (ICML), 2013. [20] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016. [21] T. Tieleman and G. Hinton. Lecture 6.5?RmsProp: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 2012. [22] Yuan Yao, Lorenzo Rosasco, and Andrea Caponnetto. On early stopping in gradient descent learning. Constructive Approximation, 26(2):289?315, 2007. [23] Sergey Zagoruyko. Torch blog. http://torch.ch/blog/2015/07/30/cifar.html, 2015. 11
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Aggressive Sampling for Multi-class to Binary Reduction with Applications to Text Classification Bikash Joshi Univ. Grenoble Alps, LIG Grenoble, France [email protected] Massih-Reza Amini Univ. Grenoble Alps, LIG Grenoble, France [email protected] Franck Iutzeler Univ. Grenoble Alps, LJK Grenoble, France [email protected] Ioannis Partalas Expedia EWE Geneva, Switzerland [email protected] Yury Maximov Los Alamos National Laboratory and Skolkovo IST, USA and Moscow, Russia [email protected] Abstract We address the problem of multi-class classification in the case where the number of classes is very large. We propose a double sampling strategy on top of a multi-class to binary reduction strategy, which transforms the original multi-class problem into a binary classification problem over pairs of examples. The aim of the sampling strategy is to overcome the curse of long-tailed class distributions exhibited in majority of large-scale multi-class classification problems and to reduce the number of pairs of examples in the expanded data. We show that this strategy does not alter the consistency of the empirical risk minimization principle defined over the double sample reduction. Experiments are carried out on DMOZ and Wikipedia collections with 10,000 to 100,000 classes where we show the efficiency of the proposed approach in terms of training and prediction time, memory consumption, and predictive performance with respect to state-of-the-art approaches. 1 Introduction Large-scale multi-class or extreme classification involves problems with extremely large number of classes as it appears in text repositories such as Wikipedia, Yahoo! Directory (www.dir.yahoo.com), or Directory Mozilla DMOZ (www.dmoz.org); and it has recently evolved as a popular branch of machine learning with many applications in tagging, recommendation and ranking. The most common and popular baseline in this case is the one-versus-all approach (OVA) [18] where one independent binary classifier is learned per class. Despite its simplicity, this approach suffers from two main limitations; first, it becomes computationally intractable when the number of classes grow large, affecting at the same time the prediction. Second, it suffers from the class imbalance problem by construction.Recently, two main approaches have been studied to cope with these limitations. The first one, broadly divided in tree-based and embedding-based methods, have been proposed with the aim of reducing the effective space of labels in order to control the complexity of the learning problem. Tree-based methods [4, 3, 6, 7, 9, 21, 5, 15] rely on binary tree structures where each leaf corresponds to a class and inference is performed by traversing the tree from top to bottom; a binary classifier being used at each node to determine the child node to develop. These methods have logarithmic time complexity with the drawback that it is a challenging task to find a balanced tree structure which can partition the class labels. Further, even though different heuristics have been developed to address the unbalanced problem, these methods suffer from the drawback that they have to make several decisions prior to reaching a final category, which leads to error propagation and 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. thus a decrease in accuracy. On the other hand, label embedding approaches [11, 5, 19] first project the label-matrix into a low-dimensional linear subspace and then use an OVA classifier. However, the low-rank assumption of the label-matrix is generally transgressed in the extreme multi-class classification setting, and these methods generally lead to high prediction error.The second type of approaches aim at reducing the original multi-class problem into a binary one by first expanding the original training set using a projection of pairs of observations and classes into a low dimensional dyadic space, and then learning a single classifier to separate between pairs constituted with examples and their true classes and pairs constituted with examples with other classes [1, 28, 16]. Although prediction in the new representation space is relatively fast, the construction of the dyadic training observations is generally time consuming and prevails over the training and prediction times. Contributions. In this paper, we propose a scalable multi-class classification method based on an aggressive double sampling of the dyadic output prediction problem. Instead of computing all possible dyadic examples, our proposed approach consists first in drawing a new training set of much smaller size from the original one by oversampling the most small size classes and by sub-sampling the few big size classes in order to avoid the curse of long-tailed class distributions common in the majority of large-scale multi-class classification problems [2]. The second goal is to reduce the number of constructed dyadic examples. Our reduction strategy brings inter-dependency between the pairs containing the same observation and its true class in the original training set. Thus, we derive new generalization bounds using local fractional Rademacher complexity showing that even with a shift in the original class distribution and also the inter-dependency between the pairs of example, the empirical risk minimization principle over the transformation of the sampled training set remains consistent. We validate our approach by conducting a series of experiments on subsets of DMOZ and the Wikipedia collections with up to 100,000 target categories. 2 A doubly-sampled multi-class to binary reduction strategy We address the problem of monolabel multi-class classification defined on joint space X ? Y . where X ? Rd is the input space and Y = {1, . . . , K} = [K] the output space, made of K y classes. Elements of X ? Y are denoted as x = (x, y). Furthermore, we assume the training set S = (xyi i )m i=1 is made of m i.i.d examples/class pairs distributed according to a fixed but unknown probability distribution D, and we consider a class of predictor functions G = {g : X ? Y ? R}. We define the instantaneous loss for predictor g ? G on example xy as: e(g, xy ) = 1 K ?1 X 1g(xy )?g(xy0 ) , (1) y 0 ?Y\{y} where 1? is the indicator function equal to 1 if the predicate ? is true and 0 otherwise. Compared to the classical multi-class error, e0 (g, xy ) = 1y6=argmaxy0 ?Y g(xy0 ) , the loss of (1) estimates the average number of classes, given any input data, that get a greater scoring by g than the correct class. The loss (1) is hence a ranking criterion, and the multi-class SVM of [29] and AdaBoost.MR [24] optimize convex surrogate functions of this loss. It is also used in label ranking [12]. Our objective is to find a function g ? G with a small expected risk R(g) = Exy ?D [e(g, xy )], by minimizing the empirical error defined as the average number of training examples xyi i ? S which, in mean, are scored lower 0 than xyi , for y 0 ? Y\{yi } : m m X X 1 ? m (g, S) = 1 R e(g, xyi i ) = m i=1 m(K ? 1) i=1 2.1 X y 0 ?Y\{yi } 1g(xyi )?g(xy0 )?0 . i (2) i Binary reduction based on dyadic representations of examples and classes In this work, we consider prediction functions of the form g = f ? ?, where ? : X ? Y ? Rp is a projection of the input and the output space into a joint feature space of dimension p; and f ? F = {f : Rp ? R} is a function that measures the adequacy between an observation x and a class y using their corresponding representation ?(xy ). The projection function ? is applicationdependent and it can either be learned [28], or defined using some heuristics [27, 16]. 2 Further, consider the following dyadic transformation    zj = ?(xki ), ?(xyi i ) , y?j = ?1 T (S) = zj = ?(xyi i ), ?(xki ) , y?j = +1 if k < yi elsewhere  , . (3) j =(i?1)(K?1)+k where j = (i ? 1)(K ? 1) + k with i ? [m], k ? [K ? 1]; that expands a K-class labeled set S of size m into a binary labeled set T (S) of size N = m(K ? 1) (e.g. Figure 1 over a toy problem). With the class of functions 0 0 H = {h : Rp ? Rp ? R; (?(xy ), ?(xy )) 7? f (?(xy )) ? f (?(xy )), f ? F}, (4) the empirical loss (Eq. (2)) can be rewritten as : N X ? T (S) (h) = 1 R 1y? h(z )?0 . N j=1 j j (5) Hence, the minimization of Eq. (5) over the transformation T (S) of a training set S defines a binary classification over the pairs of x x x x S dyadic examples. However, this binary problem takes as examples dependent random variables, T as for each original example xy ? S, the K ? 1 0 ), ?(x )), +1) (z = (?(x ), ?(x )), +1) (z = (?(x ), ?(x )), +1) pairs in {(?(xy ), ?(xy )); y?} ? T (S) all de- (z(z == (?(x (?(x ), ?(x )), ?1) (z = (?(x ), ?(x )), +1) (z = (?(x ), ?(x )), +1) y pend on x . In [16] this problem is studied by (z = (?(x ), ?(x )), ?1) (z = (?(x ), ?(x )), ?1) (z = (?(x ), ?(x )), +1) bounding the generalization error associated to (z = (?(x ), ?(x )), ?1) (z = (?(x ), ?(x )), ?1) (z = (?(x ), ?(x )), ?1) (5) using the fractional Rademacher complexity proposed in [25]. In this work, we derive Figure 1: A toy example depicting the transformaa new generalization bounds based on Local tion T (Eq. (3)) applied to a training set S of size Rademacher Complexities introduced in [22] m = 4 and K = 4. that implies second-order (i.e. variance) information inducing faster convergence rates (Theorem 1). Our analysis relies on the notion of graph covering introduced in [14] and defined as : Definition 1 (Exact proper fractional cover of G, [14]). Let G = (V, E) be a graph. C = {(Ck , ?k )}k?[J] , for some positive integer J, with Ck ? V and ?k ? [0, 1] is an exact proper fractional cover of G, if: i) it is proper: ?k, Ck is an independent set, i.e., P there is no connections between vertices in Ck ; ii) it is an exact fractional cover of G: ?v ? V, k:v?Ck ?k = 1. . P The weight W (C) of C is given by: W (C) = k?[J] ?k and the minimum weight ?? (G) = minC?K(G) W (C) over the set K(G) of all exact proper fractional covers of G is the fractional chromatic number of G. From this statement, [14] extended Hoeffding?s inequality and proposed large deviation bounds for sums of dependent random variables which was the precursor of new generalisation bounds, including a Talagrand?s type inequality for empirical processes in the dependent case presented in [22]. With the classes of functions G and H introduced previously, consider the parameterized family Hr which, for r > 0, is defined as: . Hr = {h : h ? H, V[h] = Vz,?y [1y?h(z) ] ? r}, where V denotes the variance. y1 1 1 4 7 10 y1 1 y1 2 y1 3 y1 4 y2 1 y2 2 y3 3 y4 4 2 5 8 11 y2 2 y1 1 y2 2 y2 3 y2 4 y3 3 y3 1 y3 2 y3 3 y4 4 y4 4 3 6 9 12 y1 1 y2 2 y3 3 y3 4 y4 1 y4 2 y4 3 y4 4 The fractional Rademacher complexity introduced in [25] entails our analysis : X X . 2 RT (S) (H)= E? ?k ECk sup ?? h(z? ), N h?H ??C k?[K?1] k z? ?T (S) Figure 2: The dependency graph G = {1, . . . , 12} corresponding to the toy problem of Figure 1, with (?i )N a sequence of independent i=1 Rademacher variables verifying P(?n = 1) = where dependent nodes are connected with verP(?n=?1) = 12 . If other is not specified explic- tices in blue double-line. The exact proper fracitly we assume below all ?k = 1. Our first result tional cover C1 , C2 and C3 is shown in dashed. that bounds the generalization error of a function The fractional chromatic number is in this case ? T (S) (h)], with respect ?? (G) = K ? 1 = 3. h ? H; R(h) = ET (S) [R ? T (S) (h) over a transformed training set, T (S), and the fractional Rademacher to its empirical error R complexity, RT (S) (H), is stated below. 3 m Theorem 1. Let S = (xyi i )m i=1 ? (X ? Y) be a dataset of m examples drawn i.i.d. according to a probability distribution D over X ? Y and T (S) = ((zi , y?i ))N i=1 the transformed set obtained as in Eq. (3). Then for any 1 > ? > 0 and 0/1 loss ` : {?1, +1} ? R ? [0, 1], with probability at least (1 ? ?) the following generalization bound holds for all h ? Hr : s r   log 1? 5 q r 25 log 1? ? RT (S) (` ? Hr ) + R(h) ? RT (S) (h) + RT (S) (` ? Hr ) + + . 2 2 m 48 m The proof is provided in the supplementary material, and it relies on the idea of splitting up the sum (5) into several parts, each part being a sum of independent variables. 2.2 Aggressive Double Sampling Even-though the previous multi-class to binary transformation T with a proper projection function ? allows to redefine the learning problem in a dyadic feature space of dimension p  d, the increased number of examples can lead to a large computational overhead. In order to cope with this problem, we propose a (?, ?)-double subsampling of T (S), which first aims to balance the presence of classes by constructing a new training set S? from S with probabilities ? = (?k )K k=1 . The idea here is to overcome the curse of long-tailed class Algorithm: (?, ?)-DS distributions exhibited in ma- Input: Labeled training set S = (xyi )m i i=1 jority of large-scale multi- initialization: S ? ?; ? class classification problems T (S ) ? ? ; ? ? [2] by oversampling the most for k = 1..K do small size classes and by subDraw randomly a set S?k of examples of class k from S with sampling the few big size probability ?k ; classes in S. The hyperpaS? ? S? ? S?k ; rameters ? are formally dey fined as ?k, ?k = P (xy ? forall x ? S? do Draw uniformly a set Yxy of ? classes from Y\{y} . ?  K; S? |xy ? S). In practice forall k ? Yxy do we set them inversely proif k < y then   portional to the size of each T? (S? ) ? T? (S? ) ? z = ?(xk ), ?(xy ) , y? = ?1 ; class in the original training set; ?k, ?k ? 1/?k where else   ?k is the proportion of class T? (S? ) ? T? (S? ) ? z = ?(xy ), ?(xk ) , y? = +1 ; k in S. The second aim is to reduce the number of dyadic return T? (S? ) examples controlled by the hyperparameter ?. The pseudo-code of this aggressive double sampling procedure, referred to as (?, ?)-DS, is depicted above and it is composed of two main steps. 1. For each class k ? {1, . . . , K}, draw randomly a set S?k of examples from S of that class K [ with probability ?k , and let S? = S?k ; k=1 2. For each example xy in S? , draw uniformly ? adversarial classes in Y\{y}. After this double sampling, we apply the transformation T defined as in Eq. (3), leading to a set T? (S? ) of size ?|S? |  N . In Section 3, we will show that this procedure practically leads to dramatic improvements in terms of memory consumption, computational complexity, and a higher multi-class prediction accuracy when the number of classes is very large. The empirical loss over the transformation of the new subsampled training set S? of size M , outputted by the (?, ?)-DS algorithm is : X 1 X X ? T (S ) (h) = 1 R 1y?? h(z? )?0 = 1g(xy )?g(xy0 )?0 , (6) ? ? ?M ?M y 0 y x ?S? y ?Yx (? y ? ,z? )?T? (S? ) which is essentially the same empirical risk as the one defined in Eq. (2) but taken with respect to the training set outputted by the (?, ?)-DS algorithm. Our main result is the following theorem which ? T (S ) . bounds the generalization error of a function h ? H learned by minimizing R ? ? 4 m Theorem 2. Let S = (xyi i )m be a training set of size m i.i.d. according to a i=1 ? (X ? Y) probability distribution D over X ? Y, and T (S) = ((zi , y?i ))N i=1 the transformed set obtained with the transformation function T defined as in Eq. (3). Let S? ? S, |S? | = M , be a training set outputted by the algorithm (?, ?)-DS and T (S? ) ? T (S) its corresponding transformation. Then for any 1 > ? > 0 with probability at least (1 ? ?) the following risk bound holds for all h ? H : s s 2 (K ? 1) log 2? log 4K 7? log 4K ? ? ? ? T (S ) (h) + ?RT (S ) (` ? H) + ? R(h) ? ?R + + . ? ? ? ? 2M ? ?(m ? 1) 3(m ? 1) Furthermore, for all functions in the class Hr , we have the following generalization bound that holds with probability at least (1 ? ?) : s 4K 4K ? T (S ) (h) + ?RT (S ) (` ? Hr ) + 2? log ? + 7? log ? R(h) ??R ? ? ? ? ?(m ? 1) 3(m ? 1) s r  q (K ? 1) log 2? 25? log 2? 5? r RT? (S? ) (` ? Hr ) + + , + 2 2 M? 48 M where ` : {?1, +1} ? R ? [0, 1] 0/1 is an instantaneous loss, and ? = maxy: 1?y?K ?y /?y , ? = maxy: 1?y?K 1/?y and ?y > 0 is the proportion of class y in S. The proof is provided in the supplementary material. This theorem hence paves the way for the consistency of the empirical risk minimization principle [26, Th. 2.1, p. 38] defined over the double sample reduction strategy we propose. 2.3 Prediction with Candidate Selection The prediction is carried out in the dyadic feature space, by first considering the pairs constituted by a test observation and all the classes, and then choosing the class that leads to the highest score by the learned classifier. In the large scale scenario, comAlgorithm: Prediction with Candidate Selection Algorithm puting the feature representations for all classes may require a huge Input: Unlabeled test set T ; ? p amount of time. To overcome this Learned function f : R ? R; initialization: ? ? ?; problem we sample over classes by choosing just those that are the forall x ? T do Select Yx ? Y candidate set of q nearest-centroid classes; nearest to a test example, based on ? ? ? ? argmaxk?Yx f ? (?(xk )) ; its distance with class centroids. Here we propose to consider class return predicted classes ? centroids as the average of vectors within that class. Note that class centroids are computed once in the preliminary projection of training examples and classes in the dyadic feature space and thus represent no additional computation at this stage. The algorithm above presents the pseudocode of this candidate based selection strategy 1 . 3 Experiments In this section, we provide an empirical evaluation of the proposed reduction approach with the (?, ?)DS sampling strategy for large-scale multi-class classification of document collections. First, we present the mapping ? : X ? Y ? Rp . Then, we provide a statistical and computational comparison of our method with state-of-the-art large-scale approaches on popular datasets. 3.1 a Joint example/class representation for text classification The particularity of text classification is that documents are represented in a vector space induced by the vocabulary of the corresponding collection [23]. Hence each class can be considered as a megadocument, constituted by the concatenation of all of the documents in the training set belonging to it, 1 The number of classes pre-selected can be tuned to offer a prediction time/accuracy tradeoff if the prediction time is more critical. 5 Features in the joint example/class representation ?(xy ).   representation X X lS 1. log(1 + yt ) 2. log 1 + 3. It Ft t?y?x t?y?x t?y?x     X yt X X yt yt 4. .It 5. log 1 + 6. log 1 + .It |y| |y| |y| t?y?x t?y?x t?y?x   X X y t lS 7. log 1 + . 8. 1 9. d(xy , centroid(y)) |y| F t t?y?x t?y?x X 10. BM25 = P t?y?x 2?yt It . yt +(0.25+0.75?len(y)/avg(len(y)) Table 1: Joint example/class representation for text classification, where t ? y ? x are terms that are present in both the class y?s mega-document and document P x. V represents P the set of distinct P terms within S, and xt is the frequency of term t in x, yt = x?y xt , |y| = t?V yt , Ft = x?S xt , P lS = t?V St . Finally, It is the inverse document frequency of term t, len(y) is number of terms of documents in class y, and avg(len(y)) is the average of document lengths for all the classes. and simple feature mapping of examples and classes can be defined over their common words. Here we used p = 10 features inspired from learning to rank [17] by resembling a class and a document to respectively a document and a query (Table 1). All features except feature 9, that is the distance of an example x to the centroid of all examples of a particular class y, are classical. In addition to its predictive interest, the latter is also used in prediction for performing candidate preselection. Note that for other large-scale multi-class classification applications like recommendation with extremely large number of offer categories or image classification, a same kind of mapping can either be learned or defined using their characteristics [27, 28]. 3.2 Experimental Setup Datasets. We evaluate the proposed method using popular datasets from the Large Scale Hierarchical Text Classification challenge (LSHTC) 1 and 2 [20]. These datasets are provided in a pre-processed format using stop-word removal and stemming. Various characteristics of these datesets including the statistics of train, test and heldout are listed in Table 2. Since, the datasets used in LSHTC2 challenge were in multi-label format, we converted them to multi-class format by replicating the instances belonging to different class labels. Also, for the largest dataset (WIKI-large) used in LSHTC2 challenge, we used samples with 50,000 and 100,000 classes. The smaller dataset of LSHTC2 challenge is named as WIKI-Small, whereas the two 50K and 100K samples of large dataset are named as WIKI-50K and WIKI-100K in our result section. Datasets LSHTC1 DMOZ WIKI-Small WIKI-50K WIKI-100K # of classes, K Train Size Test Size Heldout Size Dimension, d 12294 126871 31718 5000 409774 27875 381149 95288 34506 594158 36504 796617 199155 5000 380078 50000 1102754 276939 5000 951558 100000 2195530 550133 5000 1271710 Table 2: Characteristics of the datasets used in our experiments Baselines. We compare the proposed approach,2 denoted as the sampling strategy by (?, ?)-DS, with popular baselines listed below: ? OVA: LibLinear [10] implementation of one-vs-all SVM. ? M-SVM: LibLinear implementation of multi-class SVM proposed in [8]. ? RecallTree [9]: A recent tree based multi-class classifier implemented in Vowpal Wabbit. 2 Source code and datasets can be found in the following repository https://github.com/bikash617/AggressiveSampling-for-Multi-class-to-BinaryReduction 6 Data LSHTC1 m = 163589 d = 409774 K = 12294 DMOZ m = 510943 d = 594158 K = 27875 WIKI-Small m = 1000772 d = 380078 K = 36504 WIKI-50K m = 1384693 d = 951558 K = 50000 WIKI-100K m = 2750663 d = 1271710 K = 100000 train time predict time total memory Accuracy MaF1 train time predict time total memory Accuracy MaF1 train time predict time total memory Accuracy MaF1 train time predict time total memory Accuracy MaF1 train time predict time total memory Accuracy MaF1 OVA 23056s 328s 40.3G 44.1% 27.4% 180361s 2797s 131.9G 37.7% 22.2% 212438s 2270s 109.1G 15.6% 8.8 % NA NA 330G NA NA NA NA 1017G NA NA M-SVM 48313s 314s 40.3G 36.4% 18.8% 212356s 3981s 131.9G 32.2% 14.3% >4d NA 109.1G NA NA NA NA 330G NA NA NA NA 1017G NA NA RecallTree 701s 21s 122M 18.1% 3.8% 2212s 47s 256M 16.9% 1.75% 1610s 24s 178M 7.9% <1% 4188s 45s 226M 17.9% 5.5% 8593s 90s 370M 8.4% 1.4% FastXML 8564s 339s 470M 39.3% 21.3% 14334s 424s 1339M 33.4% 15.1% 10646s 453s 949M 11.1% 4.6% 30459s 1110s 1327M 25.8% 14.6% 42359s 1687s 2622M 15% 8% PfastReXML 3912s 164s 471M 39.8% 22.4% 15492s 505s 1242M 33.7% 15.9% 21702s 871s 947M 12.1% 5.63% 48739s 2461s 1781M 27.3% 16.3% 73371s 3210s 2834M 16.1% 9% PD-Sparse 5105s 67s 10.5G 45.7% 27.7% 63286s 482s 28.1G 40.8% 22.7% 16309s 382s 12.4G 15.6% 9.91% 41091s 790s 35G 33.8% 23.4% 155633s 3121s 40.3G 22.2% 15.1% (?, ?)-DS 321s 544s 2G 37.4% 26.5% 1060s 2122s 5.3G 27.8% 20.5% 1290s 2577s 3.6G 21.5% 13.3% 3723s 4083s 5G 33.4% 24.5% 9264s 20324s 9.8G 25% 17.8% Table 3: Comparison of the result of various baselines in terms of time, memory, accuracy, and macro F1-measure ? FastXML [21]: An extreme multi-class classification method which performs partitioning in the feature space for faster prediction. ? PfastReXML [13]: Tree ensemble based extreme classifier for multi-class and multilabel problems. ? PD-Sparse [30]: A recent approach which uses multi-class loss with `1 -regularization. Referring to the work [30], we did not consider other recent methods SLEEC [5] and LEML [31] in our experiments, since they have been shown to be consistently outperformed by the above mentioned state-of-the-art approaches. Platform and Parameters. In all of our experiments, we used a machine with an Intel Xeon 2.60GHz processor with 256 GB of RAM. Each of these methods require tuning of various hyper-parameters that influence their performance. For each methods, we tuned the hyperparameters over a heldout set and used the combination which gave best predictive performance. The list of used hyperparameters for the results we obtained are reported in the supplementary material (Appendix B). Evaluation Measures. Different approaches are evaluated over the test sets using accuracy and the macro F1 measure (MaF1 ), which is the harmonic average of macro precision and macro recall; higher MaF1 thus corresponds to better performance. As opposed to accuracy, macro F1 measure is not affected by the class imbalance problem inherent to multi-class classification, and is commonly used as a robust measure for comparing predictive performance of classification methods. 4 Results The parameters of the datasets along with the results for compared methods are shown in Table 3. The results are provided in terms of train and predict times, total memory usage, and predictive performance measured with accuracy and macro F1-measure (MaF1 ). For better visualization and comparison, we plot the same results as bar plots in Fig. 3 keeping only the best five methods while comparing the total runtime and memory usage. First, we observe that the tree based approaches (FastXML, PfastReXML and RecallTree) have worse predictive performance compared to the other methods. This is due to the fact that the prediction error made at the top-level of the tree cannot be corrected at lower levels, also known as cascading effect. Even though they have lower runtime and memory usage, they suffer from this side effect. For large scale collections (WIKI-Small, WIKI-50K and WIKI-100K), the solvers with competitive predictive performance are OVA, M-SVM, PD-Sparse and (?, ?)-DS. However, standard OVA and 7 LSHTC1 1200 DMOZ 450 135 900 90 600 45 300 0 0 0 12 30 10.0 8 20 MaF (%) Accuracy (%) Total Memory (GB) Time (min.) 180 WIKI-Small 1200 900 300 WIKI-50K 3000 600 150 300 1000 0 0 36 42 24 28 2.5 12 14 7.5 5.0 4 10 0 0 0.0 0 0 45 45 45 45 45 30 30 30 30 30 15 15 15 15 15 0 0 0 0 0 30 30 30 30 30 20 20 20 20 20 10 10 10 10 10 0 0 0 0 RecallTree WIKI-100K 2000 FastXML PfastReXML PD-Sparse 0 Proposed (?, ?)-DS Figure 3: Comparisons in Total (Train and Test) Time (min.), Total Memory usage (GB), and MaF1 of the five best performing methods on LSHTC1, DMOZ, WIKI-Small, WIKI-50K and WIKI-100K. M-SVM have a complexity that grows linearly with K thus the total runtime and memory usage explodes on those datasets, making them impossible. For instance, on Wiki large dataset sample of 100K classes, the memory consumption of both approaches exceeds the Terabyte and they take several days to complete the training. Furthermore, on this data set and the second largest Wikipedia collection (WIKI-50K and WIKI-100K) the proposed approach is highly competitive in terms of Time, Total Memory and both performance measures comparatively to all the other approaches. These results suggest that the method least affected by long-tailed class distributions is (?, ?)-DS, mainly because of two reasons: first, the sampling tends to make the training set balanced and second, the reduced binary dataset contains similar number of positive and negative examples. Hence, for the proposed approach, there is an improvement in both accuracy and MaF1 measures. The recent PD-Sparse method also enjoys a competitive predictive performance but it requires to store intermediary weight vectors during optimization which prevents it from scaling well. The PD-Sparse solver provides an option for hashing leading to fewer memory usage during training which we used in the experiments; however, the memory usage is still significantly high for large datasets and at the same time this option slows down the training process considerably. In overall, among the methods with competitive predictive performance, (?, ?)-DS seems to present the best runtime and memory usage; its runtime is even competitive with most of tree-based methods, leading it to provide the best compromise among the compared methods over the three time, memory and performance measures. 5 Conclusion We presented a new method for reducing a multiclass classification problem to binary classification. We employ similarity based feature representation for class and examples and a double sampling stochastic scheme for the reduction process. Even-though the sampling scheme shifts the distribution of classes and that the reduction of the original problem to a binary classification problem brings inter-dependency between the dyadic examples; we provide generalization error bounds suggesting that the Empirical Risk Minimization principle over the transformation of the sampled training set still remains consistent. Furthermore, the characteristics of the algorithm contribute for its excellent performance in terms of memory usage and total runtime and make the proposed approach highly suitable for large class scenario. Acknowledgments This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d?avenir, and by the U.S. Department of Energy?s Office of Electricity as part of the DOE Grid Modernization Initiative. 8 References [1] Naoki Abe, Bianca Zadrozny, and John Langford. An iterative method for multi-class cost-sensitive learning. In Proceedings of the 10th ACM SIGKDD, KDD ?04, pages 3?11, 2004. [2] Rohit Babbar, Cornelia Metzig, Ioannis Partalas, Eric Gaussier, and Massih R. Amini. On power law distributions in large-scale taxonomies. SIGKDD Explorations, 16(1), 2014. [3] Samy Bengio, Jason Weston, and David Grangier. Label embedding trees for large multi-class tasks. In Advances in Neural Information Processing Systems, pages 163?171, 2010. [4] Alina Beygelzimer, John Langford, and Pradeep Ravikumar. Error-correcting tournaments. In Proceedings of the 20th International Conference on Algorithmic Learning Theory, ALT?09, pages 247?262, 2009. [5] Kush Bhatia, Himanshu Jain, Purushottam Kar, Manik Varma, and Prateek Jain. Sparse local embeddings for extreme multi-label classification. In Advances in Neural Information Processing Systems, pages 730?738, 2015. [6] Anna Choromanska, Alekh Agarwal, and John Langford. Extreme multi class classification. In NIPS Workshop: eXtreme Classification, submitted, 2013. [7] Anna Choromanska and John Langford. abs/1406.1822, 2014. Logarithmic time online multiclass prediction. CoRR, [8] Koby Crammer and Yoram Singer. On the algorithmic implementation of multiclass kernel-based vector machines. J. Mach. Learn. Res., 2:265?292, 2002. [9] Hal Daume III, Nikos Karampatziakis, John Langford, and Paul Mineiro. Logarithmic time one-againstsome. arXiv preprint arXiv:1606.04988, 2016. [10] Rong-En Fan, Kai-Wei Chang, Cho-Jui Hsieh, Xiang-Rui Wang, and Chih-Jen Lin. Liblinear: A library for large linear classification. J. Mach. Learn. Res., 9:1871?1874, 2008. [11] Daniel J Hsu, Sham M Kakade, John Langford, and Tong Zhang. Multi-label prediction via compressed sensing. In Advances in Neural Information Processing Systems 22 (NIPS), pages 772?780, 2009. [12] Eyke H?llermeier and Johannes F?rnkranz. On minimizing the position error in label ranking. In Machine Learning: ECML 2007, pages 583?590. Springer, 2007. [13] Himanshu Jain, Yashoteja Prabhu, and Manik Varma. Extreme multi-label loss functions for recommendation, tagging, ranking & other missing label applications. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 935?944. ACM, 2016. [14] S. Janson. Large deviations for sums of partly dependent random variables. Random Structures and Algorithms, 24(3):234?248, 2004. [15] Kalina Jasinska and Nikos Karampatziakis. Log-time and log-space extreme classification. arXiv preprint arXiv:1611.01964, 2016. [16] Bikash Joshi, Massih-Reza Amini, Ioannis Partalas, Liva Ralaivola, Nicolas Usunier, and ?ric Gaussier. On binary reduction of large-scale multiclass classification problems. In Advances in Intelligent Data Analysis XIV - 14th International Symposium, IDA 2015, pages 132?144, 2015. [17] Tie-Yan Liu, Jun Xu, Tao Qin, Wenying Xiong, and Hang Li. Letor: Benchmark dataset for research on learning to rank for information retrieval. In Proceedings of SIGIR 2007 workshop on learning to rank for information retrieval, pages 3?10, 2007. [18] Ana Carolina Lorena, Andr? C. Carvalho, and Jo?o M. Gama. A review on the combination of binary classifiers in multiclass problems. Artif. Intell. Rev., 30(1-4):19?37, 2008. [19] Paul Mineiro and Nikos Karampatziakis. Fast label embeddings via randomized linear algebra. In Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2015, Porto, Portugal, September 7-11, 2015, Proceedings, Part I, pages 37?51, 2015. [20] I. Partalas, A. Kosmopoulos, N. Baskiotis, T. Artieres, G. Paliouras, E. Gaussier, I. Androutsopoulos, M.-R. Amini, and P. Galinari. LSHTC: A Benchmark for Large-Scale Text Classification. ArXiv e-prints, March 2015. [21] Yashoteja Prabhu and Manik Varma. Fastxml: A fast, accurate and stable tree-classifier for extreme multi-label learning. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 263?272. ACM, 2014. [22] Liva Ralaivola and Massih-Reza Amini. Entropy-based concentration inequalities for dependent variables. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pages 2436?2444, 2015. [23] G. Salton, A. Wong, and C. S. Yang. A vector space model for automatic indexing. Commun. ACM, 18(11):613?620, November 1975. 9 [24] Robert E Schapire and Yoram Singer. Improved boosting algorithms using confidence-rated predictions. Machine learning, 37(3):297?336, 1999. [25] Nicolas Usunier, Massih-Reza Amini, and Patrick Gallinari. Generalization error bounds for classifiers trained with interdependent data. In Advances in Neural Information Processing Systems 18 (NIPS), pages 1369?1376, 2005. [26] Vladimir N. Vapnik. Statistical Learning Theory. Wiley-Interscience, 1998. [27] Maksims Volkovs and Richard S. Zemel. Collaborative ranking with 17 parameters. In Advances in Neural Information Processing Systems 25, pages 2294?2302, 2012. [28] Jason Weston, Samy Bengio, and Nicolas Usunier. Wsabie: Scaling up to large vocabulary image annotation. In Proceedings of the International Joint Conference on Artificial Intelligence, IJCAI, 2011. [29] Jason Weston and Chris Watkins. Multi-class support vector machines. Technical report, Technical Report CSD-TR-98-04, Department of Computer Science, Royal Holloway, University of London, 1998. [30] Ian EH Yen, Xiangru Huang, Kai Zhong, Pradeep Ravikumar, and Inderjit S Dhillon. Pd-sparse: A primal and dual sparse approach to extreme multiclass and multilabel classification. In Proceedings of the 33nd International Conference on Machine Learning, 2016. [31] Hsiang-Fu Yu, Prateek Jain, Purushottam Kar, and Inderjit Dhillon. Large-scale multi-label learning with missing labels. In International Conference on Machine Learning, pages 593?601, 2014. 10
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Deconvolutional Paragraph Representation Learning Yizhe Zhang Dinghan Shen Guoyin Wang Zhe Gan Ricardo Henao Lawrence Carin Department of Electrical & Computer Engineering, Duke University Abstract Learning latent representations from long text sequences is an important first step in many natural language processing applications. Recurrent Neural Networks (RNNs) have become a cornerstone for this challenging task. However, the quality of sentences during RNN-based decoding (reconstruction) decreases with the length of the text. We propose a sequence-to-sequence, purely convolutional and deconvolutional autoencoding framework that is free of the above issue, while also being computationally efficient. The proposed method is simple, easy to implement and can be leveraged as a building block for many applications. We show empirically that compared to RNNs, our framework is better at reconstructing and correcting long paragraphs. Quantitative evaluation on semi-supervised text classification and summarization tasks demonstrate the potential for better utilization of long unlabeled text data. 1 Introduction A central task in natural language processing is to learn representations (features) for sentences or multi-sentence paragraphs. These representations are typically a required first step toward more applied tasks, such as sentiment analysis [1, 2, 3, 4], machine translation [5, 6, 7], dialogue systems [8, 9, 10] and text summarization [11, 12, 13]. An approach for learning sentence representations from data is to leverage an encoder-decoder framework [14]. In a standard autoencoding setup, a vector representation is first encoded from an embedding of an input sequence, then decoded to the original domain to reconstruct the input sequence. Recent advances in Recurrent Neural Networks (RNNs) [15], especially Long Short-Term Memory (LSTM) [16] and variants [17], have achieved great success in numerous tasks that heavily rely on sentence-representation learning. RNN-based methods typically model sentences recursively as a generative Markov process with hidden units, where the one-step-ahead word from an input sentence is generated by conditioning on previous words and hidden units, via emission and transition operators modeled as neural networks. In principle, the neural representations of input sequences aim to encapsulate sufficient information about their structure, to subsequently recover the original sentences via decoding. However, due to the recursive nature of the RNN, challenges exist for RNN-based strategies to fully encode a sentence into a vector representation. Typically, during training, the RNN generates words in sequence conditioning on previous ground-truth words, i.e., teacher forcing training [18], rather than decoding the whole sentence solely from the encoded representation vector. This teacher forcing strategy has proven important because it forces the output sequence of the RNN to stay close to the ground-truth sequence. However, allowing the decoder to access ground truth information when reconstructing the sequence weakens the encoder?s ability to produce self-contained representations, that carry enough information to steer the decoder through the decoding process without additional guidance. Aiming to solve this problem, [19] proposed a scheduled sampling approach during training, which gradually shifts from learning via both latent representation and ground-truth signals to solely use the encoded latent representation. Unfortunately, [20] showed that scheduled sampling is a fundamentally inconsistent 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. training strategy, in that it produces largely unstable results in practice. As a result, training may fail to converge on occasion. During inference, for which ground-truth sentences are not available, words ahead can only be generated by conditioning on previously generated words through the representation vector. Consequently, decoding error compounds proportional to the length of the sequence. This means that generated sentences quickly deviate from the ground-truth once an error has been made, and as the sentence progresses. This phenomenon was coined exposure bias in [19]. We propose a simple yet powerful purely convolutional framework for learning sentence representations. Conveniently, without RNNs in our framework, issues connected to teacher forcing training and exposure bias are not relevant. The proposed approach uses a Convolutional Neural Network (CNN) [21, 22, 23] as encoder and a deconvolutional (i.e., transposed convolutional) neural network [24, 25] as decoder. To the best of our knowledge, the proposed framework is the first to force the encoded latent representation to capture information from the entire sentence via a multi-layer CNN specification, to achieve high reconstruction quality without leveraging RNN-based decoders. Our multi-layer CNN allows representation vectors to abstract information from the entire sentence, irrespective of order or length, making it an appealing choice for tasks involving long sentences or paragraphs. Further, since our framework does not involve recursive encoding or decoding, it can be very efficiently parallelized using convolution-specific Graphical Process Unit (GPU) primitives, yielding significant computational savings compared to RNN-based models. 2 2.1 Convolutional Auto-encoding for Text Modeling Convolutional encoder Let wt denote the t-th word in a given sentence. Each word wt is embedded into a k-dimensional word vector xt = We [wt ], where We ? Rk?V is a (learned) word embedding matrix, V is the vocabulary size, and We [v] denotes the v-th column of We . All columns of We are normalized to have unit `2 -norm, i.e., ||We [v]||2 = 1, ?v, by dividing each column with its `2 -norm. After embedding, a sentence of length T (padded where necessary) is represented as X ? Rk?T , by concatenating its word embeddings, i.e., xt is the t-th column of X. For sentence encoding, we use a CNN architecture similar to [26], though originally proposed for image data. The CNN consists of L layers (L ? 1 convolutional, and the Lth fully-connected) that ultimately summarize an input sentence into a (fixed-length) latent representation vector, h. Layer l ? {1, . . . , L} consists of pl filters, learned from data. For the i-th filter in layer 1, a convolutional (i,1) operation with stride length r(1) applies filter Wc ? Rk?h to X, where h is the convolution filter (1) (i,1) (i,1) size. This yields latent feature map, c = ?(X ? Wc + b(i,1) ) ? R(T ?h)/r +1 , where ?(?) is (1) a nonlinear activation function, b(i,1) ? R(T ?h)/r +1 , and ? denotes the convolutional operator. In our experiments, ?(?) is represented by a Rectified Linear Unit (ReLU) [27]. Note that the original (1) embedding dimension, k, changes after the first convolutional layer, as c(i,1) ? R(T ?h)/r +1 , for i = 1, . . . , p1 . Concatenating the results from p1 filters (for layer 1), results in feature map, (1) C(1) = [c(1,1) . . . c(p1 ,1) ] ? Rp1 ?[(T ?h)/r +1] . After this first convolutional layer, we apply the convolution operation to the feature map, C(1) , using the same filter size, h, with this repeated in sequence for L ? 1 layers. Each time, the length along the spatial coordinate is reduced to T (l+1) = b(T (l) ? h)/r(l) + 1c, where r(l) is the stride length, T (l) is the spatial length, l denotes the l-th layer and b?c is the floor function. For the final layer, L, the feature map C(L?1) is fed into a fully-connected layer, to produce the latent representation h. Implementation-wise, we use a convolutional layer with filter size equals to T (L?1) (regardless of h), which is equivalent to a fully-connected layer; this implementation trick has been also utilized in [26]. This last layer summarizes all remaining spatial coordinates, T (L?1) , into scalar features that (i,l) encapsulate sentence sub-structures throughout the entire sentence characterized by filters, {Wc } (i,l) for i = 1, . . . , pl and l = 1, . . . , L, where Wc denotes filter i for layer l. This also implies that the extracted feature is of fixed-dimensionality, independent of the length of the input sentence. 2 (k,h,p1,r(1)) (300, 5, 300, 2) (k,h,p2,r(2)) (1, 5, 600, 2) C(1) C(2) 300 (k x T) 300 x 60 (T(1) x p1) 28 x 300 (k,T(2),p3,r(3)) (1, 12, 500, 1) 500 600 (T(2) x p2) 12 x 600 600 300 (T(2) x p2) 12 x 600 (T(1) x p1) 28 x 300 (k x T) 300 x 60 Deconvolution Layers Convolution Layers Figure 1: Convolutional auto-encoding architecture. Encoder: the input sequence is first expanded to an embedding matrix, X, then fully compressed to a representation vector h, through a multi-layer convolutional encoder with stride. In the last layer, the spatial dimension is collapsed to remove the spatial dependency. Decoder: the latent vector h is fed through a multi-layer deconvolutional decoder with stride to reconstruct X as ? via cosine-similarity cross-entropy loss. X, Having pL filters on the last layer, results in pL -dimensional representation vector, h = C(L) , for the input sentence. For example, in Figure 1, the encoder consists of L = 3 layers, which for a sentence of length T = 60, embedding dimension k = 300, stride lengths {r(1) , r(2) , r(3) } = {2, 2, 1}, filter sizes h = {5, 5, 12} and number of filters {p1 , p2 , p3 } = {300, 600, 500}, results in intermediate feature maps, C(1) and C(2) of sizes {28 ? 300, 12 ? 600}, respectively. The last feature map of size 1 ? 500, corresponds to latent representation vector, h. Conceptually, filters from the lower layers capture primitive sentence information (h-grams, analogous to edges in images), while higher level filters capture more sophisticated linguistic features, such as semantic and syntactic structures (analogous to image elements). Such a bottom-up architecture models sentences by hierarchically stacking text segments (h-grams) as building blocks for representation vector, h. This is similar in spirit to modeling linguistic grammar formalisms via concrete syntax trees [28], however, we do not pre-specify a tree structure based on some syntactic structure (i.e., English language), but rather abstract it from data via a multi-layer convolutional network. 2.2 Deconvolutional decoder We apply the deconvolution with stride (i.e., convolutional transpose), as the conjugate operation of convolution, to decode the latent representation, h, back to the source (discrete) text domain. As the deconvolution operation proceeds, the spatial resolution gradually increases, by mirroring the convolutional steps described above, as illustrated in Figure 1. The spatial dimension is first expanded to match the spatial dimension of the (L ? 1)-th layer of convolution, then progressively expanded as T (l+1) = (T (l) ? 1) ? r(l) + h, for l = 1, ? ? ? up to L-th deconvolutional layer (which corresponds to the input layer of the convolutional encoder). The output of the L-layer deconvolution operation aims ? In line with word embedding to reconstruct the word embedding matrix, which we denote as X. ? matrix We , columns of X are normalized to have unit `2 -norm. Denoting w ? t as the t-th word in reconstructed sentence s?, the probability of w ? t to be word v is specified as exp[? ?1 Dcos (? xt , We [v])] , ?1 D xt , We [v 0 ])] cos (? v 0 ?V exp[? p(w ? t = v) = P (1) hx,yi where Dcos (x, y) is the cosine similarity defined as, ||x||||y|| , We [v] is the v-th column of We , t ? ? is the t-th column of X, ? is a positive number we denote as temperature parameter [29]. This x parameter is akin to the concentration parameter of a Dirichlet distribution, in that it controls the spread of probability vector [p(w ? t = 1) . . . p(w ? t = V )], thus a large ? encourages uniformly distributed probabilities, whereas a small ? encourages sparse, concentrated probability values. In the experiments we set ? = 0.01. Note that in our setting, the cosine similarity can be obtained ? have unit `2 -norm by specification. This as an inner product, provided that columns of We and X deconvolutional module can also be leveraged as building block in VAE[30, 31] or GAN[32, 33] 3 2.3 Model learning The objective of the convolutional autoencoder described above can be written as the word-wise log-likelihood for all sentences s ? D, i.e., P P Lae = d?D t log p(w ?dt = wdt ) , (2) where D denotes the set of observed sentences. The simple, maximum-likelihood objective in (2) is optimized via stochastic gradient descent. Details of the implementation are provided in the experiments. Note that (2) differs from prior related work in two ways: i) [22, 34] use pooling and un-pooling operators, while we use convolution/deconvolution with stride; and ii) more importantly, [22, 34] do not use a cosine similarity reconstruction as in (1), but a RNN-based decoder. A further discussion of related work is provided in Section 3. We could use pooling and un-pooling instead of striding (a particular case of deterministic pooling/un-pooling), however, in early experiments (not shown) we did not observe significant performance gains, while convolution/deconvolution operations with stride are considerably more efficient in terms of memory footprint. Compared to a standard LSTM-based RNN sequence autoencoders with roughly the same number of parameters, computations in our case are considerably faster (see experiments) using single NVIDIA TITAN X GPU. This is due to the high parallelization efficiency of CNNs via cuDNN primitives [35]. Comparison between deconvolutional and RNN Decoders The proposed framework can be seen as a complementary building block for natural language modeling. Contrary to the standard LSTMbased decoder, the deconvolutional decoder imposes in general a less strict sequence dependency compared to RNN architectures. Specifically, generating a word from an RNN requires a vector of hidden units that recursively accumulate information from the entire sentence in an order-preserving manner (long-term dependencies are heavily down-weighted), while for a deconvolutional decoder, the generation only depends on a representation vector that encapsulates information from throughout the sentence without a pre-specified ordering structure. As a result, for language generation tasks, a RNN decoder will usually generate more coherent text, when compared to a deconvolutional decoder. On the contrary, a deconvolutional decoder is better at accounting for distant dependencies in long sentences, which can be very beneficial in feature extraction for classification and text summarization tasks. 2.4 Semi-supervised classification and summarization Identifying related topics or sentiments, and abstracting (short) summaries from user generated content such as blogs or product reviews, has recently received significant interest [1, 3, 4, 36, 37, 13, 11]. In many practical scenarios, unlabeled data are abundant, however, there are not many practical cases where the potential of such unlabeled data is fully realized. Motivated by this opportunity, here we seek to complement scarcer but more valuable labeled data, to improve the generalization ability of supervised models. By ingesting unlabeled data, the model can learn to abstract latent representations that capture the semantic meaning of all available sentences irrespective of whether or not they are labeled. This can be done prior to the supervised model training, as a two-step process. Recently, RNN-based methods exploiting this idea have been widely utilized and have achieved state-of-the-art performance in many tasks [1, 3, 4, 36, 37]. Alternatively, one can learn the autoencoder and classifier jointly, by specifying a classification model whose input is the latent representation, h; see for instance [38, 31]. In the case of product reviews, for example, each review may contain hundreds of words. This poses challenges when training RNN-based sequence encoders, in the sense that the RNN has to abstract information on-the-fly as it moves through the sentence, which often leads to loss of information, particularly in long sentences [39]. Furthermore, the decoding process uses ground-truth information during training, thus the learned representation may not necessarily keep all information from the input text that is necessary for proper reconstruction, summarization or classification. We consider applying our convolutional autoencoding framework to semi-supervised learning from long-sentences and paragraphs. Instead of pre-training a fully unsupervised model as in [1, 3], we cast the semi-supervised task as a multi-task learning problem similar to [40], i.e., we simultaneously train a sequence autoencoder and a supervised model. In principle, by using this joint training strategy, the learned paragraph embedding vector will preserve both reconstruction and classification ability. 4 Specifically, we consider the following objective: P P P Lsemi = ? d?{Dl +Du } t log p(w ?dt = wdt ) + d?Dl Lsup (f (hd ), yd ) , (3) where ? > 0 is an annealing parameter balancing the relative importance of supervised and unsupervised loss; Dl and Du denote the set of labeled and unlabeled data, respectively. The first term in (3) is the sequence autoencoder loss in (2) for the d-th sequence. Lsup (?) is the supervision loss for the d-th sequence (labeled only). The classifier function, f (?), that attempts to reconstruct yd from hd can be either a Multi-Layer Perceptron (MLP) in classification tasks, or a CNN/RNN in text summarization tasks. For the latter, we are interested in a purely convolutional specification, however, we also consider an RNN for comparison. For classification, we use a standard cross-entropy loss, and for text summarization we use either (2) for the CNN or the standard LSTM loss for the RNN. In practice, we adopt a scheduled annealing strategy for ? as in [41, 42], rather than fixing it a priori as in [1]. During training, (3) gradually transits from focusing solely on the unsupervised sequence autoencoder to the supervised task, by annealing ? from 1 to a small positive value ?min . We set ?min = 0.01 in the experiments. The motivation for this annealing strategy is to first focus on abstracting paragraph features, then to selectively refine learned features that are most informative to the supervised task. 3 Related Work Previous work has considered leveraging CNNs as encoders for various natural language processing tasks [22, 34, 21, 43, 44]. Typically, CNN-based encoder architectures apply a single convolution layer followed by a pooling layer, which essentially acts as a detector of specific classes of h-grams, given a convolution filter window of size h. The deep architecture in our framework will, in principle, enable the high-level layers to capture more sophisticated language features. We use convolutions with stride rather than pooling operators, e.g., max-pooling, for spatial downsampling following [26, 45], where it is argued that fully convolutional architectures are able to learn their own spatial downsampling. Further, [46] uses a 29-layer CNN for text classification. Our CNN encoder is considerably simpler in structure (convolutions with stride and no more than 4 layers) while still achieving good performance. Language decoders other than RNNs are less well studied. Recently, [47] proposed a hybrid model by coupling a convolutional-deconvolutional network with an RNN, where the RNN acts as decoder and the deconvolutional model as a bridge between the encoder (convolutional network) and decoder. Additionally, [42, 48, 49, 50] considered CNN variants, such as pixelCNN [51], for text generation. Nevertheless, to achieve good empirical results, these methods still require the sentences to be generated sequentially, conditioning on the ground truth historical information, akin to RNN-based decoders, thus still suffering from the exposure bias. Other efforts have been made to improve embeddings from long paragraphs using unsupervised approaches [2, 52]. The paragraph vector [2] learns a fixed length vector by concatenating it with a word2vec [53] embedding of history sequence to predict future words. The hierarchical neural autoencoder [52] builds a hierarchical attentive RNN, then it uses paragraph-level hidden units of that RNN as embedding. Our work differs from these approaches in that we force the sequence to be fully restored from the latent representation, without aid from any history information. Previous methods have considered leveraging unlabeled data for semi-supervised sequence classification tasks. Typically, RNN-based methods consider either i) training a sequence-to-sequence RNN autoencoder, or a RNN classifier that is robust to adversarial perturbation, as initialization for the encoder in the supervised model [1, 4]; or, ii) learning latent representation via a sequence-to-sequence RNN autoencoder, and then using them as inputs to a classifier that also takes features extracted from a CNN as inputs [3]. For summarization tasks, [54] has considered a semi-supervised approach based on support vector machines, however, so far, research on semi-supervised text summarization using deep models is scarce. 4 Experiments Experimental setup For all the experiments, we use a 3-layer convolutional encoder followed by a 3-layer deconvolutional decoder (recall implementation details for the top layer). Filter size, stride 5 Ground-truth: Hier. LSTM [52] Our LSTM-LSTM Our CNN-DCNN on every visit to nyc , the hotel beacon is the place we love to stay . so conveniently located to central park , lincoln center and great local restaurants . the rooms are lovely . beds so comfortable , a great little kitchen and new wizz bang coffee maker . the staff are so accommodating and just love walking across the street to the fairway supermarket with every imaginable goodies to eat . every time in new york , lighthouse hotel is our favorite place to stay . very convenient , central park , lincoln center , and great restaurants . the room is wonderful , very comfortable bed , a kitchenette and a large explosion of coffee maker . the staff is so inclusive , just across the street to walk to the supermarket channel love with all kinds of what to eat . on every visit to nyc , the hotel beacon is the place to relax and wanting to become conveniently located . hotel , in the evenings out good budget accommodations . the views are great and we were more than two couples . manny the doorman has a great big guy come and will definitly want to leave during my stay and enjoy a wonderfully relaxing wind break in having for 24 hour early rick?s cafe . oh perfect ! easy easy walking distance to everything imaginable groceries . if you may want to watch yours ! on every visit to nyc , the hotel beacon is the place we love to stay . so closely located to central park , lincoln center and great local restaurants . biggest rooms are lovely . beds so comfortable , a great little kitchen and new UNK suggestion coffee maker . the staff turned so accommodating and just love walking across the street to former fairway supermarket with every food taxes to eat . Table 1: Reconstructed paragraph of the Hotel Reviews example, used in [52]. and word embedding are set to h = 5, rl = 2, for l = 1, . . . , 3 and k = 300, respectively. The dimension of the latent representation vector varies for each experiment, thus is reported separately. For notational convenience, we denote our convolutional-deconvolutional autoencoder as CNNDCNN. In most comparisons, we also considered two standard autoencoders as baselines: a) CNNLSTM: CNN encoder coupled with LSTM decoder; and b) LSTM-LSTM: LSTM encoder with LSTM decoder. An LSTM-DCNN configuration is not included because it yields similar performance to CNN-DCNN while being more computationally expensive. The complete experimental setup and baseline details is provided in the Supplementary Material (SM). CNN-DCNN has the least number of parameters. For example, using 500 as the dimension of h results in about 9, 13, 15 million total trainable parameters for CNN-DCNN, CNN-LSTM and LSTM-LSTM, respectively. BLEU 24.1 26.7 28.5 18.3 94.2 ROUGE-1 57.1 59.0 62.4 56.6 97.0 ROUGE-2 30.2 33.0 35.5 28.2 94.2 100 Bleu score Model LSTM-LSTM [52] Hier. LSTM-LSTM [52] Hier. + att. LSTM-LSTM [52] CNN-LSTM CNN-DCNN 80 60 40 CNN-DCNN CNN-LSTM LSTM-LSTM 20 0 60 80 100 120 140 160 180 200 Sentence length Table 2: Reconstruction evaluation results on the Hotel Reviews Figure 2: BLEU score vs. sentence Dataset. length for Hotel Review data. Paragraph reconstruction We first investigate the performance of the proposed autoencoder in terms of learning representations that can preserve paragraph information. We adopt evaluation criteria from [52], i.e., ROUGE score [55] and BLEU score [56], to measure the closeness of the reconstructed paragraph (model output) to the input paragraph. Briefly, ROUGE and BLEU scores measures the n-gram recall and precision between the model outputs and the (ground-truth) references. We use BLEU-4, ROUGE-1, 2 in our evaluation, in alignment with [52]. In addition to the CNNLSTM and LSTM-LSTM autoencoder, we also compared with the hierarchical LSTM autoencoder [52]. The comparison is performed on the Hotel Reviews datasets, following the experimental setup from [52], i.e., we only keep reviews with sentence length ranging from 50 to 250 words, resulting in 348,544 training data samples and 39,023 testing data samples. For all comparisons, we set the dimension of the latent representation to h = 500. From Table 1, we see that for long paragraphs, the LSTM decoder in CNN-LSTM and LSTM-LSTM suffers from heavy exposure bias issues. We further evaluate the performance of each model with different paragraph lengths. As shown in Figure 2 and Table 2, on this task CNN-DCNN demonstrates a clear advantage, meanwhile, as the length of the sentence increases, the comparative advantage becomes more substantial. For LSTM-based methods, the quality of the reconstruction deteriorates quickly as sequences get longer. In constrast, the reconstruction quality of CNN-DCNN is stable and consistent regardless of sentence length. Furthermore, the computational cost, evaluated as wall-clock, is significantly lower in CNN-DCNN. Roughly, CNN-LSTM is 3 times slower than CNN-DCNN, and LSTM-LSTM is 5 times slower on a single GPU. Details are reported in the SM. Character-level and word-level correction This task seeks to evaluate whether the deconvolutional decoder can overcome exposure bias, which severely limits LSTM-based decoders. We consider 6 a denoising autoencoder where the input is tweaked slightly with certain modifications, while the model attempts to denoise (correct) the unknown modification, thus recover the original sentence. Character Error Rate (CER) For character-level correction, we consider the Yahoo! Answer dataset [57]. The dataset description and setup for word-level correction is provided in the SM. We follow the experimental setup in [58] for word-level and character-level spelling correction (see details in the SM). We considered substituting each word/character with a different one at random with probability ?, with ? = 0.30. For character-level analysis, we first map all characters into a 40 dimensional embedding vector, with the network structure for word- and character-level models kept the same. 1.0 0.8 0.6 Original c a n a n y o n e s u g g e s t s o m e g o o d b o o k s ? CNN-DCNN CNN-LSTM LSTM-LSTM LSTM-LSTM c a n a n y o n e s u g g e s t j o k e f o o d y o u n g ? CNN-LSTM c a n a n y o n e g u i t e s s o m e o w e p o o k s ? CNN-DCNN c a n a n y o n e s u g g e s t s o m e w o o d b o o k s ? 0.4 Original w h a t s y o u r i d e a o f a s t e p p i n g s t o n e t o b e t t e r t h i n g s t o c o m e ? Modified w u a t s y o g r i d e m o f t s t e p u k n g j t z n e t i b e t t e r t h i n g z t t c o e e ? 0.2 0.0 0 Modified c a p a n y o n k w u g g e s t x o h e i o r d y o o k u ? ActorCritic c a n a n y o n e w i t h e s t t o e f o r d y o u u ? ActorCritic w h a t s y o u r i d e m o f t s t e p u a n g j o k n e t i b e t t e r t h i n g i t t c o m e ? LSTM-LSTM w h a t s y o u r i d e a o f a s p e a k i n g s t a n d t o b e t t e r t h i n g s t o c o m e ? 10 20 30 40 50 60 70 Time (hour) CNN-LSTM w h a t s y o u r i d e m o f a s t e p p i n g s t a r t t o b e t t e r t h i n g t o c o m e ? CNN-DCNN w h a t s y o u r i d e a o f a s t e p p i n g s t o n e t o b e t t e r t h i n g s t o c o m e ? Model Actor-critic[58] LSTM-LSTM CNN-LSTM CNN-DCNN Model LSTM-LSTM CNN-LSTM CNN-DCNN Yahoo(CER) 0.2284 0.2621 0.2035 0.1323 ArXiv(WER) 0.7250 0.3819 0.3067 Figure 3: CER comparison. Figure 4: Spelling error denoising compar- Table 3: CER and WER comBlack triangles indicate the end ison. Darker colors indicate higher uncer- parison on Yahoo and ArXiv of an epoch. tainty. Trained on modified sentences. data. We employ Character Error Rate (CER) [58] and Word Error Rate (WER) [59] for evaluation. The WER/CER measure the ratio of Levenshtein distance (a.k.a., edit distance) between model predictions and the ground-truth, and the total length of sequence. Conceptually, lower WER/CER indicates better performance. We use LSTM-LSTM and CNN-LSTM denoising autoencoders for comparison. The architecture for the word-level baseline models is the same as in the previous experiment. For character-level correction, we set dimension of h to 900. We also compare to actor-critic training [58], following their experimental guidelines (see details in the SM). As shown in Figure 3 and Table 3, we observed CNN-DCNN achieves both lower CER and faster convergence. Further, CNN-DCNN delivers stable denoising performance irrespective of the noise location within the sentence, as seen in Figure 4. For CNN-DCNN, even when an error is detected but not exactly corrected (darker colors in Figure 4 indicate higher uncertainty), denoising with future words is not effected, while for CNN-LSTM and LSTM-LSTM the error gradually accumulates with longer sequences, as expected. For word-level correction, we consider word substitutions only, and mixed perturbations from three kinds: substitution, deletion and insertion. Generally, CNN-DCNN outperforms CNN-LSTM and LSTM-LSTM, and is faster. We provide experimental details and comparative results in the SM. Semi-supervised sequence classification & summarization We investigate whether our CNNDCNN framework can improve upon supervised natural language tasks that leverage features learned from paragraphs. In principle, a good unsupervised feature extractor will improve the generalization ability in a semi-supervised learning setting. We evaluate our approach on three popular natural language tasks: sentiment analysis, paragraph topic prediction and text summarization. The first two tasks are essentially sequence classification, while summarization involves both language comprehension and language generation. We consider three large-scale document classification datasets: DBPedia, Yahoo! Answers and Yelp Review Polarity [57]. The partition of training, validation and test sets for all datasets follows the settings from [57]. The detailed summary statistics of all datasets are shown in the SM. To demonstrate the advantage of incorporating the reconstruction objective into the training of text classifiers, we further evaluate our model with different amounts of labeled data (0.1%, 0.15%, 0.25%, 1%, 10% and 100%, respectively), and the whole training set as unlabeled data. For our purely supervised baseline model (supervised CNN), we use the same convolutional encoder architecture described above, with a 500-dimensional latent representation dimension, followed by a MLP classifier with one hidden layer of 300 hidden units. The dropout rate is set to 50%. Word embeddings are initialized at random. As shown in Table 4, the joint training strategy consistently and significantly outperforms the purely supervised strategy across datasets, even when all labels are available. We hypothesize that during the early phase of training, when reconstruction is emphasized, features from text fragments can be readily 7 Model ngrams TFIDF Large Word ConvNet Small Word ConvNet Large Char ConvNet Small Char ConvNet SA-LSTM (word-level) Deep ConvNet Ours (Purely supervised) Ours (joint training with CNN-LSTM) Ours (joint training with CNN-DCNN) DBpedia 1.31 1.72 1.85 1.73 1.98 1.40 1.29 1.76 1.36 1.17 Yelp P. 4.56 4.89 5.54 5.89 6.53 4.28 4.62 4.21 3.96 Yahoo 31.49 29.06 30.02 29.55 29.84 26.57 27.42 26.32 25.82 Accuracy (%) learned. As the training proceeds, the most discriminative text fragment features are selected. Further, the subset of features that are responsible for both reconstruction and discrimination presumably encapsulate longer dependency structure, compared to the features using a purely supervised strategy. Figure 5 demonstrates the behavior of our model in a semi-supervised setting on Yelp Review dataset. The results for Yahoo! Answer and DBpedia are provided in the SM. 100 95 90 85 80 Supervised 75 70 Semi (CNN-DCNN) 65 Semi (CNN-LSTM) 60 55 0.1 1 10 100 Proportion (%) of labeled data Table 4: Test error rates of document classification (%). Results Figure 5: Semi-supervised classifica- from other methods were obtained from [57]. tion accuracy on Yelp review data. For summarization, we used a dataset composed of 58,000 abstract-title pairs, from arXiv. Abstracttitle pairs are selected if the length of the title and abstract do not exceed 50 and 500 words, respectively. We partitioned the training, validation and test sets into 55000, 2000, 1000 pairs each. We train a sequence-to-sequence model to generate the title given the abstract, using a randomly selected subset of paired data with proportion ? = (5%, 10%, 50%, 100%). For every value of ?, we considered both purely supervised summarization using just abstract-title pairs, and semisupervised summarization, by leveraging additional abstracts without titles. We compared LSTM and deconvolutional network as the decoder for generating titles for ? = 100%. Table 5 summarizes quantitative results Obs. proportion ? 5% 10% 50% 100% DCNN dec. using ROUGE-L (longest common subSupervised 12.40 13.07 15.87 16.37 14.75 Semi-sup. 16.04 16.62 17.64 18.14 16.83 sequence) [55]. In general, the additional abstracts without titles improve the gen- Table 5: Summarization task on arXiv data, using ROUGE-L eralization ability on the test set. Inter- metric. First 4 columns are for the LSTM decoder, and the last estingly, even when ? = 100% (all titles column is for the deconvolutional decoder (100% observed). are observed), the joint training objective still yields a better performance than using Lsup alone. Presumably, since the joint training objective requires the latent representation to be capable of reconstructing the input paragraph, in addition to generating a title, the learned representation may better capture the entire structure (meaning) of the paragraph. We also empirically observed that titles generated under the joint training objective are more likely to use the words appearing in the corresponding paragraph (i.e., more extractive), while the the titles generated using the purely supervised objective Lsup , tend to use wording more freely, thus more abstractive. One possible explanation is that, for the joint training strategy, since the reconstructed paragraph and title are all generated from latent representation h, the text fragments that are used for reconstructing the input paragraph are more likely to be leveraged when ?building? the title, thus the title bears more resemblance to the input paragraph. As expected, the titles produced by a deconvolutional decoder are less coherent than an LSTM decoder. Presumably, since each paragraph can be summarized with multiple plausible titles, the deconvolutional decoder may have trouble when positioning text segments. We provide discussions and titles generated under different setups in the SM. Designing a framework which takes the best of these two worlds, LSTM for generation and CNN for decoding, will be an interesting future direction. 5 Conclusion We proposed a general framework for text modeling using purely convolutional and deconvolutional operations. The proposed method is free of sequential conditional generation, avoiding issues associated with exposure bias and teacher forcing training. Our approach enables the model to fully encapsulate a paragraph into a latent representation vector, which can be decompressed to reconstruct the original input sequence. Empirically, the proposed approach achieved excellent long paragraph reconstruction quality and outperforms existing algorithms on spelling correction, and semi-supervised sequence classification and summarization, with largely reduced computational cost. 8 Acknowledgements This research was supported in part by ARO, DARPA, DOE, NGA and ONR. References [1] Andrew M Dai and Quoc V Le. Semi-supervised sequence learning. In NIPS, 2015. [2] Quoc Le and Tomas Mikolov. Distributed representations of sentences and documents. In ICML, 2014. [3] Rie Johnson and Tong Zhang. Supervised and Semi-Supervised Text Categorization using LSTM for Region Embeddings. arXiv, February 2016. [4] Takeru Miyato, Andrew M Dai, and Ian Goodfellow. Adversarial Training Methods for Semi-Supervised Text Classification. In ICLR, May 2017. [5] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural Machine Translation by Jointly Learning to Align and Translate. In ICLR, 2015. 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Random Permutation Online Isotonic Regression Wojciech Kot?owski Pozna?n University of Technology Poland [email protected] Wouter M. Koolen Centrum Wiskunde & Informatica Amsterdam, The Netherlands [email protected] Alan Malek MIT Cambridge, MA [email protected] Abstract We revisit isotonic regression on linear orders, the problem of fitting monotonic functions to best explain the data, in an online setting. It was previously shown that online isotonic regression is unlearnable in a fully adversarial model, which lead to its study in the fixed design model. Here, we instead develop the more practical random permutation model. We show that the regret is bounded above by the excess leave-one-out loss for which we develop efficient algorithms and matching lower bounds. We also analyze the class of simple and popular forward algorithms and recommend where to look for algorithms for online isotonic regression on partial orders. 1 Introduction A function f : R ? R is called isotonic (non-decreasing) if x ? y implies f (x) ? f (y). Isotonic functions model monotonic relationships between input and output variables, like those between drug dose and response [25] or lymph node condition and survival time [24]. The problem of isotonic regression is to find the isotonic function that best explains a given data set or population distribution. The isotonic regression problem has been extensively studied in statistics [1, 24], which resulted in efficient optimization algorithms for fitting isotonic functions to the data [7, 16] and sharp convergence rates of estimation under various model assumptions [26, 29]. In online learning problems, the data arrive sequentially, and the learner is tasked with predicting each subsequent data point as it arrives [6]. In online isotonic regression, the natural goal is to predict the incoming data points as well as the best isotonic function in hindsight. Specifically, for time steps t = 1, . . . , T , the learner observes an instance xi ? R, makes a prediction ybi of the true label yi , which is assumed to lie in [0, 1]. There is no restriction that the labels or predictions be isotonic. We evaluate a prediction ybi by its squared loss (b yi ? yi )2 . The quality of an algorithm is measured by its PT yi ? yi )2 ? L?T , where L?T is the loss of the best isotonic function on the entire data regret, t=1 (b sequence. Isotonic regression is nonparametric: the number of parameters grows linearly with the number of data points. It is thus natural to ask whether there are efficient, provably low regret algorithms for online isotonic regression. As of yet, the picture is still very incomplete in the online setting. The first online results were obtained in the recent paper [14] which considered linearly ordered domains in the adversarial fixed design model, i.e. a model in which all the inputs x1 , . . . , xT are given to the learner before the start of prediction. The authors show that, due to the nonparametric nature of the problem, many textbook online learning algorithms fail to learn at all (including Online Gradient Descent, Follow the Leader and Exponential Weights) in the sense that their worst-case regret grows 1 linearly with the number of data points. They prove a ?(T 3 ) worst case regret lower bound, and 1 ? 3 ) regret. Unfortunately, the fixed design develop a matching algorithm that achieves the optimal O(T assumption is often unrealistic. This leads us to our main question: Can we design methods for online isotonic regression that are practical (do not hinge on fixed design)? 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Our contributions Our long-term goal is to design practical and efficient methods for online isotonic regression, and in this work we move beyond the fixed design model and study algorithms that do not depend on future instances. Unfortunately, the completely adversarial design model (in which the instances are selected by an adaptive adversary) is impossibly hard: every learner can suffer linear regret in this model [14]. So in order to drop the fixed design assumption, we need to constrain the adversary in some other way. In this paper we consider the natural random permutation model, in which all T instances and labels are chosen adversarially before the game begins but then are presented to the learner in a random order. This model corresponds with the intuition that the data gathering process (which fixes the order) is independent of the underlying data generation mechanism (which fixes instances and labels). We will show that learning is possible in the random permutation model (in fact we present a reduction ? 13 ) upper bound on showing that it is not harder than adversarial fixed design) by proving an O(T regret for an online-to-batch conversion of the optimal fixed design algorithm from [14] (Section 3). Our main tool for analyzing the random permutation model is the leave-one-out loss, drawing interesting connections with cross-validation and calibration. The leave-one-out loss on a set of t labeled instances is the error of the learner predicting the i-th label after seeing all remaining t ? 1 labels, averaged uniformly over i = 1, . . . , t. We begin by proving a general correspondence between regret and leave-one-out loss for the random permutation model in Section 2.1, which allows us to use excess leave-one-out loss as a proxy for regret. We then describe a version of online-to-batch conversion that relates the fixed design model with the random permutation model, resulting in an ? 31 ) regret. algorithm that attains the optimal O(T Section 4 then turns to the computationally efficient and natural class of forward algorithms that use an offline optimization oracle to form their prediction. This class contains most common online 1 isotonic regression algorithms. We then show a O(T 2 ) upper bound on the regret for the entire class, 1 which improves to O(T 3 ) for the well-specified case where the data are in fact generated from an isotonic function plus i.i.d. noise (the most common model in the statistics literature). 1 While forward algorithms match the lower bound for the well-specified case, there is a factor T 6 gap in the random permutation case. Section 4.6 proposes a new algorithm that calls a weighted offline oracle with a large weight on the current instance. This algorithm can be efficiently computed via [16]. We prove necessary bounds on the weight. Related work Offline isotonic regression has been extensively studied in statistics starting from work by [1, 4]. Applications range across statistics, biology, medicine, psychology, etc. [24, 15, 25, 22, 17]. In statistics, isotonic regression is studied in generative models [26, 3, 29]. In machine learning, isotonic regression is used for calibrating class probability estimates [28, 21, 18, 20, 27], ROC analysis [8], training Generalized Linear Models and Single Index Models[12, 11], data cleaning [13], and ranking [19]. Fast algorithms for partial ordering are developed in [16]. In the online setting, [5] bound the minimax regret for monotone predictors under logarithmic loss and [23, 10] study online nonparametric regression in general. Efficient algorithms and worst-cases regret bounds for fixed design online isotonic regression are studied in [14]. Finally, the relation between regret and leave-one-out loss was pioneered by [9] for linear regression. 2 Problem Setup Given a finite set of instances {x1 , . . . , xt } ? R, a function f : {x1 , . . . , xt } ? [0, 1] is isotonic (non-decreasing) if xi ? xj implies f (xi ) ? f (xj ) for all i, j ? {1, . . . , t}. Given a set of labeled instances D = {(x1 , y1 ), . . . , (xt , yt )} ? R ? [0, 1], let L? (D) denote the total squared loss of the best isotonic function on D, t X L? (D) := min (yi ? f (xi ))2 . isotonic f i=1 This convex optimization problem can be solved by the celebrated Pool Adjacent Violators Algorithm (PAVA) in time linear in t [1, 7]. The optimal solution, called the isotonic regression function, is piecewise constant and its value on any of its levels sets equals the average of labels within that set [24]. 2 Online isotonic regression in the random permutation model is defined as follows. At the beginning of the game, the adversary chooses data instances x1 < . . . < xT 1 and labels y1 , . . . , yT . A permutation ? = (?1 , . . . , ?T ) of {1, . . . , T } is then drawn uniformly at random and used to determine the order in which the data will be revealed. In round t, the instance x?t is revealed to the learner who then predicts yb?t . Next, the learner observes the true label y?t and incurs the squared loss (b y?t ? y?t )2 . ? ? For a fixed permutation ?, we use the shorthand notation Lt = L ({(x?1 , y?1 ), . . . , (x?t , y?t )}) to denote the optimal isotonic regression loss of the first t labeled instances (L?t will clearly depend on ?, except for the case t = T ). The goal of the learner is to minimize the expected regret, X  T T X RT := E? (y?t ? yb?t )2 ? L?T = rt , t=1 t=1 where we have decomposed the regret into its per-round increase, h i rt := E? (y?t ? yb?t )2 ? L?t + L?t?1 , (1) L?0 := 0. To simplify the analysis, let us assume that the prediction strategy does not depend with on the order in which the past data were revealed (which is true for all algorithms considered in this paper). Fix t and define D = {(x?1 , y?1 ), . . . , (x?t , y?t )} to be the set of first t labeled instances. Furthermore, let D?i = D\{(x?i , y?i )} denote the set D with the instance from round i removed. Using this notation, the expression under the expectation in (1) can be written as 2 y?t ? yb?t (D?t ) ? L? (D) + L? (D?t ), where we made the dependence of yb?t on D?t explicit (and used the fact that it only depends on the set of instances, not on their order). By symmetry of the expectation over permutations with respect to the indices, we have h h 2 i 2 i     E? y?t ? yb?t (D?t ) = E? y?i ? yb?i (D?i ) , and E? L? (D?t ) = E? L? (D?i ) , for all i = 1, . . . , t. Thus, (1) can as well be rewritten as:   X t   2 1 y?i ? yb?i (D?i ) + L? (D?i ) ? L? (D) . rt = E? t i=1 Let us denote the expression inside the expectation by rt (D) to stress its dependence on the set of instances D, but not on the order in which they were revealed. If we can show that rt (D) ? Bt holds PT for all t, then its expectation has the same bound, so RT ? t=1 Bt . 2.1 Excess Leave-One-Out Loss and Regret Our main tool for analyzing the random permutation model is the leave-one-out loss. In the leave-one-out model, there is no sequential structure. The adversary picks a data set D = {(x1 , y1 ), . . . , (xt , yt )} with x1 < . . . < xt . An index i is sampled uniformly at random, the learner is given D?i , the entire data set except (xi , yi ), and predicts ybi (as a function of D?i ) on instance xi . We call the difference between the expected loss of the learner and L? (D) the expected excess leave-one-out loss: ! X  t 2 1 ? `oot (D) := yi ? ybi (D?i ) ? L (D) . (2) t i=1 The random permutation model has the important property that the bound on the excess leave-one-out loss of a prediction algorithm translates into a regret bound. A similar result has been shown by [9] for expected loss in the i.i.d. setting. Lemma 2.1. rt (D) ? `oot (D) for any t and any data set D = {(x1 , y1 ), . . . , (xt , yt )}. Pt 2 Proof. As x1 < . . . < xt , let (y1? , . . . , yt? ) = argminf1 ?...?ft i=1 (yi ? fi ) be the isotonic Pt ? ? regression function on D. From the definition of L , we can see that L (D) = i=1 (yi? ? yi )2 ? L? (D?i ) + (yi ? yi? )2 . Thus, the regret increase rt (D) is bounded by rt (D) = t X (yi ? ybi )2 + L? (D?i ) i=1 t ? L? (D) ? t X (yi ? ybi )2 ? (yi ? y ? )2 i i=1 1 t = `oot (D). We assume all points xt are distinct as it will significantly simplify the presentation. All results in this paper are also valid for the case x1 ? . . . ? xT . 3 However, we note that lower bounds for `oot (D) do not imply lower bounds on regret. In what follows, our strategy will be to derive bounds `oot (D) ? Bt for various algorithms, from PT which the regret bound RT ? t=1 Bt can be immediately obtained. From now on, we abbreviate `oot (D) to `oot , (as D is clear from the context); we will also consistently assume x1 < . . . < xt . 2.2 Noise free case As a warm-up, we analyze the noise-free case (when the labels themselves are isotonic) and demonstrate that analyzing `oot easily results in an optimal bound for this setting. Proposition 2.2. Assume that the labels satisfy y1 ? y2 ? . . . ? yt . The prediction ybi that is the linear interpolation between adjacent labels ybi = 12 (yi?1 + yi+1 ), has `oot ? 1 1 , and thus RT ? log(T + 1). 2t 2 Pt 1 2 ? Proof. For ?i := yi ? yi?1 , it is easy to check that `oot = 4t i=1 (?i+1 ? ?i ) because the L (D) term is zero. This expression is a convex function of ?1 , . . . , ?t+1 . Note that ?i ? 0 for each Pt+1 i = 1, . . . , t + 1, and i=1 ?i = 1. Since the maximum of a convex function is at the boundary of the feasible region, the maximizer is given by ?i = 1 for some i ? {1, . . . , t + 1}, and ?j = 0 for all j ? {1, . . . , t + 1}, j 6= i. This implies that `oot ? (2t)?1 . 2.3 General Lower Bound In [14], a general lower bound was derived showing that the regret of any online isotonic regression 1 procedure is at least ?(T 3 ) for the adversarial setup (when labels and the index order were chosen adversarially). This lower bound applies regardless of the order of outcomes, and hence it is also a lower bound for the random permutation model. This bound translates into `oot = ?(t?2/3 ). 3 Online-to-batch for fixed design Here, we describe an online-to-batch conversion that relates the adversarial fixed design model with the random permutation model considered in this paper. In the fixed design model with time horizon Tfd the learner is given the points x1 , . . . , xTfd in advance (which is not the case in the random permutation model), but the adversary chooses the order ? in which the labels are revealed (as opposed to ? being drawn at random). We can think of an algorithm for fixed design as a prediction function  ybfd x?t y?1 , . . . , y?t?1 , {x1 , . . . , xTfd } , for any order ?, any set {x1 , . . . , xTfd } (and hence any time horizon Tfd ), and any time t. This notation is quite heavy, but makes it explicit that the learner, while predicting at point x?t , knows the previously revealed labels and the whole set of instances. In the random permutation model, at trial t, the learner only knows the previously revealed t ? 1 labeled instances and predicts on the new instance. Without loss of generality, denote the past instances by D?i = {(x1 , y1 ), . . . , (xi?1 , yi?1 ), (xi+1 , yi+1 ), . . . (xt , yt )}, and the new instance by xi , for some i ? {1, . . . , t}. Given an algorithm for fixed design ybfd , we construct a prediction ybt = ybt (D?i , xi ) of the algorithm in the random permutation model. The reduction goes through an online-to-batch conversion. Specifically, at trial t, given past labeled instances D?i , and a new point xi , the learner plays the expectation of the prediction of the fixed design algorithm with time horizon T fd = t and points {x1 , . . . , xt } under a uniformly random time from the past j ? {1, . . . , t} and a random permutation ? on {1, . . . , t}, with ?t = i, i.e.2  X  t  1 fd ybt = E{?:?t =i} yb (xi |y?1 , . . . , y?j?1 , {x1 , . . . , xt } . (3) t j=1 2 Choosing the prediction as an expectation is elegant but inefficient. However, the proof indicates that we might as well sample a single j and a single random permutation ? to form the prediction and the reduction would also work in expectation. 4 Note that this is a valid construction, as the right hand side only depends on D?i and xi , which are known to the learner in a random permutation model at round t. We prove (in Appendix A) that the excess leave-one-out loss of yb at trial t is upper bounded by the expected regret (over all permutations) of ybfd in trials 1, . . . , t divided by t: Theorem 3.1. Let D = {(x1 , y1 ), . . . , (xt , yt )} be a set of t labeled instances. Fix any algorithm ybfd for online adversarial isotonic regression with fixed design, and let Regt (b y fd | ?) denote its regret on D when the labels are revealed in order ?. The random permutation learner yb from (3) ensures y fd | ?)]. `oot (D) ? 1t E? [Regt (b 1 ? 3 ) fixed design regret result from [14]. This constructions allows immediate transport of the O(T fd Theorem 3.2. There is an algorithm for the random-permutation model with excess leave-one-out ? ? 23 ) and hence expected regret RT ? P O(t ? 13 ). ? ? 32 ) = O(T loss `oot = O(t t 4 Forward Algorithms For clarity of presentation, we use vector notation in this section: y = (y1 , . . . , yt ) is the label vector, y ? = (y1? , . . . , yt? ) is the isotonic regression function, and y?i = (y1 , . . . , yi?1 , yi+1 , . . . , yt ) is y with i-th label removed. Moreover, keeping in mind that x1 < . . . < xt , we can drop xi ?s entirely from the notation and refer to an instance xi simply by its index i. Given labels y?i and some index i to predict on, we want a good prediction for yi . Follow the Leader (FL) algorithms, which predict using the best isotonic function on the data seen so far, are not directly applicable to online isotonic regression: the best isotonic function is only defined at the observed data instances and can be arbitrary (up to monotonicity constraint) otherwise. Instead, we analyze a simple and natural class of algorithms which we dub forward algorithms3 . We define a forward algorithm, or FA, to be any algorithm that estimates a label yi0 ? [0, 1] (possibly dependent on i and y?i ), and plays with the FL strategy on the sequence of past data including the new instance with the estimated label, i.e. performs offline isotonic regression on y 0 , X  t yb = argmin (yj0 ? fj )2 , where y 0 = (y1 , . . . , yi?1 , yi0 , yi+1 , . . . , yt ). f1 ?...?ft j=1 Then, FA predicts with ybi , the value at index i of the offline function of the augmented data. Note that if the estimate turned out to be correct (yi0 = yi ), the forward algorithm would suffer no additional loss for that round. Forward algorithms are practically important: we will show that many popular algorithms can be cast as FA with a particular estimate. FA automatically inherit any computational advances for offline isotonic regression; in particular, they scale efficiently to partially ordered data [16]. To our best knowledge, we are first to give bounds on the performance of these algorithms in the online setting. Alternative formulation We can describe a FA using a minimax representation of the isotonic regression [see, e.g., 24]: the optimal isotonic regression y ? satisfies yi? = min max y `,r = max min y `,r , r?i `?i Pr `?i r?i (4) yj j=` where y `,r = r?`+1 . The ?saddle point? (`i , ri ) for which yi? = y `i ,ri , specifies the boundaries of ? the level set {j : yj = yi? } of the isotonic regression function that contains i. It follows from (4) that isotonic regression is monotonic with respect to labels: for any two label sequences y and z such that yi ? zi for all i, we have yi? ? zi? for all i. Thus, if we denote the predictions for label estimates yi0 = 0 and yi0 = 1 by ybi0 and ybi1 , respectively, the monotonicity implies that any FA has ybi0 ? ybi ? ybi1 . Conversely, using the continuity of isotonic regression y ? as a function of y, (which follows from (4)), we can show that for any prediction ybi with ybi0 ? ybi ? ybi1 , there exists an estimate yt0 ? [0, 1] that could generate this prediction. Hence, we can equivalently interpret FA as an algorithm which in each trial predicts with some ybi in the range [b yi0 , ybi1 ]. 3 The name highlights resemblance to the Forward algorithm introduced by [2] for exponential family models. 5 4.1 Instances With the above equivalence between forward algorithms and algorithms that predict in [b yi0 , ybi1 ], we can show that many of the well know isotonic regression algorithms are forward algorithms and thereby add weight to our next section where we prove regret bounds for the entire class. Isotonic regression with interpolation (IR-Int)[28] Given y?i and index i, the algorithm  first ? ? computes f ? , the isotonic regression of y?i , and then predicts with ybiint = 21 fi?1 + fi+1 , where ? we used f0? = 0 and ft+1 = 1. To see that this is a FA, note that if we use estimate yi0 = ybiint , the ? ? , yi0 , fi+1 , . . . , ft? ). isotonic regression of y 0 = (y1 , . . . , yi?1 , yi0 , yi+1 , . . . , yt ) is yb = (f1? , . . . , fi?1 ? This is because: i) yb is isotonic by construction; ii) f has the smallest squared error loss for y?t among isotonic functions; and iii) the loss of yb on point yi0 is zero, and the loss of yb on all other points is equal to the loss of f ? . Direct combination of ybi0 and ybi1 . It is clear from Section 4, that any algorithm that predicts ybi = ?i ybi0 + (1 ? ?i )b yi1 for some ?i ? [0, 1] is a FA. The weight ?i can be set to a constant (e.g., ?i = 1/2), or can be chosen depending on ybi1 and ybi0 . Such algorithms were considered by [27]: log-IVAP : ybilog = ybi1 ybi1 , + 1 ? ybi0 Brier-IVAP : ybiBrier = 1 + (b yi0 )2 ? (1 ? ybi1 )2 . 2 It is straightforward to show that both algorithms satisfy ybi0 ? ybi ? ybi1 and are thus instances of FA. Last-step minimax (LSM). LSM plays the minimax strategy with one round remaining, n o ybi = argmin max (b y ? yi )2 ? L? (y) , y b?[0,1] yi ?[0,1] where L? (y) is the isotonic regression loss on y. Define L?b = L? (y1 , . . . , yi?1 , b, yi+1 , . . . , yt ) for b ? {0, 1}, i.e. L?b is the loss of isotonic regression function with label estimate yi0 = b. In ? 1+L? 0 ?L1 Appendix B we show that ybi = and it is also an instance of FA. 2 4.2 Bounding the leave-one-out loss q We now give a O( logt t ) bound on the leave-one-out loss for forward algorithms. Interestingly, the bound holds no matter what label estimate the algorithm uses. The proof relies on the stability of isotonic regression with respect to a change of a single label. While the bound looks suboptimal in light of Section 2.3, we will argue in Section 4.5 that the bound is actually tight (up to a logarithmic factor) for one FA and experimentally verify that all other mentioned forward algorithms also have a tight lower bound of that form for the same sequence of outcomes. We will bound `oot by defining ?i = ybi ? yi? and using the following simple inequality: t t t  1X 1 X 2X (b yi ? yi )2 ? (yi? ? yi )2 = (b yi ? yi? )(b yi + yi? ? 2yi ) ? |?i |. t i=1 t i=1 t i=1 q  log t Theorem 4.1. Any forward algorithm has `oot = O . t `oot = Proof. Fix some forward algorithm. For any i, let {j : yj? = yi? } = {`i , . . . , ri }, for some `i ? i ? ri , be the level set of isotonic regression at level yi? . We need the stronger version of the minimax representation, shown in Appendix C: yi? = min y `i ,r = max y `,ri . r?i `?i (5) n h o k We partition the points {1, . . . , t} into K consecutive segments: Sk = i : yi? ? k?1 , for K K  K?1 ? ? k = 1, . . . , K ? 1 and SK = i : yi ? K . Due to monotonicity of y , Sk are subsets of the form {`k , . . . , rk } (where we use rk = `k ? 1 if Sk is empty). From the definition, every level set of y ? is contained in Sk for some k, and each `k (rk ) is a left-end (right-end) of some level set. 6 Now, choose some index i, and let Sk be such that i ? Sk . Let yi0 be the estimate of the FA, and let y 0 = (y1 , . . . , yi?1 , yi0 , yi+1 , . . . , yt ). The minimax representation (4) and definition of FA imply yi0 ? yi o r?i r?i `?i r?i r ? `k + 1 yi0 ? yi yi0 ? yi ? min y `k ,r + min ? min y `k ,r + min r?i r?i r ? `k + 1 r?i r ? `k + 1 r?`k by (5) 1 ?1 1 1 ? y`?k + min ? y`?k ? ? yi? ? ? . r?i r ? `k + 1 i ? `k + 1 K i ? `k + 1 1 1 + rk ?i+1 . Hence, we can bound |?i | = |b A symmetric argument gives ybi ? yi? + K yi ? yi? | ?   P |Sk | 1 1 1 i?Sk |?i | ? K + 2 1 + log |Sk | , K + max i?`k +1 , rk ?i+1 . Summing over i ? Sk yields which allows the bound K 2X 2 `oot ? |?i | ? + 4 (1 + log t). t i K t p The theorem follows from setting K = ?( t/ log t). n ybi = max min y 0`,r ? min y 0`k ,r = min y `k ,r + 4.3 Forward algorithms for the well-specified case p P While the `oot upper bound of the previous section yields a regret bound RT ? t O( log t/t) = ? 12 ) that is a factor O(T 16 ) gap from the lower bound in Section 2.3, there are two pieces of good O(T news. First, forward algorithms do get the optimal rate in the well-specified setting, popular in the classical statistics literature, where the labels are generated i.i.d. such that E[yi ] = ?i with isotonic 1 ?1 ? . . . ? ?t .4 Second, there is a ?(t? 2 ) lower bound for forward algorithms as proven in the next section. Together, these results imply that the random permutation model in indeed harder than the well-specified case: forward algorithms are sufficient for the latter but not the former. Theorem 4.2. For data generated from the well-specified setting (monotonic means with i.i.d. noise), ? 13 ) bound on the regret. ? ? 23 ), which translates to a O(T any FA has `oot = O(t The proof is given in Appendix D. Curiously, the proof makes use of the existence of the seemingly ? ? 32 ) excess leave-one-out loss from Theorem 3.2. unrelated optimal algorithm with O(t 4.4 Entropic loss We now abandon the squared loss for a moment and analyze how a FA performs when the loss function is the entropic loss, defined as ?y log yb ? (1 ? y) log(1 ? yb) for y ? [0, 1]. Entropic loss (precisely: its binary-label version known as log-loss) is extensively used in the isotonic regression context for maximum likelihood estimation [14] or for probability calibration [28, 21, 27]. A surprising fact in isotonic regression is that minimizing entropic loss5 leads to exactly the same optimal solution as in the squared loss case, the isotonic regression function y ? [24]. Not every FA is appropriate for entropic loss, as recklessly choosing the label estimate might result in an infinite loss in just a single trial (as noted by [27]). Indeed, consider a sequence of outcomes with y1 = 0 and yi = 1 for i > 1. While predicting on index i = 1, choosing y10 = 1 results in yb1 = 1, for which the entropic loss is infinite (as y1 = 0). Does there exists a FA which achieves a meaningful bound on `oot in the entropic loss setup? We answer this question in the affirmative, showing that the log-IVAP predictor FA gets the same excess-leave-one-out loss bound as given in Theorem 4.1. As the reduction from the regret to leaveone-out loss (Lemma 2.1) does not use any properties of the loss function, this immediately implies a bound on the expected regret. Interestingly, the proof (given in Appendix G) uses as an intermediate step the bound on |?i | for the worst possible forward algorithm which always produces the estimate being the opposite of the actual label. q  log t Theorem 4.3. The log-IVAP algorithm has `oot = O for the entropic loss. t 4 5 The ?(T 1/3 ) regret lower bound in [14] uses a mixture of well-specified distributions and still applies. In fact, this statement applies to any Bregman divergence [24]. 7 4.5 Lower bound 1 The last result of this section is that FA can be made to have `oot = ?(t? 2 ). We show this by means of a counterexample. Assume t = n2 for some integer n > 0 and let the labels ? be binary, yi ? {0, 1}. We split the set {1, . . . , t} into n consecutive segments, each of size n = t. The proportion of ones (yi = 1) in the k-th segment is equal to nk , but within each segment all ones precede all zeros. For instance, when t = 25, the label sequence is: 10000 | {z } 11100 | {z } 11110 | {z } 11111 | {z }, | {z } 11000 1/5 2/5 3/5 4/5 5/5 One can use the minimax formulation (4) to verify that the segments will correspond to the level sets of the isotonic regression and that yi? = nk for any i in the k-th segment. This sequence is hard: 1 Lemma 4.4. The IR-Int algorithm run on the sequence described above has `oot = ?(t? 2 ). We prove the lower bound for IR-Int, since the presentation (in Appendix E) is clearest. Empirical simulations showing that the other forward algorithms also suffer this regret are in Appendix F. 4.6 Towards optimal forward algorithms An attractive feature of forward algorithms is that they generalize to partial orders, for which efficient ? ? 12 ) offline optimization algorithms exist. However, in Section 4 we saw that FAs only give a O(t ? ? 23 ) is possible (with an algorithm that is not known to scale rate, while in Section 3 we saw that O(t to partial orders). Is there any hope of an algorithm that both generalizes and has the optimal rate? In this section, we propose the Heavy-? algorithm, a slight modification of the forward algorithm that plugs in label estimate yi0 = ? ? [0, 1] with weight c (with unit weight on all other points), then plays the value of the isotonic regression function. Implementation is straightforward for offline isotonic regression algorithms that permit the specification of weights (such as [16]). Otherwise, we might simulate such weighting by plugging in c copies of the estimated label ? at location xi . What label estimate ? and weight c should we use? We show that the choice of ? is not very sensitive, 1 but it is crucial to tune the weight to c = ?(t 3 ). Lemmas H.1 and H.2 show that higher and lower c are necessarily sub-optimal for `oot . This leaves only one choice for c, for which we believe 1 ? ? 23 ). Conjecture 4.5. Heavy-? with weight c = ?(t 3 ) has `oot = O(t We cannot yet prove this conjecture, although numerical experiments strongly suggest it. We do not believe that picking a constant label ? is special. For example, we might alternatively predict with the average of the predictions of Heavy-1 and Heavy-0. Yet not any label estimate works. In particular, if we estimate the label that would be predicted by IR-Int (see 4.1) and the discussion below it), and we plug that in with any weight c ? 0, then the isotonic regression function will still have that same 1 label estimate as its value. This means that the ?(t? 2 ) lower bound of Section 4.5 applies. 5 Conclusion We revisit the problem of online isotonic regression and argue that we need a new perspective to design practical algorithms. We study the random permutation model as a novel way to bypass the stringent fixed design requirement of previous work. Our main tool in the design and analysis of algorithms is the leave-one-out loss, which bounds the expected regret from above. We start by observing that the adversary from the adversarial fixed design setting also provides a lower bound here. We then show that this lower bound can be matched by applying online-to-batch conversion to the optimal algorithm for fixed design. Next we provide an online analysis of the natural, popular and practical class of Forward Algorithms, which are defined in terms of an offline optimization oracle. We show that Forward algorithms achieve a decent regret rate in all cases, and match the optimal rate in special cases. We conclude by sketching the class of practical Heavy algorithms and conjecture that a specific parameter setting might guarantee the correct regret rate. Open problem The next major challenge is the design and analysis of efficient algorithms for online isotonic regression on arbitrary partial orders. Heavy-? is our current best candidate. We pose ? 13 ) regret on linear orders as an open problem. deciding if it in fact even guarantees O(T 8 Acknowledgments Wojciech Kot?owski acknowledges support from the Polish National Science Centre (grant no. 2016/22/E/ST6/00299). Wouter Koolen acknowledges support from the Netherlands Organization for Scientific Research (NWO) under Veni grant 639.021.439. This work was done in part while Koolen was visiting the Simons Institute for the Theory of Computing. References [1] M. Ayer, H. D. Brunk, G. M. Ewing, W. T. Reid, and E. Silverman. An empirical distribution function for sampling with incomplete information. Annals of Mathematical Statistics, 26(4): 641?647, 1955. [2] K. Azoury and M. Warmuth. Relative loss bounds for on-line density estimation with the exponential family of distributions. Journal of Machine Learning, 43(3):211?246, 2001. [3] Lucien Birg? and Pascal Massart. Rates of convergence for minimum contrast estimators. Probability Theory and Related Fields, 97:113?150, 1993. [4] H. D. Brunk. Maximum likelihood estimates of monotone parameters. Annals of Mathematical Statistics, 26(4):607?616, 1955. [5] Nicol? Cesa-Bianchi and G?bor Lugosi. Worst-case bounds for the logarithmic loss of predictors. Machine Learning, 43(3):247?264, 2001. [6] Nicol? Cesa-Bianchi and G?bor Lugosi. Prediction, learning, and games. Cambridge University Press, 2006. [7] Jan de Leeuw, Kurt Hornik, and Patrick Mair. Isotone optimization in R: Pool-adjacent-violators algorithm (PAVA) and active set methods. Journal of Statistical Software, 32:1?24, 2009. [8] Tom Fawcett and Alexandru Niculescu-Mizil. PAV and the ROC convex hull. Machine Learning, 68(1):97?106, 2007. [9] J?rgen Forster and Manfred K Warmuth. Relative expected instantaneous loss bounds. Journal of Computer and System Sciences, 64(1):76?102, 2002. [10] Pierre Gaillard and S?bastien Gerchinovitz. A chaining algorithm for online nonparametric regression. In Conference on Learning Theory (COLT), pages 764?796, 2015. [11] Sham M Kakade, Varun Kanade, Ohad Shamir, and Adam Kalai. Efficient learning of generalized linear and single index models with isotonic regression. In Neural Information Processing Systems (NIPS), pages 927?935, 2011. [12] Adam Tauman Kalai and Ravi Sastry. The isotron algorithm: High-dimensional isotonic regression. In COLT, 2009. [13] Wojciech Kot?owski and Roman S?owi?nski. Rule learning with monotonicity constraints. In International Conference on Machine Learning (ICML), pages 537?544, 2009. [14] Wojciech Kot?owski, Wouter M. Koolen, and Alan Malek. Online isotonic regression. In Vitaly Feldman and Alexander Rakhlin, editors, Proceedings of the 29th Annual Conference on Learning Theory (COLT), pages 1165?1189, June 2016. [15] J. B. Kruskal. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29(1):1?27, 1964. [16] Rasmus Kyng, Anup Rao, and Sushant Sachdeva. Fast, provable algorithms for isotonic regression in all `p -norms. In Neural Information Processing Systems (NIPS), 2015. [17] Ronny Luss, Saharon Rosset, and Moni Shahar. Efficient regularized isotonic regression with application to gene?gene interaction search. Annals of Applied Statistics, 6(1):253?283, 2012. 9 [18] Aditya Krishna Menon, Xiaoqian Jiang, Shankar Vembu, Charles Elkan, and Lucila OhnoMachado. Predicting accurate probabilities with a ranking loss. In Interantional Conference on Machine Learning (ICML), 2012. [19] T. Moon, A. Smola, Y. Chang, and Z. Zheng. Intervalrank: Isotonic regression with listwise and pairwise constraint. In WSDM, pages 151?160. ACM, 2010. [20] Harikrishna Narasimhan and Shivani Agarwal. On the relationship between binary classification, bipartite ranking, and binary class probability estimation. In Neural Information Processing Systems (NIPS), pages 2913?2921, 2013. [21] Alexandru Niculescu-Mizil and Rich Caruana. Predicting good probabilities with supervised learning. In ICML, volume 119, pages 625?632. ACM, 2005. [22] G. Obozinski, C. E. Grant, G. R. G. Lanckriet, M. I. Jordan, and W. W. Noble. Consistent probabilistic outputs for protein function prediction. Genome Biology, 2008 2008. [23] Alexander Rakhlin and Karthik Sridharan. Online nonparametric regression. In Conference on Learning Theory (COLT), pages 1232?1264, 2014. [24] T. Robertson, F. T. Wright, and R. L. Dykstra. Order Restricted Statistical Inference. John Wiley & Sons, 1998. [25] Mario Stylianou and Nancy Flournoy. Dose finding using the biased coin up-and-down design and isotonic regression. Biometrics, 58(1):171?177, 2002. [26] Sara Van de Geer. Estimating a regression function. Annals of Statistics, 18:907?924, 1990. [27] Vladimir Vovk, Ivan Petej, and Valentina Fedorova. Large-scale probabilistic predictors with and without guarantees of validity. In Neural Information Processing Systems (NIPS), pages 892?900, 2015. [28] Bianca Zadrozny and Charles Elkan. Transforming classifier scores into accurate multiclass probability estimates. In International Conference on Knowledge Discovery and Data Mining (KDD), pages 694?699, 2002. [29] Cun-Hui Zhang. Risk bounds in isotonic regression. The Annals of Statistics, 30(2):528?555, 2002. 10
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A Unified Game-Theoretic Approach to Multiagent Reinforcement Learning Marc Lanctot DeepMind lanctot@ Karl Tuyls DeepMind karltuyls@ Vinicius Zambaldi DeepMind vzambaldi@ ? Audrunas Gruslys DeepMind audrunas@ Julien P?rolat DeepMind perolat@ David Silver DeepMind davidsilver@ Angeliki Lazaridou DeepMind angeliki@ Thore Graepel DeepMind thore@ [email protected] Abstract To achieve general intelligence, agents must learn how to interact with others in a shared environment: this is the challenge of multiagent reinforcement learning (MARL). The simplest form is independent reinforcement learning (InRL), where each agent treats its experience as part of its (non-stationary) environment. In this paper, we first observe that policies learned using InRL can overfit to the other agents? policies during training, failing to sufficiently generalize during execution. We introduce a new metric, joint-policy correlation, to quantify this effect. We describe an algorithm for general MARL, based on approximate best responses to mixtures of policies generated using deep reinforcement learning, and empirical game-theoretic analysis to compute meta-strategies for policy selection. The algorithm generalizes previous ones such as InRL, iterated best response, double oracle, and fictitious play. Then, we present a scalable implementation which reduces the memory requirement using decoupled meta-solvers. Finally, we demonstrate the generality of the resulting policies in two partially observable settings: gridworld coordination games and poker. 1 Introduction Deep reinforcement learning combines deep learning [59] with reinforcement learning [94, 64] to compute a policy used to drive decision-making [73, 72]. Traditionally, a single agent interacts with its environment repeatedly, iteratively improving its policy by learning from its observations. Inspired by recent success in Deep RL, we are now seeing a renewed interest in multiagent reinforcement learning (MARL) [90, 17, 99]. In MARL, several agents interact and learn in an environment simultaneously, either competitively such as in Go [91] and Poker [39, 105, 74], cooperatively such as when learning to communicate [23, 93, 36], or some mix of the two [60, 95, 35]. The simplest form of MARL is independent RL (InRL), where each learner is oblivious to the other agents and simply treats all the interaction as part of its (?localized?) environment. Aside from the problem that these local environments are non-stationary and non-Markovian [57] resulting in a loss of convergence guarantees for many algorithms, the policies found can overfit to the other agents? policies and hence not generalize well. There has been relatively little work done in RL community on overfitting to the environment [102, 69], but we argue that this is particularly important in multiagent settings where one must react dynamically based on the observed behavior of others. Classical techniques collect or approximate extra information such as the joint values [62, 19, 29, 56], 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. use adaptive learning rates [12], adjust the frequencies of updates [48, 81], or dynamically respond to the other agents actions online [63, 50]. However, with the notable exceptions of very recent work [22, 80], they have focused on (repeated) matrix games and/or the fully-observable case. There have several proposals for treating partial observability in the multiagent setting. When the model is fully known and the setting is strictly adversarial with two players, there are policy iteration methods based on regret minimization that scale very well when using domain-specific abstractions [27, 14, 46, 47], which was a major component of the expert no-limit poker AI Libratus [15]; recently these methods were combined with deep learning to create an expert no-limit poker AI called DeepStack [74]. There is a significant amount of work that deals with the case of decentralized cooperative problems [76, 79], and in the general setting by extending the notion of belief states and Bayesian updating from POMDPs [28]. These models are quite expressive, and the resulting algorithms are fairly complex. In practice, researchers often resort to approximate forms, by sampling or exploiting structure, to ensure good performance due to intractability [41, 2, 68]. In this paper, we introduce a new metric for quantifying the correlation effects of policies learned by independent learners, and demonstrate the severity of the overfitting problem. These coordination problems have been well-studied in the fully-observable cooperative case [70]: we observe similar problems in a partially-observed mixed cooperative/competitive setting and, and we show that the severity increases as the environment becomes more partially-observed. We propose a new algorithm based on economic reasoning [82], which uses (i) deep reinforcement learning to compute best responses to a distribution over policies, and (ii) empirical game-theoretic analysis to compute new meta-strategy distributions. As is common in the MARL setting, we assume centralized training for decentralized execution: policies are represented as separate neural networks and there is no sharing of gradients nor architectures among agents. The basic form uses a centralized payoff table, which is removed in the distributed, decentralized form that requires less space. 2 Background and Related Work In this section, we start with basic building blocks necessary to describe the algorithm. We interleave this with the most relevant previous work for our setting. Several components of the general idea have been (re)discovered many times across different research communities, each with slightly different but similar motivations. One aim here is therefore to unify the algorithms and terminology. A normal-form game is a tuple (?, U, n) where n is the number of players, ? = (?1 , ? ? ? , ?n ) is the set of policies (or strategies, one for each player i ? [[n]], where [[n]] = {1, ? ? ? , n}), and U : ? ? <n is a payoff table of utilities for each joint policy played by all players. Extensive-form games extend these formalisms to the multistep sequential case (e.g. poker). Players try to maximize their own expected utility. Each player does this by choosing a policy from ?i , or by sampling from a mixture (distribution) over them ?i ? ?(?i ). In this multiagent setting, the quality of ?i depends on other players? strategies, and so it cannot be found nor assessed independently. Every finite extensive-form game has an equivalent normal-form [53], but since it is exponentially larger, most algorithms have to be adapted to handle the sequential setting directly. There are several algorithms for computing strategies. In zero-sum games (where ?? ? ?, ~1 ? U (?) = 0), one can use e.g. linear programming, fictitious play [13], replicator dynamics [97], or regret minimization [8]. Some of these techniques have been extended to extensive (sequential) form [39, 25, 54, 107] with an exponential increase in the size of the state space. However, these extensions have almost exclusively treated the two-player case, with some notable exceptions [54, 26]. Fictitious play also converges in potential games which includes cooperative (identical payoff) games. The double oracle (DO) algorithm [71] solves a set of (two-player, normal-form) subgames induced by subsets ?t ? ? at time t. A payoff matrix for the subgame Gt includes only those entries corresponding to the strategies in ?t . At each time step t, an equilibrium ? ?,t is obtained for Gt , and ?,t to obtain Gt+1 each player adds a best response ?it+1 ? BR(??i ) from the full space ?i , so for all i, t+1 t ?t+1 = ? ? {? }. The algorithm is illustrated in Figure 1. Note that finding an equilibrium in a i i i zero-sum game takes time polynomial in |?t |, and is PPAD-complete for general-sum [89]. Clearly, DO is guaranteed to converge to an equilibrium in two-player games. But, in the worst-case, the entire strategy space may have to be enumerated. For example, this is necessary for Rock-Paper2 Figure 1: The Double Oracle Algorithm. Figure taken from [10] with authors? permission. Scissors, whose only equilibrium has full support ( 13 , 13 , 13 ). However, there is evidence that support sizes shrink for many games as a function of episode length, how much hidden information is revealed and/or affects it has on the payoff [65, 86, 10]. Extensions to the extensive-form games have been developed [67, 9, 10] but still large state spaces are problematic due to the curse of dimensionality. Empirical game-theoretic analysis (EGTA) is the study of meta-strategies obtained through simulation in complex games [100, 101]. An empirical game, much smaller in size than the full game, is constructed by discovering strategies, and meta-reasoning about the strategies to navigate the strategy space. This is necessary when it is prohibitively expensive to explicitly enumerate the game?s strategies. Expected utilities for each joint strategy are estimated and recorded in an empirical payoff table. The empirical game is analyzed, and the simulation process continues. EGTA has been employed in trading agent competitions (TAC) and automated bidding auctions. One study used evolutionary dynamics in the space of known expert meta-strategies in Poker [83]. Recently, reinforcement learning has been used to validate strategies found via EGTA [104]. In this work, we aim to discover new strategies through learning. However, instead of computing exact best responses, we compute approximate best responses using reinforcement learning. A few epochs of this was demonstrated in continuous double auctions using tile coding [87]. This work follows up in this line, running more epochs, using modern function approximators (deep networks), a scalable implementation, and with a focus on finding policies that can generalize across contexts. A key development in recent years is deep learning [59]. While most work in deep learning has focused on supervised learning, impressive results have recently been shown using deep neural networks for reinforcement learning, e.g. [91, 38, 73, 77]. For instance, Mnih et al. [73] train policies for playing Atari video games and 3D navigation [72], given only screenshots. Silver et al. introduced AlphaGo [91, 92], combining deep RL with Monte Carlo tree search, outperforming human experts. Computing approximate responses is more computationally feasible, and fictitious play can handle approximations [42, 61]. It is also more biologically plausible given natural constraints of bounded rationality. In behavioral game theory [103], the focus is to predict actions taken by humans, and the responses are intentionally constrained to increase predictive ability. A recent work uses a deep learning architecture [34]. The work that closely resembles ours is level-k thinking [20] where level k agents respond to level k ? 1 agents, and more closely cognitive hierarchy [18], in which responses are to distributions over levels {0, 1, . . . , k ? 1}. However, our goals and motivations are very different: we use the setup as a means to produce more general policies rather than to predict human behavior. Furthermore, we consider the sequential setting rather than normal-form games. Lastly, there has been several studies from the literature on co-evolutionary algorithms; specifically, how learning cycles and overfitting to the current populations can be mitigated [78, 85, 52]. 3 Policy-Space Response Oracles We now present our main conceptual algorithm, policy-space response oracles (PSRO). The algorithm is a natural generalization of Double Oracle where the meta-game?s choices are policies rather than actions. It also generalizes Fictitious Self-Play [39, 40]. Unlike previous work, any meta-solver can be plugged in to compute a new meta-strategy. In practice, parameterized policies (function approximators) are used to generalize across the state space without requiring any domain knowledge. The process is summarized in Algorithm 1. The meta-game is represented as an empirical game, starting with a single policy (uniform random) and growing, each epoch, by adding policies (?oracles?) 3 Algorithm 2: Deep Cognitive Hierarchies input :player number i, level k while not terminated do C HECK L OAD MS({j|j ? [[n]], j 6= i}, k) C HECK L OAD O RACLES(j ? [[n]], k 0 ? k) C HECK S AVE MS(?i,k ) C HECK S AVE O RACLE(?i,k ) Sample ??i ? ??i,k Train oracle ?i,k over ?1 ? (?i,k , ??i ) if iteration number mod Tms = 0 then Sample ?i ? ?i,k Compute ui (?2 ), where ?2 ? (?i , ??i ) Update stats for ?i and update ?i,k Output ?i,k for player i at level k Algorithm 1: Policy-Space Response Oracles input :initial policy sets for all players ? Compute exp. utilities U ? for each joint ? ? ? Initialize meta-strategies ?i = U NIFORM(?i ) while epoch e in {1, 2, ? ? ? } do for player i ? [[n]] do for many episodes do Sample ??i ? ??i Train oracle ?i0 over ? ? (?i0 , ??i ) ?i = ?i ? {?i0 } Compute missing entries in U ? from ? Compute a meta-strategy ? from U ? Output current solution strategy ?i for player i that approximate best responses to the meta-strategy of the other players. In (episodic) partially observable multiagent environments, when the other players are fixed the environment becomes Markovian and computing a best response reduces to solving a form of MDP [30]. Thus, any reinforcement learning algorithm can be used. We use deep neural networks due to the recent success in reinforcement learning. In each episode, one player is set to oracle(learning) mode to train ?i0 , and a fixed policy is sampled from the opponents? meta-strategies (??i ? ??i ). At the end of the epoch, the new oracles are added to their policy sets ?i , expected utilities for new policy combinations are computed via simulation and added to the empirical tensor U ? , which takes time exponential in |?|. Define ?T = ?T ?1 ? ? 0 as the policy space including the currently learning oracles, and |?i | = |?Ti | for all i ? [[n]]. Iterated best response is an instance of PSRO with ??i = (0, 0, ? ? ? , 1, 0). Similarly, Independent RL and fictitious play are instances of PSRO with ??i = (0, 0, ? ? ? , 0, 1) and ??i = (1/K, 1/K, ? ? ? , 1/K, 0), respectively, where K = |?T?i?1 |. Double Oracle is an instance of PSRO T ?1 with n = 2 and ? T set to a Nash equilibrium profile of the meta-game (?T ?1 , U ? ). An exciting question is what can happen with (non-fixed) meta-solvers outside this known space? Fictitious play is agnostic to the policies it is responding to; hence it can only sharpen the metastrategy distribution by repeatedly generating the same best responses. On the other hand, responses to equilibrium strategies computed by Double Oracle will (i) overfit to a specific equilibrium in the n-player or general-sum case, and (ii) be unable to generalize to parts of the space not reached by any equilibrium strategy in the zero-sum case. Both of these are undesirable when computing general policies that should work well in any context. We try to balance these problems of overfitting with a compromise: meta-strategies with full support that force (mix in) ? exploration over policy selection. 3.1 Meta-Strategy Solvers A meta-strategy solver takes as input the empirical game (?, U ? ) and produces a meta-strategy ?i for each player i. We try three different solvers: regret-matching, Hedge, and projected replicator dynamics. These specific meta-solvers accumulate values for each policy (?arm?) and an aggregate value based on all players? meta-strategies. We refer to ui (?) as player i?s expected value given all players? meta-strategies and the current empirical payoff tensor U ? (computed via multiple tensor dot products.) Similarly, denote ui (?i,k , ??i ) as the expected utility if player i plays their k th ? [[K]] ? {0} policy and the other players play with their meta-strategy ??i . Our strategies use ? an exploration parameter ?, leading to a lower bound of K+1 on the probability of selecting any ?i,k . The first two meta-solvers (Regret Matching and Hedge) are straight-forward applications of previous algorithms, so we defer the details to Appendix A.1 Here, we introduce a new solver we call projected replicator dynamics (PRD). From Appendix A, when using the asymmetric replicator dynamics, e.g. with two players, where U ? = (A, B), the change in probabilities for the k th component (i.e., the policy ?i,k ) of meta-strategies (?1 , ?2 ) = (x, y) are: dyk = yk [(xT B)k ? xT By], dt dxk = xk [(Ay)k ? xT Ay], dt 1 Appendices are available in the longer technical report version of the paper, see [55]. 4 To simulate the replicator dynamics in practice, discretized updates are simulated using a step-size of ?. We add a projection operator P (?) to these equations that guarantees exploration: x ? P (x + ? dx dt ), 0 K+1 y ? P (y + ? dy ), where P (x) = argmin {||x ? x||}, if any x < ?/(K + 1) or x 0 k dt P x ??? ? K+1 otherwise, and ?? = {x | xk ? K+1 , k xk = 1} is the ?-exploratory simplex of size K + 1. This enforces exploratory ?i (?i,k ) ? ?/(K + 1). The PRD approach can be understood as directing exploration in comparison to standard replicator dynamics approaches that contain isotropic diffusion or mutation terms (which assume undirected and unbiased evolution), for more details see [98]. 3.2 Deep Cognitive Hierarchies K + 1 levels While the generality of PSRO is clear and appealing, the RL step can take a long time to converge to a good response. In complex environments, much of the basic behavior that was learned in one epoch may need to be relearned when starting again from scratch; also, it may be desirable to run many epochs to get oracle policies that can recursively reason through deeper levels of contingencies. rand ?1,1 ?1,1 . . . . . rand . . . . . rand . . . . . rand . . . . . N players ?i,k ? To overcome these problems, we introduce a practical parallel form of PSRO. Instead of an unbounded number Figure 2: Overview of DCH of epochs, we choose a fixed number of levels in advance. Then, for an n-player game, we start nK processes in parallel (level 0 agents are uniform random): each one trains a single oracle policy ?i,k for player i and level k and updates its own meta-strategy ?i,k , saving each to a central disk periodically. Each process also maintains copies of all the other oracle policies ?j,k0 ?k at the current and lower levels, as well as the meta-strategies at the current level ??i,k , which are periodically refreshed from a central disk. We circumvent storing U ? explicitly by updating the meta-strategies online. We call this a Deep Cognitive Hierarchy (DCH), in reference to Camerer, Ho, & Chong?s model augmented with deep RL. Example oracle response dynamics shown in Figure 2, and the pseudo-code in Algorithm 2. i,k Since each process uses slightly out-dated copies of the other process?s policies and meta-strategies, DCH approximates PSRO. Specifically, it trades away accuracy of the correspondence to PSRO for practical efficiency and, in particular, scalability. Another benefit of DCH is an asymptotic reduction in total space complexity. In PSRO, for K policies and n players, the space required to store the empirical payoff tensor is K n . Each process in DCH stores nK policies of fixed size, and n meta-strategies (and other tables) of size bounded by k ? K. Therefore the total space required is O(nK ? (nK + nK)) = O(n2 K 2 ). This is possible is due to the use of decoupled meta-solvers, which compute strategies online without requiring a payoff tensor U ? , which we describe now. 3.2.1 Decoupled Meta-Strategy Solvers In the field of online learning, the experts algorithms (?full information? case) receive information about each arm at every round. In the bandit (?partial information?) case, feedback is only given for the arm that was pulled. Decoupled meta-solvers are essentially sample-based adversarial bandits [16] applied to games. Empirical strategies are known to converge to Nash equilibria in certain classes of games (i.e. zero-sum, potential games) due to the folk theorem [8]. We try three: decoupled regret-matching [33], Exp3 (decoupled Hedge) [3], and decoupled PRD. Here again, we use exploratory strategies with ? of the uniform strategy mixed in, which is also necessary to ensure that the estimates are unbiased. For decoupled PRD, we maintain running averages for the overall average value an value of each arm (policy). Unlike in PSRO, in the case of DCH, one sample is obtained at a time and the meta-strategy is updated periodically from online estimates. 4 Experiments In all of our experiments, oracles use Reactor [31] for learning, which has achieved state-of-the-art results in Atari game-playing. Reactor uses Retrace(?) [75] for off-policy policy evaluation, and ?-Leave-One-Out policy gradient for policy updates, and supports recurrent network training, which could be important in trying to match online experiences to those observed during training. 5 The action spaces for each player are identical, but the algorithms do not require this. Our implementation differs slightly from the conceptual descriptions in Section 3; see App. C for details. First-Person Gridworld Games. Each agent has a local field-of-view (making the world partially observable), sees 17 spaces in front, 10 to either side, and 2 spaces behind. Consequently, observations are encoded as 21x20x3 RGB tensors with values 0 ? 255. Each agent has a choice of turning left or right, moving forward or backward, stepping left or right, not moving, or casting an endless light beam in their current direction. In addition, the agent has two composed actions of moving forward and turning. Actions are executed simultaneously, and order of resolution is randomized. Agents start on a random spawn point at the beginning of each episode. If an agent is touched (?tagged?) by another agent?s light beam twice, then the target agent is immediately teleported to a spawn point. In laser tag, the source agent then receives 1 point of reward for the tag. In another variant, gathering, there is no tagging but agents can collect apples, for 1 point per apple, which refresh at a fixed rate. In pathfind, there is no tagging nor apples, and both agents get 1 point reward when both reach their destinations, ending the episode. In every variant, an episode consists of 1000 steps of simulation. Other details, such as specific maps, can be found in Appendix D. Leduc Poker is a common benchmark in Poker AI, consisting of a six-card deck: two suits with three cards (Jack, Queen, King) each. Each player antes 1 chip to play, and receives one private card. There are two rounds of betting, with a maximum of two raises each, whose values are 2 and 4 chips respectively. After the first round of betting, a single public card is revealed. The input is represented as in [40], which includes one-hot encodings of the private card, public card, and history of actions. Note that we use a more difficult version than in previous work; see Appendix D.1 for details. 4.1 Joint Policy Correlation in Independent Reinforcement Learning To identify the effect of overfitting in independent reinforcement learners, we introduce joint policy correlation (JPC) matrices. To simplify the presentation, we describe here the special case of symmetric two-player games with non-negative rewards; for a general description, see Appendix B.2. Values are obtained by running D instances of the same experiment, differing only in the seed used to initialize the random number generators. Each experiment d ? [[D]] (after many training episodes) produces policies (?1d , ?2d ). The entries of each D ? D matrix shows the mean return over T = 100 PT episodes, t=1 T1 (R1t + R2t ), obtained when player 1 uses row policy ?1di and and player 2 uses d column policy ?2 j . Hence, entries on the diagonals represent returns for policies that learned together (i.e., same instance), while off-diagonals show returns from policies that trained in separate instances. 11.0 15.5 24 2 14.6 12.9 30.3 7.3 23.8 18 3 27.3 31.7 27.6 30.6 26.2 12 25.1 27.3 29.6 5.3 29.8 6 0 1 2 3 4 Player #2 0 17.8 18.5 7.1 0.6 1 30.8 2.9 12.2 18.2 11.3 0.8 8.2 16 2 1 29.9 23.0 18.2 2.3 20.0 6.0 2.6 12 3.6 2.6 4.1 20.5 4.7 8 0.5 3.5 0.6 3.9 19.2 0 1 2 3 4 Player #1 9.2 3 3.7 4 0 23.9 Player #1 30.9 4 30 30.7 Player #2 20 4 Figure 3: Example JPC matrices for InRL on Laser Tag small2 map (left) and small4 (right). ? ? O)/ ? D ? where From a JPC matrix, we compute an average proportional loss in reward as R? = (D ? is the mean value of the diagonals and O ? is the mean value of the off-diagonals. E.g. in Figure 3: D D = 30.44, O = 20.03, R? = 0.342. Even in a simple domain with almost full observability (small2), an independently-learned policy could expect to lose 34.2% of its reward when playing with another independently-learned policy even though it was trained under identical circumstances! This clearly demonstrates an important problem with independent learners. In the other variants (gathering and pathfind), we observe no JPC problem, presumably because coordination is not required and the policies are independent. Results are summarized in Table 1. We have also noticed similar effects when using DQN [73] as the oracle training algorithm; see Appendix B.1 for example videos. 6 Environment Map Laser Tag Laser Tag Laser Tag Gathering Pathfind small2 small3 small4 field merge Table 1: InRL DCH(Reactor, 2, 10) JPC Reduction ? ? ? ? D O R? D O R? 30.44 20.03 0.342 28.20 26.63 0.055 28.7 % 23.06 9.06 0.625 27.00 23.45 0.082 54.3 % 20.15 5.71 0.717 18.72 15.90 0.150 56.7 % 147.34 146.89 0.003 139.70 138.74 0.007 ? 108.73 106.32 0.022 90.15 91.492 <0 ? Summary of JPC results in first-person gridworld games. We see that a (level 10) DCH agent reduces the JPC problem significantly. On small2, DCH reduces the expected loss down to 5.5%, 28.7% lower than independent learners. The problem gets larger as the map size grows and problem becomes more partially observed, up to a severe 71.7% average loss. The reduction achieved by DCH also grows from 28.7% to 56.7%. Is the Meta-Strategy Necessary During Execution? The figures above represent the fully-mixed strategy ?i,10 . We also analyze JPC for only the highest-level policy ?i,10 in the laser tag levels. The values are larger here: R? = 0.147, 0.27, 0.118 for small2-4 respectively, showing the importance of the meta-strategy. However, these are still significant reductions in JPC: 19.5%, 36.5%, 59.9%. How Many Levels? On small4, we also compute values for level 5 and level 3: R? = 0.156 and R? = 0.246, corresponding to reductions in JPC of 56.1% and 44%, respectively. Level 5 reduces JPC by a similar amount as level 10 (56.1% vs 56.7%), while level 3 less so (44% vs. 56.1%.) 4.2 Learning to Safely Exploit and Indirectly Model Opponents in Leduc Poker We now show results for a Leduc poker where strong benchmark algorithms exist, such as counterfactual regret (CFR) minimization [107, 11]. We evaluate our policies using two metrics: the first is performance against fixed players (random, CFR?s average strategy after 500 iterations ?cfr500?, and a purified version of ?cfr500pure? that chooses the with highest probability.) The second Paction n is commonly used in poker AI: NASH C ONV(?) = i max?i0 ??i ui (?i0 , ??i ) ? ui (?), representing how much can be gained by deviating to their best response (unilaterally), a value that can be interpreted as a distance from a Nash equilibrium (called exploitability in the two-player setting). NashConv is easy to compute in small enough games [45]; for CFR?s values see Appendix E.1. Effect of Exploration and Meta-Strategy Overview. We now analyze the effect of the various meta-strategies and exploration parameters. In Figure 4, we measure the mean area-under-the-curve (MAUC) of the NashConv values for the last (right-most) 32 values in the NashConv graph, and exploration rate of ? = 0.4. Figures for the other values of ? are in Appendix E, but we found this value of ? works best for minimizing NashConv. Also, we found that decoupled replicator dynamics works best, followed by decoupled regret-matching and Exp3. Also, it seems that the higher the level, the lower the resulting NashConv value is, with diminishing improvements. For exploitation, we found that ? = 0.1 was best, but the meta-solvers seemed to have little effect (see Figure 10.) Comparison to Neural Fictitious Self-Play. We now compare to Neural Fictitious Self-Play (NFSP) [40], an implementation of fictitious play in sequential games using reinforcement learning. Note that NFSP, PSRO, and DCH are all sample-based learning algorithms that use general function approximation, whereas CFR is a tabular method that requires a full game-tree pass per iteration. NashConv graphs are shown for {2,3}-player in Figure 5, and performance vs. fixed bots in Figure 6. 0.2 3.0 level 2.0 1.5 1.0 0.5 0.0 uprd urm metasolver exp3 (a) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.4 0.6 0.8 MAUC MAUC (NashConv) 2.5 1.0 1.2 1.4 min_exploration_weight 0.1 0.25 0.4 1.6 1.8 2.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 level (b) Figure 4: (a) Effect of DCH parameters on NashConv in 2 player Leduc Poker. Left: decoupled PRD, Middle: decoupled RM, Right: Exp3, and (b) MAUC of the exploitation graph against cfr500. 7 5 NFSP DCH PSRO 12 10 3 NashConv NashConv 14 NFSP DCH PSRO 4 2 8 6 1 0 4 0 500 1000 1500 Episodes (in thousands) 2000 2 2500 0 100 200 300 400 500 Episodes (in thousands) (a) 2 players 600 700 (b) 3 players Figure 5: Exploitability for NFSP x DCH x PSRO. 1 0 2 0 1 0 Mean score 1 Mean score Mean score 1 2 3 1 NFSP DCH PSRO 2 200 400 600 800 1000 Episodes (in thousands) 1200 1400 (a) Random bots as ref. set 1600 NFSP DCH PSRO 4 5 500 1000 1500 2000 Episodes (in thousands) 2500 3000 2 3 4 NFSP DCH PSRO 5 100 200 300 400 500 Episodes (in thousands) 600 700 800 (b) 2-player CFR500 bots as ref. set (c) 3-player CFR500 bots as ref. set Figure 6: Evaluation against fixed set of bots. Each data point is an average of the four latest values. We observe that DCH (and PSRO) converge faster than NFSP at the start of training, possibly due to a better meta-strategy than the uniform random one used in fictitious play. The convergence curves eventually plateau: DCH in two-player is most affected, possibly due to the asynchronous nature of the updates, and NFSP converges to a lower exploitability in later episodes. We believe that this is due to NFSP?s ability to learn a more accurate mixed average strategy at states far down in the tree, which is particularly important in poker, whereas DCH and PSRO mix at the top over full policies. On the other hand, we see that PSRO/DCH are able to achieve higher performance against the fixed players. Presumably, this is because the policies produced by PSRO/DCH are better able to recognize flaws in the weaker opponent?s policies, since the oracles are specifically trained for this, and dynamically adapt to the exploitative response during the episode. So, NFSP is computing a safe equilibrium while PSRO/DCH may be trading convergence precision for the ability to adapt to a range of different play observed during training, in this context computing a robust counter-strategy [44, 24]. 5 Conclusion and Future Work In this paper, we quantify a severe problem with independent reinforcement learners, joint policy correlation (JPC), that limits the generality of these approaches. We describe a generalized algorithm for multiagent reinforcement learning that subsumes several previous algorithms. In our experiments, we show that PSRO/DCH produces general policies that significantly reduce JPC in partially-observable coordination games, and robust counter-strategies that safely exploit opponents in a common competitive imperfect information game. The generality offered by PSRO/DCH can be seen as a form of ?opponent/teammate regularization?, and has also been observed recently in practice [66, 5]. We emphasize the game-theoretic foundations of these techniques, which we hope will inspire further investigation into algorithm development for multiagent reinforcement learning. In future work, we will consider maintaining diversity among oracles via loss penalties based on policy dissimilarity, general response graph topologies, environments such as emergent language games [58] and RTS games [96, 84], and other architectures for prediction of behavior, such as opponent modeling [37] and imagining future states via auxiliary tasks [43]. We would also like to investigate fast online adaptation [1, 21] and the relationship to computational Theory of Mind [106, 4], as well as generalized (transferable) oracles over similar opponent policies using successor features [6]. Acknowledgments. We would like to thank DeepMind and Google for providing an excellent research environment that made this work possible. 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Inverse Filtering for Hidden Markov Models Robert Mattila Department of Automatic Control KTH Royal Institute of Technology [email protected] Vikram Krishnamurthy Cornell Tech Cornell University [email protected] Cristian R. Rojas Department of Automatic Control KTH Royal Institute of Technology [email protected] Bo Wahlberg Department of Automatic Control KTH Royal Institute of Technology [email protected] Abstract This paper considers a number of related inverse filtering problems for hidden Markov models (HMMs). In particular, given a sequence of state posteriors and the system dynamics; i) estimate the corresponding sequence of observations, ii) estimate the observation likelihoods, and iii) jointly estimate the observation likelihoods and the observation sequence. We show how to avoid a computationally expensive mixed integer linear program (MILP) by exploiting the algebraic structure of the HMM filter using simple linear algebra operations, and provide conditions for when the quantities can be uniquely reconstructed. We also propose a solution to the more general case where the posteriors are noisily observed. Finally, the proposed inverse filtering algorithms are evaluated on real-world polysomnographic data used for automatic sleep segmentation. 1 Introduction The hidden Markov model (HMM) is a cornerstone of statistical modeling [1?4]. In it, a latent (i.e., hidden) state evolves according to Markovian dynamics. The state of the system is only indirectly observed via a sensor that provides noisy observations. The observations are sampled independently, conditioned on the state of the system, according to observation likelihood probabilities. Of paramount importance in many applications of HMMs is the classical stochastic filtering problem, namely: Given observations from an HMM with known dynamics and observation likelihood probabilities, compute the posterior distribution of the latent state. Throughout the paper, we restrict our attention to discrete-time finite observation-alphabet HMMs. For such HMMs, the solution to the filtering problem is a recursive algorithm known as the HMM filter [1, 4]. In this paper, we consider the inverse of the above problem. In particular, our aim is to provide solutions to the following inverse filtering problems: Given a sequence of posteriors (or, more generally, noisily observed posteriors) from an HMM with known dynamics, compute (estimate) the observation likelihood probabilities and/or the observations that generated the posteriors. To motivate these problems, we give several possible applications of our results below. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Applications The underlying idea of inverse filtering problems (?inform me about your state estimate and I will know your sensor characteristics, including your measurements?) has potential applications in, e.g., autonomous calibration of sensors, fault diagnosis, and detecting Bayesian behavior in agents. In model-based fault-detection [5, 6], sensor information together with solutions to related inverse filtering problems are used to detect abnormal behavior. (As trivial examples; i) if the true sequence of observations is known from a redundant sensor, it can be compared to the reconstructed sequence; if there is a miss-match, something is wrong, or ii) if multiple data batches are available, then change detection can be performed on the sequence of reconstructed observation likelihoods.) They are also of relevance in a revealed preference context in microeconomics where the aim is to detect expected utility maximization behavior of an agent; estimating the posterior given the agent?s actions is a crucial step, see, e.g., [7]. Recent advances in wearables and smart-sensor technology have led to consumer grade products (smart watches with motion and heart-beat monitoring, sleep trackers, etc.) that produce vast amounts of personal data by performing state estimation. This information can serve as an indicator of health, fitness and stress. It may be very difficult, or even impossible, to access the raw sensor data since the sensor and state estimator usually are tightly integrated and encapsulated in intelligent sensor systems. Inverse filtering provides a framework for reverse engineering and performing fault detection of such sensors. In Section 5, we demonstrate our proposed solutions on a system that performs automatic sequencing of sleep stages based on electroencephalogram (EEG) data ? the outputs of such an automatic system are exactly posteriors over the different sleep stages [8]. Another important application of the inverse filtering problem arises in electronic warfare and cyberphysical security. How can one determine how accurate an enemy?s sensors are? In such problems, the state of the underlying Markov chain is usually known (a probing sequence), and one observes actions taken by the enemy which are based on filtered posterior distributions. The aim is to estimate the observation likelihood probabilities of the enemy, i.e., determine how accurate its sensors are. Our contributions It is possible to obtain a solution to the inverse filtering problem for HMMs by employing a brute-force approach (see Section 2.3) ? essentially by testing observations from the alphabet, and at the same time finding system parameters consistent with the data. However, this leads to a computationally expensive combinatorial optimization problem. Instead, we demonstrate in this paper an efficient solution based on linear algebra by exploiting the inherent structure of the problem and the HMM filter. In particular, the contributions of this paper are three-fold: 1. We propose analytical solutions to three inverse filtering problems for HMMs that avoid computationally expensive mixed integer linear program (MILP) formulations. Moreover, we establish theorems guaranteeing unique identifiability. 2. We consider the setting where the output of the HMM filter is corrupted by noise, and propose an inverse filtering algorithm based on clustering. 3. We evaluate the algorithm on real-world data for automatic segmentation of the sleep cycle. Related work There are only two known cases where the optimal filter allows a finite dimensional characterization: the HMM filter for (discrete) HMMs, and the Kalman filter [9, 10] for linear Gaussian state-space models. Inverse filtering problems for the Kalman filter have been considered in, e.g., [5, 6, 10], however, inverse filtering for HMMs has, to the best knowledge of the authors, received much less attention. The inverse filtering problem has connections to a number of other inverse problems in various fields. For example, in control theory, the fundamental inverse optimal control problem, whose formulation dates back to 1964 [11], studies the question: given a system and a policy, for what cost criteria is the policy optimal? In microeconomic theory, the related problem of revealed preferences [12] asks the question: given a set of decisions made by an agent, is it possible to determine if a utility is being maximized, and if so, which? In machine learning, there are clear connections to, e.g., apprenticeship learning, imitation learning and inverse reinforcement learning, see, e.g., [13?17], which recently have received much attention. In these, the reward function of a Markov decision process (MDP) is learned by observing an expert demonstrating the task that an agent wants to learn to perform. The key difference between these works and our work is the set of system parameters we aim to learn. 2 2 Preliminaries In this section, we formulate the inverse filtering problems, discuss how these can be solved using combinatorial optimization, and state our assumptions formally. With regards to notation, all vectors are column vectors, unless transposed. The vector 1 is the vector of all ones. ? denotes the Moore?Penrose pseudoinverse. 2.1 Hidden Markov models (HMMs) and the HMM filter We consider a discrete-time finite observation-alphabet HMM. Denote its state at time k as xk ? {1, . . . , X} and the corresponding observation yk ? {1, . . . , Y }. The underlying Markov chain xk evolves according to the row-stochastic transition probability matrix P ? RX?X , where [P ]ij = Pr[xk+1 = j|xk = i]. The initial state x0 is sampled from the probability distribution ?0 ? RX , where [?0 ]i = Pr[x0 = i]. The noisy observations of the underlying Markov chain are obtained from the row-stochastic observation likelihood matrix B ? RX?Y , where [B]ij = Pr[yk = j|xk = i] are the observation likelihood probabilities. We denote the columns of the observation likelihood matrix as {bi }Yi=1 , i.e., B = [b1 . . . bY ]. In the classical stochastic filtering problem, the aim is to compute the posterior distribution ?k ? RX of the latent state (Markov chain, in our case) at time k, given observations from the system up to time k. The HMM filter [1, 4] computes these posteriors via the following recursive update: ?k = Byk P T ?k?1 , 1T Byk P T ?k?1 (1) initialized by ?0 , where [?k ]i = Pr[xk = i|y1 , . . . , yk ] is the posterior distribution at time k, Byk = diag(byk ) ? RX?X , and {yk }N k=1 is a set of observations. 2.2 Inverse HMM filtering problem formulations The inverse filtering problem for HMMs is not a single problem ? multiple variants can be formulated depending on what information is available a priori. We pose and consider a number of variations of increasing levels of generality depending on what data we can extract from the sensor system. To restrict the scope of the paper, we assume throughout that the transition matrix P is known, and is the same in both the system and the HMM filter (i.e, we do not consider miss-matched HMM filtering problems). Formally, the inverse filtering problems considered in this paper are as follows: Problem 1 (Inverse  filtering problem with unknown observations). Consider the known data D = P, B, {?k }N k=0 , where the posteriors have been generated by an HMM-filter sensor. Reconstruct the observations {yk }N k=1 . Problem 2 (Inverse filtering problem with unknown sensor). Consider the known data D =  N , where the posteriors have been generated by an HMM-filter sensor. ReconP, {yk }N , {? } k k=1 k=0 struct the observation likelihood matrix B. Combining these two formulations yields the general problem: Problem 3 (Inverse filtering  problem with unknown sensor and observations). Consider the known data D = P, {?k }N k=0 , where the posteriors have been generated by an HMM-filter sensor. Reconstruct the observations {yk }N k=1 and the observation likelihood matrix B. Finally, we consider the more general setting where the posteriors we obtain are corrupted by noise (due to, e.g., quantization, measurement or model uncertainties). In particular, we consider the case where the following sequence of noisy posteriors is obtained over time: ? ?k = ?k + noise, (2) from the sensor system. We state directly the generalization of Problem 3 (the corresponding generalizations of Problems 1 and 2 follow as special-cases): Problem 4 (Noise-corrupted inverse filtering problem with unknown sensor and observations).  Consider the data D = P, {? ?k }N k=0 , where the posteriors ?k have been generated by an HMMfilter sensor, but we obtain noise-corrupted measurements ? ?k . Estimate the observations {yk }N k=1 and the observation likelihood matrix B. 3 2.3 Inverse filtering as an optimization problem It is possible to formulate Problems 1-4 as optimization problems of increasing levels of generality. As a first step, rewrite the HMM filter equation (1) as:1 (1) ?? bTyk P T ?k?1 ?k = diag(byk )P T ?k?1 . (3) In Problem 3 we need to find what observation occurred at each time instant (a combinatorial problem), and at the same time reconstruct an observation likelihood matrix consistent with the data. To be consistent with the data, equation (3) has to be satisfied. This feasibility problem can be formulated as the following mixed-integer linear program (MILP): min Y {yk }N k=1 ,{bi }i=1 s.t. N X kbTyk P T ?k?1 ?k ? diag(byk )P T ?k?1 k? k=1 yk ? {1, . . . , Y }, bi ? 0, [b1 . . . bY ]1 = 1, for k = 1, . . . , N, for i = 1, . . . , Y, (4) where the choice of norm is arbitrary since for noise-free data it is possible to exactly fit observations and an observation likelihood matrix. In Problem 1, the bi :s are dropped as optimization variables and the problem reduces to an integer program (IP). In Problem 2, where the sequence of observations is known, the problem reduces to a linear program (LP) . Despite the ease of formulation, the down-side of this approach is that, even though Problems 1 and 2 are computationally tractable, the MILP-formulation of Problem 3 can become computationally very expensive for larger data sets. In the following sections, we will outline how the problems can be solved efficiently by exploiting the structure of the HMM filter. 2.4 Assumptions Before providing solutions to Problems 1-4, we state the assumptions that the HMMs in this paper need to satisfy to guarantee unique solutions. The first assumption serves as a proxy for ergodicity of the HMM and the HMM filter ? it is a common assumption in statistical inference for HMMs [18, 4]. Assumption 1 (Ergodicity). The transition matrix P and the observation matrix B are elementwise (strictly) positive. The second assumption is a natural rank assumption on the observation likelihoods. The assumption says that the conditional distribution of any observation is not a linear combination of the conditional distributions of any other observations. Assumption 2 (Distinguishable observation likelihoods). The observation likelihood matrix B is full column rank. We will see that this assumption can be relaxed to the following assumption in problems where only the sequence of observations is to be reconstructed: Assumption 3 (Non-parallel observation likelihoods). No pair of columns of the observation likelihood matrix B is colinear, i.e., bi 6= ?bj for any real number ? and any i 6= j. Without Assumption 3, it is impossible to distinguish between observation i and observation j. Note also that Assumption 2 implies Assumption 3. 3 Solution to the inverse filtering problem for HMMs in absence of noise In this section, we detail our solutions to Problems 1-3. We first provide the following two useful lemmas that will be key to the solutions for Problems 1-4. They give an alternative characterization of the HMM-filter update equation. (Note that all proofs are in the supplementary material.) 1 Multiplication by the denominator is allowed under Assumption 1 ? see below. 4 Lemma 1. The HMM-filter update equation (3) can equivalently be written   ?k (P T ?k?1 )T ? diag(P T ?k?1 ) byk = 0. (5) The second lemma characterizes the solutions to (5). Lemma 2. Under Assumption 1, the nullspace of the X ? X matrix ?k (P T ?k?1 )T ? diag(P T ?k?1 ) (6) is of dimension one for k > 1. 3.1 Solution to the inverse filtering problem with unknown observations In the formulation of Problem 1, we assumed that the observation likelihoods B were known, and aimed to reconstruct the sequence of observations from the posterior data. Equation (5) constrains which columns of the observation matrix B that are consistent with the update of the posterior vector at each time instant. Formally, any sequence   y?k ? y ? {1, . . . , Y } : ?k (P T ?k?1 )T ? diag(P T ?k?1 ) by = 0 , (7) for k = 1, . . . , N , is consistent with the HMM filter posterior updates. (Recall that by denotes column y of the observation matrix B.) Since the problems (7) are decoupled in time k, they can trivially be solved in parallel. Theorem 1. Under Assumptions 1 and 3, the set in the right-hand side of equation (7) is a singleton, and is equal to the true observation, i.e., y?k = yk , (8) for k > 1. 3.2 Solution to the inverse filtering problem with unknown sensor The second inverse filtering problem we consider is when the sequence of observations is known, but the observation likelihoods B are unknown (Problem 2). This problem can be solved by exploiting Lemmas 1 and 2. Computing a basis for the nullspace of the coefficient matrix in formulation (5) of the HMM filter recovers, according to Lemmas 1 and 2, the direction of one column of B. In particular, the direction of the column corresponding to observation yk , i.e., byk . From such basis vectors, we can construct a matrix C ? RX?Y where the yth column is aligned with by . Note that to be able to fully construct this matrix, every observation from the set {1, . . . , Y } needs to have been observed at least once. Due to being basis vectors for nullspaces, the columns of C are only determined up to scalings, so we need to exploit the structure of the observation matrix B to properly normalize them. To form an ? from C, we employ that the observation likelihood matrix is row-stochastic. This means estimate B that we should rescale each column: ? = C diag(?) B (9) ? = 1. Details are provided in the following theorem. for some ? ? RY , such that B1 Theorem 2. If Assumption 1 holds, and every possible observation has been observed (i.e., that {1, . . . , Y } ? {yk }N k=1 ), then: ? = B, i) there exists ? ? RY such that B ? is equal to B. In particular, ii) if Assumption 2 holds, then the choice of ? is unique, and B ? = C ? 1. 5 3.3 Solution to the inverse filtering problem with unknown sensor and observations Finally, we turn to the general formulation in which we consider the combination of the previous two problems: both the sequence of observations and the observation likelihoods are unknown (Problem 3). Again, the solution follows from Lemmas 1 and 2. Note that there will be a degree of freedom since we can arbitrarily relabel each observation and correspondingly permute the columns of the observation likelihood matrix. As in the solution to Problem 2, computing a basis vector, say c?k , for the nullspace of the coefficient matrix in equation (5) recovers the direction of one column of the B matrix. However, since the sequence of observations is unknown, we do not know which column. To circumvent this, we concatenate such basis vectors in a matrix2 C? = [? c2 . . . c?N ] ? RX?(N ?1) . (10) For sufficiently large N ? essentially when every possible observation has been processed by the HMM filter ? the matrix C? in (10) will contain Y columns out of which no pair is colinear (due to Assumption 3). All the columns that are parallel correspond to one particular observation. Let {?1 , . . . , ?Y } be the indices of Y such columns, and construct ? C = C? (11) ? = [e?1 . . . e?Y ] ? R(N ?1)?Y , (12) using the selection matrix where ei is the ith Cartesian basis vector. Lemma 3. Under Assumption 1 and Assumption 3, the expected number of samples needed to be able to construct the selection matrix ? is upper-bounded by ? ?1 (1 + 1/2 + ? ? ? + 1/Y ) , (13) where B ? ? > 0 elementwise. With C constructed in (11), we have obtained the direction of each column of the observation matrix. However, as before, they need to be properly normalized. For this, we exploit the sum-to-one property of the observation matrix as in the previous section. Let ? = C diag(?), B (14) ? = 1. Details on how to find ? are provided in the theorem below. for ? ? RY , such that B1 This solves the first part of the problem, i.e., reconstructing the observation matrix. Secondly, to recover the sequence of observations, take n o y?k ? y ? {1, . . . , Y } : ?by = ?? ck for some real number ? , (15) ? that the nullspace of the HMM filter coefficientfor k > 1. In words; check which columns of B matrix (6) is colinear with at each time instant. Theorem 3. If Assumptions 1 and 3 hold, and the number of samples N is sufficiently large ? see Lemma 3 ? then: ? = BP, where P is a permutation matrix. i) there exists ? ? RY in equation (14) such that B ii) the set on the right-hand side of equation (15) is a singleton. Moreover, the reconstructed observations y?k are, up to relabellings corresponding to P, equal to the true observations yk . ? = BP. In particular, ? = C ? 1. iii) if Assumption 2 holds, then the choice of ? is unique, and B 2 We start with c?2 , since we make no assumption on the positivity of ?0 ? see the proof of Lemma 2. 6 4 Solution to the inverse filtering problem for HMMs in presence of noise In this section, we discuss the more general setting where the posteriors obtained from the sensor system are corrupted by noise. We will see that this problem naturally fits in a clustering framework since every posterior update will provide us with a noisy estimate of the direction of one column of the observation likelihood matrix. We consider an additive noise model of the following form: Assumption 4 (Noise model). The posteriors are corrupted by additive noise wk : ? ?k = ?k + wk , (16) T such that 1 ? ?k = 1 and ? ?k > 0. This noise model is valid, for example, when each observed posterior vector has been subsequently renormalized after noise that originates from quantization or measurement errors has been added. In the solution proposed in Section 3.3 for the noise-free case, the matrix C? in equation (10) was constructed by concatenating basis vectors for the nullspaces of the coefficient matrix in equation (5). With perturbed posterior vectors, the corresponding system of equations becomes   ? ?k (P T ? ?k?1 )T ? diag(P T ? ?k?1 ) c?k = 0, (17) where c?k is now a perturbed (and scaled) version of byk . That this equation is valid is guaranteed by the generalization of Lemma 2: Lemma 4. Under Assumptions 1 and 4, the nullspace of the matrix ? ?k (P T ? ?k?1 )T ? diag(P T ? ?k?1 ) (18) is of dimension one for k > 1. Remark 1. In case Assumption 4 does not hold, the problem can instead be interpreted as a perturbed eigenvector problem. The vector c?k should then be taken as the eigenvector corresponding to the smallest eigenvalue. Lemma 4 says that we can construct a matrix C? (analogous to C? in Section 3.3) by concatenating the basis vectors from the one-dimensional nullspaces in (17). Due to the perturbations, every solution to equation (17) will be a perturbed version of the solution to the corresponding noise-free version of the equation. This means that it will not be possible to construct a selection matrix ? as was done for C? in equation (12). However, because there are only Y unique solutions to the noise-free equations (5), it is natural to circumvent this (assuming that the perturbations are small) by clustering the columns of C? into Y clusters. As the columns of C? are only unique up to scaling, the clustering has to be performed with respect to their angular separations (using, e.g., the spherical k-means algorithm [19]). Let C ? RX?Y be the matrix of the Y centroids resulting from running a clustering algorithm on the ? Each centroid can be interpreted as a noisy estimate of one column of the observation columns of C. likelihood matrix. To obtain a properly normalized estimate of the observation likelihood matrix, we take ? = CA, B (19) where A ? RY ?Y . Note that, since C now contains noisy estimates of the directions of the columns of the observation likelihood matrix, we are not certain to be able to properly normalize it by purely rescaling each column (i.e., taking A to be a diagonal matrix as was done in Sections 3.2 and 3.3). A logical choice is the solution to the following LP, min max [A]ij A?RY ?Y i6=j CA ? 0, CA1 = 1, (20) which tries to minimize the off-diagonal elements of A. The resulting rescaling matrix A guarantees ? = CA is a proper stochastic matrix (non-negative and has row-sum equal to one), as well as that B ? are minimized. that the discrepancy between the directions of the columns of C and B s.t. The second part of the problem ? reconstructing the sequence of observations ? follows naturally from the clustering algorithm; an estimate of the sequence is obtained by checking to what cluster the solution c?k of equation (17) belongs in for each time instant. 7 5 Experimental results for sleep segmentation In this section, we illustrate the inverse filtering problem on real-world data. Background Roughly one third of a person?s life is spent sleeping. Sleep disorders are becoming more prevalent and, as public awareness has increased, the usage of sleep trackers is becoming wide-spread. The example below illustrates how the inverse filtering formulation and associated algorithms can be used as a step in real-time diagnosis of failure of sleep-tracking medical equipment. During the course of sleep, a human transitions through five different sleep stages [20]: wake, S1, S2, slow wave sleep (SWS) and rapid eye movement (REM). An important part of sleep analysis is obtaining a patient?s evolution over these sleep stages. Manual sequencing from all-night polysomnographic (PSG) recordings (including, e.g., electroencephalogram (EEG) readings) can be performed according to the Rechtschaffen and Kales (R&K) rules by well-trained experts [8, 20]. However, this is costly and laborious, so several works, e.g., [8, 20, 21], propose automatic sequencing based on HMMs. These systems usually output a posterior distribution over the sleep stages, or provide a Viterbi path. A malfunction of such an automatic system could have problematic consequences since medical decisions would be based on faulty information. The inverse filtering problem arises naturally for such reasons of fault-detection. Joint knowledge of the transition matrix can be assumed, since it is possible to obtain, from public sources, manually labeled data from which an estimate of P can be computed. Setup A version of the automatic sleep-staging system in [8, 20] was implemented. The mean frequency over the 0-30 Hz band of the EEG (over C3-A2 or C4-A1, according to the international 10-20 system) was used as observations. These readings were encoded to five symbols using a vectorquantization based codebook. The model was trained on data from nine patients in the PhysioNet CAP Sleep Database [22, 23]. The model was then evaluated on another patient ? see Fig. 1 ? over one full-night of sleep. The manually labeled stages according to K&R-rules are dashed-marked in the figure. To summarize the resulting posterior distributions over the sleep stages, we plot the mean state estimate when equidistant numbers have been assigned to each state. For the inverse filtering, the full posterior vectors were elementwise corrupted by Gaussian noise of standard deviation ?, and projected back to the simplex (to ensure a valid posterior probability vector) ? simulating a noisy reading from the automatic system. A total of one hundred noise realizations were simulated. The noise can be a manifestation of measurement or quantization noise in the sensor system, or noise related to model uncertainties (in this case, an error in the transition probability matrix P ). Results After permuting the labels of the observations, the error in the reconstructed observation likelihood matrix, as well as the fraction of correctly reconstructed observations, were computed. This is illustrated in Fig. 2. For the 1030 quantized EEG samples from the patient, the entire procedure takes less than one second on a 2.0 Ghz Intel Core 2 Duo processor system. REM SWS S2 S1 WAKE 0 1 2 3 4 5 6 7 8 hours since bedtime Figure 1: One night of sleep in which polysomnographic (PSG) observation data has been manually processed by an expert sleep analyst according to the R&K rules to obtain the sleep stages ( ). The posterior distribution over the sleep stages, resulting from an automatic sleep-staging system, has been summarized to a mean state estimate ( ). 8 0.5 0.2 10 ?8 10 ?6 ?4 10 noise ? 10 ?2 10 0 P 1 ? ? BPkF min kB fraction Correctly recovered observations Error in B 100 10?2 10?4 10?8 10?6 10?4 noise ? 10?2 100 Figure 2: Result of inverse filtering for various noise standard deviations ?. The vector of posterior probabilities is perturbed elementwise with Gaussian noise. Right: Error in the recovered observation likelihood matrix after permuting the columns to find the best match to the true matrix. Left: Fraction of correctly reconstructed observations. As the signal-to-noise ratio increases, the inverse filtering algorithm successfully reconstructs the sequence of observations and estimates the observation likelihoods. From Fig. 2, we can see that as the variance of the noise decreases, the left hand side of equation (17) converges to that of equation (5) and the true quantities are recovered. On the other extreme, as the signal-to-noise ratio becomes small, the estimated sequence of observations tends to that of a uniform distribution at 1/Y = 0.2. This is because the clusters in C? become heavily intertwined. The discontinuous nature of the solution of the clustering algorithm is apparent by the plateau-like behaviour in the middle of the scale ? a few observations linger on the edge of being assigned to the correct clusters. In conclusion, the results show that it is possible to estimate the observation sequence processed by the automatic sleep-staging system, as well as, its sensor?s specifications. This is an important step in performing fault detection for such a device: for example, using several nights of data, it is possible to perform change detection on the observation likelihoods to detect if the sleep monitoring device has failed. 6 Conclusions In this paper, we have considered several inverse filtering problems for HMMs. Given posteriors from an HMM filter (or more generally, noisily observed posteriors), the aim was to reconstruct the observation likelihoods and also the sample path of observations. It was shown that a computationally expensive solution based on combinatorial optimization can be avoided by exploiting the algebraic structure of the HMM filter. We provided solutions to the inverse filtering problems, as well as theorems guaranteeing unique identifiability. The more general case of noise-corrupted posteriors was also considered. A solution based on clustering was proposed and evaluated on real-world data based on a system for automatic sleep-staging from EEG readings. In the future, it would be interesting to consider other variations and generalizations of inverse filtering. For example, the case where the system dynamics are unknown and need to be estimated, or when only actions based on the filtered distribution can be observed. Acknowledgments This work was partially supported by the Swedish Research Council under contract 2016-06079, the U.S. Army Research Office under grant 12346080 and the National Science Foundation under grant 1714180. The authors would like to thank Alexandre Proutiere for helpful comments during the preparation of this work. References [1] V. Krishnamurthy, Partially Observed Markov Decision Processes. Cambridge, UK: Cambridge University Press, 2016. 9 [2] L. Rabiner, ?A tutorial on hidden Markov models and selected applications in speech recognition,? Proceedings of the IEEE, vol. 77, pp. 257?286, Feb. 1989. [3] R. J. Elliott, J. B. Moore, and L. Aggoun, Hidden Markov Models: Estimation and Control. New York, NY: Springer, 1995. [4] O. Capp?, E. Moulines, and T. Ryd?n, Inference in Hidden Markov Models. New York, NY: Springer, 2005. [5] F. Gustafsson, Adaptive filtering and change detection. New York: Wiley, 2000. [6] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems. Boston, MA: Springer, 1999. [7] A. Caplin and M. Dean, ?Revealed preference, rational inattention, and costly information acquisition,? The American Economic Review, vol. 105, no. 7, pp. 2183?2203, 2015. [8] A. Flexerand, G. Dorffner, P. Sykacekand, and I. Rezek, ?An automatic, continuous and probabilistic sleep stager based on a hidden Markov model,? Applied Artificial Intelligence, vol. 16, pp. 199?207, Mar. 2002. [9] D. Koller and N. Friedman, Probabilistic graphical models: principles and techniques. Cambridge, MA: MIT Press, 2009. [10] B. Anderson and J. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [11] R. E. Kalman, ?When is a linear control system optimal,? Journal of Basic Engineering, vol. 86, no. 1, pp. 51?60, 1964. [12] H. R. Varian, Microeconomic analysis. New York: Norton, 3rd ed., 1992. [13] D. Hadfield-Menell, S. J. Russell, P. Abbeel, and A. Dragan, ?Cooperative inverse reinforcement learning,? in Advances in Neural Information Processing Systems, 2016. [14] J. Choi and K.-E. Kim, ?Nonparametric Bayesian inverse reinforcement learning for multiple reward functions,? in Advances in Neural Information Processing Systems, 2012. [15] E. Klein, M. Geist, B. Piot, and O. Pietquin, ?Inverse Reinforcement Learning through Structured Classification,? in Advances in Neural Information Processing Systems, 2012. [16] S. Levine, Z. Popovic, and V. Koltun, ?Nonlinear inverse reinforcement learning with gaussian processes,? in Advances in Neural Information Processing Systems, 2011. [17] A. Ng, ?Algorithms for inverse reinforcement learning,? in Proceedings of the 17th International Conference on Machine Learning (ICML?00), pp. 663?670, 2000. [18] L. E. Baum and T. Petrie, ?Statistical inference for probabilistic functions of finite state Markov chains,? The annals of mathematical statistics, vol. 37, no. 6, pp. 1554?1563, 1966. [19] C. Buchta, M. Kober, I. Feinerer, and K. Hornik, ?Spherical k-means clustering,? Journal of Statistical Software, vol. 50, no. 10, pp. 1?22, 2012. [20] S.-T. Pan, C.-E. Kuo, J.-H. Zeng, and S.-F. Liang, ?A transition-constrained discrete hidden Markov model for automatic sleep staging,? BioMedical Engineering OnLine, vol. 11, no. 1, p. 52, 2012. [21] Y. Chen, X. Zhu, and W. Chen, ?Automatic sleep staging based on ECG signals using hidden Markov models,? in Proceedings of the 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 530?533, 2015. [22] A. L. Goldberger, L. A. Amaral, L. Glass, J. M. Hausdorff, P. C. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E. Stanley, ?Physiobank, physiotoolkit, and physionet,? Circulation, vol. 101, no. 23, pp. e215?e220, 2000. [23] M. G. Terzano, L. Parrino, A. Sherieri, R. Chervin, S. Chokroverty, C. Guilleminault, M. Hirshkowitz, M. Mahowald, H. Moldofsky, A. Rosa, and others, ?Atlas, rules, and recording techniques for the scoring of cyclic alternating pattern (CAP) in human sleep,? Sleep medicine, vol. 2, no. 6, pp. 537?553, 2001. 10
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Non-parametric Structured Output Networks Andreas M. Lehrmann Disney Research Pittsburgh, PA 15213 [email protected] Leonid Sigal Disney Research Pittsburgh, PA 15213 [email protected] Abstract Deep neural networks (DNNs) and probabilistic graphical models (PGMs) are the two main tools for statistical modeling. While DNNs provide the ability to model rich and complex relationships between input and output variables, PGMs provide the ability to encode dependencies among the output variables themselves. End-to-end training methods for models with structured graphical dependencies on top of neural predictions have recently emerged as a principled way of combining these two paradigms. While these models have proven to be powerful in discriminative settings with discrete outputs, extensions to structured continuous spaces, as well as performing efficient inference in these spaces, are lacking. We propose non-parametric structured output networks (NSON), a modular approach that cleanly separates a non-parametric, structured posterior representation from a discriminative inference scheme but allows joint end-to-end training of both components. Our experiments evaluate the ability of NSONs to capture structured posterior densities (modeling) and to compute complex statistics of those densities (inference). We compare our model to output spaces of varying expressiveness and popular variational and sampling-based inference algorithms. 1 Introduction In recent years, deep neural networks have led to tremendous progress in domains such as image classification [1, 2] and segmentation [3], object detection [4, 5] and natural language processing [6, 7]. These achievements can be attributed to their hierarchical feature representation, the development of effective regularization techniques [8, 9] and the availability of large amounts of training data [10, 11]. While a lot of effort has been spent on identifying optimal network structures and trainings schemes to enable these advances, the expressiveness of the output space has not evolved at the same rate. Indeed, it is striking that most neural architectures model categorical posterior distributions that do not incorporate any structural assumptions about the underlying task; they are discrete and global (Figure 1a). However, many tasks are naturally formulated as structured problems or would benefit from continuous representations due to their high cardinality. In those cases, it is desirable to learn an expressive posterior density reflecting the dependencies in the underlying task. As a simple example, consider a stripe of n noisy pixels in a natural image. If we want to learn a neural network that encodes the posterior distribution p? (y | x) of the clean output y given the noisy input x, we must ensure that p? is expressive enough to represent potentially complex noise distributions and structured enough to avoid modeling spurious dependencies between the variables. Probabilistic graphical models [12], such as Bayesian networks or Markov random fields, have a long history in machine learning and provide principled frameworks for such structured data. It is therefore natural to use their factored representations as a means of enforcing structure in a deep neural network. While initial results along this line of research have been promising [13, 14], they focus exclusively on the discrete case and/or mean-field inference. Instead, we propose a deep neural network that encodes a non-parametric posterior density that factorizes over a graph (Figure 1b). We perform recurrent inference inspired by message-passing in this structured output space and show how to learn all components end-to-end. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. (i) Deep Neural Network Classification Neural Network Dropout FC U V U ReLU ReLU ?= |Y | {pi }i=1 Pooling p? Convolution p? (Y = y) Dropout FC ReLU ReLU Pooling x Y ? (a) Traditional Neural Network: discrete, global, parametric. ?3|2 ?en? ?n|? ?e32 V Y1 Y1 ?e21 ?2|1 e e, B} e ? ?e = {w, p?2|1 Y3 Y2 Y4 ??? Convolution ??? p? (y | x) ?1 ??? y ?e1 ??? x (ii) Non-parametric Graphical Model ?? (?, y) Yn-1 LM Y5 Yn Yi | pa(Yi ) ? p?i|pa(i) Recurrent Inference Network Fig. 2 LI (b) Non-parametric Structured Output Network: continuous, structured, non-parametric. Figure 1: Overview: Non-parametric Structured Output Networks. (a) Traditional neural networks use a series of convolution and inner product modules to predict a discrete posterior without graphical structure (e.g., VGG [15]). [grey = b optional] (b) Non-parametric structured output networks use a deep neural network to predict a non-parametric graphical model p? (x) (y) (NGM) that factorizes over a graph. A recurrent inference network (RIN) computes statistics t[p? (x) (y)] from this structured output density. At training time, we propagate stochastic gradients from both NGM and RIN back to the inputs. 1.1 Related Work Our framework builds upon elements from neural networks, structured models, non-parametric statistics, and approximate inference. We will first present prior work on structured neural networks and then discuss the relevant literature on approximate non-parametric inference. 1.1.1 Structured Neural Networks Structured neural networks combine the expressive representations of deep neural networks with the structured dependencies of probabilistic graphical models. Early attempts to combine both frameworks used high-level features from neural networks (e.g., fc7) to obtain fixed unary potentials for a graphical model [18]. More recently, statistical models and their associated inference tasks have been reinterpreted as (layers in) neural networks, which has allowed true end-to-end training and blurred the line between both paradigms: [13, 14] express the classic mean-field update equations as a series of layers in a recurrent neural network (RNN). Structure inference machines [17] use an RNN to simulate message-passing in a graphical model with soft-edges for activity recognition. A full backward-pass through loopy-BP was proposed in [19]. The structural-RNN [16] models all node and edge potentials in a spatio-temporal factor graph as RNNs that are shared among groups of nodes/edges with similar semantics. Table 1 summarizes some important properties of these methods. Notably, all output spaces except for the non-probabilistic work [16] are discrete. 1.1.2 Inference in Structured Neural Networks In contrast to a discrete and global posterior, which allows inference of common statistics (e.g., its mode) in linear time, expressive output spaces, as in Figure 1b, require message-passing schemes [20] Output Space Related Work Continuous Non-parametric Structured End-to-end Training Prob. Inference Posterior Sampling VGG MRF-RNN [15] [14] 7 7 7 X X MF 7 D X Structural RNN [16] Structure Inference Machines [17] Deep Structured Models [13] X 7 X X 7 7 7 7 X 7 MP X X X MF 7 Table 1: Output Space Properties Across Models. [MF: mean-field; MP: message passing; D: direct; ? ?: not applicable] 2 NSON (ours) X X X X MP X to propagate and aggregate information. Local potentials outside of the exponential family, such as non-parametric distributions, lead to intractable message updates, so one needs to resort to approximate inference methods, which include the following two popular groups: Variational Inference. Variational methods, such as mean-field and its structured variants [12], approximate an intractable target distribution with a tractable variational distribution by maximizing the evidence lower bound (ELBO). Stochastic extensions allow the use of this technique even on large datasets [21]. If the model is not in the conjugate-exponential family [22], as is the case for non-parametric graphical models, black box methods must be used to approximate an intractable expectation in the ELBO [23]. For fully-connected graphs with Gaussian pairwise potentials, the dense-CRF model [24] proposes an efficient way to perform the variational updates using the permutohedral lattice [25]. For general edge potentials, [26] proposes a density estimation technique that allows the use of non-parametric edge potentials. Sampling-based Inference. This group of methods employs (sets of) samples to approximate intractable operations when computing message updates. Early works use iterative refinements of approximate clique potentials in junction trees [27]. Non-parametric belief propagation (NBP) [28, 29] represents each message as a kernel density estimate and uses Gibbs sampling for propagation. Particle belief propagation [30] represents each message as a set of samples drawn from an approximation to the receiving node?s marginal, effectively circumventing the kernel smoothing required in NBP. Diverse particle selection [31] keeps a diverse set of hypothesized solutions at each node that pass through an iterative augmentation-update-selection scheme that preserves message values. Finally, a mean shift density approximation has been used as an alternative to sampling in [32]. 1.2 Contributions Our NSON model is inspired by the structured neural architectures (Section 1.1.1). However, in contrast to those approaches, we model structured dependencies on top of expressive non-parametric densities. In doing so, we build an inference network that computes statistics of these non-parametric output densities, thereby replacing the need for more conventional inference (Section 1.1.2). In particular, we make the following contributions: (1) We propose non-parametric structured output networks, a novel approach combining the predictive power of deep neural networks with the structured representation and multimodal flexibility of non-parametric graphical models; (2) We show how to train the resulting output density together with recurrent inference modules in an end-to-end way; (3) We compare non-parametric structured output networks to a variety of alternative output densities and demonstrate superior performance of the inference module in comparison to variational and sampling-based approaches. 2 Non-parametric Structured Output Networks Traditional neural networks (Figure 1a; [15]) encode a discrete posterior distribution by predicting e an input-conditioned parameter vector ?(x) of a categorical distribution, i.e., Y | X = x ? p?(x) . e ? (x) parameterizes a Non-parametric structured output networks (Figure 1b) do the same, except that e continuous graphical model with non-parametric potentials. It consists of three components: A deep neural network (DNN), a non-parametric graphical model (NGM), and a recurrent inference network (RIN). While the DNN+NGM encode a structured posterior (= b model), the RIN computes complex statistics in this output space (= b inference). At a high level, the DNN, conditioned on an input x, predicts the parameters ?e = {?eij } (e.g., kernel weights, centers and bandwidths) of local non-parametric distributions over a node and its parents according to the NGM?s graph structure (Figure 1b). Using a function ?? , these local joint distributions are then transformed to conditional distributions parameterized by ? = {?i|j } (e.g., through a closed-form conditioning operation) and assembled into a structured joint density p? (x) (y) with conditional (in)dependencies prescribed by the graphical model. Parameters of the DNN are optimized with respect to a maximum-likelihood loss LM . Simultaneously, a recurrent inference network (detailed in Figure 2) that takes ?e as input, is trained to compute statistics of the structured distribution (e.g., marginals) using a separate inference loss LI . The following two paragraphs discuss these elements in more detail. 3 Model (DNN+NGM). The DNN is parameterized by a weight vector from a generic input space X to a Cartesian parameter space ?n , M and encodes a function M e x7 ! ? (x) = (?ei,pa(i) (x))ni=1 , (1) each of whose components models a joint kernel density (Yi , pa(Yi )) ? p?ei,pa(i) (x) and thus, implicitly, the local conditional distribution Yi | pa(Yi ) ? p?i|pa(i) (x) of a non-parametric graphical model p? (x) (y) = n Y i=1 (2) p?i|pa(i) (x) (yi | pa(yi )) over a structured output space Y with directed, acyclic graph G = (Y, E). Here, pa(?) denotes the set of parent nodes w.r.t. G, which we fix in advance based on prior knowledge or structure learning [12]. The conditional density of a node Y = Yi with parents Y 0 = pa(Yi ) and parameters ? = ?i|pa(i) (x) is thus given by1 N X p? (y | y 0 ) = w(j) ? |B(j) | 1 ?(B( j) (y ?(j) )), (3) j=1 Q where the differentiable kernel ?(u) = i q(ui ) is defined in terms of a symmetric, zero-mean density q with positive variance and the conditional parameters ? = (w, ? , B) 2 ? correspond to the full set of kernel weights, kernel centers, and kernel bandwidth matrices, respectively.2 The functional relationship between ? and its joint counterpart ?e = ?ei,pa(i) (x) is mediated through a e = ?? (w, e = ? and can be computed in closede, B) e ? kernel-dependent conditioning operation ?? (?) form for a wide range of kernels, including Gaussian, cosine, logistic and other kernels with sigmoid e (j) ? e(j) y e (j) = By 0(j) and ? CDF. In particular, for block decompositions B e(j) = (j) , we obtain e =?= ?? (?) 8 (j) e (j) > e(j) ? |B >w /w y0 | < 0 1 ?(j) = ? e(j) y , > > : B(j) = B e (j) . y ( j) e 0 ?(B y e B y0 (y 0 (j) ? ey0 )), ? ey0 1?j?N (4) See Appendix A.1 for a detailed derivation. We refer to the structured posterior density in Eq. (2) with the non-parametric local potentials in Eq. (3) as a non-parametric structured output network. 0 Given an output training set DY = {y(i) 2 Y}N i=1 , traditional kernel density estimation [33] can be viewed as an extreme special case of this architecture in which the discriminative, trainable DNN is replaced with a generative, closed-form estimator and n := 1 (no structure), N := N 0 (#kernels = #training points), w(i) := (N 0 ) 1 (uniform weights), B(i) := B(0) (shared covariance) and ?(i) := y(i) (fixed centers). When learning M from data, we can easily enforce parts or all of those restrictions in our model (see Section 5), but Section 3 will provide all necessary derivations for the more general case shown above. Inference (RIN). In contrast to traditional classification networks with discrete label posterior, non-parametric structured output networks encode a complex density with rich statistics. We employ a recurrent inference network with parameters I to compute such statistics t from the predicted parameters ?e(x) 2 ?n , I ?e(x) 7 ! t[p? (x) ]. (5) Similar to conditional graphical models, the underlying assumption is that the input-conditioned density p? (x) contains all information about the semantic entities of interest and that we can infer whichever statistic we are interested in from it. A popular example of a statistic is a summary statistic, t[p? (x) ](yi ) = opy\yi p? (x) (y) d(y\yi ), (6) R which is known as sum-product BP (op = ; computing marginals) and max-product BP (op = max; computing max-marginals). Note, however, that we can attach recurrent inference networks corresponding to arbitrary tasks to this meta representation. Section 4 discusses the necessary details. ? ? := B(j) 1 1 > 1 We write B( 2 Note that ? represents the parameters of a specific node; different nodes may have different parameters. j) and B T := B 4 to avoid double superscripts. 3 Learning Structured Densities using Non-Parametric Back-Propagation The previous section introduced the model and inference components of a non-parametric structured output network. We will now describe how to learn the model (DNN+NGM) from a supervised training set (x(i) , y(i) ) ? pD . 3.1 Likelihood Loss ? (x; M )) to explicitly refer to the weights M of the deep neural network We write ? (x; M ) = ?? (e predicting the non-parametric graphical model (Eq. (1)). Since the parameters of p? (x) are deterministic predictions from the input x, the only free and learnable parameters are the components of M . We train the DNN via empirical risk minimization with a negative log-likelihood loss LM , ? M = argmin E(x,y)?bpD [LM (?? (x; M ), y)] M = argmax E(x,y)?bpD [log p? (x; M) (7) (y)], M where pbD refers to the empirical distribution and the expectation in Eq. (7) is taken over the factorization in Eq. (2) and the local distributions in Eq. (3). Note the similarities and differences between a non-parametric structured output network and a non-parametric graphical model with unary potentials from a neural network: Both model classes describe a structured posterior. However, while the unaries in the latter perform a reweighting of the potentials, a non-parametric structured output network predicts those potentials directly and allows joint optimization of its DNN and NGM components by back-propagating the structured loss first through the nodes of the graphical model and then through the layers of the neural network all the way back to the input. 3.2 Topological Non-parametric Gradients We optimize Eq. (7) via stochastic gradient descent of the loss LM w.r.t. the deep neural network weights M using Adam [34]. Importantly, the gradients r M LM (?? (x; M ), y) decompose into a factor from the deep neural network and a factor from the non-parametric graphical model, r M LM (?? (x; M ), y) = @ log p? (x; e @ ??(x; e (y) @ ??(x; M) , ? @ M M) M) (8) where the partial derivatives of the second factor can be obtained via standard back-propagation and the first factor decomposes according to the graphical model?s graph structure G, @ log p? (x; e @ ??(x; M) (y) M) n X @ log p?i|pa(i) (x; M ) (yi | pa(yi )) = . e @ ??(x; M) (9) i=1 e The gradient of a local model w.r.t. the joint parameters ??(x; M ) is given by two factors accounting for the gradient w.r.t. the conditional parameters and the Jacobian of the conditioning operation, @ log p?i|pa(i) (x; M ) (yi | pa(yi )) @ log p?i|pa(i) (x; M ) (yi | pa(yi )) @ ? (x; = ? e @ ? (x; M ) @ ??(x; @ ?e(x; M) M) M) . (10) Note that the Jacobian takes a block-diagonal form, because ? = ?i|pa(i) (x; M ) is independent of ?e = ?ej,pa(j) (x; M ) for i 6= j. Each block constitutes the backward-pass through a node Yi ?s conditioning operation, 2 3 @w @w @w e @B e @e ? 6 @w @? @ (w, ? , B) 6 ? @? 6 = = 6 0 @e ? e e, B) e ? @ ?e @ (w, 4 0 0 7 7 0 7 7, 5 (11) @B e @B where the individual entries are given by the derivatives of Eq. (4), e.g., @w e = (w ? w + diag(w)) ? diag(w) e @w 5 1 . (12) Similar equations exist for the derivatives of the weights w.r.t. the kernel locations and kernel bandwidth matrices; the remaining cases are simple projections. In practice, we may be able to group the potentials p?i|pa(i) according to their semantic meaning, in which case we can train one potential per group instead of one potential per node by sharing the corresponding parameters in Eq. (9). All topological operations can be implemented as separate layers in a deep neural network and the corresponding gradients can be obtained using automatic differentiation. 3.3 Distributional Non-parametric Gradients We have shown how the gradient of the loss factorizes over the graph of the output space. Next, we will provide the gradients of those local factors log p? (y | y 0 ) (Eq. (3)) w.r.t. the local parameters ? = ?i|pa(i) . To reduce notational clutter, we introduce the shorthand yb(k) := B( k) (y ?(k) ) to refer to the normalized input and provide only final results; detailed derivations for all gradients and worked out examples for specific kernels can be found in Appendix A.2. Kernel Weights. ? rw log p? (y | y ) = > , w ? 0 ? := ? |B ( k) (k) |?(b y ) ?N . (13) k=1 Note that w is required to lie on the standard (N 1)-simplex (N 1) . Different normalizations are possible, including a softmax or a projection onto the simplex, i.e., ? (N 1) (w(i) ) = max(0, w(i) + u) and u is the unique translation such that the positive points sum to 1 [35]. Kernel Centers. ? ?N B( >k) @?(b y (k) ) := ? . (14) |B(k) | @b y (k) k=1 The kernel centers do not underlie any spatial restrictions, but proper initialization is important. Typically, we use the centers of a k-means clustering with k := N to initialize the kernel centers. w r? log p? (y | y ) = , w> ? 0 Kernel Bandwidth Matrices. rB log p? (y | y 0 ) = w , w> ? := ? ? ??N B( >k) @?(b y (k) ) (>k) (k) ? ?(b y ) + y b . |B(k) | @b y (k) k=1 (15) While computation of the gradient w.r.t. B is a universal approach, specific kernels may allow alternative gradients: In a Gaussian kernel, for instance, the Gramian of the bandwidth matrix acts as a covariance matrix. We can thus optimize B(k) B(>k) in the interior of the cone of positive-semidefinite matrices by computing the gradients w.r.t. the Cholesky factor of the inverse covariance matrix. 4 Inferring Complex Statistics using Neural Belief Propagation The previous sections introduced non-parametric structured output networks and showed how their components, DNN and NGM, can be learned from data. Since the resulting posterior density p? (x) (y) (Eq. (2)) factorizes over a graph, we can, in theory, use local messages to propagate beliefs about statistics t[p? (x) (y)] along its edges (BP; [20]). However, special care must be taken to handle intractable operations caused by non-parametric local potentials and to allow an end-to-end integration. For ease of exposition, we assume that we can represent the local conditional distributions as a set of pairwise potentials { (yi , yj )}, effectively converting our directed model to a normalized MRF. This is not limiting, as we can always convert a factor graph representation of Eq. (2) into an equivalent pairwise MRF [36]. In this setting, a BP message ?i!j (yj ) from Yi to Yj takes the form ?i!j (yj ) = opyi (yi , yj ) ? ??!i (yi ), (16) where the operator opy computes a summary statistic, such as integration or maximization, and ??!i (yi ) is the product of all incoming messages at Yi . In case of a graphical model with nonparametric local distributions (Eq. (3)), this computation is not feasible for two reasons: (1) the premessages ??!i (yi ) are products of sums, which means that the number of kernels grows exponentially in the number of incoming messages; (2) the functional opy does not usually have an analytic form. 6 LM ?e32 1 k 2 ne(i)\j ?eij FC+ReLU (t 1) ? bk!i Stacking 1 (t) ? bi!j t = 1, . . . , T bbi (i) LI ?e21 FC+ReLU 1 1 (T ) ? bk!i k 2 ne(i) (t 1) ? b3!2 (t 1) ? b1!2 ?e42 (t 1) ? b3!2 (t 1) ? b1!2 FC Non-parametric Graphical Model Fig. 1(b)(ii) FC {?eij } ?e42 FC Deep Neural Network Fig. 1(b)(i) ?e32 (t) ? b2!4 ?e21 i = 1, . . . , n (a) Recurrent Inference Network. (b) Partially Unrolled Inference Network. Figure 2: Inferring Complex Statistics. Expressive output spaces require explicit inference procedures to obtain posterior statistics. We use an inference network inspired by message-passing schemes in non-parametric graphical models. (a) An RNN iteratively computes outgoing messages from incoming messages and the local potential. (b) Unrolled inference network illustrating the computation of ? b2!4 in the graph shown in Figure 1b. Inspired by recent results in imitation learning [37] and inference machines for classification [17, 38], we take an alternate route and use an RNN to model the exchange of information between nonparametric nodes. In particular, we introduce an RNN node ? bi!j for each message and connect them in time according to Eq. (16), i.e., each node has incoming connections from its local potential ?eij , predicted by the DNN, and the nodes {b ?k!i : k 2 neG (i)\j}, which correspond to the incoming messages. The message computation itself is approximated through an FC+ReLU layer with weights i!j . An approximate message ? bi!j from Yi to Yj can thus be written as I ? bi!j = ReLU(FC i!j I (Stacking(?eij , {b ?k!i : k 2 neG (i)\j}))), (17) where neG (?) returns the neighbors of a node in G. The final beliefs bbi = ? b?!i ? ? bi!j can be implemented analogously. Similar to (loopy) belief updates in traditional message-passing, we run the RNN for a fixed number of iterations, at each step passing all neural messages. Furthermore, using the techniques discussed in Section 3.3, we can ensure that the messages are valid non-parametric distributions. All layers in this recurrent inference network are differentiable, so that we can propagate Pn (i) a decomposable inference loss LI = i=1 LI end-to-end back to the inputs. In practice, we find that generic loss functions work well (see Section 5) and that canonic loss functions can often be obtained directly from the statistic. The DNN weights M are thus updated so as to do both predict the right posterior density and, together with the RIN weights I , perform correct inference in it (Figure 2). 5 Experiments We validate non-parametric structured output networks at both the model (DNN+NGM) and the inference level (RIN). Model validation consists of a comparison to baselines along two binary axes, structuredness and non-parametricity. Inference validation compares our RIN unit to the two predominant groups of approaches for inference in structured non-parametric densities, i.e., sampling-based and variational inference (Section 1.1.2). 5.1 Dataset We test our approach on simple natural pixel statistics from Microsoft COCO [11] by sampling stripes y = (yi )ni=1 2 [0, 255]n of n = 10 pixels. Each pixel yi is corrupted by a linear noise model, leading to the observable output xi = ? yi + ?, with ? ? N (255 ? , 1 , 2 ) and ? Ber( ), where the target space of the Bernoulli trial is { 1, +1}. For our experiments, we set 2 = 100 and = 0.5. Using this noise process, we generate training and test sets of sizes 100,000 and 1,000, respectively. 5.2 Model Validation The distributional gradients (Eq. (9)) comprise three types of parameters: Kernel locations, kernel weights, and kernel bandwidth matrices. Default values for the latter two exist in the form of uniform weights and plug-in bandwidth estimates [33], respectively, so we can turn optimization of those 7 Neural Network Structured Gaussian Kernel Density Non-param. Model 7 X 7 7 Parameter Group Estimation W +W B +B B +B 1.13 (ML estimation) +6.66 (Plug-in bandwidth estimation) Gaussian 7 7 0.90 +2.54 0.88 +2.90 GGM [39] 7 X 0.85 +1.55 0.93 +1.53 + Mixture Density [40] X 7 +9.22 +6.87 +11.18 +11.51 NGM-100 (ours) X X +15.26 +15.30 +16.00 +16.46 (a) Model Validation Inference Particles Performance Runtime (marg. log-lik.) (sec) BB-VI [23] 400 800 +2.30 +3.03 660.65 1198.08 P-BP [30] 50 100 200 400 +2.91 +6.13 +7.01 +8.85 0.49 2.11 6.43 21.13 +16.62 0.04 RIN-100 (ours) (b) Inference Validation Table 2: Quantitative Evaluation. (a) We report the expected log-likelihood of the test set under the predicted posterior p? (x) (y), showing the need for a structured and non-parametric approach to model rich posteriors. (b) Inference using our RIN architecture is much faster than sampling-based or variational inference while still leading to accurate marginals. [(N/G)GM: Non-parametric/Gaussian Graphical Model; RIN-x: Recurrent Inference Network with x kernels; P-BP: Particle Belief Propagation; BB-VI: Black Box Variational Inference] parameter groups on/off as desired.3 In addition to those variations, non-parametric structured output networks with a Gaussian kernel ? = N (? | ~0, I) comprise a number of popular baselines as special cases, including neural networks predicting a Gaussian posterior (n = 1, N = 1), mixture density networks (n = 1, N > 1; [40]), and Gaussian graphical models (n > 1, N = 1; [39]). For the sake of completeness, we also report the performance of two basic posteriors without preceding neural network, namely a pure Gaussian and traditional kernel density estimation (KDE). We compare our approach to those baselines in terms of the expected log-likelihood on the test set, which is a relative measure for the KL-divergence to the true posterior. Setup and Results. For the two basic models, we learn a joint density p(y, x) by maximum likelihood (Gaussian) and plug-in bandwidth estimation (KDE) and condition on the inputs x to infer the labels y. We train the other 4 models for 40 epochs using a Gaussian kernel and a diagonal bandwidth matrix for the non-parametric models. The DNN consists of 2 fully-connected layers with 256 units and the kernel weights are constrained to lie on a simplex with a softmax layer. The NGM uses a chain-structured graph that connects each pixel to its immediate neighbors. Table 2a shows our results. Ablation study: unsurprisingly, a purely Gaussian posterior cannot represent the true posterior appropriately. A multimodal kernel density works better than a neural network with parametric posterior but cannot compete with the two non-parametric models attached to the neural network. Among the methods with a neural network, optimization of kernel locations only (first column) generally performs worst. However, the W + B setting (second column) gets sometimes trapped in local minima, especially in case of global mixture densities. If we decide to estimate a second parameter group, weights (+W ) should therefore be preferred over bandwidths (+B). Best results are obtained when estimation is turned on for all three parameter groups. Baselines: the two non-parametric methods consistently perform better than the parametric approaches, confirming our claim that non-parametric densities are a powerful alternative to a parametric posterior. Furthermore, a comparison of the last two rows shows a substantial improvement due to our factored representation, demonstrating the importance of incorporating structure into high-dimensional, continuous estimation problems. Learned Graph Structures. While the output variables in our experiments with one-dimensional pixel stripes have a canonical dependence structure, the optimal connectivity of the NGM in tasks with complex or no spatial semantics might be less obvious. As an example, we consider the case of twodimensional image patches of size 10 ? 10, which we extract and corrupt following the same protocol and noise process as above. Instead of specifying the graph by hand, we use a mutual information criterion [41] to learn the optimal arborescence from the training labels. With estimation of all parameter groups turned on (+W + B), we obtain results that are fully in line with those above: the expected test log-likelihood of NSONs (+153.03) is again superior to a global mixture density (+76.34), which in turn outperforms the two parametric approaches (GGM: +18.60; Gaussian: 19.03). A full ablation study as well as a visualization of the inferred graph structure are shown in Appendix A.3. 3 Since plug-in estimators depend on the kernel locations, the gradient w.r.t. the kernel locations needs to take these dependencies into account by backpropagating through the estimator and computing the total derivative. 8 5.3 Inference Validation Section 4 motivated the use of a recurrent inference network (RIN) to infer rich statistics from structured, non-parametric densities. We compare this choice to the other two groups of approaches, i.e., variational and sampling-based inference (Section 1.1.2), in a marginal inference task. To this end, we pick one popular member from each group as baselines for our RIN architecture. Particle Belief Propagation (P-BP; [30]). Sum-product particle belief propagation approximates R (s) a BP-message (Eq. (16); op := ) with a set of particles {yj }Ss=1 per node Yj by computing (k) ? bi!j (yj ) = (s) (k) (s) S X (yi , yj ) ? ? b?!i (yi ) (s) S?(yi ) s=1 , (18) where the particles are sampled from a proposal distribution ? that approximates the true marginal by running MCMC on the beliefs ? b?!i (yi ) ? ? bi!j (yi ). Similar versions exist for other operators [42]. Black Box Variational Inference (BB-VI; [23]). Black box variational inference maximizes the ELBO LV I [q ] with respect to a variational distribution q by approximating its gradient through a set of samples {y(s) }Ss=1 ? q and performing stochastic gradient ascent, ? S X p? (y) p? (y(s) ) r LV I [q ] = r Eq (y) log ?S 1 r log q (y(s) ) log . (19) q (y) q (y(s) ) s=1 A statistic t (Eq. (5)) can then be estimated from the tractable variational distribution q (y) instead of the complex target distribution p? (y). We useQan isotropic Gaussian kernel ? = N (? | ~0, I) n together with the traditional factorization q (y) = i=1 q i (yi ), in which case variational sampling is straighforward and the (now unconditional) gradients are given directly by Section 3.3. 5.3.1 Setup and Results. We train our RIN architecture with a negative log-likelihood loss attached to each belief node, (i) LI = log p?i (yi ), and compare its performance to the results obtained from P-BP and BB-VI by calculating the sum of marginal log-likelihoods. For the baselines, we consider different numbers of particles, which affects both performance and speed. Additionally, for BB-VI we track the performance across 1024 optimization steps and report the best results. Table 2b summarizes our findings. Among the baselines, P-BP performs better than BB-VI once a required particle threshold is exceeded. We believe this is a manifestation of the special requirements associated with inference in non-parametric densities: while BB-VI needs to fit a high number of parameters, which poses the risk of getting trapped in local minima, P-BP relies solely on the evaluation of potentials. However, both methods are outperformed by a significant margin by our RIN, which we attribute to its end-to-end training in accordance with DNN+NGM and its ability to propagate and update full distributions instead of their mere value at a discrete set of points. In addition to pure performance, a key advantage of RIN inference over more traditional inference methods is its speed: our RIN approach is over 50? faster than P-BP with 100 particles and orders of magnitude faster than BB-VI. This is significant, even when taking dependencies on hardware and implementation into account, and allows the use of expressive non-parametric posteriors in time-critical applications. 6 Conclusion We proposed non-parametric structured output networks, a highly expressive framework consisting of a deep neural network predicting a non-parametric graphical model and a recurrent inference network computing statistics in this structured output space. We showed how all three components can be learned end-to-end by backpropagating non-parametric gradients through directed graphs and neural messages. Our experiments showed that non-parametric structured output networks are necessary for both effective learning of multimodal posteriors and efficient inference of complex statistics in them. 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Time Warping Invariant Neural Networks Guo-Zheng Sun, Hsing-Hen Chen and Yee-Chun Lee Institute for Advanced Computer Studies and Laboratory for Plasma Research, University of Maryland College Park, MD 20742 Abstract We proposed a model of Time Warping Invariant Neural Networks (TWINN) to handle the time warped continuous signals. Although TWINN is a simple modification of well known recurrent neural network, analysis has shown that TWINN completely removes time warping and is able to handle difficult classification problem. It is also shown that TWINN has certain advantages over the current available sequential processing schemes: Dynamic Programming(DP)[I], Hidden Markov Model(HMM)[2], Time Delayed Neural Networks(TDNN) [3] and Neural Network Finite Automata(NNFA)[4]. We also analyzed the time continuity employed in TWINN and pointed out that this kind of structure can memorize longer input history compared with Neural Network Finite Automata (NNFA). This may help to understand the well accepted fact that for learning grammatical reference with NNF A one had to start with very short strings in training set. The numerical example we used is a trajectory classification problem. This problem, making a feature of variable sampling rates, having internal states, continuous dynamics, heavily time-warped data and deformed phase space trajectories, is shown to be difficult to other schemes. With TWINN this problem has been learned in 100 iterations. For benchmark we also trained the exact same problem with TDNN and completely failed as expected. I. INTRODUCTION In dealing with the temporal pattern classification or recognition, time warping of input signals is one of the difficult problems we often encounter. Although there are a number of schemes available to handle time warping, e.g. Dynamic Programming (DP) and Hidden Markov Model(HMM), these schemes also have their own shortcomings in certain aspects. More depressing is that, as far as we know, there are no efficient neural network schemes to handle time warping. In this paper we proposed a model of Time Warping Invariant Neural Networks (TWINN) as a solution. Although TWINN is only a simple modification to the well known neural net structure, analysis shows that TWINN has the built-in ability to remove time warping completely. The basic idea ofTWINN is straightforward. If one plots the state trajectories of a continuous 180 Time Warping Invariant Neural Networks dynamical system in its phase space, these trajectory curves are independent of time warping because time warping can only change the time duration when traveling along these trajectories and does not affect their shapes and structures. Therefore, if we normalize the time dependence of the state variables with respect to any phase space variable, say the length of trajectory, the neural network dynamics becomes time warping invariant. To illustrate the power of the TWINN we tested it with a numerical example of trajectory classification. This problem, chosen as a typical problem that the TWINN could handle, has the following properties: (1). The input signals obey a continuous time dynamics and are sampled with various sampling rates. (2). The dynamics of the de-warped signals has internal states. (3). The temporal patterns consist of severely time warped signals. To our knowledge there have not been any neural network schemes which can deal with this case effectively. We tested it with TDNN and failed to learn. In the next section we will introduce the TWINN and prove its time warping invariance. In Section III we analyze its features and identify the advantages over other schemes. The numerical example of the trajectory classification with TWINN is presented in Section IV. II. TIME WARPING INVARIANT NEURAL NETWORKS (TWINN) To process temporal signals, we consider a fully recurrent network, which consists of two groups of neurons: the state neurons (or recurrent units) represented by vector S(t) and the input neurons that are clamped to the external input signals {I(t), t = 0, I, 2, ...... , T-l). The Time Warping Invariant Neural Networks (TWINN) is simply defined as: S(t+ 1) = S(t) +1(t)F(S(t), W,/(t? (1) where W is the weight matrix, [(t) is the distance between two consecutive input vectors defined by the norm l(t) = 11/(t+ 1) -/(t) II (2) and the mapping function F is a nonlinear function usually referred as neural activity function. For example of first order networks, it could take the form: (3) Fj(S(t), W,/(t? = Tanh(~Wij(S(t) EfH(t?) J where Tanh(x) is Hyperbolic Tangent function and symbol Ef> stands for the vector concatenation. For the purpose of classification (or recognition), we assign the target final state Sk> (k= 1,2,3, ... K), for each category of patterns. After we feed into the TWINN the whole sequence {J(O), 1(1), 1(2), ...... ,/(T-l)}, the state vector S(t) will reach the final state SeT). We then need to compare S(n with the target final state Sk for each category k, (k=I,2,3, ... K), and calculate the error: (4) The one with minimal error will be classified as such. The ideal error is zero. For the purpose of training, we are given a set of training examples for each category. We then minimize the error functions given by Eq. (4) using either back-propagation[7] or forward propagation algorithm[8]. The training process can be terminated when the total error reach its minimum. The formula of TWINN as shown in Eq. (1) does not look like new. The subtle difference from wildly used models is the introduction of normalization factor let) as in Eq. (1). The main advantage by doing this lies in its built-in time warping ability. This can be directly seen from its continuous version. As Eq. (1) is the discrete implementation of continuous dynamics, we can easily convert it into a continuous version by replacing "t +1" by "t+~t" and let ~t --? O. By doing so, we get 181 182 Sun, Chen, and Lee S(t+~t) . -Set) 11m - - - - - - - 61-.01l/(t+M) -/(t) II - dS - (5) dL where L is the input trajectory length, which can be expressed as an integral I L (t) III = ~~II dt (6) o or summation (as in discrete version) L(t) = I L II/(t+ 1) - ?/(t) II (7) 1:=0 For deterministic dynamics, the distance L(t) is a single-valued function. Therefore, we can make a unique mapping from t to L, TI: t --7 L, and any function of t can be transformed into a function of L in terms of this mapping. For instance, the input trajectory I(t) and the state trajectory Set) can be transformed into I(L) and S(L). By doing so, discrete dynamics of Eq. (1) becomes, in the continuous limit, ~~ = F (S (L), W, I (L) ) ( 8) It is obvious that there is no explicit time dependence in Eq. (8) and therefore the dynamics represented by Eq. (8) is time warping independent. To be more specific, if we draw the trajectory curves of l(t) and S(t) in their phase spaces respectively, these two curves would not be deformed if we only change the time duration when traveling along the curves. Therefore, if we generate several input sequences {J(t)} using different time warping functions and feed them into TWINN, represented by Eq. (8) or Eq. (1), the induced state dynamics of S(L) would be the same. Meanwhile, the final state is the solo criterion for classification. Therefore, any time warped signals would be classified by the TWINN as the same. This is the so called "time warping invariant". III. ANALYSIS OF TWINN VS. OTHER SCHEMES We emphasize two points in this section. First, we would analyze the advantages of the TWINN over the other neural network structures, like TDNN, and other mature and well known algorithms for time warping, such as HMM and Dynamics Programming. Second, we would analyze the memory capacity of input history for both the continuous dynamical networks as illustrated in Eq. (1) and its discrete companion, Neural Network Finite Automata used in grammatical inference by Liu [3], Sun [4] and Giles [5]. And, we will show by mathematical estimation that the continuity employed in TWINN increases the power of memorizing history compared with NNFA The Time Delayed Neural Networks (TDNN)[3] has been a useful neural network structure in processing temporal signals and achieves successes in several applications, e.g. speech recognition. The traditional neural network structures are either feedforward or recurrent. The TDNN is something in between. The power of TDNN is in its dynamic combination of the spatial processing (as in a feedforward net) and sequential processing (as in a recurrent net with short time memory). Therefore, the TDNN could detect the local features within each windowed frame and store their voting scores into the short time memory neurons, and then make a final decision at the end of input sequence. This technique is suitable for processing the temporal patterns where the classification is decided by the integration of local features. But, it could not handle the long time correlation across time frames like a state machine. It also does not tolerate time warping effectively. Each of time warped patterns will be treated as a new feature. ThereforG, TDNN would not be able to handle the numerical example given in this paper which has both the severe time warping and the internal states (long time correlation). The benchmark test has been performed and it proved our prediction. Actually, it can be seen later that in our exam- Time Warping Invariant Neural Networks pIes, no matter which category they belong to, all windowed frames would contain similar local features, the simple integration of local features do not contribute directly to the final classification, rather the whole sinal history will decide the classification. As for the Dynamic Programming, it is to date the most efficient way to cope with time warping problem. The most impressing feature of dynamic programming is that it accomplishes a global search among all NN possible paths using only -0(N2) operations, where N is the length of the input time series and, of course, one operation here represents all calculations involved in evaluating the 'score" of one path. But, on the other hand this is not ideal. If we can do the time warping using recurrent network, the number of uperations will be reduced to -O(N). This is a dramatic saving. Another undesirable feature of current dynamic warping scheme is that the recognition or classification result heavily depends on the pre-selected template and therefore one may need a large number of templates for a better classification rate. By adding one or two template we actually double or triple the number of operations. Therefore, search for a neural network time warping scheme is a pressing task. Another available technique for time warping is Hidden Markov Model (HMM), which has been successfully applied in speech recognition. The way for HMM to deal with time warping is in terms of statistical behavior of its hidden state transition. Starting from one state qj, HMM allows a certain probability ~j to forward to another state qj. Therefore, for any given HMM one could generate various state sequences, say, qlq2q2q3q4q4qS' QlQ2Q2Q2q3Q3q4q4qS' etc., each with a certain occurrence probability. But, these state sequences are "hidden", the observed part is a set of speech data or symbol represented by {Sk} for example. HMM also includes a set of observation probability B=={bjk }, so that when it is in a certain state, say Qj' HMM allows each symbol from the set {sk} to occur with the probability bjk . Therefore, for any state sequence one can generate various series of symbols. As an example, let us consider one simple way to generate symbols: in state Qj we generate symbol Sj (with probability bjj ). By doing so, the two state sequences mentioned above would correspond to two possible symbol sequences: sl s2s2s3s4s4sS and sl s2s2s2s3s3s4S4sS' Examining the two strings closely, we find that the second one may be considered as the time warped version of the first one, or vice versa. If we present these two strings to the HMM for testing, it will accept them with similar probabilities. This is the way that HMM tolerates time warping. And, these state transition probabilities of HMM are learned from the statistics of training set by using re-estimation formula. In this sense, HMM does not deal with time warping directly, instead, it learns statistical distribution of training set which contains time warped patterns. Consequently, if one presents a test pattern with time warped signals which is far away from the statistical distribution of training set, it is very unlikely for a HMM to recognize this pattern. On the contrary, the model of TWINN we proposed here has intrinsic built-in time warping nature. Although the TWINN itself has internal states, these internal states are not used for tolerating time warping. Instead, they are used to learn more complex behavior of the "de-warped" trajectories. In this sense, TWINN could be more powerful than HMM. Another feature ofTWINN needs be mention is its explicit expression of continuous mapping from S(t) to S(t+1) as shown in Eq. (1). In our early work of [4,5,6], to train a NNFA (Neural Network Finite Automaton), we used a discrete mapping S(t+ 1) = F(S(t), W,/(t? (9) where F is a nonlinear function, say Sigmoid function g(x) == 1 l(l+e' X). This model has been successfully applied into the grammatical inference. The reason we call Eq. (1) a continuous mapping but Eq. (9) a discrete one, even though both of them are implemented in discrete time steps, is because there is an explicit infinitesimal factor let) used in Eq. (1). Due to this factor the continuous state dynamics is guaranteed, by which we mean that the state variation S(t+ I) - S(t+1) approaches zero if the input variation 1(t+l) -1{t+I) does so. But, In general, the state 183 184 Sun, Chen, and Lee variation S(t+ 1) - S(t+ 1) generated by Eq. (9) is of order of one, regardless of what input variations are. If one starts from random initial weights, Eq. (9) provides a discrete jump between different, randomly distributed states, which is far away from any continuous dynamics. We did numerical test using NNFA of Eq. (9) to learn the classification problem of continuous trajectories as shown in Section V. For simplicity we did not include time warping, but the NNFA still failed to learn. The reason is that when we tried to train a NNF A to learning the continuous dynamics, we were actually forcing the weights to generate an almost identical mapping F from Set) to S(t+ 1). This is a very strong constrain on the weight parameters, such that it drives the diagonal terms to positive infinity and off-diagonal terms to negative infinity (Sigmoid function is used). When this happens, the learning is stuck due to the saturation effect. The failure of NNF A may also comes from the short history memory capacity compared to the continuous mapping ofEq. (1). It has been shown by many numerical experiments on grammatical inference [3,4,5] that to train an NNFA as in Eq. (9) effectively, one has to start with short training patterns (usually, the sentence length ~ 4). Otherwise, learning will fail or be very slow. This is exactly what happened to learning the trajectory classification using NNFA, where the lengths of our training patterns are in general considerably long (normally,- 60). But, TWINN learned it easily. To understand the NNFA's failure and TWINN's success, in the following, we will analyze how the history information enters the learning process. Consider the example of learning grammatical inference. Before training since we have no (I priori knowledge about the target values of weights, we normally start with random initial values. On the other hand, during training the credit assignment (or the weight correction ~ W) can only be done at the end of each input sequence. Consequently, each ~W should explicitly contain the information about all symbols contained in that string, otherwise the learning is meaningless. But, in numerical implementation, every variable, including both ~W and W, has a finite precision and any information beyond the precision range will be lost. Therefore, to compare which model has the longer history memory we need to examine how the history information relates to the finite precisions of ~ Wand W. Let us illustrate this point with a simple second-order connected fully recurrent network and write both Eq. (1) and Eq. (9) in a unified form S(t+l) =G,+l (10) such that Eq. (1) is represented by G' + I = S (1) + I (1) g (K (1) ) (11 ) and Eq. (9) is just G,+l = g(K(t)) where K(t) is the weighted sum of concatenation of vectors Set) and /(t) Kj(t) = LWjj(S(t) EfH(t?j (12) (13) j For a grammatical inference problem the error is calculated from the final state S(I) as E= (S(T)-Starget)2 (14) Learning is to minimize this error function. According to the standard error back-propagation scheme, the recurrent net can be viewed as a multi-layered net with identical weights between neurons at adjacent time step: w(t) = W, where w(t) is the "till layer" weights connecting input S(t-I) to output S(t). The total weight correction is the summation of all weight corrections at each layer. By using the gradient descent scheme one immediately has ~W= aE aE aG I LOW(t) =-llLaW(t) =-llL aS(t) ? aW(t) T T 1=1 1=1 T (15) 1=1 If we define new symbols: vector u(t), second-order tensor A(t) and third-order tensor B(t) as Time Warping Invariant Neural Networks aG~+ I A .. (t) IJ == a I S.(t) (16) J the weight correction can be simply written as T ~W=-1\~U(t).B(t) (17) and the "error rate" u(t) can be back-propagated using the Derivative Chain Rule u (t) = u (t + 1) . A (t) (18) t = 1, 2, ... , T - 1 ; so that it is easy to have u(t) = u(n ?A(T-I) ? A(T-2) ? ... ?A(t) ==u(n'tJ~t~(t) t = 1,2, ... ,T-I; (19) First, let us examine the model ofNNFA in Eq. (9). Using Eqs. (12), (13) and (16), Ai/t) and Bijk(t) can be written as A lj.. (t) = g' (K. (t) ) W .. I ~ B??k(f) IJ = aIJ.. (S(t-I) El)/(t-I?k (20) = where g'(x) == dg/dx gO-g) is the derivative of Sigmoid function and 8ij is Kronecker delta function. If we substitute Bijk(t) into Eq. (17), ~Wbecomes a weighted sum of all input symbols {/(O), 1(1), 1(2), ...... J(T-I)}, each with different weighting factor u(t). Therefore, to guarantee that ~ W contain the information of all input symbols {/(O), 1(1), 1(2), ...... J(T-I)}, the ratio of lu(t)lmaxllu(t)lmin should be within the range of precision of ~W. This is the main point. The exact mathematical analysis has not been done, but from a rough estimate we can gain some good understanding. From Eq. (9), u(t) is a matrices product of Aij(t), and u(1) the coefficient of 1(0) contains the highest order product of Ai/t). The key point is that the coefficient ratio between the adjacent symbols: lu(t)"lu(t+l) is of the order of lAi/t)I, which is a small value, therefore the earlier symbol information could be lost from ~ W due to its finite precision. It can be shown that xg'(x) =x g(x)( l-g(x)< 0.25 for any real value of x. Then, we roughly have lAij(t)1 =Ig' Wijl Ig(l-g)Wij 1< 0.25, if we assume the values of weights Wij to be order 1. Thus, the ratio R=lu(t)lmax"u(t)lmin is estimated as = 1 R- IU(1)l/lu(nl-"p_ 1 IA(f')1 <2- 2. (T-l) (21) From Eq. (21) we see that if the input pattern length is T= lOwe need at least 2(T-1) == 18 bits computer memory to store weight variables (including u, W and ~ W). If T= 60, as in the trajectory classification problem, it requires at least 128 bit weight variables. This is why the NNFA Eq. (9) could not work. Similarly, for the dynamics of Eq. 0), we use Eqs. (11), (13) and (16), and obtain Aij(l) = 1+1(t) (g'(Kj(t?W i} Bijk(l) = 1(1) (aij(S(t-l) GH(t-l?k) (22) From Eq. (22) we see that no matter how small the factor let) will be, lAi/!)1 remains a value of order of one, therefore the ratio R=lu(t)lmax"u(t)lmin which is estimated as a product of lAij(1)1 would be of order of one compared with result of discrete case as in Eq. (21).Therefore, the contributions from all {I(O), 1(1), 1(2), ...... J(T-I)} to the weight correction ~ Ware of the same order. This prevents the information loss during learning. IV NUMERICAL SIMULATION We demonstrate the power of TWINN with a trajectory classification problem. The three 2- 185 186 Sun, Chen, and Lee D trajectory equations are artificially given by (xCt) =sin(t+~)lsinCt)1 (xCt) =sin(O.5t+~)sin(1.5t) (xU) =sinU+~)sin(2t) \y (t) = cos (t+~) Isin (t) I \y (t) = cos (O.5t +~) sin (1.5t) ~ (1) = cos (t +~) sin (2t) (23) where ~ is a unifonnly distributed random parameter. When ~ is changed, these trajectories are distorted accordingly. Some examples (three for each class) are shown in Fig. I. Class 1 Class 2 - -G .7 !. -0.5 Class 3 - Fig.l PHASE SPACE TRAJECTORIES Three different shapes of 2-D trajecwry, each is shown in one column with three examples. Recurrent neural networks are trained to recognize the different shapes of trajectory. The trajectory data are the time series of two dimensional coordinate pairs {x(t), y(t)} sampled along three different types of curves in the phase space. The neural net dynamics of TWINN is (24) where we used 6 input neurons I ={I, x(t), y(t), .?(t), y2(t), x(t)y(t)} (normalized to norm = 1.0) and 4 (N=4) state neurons S ={ Sl' S2' S3' S4}. The neural network structure is shown in Fig. 2. Fig.2 Time Warping Invariant Neural Network for Trajectory Classification Fig.3 Time Delayed Neural Network for Trajectory ClassificatiolJ For training, we assign the desired final output for the three trajectory classes to be (1,0,0), Time Warping Invariant Neural Networks (0,1,0) and (0,0,1) respectively. For recognition, each trajectory data sequence needs to be fed to the input neurons and the state neurons evolve according to the dynamics in Eq. (24). At the end of input series we check the last three state neurons and classify the input trajectory according to the "winner-take-all" rule. In each iteration of training we randomly picked up 150 deformed trajectories, 50 for each of the three categories, by choosing different values of ~ within O$~ $27t. To simulate time warping we randomly sampled the data by choosing the random time step ~t = 27trff along each trajectory, where r is a random number between 0 and 2 and the sampling rate T=60 for training patterns, and T=20 to 200 for testing patterns. Therefore, each training pattern is a time warped trajectory data with averaged length = 60. Using RTRL algorithm[8] to minimize the error function, after 100 iterations of training it converged to Mean Square Error of == 0.03. We tested the trained network with hundreds of randomly picked input sequences with different sampling rate (from 20/27t to 200127t) and different wrapping functions (non-uniform step length). All input trajectories are classified correctly. If the sampling rates are too large (>200) or too small( <20), some classification errors will occur. We test the same example with TDNN. See Fig.3 for its parameters. The top layer contains three output neurons for the three classes of trajectories. The classification rules, error function and training patterns are the same as those ofTWINN. After three days of training with DEC3100 Workstation the training error (MSE) approaches 0.5 and in testing the error rate is 70%. V. CONCLUSION We have proposed a model of Time Warping Invariant Neural Network to handle temporal pattern classification where the severely time warped and deformed data may occur. This model is shown to have built-in time warping ability. We have analyzed the properties ofTWINN and shown that for trajectory classification it has several advantages over other schemes: HMM, DP, TDNN and NNFA. We also numerically implemented the TWINN and trained a trajectory classification easily. This problem is shown by analysis to be difficult to other schemes. It has been trained with TDNN but failed. References [1] H.Sakoe and S. Chiba, "Dynamic Programming Algorithm Optimization for Spoken Word Recognition", IEEE Transactions on Acoustics Speech and Signal Processing, Vol. ASSP-26, pp.43-49, Feb. 1978. [2] L.R.Rabiner and B.H.Juang, "An Introduction to Hidden Markov Models", IEEE, ASSP Mag., Vol.3, No.1, pp. 4-16, 1986. [3]A. Weibel, T. Hanazawa, G. Hinton, K.shikano and K. Lang, "Phoneme Recognition Using Time-Delay Neural Networks", IEEE Transactions on Acoustics Speech and Signal Processing, March, 1989. [4]. Y.D. Liu, G.Z. Sun, H.H. Chen, c.L. Giles and Y.c. Lee, "Grammatic Inference and Neural Network State Machine", Proceedings of the International Joint Conference on Neural Networks, pp. 1-285, Washington D.C. (1990). [5]. G.Z. Sun, H.H. Chen, c.L. Giles, Y.c. Lee and D. Chen, "Connectionist Pushdown Automata that Learn Context-Free Grammars", Proceedings of the International Joint Conference on Neural networks, pp. 1-577, Washington D.C. (1990). [6]Giles, C.L., Sun, G.Z., Chen, H.H., Lee,Y.C., and Chen, D. (1990). "Higher Order Recurrent Networks & Grammatical Inference". Advances in Neurallnformation Processing Systems 2, D.S. Touretzky (editor), 380-386, Morgan Kaufmann, San Mateo, c.A. (7) [7] D.Rumelhart, G. Hinton, and R. Williams. "Learning internal representations by error propagation", In PDP: VoU MIT press 1986. P. Werbos, "Beyond Regression: New tools for prediction and analysis in the behavior sciences", Ph.D. thesis, Harvard university, 1974. [8] R. Williams and D. Zipser, "A learning algorithm for continually running fully recurrent neural networks", Neural Computation 1(1989), pp.270-280. 187
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Learning Active Learning from Data Ksenia Konyushkova? CVLab, EPFL Lausanne, Switzerland [email protected] Sznitman Raphael ARTORG Center, University of Bern Bern, Switzerland [email protected] Pascal Fua CVLab, EPFL Lausanne, Switzerland [email protected] Abstract In this paper, we suggest a novel data-driven approach to active learning (AL). The key idea is to train a regressor that predicts the expected error reduction for a candidate sample in a particular learning state. By formulating the query selection procedure as a regression problem we are not restricted to working with existing AL heuristics; instead, we learn strategies based on experience from previous AL outcomes. We show that a strategy can be learnt either from simple synthetic 2D datasets or from a subset of domain-specific data. Our method yields strategies that work well on real data from a wide range of domains. 1 Introduction Many modern machine learning techniques require large amounts of training data to reach their full potential. However, annotated data is hard and expensive to obtain, notably in specialized domains where only experts whose time is scarce and precious can provide reliable labels. Active learning (AL) aims to ease the data collection process by automatically deciding which instances an annotator should label to train an algorithm as quickly and effectively as possible. Over the years many AL strategies have been developed for various classification tasks, without any one of them clearly outperforming others in all cases. Consequently, a number of meta-AL approaches have been proposed to automatically select the best strategy. Recent examples include bandit algorithms [2, 11, 3] and reinforcement learning approaches [5]. A common limitation of these methods is that they cannot go beyond combining pre-existing hand-designed heuristics. Besides, they require reliable assessment of the classification performance which is problematic because the annotated data is scarce. In this paper, we overcome these limitations thanks to two features of our approach. First, we look at a whole continuum of AL strategies instead of combinations of pre-specified heuristics. Second, we bypass the need to evaluate the classification quality from application-specific data because we rely on experience from previous tasks and can seamlessly transfer strategies to new domains. More specifically, we formulate Learning Active Learning (LAL) as a regression problem. Given a trained classifier and its output for a specific sample without a label, we predict the reduction in generalization error that can be expected by adding the label to that datapoint. In practice, we show that we can train this regression function on synthetic data by using simple features, such as the variance of the classifier output or the predicted probability distribution over possible labels for a ? http://ksenia.konyushkova.com 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. specific datapoint. The features for the regression are not domain-specific and this enables to apply the regressor trained on synthetic data directly to other classification problems. Furthermore, if a sufficiently large annotated set can be provided initially, the regressor can be trained on it instead of on synthetic data. The resulting AL strategy is then tailored to the particular problem at hand. We show that LAL works well on real data from several different domains such as biomedical imaging, economics, molecular biology and high energy physics. This query selection strategy outperforms competing methods without requiring hand-crafted heuristics and at a comparatively low computational cost. 2 Related work The extensive development of AL in the last decade has resulted in various strategies. They include uncertainty sampling [32, 15, 27, 34], query-by-committee [7, 13], expected model change [27, 30, 33], expected error or variance minimization [14, 9] and information gain [10]. Among these, uncertainty sampling is both simple and computationally efficient. This makes it one of the most popular strategies in real applications. In short, it suggests labeling samples that are the most uncertain, i.e., closest to the classifier?s decision boundary. The above methods work very well in cases such as the ones depicted in the top row of Fig. 2, but often fail in the more difficult ones depicted in the bottom row [2]. Among AL methods, some cater to specific classifiers, such as those relying on Gaussian processes [16], or to specific applications, such as natural language processing [32, 25], sequence labeling tasks [28], visual recognition [21, 18], semantic segmentation [33], foreground-background segmentation [17], and preference learning [29, 22]. Moreover, various query strategies aim to maximize different performance metrics, as evidenced in the case of multi-class classification [27]. However, there is no one algorithm that consistently outperforms all others in all applications [28]. Meta-learning algorithms have been gaining in popularity in recent years [31, 26], but few of them tackle the problem of learning AL strategies. Baram et al. [2] combine several known heuristics with the help of a bandit algorithm. This is made possible by the maximum entropy criterion, which estimates the classification performance without labels. Hsu et al. [11] improve it by moving the focus from datasamples as arms to heuristics as arms in the bandit and use a new unbiased estimator of the test error. Chu and Lin [3] go further and transfer the bandit-learnt combination of AL heuristics between different tasks. Another approach is introduced by Ebert et al. [5]. It involves balancing exploration and exploitation in the choice of samples with a Markov decision process. The two main limitations of these approaches are as follows. First, they are restricted to combining already existing techniques and second, their success depends on the ability to estimate the classification performance from scarce annotated data. The data-driven nature of LAL helps to overcome these limitations. Sec. 5 shows that it outperforms several baselines including those of Hsu et al. [11] and Kapoor et al. [16]. 3 Towards data-driven active learning In this section we briefly introduce the active leaning framework along with uncertainty sampling (US), the most frequently-used AL heuristic. Then, we motivate why a data-driven approach can improve AL strategies and how it can deal with the situations where US fails. We select US as a representative method because it is popular and widely applicable, however the behavior that we describe is typical for a wide range of AL strategies. 3.1 Active learning (AL) Given a machine learning model and a pool of unlabeled data, the goal of AL is to select which data should be annotated in order to learn the model as quickly as possible. In practice, this means that instead of asking experts to annotate all the data, we select iteratively and adaptively which datapoints should be annotated next. In this paper we are interested in classifying datapoints from a target dataset Z = {(x1 , y1 ), . . . , (xN , yN )}, where xi is a D-dimensional feature vector and yi 2 {0, 1} is its binary label. We choose a probabilistic classifier f that can be trained on some Lt ? Z to map 2 features to labels, ft (xi ) = y?i , through the predicted probability pt (yi = y | xi ). The standard AL procedure unfolds as follows. 1. The algorithm starts with a small labeled training dataset Lt ? Z and large pool of unannotated data Ut = Z \ Lt with t = 0. 2. A classifier ft is trained using Lt . 3. A query selection procedure picks an instance x? 2 Ut to be annotated at the next iteration. 4. x? is given a label y ? by an oracle. The labeled and unlabeled sets are updated. 5. t is incremented, and steps 2?5 iterate until the desired accuracy is achieved or the number of iterations has reached a predefined limit. Uncertainty sampling (US) US has been reported to be successful in numerous scenarios and settings and despite its simplicity, it often works remarkably well [32, 15, 27, 34, 17, 24]. It focuses its selection on samples which the current classifier is the least certain about. There are several definitions of maximum uncertainty but one of the most widely used ones is to select a sample x? that maximizes the entropy H over the probability of predicted classes: x? = arg max H[pt (yi = y | xi )] . (1) xi 2Ut 3.2 Success, failure, and motivation We now motivate the need for LAL by presenting two toy examples. In the first one, US is empirically observed to be the best greedy approach, but in the second it makes suboptimal decisions. Let us consider simple two-dimensional datasets Z and Z 0 drawn from the same distribution with an equal number of points in each class (Fig. 1, left). The data in each class comes from a Gaussian distribution with a different mean and the same isotropic covariance. We can initialize the AL procedure of Sec. 3.1 with one sample from each class and its respective label: L0 = {(x1 , 0), (x2 , 1)} ? Z and U0 = Z \ L0 . Here we train a simple logistic regression classifier f on L0 and then test it on Z 0 . 0 If |Z P | is large, the test error can be considered as a good approximation of the generalization error: `0 = (x0 ,y0 )2Z 0 `(? y , y 0 ), where y? = f0 (x0 ). Let us try to label every point x from U0 one by one, form a new labeled P set Lx = L00 [ (x, y) and check what error a new classifier fx yields on Z 0 , that is, `x = (x0 ,y0 )2Z 0 `(? y , y ), where 0 y? = fx (x ). The difference between errors obtained with classifiers constructed on L0 and Lx indicates how much the addition of a new datapoint x reduces the generalization error: x = `0 `x . We plot x for the 0/1 loss function, averaged over 10 000 experiments as a function of the predicted probability p0 (Fig. 1, left). By design, US would select a datapoint with probability of class 0 close to 0.5. We observe that in this experiment, the datasample with p0 closest to 0.5 is indeed the one that yields the greatest error reduction. 0.02 0.03 0.01 0.00 0.00 ?0.03 0 0 1 1 Figure 1: Balanced vs unbalanced. Left: two Gaussian clouds of the same size. Right: two Gaussian clouds with the class 0 twice bigger than class 1. The test error reduction as a function of predicted probability of class 0 in the respective datasets. In the next experiment, the class 0 contains twice as many datapoints as the other class, see Fig. 1 (right). As before, we plot the average error reduction as a function of p0 . We observe this time that the value of p0 that corresponds to the largest expected error reduction is different from 0.5 and thus the choice of US becomes suboptimal. Also, the reduction in error is no longer symmetric for the two classes. The more imbalanced the two classes are, the further from the optimum the choice made by 3 US is. In a complex realistic scenario, there are many other factors such as label noise, outliers and shape of distribution that further compound the problem. Although query selection procedures can take into account statistical properties of the datasets and classifier, there is no simple way to foresee the influence of all possible factors. Thus, in this paper, we suggest Learning Active Learning (LAL). It uses properties of classifiers and data to predict the potential error reduction. We tackle the query selection problem by using a regression model; this perspective enables us to construct new AL strategies in a flexible way. For instance, in the example of Fig. 1 (right) we expect LAL to learn a model that automatically adapts its selection to the relative prevalence of the two classes without having to explicitly state such a rule. Moreover, having learnt the error reduction prediction function, we can seamlessly transfer LAL strategy to other domains with very little annotated data. 4 Monte-Carlo LAL Our approach to AL is data-driven and can be formulated as a regression problem. Given a representative dataset with ground truth, we simulate an online learning procedure using a Monte-Carlo technique. We propose two versions of AL strategies that differ in the way how datasets for learning a regressor are constructed. When building the first one, LAL INDEPENDENT, we incorporate unused labels individually and at random to retrain the classifier. Our goal is to correlate the change in test performance with the properties of the classifier and of newly added datapoint. To build the LAL ITERATIVE strategy, we further extend our method by a sequential procedure to account for selection bias caused by AL. We formalize our LAL procedures in the remainder of the section. 4.1 Independent LAL Let the representative dataset2 consist of a training set D and a testing set D0 . Let f be a classifier with a given training procedure. We start collecting data for the regressor by splitting D into a labeled set L? of size ? and an unlabeled set U? containing the remaining points (Alg. 1 DATA M ONTE C ARLO). We then train a classifier f on L? , resulting in a function f? that we use to predict class labels for elements x0 from the test set D0 and estimate the test classification loss `? . We characterize the classifier state by K parameters ? = { 1? , . . . , K ? }, which are specific to the particular classifier type and are sensitive to the change in the training set while being relatively invariant to the stochasticity of the optimization procedure. For example, they can be the parameters of the kernel function if f is kernel-based, the average depths of the trees if f is a tree-based method, or prediction variability if f is an ensemble classifier. The above steps are summarized in lines 3?5 of Alg. 1. Algorithm 1 DATA M ONTE C ARLO 1: Input: training set D and test set D 0 , classification procedure f , partitioning function S PLIT, 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: size ? Initialize: L? , U? S PLIT(D, ? ) train a classifier f? estimate the test set loss `? compute the classification state parameters { 1? , . . . , K ? } for m = 1 to M do select x 2 U? at random form a new labeled dataset Lx L? [ {x} compute the datapoint parameters { x1 , . . . , xR } train a classifier fx estimate the new test loss `x compute loss reduction x `? ?`x ? the 1 K 1 R ?m ??? ??? x ? ? x x , m ? {?m } , { m} : 1 ? m ? M Return: matrix of learning states ? 2 RM ?(K+R) , vector of reductions in error 2 RM 2 The representative dataset is an annotated dataset that does not need to come from the domain of interest. In Sec. 5 we show that a simple synthetic dataset is sufficient for learning strategies that can be applied to various real tasks across various domains. 4 Algorithm 2 BUILD LAL INDEPENDENT Algorithm 3 BUILD LAL ITERATIVE 1: Input: iteration range {?min , . . . , ?max }, 1: Input: iteration range {?min , . . . , ?max }, 2: 3: 2: S PLIT random partitioning function 3: Initialize: generate train set D and test 4: 5: 6: 7: 8: 9: 10: classification procedure f S PLIT random partitioning function Initialize: generate train set D and test dataset D0 for ? in {?min , . . . , ?max } do for q = 1 to Q do ?? q , ? q DATA M ONTE C ARLO (D, D0 , f, S PLIT, ? ) ?, {?? q }, { ? q } train a regressor g : ? 7! on data ?, construct LAL INDEPENDENT A(g): x? = arg maxx2Ut g[?t,x )] Return: LAL INDEPENDENT classification procedure f 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: dataset D0 for ? in {?min , . . . , ?max } do for q = 1 to Q do ?? q , ? q DATA M ONTE C ARLO (D, D0 , f, S PLIT, ? ) ?? , ? {?? q , ? q } train regressor g? : ? 7! on ?? , ? S PLIT A(g? ) ?, {?? , ? } train a regressor g : ? 7! on ?, construct LAL ITERATIVE A(g) Return: LAL ITERATIVE Next, we randomly select a new datapoint x from U? which is characterized by R parameters 1 R x = { x , . . . , x }. For example, they can include the predicted probability to belong to class y, the distance to the closest point in the dataset or the distance to the closest labeled point, but they do not include the features of x. We form a new labeled set Lx = L? [ {x} and retrain f (lines 7?13 of Alg. 1). The new classifier fx results in the test-set loss `x . Finally, we record the difference between previous and new loss x = `? `x which is associated ?to the learning state in which it was ? received. K 1 R The learning state is characterized by a vector ??x = 1? ? ? ? 2 RK+R , ??? ? x x whose elements depend both on the state of the current classifier f? and on the datapoint x. To build an AL strategy LAL INDEPENDENT we repeat the DATA M ONTE C ARLO procedure for Q different initializations L1? , L2? , . . . , LQ ? and T various labeled subset sizes ? = 2, . . . , T + 1 (Alg. 2 lines 4 and 5). For each initialization q and iteration ? , we sample M different datapoints x each of which yields classifier/datapoint state pairs with an associated reduction in error (Alg. 1, line 13). This results in a matrix ? 2 R(QM T )?(K+R) of observations ? and a vector 2 RQM T of labels (Alg. 2, line 9). Our insight is that observations ? should lie on a smooth manifold and that similar states of the classifier result in similar behaviors when annotating similar samples. From this, a regression function can predict the potential error reduction of annotating a specific sample in a given classifier state. Line 10 of the BUILD LAL INDEPENDENT algorithm looks for a mapping g : ? 7! . This mapping is not specific to the dataset D, and thus can be used to detect samples that promise the greatest increase in classifier performance in other target domains Z. The resulting LAL INDEPENDENT strategy greedily selects a datapoint with the highest potential error reduction at iteration t by taking the maximum of the value predicted by the regressor g: x? = arg max g( t , x2Ut 4.2 x ). (2) Iterative LAL For any AL strategy at iteration t > 0, the labeled set Lt consists of samples selected at previous iterations, which is clearly not random. However, in Sec. 4.1 the dataset D is split into L? and U? randomly no matter how many labeled samples ? are available. To account for this, we modify the approach of Section 4.1 in Alg. 3 BUILD LAL ITERATIVE. Instead of partitioning the dataset D into L? and U? randomly, we suggest simulating the AL procedure which selects datapoints according to the strategy learnt on the previously collected data (Alg. 3, line 10). It first learns a strategy A(g2 ) based on a regression function g2 which selects the most promising 3rd datapoint when 2 random points are available. In the next iteration, it learns a strategy A(g3 ) that selects 4th datapoint given 2 random points and 1 selected by A(g2 ) etc. In this way, 5 samples at each iteration depend on the samples at the previous iteration and the sampling bias of AL is represented in the data ?, from which the final strategy LAL ITERATIVE is learnt. The resulting strategies LAL INDEPENDENT and LAL ITERATIVE are both reasonably fast during the online steps of AL: they just require evaluating the RF regressor. The offline part, generating a datasets to learn a regression function, can induce a significant computational cost depending on the parameters of the algorithm. For this reason, LAL INDEPENDENT is preferred to LAL ITERATIVE when an application-specific strategy is needed. 5 Experiments Implementation details We test AL strategies in two possible settings: a) cold start, where we start with one sample from each of two classes and b) warm start, where a larger dataset of size N0 ? N is available to train the initial classifier. In cold start we take the representative dataset to be a 2D synthetic dataset where class-conditional data distributions are Gaussian and we use the same LAL regressor in all 7 classification tasks. While we mostly concentrate on cold start scenario, we look at a few examples of warm start because we believe that it is largely overloooked in the litterature, but it has a significant practical interest. Learning a classifier for a real-life application with AL rarely starts from scratch, but a small initial annotated set is provided to understand if a learning-based approach is applicable at all. While a small set is good to provide an initial insight, a real working prototype still requires much more training data. In this situation, we can benefit from the available training data to learn a specialized AL strategy for an application. In most of the experiments, we use Random Forest (RF) classifiers for f and a RF regressor for g. The state of the learning process ?t at time t consists of the following features: a) predicted probability p(y = 0|Lt , x); b) proportion of class 0 in Lt ; c) out-of-bag cross-validated accuracy of ft ; d) variance of feature importances of ft ; e) forest variance computed as variance of trees? predictions on Ut ; f) average tree depth of the forest; g) size of Lt . For additional implementational details, including examples of the synthetic datasets, parameters of the data generation algorithm and features in the case of GP classification, we refer the reader to the supplementary material. The code is made available at https://github.com/ksenia-konyushkova/LAL. Baselines and protocol We consider the three versions of our approach: a) LAL-independent-2D, LAL INDEPENDENT strategy trained on a synthetic dataset of cold start; b) LAL-iterative-2D, LAL ITERATIVE strategy trained on a synthetic dataset of cold start; c) LAL-independent-WS, LAL INDEPENDENT strategy trained on warm start representative data. We compare them against the following 4 baselines: a) Rs, random sampling; b) Us, uncertainty sampling; c) Kapoor [16], an algorithm that balances exploration and exploitation by incorporating mean and variance estimation of the GP classifier; d) ALBE [11], a recent example of meta-AL that adaptively uses a combination of strategies, including Us, Rs and that of Huang et al. [12] (a strategy that uses the topology of the feature space in the query selection). The method of Hsu et al. [11] is chosen as a our main baseline because it is a recent example of meta AL and is known to outperform several benchmarks. In all AL experiments we select samples from a training set and report the classification performance on an independent test set. We repeat each experiment 50?100 times with random permutations of training and testing splits and different initializations. Then we report the average test performance as a function of the number of labeled samples. The performance metrics are task-specific and include classification accuracy, IOU [6], dice score [8], AMS score [1], as well as area under the ROC curve (AUC). 5.1 Synthetic data Two-Gaussian-clouds experiments In this dataset we test our approach with two classifiers: RF and Gaussian Process classifier (GPC). Due to the the computational cost of GPC, it is only tested in this experiment. We generate 100 new unseen synthetic datasets of the form as shown in the top row of Fig. 2 and use them for testing AL strategies. In both cases the proposed LAL strategies select datapoints that help to construct better classifiers faster than Rs, Us, Kapoor and ALBE. XOR-like experiments XOR-like datasets are known to be challenging for many machine learning methods and AL is no exception. It was reported in Baram et al. [2] that various AL algorithms 6 accuracy Gaussian clouds, RF Gaussian clouds, GP 0.9 0.9 0.8 0.8 0.7 0.7 0.6 Rs Us ALBE Kapoor LAL-independent-2D LAL-iterative-2D 0.6 0 50 100 0 100 Checkerboard 4x4 Checkerboard 2x2 1.0 accuracy 50 0.8 0.6 Rotated checkerboard 2x2 0.85 1.0 0.75 0.9 0.65 0.8 0.55 0.7 0.6 0.45 0 100 # labelled points 200 0 100 # labelled points 200 0 100 # labelled points 200 Figure 2: Experiments on the synthetic data. Top row: RF and GP on 2 Gaussian clouds. Bottom row from left to right: experiments on Checkerboard 2 ? 2, Checkerboard 4 ? 4, and Rotated Checkerboard 2 ? 2 datasets. struggle with tasks such as those depicted in the bottom row of Fig. 2, namely Checkerboard 2 ? 2 and Checkerboard 4 ? 4. Additionally, we consider Rotated Checkerboard 2 ? 2 dataset (Fig. 2, bottom row, right). The task for RF becomes more difficult in this case because the discriminating features are no longer aligned to the axis. As previously observed [2], Us loses to Rs in these cases. ALBE does not suffer from such adversarial conditions as much as Us, but LAL-iterative-2D outperforms it on all XOR-like datasets. 5.2 Real data We now turn to real data from domains where annotating is hard because it requires special training to do it correctly: Striatum, 3D Electron Microscopy stack of rat neural tissue, the task is to detect and segment mitochondria [20, 17]; MRI, brain scans obtained from the BRATS competition [23], the task is to segment brain tumor in T1, T2, FLAIR, and post-Gadolinium T1 MR images; Credit card [4], a dataset of credit card transactions made in 2013 by European cardholders, the task is to detect fraudulent transactions; Splice, a molecular biology dataset with the task of detecting splice junctions in DNA sequences [19]; Higgs, a high energy physics dataset that contains measurements simulating the ATLAS experiment [1], the task is to detect the Higgs boson in the noise signal. Additional details about the above datasets including sizes, dimensionalities and preprocessing techniques can be found in the supplementary materials. Cold Start AL Top row of Fig. 3 depicts the results of applying Rs, Us, LAL-independent2D, and LAL-iterative-2D on the Striatum, MRI, and Credit card datasets. Both LAL strategies outperform Us, with LAL-iterative-2D being the best of the two. The best score of Us in these complex real-life tasks is reached 2.2?5 times faster by the LAL-iterative-2D. Considering that the LAL regressor was learned using a simple synthetic 2D dataset, it is remarkable that it works effectively on such complex and high-dimensional tasks. Due to the high computational cost of ALBE, we downsample Striatum and MRI datasets to 2000 datapoints (referred to as Striatum mini and MRI mini). Downsampling was not possible for the Credit card dataset due to the sparsity of positive labels (0.17%). We see in the bottom row of Fig. 3 that ALBE performs worse than 7 0.50 0.6 0.93 0.35 AUC 0.96 0.4 0.2 0.20 0 250 0.8 0.55 0.6 dice 0.70 0.40 0.84 0 500 100 200 0.10 0 150 300 Rs Us ALBE LAL-independent-2D LAL-iterative-2D 0.4 0.2 0.25 0.90 0.87 0.0 0.05 IOU Credit card MRI 0.8 dice IOU Striatum 0.65 0.0 0 100 # labelled points 200 0 100 200 # labelled points Figure 3: Experiments on real data. Top row: IOU for Striatum, dice score for MRI and AUC for Credit card as a function of a number of labeled points. Bottom row: Comparison with ALBE on the Striatum mini and MRI mini datasets. Us but better than Rs. We ascribe this to the lack of labeled data, which ALBE needs to estimate classification accuracy (see Sec. 2). Warm Start AL In Fig. 4 we compare LAL-independent-WS on the Splice and Higgs datasets by initializing BUILD LAL INDEPENDENT with 100 and 200 datapoints from the corresponding tasks. Notice that this is the only experiment where a significant amount of labelled data in the domain of interest is available prior to AL. We tested ALBE on the Splice dataset, however in the Higgs dataset the number of iterations in the experiment is too big. LAL-independent-WS outperforms other methods with ALBE delivering competitive performance?yet, at a high computational cost?only after many AL iterations. Splice Higgs 300 AMS accuracy 0.95 0.92 0.89 Rs Us ALBE LAL-independent-WS 270 240 100 200 labelled points 300 210 1000 2000 labelled points Figure 4: Experiments on the real datasets in warm start scenario. Accuracy for Splice is on the left, AMS score for Higgs is on the right. 5.3 Analysis of LAL strategies and time comparison To better understand LAL strategies, we show in Fig. 5 (left) the relative importance of the features of the regressor g for LAL ITERATIVE. We observe that both classifier state parameters and datapoint parameters influence the AL selection giving evidence that both of them are important for selecting a point to label. In order to understand what kind of selection LAL INDEPENDENT and LAL ITERATIVE do, we record the predicted probability of the chosen datapoint p(y ? = 0|Dt , x? ) in 10 cold start experiments with the same initialization on the MRI dataset. Fig. 5 (right) shows the histograms of these probabilities for Us, LAL-independent-2D and LAL-iterative-2D. LAL strategies have 8 1000 Us LAL-independent-2D LAL-iterative-2D forest variance feature importance tree depth probability out-of-bag proportion size 500 0 0.0 0.1 0.2 0.3 0.0 0.4 0.2 0.4 0.6 0.8 1.0 ? Relative Importance probability p Figure 5: Left: feature importances of the RF regressor representing LAL ITERATIVE strategy. Right: histograms of the selected probability for different AL strategies in experiments with MRI dataset. high variance and modes different from 0.5. Not only does the selection by LAL strategies differ significantly from standard Us, but also the independent and iterative approaches differ from each other. Computational costs While collecting synthetic data can be slow, it must only be done once, offline, for all applications. Besides, Alg. 1, 2 and 3 can be trivially parallelised thanks to a number of independent loops. Collecting data offline for warm start, that is application specific, took us approximately 2.7h and 1.9h for Higgs and Splice datasets respectively. By contrast, the online user-interaction part is fast: it simply consists of learning ft , extracting learning state parameters and evaluating the regressor g. The LAL run time depends on the parameters of the random forest regressor which are estimated via cross-validation (discussed in the supplementary materials). Run times of a Python-based implementation running on 1 core are given in Tab. 1 for a typical parameter set (? 20% depending on exact parameter values). Real-time performance can be attained by parallelising and optimising the code, even in applications with large amounts of high-dimensional data. Table 1: Time in seconds for one iteration of AL for various strategies and tasks. Dataset Dimensions # samples Us ALBE LAL Checkerboard MRI mini MRI Striatum mini Striatum Credit 6 2 188 188 272 272 30 1000 2000 22 934 2000 276 130 142 404 0.11 0.11 0.12 0.11 2.05 0.43 13.12 64.52 ? 75.64 ? ? 0.54 0.55 0.88 0.59 19.50 4.73 Conclusion In this paper we introduced a new approach to AL that is driven by data: Learning Active Learning. We found out that Learning Active Learning from simple 2D data generalizes remarkably well to challenging new domains. Learning from a subset of application-specific data further extends the applicability of our approach. Finally, LAL demonstrated robustness to the choice of type of classifier and features. In future work we would like to address issues of multi-class classification and batch-mode AL. Also, we would like to experiment with training the LAL regressor to predict the change in various performance metrics and with different families of classifiers. Another interesting direction is to transfer a LAL strategy between different real datasets, for example, by training a regressor on multiple real datasets and evaluating its performance on unseen datasets. Finally, we would like to go beyond constructing greedy strategies by using reinforcement learning. 9 Acknowledgements This project has received funding from the European Union?s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 720270 (HBP SGA1). We would like to thank Carlos Becker and Helge Rhodin for their comments on the text, and Lucas Maystre for his discussions and attention to details. References [1] C. Adam-Bourdarios, G. Cowan, C. Germain, I. Guyon, B. K?gl, and D. Rousseau. The higgs boson machine learning challenge. In NIPS 2014 Workshop on High-energy Physics and Machine Learning, 2015. [2] Y. Baram, R. El-Yaniv, and K. Luz. Online choice of active learning algorithms. Journal of Machine Learning Research, 2004. [3] H.-M. Chu and H.-T. Lin. Can active learning experience be transferred? arXiv preprint arXiv:1608.00667, 2016. [4] A. Dal Pozzolo, O. Caelen, R. A. Johnson, and G. Bontempi. Calibrating probability with undersampling for unbalanced classification. In IEEE Symposium Series on Computational Intelligence, 2015. [5] S. Ebert, M. Fritz, and B. Schiele. RALF: A reinforced active learning formulation for object class recognition. In Conference on Computer Vision and Pattern Recognition, 2012. [6] M. Everingham, L. Van Gool, C. Williams, J. Winn, and A. Zisserman. The pascal visual object classes (voc) challenge. International journal of computer vision, 2010. [7] R. 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Combining generative and discriminative models for semantic segmentation. In Information Processing in Medical Imaging, 2011. [14] A. J. Joshi, F. Porikli, and N. P. Papanikolopoulos. Scalable active learning for multiclass image classification. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2012. [15] A.J. Joshi, F. Porikli, and N. Papanikolopoulos. Multi-class active learning for image classification. In Conference on Computer Vision and Pattern Recognition, 2009. [16] A. Kapoor, K. Grauman, R. Urtasun, and T. Darrell. Active learning with Gaussian Processes for object categorization. In International Conference on Computer Vision, 2007. [17] K. Konyushkova, R. Sznitman, and P. Fua. Introducing geometry into active learning for image segmentation. In International Conference on Computer Vision, 2015. 10 [18] J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In Conference on Computer Vision and Pattern Recognition, 2015. [19] A. C. Lorena, G. E. A. P. A. Batista, A. C. P. L. F. de Carvalho, and M. C. Monard. Splice junction recognition using machine learning techniques. In Brazilian Workshop on Bioinformatics, 2002. [20] A. Lucchi, Y. Li, K. Smith, and P. Fua. Structured image segmentation using kernelized features. In European Conference on Computer Vision, 2012. [21] T. Luo, K. Kramer, S. Samson, A. Remsen, D. B. Goldgof, L. O. Hall, and T. Hopkins. Active learning to recognize multiple types of plankton. In International Conference on Pattern Recognition, 2004. [22] L. Maystre and M. Grossglauser. Just sort it! A simple and effective approach to active preference learning. In International Conference on Machine Learning, 2017. [23] B. Menza, A. Jacas, et al. The multimodal brain tumor image segmentation benchmark (BRATS). IEEE Transactions on Medical Imaging, 2014. [24] A. Mosinska, R. Sznitman, P. Glowacki, and P. Fua. Active learning for delineation of curvilinear structures. 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VAE Learning via Stein Variational Gradient Descent Yunchen Pu, Zhe Gan, Ricardo Henao, Chunyuan Li, Shaobo Han, Lawrence Carin Department of Electrical and Computer Engineering, Duke University {yp42, zg27, r.henao, cl319, shaobo.han, lcarin}@duke.edu Abstract A new method for learning variational autoencoders (VAEs) is developed, based on Stein variational gradient descent. A key advantage of this approach is that one need not make parametric assumptions about the form of the encoder distribution. Performance is further enhanced by integrating the proposed encoder with importance sampling. Excellent performance is demonstrated across multiple unsupervised and semi-supervised problems, including semi-supervised analysis of the ImageNet data, demonstrating the scalability of the model to large datasets. 1 Introduction There has been significant recent interest in the variational autoencoder (VAE) [11], a generalization of the original autoencoder [33]. VAEs are typically trained by maximizing a variational lower bound of the data log-likelihood [2, 10, 11, 12, 18, 21, 22, 23, 30, 34, 35]. To compute the variational expression, one must be able to explicitly evaluate the associated distribution of latent features, i.e., the stochastic encoder must have an explicit analytic form. This requirement has motivated design of encoders in which a neural network maps input data to the parameters of a simple distribution, e.g., Gaussian distributions have been widely utilized [1, 11, 27, 25]. The Gaussian assumption may be too restrictive in some cases [28]. Consequently, recent work has considered normalizing flows [28], in which random variables from (for example) a Gaussian distribution are fed through a series of nonlinear functions to increase the complexity and representational power of the encoder. However, because of the need to explicitly evaluate the distribution within the variational expression used when learning, these nonlinear functions must be relatively simple, e.g., planar flows. Further, one may require many layers to achieve the desired representational power. We present a new approach for training a VAE. We recognize that the need for an explicit form for the encoder distribution is only a consequence of the fact that learning is performed based on the variational lower bound. For inference (e.g., at test time), we do not need an explicit form for the distribution of latent features, we only require fast sampling from the encoder. Consequently, rather than directly employing the traditional variational lower bound, we seek to minimize the KullbackLeibler (KL) distance between the true posterior of model and latent parameters. Learning then becomes a novel application of Stein variational gradient descent (SVGD) [15], constituting its first application to training VAEs. We extend SVGD with importance sampling [1], and also demonstrate its novel use in semi-supervised VAE learning. The concepts developed here are demonstrated on a wide range of unsupervised and semi-supervised learning problems, including a large-scale semi-supervised analysis of the ImageNet dataset. These experimental results illustrate the advantage of SVGD-based VAE training, relative to traditional approaches. Moreover, the results demonstrate further improvements realized by integrating SVGD with importance sampling. Independent work by [3, 6] proposed the similar models, in which the aurthers incorporated SVGD with VAEs [3] and importance sampling [6] for unsupervised learning tasks. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 2 2.1 Stein Learning of Variational Autoencoder (Stein VAE) Review of VAE and Motivation for Use of SVGD Consider data D = {xn }N n=1 , where xn are modeled via decoder xn |z n ? p(x|z n ; ?). A prior p(z) is placed on the latent codes. To learn parameters ?, one typically is interested in maximizing PN the empirical expected log-likelihood, N1 n=1 log p(xn ; ?). A variational lower bound is often employed: h p(x|z; ?)p(z) i L(?, ?; x) = Ez|x;? log = ?KL(q(z|x; ?)kp(z|x; ?)) + log p(x; ?) , (1) q(z|x; ?) with log p(x; ?) ? L(?, ?; x), and where Ez|x;? [?] is approximated by averaging over a finite number of samples drawn from encoder q(z|x; ?). Parameters ? and ? are typically iteratively PN optimized via stochastic gradient descent [11], seeking to maximize n=1 L(?, ?; xn ). To evaluate the variational expression in (1), we require the ability to sample efficiently from q(z|x; ?), to approximate the expectation. We also require a closed form for this encoder, to evaluate log[p(x|z; ?)p(z)/q(z|x; ?)]. In the proposed VAE learning framework, rather than maximizing the variational lower bound explicitly, we focus on the term KL(q(z|x; ?)kp(z|x; ?)), which we seek to minimize. This can be achieved by leveraging Stein variational gradient descent (SVGD) [15]. Importantly, for SVGD we need only be able to sample from q(z|x; ?), and we need not possess its explicit functional form. In the above discussion, ? is treated as a parameter; below we treat it as a random variable, as was considered in the Appendix of [11]. Treatment of ? as a random variable allows for model averaging, and a point estimate of ? is revealed as a special case of the proposed method. The set of codes associated with all xn ? D is represented Z = {z n }N n=1 . The prior on {?, Z} is QN here represented as p(?, Z) = p(?) n=1 p(z n ). We desire the posterior p(?, Z|D). Consider the revised variational expression h p(D|Z, ?)p(?, Z) i L1 (q; D) = Eq(?,Z) log = ?KL(q(?, Z)kp(?, Z|D)) + log p(D; M) , (2) q(?, Z) where p(D; M) is the evidence for the underlying model M. Learning q(?, Z) such that L1 is maximized is equivalent to seeking q(?, Z) that minimizes KL(q(?, Z)kp(?, Z|D)). By leveraging and generalizing SVGD, we will perform the latter. 2.2 Stein Variational Gradient Descent (SVGD) Rather than explicitly specifying a form for p(?, Z|D), we sequentially refine samples of ? and Z, such that they are better matched to p(?, Z|D). We alternate between updating the samples of ? and samples of Z, analogous to how ? and ? are updated alternatively in traditional VAE optimization of (1). We first consider updating samples of ?, with the samples of Z held fixed. Specifically, M assume we have samples {? j }M j=1 drawn from distribution q(?), and samples {z jn }j=1 drawn from M distribution q(Z). We wish to transform {? j }j=1 by feeding them through a function, and the corresponding (implicit) transformed distribution from which they are drawn is denoted as qT (?). It is desired that, in a KL sense, qT (?)q(Z) is closer to p(?, Z|D) than was q(?)q(Z). The following theorem is useful for defining how to best update {? j }M j=1 . Theorem 1 Assume ? and Z are Random Variables (RVs) drawn from distributions q(?) and q(Z), respectively. Consider the transformation T (?) = ? + ?(?; D) and let qT (?) represent the distribution of ? 0 = T (?). We have    ? KL(qT kp) |=0 = ?E??q(?) trace(Ap (?; D)) , (3) where qT = qT (?)q(Z), p = p(?, Z|D), Ap (?; D) = ?? log p?(?; D)?(?; D)T + ?? ?(?; D), log p?(?; D) = EZ?q(Z) [log p(D, Z, ?)], and p(D, Z, ?) = p(D|Z, ?)p(?, Z). The proof is provided in Appendix A. Following [15], we assume ?(?; D) lives in a reproducing kernel Hilbert space (RKHS) with kernel k(?, ?). Under this assumption, the solution for ?(?; D) 2 that maximizes the decrease in the KL distance (3) is ? ? (?; D) = Eq(?) [k(?, ?)?? log p?(?; D) + ?? k(?, ?)] . (4) Theorem 1 concerns updating samples from q(?) assuming fixed q(Z). Similarly, to update q(Z) with q(?) fixed, we employ a complementary form of Theorem 1 (omitted for brevity). In that case, we consider transformation T (Z) = Z + ?(Z; D), with Z ? q(Z), and function ?(Z; D) is also assumed to be in a RKHS. (t+1) (t) (t) The expectations in (3) and (4) are approximated by samples ? j = ? j + ?? j , with   PM (t) (t) (t) (t) (t) (t) 1 ?? j ? M j 0 =1 k? (? j 0 , ? j )??(t) log p?(? j 0 ; D) + ??(t) k? (? j 0 , ? j )) , j0 j0 with ?? log p?(?; D) ? 1 M PN n=1 PM j=1 ?? log p(xn |z jn , ?)p(?). (t+1) (t) (t) z jn = z jn + ?z jn : A similar update of samples is manifested for the latent variables   PM (t) (t) (t) (t) (t) (t) 1 ?z jn = M j 0 =1 kz (z j 0 n , z jn )?z(t) log p?(z j 0 n ; D) + ?z(t) kz (z j 0 n , z jn ) , j0 n (5) j0 n (6) PM 0 1 where ?zn log p?(z n ; D) ? M j=1 ?z n log p(xn |z n , ? j )p(z n ). The kernels used to update samples of ? and z n are in general different, denoted respectively k? (?, ?) and kz (?, ?), and  is a small step size. For notational simplicity, M is the same in (5) and (6), but in practice a different number of samples may be used for ? and Z. If M = 1 for parameter ?, indices j and j 0 are removed in (5). Learning then reduces to gradient descent and a point estimate for ?, identical to the optimization procedure used for the traditional VAE expression in (1), but with the (multiple) samples associated with Z sequentially transformed via SVGD (and, importantly, without the need to assume a form for q(z|x; ?)). Therefore, if only a point estimate of ? is desired, (1) can be optimized wrt ?, while for updating Z SVGD is applied. 2.3 Efficient Stochastic Encoder At iteration t of the above learning procedure, we realize a set of latent-variable (code) samples (t) {z jn }M j=1 for each xn ? D under analysis. For large N , training may be computationally expensive. Further, the need to evolve (learn) samples {z j? }M j=1 for each new test sample, x? , is undesirable. We therefore develop a recognition model that efficiently computes samples of latent codes for a data sample of interest. The recognition model draws samples via z jn = f ? (xn , ? jn ) with ? jn ? q0 (?). Distribution q0 (?) is selected such that it may be easily sampled, e.g., isotropic Gaussian. After each iteration of updating the samples of Z, we refine recognition model f ? (x, ?) to mimic the Stein sample dynamics. Assume recognition-model parameters ? (t) have been learned thus far. (t) Using ? (t) , latent codes for iteration t are constituted as z jn = f ?(t) (xn , ? jn ), with ? jn ? q0 (?). These codes are computed for all data xn ? Bt , where Bt ? D is the minibatch of data at iteration (t) t. The change in the codes is ?z jn , as defined in (6). We then update ? to match the refined codes, as P PM (t+1) ? (t+1) = arg min? xn ?Bt j=1 kf ? (xn , ? jn ) ? z jn k2 . (7) The analytic solution of (7) is intractable. We update ? with K steps of gradient descent as ? (t,k) = P PM (t,k?1) (t,k?1) ? (t,k?1) ? ? xn ?Bt j=1 ?? jn , where ?? jn = ?? f ? (xn , ? jn )(f ? (xn , ? jn ) ? (t+1) z jn )|?=?(t,k?1) , ? is a small step size, ? (t) = ? (t,0) , ? (t+1) = ? (t,K) , and ?? f ? (xn , ? jn ) is the transpose of the Jacobian of f ? (xn , ? jn ) wrt ?. Note that the use of minibatches mitigates challenges of training with large training sets, D. The function f ? (x, ?) plays a role analogous to q(z|x; ?) in (1), in that it yields a means of efficiently drawing samples of latent codes z, given observed x; however, we do not impose an explicit functional form for the distribution of these samples. 3 3 3.1 Stein Variational Importance Weighted Autoencoder (Stein VIWAE) Multi-sample importance-weighted KL divergence Recall the variational expression in (1) employed in conventional VAE learning. Recently, [1, 19] showed that the multi-sample (k samples) importance-weighted estimator i h i Pk ) (8) Lk (x) = Ez1 ,...,zk ?q(z|x) log k1 i=1 p(x,z q(z i |x) , provides a tighter lower bound and a better proxy for the log-likelihood, where z 1 , . . . , z k are random variables sampled independently from q(z|x). Recall from (3) that the KL divergence played a key role in the Stein-based learning of Section 2. Equation (8) motivates replacement of the KL objective function with the multi-sample importance-weighted KL divergence h i i Pk |D) KLkq,p (?; D) , ?E?1:k ?q(?) log k1 i=1 p(? , (9) i q(? ) where ? = (?, Z) and ?1:k = ?1 , . . . , ?k are independent samples from q(?, Z). Note that the special case of k = 1 recovers the standard KL divergence. Inspired by [1], the following theorem (proved in Appendix A) shows that increasing the number of samples k is guaranteed to reduce the KL divergence and provide a better approximation of target distribution. Theorem 2 For any natural number k, we have KLkq,p (?; D) ? KLk+1 q,p (?; D) ? 0, and if q(?)/p(?|D) is bounded, then limk?? KLkq,p (?; D) = 0. We minimize (9) with a sample transformation based on a generalization of SVGD and the recognition model (encoder) is trained in the same way as in Section 2.3. Specifically, we first draw samples M 1:k M {? 1:k j }j=1 and {z jn }j=1 from a simple distribution q0 (?), and convert these to approximate draws 1:k from p(? , Z 1:k |D) by minimizing the multi-sample importance weighted KL divergence via nonlinear functional transformation. 3.2 Importance-weighted SVGD for VAEs The following theorem generalizes Theorem 1 to multi-sample weighted KL divergence. Theorem 3 Let ?1:k be RVs drawn independently from distribution q(?) and KLkq,p (?, D) is the multi-sample importance weighted KL divergence in (9). Let T (?) = ? + ?(?; D) and qT (?) represent the distribution of ?0 = T (?). We have   ? KLkq,p (?0 ; D) |=0 = ?E?1:k ?q(?) (Akp (?1:k ; D)) . (10) The proof and detailed definition is provided in Appendix A. The following corollaries generalize Theorem 1 and (4) via use of importance sampling, respectively. Corollary 3.1 ? 1:k and Z 1:k are RVs drawn independently from distributions q(?) and q(Z), respectively. Let T (?) = ? + ?(?; D), qT (?) represent the distribution of ? 0 = T (?), and ?0 = (? 0 , Z) . We have   ? KLkqT ,p (?0 ; D) |=0 = ?E?1:k ?q(?) (Akp (? 1:k ; D)) , (11) where Akp (? 1:k ; D) = 1 ? ? Pk i i=1 ?i Ap (? ; D), ?i = EZ i ?q(Z) h p(? i ,Z i ,D) q(? i )q(Z i ) i , ? ? = Pk i=1 ?i ; Ap (?; D) and log p?(?; D) are as defined in Theorem 1. Corollary 3.2 Assume ?(?; D) lives in a reproducing kernel Hilbert space (RKHS) with kernel k? (?, ?). The solution for ?(?; D) that maximizes the decrease in the KL distance (11) is ? ? (?; D) = E?1:k ?q(?) h P k 1 ? ? i=1 i ?i ??i k? (? i , ?) + k? (? i , ?)??i log p?(? i ; D) . 4 (12) M Corollary 3.1 and Corollary 3.2 provide a means of updating multiple samples {? 1:k j }j=1 from q(?) i i i via T (? ) = ? + ?(? ; D). The expectation wrt q(Z) is approximated via samples drawn from q(Z). Similarly, we can employ a complementary form of Corollary 3.1 and Corollary 3.2 to update multiple samples {Zj1:k }M j=1 from q(Z). This suggests an importance-weighted learning procedure M 1:k M that alternates between update of particles {? 1:k j }j=1 and {Zj }j=1 , which is similar to the one in Section 2.2. Detailed update equations are provided in Appendix B. 4 Semi-Supervised Learning with Stein VAE l Consider labeled data as pairs Dl = {xn , y n }N n=1 , where the label y n ? {1, . . . , C} and the de? = p(x|z n ; ?)p(y|z n ; ?), ? where ? ? represents coder is modeled as (xn , y n |z n ) ? p(x, y|z n ; ?, ?) the parameters of the decoder for labels. The set of codes associated with all labeled data are reprel sented as Zl = {z n }N n=1 . We desire to approximate the posterior distribution on the entire dataset ? p(?, ?, Z, Zl |D, Dl ) via samples, where D represents the unlabeled data, and Z is the set of codes ? and Zl . associated with D. In the following, we will only discuss how to update the samples of ?, ? Updating samples Z is the same as discussed in Sections 2 and 3.2 for Stein VAE and Stein VIWAE, respectively. ? M ? Assume {? j }M j=1 drawn from distribution q(?), {? j }j=1 drawn from distribution q(?), and samples M {z jn }j=1 drawn from (distinct) distribution q(Zl ). The following corollary generalizes Theorem 1 and (4), which is useful for defining how to best update {? j }M j=1 . ? Z and Zl are RVs drawn from distributions q(?), q(?), ? q(Z) and Corollary 3.3 Assume ?, ?, q(Zl ), respectively. Consider the transformation T (?) = ? + ?(?; D, Dl ) where ?(?; D, Dl ) lives in a RKHS with kernel k? (?, ?). Let qT (?) represent the distribution of ? 0 = T (?). For ? and p = p(?, ?, ? Z|D, Dl ), we have qT = qT (?)q(Z)q(?)   ? KL(qT kp) |=0 = ?E??q(?) (Ap (?; D, Dl )) , (13) where Ap (?; D, Dl ) = ?? ?(?; D, Dl ) + ?? log p?(?; D, Dl )?(?; D, Dl )T , log p?(?; D, Dl ) = EZ?q(Z) [log p(D|Z, ?)] + EZl ?q(Zl ) [log p(Dl |Zl , ?)], and the solution for ?(?; D, Dl ) that maximizes the change in the KL distance (13) is ? ? (?; D, Dl ) = Eq(?) [k(?, ?)?? log p?(?; D, Dl ) + ?? k(?, ?)] . (14) Further details are provided in Appendix C. 5 Experiments For all experiments, we use a radial basis-function (RBF) kernel as in [15], i.e., k(x, x0 ) = exp(? h1 kx ? x0 k22 ), where the bandwidth, h, is the median of pairwise distances between current samples. q0 (?) and q0 (?) are set to isotropic Gaussian distributions. We share the samples of ? across data points, i.e., ? jn = ? j , for n = 1, . . . , N (this is not necessary, but it saves computation). The samples of ? and z, and parameters of the recognition model, ?, are optimized via Adam [9] with learning rate 0.0002. We do not perform any dataset-specific tuning or regularization other than dropout [32] and early stopping on validation sets. We set M = 100 and k = 50, and use minibatches of size 64 for all experiments, unless otherwise specified. 5.1 Expressive power of Stein recognition model Gaussian Mixture Model We synthesize data by (i) drawing z n ? 21 N (?1 , I) + 12 N (?2 , I), where ?1 = [5, 5]T , ?2 = [?5, ?5]T ; (ii) drawing xn ? N (?z n , ? 2 I), where ? = 21 ?1 ?2 and ? = 0.1. The recognition model f? (xn , ? j ) is specified as a multi-layer perceptron (MLP) with 100 hidden units, by first concatenating ? j and xn into a long vector. The dimension of ? j is set to 2. The recognition model for standard VAE is also an MLP with 100 hidden units, and with the assumption of a Gaussian distribution for the latent codes [11]. 5 Figure 1: Approximation of posterior distribution: Stein VAE vs. VAE. The figures represent different samples of Stein VAE. (left) 10 samples, (center) 50 samples, and (right) 100 samples. We generate N = 10, 000 data points for training and 10 data points for testing. The analytic form of true posterior distribution is provided in Appendix D. Figure 1 shows the performance of Stein VAE approximations for the true posterior; other similar examples are provided in Appendix F. The Stein recognition model is able to capture the multi-modal posterior and produce accurate density approximation. Poisson Factor Analysis Given a discrete vector xn ? ZP + , Poisson factor analysis [36] assumes xn is a weighted combination of V latent factors xn ? ?V Pois(?z n ), where ? ? RP is the factor loadings + matrix and z n ? RV+ is the vector of factor scores. We consider topic modeling with Dirichlet priors on ? v (v-th column of ?) and gamma priors on each component of z n . We evaluate our model on the 20 Newsgroups dataset containing N = 18, 845 documents with a vocabulary of P = 2, 000. The data are partitioned into 10,314 training, 1,000 validation and 7,531 test documents. The number of factors (topics) is set to V = 128. ? is first learned by Markov chain Monte Carlo (MCMC) [4]. We then fix ? at its MAP value, and only learn the recognition model ? using standard VAE and Stein VAE; this is done, as in the previous example, to examine the accuracy of the recognition model to estimate the posterior of the latent factors, isolated from estimation of ?. The recognition model is an MLP with 100 hidden units. Figure 2: Univariate marginals and pairwise posteriors. Purple, red and green represent the distribution inferred from MCMC, standard VAE and Stein VAE, respectively. An analytic form of the true posterior distribution Table 1: Negative log-likelihood (NLL) on p(z n |xn ) is intractable for this problem. Consequently, MNIST. ? Trained with VAE and tested with we employ samples collected from MCMC as ground IWAE. ? Trained and tested with IWAE. truth. With ? fixed, we sample z n via Gibbs sampling, usMethod NLL ing 2,000 burn-in iterations followed by 2,500 collection DGLM [27] 89.90 draws, retaining every 10th collection sample. We show Normalizing flow [28] 85.10 the marginal and pairwise posterior of one test data point ? VAE + IWAE [1] 86.76 in Figure 2. Additional results are provided in Appendix IWAE + IWAE [1]? 84.78 F. Stein VAE leads to a more accurate approximation than standard VAE, compared to the MCMC samples. ConsidStein VAE + ELBO 85.21 ering Figure 2, note that VAE significantly underestimates Stein VAE + S-ELBO 84.98 the variance of the posterior (examining the marginals), a Stein VIWAE + ELBO 83.01 Stein VIWAE + S-ELBO 82.88 well-known problem of variational Bayesian analysis [7]. In sharp contrast, Stein VAE yields highly accurate approximations to the true posterior. 5.2 Density estimation Data We consider five benchmark datasets: MNIST and four text corpora: 20 Newsgroups (20News), New York Times (NYT), Science and RCV1-v2 (RCV2). For MNIST, we used the standard split of 50K training, 10K validation and 10K test examples. The latter three text corpora 6 consist of 133K, 166K and 794K documents. These three datasets are split into 1K validation, 10K testing and the rest for training. Evaluation Given new data x? (testing data), the marginal log-likelihood/perplexity values are estimated by the variational evidence lower bound (ELBO) while integrating the decoder parameters ? out, log p(x? ) ? Eq(z? ) [log p(x? , z ? )] + H(q(z ? )) = ELBO(q(z ? )), where p(x? , z ? ) = Eq(?) [log p(x? , ?, z ? )] and H(q(?)) = ?Eq (log q(?)) is the entropy. The expectation is approxiM mated with samples {? j }M j=1 and {z ?j }j=1 with z ?j = f ? (x? , ? j ), ? j ? q0 (?). Directly evaluating ?f (x,?) ?1 q(z ? ) is intractable, thus it is estimated via density transformation q(z) = q0 (?) det ??? . We further estimate the marginal loglikelihood/perplexity values via the Method 20News NYT Science RCV2 stochastic variational lower bound, as DocNADE [14] 896 2496 1725 742 the mean of 5K-sample importance DEF [24] ?2416 1576 ?weighting estimate [1]. Therefore, for NVDM [17] 852 ??550 each dataset, we report four results: (i) Stein VAE + ELBO 849 2402 1499 549 Stein VAE + ELBO, (ii) Stein VAE + SStein VAE + S-ELBO 845 2401 1497 544 ELBO, (iii) Stein VIWAE + ELBO and Stein VIWAE + ELBO 837 2315 1453 523 (iv) Stein VIWAE + S-ELBO; the first 829 2277 1421 518 Stein VIWAE + S-ELBO term denotes the training procedure is employed as Stein VAE in Section 2 or Stein VIWAE in Section 3; the second term denotes the testing log-likelihood/perplexity is estimated by the ELBO or the stochastic variational lower bound, S-ELBO [1]. Table 2: Test perplexities on four text corpora. Model For MNIST, we train the model with one stochastic layer, z n , with 50 hidden units and two deterministic layers, each with 200 units. The nonlinearity is set as tanh. The visible layer, xn , follows a Bernoulli distribution. For the text corpora, we build a three-layer deep Poisson network [24]. The sizes of hidden units are 200, 200 and 50 for the first, second and third layer, respectively (see [24] for detailed architectures). Time (s) 5.3 Negative Log?likelihood (nats) 6 Results The log-likelihood/perplexity results 88 Negative Log?likelihood are summarized in Tables 1 and 2. On MNIST, 5 Testing Time for Entire Dataset our Stein VAE achieves a variational lower bound Training Time for Each Epoch 87 4 of -85.21 nats, which outperforms standard VAE with the same model architecture. Our Stein VI3 86 WAE achieves a log-likelihood of -82.88 nats, 2 exceeding normalizing flow (-85.1 nats) and im1 portance weighted autoencoder (-84.78 nats), 85 which is the best prior result obtained by feed1 5 10 20 40 60 100 200 300 forward neural network (FNN). DRAW [5] and Number of Samples (M) PixelRNN [20], which exploit spatial structure, Figure 3: NLL vs. Training/Testing time on MNIST achieved log-likelihoods of around -80 nats. Our with various numbers of samples for ?. model can also be applied on these models, but this is left as interesting future work. To further illustrate the benefit of model averaging, we vary the number of samples for ? (while retaining 100 samples for Z) and show the results associated with training/testing time in Figure 3. When M = 1 for ?, our model reduces to a point estimate for that parameter. Increasing the number of samples of ? (model averaging) improves the negative log-likelihood (NLL). The testing time of using 100 samples of ? is around 0.12 ms per image. Semi-supervised Classification We consider semi-supervised classification on MNIST and ImageNet [29] data. For each dataset, we report the results obtained by (i) VAE, (ii) Stein VAE, and (iii) Stein VIWAE. MNIST We randomly split the training set into a labeled and unlabeled set, and the number of labeled samples in each category varies from 10 to 300. We perform testing on the standard test ? = set with 20 different training-set splits. The decoder for labels is implemented as p(y n |z n , ?) ? n ). We consider two types of decoders for images p(xn |z n , ?) and encoder f (x, ?): softmax(?z ? 7 (i) FNN: Following [12], we use a 50-dimensional latent variables z n and two hidden layers, each with 600 hidden units, for both encoder and decoder; softplus is employed as the nonlinear activation function. (ii) All convolutional nets (CNN): Inspired by [31], we replace the two hidden layers with 32 and 64 kernels of size 5 ? 5 and a stride of 2. A fully connected layer is stacked on the CNN to produce a 50-dimensional latent variables z n . We use the leaky rectified activation [16]. The input of the encoder is formed by spatially aligning and stacking xn and ?, while the output of decoder is the image itself. Table 3 shows the classi- Table 3: Semi-supervised classification error (%) on MNIST. N? is the number fication results. Our Stein of labeled images per class. ? [12]; ? our implementation. VAE and Stein VIWAE FNN CNN N? consistently achieve betVAE? Stein VAE Stein VIWAE VAE? Stein VAE Stein VIWAE ter performance than the 10 3.33 ? 0.14 2.78 ? 0.24 2.67 ? 0.09 2.44 ? 0.17 1.94 ? 0.24 1.90 ? 0.05 VAE. We further observe 60 2.59 ?0.05 2.13 ? 0.08 2.09 ? 0.03 1.88 ?0.05 1.44 ? 0.04 1.41 ? 0.02 that the variance of Stein 100 2.40 ?0.02 1.92 ? 0.05 1.88 ? 0.01 1.47 ?0.02 1.01 ? 0.03 0.99 ? 0.02 VIWAE results is much 300 2.18 ?0.04 1.77 ? 0.03 1.75 ? 0.01 0.98 ?0.02 0.89 ? 0.03 0.86 ? 0.01 smaller than that of Stein VAE results on small labeled data, indicating the former produces more robust parameter estimates. State-ofthe-art results [26] are achieved by the Ladder network, which can be employed with our Stein-based approach, however, we will consider this extension as future work. ImageNet 2012 We Table 4: Semi-supervised classification accuracy (%) on ImageNet. consider scalability of our VAE Stein VAE Stein VIWAE DGDN [21] model to large datasets. We split the 1.3 million 1 % 35.92? 1.91 36.44 ? 1.66 36.91 ? 0.98 43.98? 1.15 2 % 40.15? 1.52 41.71 ? 1.14 42.57 ? 0.84 46.92? 1.11 training images into an 5 % 44.27? 1.47 46.14 ? 1.02 46.20 ? 0.52 47.36? 0.91 unlabeled and labeled set, 10 % 46.92? 1.02 47.83 ? 0.88 48.67 ? 0.31 48.41? 0.76 and vary the proportion 20 % 50.43? 0.41 51.62 ? 0.24 51.77 ? 0.12 51.51? 0.28 of labeled images from 30 % 53.24? 0.33 55.02 ? 0.22 55.45 ? 0.11 54.14? 0.12 1% to 40%. The classes 40 % 56.89? 0.11 58.17 ? 0.16 58.21 ? 0.12 57.34? 0.18 are balanced to ensure that no particular class is over-represented, i.e., the ratio of labeled and unlabeled images is the same for each class. We repeat the training process 10 times for the training setting with labeled images ranging from 1% to 10% , and 5 times for the the training setting with labeled images ranging from 20% to 40%. Each time we utilize different sets of images as the unlabeled ones. We employ an all convolutional net [31] for both the encoder and decoder, which replaces deterministic pooling (e.g., max-pooling) with stridden convolutions. Residual connections [8] are incorporated to encourage gradient flow. The model architecture is detailed in Appendix E. Following [13], images are resized to 256 ? 256. A 224 ? 224 crop is randomly sampled from the images or its horizontal flip with the mean subtracted [13]. We set M = 20 and k = 10. Table 4 shows classification results indicating that Stein VAE and Stein IVWAE outperform VAE in all the experiments, demonstrating the effectiveness of our approach for semi-supervised classification. When the proportion of labeled examples is too small (< 10%), DGDN [21] outperforms all the VAE-based models, which is not surprising provided that our models are deeper, thus have considerably more parameters than DGDN [21]. 6 Conclusion We have employed SVGD to develop a new method for learning a variational autoencoder, in which we need not specify an a priori form for the encoder distribution. Fast inference is manifested by learning a recognition model that mimics the manner in which the inferred code samples are manifested. The method is further generalized and improved by performing importance sampling. An extensive set of results, for unsupervised and semi-supervised learning, demonstrate excellent performance and scaling to large datasets. Acknowledgements This research was supported in part by ARO, DARPA, DOE, NGA, ONR and NSF. 8 References [1] Y. Burda, R. Grosse, and R. Salakhutdinov. Importance weighted autoencoders. In ICLR, 2016. [2] L. Chen, S. Dai, Y. Pu, C. Li, and Q. Su Lawrence Carin. Symmetric variational autoencoder and connections to adversarial learning. In arXiv, 2017. [3] Y. Feng, D. Wang, and Q. Liu. Learning to draw samples with amortized stein variational gradient descent. In UAI, 2017. [4] Z. Gan, C. Chen, R. Henao, D. Carlson, and L. Carin. Scalable deep poisson factor analysis for topic modeling. In ICML, 2015. [5] K. Gregor, I. Danihelka, A. Graves, and D. Wierstra. Draw: A recurrent neural network for image generation. In ICML, 2015. [6] J. Han and Q. Liu. Stein variational adaptive importance sampling. In UAI, 2017. [7] S. Han, X. Liao, D.B. Dunson, and L. Carin. Variational gaussian copula inference. In AISTATS, 2016. [8] K. He, X. Zhang, S. Ren, and Sun J. Deep residual learning for image recognition. In CVPR, 2016. [9] D. Kingma and J. Ba. Adam: A method for stochastic optimization. In ICLR, 2015. [10] D. P. Kingma, T. Salimans, R. Jozefowicz, X.i Chen, I. Sutskever, and M. Welling. 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Rezende. Variational inference for monte carlo objectives. In ICML, 2016. [20] A. Oord, N. Kalchbrenner, and K. Kavukcuoglu. Pixel recurrent neural network. In ICML, 2016. [21] Y. Pu, Z. Gan, R. Henao, X. Yuan, C. Li, A. Stevens, and L. Carin. Variational autoencoder for deep learning of images, labels and captions. In NIPS, 2016. [22] Y. Pu, X. Yuan, and L. Carin. Generative deep deconvolutional learning. In ICLR workshop, 2015. [23] Y. Pu, X. Yuan, A. Stevens, C. Li, and L. Carin. A deep generative deconvolutional image model. Artificial Intelligence and Statistics (AISTATS), 2016. 9 [24] R. Ranganath, L. Tang, L. Charlin, and D. M.Blei. Deep exponential families. In AISTATS, 2015. [25] R. Ranganath, D. Tran, and D. M. Blei. Hierarchical variational models. In ICML, 2016. [26] A. Rasmus, M. Berglund, M. Honkala, H. Valpola, and T. Raiko. Semi-supervised learning with ladder networks. In NIPS, 2015. [27] D. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In ICML, 2014. [28] D.J. Rezende and S. Mohamed. Variational inference with normalizing flows. In ICML, 2015. [29] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei-fei. Imagenet large scale visual recognition challenge. IJCV, 2014. [30] D. Shen, Y. Zhang, R. Henao, Q. Su, and L. Carin. Deconvolutional latent-variable model for text sequence matching. In arXiv, 2017. [31] J. T. Springenberg, A. Dosovitskiy, T. Brox, and M. Riedmiller. Striving for simplicity: The all convolutional net. In ICLR workshop, 2015. [32] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. JMLR, 2014. [33] P. Vincent, H. Larochelle, I. Lajoie, Y. Bengio, and P.-A. Manzagol. Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion. JMLR, 2010. [34] Y. Pu W. Wang, R. Henao, L. Chen, Z. Gan, C. Li, and Lawrence Carin. Adversarial symmetric variational autoencoder. In NIPS, 2017. [35] Y. Zhang, D. Shen, G. Wang, Z. Gan, R. Henao, and L. Carin. Deconvolutional paragraph representation learning. In NIPS, 2017. [36] M. Zhou, L. Hannah, D. Dunson, and L. Carin. Beta-negative binomial process and Poisson factor analysis. In AISTATS, 2012. 10
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Reconstructing perceived faces from brain activations with deep adversarial neural decoding Ya?gmur G??l?t?rk*, Umut G??l?*, Katja Seeliger, Sander Bosch, Rob van Lier, Marcel van Gerven, Radboud University, Donders Institute for Brain, Cognition and Behaviour Nijmegen, the Netherlands {y.gucluturk, u.guclu}@donders.ru.nl *Equal contribution Abstract Here, we present a novel approach to solve the problem of reconstructing perceived stimuli from brain responses by combining probabilistic inference with deep learning. Our approach first inverts the linear transformation from latent features to brain responses with maximum a posteriori estimation and then inverts the nonlinear transformation from perceived stimuli to latent features with adversarial training of convolutional neural networks. We test our approach with a functional magnetic resonance imaging experiment and show that it can generate state-of-the-art reconstructions of perceived faces from brain activations. ConvNet (adversarial training) likelihood (Gaussian) posterior (Gaussian) maximum a posteriori brain resp. ConvNet (pretrained) + PCA prior (Gaussian) latent feat. perceived stim. *reconstruction *from brain resp. Figure 1: An illustration of our approach to solve the problem of reconstructing perceived stimuli from brain responses by combining probabilistic inference with deep learning. 1 Introduction A key objective in sensory neuroscience is to characterize the relationship between perceived stimuli and brain responses. This relationship can be studied with neural encoding and neural decoding in functional magnetic resonance imaging (fMRI) [1]. The goal of neural encoding is to predict brain responses to perceived stimuli [2]. Conversely, the goal of neural decoding is to classify [3, 4], identify [5, 6] or reconstruct [7?11] perceived stimuli from brain responses. The recent integration of deep learning into neural encoding has been a very successful endeavor [12, 13]. To date, the most accurate predictions of brain responses to perceived stimuli have been achieved with convolutional neural networks [14?20], leading to novel insights about the functional organization of neural representations. At the same time, the use of deep learning as the basis for neural decoding has received less widespread attention. Deep neural networks have been used for classifying or identifying stimuli via the use of a deep encoding model [16, 21] or by predicting 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. intermediate stimulus features [22, 23]. Deep belief networks and convolutional neural networks have been used to reconstruct basic stimuli (handwritten characters and geometric figures) from patterns of brain activity [24, 25]. To date, going beyond such mostly retinotopy-driven reconstructions and reconstructing complex naturalistic stimuli with high accuracy have proven to be difficult. The integration of deep learning into neural decoding is an exciting approach for solving the reconstruction problem, which is defined as the inversion of the (non)linear transformation from perceived stimuli to brain responses to obtain a reconstruction of the original stimulus from patterns of brain activity alone. Reconstruction can be formulated as an inference problem, which can be solved by maximum a posteriori estimation. Multiple variants of this formulation have been proposed in the literature [26?30]. At the same time, significant improvements are to be expected from deep neural decoding given the success of deep learning in solving image reconstruction problems in computer vision such as colorization [31], face hallucination [32], inpainting [33] and super-resolution [34]. Here, we present a new approach by combining probabilistic inference with deep learning, which we refer to as deep adversarial neural decoding (DAND). Our approach first inverts the linear transformation from latent features to observed responses with maximum a posteriori estimation. Next, it inverts the nonlinear transformation from perceived stimuli to latent features with adversarial training and convolutional neural networks. An illustration of our model is provided in Figure 1. We show that our approach achieves state-of-the-art reconstructions of perceived faces from the human brain. 2 2.1 Methods Problem statement Let x ? Rh?w?c , z ? Rp , y ? Rq be a stimulus, feature, response triplet, and ? : Rh?w?c ? Rp be a latent feature model such that z = ?(x) and x = ??1 (z). Without loss of generality, we assume that all of the variables are normalized to have zero mean and unit variance. We are interested in solving the problem of reconstructing perceived stimuli from brain responses: ? = ??1 (arg max Pr (z | y)) x (1) z where Pr(z | y) is the posterior. We reformulate the posterior through Bayes? theorem:   ?1 ?=? x arg max [Pr(y | z) Pr(z)] (2) z where Pr(y | z) is the likelihood, and Pr(z) is the prior. In the following subsections, we define the latent feature model, the likelihood and the prior. 2.2 Latent feature model We define the latent feature model ?(x) by modifying the VGG-Face pretrained model [35]. This model is a 16-layer convolutional neural network, which was trained for face recognition. First, we truncate it by retaining the first 14 layers and discarding the last two layers of the model. At this point, the truncated model outputs 4096-dimensional latent features. To reduce the dimensionality of the latent features, we then combine the model with principal component analysis by estimating the loadings that project the 4096-dimensional latent features to the first 699 principal component scores (maximum number of components given the number of training observations) and adding them at the end of the truncated model as a new fully-connected layer. At this point, the combined model outputs 699-dimensional latent features. Following the ideas presented in [36?38], we define the inverse of the feature model ??1 (z) (i.e., the image generator) as a convolutional neural network which transforms the 699-dimensional latent variables to 64 ? 64 ? 3 images and estimate its parameters via an adversarial process. The generator comprises five deconvolution layers: The ith layer has 210?i kernels with a size of 4 ? 4, a stride of 2 ? 2, a padding of 1 ? 1, batch normalization and rectified linear units. Exceptions are the first layer which has a stride of 1 ? 1, and no padding; and the last layer which has three kernels, no batch normalization [39] and hyperbolic tangent units. Note that we do use the inverse of the loadings in the generator. 2 To enable adversarial training, we define a discriminator (?) along with the generator. The discriminator comprises five convolution layers. The ith layer has 25+i kernels with a size of 4 ? 4, a stride of 2 ? 2, a padding of 1 ? 1, batch normalization and leaky rectified linear units with a slope of 0.2 except for the first layer which has no batch normalization and last layer which has one kernel, a stride of 1 ? 1, no padding, no batch normalization and a sigmoid unit. We train the generator and the discriminator by pitting them against each other in a two-player zero-sum game, where the goal of the discriminator is to discriminate stimuli from reconstructions and the goal of the generator is to generate reconstructions that are indiscriminable from original stimuli. This ensures that reconstructed stimuli are similar to target stimuli on a pixel level and a feature level. The discriminator is trained by iteratively minimizing the following discriminator loss function:   Ldis = ?E log(?(x)) + log(1 ? ?(??1 (z))) (3) where ? is the output of the discriminator which gives the probability that its input is an original stimulus and not a reconstructed stimulus. The generator is trained by iteratively minimizing a generator loss function, which is a linear combination of an adversarial loss function, a feature loss function and a stimulus loss function:   Lgen = ??adv E log(?(??1 (z))) +?fea E[k?(x) ? ?(??1 (z))k2 ] +?sti E[kx ? ??1 (z)k2 ] (4) | {z } | {z } | {z } Lfea Ladv Lsti where ? is the relu3_3 outputs of the pretrained VGG-16 model [40, 41]. Note that the targets and the reconstructions are lower resolution (i.e., 64 ? 64) than the images that are used to obtain the latent features (i.e., 224 ? 224). 2.3 Likelihood and prior We define the likelihood as a multivariate Gaussian distribution over y: Pr(y|z) = Ny (B> z, ?) p?q diag(?12 , . . . , ?q2 ) (5) q?q where B = (? 1 , . . . , ? q ) ? R and ? = ? R . Here, the features ? voxels matrix B contains the learnable parameters of the likelihood in its columns ? i (which can also be interpreted as regression coefficients of a linear regression model, which predicts y from z). ? = arg min E[kyi ? ? > zk2 ] We estimate the parameters with ordinary least squares, such that ? i ?i i ? > zk2 ]. and ? ? 2 = E[kyi ? ? i i We define the prior as a zero mean and unit variance multivariate Gaussian distribution Pr(z) = Nz (0, I). 2.4 Posterior To derive the posterior (2), we first reformulate the likelihood as a multivariate Gaussian distribution over z. That is, after taking out constant terms with respect to z from the likelihood, it immediately becomes proportional to the canonical form Gaussian over z with ? = B??1 y and ? = B??1 B> , which is equivalent to the standard form Gaussian with mean ??1 ? and covariance ??1 . This allows us to write: Pr(z|y) ? Nz ??1 ?, ??1 )Nz (0, I  (6) Next, recall that the product of two multivariate Gaussians can be formulated in terms of one multivariate Gaussian [42]. That is, Nz (m1 , ?1 )Nz (m2 , ?2 ) ? Nz (mc , ?c ) with mc =    ?1 ?1 ?1 ?1 ?1 ??1 ??1 m1 + ??1 . By plugging this formula1 + ?2 2 m2 and ?c = ?1 + ?2 tion into Equation (6), we obtain Pr(z|y) ? Nz (mc , ?c ) with mc = (B??1 B> + I)?1 B??1 y and ?c = (B??1 B> + I)?1 . Recall that we are interested in reconstructing stimuli from responses by generating reconstructions from the features that maximize the posterior. Notice that the (unnormalized) posterior is maximized 3 at its mean mc since this corresponds to the mode for a multivariate Gaussian distribution. Therefore, the solution of the problem of reconstructing stimuli from responses reduces to the following simple expression:  ? = ??1 (B??1 B> + I)?1 B??1 y x (7) 3 3.1 Results Datasets We used the following datasets in our experiments: fMRI dataset. We collected a new fMRI dataset, which comprises face stimuli and associated bloodoxygen-level dependent (BOLD) responses. The stimuli used in the fMRI experiment were drawn from [43?45] and other online sources, and consisted of photographs of front-facing individuals with neutral expressions. We measured BOLD responses (TR = 1.4 s, voxel size = 2 ? 2 ? 2 mm3 , whole-brain coverage) of two healthy adult subjects (S1: 28-year old female; S2: 39-year old male) as they were fixating on a target (0.6 ? 0.6 degree) [46] superimposed on the stimuli (15 ? 15 degrees). Each face was presented at 5 Hz for 1.4 s and followed by a middle gray background presented for 2.8 s. In total, 700 faces were presented twice for the training set, and 48 faces were repeated 13 times for the test set. The test set was balanced in terms of gender and ethnicity (based on the norming data provided in the original datasets). The experiment was approved by the local ethics committee (CMO Regio Arnhem-Nijmegen) and the subjects provided written informed consent in accordance with the Declaration of Helsinki. Our fMRI dataset is available from the first authors on reasonable request. The stimuli were preprocessed as follows: Each image was cropped and resized to 224 ? 224 pixels. This procedure was organized such that the distance between the top of the image and the vertical center of the eyes was 87 pixels, the distance between the vertical center of the eyes and the vertical center of the mouth was 75 pixels, the distance between the vertical center of the mouth and the bottom of the image was 61 pixels, and the horizontal center of the eyes and the mouth was at the horizontal center of the image. The fMRI data were preprocessed as follows: Functional scans were realigned to the first functional scan and the mean functional scan, respectively. Realigned functional scans were slice time corrected. Anatomical scans were coregistered to the mean functional scan. Brains were extracted from the coregistered anatomical scans. Finally, stimulus-specific responses were deconvolved from the realigned and slice time corrected functional scans with a general linear model [47]. Here, deconvolution refers to estimating regression coefficients (y) of the following GLMs: y? = Xy, where y? is raw voxel responses, X is HRF-convolved design matrix (one regressor per stimulus indicating its presence), and y is deconvolved voxel responses such that y is a vector of size m ? 1 with m denoting the number of unique stimuli, and there is one y per voxel. CelebA dataset [48]. This dataset comprises 202599 in-the-wild portraits of 10177 people, which were drawn from online sources. The portraits are annotated with 40 attributes and five landmarks. We preprocessed the portraits as we preprocessed the stimuli in our fMRI dataset. 3.2 Implementation details Our implementation makes use of Chainer and Cupy with CUDA and cuDNN [49] except for the following: The VGG-16 and VGG-Face pretrained models were ported to Chainer from Caffe [50]. Principal component analysis was implemented in scikit-learn [51]. fMRI preprocessing was implemented in SPM [52]. Brain extraction was implemented in FSL [53]. We trained the discriminator and the generator on the entire CelebA dataset by iteratively minimizing the discriminator loss function and the generator loss function in sequence for 100 epochs with Adam [54]. Model parameters were initialized as follows: biases were set to zero, the scaling parameters were drawn from N (1, 2?10?2 I), the shifting parameters were set to zero and the weights were drawn from N (1, 10?2 I) [37]. We set the hyperparameters of the loss functions as follows: ?adv = 102 , ?dis = 102 , ?fea = 10?2 and ?sti = 2 ? 10?6 [38]. We set the hyperparameters of the optimizer as follows: ? = 0.001, ?1 = 0.9, ?2 = 0.999 and  = 108 [37]. We estimated the parameters of the likelihood term on the training split of our fMRI dataset. 4 3.3 Evaluation metrics We evaluated our approach on the test split of our fMRI dataset with the following metrics: First, the feature similarity between the stimuli and their reconstructions, where the feature similarity is defined as the Euclidean similarity between the features, defined as the relu7 outputs of the VGGFace pretrained model. Second, the Pearson correlation coefficient between the stimuli and their reconstructions. Third, the structural similarity between the stimuli and their reconstructions [55]. All evaluation was done on a held-out set not used at any point during model estimation or training. The voxels used in the reconstructions were selected as follows: For each test trial, n voxels with smallest residuals (on training set) were selected. n itself was selected such that reconstruction accuracy of remaining test trials was highest. We also performed an encoding analysis to see how well the latent features were predictive of voxel responses in different brain areas. The results of this analysis is reported in the supplementary material. 3.4 Reconstruction We first demonstrate our results by reconstructing the stimulus images in the test set using i) the latent features and ii) the brain responses. Figure 2 shows 4 representative examples of the test stimuli and their reconstructions. The first column of both panels show the original test stimuli. The second column of both panels show the reconstructions of these stimuli x from the latent features z obtained by ?(x). These can be considered as an upper limit for the reconstruction accuracy of the brain responses since they are the best possible reconstructions that we can expect to achieve with a perfect neural decoder that can exactly predict the latent features from brain responses. The third and fourth columns of the figure show reconstructions of brain responses to stimuli of Subject 1 and Subject 2, respectively. stim. reconstruction from: model brain 1 brain 2 stim. reconstruction from: model brain 1 brain 2 Figure 2: Reconstructions of the test stimuli from the latent features (model) and the brain responses of the two subjects (brain 1 and brain 2). Visual inspection of the reconstructions from brain responses reveals that they match the test stimuli in several key aspects, such as gender, skin color and facial features. Table 1 shows the three reconstruction accuracy metrics for both subjects in terms of ratio of the reconstruction accuracy from brain responses to the reconstruction accuracy from latent features, which were significantly (p < 0.05, permutation test) above those for randomly sampled latent features (cf. 0.5181, 0.1532 and 0.5183, respectively). Table 1: Reconstruction accuracy of the proposed decoding approach. The results are reported as the ratio of accuracy of reconstructing from brain responses and latent features. S1 S2 Feature similarity Pearson correlation coefficient Structural similarity 0.6546 ? 0.0220 0.6465 ? 0.0222 0.6512 ? 0.0493 0.6580 ? 0.0480 0.8365 ? 0.0239 0.8325 ? 0.0229 Furthermore, besides reconstruction accuracy, we tested the identification performance within and between groups that shared similar features (those that share gender or ethnicity as defined by the norming data were assumed to share similar features). Identification accuracies (which ranged between 57% and 62%) were significantly above chance-level (which ranged between 3% and 8%) in all cases (p  0.05, Student?s t-test). Furthermore, we found no significant differences between the identification accuracies when a reconstruction was identified among a group sharing similar features versus among a group that did not share similar features (p > 0.79, Student?s t-test) (cf. [56]). 5 3.5 Visualization, interpolation and sampling In the second experiment, we analyzed the properties of the stimulus features predictive of brain activations to characterize neural representations of faces. We first investigated the model representations to better understand what kind of features drive responses of the model. We visualized the features explaining the highest variance by independently setting the values of the first few latent dimensions to vary between their minimum and maximum values and generating reconstructions from these representations (Figure 3). As a result, we found that many of the latent features were coding for interpretable high level information such as age, gender, etc. For example, the first feature in Figure 3 appears to code for gender, the second one appears to code for hair color and complexion, the third one appears to code for age, and the fourth one appears to code for two different facial expressions. reconstruction (from features) feature i = min. <-> feature i = max. 4 3 feature 2 1 reconstruction (from features) feature i = min. <-> feature i = max. Figure 3: Reconstructions from features with single features set to vary between their minimum and maximum values. We then explored the feature space that was learned by the latent feature model and the response space that was learned by the likelihood by systematically traversing the reconstructions obtained from different points in these spaces. Figure 4A shows examples of reconstructions of stimuli from the latent features (rows one and four) and brain responses (rows two, three, five and six), as well as reconstructions from their interpolations between two points (columns three to nine). The reconstructions from the interpolations between two points show semantic changes with no sharp transitions. Figure 4B shows reconstructions from latent features sampled from the model prior (first row) and from responses sampled from the response prior of each subject (second and third rows). The reconstructions from sampled representations are diverse and of high quality. These results provide evidence that no memorization took place and the models learned relevant and interesting representations [37]. Furthermore, these results suggest that neural representations of faces might be embedded in a continuous and distributed space in the brain. 3.6 Comparison versus state-of-the-art In this section we qualitatively (Figure 5) and quantitatively (Table 2) compare the performance of our approach with two existing decoding approaches from the literature? . Figure 5 shows example reconstructions from brain responses with three different approaches, namely with our approach, the eigenface approach [11, 57] and the identity transform approach [58, 29]. To achieve a fair comparison, the implementations of the three approaches only differed in terms of the feature models that were used, i.e. the eigenface approach had an eigenface (PCA) feature model and the identity transform approach had simply an identity transformation in place of the feature model. Visual inspection of the reconstructions displayed in Figure 5 shows that DAND clearly outperforms the existing approaches. In particular, our reconstructions better capture the features of the stimuli ? We also experimented with the VGG-ImageNet pretrained model, which failed to match the reconstruction performance of the VGG-Face model, while their encoding performances were comparable in non-face related brain areas. We plan to further investigate other models in detail in future work. 6 A recon. recon. (from interpolated features or responses) recon. stim. brain 2 brain 1 reconstruction from: model brain 2 brain 1 model stim. B reconstruction from: brain 2 brain 1 model recon. (from sampled features or responses) Figure 4: Reconstructions from interpolated (A) and sampled (B) latent features (model) and brain responses of the two subjects (brain 1 and brain 2). such as gender, skin color and facial features. Furthermore, our reconstructions are more detailed, sharper, less noisy and more photorealistic than the eigenface and identity transform approaches. A quantitative comparison of the performance of the three approaches shows that the reconstruction accuracies achieved by our approach were significantly higher than those achieved by the existing approaches (p  0.05, Student?s t-test). Table 2: Reconstruction accuracies of the three decoding approaches. LF denotes reconstructions from latent features. Identity Eigenface DAND Feature similarity Pearson correlation coefficient Structural similarity S1 S2 LF S1 S2 LF 0.1254 ? 0.0031 0.1254 ? 0.0038 1.0000 ? 0.0000 0.1475 ? 0.0043 0.1457 ? 0.0043 0.3841 ? 0.0149 0.4194 ? 0.0347 0.4299 ? 0.0350 1.0000 ? 0.0000 0.3779 ? 0.0403 0.2241 ? 0.0435 0.9875 ? 0.0011 0.3744 ? 0.0083 0.3877 ? 0.0083 1.0000 ? 0.0000 0.3735 ? 0.0102 0.3671 ? 0.0113 0.9234 ? 0.0040 S1 S2 LF 0.1900 ? 0.0052 0.1867 ? 0.0054 0.2895 ? 0.0137 0.4679 ? 0.0358 0.4722 ? 0.0344 0.7181 ? 0.0419 0.4662 ? 0.0126 0.4676 ? 0.0130 0.5595 ? 0.0181 7 stim. deep recon. from: eigen. recon. from: identity recon. from: model brain 1 brain 2 model brain 1 brain 2 model brain 1 brain 2 Figure 5: Reconstructions from the latent features and brain responses of the two subjects (brain 1 and brain 2) using our decoding approach, as well as the eigenface and identity transform approaches for comparison. 3.7 Factors contributing to reconstruction accuracy Finally, we investigated the factors contributing to the quality of reconstructions from brain responses. All of the faces in the test set had been annotated with 30 objective physical measures (such as nose width, face length, etc.) and 14 subjective measures (such as attractiveness, gender, ethnicity, etc.). Among these measures, we identified five subjective measures that are important for face perception [59?64] as measures of interest and supplemented them with an additional measure of stimulus complexity. Complexity was included because of its important role in visual perception [65]. The selected measures were attractiveness, complexity, ethnicity, femininity, masculinity and prototypicality. Note that the complexity measure was not part of the dataset annotations and was defined as the Kolmogorov complexity of the stimuli, which was taken to be their compressed file sizes [66]. To this end, we correlated the reconstruction accuracies of the 48 stimuli in the test set (for both subjects) with their corresponding measures (except for ethnicity) and used a two-tailed Student?s t-test to test if the multiple comparison corrected (Bonferroni correction) p-value was less than the critical value of 0.05. In the case of ethnicity we used one-way analysis of variance to compare the reconstruction accuracies of faces with different ethnicities. We were able to reject the null hypothesis for the measures complexity, femininity and masculinity, but failed to do so for attractiveness, ethnicity and prototypicality. Specifically, we observed a significant negative correlation (r = -0.3067) between stimulus complexity and reconstruction accuracy. Furthermore, we found that masculinity and reconstruction accuracy were significantly positively correlated (r = 0.3841). Complementing this result, we found a negative correlation (r = -0.3961) between femininity and reconstruction accuracy. We found no effect of attractiveness, ethnicity and prototypicality on the quality of reconstructions. We then compared the complexity levels of the images of each gender and found that female face images were significantly more complex than male face images (p < 0.05, Student?s t-test), pointing to complexity as the factor underlying the relationship between reconstruction accuracy and gender. This result demonstrates the importance of taking stimulus complexity into account while making inferences about factors driving the reconstructions from brain responses. 4 Conclusion In this study we combined probabilistic inference with deep learning to derive a novel deep neural decoding approach. We tested our approach by reconstructing face stimuli from BOLD responses at an unprecedented level of accuracy and detail, matching the target stimuli in several key aspects such as gender, skin color and facial features as well as identifying perceptual factors contributing to the reconstruction accuracy. Deep decoding approaches such as the one developed here are expected to play an important role in the development of new neuroprosthetic devices that operate by reading subjective information from the human brain. 8 Acknowledgments This work has been partially supported by a VIDI grant (639.072.513) from the Netherlands Organization for Scientific Research and a GPU grant (GeForce Titan X) from the Nvidia Corporation. References [1] T. Naselaris, K. N. Kay, S. Nishimoto, and J. L. Gallant, ?Encoding and decoding in fMRI,? NeuroImage, vol. 56, no. 2, pp. 400?410, may 2011. [2] M. van Gerven, ?A primer on encoding models in sensory neuroscience,? J. Math. Psychol., vol. 76, no. 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Efficient Use of Limited-Memory Accelerators for Linear Learning on Heterogeneous Systems ? Celestine Dunner IBM Research - Zurich Switzerland [email protected] Thomas Parnell IBM Research - Zurich Switzerland [email protected] Martin Jaggi EPFL Switzerland [email protected] Abstract We propose a generic algorithmic building block to accelerate training of machine learning models on heterogeneous compute systems. Our scheme allows to efficiently employ compute accelerators such as GPUs and FPGAs for the training of large-scale machine learning models, when the training data exceeds their memory capacity. Also, it provides adaptivity to any system?s memory hierarchy in terms of size and processing speed. Our technique is built upon novel theoretical insights regarding primal-dual coordinate methods, and uses duality gap information to dynamically decide which part of the data should be made available for fast processing. To illustrate the power of our approach we demonstrate its performance for training of generalized linear models on a large-scale dataset exceeding the memory size of a modern GPU, showing an order-of-magnitude speedup over existing approaches. 1 Introduction As modern compute systems rapidly increase in size, complexity and computational power, they become less homogeneous. Today?s systems exhibit strong heterogeneity at many levels: in terms of compute parallelism, memory size and access bandwidth, as well as communication bandwidth between compute nodes (e.g., computers, mobile phones, server racks, GPUs, FPGAs, storage nodes etc.). This increasing heterogeneity of compute environments is posing new challenges for the development of efficient distributed algorithms. That is to optimally exploit individual compute resources with very diverse characteristics without suffering from the I/O cost of exchanging data between them. In this paper, we focus on the task of training large scale machine learning models in such heterogeneous compute enUnit B Unit A vironments and propose a new generic algorithmic building ???? ? block to efficiently distribute the workload between heterogeneous compute units. Assume two compute units, denoted A and B, which differ in compute power as well as memory capacity as illustrated in Figure 1. The computational power of unit A is smaller and its memory capacity is larger relative to its peer unit B (i.e., we assume that the training data fits into the memory of A, but not into B?s). Hence, on the compu- Figure 1: Compute units A, B with tationally more powerful unit B, only part of the data can be different memory size, bandwidth processed at any given time. The two units, A and B, are able and compute power. to communicate with each other over some interface, however there is cost associated with doing so. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. This generic setup covers many essential elements of modern machine learning systems. A typical example is that of accelerator units, such as a GPUs or FPGAs, augmenting traditional computers or servers. While such devices can offer a significant increase in computational power due to their massively parallel architectures, their memory capacity is typically very limited. Another example can be found in hierarchical memory systems where data in the higher level memory can be accessed and hence processed faster than data in the ? typically larger ? lower level memory. Such memory systems are spanning from, e.g., fast on-chip caches on one extreme to slower hard drives on the other extreme. The core question we address in this paper is the following: How can we efficiently distribute the workload between heterogeneous units A and B in order to accelerate large scale learning? The generic algorithmic building block we propose systematically splits the overall problem into two workloads, a more data-intensive but less compute-intensive part for unit A and a more computeintensive but less data-intensive part for B. These workloads are then executed in parallel, enabling full utilization of both resources while keeping the amount of necessary communication between the two units minimal. Such a generic algorithmic building block is useful much more widely than just for training on two heterogeneous compute units ? it can serve as a component of larger training algorithms or pipelines thereof. In a distributed training setting, our scheme allows each individual node to locally benefit from its own accelerator, therefore speeding up the overall task on a cluster, e.g., as part of [14] or another distributed algorithm. Orthogonal to such a horizontal application, our scheme can also be used as a building block vertically integrated in a system, serving the efficiency of several levels of the memory hierarchy of a given compute node. Related Work. The most popular existing approach to deal with memory limitations is to process data in batches. For example, for the special case of SVMs, [15] splits data samples into blocks which are then loaded and processed sequentially (on B), in the setting of limited RAM and the full data residing on disk. This approach enables contiguous chunks of data to be loaded which is beneficial in terms of I/O overhead; it however treats samples uniformly. Later, in [2, 7] it is proposed to selectively load and keep informative samples in memory in order to reduce disk access, but this approach is specific to support vectors and is unable to theoretically quantify the possible speedup. In this work, we propose a novel, theoretically-justified scheme to efficiently deal with memory limitations in the heterogeneous two-unit setting illustrated in Figure 1. Our scheme can be applied to a broad class of machine learning problems, including generalized linear models, empirical risk minimization problems with a strongly convex regularizer, such as SVM, as well as sparse models, such as Lasso. In contrast to the related line of research [15, 2, 7], our scheme is designed to take full advantage of both compute resources A and B for training, by systematically splitting the workload among A and B in order to adapt to their specific properties and to the available bandwidth between them. At the heart of our approach lies a smart data selection scheme using coordinate-wise duality gaps as selection criteria. Our theory will show that our selection scheme provably improves the convergence rate of training overall, by explicitly quantifying the benefit over uniform sampling. In contrast, existing work [2, 7] only showed that the linear convergence rate on SVMs is preserved asymptotically, but not necessarily improved. A different line of related research is steepest coordinate selection. It is known that steepest coordinate descent can converge much faster than uniform [8] for single coordinate updates on smooth objectives, however it typically does not perform well for general convex problems, such as those with L1 regularization. In our work, we overcome this issue by using the generalized primal-dual gaps [4] which do extend to L1 problems. Related to this notion, [3, 9, 11] have explored the use of similar information as an adaptive measure of importance, in order to adapt the sampling probabilities of coordinate descent. Both this line of research as well as steepest coordinate descent [8] are still limited to single coordinate updates, and cannot be readily extended to arbitrary accuracy updates on a larger subset of coordinates (performed per communication round) as required in our heterogeneous setting. Contributions. The main contributions of this work are summarized as follows: ? We analyze the per-iteration-improvement of primal-dual block coordinate descent and how it depends on the selection of the active coordinate block at that iteration. We extend the convergence theory to arbitrary approximate updates on the coordinate subsets, and propose a novel dynamic selection scheme for blocks of coordinates, which relies on coordinate-wise duality gaps, and we precisely quantify the speedup of the convergence rate over uniform sampling. 2 ? Our theoretical findings result in a scheme for learning in heterogeneous compute environments which is easy to use, theoretically justified and versatile in that it can be adapted to given resource constraints, such as memory, computation and communication. Furthermore our scheme enables parallel execution between, and also within, two heterogeneous compute units. ? For the example of joint training in a CPU plus GPU environment ? which is very challenging for data-intensive work loads ? we demonstrate a more than 10? speed-up over existing methods for limited-memory training. 2 Learning Problem For the scope of this work we focus on the training of convex generalized linear models of the form min ??Rn O(?) := f (A?) + g(?) (1) P where f is a smooth function and g(?) = i gi (?i ) is separable, ? ? Rn describes the parameter vector and A = [a1 , a2 , . . . , an ] ? Rd?n the data matrix with column vectors ai ? Rd . This setting covers many prominent machine learning problems, including generalized linear models as used for regression, classification and feature selection. To avoid confusion, it is important to distinguish the two main application classes: On one hand, we cover empirical risk minimization (ERM) problems with a strongly convex regularizer such as L2 -regularized SVM ? where ? then is the dual variable vector and f is the smooth regularizer conjugate, as in SDCA [13]. On the other hand, we also cover the class of sparse models such as Lasso or ERM with a sparse regularizer ? where f is the data-fit term and g takes the role of the non-smooth regularizer, so ? are the original primal parameters. Duality Gap. Through the perspective of Fenchel-Rockafellar duality, one can, for any primaldual solution pair (?, w), define the non-negative duality gap for (1) as gap(?; w) := f (A?) + g(?) + f ? (w) + g ? (?A> w) (2) where the functions f ? , g ? in (2) are defined as the convex conjugate1 of their corresponding counterparts f, g [1]. Let us consider parameters w that are optimal relative to a given ?, i.e., w := w(?) = ?f (A?), (3) which implies f (A?) + f ? (w) = hA?, wi. In this special case, the duality gap (2) simplifies and becomes separable over the columns ai of A and the corresponding parameter weights ?i given w. We will later exploit this property to quantify the suboptimality of individual coordinates. X gap(?) = gapi (?i ), where gapi (?i ) := w> ai ?i + gi (?i ) + gi? (?a> (4) i w). i?[n] Notation. For the remainder of the paper we use v[P] to denote a vector v with non-zero entries only for the coordinates i ? P ? [n] = {1, . . . , n}. Similarly we write A[P] to denote the matrix A composing only of columns indexed by i ? P. 3 Approximate Block Coordinate Descent The theory we present in this section serves to derive a theoretical framework for our heterogeneous learning scheme presented in Section 4. Therefore, let us consider the generic block minimization scheme described in Algorithm 1 to train generalized linear models of the form (1). 3.1 Algorithm Description In every round t, of Algorithm 1, a block P of m coordinates of ? is selected according to an arbitrary selection rule. Then, an update is computed on this block of coordinates by optimizing arg min ??[P] ?Rn O(? + ??[P] ) (5) where an arbitrary solver can be used to find this update. This update is not necessarily perfectly optimal but of a relative accuracy ?, in the following sense of approximation quality: 1 For h : Rd ? R the convex conjugate is defined as h? (v) := supu?Rd v> u ? h(u). 3 Algorithm 1 Approximate Block CD Algorithm 2 D U HL 1: Initialize ?(0) := 0, z := 0 2: for t = 0, 1, 2, ... 3: determine P according to (13) 4: refresh memory B to contain A[P] . 5: on B do: 6: ??[P] ? ?-approx. solution to (12) 7: in parallel on A do: 8: while B not finished 9: sample j ? [n] (t) 10: update zj := gapj (?j ) 11: ?(t+1) := ?(t) + ??[P] (0) 1: Initialize ? := 0 2: for t = 0, 1, 2, ... do 3: select a subset P with |P| = m 4: ??[P] ? ?-approx. solution to (5) 5: ?(t+1) := ?(t) + ??[P] 6: end for Definition 1 (?-Approximate Update). The block update ??[P] is ?-approximate iff ?? ? [0, 1] : O(? + ??[P] ) ? ?O(? + ???[P] ) + (1 ? ?)O(?) where 3.2 ???[P] (6) ? arg min??[P] ?Rn O(? + ??[P] ). Convergence Analysis In order to derive a precise convergence rate for Algorithm 1 we build on the convergence analysis of [4, 13]. We extend their analysis of stochastic coordinate descent in two ways: 1) to a block coordinate scheme with approximate coordinate updates, and 2) to explicitly cover the importance of each selected coordinate, as opposed to uniform sampling. We define ?t,P := P (t) 1 j?P gapj (?j ) m P (t) 1 j?[n] gapj (?j ) n (7) which quantifies how much the coordinates i ? P of ?(t) contribute to the global duality gap (2). Thus giving a measure of suboptimality for these coordinates. In Algorithm 1 an arbitrary selection scheme (deterministic or randomized) can be applied and our theory will explain how the convergence of Algorithm 1 depends on the selection through the distribution of ?t,P . That is, for strongly convex functions gi , we found that the per-step improvement in suboptimality is proportional to ?t,P of the specific coordinate block P being selected at that iteration t: (t+1) ? (1 ? ?t,P ?c) (t) (8) where (t) := O(?(t) ) ? O(?? ) measures the suboptimality of ?(t) and c > 0 is a constant which will be specified in the following theorem. A similar dependency on ?t,P can also be shown for non-strongly convex functions gi , leading to our two main convergence results for Algorithm 1: Theorem 1. For Algorithm 1 running on (1) where f is L-smooth and gi is ?-strongly convex with ? > 0 for all i ? [n], it holds that  t m ? (t) (0) EP [ | ? ] ? 1 ? ?P (0) (9) n ?L + ? where ? := kA[P] k2op and ?P := mint ? EP [?t,P | ?(t) ]. Expectations are over the choice of P. That is, for strongly convex gi , Algorithm 1 has a linear convergence rate. This was shown before in [13, 4] for the special case of exact coordinate updates. In strong contrast to earlier coordinate descent analyses which build on random uniform sampling, our theory explicitly quantifies the impact of the sampling scheme on the convergence through ?t,P . This allows one to benefit from smart selection and provably improve the convergence rate by taking advantage of the inhomogeneity of the duality gaps. The same holds for non-strongly convex functions gi : 4 Theorem 2. For Algorithm 1 running on (1) where f is L-smooth and gi has B-bounded support for all i ? [n], it holds that 1 2?n2 EP [(t) | ?(0) ] ? (10) ?P m 2n + t ? t0  n (0)  with ? := 2LB 2 ? where ? := kA[P] k2op and t ? t0 = max 0, m log 2?m where ?P := n? mint ? EP [?t,P | ?(t) ]. Expectations are over the choice of P. Remark 1. Note that for uniform selection, our proven convergence rates for Algorithm 1 recover classical primal-dual coordinate descent [4, 13] as a special case, where in every iteration a single coordinate is selected and each update is solved exactly, i.e., ? = 1. In this case ?t,P measures the contribution of a single coordinate to the duality gap. For uniform sampling, EP [?t,P | ?(t) ] = 1 and hence ?P = 1 which recovers [4, Theorems 8 and 9]. 3.3 Gap-Selection Scheme The convergence results of Theorems 1 and 2 suggest that the optimal rule for selecting the block of coordinates P in step 3 of Algorithm 1, leading to the largest improvement in that step, is the following: X (t)  P := arg max (11) gapj ?j . P?[n]:|P|=m j?P This scheme maximizes ?t,P at every iterate. Furthermore, the selection scheme (11) guarantees ?t,P ? 1 which quantifies the relative gain over random uniform sampling. In contrast to existing importance sampling schemes [16, 12, 5] which assign static probabilities to individual coordinates, our selection scheme (11) is dynamic and adapts to the current state ?(t) of the algorithm, similar to that used in [9, 11] in the standard non-heterogeneous setting. 4 Heterogeneous Training In this section we build on the theoretical insight of the previous section to tackle the main objective of this work: How can we efficiently distribute the workload between two heterogeneous compute units A and B to train a large-scale machine learning problem where A and B fulfill the following two assumptions: Assumption 1 (Difference in Memory Capacity). Compute unit A can fit the whole dataset in its memory and compute unit B can only fit a subset of the data. Hence, B only has access to A[P] , a subset P of m columns of A, where m is determined by the memory size of B. Assumption 2 (Difference in Computational Power). Compute unit B can access and process data faster than compute unit A. 4.1 D U HL: A Duality Gap-Based Heterogeneous Learning Scheme We propose a duality gap-based heterogeneous learning scheme, henceforth referring to as D U HL, for short. D U HL is designed for efficient training on heterogeneous compute resources as described above. The core idea of D U HL is to identify a block P of coordinates which are most relevant to improving the model at the current stage of the algorithm, and have the corresponding data columns, A[P] , residing locally in the memory of B. Compute unit B can then exploit its superior compute power by using an appropriate solver to locally find a block coordinate update ??[P] . At the same time, compute unit A, is assigned the task of updating the block P of important coordinates as the algorithm proceeds and the iterates change. Through this split of workloads D U HL enables full utilization of both compute units A and B. Our scheme, summarized in Algorithm 2, fits the theoretical framework established in the previous section and can be viewed as an instance of Algorithm 1, implementing a time-delayed version of the duality gap-based selection scheme (11). Local Subproblem. In the heterogeneous setting compute unit B only has access to its local data A[P] and some current state v := A? ? Rd in order to compute a block update ??[P] in Step 4 of Algorithm 1. While for quadratic functions f this information is sufficient to optimize (5), for non-quadratic functions f we consider the following modified local optimization problem instead: X L arg min f (v) + h?f (v), A??[P] i + kA??[P] k22 + gi ((? + ??[P] )i ). (12) 2 ??[P] ?Rn i?P 5 Figure 2: Illustration of one round of DUHL as described in Algorithm 2. It can be shown that the convergence guarantees of Theorems 1 and 2 similarly hold if the block coordinate update in Step 4 of Algorithm 1 is computed on (12) instead of (5) (see Appendix C for more details). A Time-Delayed Gap Measure. Motivated by our theoretical findings, we use the duality gap as a measure of importance for selecting which coordinates unit B is working on. However, a scheme as suggested in (11) is not suitable for our purpose since it requires knowledge of the duality gaps (4) for every coordinate i at a given iterate ?(t) . For our scheme this would imply a computationally expensive selection step at the beginning of every round which has to be performed in sequence to the update step. To overcome this and enable parallel execution of the two workloads on A and B, we propose to introduce a gap memory. This is an n-dimensional vector z where zi measures the (t0 ) importance of coordinate ?i . We have zi := gap(?i ) where t0 ? [0, t] and the different elements 0 of z are allowed to be based on different, possibly stale iterates ?(t ) . Thus, the entries of z can be continuously updated during the course of the algorithm. Then, at the beginning of every round the new block P is chosen based on the current state of z as follows: X zj . (13) P := arg max P?[n]:|P|=m j?P In DUHL, keeping z up to date is the job of compute unit A. Hence, while B is computing a block coordinate update ??[P] , A updates z by randomly sampling from the entire training data. Then, as soon as B is done, the current state of z is used to determine P for the next round and data columns on B are replaced if necessary. The parallel execution of the two workloads during a single round of DUHL is illustrated in Figure 2. Note, that the freshness of the gap-memory z depends on the relative compute power of A versus B, as well as ? which controls the amount of time spent computing on unit B in every round. In Section 5.2 we will experimentally investigate the effect of staleness of the values zi on the convergence behavior of our scheme. 5 Experimental Results For our experiments we have implemented D U HL for the particular use-case where A corresponds to a CPU with attached RAM and B corresponds to a GPU ? A and B communicate over the PCIe bus. We use an 8-core Intel Xeon E5 x86 CPU with 64GB of RAM which is connected over PCIe Gen3 to an NVIDIA Quadro M4000 GPU which has 8GB of RAM. GPUs have recently experience a widespread adoption in machine learning systems and thus this hardware scenario is timely and highly relevant. In such a setting we wish to apply D U HL to efficiently populate the GPU memory and thereby making this part of the data available for fast processing. GPU solver. In order to benefit from the enormous parallelism offered by GPUs and fulfill Assumption 2, we need a local solver capable of exploiting the power of the GPU. Therefore, we have chosen to implement the twice parallel, asynchronous version of stochastic coordinate descent 6 (a) (b) (a) Figure 3: Validation of faster convergence: (a) theoretical quantity ?t,P (orange), versus the practically observed speedup (green) ? both relative to the random scheme baseline, (b) convergence of gap selection compared to random selection. (b) Figure 4: Effect of stale entries in the gap memory of D U HL: (a) number of rounds needed to reach suboptimality 10?4 for different update frequencies compared to o-D U HL, (b) the number of data columns that are replaced per round for update frequency of 5%. (TPA-SCD) that has been proposed in [10] for solving ridge regression. In this work we have generalized the implementation further so that it can be applied in a similar manner to solve the Lasso, as well as the SVM problem. For more details about the algorithm and how to generalize it we refer the reader to Appendix D. 5.1 Algorithm Behavior Firstly, we will use the publicly available epsilon dataset from the LIBSVM website (a fully dense dataset with 400?000 samples and 2?000 features) to study the convergence behavior of our scheme. For the experiments in this section we assume that the GPU fits 25% of the training data, i.e., m = n4 and show results for training the sparse Lasso as well as the ridge regression model. For the Lasso case we have chosen the regularizer to obtain a support size of ? 12% and we apply the coordinatewise Lipschitzing trick [4] to the L1 -regularizer in order to allow the computation of the duality gaps. For computational details we refer the reader to Appendix E. Validation of Faster Convergence. From our theory in Section 3.2 we expect that during any given round t of Algorithm 1, the relative gain in convergence rate of one sampling scheme over the other should be quantified by the ratio of the corresponding values of ?t,P := ??t,P (for the respective block of coordinates processed in this round). To verify this, we trained a ridge regression model on the epsilon dataset implementing a) the gap-based selection scheme, (11), and b) random selection, fixing ? for both schemes. Then, in every round t of our experiment, we record the value of ?t,P as defined in (7) and measure the relative gain in convergence rate of the gap-based scheme over the random scheme. In Figure 3(a) we plot the effective speedup of our scheme, and observe that this speedup almost perfectly matches the improvement predicted by our theory as measured by ?t,P - we observe a relative measurement error of 0.42. Both speedup numbers are calculated relative to plain random selection. In Figure 3(b) we see that the gap-based selection can achieve a remarkable 10? improvement in convergence over the random reference scheme. When running on sparse problems instead of ridge regression, we have observed ?t,P of the oracle scheme converging n to m within only a few iterations if the support of the problem is smaller than m and fits on the GPU. Effect of Gap-Approximation. In this section we study the effect of using stale, inconsistent gapmemory entries for selection on the convergence of D U HL. While the freshness of the memory entries is, in reality, determined by the relative compute power of unit B over unit A and the relative accuracy ?, in this experiment we artificially vary the number of gap updates performed during each round while keeping ? fixed. We train the Lasso model and show, in Figure 4(a), the number of rounds needed to reach a suboptimality of 10?4 , as a function of the number of gap entries updated per round. As a reference we show o-D U HL which has access to an oracle providing the true duality gaps. We observe that our scheme is quite robust to stale gap values and can achieve performance within a factor of two over the oracle scheme up to an average delay of 20 iterations. As the update frequency decreases we observed that the convergence slows down in the initial rounds because the algorithm needs more rounds until the active set of the sparse problem is correctly detected. 7 (d) Lasso (e) SVM (f) ridge regression Figure 5: Performance results of DUHL on the 30GB ImageNet dataset. I/O cost (top) and convergence behavior (bottom) for Lasso, SVM and ridge regression. Reduced I/O operations. The efficiency of our scheme regarding I/O operations is demonstrated in Figure 4(b), where we plot the number of data columns that are replaced on B in every round of Algorithm 2. Here the Lasso model is trained assuming a gap update frequency of 5%. We observe that the number of required I/O operations of our scheme is decreasing over the course of the algorithm. When increasing the freshness of the gap memory entries we could see the number of swaps go to zero faster. 5.2 Reference Schemes In the following we compare the performance of our scheme against four reference schemes. We compare against the most widely-used scheme for using a GPU to accelerate training when the data does not fit into the memory of the GPU, that is the sequential block selection scheme presented in [15]. Here the data columns are split into blocks of size m which are sequentially put on the GPU and operated on (the data is efficiently copied to the GPU as a contiguous memory block). We also compare against importance sampling as presented in [16], which we refer to as IS. Since probabilities assigned to individual data columns are static we cannot use them as importance measures in a deterministic selection scheme. Therefore, in order to apply importance sampling in the heterogeneous setting, we non-uniformly sample m data-columns to reside inside the GPU memory in every round of Algorithm 2 and have the CPU determine the new set in parallel. As we will see, data column norms often come with only small variance, in particular for dense datasets. Therefore, importance sampling often fails to give a significant gain over uniformly random selection. Additionally, we compare against a single-threaded CPU implementation of a stochastic coordinate descent solver to demonstrate that with our scheme, the use of a GPU in such a setting indeed yields a significant speedup over a basic CPU implementation despite the high I/O cost of repeatedly copying data on and off the GPU memory. To the best of our knowledge, we are the first to demonstrate this. For all competing schemes, we use TPA-SCD as the solver to efficiently compute the block update ??[P] on the GPU. The accuracy ? of the block update computed in every round is controlled by the number of randomized passes of TPA-SCD through the coordinates of the selected block P. For a fair comparison we optimize this parameter for the individual schemes. 5.3 Performance Analysis of DUHL For our large-scale experiments we use an extended version of the Kaggle Dogs vs. Cats ImageNet dataset as presented in [6], where we additionally double the number of samples, while using single precision floating point numbers. The resulting dataset is fully dense and consists of 40?000 samples and 200?704 features, resulting in over 8 billion non-zero elements and a data size of 30GB. Since the memory capacity of our GPU is 8GB, we can put ? 25% of the data on the GPU. We will show 8 results for training a sparse Lasso model, ridge regression as well as linear L2 -regularized SVM. For Lasso we choose the regularization to achieve a support size of 12%, whereas for SVM the regularizer was chosen through cross-validation. For all three tasks, we compare the performance of D U HL to sequential block selection, random selection, selection through importance sampling (IS) all on GPU, as well as a single-threaded CPU implementation. In Figure 5(d) and 5(e) we demonstrate that for Lasso as well as SVM, D U HL converges 10? faster than any reference scheme. This gain is achieved by improved convergence ? quantified through ?t,P ? as well as through reduced I/O cost, as illustrated in the top plots of Figure 5, which show the number of data columns replaced per round. The results in Figure 5(f) show that the application of D U HL is not limited to sparse problems and SVMs. Even for ridge regression D U HL significantly outperforms all the reference scheme considered in this study. 6 Conclusion We have presented a novel theoretical analysis of block coordinate descent, highlighting how the performance depends on the coordinate selection. These results prove that the contribution of individual coordinates to the overall duality gap is indicative of their relevance to the overall model optimization. Using this measure we develop a generic scheme for efficient training in the presence of high performance resources of limited memory capacity. We propose D U HL, an efficient gap memory-based strategy to select which part of the data to make available for fast processing. On a large dataset which exceeds the capacity of a modern GPU, we demonstrate that our scheme outperforms existing sequential approaches by over 10? for Lasso and SVM models. Our results show that the practical gain matches the improved convergence predicted by our theory for gap-based sampling under the given memory and communication constraints, highlighting the versatility of the approach. References [1] Heinz H Bauschke and Patrick L Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer New York, New York, NY, 2011. [2] Kai-Wei Chang and Dan Roth. Selective block minimization for faster convergence of limited memory large-scale linear models. In Proceedings of the 17th ACM SIGKDD international conference on Knowledge Discovery and Data Mining, pages 699?707, New York, USA, August 2011. ACM. [3] Dominik Csiba, Zheng Qu, and Peter Richt?arik. Stochastic Dual Coordinate Ascent with Adaptive Probabilities. In ICML 2015 - Proceedings of the 32th International Conference on Machine Learning, February 2015. [4] Celestine D?unner, Simone Forte, Martin Tak?ac, and Martin Jaggi. Primal-Dual Rates and Certificates. In Proceedings of the 33th International Conference on Machine Learning (ICML) - Volume 48, pages 783?792, 2016. [5] Olivier Fercoq and Peter Richt?arik. Optimization in High Dimensions via Accelerated, Parallel, and Proximal Coordinate Descent. SIAM Review, 58(4):739?771, January 2016. [6] Christina Heinze, Brian McWilliams, and Nicolai Meinshausen. DUAL-LOCO: Distributing Statistical Estimation Using Random Projections. In AISTATS - Proceedings of the th International Conference on Artificial Intelligence and Statistics, pages 875?883, 2016. [7] Shin Matsushima, SVN Vishwanathan, and Alex J Smola. Linear support vector machines via dual cached loops. In Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 177?185, New York, USA, 2012. ACM Press. [8] Julie Nutini, Mark Schmidt, Issam Laradji, Michael Friedlander, and Hoyt Koepke. Coordinate Descent Converges Faster with the Gauss-Southwell Rule Than Random Selection. In ICML 2015 - Proceedings of the 32th International Conference on Machine Learning, pages 1632?1641, 2015. [9] Anton Osokin, Jean-Baptiste Alayrac, Isabella Lukasewitz, Puneet K. Dokania, and Simon LacosteJulien. Minding the gaps for block frank-wolfe optimization of structured svms. In Proceedings of the 33rd International Conference on Machine Learning (ICML) - Volume 48, pages 593?602. JMLR.org, 2016. [10] Thomas Parnell, Celestine D?unner, Kubilay Atasu, Manolis Sifalakis, and Haris Pozidis. Large-Scale Stochastic Learning using GPUs. In Proceedings of the 6th International Workshop on Parallel and Distributed Computing for Large Scale Machine Learning and Big Data Analytics (IPDPSW), IEEE, 2017. 9 [11] Dmytro Perekrestenko, Volkan Cevher, and Martin Jaggi. Faster Coordinate Descent via Adaptive Importance Sampling. In AISTATS - Artificial Intelligence and Statistics, pages 869?877. April 2017. [12] Zheng Qu and Peter Richt?arik. Coordinate descent with arbitrary sampling I: algorithms and complexity. Optimization Methods and Software, 31(5):829?857, April 2016. [13] Shai Shalev-Shwartz and Tong Zhang. Stochastic dual coordinate ascent methods for regularized loss. J. Mach. Learn. Res., 14(1):567?599, February 2013. [14] Virginia Smith, Simone Forte, Chenxin Ma, Martin Tak?ac? , Michael I Jordan, and Martin Jaggi. CoCoA: A General Framework for Communication-Efficient Distributed Optimization. arXiv, November 2016. [15] Hsiang-Fu Yu, Cho-Jui Hsieh, Kai-Wei Chang, and Chih-Jen Lin. Large Linear Classification When Data Cannot Fit in Memory. ACM Transactions on Knowledge Discovery from Data, 5(4):1?23, February 2012. [16] Peilin Zhao and Tong Zhang. Stochastic Optimization with Importance Sampling for Regularized Loss Minimization. In ICML 2015 - Proceedings of the 32th International Conference on Machine Learning, pages 1?9, 2015. 10
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Temporal Coherency based Criteria for Predicting Video Frames using Deep Multi-stage Generative Adversarial Networks Prateep Bhattacharjee1 , Sukhendu Das2 Visualization and Perception Laboratory Department of Computer Science and Engineering Indian Institute of Technology Madras, Chennai, India 1 [email protected], 2 [email protected] Abstract Predicting the future from a sequence of video frames has been recently a sought after yet challenging task in the field of computer vision and machine learning. Although there have been efforts for tracking using motion trajectories and flow features, the complex problem of generating unseen frames has not been studied extensively. In this paper, we deal with this problem using convolutional models within a multi-stage Generative Adversarial Networks (GAN) framework. The proposed method uses two stages of GANs to generate crisp and clear set of future frames. Although GANs have been used in the past for predicting the future, none of the works consider the relation between subsequent frames in the temporal dimension. Our main contribution lies in formulating two objective functions based on the Normalized Cross Correlation (NCC) and the Pairwise Contrastive Divergence (PCD) for solving this problem. This method, coupled with the traditional L1 loss, has been experimented with three real-world video datasets viz. Sports-1M, UCF-101 and the KITTI. Performance analysis reveals superior results over the recent state-of-the-art methods. 1 Introduction Video frame prediction has recently been a popular problem in computer vision as it caters to a wide range of applications including self-driving cars, surveillance, robotics and in-painting. However, the challenge lies in the fact that, real-world scenes tend to be complex, and predicting the future events requires modeling of complicated internal representations of the ongoing events. Past approaches on video frame prediction include the use of recurrent neural architectures [19], Long Short Term Memory [8] networks [22] and action conditional deep networks [17]. Recently, the work of [14] modeled the frame prediction problem in the framework of Generative Adversarial Networks (GAN). Generative models, as introduced by Goodfellow et. al., [5] try to generate images from random noise by simultaneously training a generator (G) and a discriminator network (D) in a process similar to a zero-sum game. Mathieu et. al. [14] shows the effectiveness of this adversarial training in the domain of frame prediction using a combination of two objective functions (along with the basic adversarial loss) employed on a multi-scale generator network. This idea stems from the fact that the original L2-loss tends to produce blurry frames. This was overcome by the use of Gradient Difference Loss (GDL) [14], which showed significant improvement over the past approaches when compared using similarity and sharpness measures. However, this approach, although producing satisfying results for the first few predicted frames, tends to generate blurry results for predictions far away (?6) in the future. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Figure 1: The proposed multi-stage GAN framework. The stage-1 generator network produces a low-resolution version of predicted frames which are then fed to the stage-2 generator. Discriminators at both the stages predict 0 or 1 for each predicted frame to denote its origin: synthetic or original. In this paper, we aim to get over this hurdle of blurry predictions by considering an additional constraint between consecutive frames in the temporal dimension. We propose two objective functions: (a) Normalized Cross-Correlation Loss (NCCL) and (b) Pairwise Contrastive Divergence Loss (PCDL) for effectively capturing the inter-frame relationships in the GAN framework. NCCL maximizes the cross-correlation between neighborhood patches from consecutive frames, whereas, PCDL applies a penalty when subsequent generated frames are predicted wrongly by the discriminator network (D), thereby separating them far apart in the feature space. Performance analysis over three real world video datasets shows the effectiveness of the proposed loss functions in predicting future frames of a video. The rest of the paper is organized as follows: section 2 describes the multi-stage generative adversarial architecture; sections 3 - 6 introduce the different loss functions employed: the adversarial loss (AL) and most importantly NCCL and PCDL. We show the results of our experiments on Sports-1M [10], UCF-101 [21] and KITTI [4] and compare them with state-of-the-art techniques in section 7. Finally, we conclude our paper highlighting the key points and future direction of research in section 8. 2 Multi-stage Generative Adversarial Model Generative Adversarial Networks (GAN) [5] are composed of two networks: (a) the Generator (G) and (b) the Discriminator (D). The generator G tries to generate realistic images by learning to model the true data distribution pdata and thereby trying to make the task of differentiating between original and generated images by the discriminator difficult. The discriminator D, in the other hand, is optimized to distinguish between the synthetic and the real images. In essence, this procedure of alternate learning is similar to the process of two player min-max games [5]. Overall, the GANs minimize the following objective function: min max v(D, G) = Ex?pdata [log(D(x))] + Ez?pz [log(1 ? D(G(z)))] G D (1) where, x is a real image from the true distribution pdata and z is a vector sampled from the distribution pz , usually to be uniform or Gaussian. The adversarial loss employed in this paper is a variant of that in equation 1, as the input to our network is a sequence of frames of a video, instead of a vector z. As convolutions account only for short-range relationships, pooling layers are used to garner information from wider range. But, this process generates low resolution images. To overcome this, Mathieu et. al. [14] uses a multi-scale generator network, equivalent to the reconstruction process of a Laplacian pyramid [18], coupled with discriminator networks to produce high-quality output frames of size 32 ? 32. There are two shortcomings of this approach: 2 a. Generating image output at higher dimensions viz. (128 ? 128) or (256 ? 256), requires multiple use of upsampling operations applied on the output of the generators. In our proposed model, this upsampling is handled by the generator networks itself implicitly through the use of consecutive unpooling operations, thereby generating predicted frames at much higher resolution in lesser number of scales. b. As the generator network parameters are not learned with respect to any objective function which captures the temporal relationship effectively, the output becomes blurry after ? 4 frames. To overcome the first issue, we propose a multi-stage (2-stage) generative adversarial network (MS-GAN). 2.1 Stage-1 Generating the output frame(s) directly often produces blurry outcomes. Instead, we simplify the process by first generating crude, low-resolution version of the frame(s) to be predicted. The stage-1 generator (G1 ) consists of a series of convolutional layers coupled with unpooling layers [25] which upsample the frames. We used ReLU non-linearity in all but the last layer, in which case, hyperbolic tangent (tanh) was used following the scheme of [18]. The inputs to G1 are m number of consecutive frames of dimension W0 ? H0 , whereas the outputs are n predicted frames of size W1 ? H1 , where, W1 = W0 ? 2 and H1 = H0 ? 2. These outputs, stacked with the upsampled version of the original input frames, produce the input of dimension (m + n) ? W1 ? H1 for the stage-1 discriminator (D1 ). D1 applies a chain of convolutional layers followed by multiple fully-connected layers to finally produce an output vector of dimension (m + n), consisting of 0?s and 1?s. One of the key differences of our proposed GAN framework with the conventional one [5]is that, the discriminator network produces decision output for multiple frames, instead of a single 0/1 outcome. This is exploited by one of the proposed objective functions, the PCDL, which is described later in section 4. 2.2 Stage-2 The second stage network closely resembles the stage-1 architecture, differing only in the input and output dimensions. The input to the stage-2 generator (G2 ) is formed by stacking the predicted frames and the upsampled inputs of G1 , thereby having dimension of (m + n) ? W1 ? H1 . The output of G2 are n predicted high-resolution frames of size W2 ? H2 , where, W2 = W1 ? 4 and H2 = H1 ? 4. The stage-2 discriminator (D2 ), works in a similar fashion as D1 , producing an output vector of length (m + n). Effectively, the multi-stage model can be represented by the following recursive equations: ( Gk (Y?k?1 , Xk?1 ), f or k ? 2 Y?k = Gk (Xk?1 ) f or k = 1 (2) where, Y?k is the set of predicted frames and Xk are the input frames at the kth stage of the generator network Gk . 2.3 Training the multi-stage GAN The training procedure of the multi-stage GAN model follows that of the original generative adversarial networks with minor variations. The training of the discriminator and the generator are described as follows: Training of the discriminator Considering the input to the discriminator (D) as X (series of m frames) and the target output to be Y (series of n frames), D is trained to distinguish between synthetic and original inputs by classifying (X, Y ) into class 1 and (X, G(X)) into class 0. Hence, for each of the k stages, we train D with target ~1 (Vector of 1?s with dimension m) for (X, Y ) and 3 target ~0 (Vector of 0?s with dimension n) for (X, G(X)). The loss function for training D is: Nstages LD adv = X Lbce (Dk (Xk , Yk ), ~1) + Lbce (Dk (Xk , Gk (Xk )), ~0) (3) k=1 where, Lbce , the binary cross-entropy loss is defined as: 0 Lbce (A, A ) = ? |A| X 0 0 0 A i log(Ai ) + (1 ? A i )log(1 ? Ai ), Ai ? {0, 1}, A i ? [0, 1] (4) i=1 where, A and A0 are the target and discriminator outputs respectively. Training of the generator We perform an optimization step on the generator network (G), keeping the weights of D fixed, by feeding a set of consecutive frames X sampled from the training data with target Y (set of ground-truth output frames) and minimize the following adversarial loss: Nstages LG adv (X) = X Lbce (Dk (Xk , Gk (Xk )), ~1) (5) k=1 By minimizing the above two loss criteria (eqns. 3, 5), G makes the discriminator believe that, the source of the generated frames is the input data space itself. Although this process of alternate optimization of D and G is reasonably well designed formulation, in practical purposes, this produces an unstable system where G generates samples that consecutively move far away from the original input space and in consequence D distinguishes them easily. To overcome this instability inherent in the GAN principle and the issue of producing blurry frames defined in section 2, we formulate a pair of objective criteria: (a) Normalized Cross Correlation Loss (NCCL) and (b)Pairwise Contrastive Divergence Loss (PCDL), to be used along with the established adversarial loss (refer eqns. 3 and 5). 3 Normalized Cross-Correlation Loss (NCCL) The main advantage of video over image data is the fact that, it offers a far richer space of data distribution by adding the temporal dimension along with the spatial one. Convolutional Neural Networks (CNN) can only capture short-range relationships, a small part of the vast available information, from the input video data, that too in the spatial domain. Although this can be somewhat alleviated by the use of 3D convolutions [9], that increases the number of learn-able parameters immensely. Normalized cross-correlation has been used since long time in the field of video analytics [1, 2, 16, 13, 23] to model the spatial and temporal relationships present in the data. Normalized cross correlation (NCC) measures the similarity of two image patches as a function of the displacement of one relative to the other. This can be mathematically defined as: X (f (x, y) ? ?f )(g(x, y) ? ?g ) (6) N CC(f, g) = ?f ?g x,y where, f (x, y) is a sub-image, g(x, y) is the template to be matched, ?f , ?g denotes the mean of the sub-image and the template respectively and ?f , ?g denotes the standard deviation of f and g respectively. In the domain of video frame(s) prediction, we incorporate the NCC by first extracting small nonoverlapping square patches of size h ? h (1 < h ? 4), denoted by a 3-tuple Pt {x, y, h}, where, x and y are the co-ordinates of the top-left pixel of a particular patch, from the predicted frame at time t and then calculating the cross-correlation score with the patch extracted from the ground truth frame at time (t ? 1), represented by P?t?1 {x ? 2, y ? 2, h + 4}. In simpler terms, we estimate the cross-correlation score between a small portion of the current predicted frame and the local neighborhood of that in the previous ground-truth frame. We assume that, the motion features present in the entire scene (frame) be effectively approximated by adjacent spatial blocks of lower resolution,using small local neighborhoods in the temporal dimension. This stems from the fact that, unless the video contains significant jitter or unexpected random events like 4 Algorithm 1: Normalized cross-correlation score for estimating similarity between a set of predicted frame(s) and a set of ground-truth frame(s). Input: Ground-truth frames (GT ), Predicted frames (P RED) Output: Cross-correlation score (ScoreN CC ) // h = height and width of an image patch // H = height and width of the predicted frames // t = current time // T = Number of frames predicted Initialize: ScoreN CC = 0; for t = 1 to T do for i = 0 to H, i ? i + h do for j = 0 to H, j ? j + h do Pt ? extract_patch(P REDt , i, j, h); /* Extracts a patch from the predicted frame at time t of dimension h ? h starting from the top-left pixel index (i, j) */ P?t?1 ? extract_patch(GTt?1 , i ? 2, j ? 2, h + 4); /* Extracts a patch from the ground-truth frame at time (t ? 1) of dimension (h + 4) ? (h + 4) starting from the top-left pixel index (i ? 2, j ? 2) */ ?Pt ? avg(Pt ); ?P?t?1 ? avg(P?t?1 ); ?Pt ? standard_deviation(Pt ); ?P?t?1 ? standard_deviation(P?t?1 ); P (Pt (x,y)??Pt )(P?t?1 (x,y)??P?t?1 )  ; ScoreN CC ? ScoreN CC + max 0, x,y ?P ? ? t end end ScoreN CC ? ScoreN CC/bH/hc2 ; end ScoreN CC ? ScoreN CC/(T ?1); Pt?1 // Average over all the patches // Average over all the frames scene change, the motion features remain smooth over time. The step-by-step process for finding the cross-correlation score by matching local patches of predicted and ground truth frames is described in algorithm 1. The idea of calculating the NCC score is modeled into an objective function for the generator network G, where it tries to maximize the score over a batch of inputs. In essence, this objective function models the temporal data distribution by smoothing the local motion features generated by the convolutional model. This loss function, LN CCL , is defined as: LN CCL (Y, Y? ) = ?ScoreN CC (Y, Y? ) (7) where, Y and Y? are the ground truth and predicted frames and ScoreN CC is the average normalized cross-correlation score over all the frames, obtained using the method as described in algorithm 1. The generator tries to minimize LN CCL along with the adversarial loss defined in section 2. We also propose a variant of this objective function, termed as Smoothed Normalized CrossCorrelation Loss (SNCCL), where the patch similarity finding logic of NCCL is extended by convolving with Gaussian filters to suppress transient (sudden) motion patterns. A detailed discussion of this algorithm is given in sec. A of the supplementary document. 4 Pairwise Contrastive Divergence Loss (PCDL) As discussed in sec. 3, the proposed method captures motion features that vary slowly over time. The NCCL criteria aims to achieve this using local similarity measures. To complement this in a global scale, we use the idea of pairwise contrastive divergence over the input frames. The idea of exploiting this temporal coherence for learning motion features has been studied in the recent past [6, 7, 15]. 5 By assuming that, motion features vary slowly over time, we describe Y?t and Y?t+1 as a temporal pair, where, Y?i and Y?t+1 are the predicted frames at time t and (t + 1) respectively, if the outputs of the discriminator network D for both these frames are 1. With this notation, we model the slowness principle of the motion features using an objective function as: T ?1 X ? LP CDL (Y , p~) = D? (Y?i , Y?i+1 , pi ? pi+1 ) i=0 = T ?1 X (8) pi ? pi+1 ? d(Y?i , Y?i+1 ) + (1 ? pi ? pi+1 ) ? max(0, ? ? d(Y?i , Y?i+1 )) i=0 where, T is the time-duration of the frames predicted, pi is the output decision (pi ? {0, 1}) of the discriminator, d(x, y) is a distance measure (L2 in this paper) and ? is a positive margin. Equation 8 in simpler terms, minimizes the distance between frames that have been predicted correctly and encourages the distance in the negative case, up-to a margin ?. 5 Higher Order Pairwise Contrastive Divergence Loss The Pairwise Contrastive Divergence Loss (PCDL) discussed in the previous section takes into account (dis)similarities between two consecutive frames to bring them further (or closer) in the spatio-temporal feature space. This idea can be extended for higher order situations involving three or more consecutive frames. For n = 3, where n is the number of consecutive frames considered, PCDL can be defined as: T ?2 X L3?P CDL = D? (|Y?i ? Y?i+1 |, |Y?i+1 ? Y?i+2 |, pi,i+1,i+2 ) i=0 = T ?2 X pi,i+1,i+2 ? d(|Y?i ? Y?i+1 |, |Y?i+1 ? Y?i+2 |) (9) i=0 + (1 ? pi,i+1,i+2 ) ? max(0, ? ? d(|(Y?i ? Y?i+1 )|, |(Y?i+1 ? Y?i+2 )|)) where, pi,i+1,i+2 = 1 only if pi , pi+1 and pi+2 - all are simultaneously 1, i.e., the discriminator is very sure about the predicted frames, that they are from the original data distribution. All the other symbols bear standard representations defined in the paper. This version of the objective function, in essence shrinks the distance between the predicted frames occurring sequentially in a temporal neighborhood, thereby increasing their similarity and maintaining the temporal coherency. 6 Combined Loss Finally, we combine the objective functions given in eqns. 5 - 8 along with the general L1-loss with different weights as: ? LCombined =?adv LG adv (X) + ?L1 LL1 (X, Y ) + ?N CCL LN CCL (Y, Y ) (10) + ?P CDL LP CDL (Y? , p~) + ?P CDL L3?P CDL (Y? , p~) All the weights viz. ?L1 , ?N CCL , ?P CDL and ?3?P CDL have been set as 0.25, while ?adv equals 0.01. This overall loss is minimized during the training stage of the multi-stage GAN using Adam optimizer [11]. We also evaluate our models by incorporating another loss function described in section A of the supplementary document, the Smoothed Normalized Cross-Correlation Loss (SNCCL). The weight for SNCCL, ?SN CCL equals 0.33 while ?3?P CDL and ?P CDL is kept at 0.16. 7 Experiments Performance analysis with experiments of our proposed prediction model for video frame(s) have been done on video clips from Sports-1M [10], UCF-101 [21] and KITTI [4] datasets. The inputoutput configuration used for training the system is as follows: input: 4 frames and output: 4 frames. 6 We compare our results with recent state-of-the-art methods using two popular metrics: (a) Peak Signal to Noise Ratio (PSNR) and (b) Structural Similarity Index Measure (SSIM) [24]. 7.1 Datasets Sports-1M A large collection of sports videos collected from YouTube spread over 487 classes. The main reason for choosing this dataset is the amount of movement in the frames. Being a collection of sports videos, this has sufficient amount of motion present in most of the frames, making it an efficient dataset for training the prediction model. Only this dataset has been used for training all throughout our experimental studies. UCF-101 This dataset contains 13320 annotated videos belonging to 101 classes having 180 frames/video on average. The frames in this video do not contain as much movement as the Sports1m and hence this is used only for testing purpose. KITTI This consists of high-resolution video data from different road conditions. We have taken raw data from two categories: (a) city and (b) road. 7.2 Architecture of the network Table 1: Network architecture details; G and D represents the generator and discriminator networks respectively. U denotes an unpooling operation which upsamples an input by a factor of 2. Network Number of feature maps Stage-1 (G) 64, 128, 256U, 128, 64 Kernel sizes Fully connected 5, 3, 3, 3, 5 N/A Stage-2 (G) 64, 128U, 256, 512U, 256, 128, 64 5, 5, 5, 5, 5, 5, 5 N/A Stage-1 (D) 64, 128, 256 Stage-2 (D) 128, 256, 512, 256, 128 3, 5, 5 1024, 512 7, 5, 5, 5, 5 1024, 512 The architecture details for the generator (G) and discriminator (D) networks used for experimental studies are shown in table 1. All the convolutional layers except the terminal one in both stages of G are followed by ReLU non-linearity. The last layer is tied with tanh activation function. In both the stages of G, we use unpooling layers to upsample the image into higher resolution in magnitude of 2 in both dimensions (height and width). The learning rate is set to 0.003 for G, which is gradually decreased to 0.0004 over time. The discriminator (D) uses ReLU non-linearities and is trained with a learning rate of 0.03. We use mini-batches of 8 clips for training the overall network. 7.3 Evaluation metric for prediction Assessment of the quality of the predicted frames is done by two methods: (a) Peak Signal to Noise Ratio (PSNR) and (b) Structural Similarity Index Measure (SSIM). PSNR measures the quality of the reconstruction process through the calculation of Mean-squared error between the original and the reconstructed signal in logarithmic decibel scale [1]. SSIM is also an image similarity measure where, one of the images being compared is assumed to be of perfect quality [24]. As the frames in videos are composed of foreground and background, and in most cases the background is static (not the case in the KITTI dataset, as it has videos taken from camera mounted on a moving car), we extract random sequences of 32 ? 32 patches from the frames with significant motion. Calculation of motion is done using the optical flow method of Brox et. al. [3]. 7.4 Comparison We compare the results on videos from UCF-101, using the model trained on the Sports-1M dataset. Table 2 demonstrates the superiority of our method over the most recent work [14]. We followed similar choice of test set videos as in [14] to make a fair comparison. One of the impressive facts in our model is that, it can produce acceptably good predictions even in the 4th frame, which is a significant result considering that [14] uses separate smaller multi-scale models for achieving this 7 Figure 2: Qualitative results of using the proposed framework for predicting frames in UCF-101 with the three rows representing (a) Ground-truth, (b) Adv + L1 and (c) Combined (section 6) respectively. ?T? denotes the time-step. Figures in insets show zoomed-in patches for better visibility of areas involving motion (Best viewed in color). feat. Also note that, even though the metrics for the first predicted frame do not differ by a large margin compared to the results from [14] for higher frames, the values decrease much slowly for the models trained with the proposed objective functions (rows 8-10 of table 2). The main reason for this phenomenon in our proposed method is the incorporation of the temporal relations in the objective functions, rather than learning only in the spatial domain. Similar trend was also found in case of the KITTI dataset. We could not find any prior work in the literature reporting findings on the KITTI dataset and hence compared only with several of our proposed models. In all the cases, the performance gain with the inclusion of NCCL and PCDL is evident. Finally, we show the prediction results obtained on both the UCF-101 and KITTI in figures 2 and 3. It is evident from the sub-figures that, our proposed objective functions produce impressive quality frames while the models trained with L1 loss tends to output blurry reconstruction. The supplementary document contains visual results (shown in figures C.1-C.2) obtained in case of predicting frames far-away from the current time-step (8 frames). 8 Conclusion In this paper, we modified the Generative Adversarial Networks (GAN) framework with the use of unpooling operations and introduced two objective functions based on the normalized crosscorrelation (NCCL) and the contrastive divergence estimate (PCDL), to design an efficient algorithm for video frame(s) prediction. Studies show significant improvement of the proposed methods over the recent published works. Our proposed objective functions can be used with more complex networks involving 3D convolutions and recurrent neural networks. In the future, we aim to learn weights for the cross-correlation such that it focuses adaptively on areas involving varying amount of motion. 8 Table 2: Comparison of performance for different methods using PSNR/SSIM scores for the UCF-101 and KITTI datasets. The first five rows report the results from [14]. (*) indicates models fine tuned on patches of size 64 ? 64 [14]. (-) denotes unavailability of data. GDL stands for Gradient Difference Loss [14]. SNCCL is discussed in section A of the supplementary document. Best results in bold. Methods L1 GDL L1 GDL L1* Adv + GDL fine-tuned* Optical flow Next-flow [20] Deep Voxel Flow [12] Adv + NCCL + L1 Combined Combined + SNCCL Combined + SNCCL (full frame) 1st frame prediction score UCF KITTI 28.7/0.88 29.4/0.90 29.9/0.90 32.0/0.92 31.6/0.93 31.9/35.8/0.96 35.4/0.94 37.1/0.91 37.3/0.95 39.7/0.93 38.2/0.95 40.2/0.94 37.3/0.94 39.4/0.94 2nd frame prediction score UCF KITTI 23.8/0.83 24.9/0.84 26.4/0.87 28.9/0.89 28.2/0.90 33.9/0.92 35.4/0.90 35.7/0.92 37.1/0.91 36.8/0.93 37.7/0.91 35.1/0.91 36.4/0.91 4th frame prediction score UCF KITTI 28.7/0.75 27.8/0.75 30.2/0.76 29.6/0.76 30.9/0.77 30.4/0.77 29.5/0.75 29.1/0.76 Figure 3: Qualitative results of using the proposed framework for predicting frames in the KITTI Dataset, for (a) L1, (b) NCCL (section 3), (c) Combined (section 6) and (d) ground-truth (Best viewed in color). 9 References [1] A. C. Bovik. The Essential Guide to Video Processing. Academic Press, 2nd edition, 2009. [2] K. Briechle and U. D. Hanebeck. 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Sobolev Training for Neural Networks Wojciech Marian Czarnecki, Simon Osindero, Max Jaderberg Grzegorz Swirszcz, and Razvan Pascanu DeepMind, London, UK {lejlot,osindero,jaderberg,swirszcz,razp}@google.com Abstract At the heart of deep learning we aim to use neural networks as function approximators ? training them to produce outputs from inputs in emulation of a ground truth function or data creation process. In many cases we only have access to input-output pairs from the ground truth, however it is becoming more common to have access to derivatives of the target output with respect to the input ? for example when the ground truth function is itself a neural network such as in network compression or distillation. Generally these target derivatives are not computed, or are ignored. This paper introduces Sobolev Training for neural networks, which is a method for incorporating these target derivatives in addition the to target values while training. By optimising neural networks to not only approximate the function?s outputs but also the function?s derivatives we encode additional information about the target function within the parameters of the neural network. Thereby we can improve the quality of our predictors, as well as the data-efficiency and generalization capabilities of our learned function approximation. We provide theoretical justifications for such an approach as well as examples of empirical evidence on three distinct domains: regression on classical optimisation datasets, distilling policies of an agent playing Atari, and on large-scale applications of synthetic gradients. In all three domains the use of Sobolev Training, employing target derivatives in addition to target values, results in models with higher accuracy and stronger generalisation. 1 Introduction Deep Neural Networks (DNNs) are one of the main tools of modern machine learning. They are consistently proven to be powerful function approximators, able to model a wide variety of functional forms ? from image recognition [8, 24], through audio synthesis [27], to human-beating policies in the ancient game of GO [22]. In many applications the process of training a neural network consists of receiving a dataset of input-output pairs from a ground truth function, and minimising some loss with respect to the network?s parameters. This loss is usually designed to encourage the network to produce the same output, for a given input, as that from the target ground truth function. Many of the ground truth functions we care about in practice have an unknown analytic form, e.g. because they are the result of a natural physical process, and therefore we only have the observed input-output pairs for supervision. However, there are scenarios where we do know the analytic form and so are able to compute the ground truth gradients (or higher order derivatives), alternatively sometimes these quantities may be simply observable. A common example is when the ground truth function is itself a neural network; for instance this is the case for distillation [9, 20], compressing neural networks [7], and the prediction of synthetic gradients [12]. Additionally, if we are dealing with an environment/data-generation process (vs. a pre-determined set of data points), then even though we may be dealing with a black box we can still approximate derivatives using finite differences. In this work, we consider how this additional information can be incorporated in the learning process, and what advantages it can provide in terms of data efficiency and performance. We 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. D_{\mathbf{x}} f Dx hDx hm, v1 i, v2 i Dx2 m l2 @ @x Dx m @ @x l1 Dx f l @ @x @ @x l1 Dx hf, v1 i @ @x v1 m f Dx hDx hf, v1 i, v2 i v2 mv1 = Dx hm, v1 i @ @x @ @x m @ @x Dx2 f l2 l x x a) b) f Figure 1: a) Sobolev Training of order 2. Diamond nodes m and f indicate parameterised functions, where m is trained to approximate f . Green nodes receive supervision. Solid lines indicate connections through which error signal from loss l, l1 , and l2 are backpropagated through to train m. b) Stochastic Sobolev Training of order 2. If f and m are multivariate functions, the gradients are Jacobian matrices. To avoid computing these high dimensional objects, we can efficiently compute and fit their projections on a random vector vj sampled from the unit sphere. D_{\mathbf{x}} \langle D_{\mathbf{x}} \langle m, v_1 \rangle, v_2 \rangle propose Sobolev Training (ST) for neural networks as a simple and efficient technique for leveraging derivative information about the desired function in a way that can easily be incorporated into any training pipeline using modern machine learning libraries. The approach is inspired by the work of Hornik [10] which proved the universal approximation theorems for neural networks in Sobolev spaces ? metric spaces where distances between functions are defined both in terms of their differences in values and differences in values of their derivatives. In particular, it was shown that a sigmoid network can not only approximate a function?s value arbitrarily well, but that the network?s derivatives with respect to its inputs can approximate the corresponding derivatives of the ground truth function arbitrarily well too. Sobolev Training exploits this property, and tries to match not only the output of the function being trained but also its derivatives. There are several related works which have also exploited derivative information for function approximation. For instance Wu et al. [30] and antecedents propose a technique for Bayesian optimisation with Gaussian Processess (GP), where it was demonstrated that the use of information about gradients and Hessians can improve the predictive power of GPs. In previous work on neural networks, derivatives of predictors have usually been used either to penalise model complexity (e.g. by pushing Jacobian norm to 0 [19]), or to encode additional, hand crafted invariances to some transformations (for instance, as in Tangentprop [23]), or estimated derivatives for dynamical systems [6] and very recently to provide additional learning signal during attention distillation [31]1 . Similar techniques have also been used in critic based Reinforcement Learning (RL), where a critic?s derivatives are trained to match its target?s derivatives [29, 15, 5, 4, 26] using small, sigmoid based models. Finally, Hyv?rinen proposed Score Matching Networks [11], which are based on the somewhat surprising observation that one can model unknown derivatives of the function without actual access to its values ? all that is needed is a sampling based strategy and specific penalty. However, such an estimator has a high variance [28], thus it is not really useful when true derivatives are given. To the best of our knowledge and despite its simplicity, the proposal to directly match network derivatives to the true derivatives of the target function has been minimally explored for deep networks, especially modern ReLU based models. In our method, we show that by using the additional knowledge of derivatives with Sobolev Training we are able to train better models ? models which achieve lower approximation errors and generalise to test data better ? and reduce the sample complexity of learning. The contributions of our paper are therefore threefold: (1): We introduce 1 Please relate to Supplementary Materials, section 5 for details 2 Sobolev Training ? a new paradigm for training neural networks. (2): We look formally at the implications of matching derivatives, extending previous results of Hornik [10] and showing that modern architectures are well suited for such training regimes. (3): Empirical evidence demonstrating that Sobolev Training leads to improved performance and generalisation, particularly in low data regimes. Example domains are: regression on classical optimisation problems; policy distillation from RL agents trained on the Atari domain; and training deep, complex models using synthetic gradients ? we report the first successful attempt to train a large-scale ImageNet model using synthetic gradients. 2 Sobolev Training We begin by introducing the idea of training using Sobolev spaces. When learning a function f , we may have access to not only the output values f (xi ) for training points xi , but also the values of its j-th order derivatives with respect to the input, Dxj f (xi ). In other words, instead of the typical training set consisting of pairs {(xi , f (xi ))}N i=1 we have access to (K + 2)-tuples {(xi , f (xi ), Dx1 f (xi ), ..., DxK f (xi ))}N i=1 . In this situation, the derivative information can easily be incorporated into training a neural network model of f by making derivatives of the neural network match the ones given by f . Considering a neural network model m parameterised with ?, one typically seeks to minimise the empirical error in relation to f according to some loss function ` N X i=1 `(m(xi |?), f (xi )). When learning in Sobolev spaces, this is replaced with: ? ? N K X X  ?`(m(xi |?), f (xi )) + `j Dxj m(xi |?), Dxj f (xi ) ? , i=1 (1) j=1 where `j are loss functions measuring error on j-th order derivatives. This causes the neural network to encode derivatives of the target function in its own derivatives. Such a model can still be trained using backpropagation and off-the-shelf optimisers. A potential concern is that this optimisation might be expensive when either the output dimensionality of f or the order K are high, however one can reduce this cost through stochastic approximations. Specifically, if f is a multivariate function, instead of a vector gradient, one ends up with a full Jacobian matrix which can be large. To avoid adding computational complexity to the training process, one can use an efficient, stochastic version of Sobolev Training: instead of computing a full Jacobian/Hessian, one just computes its projection onto a random vector (a direct application of a known estimation trick [19]). In practice, this means that during training we have a random variable v sampled uniformly from the unit sphere, and we match these random projections instead: ? ? N K X X  j  ?`(m(xi |?), f (xi )) + Evj `j Dx m(xi |?), v j , Dxj f (xi ), v j ? . (2) i=1 j=1 Figure 1 illustrates compute graphs for non-stochastic and stochastic Sobolev Training of order 2. 3 Theory and motivation While in the previous section we defined Sobolev Training, it is not obvious that modeling the derivatives of the target function f is beneficial to function approximation, or that optimising such an objective is even feasible. In this section we motivate and explore these questions theoretically, showing that the Sobolev Training objective is a well posed one, and that incorporating derivative information has the potential to drastically reduce the sample complexity of learning. Hornik showed [10] that neural networks with non-constant, bounded, continuous activation functions, with continuous derivatives up to order K are universal approximators in the Sobolev spaces of order K, thus showing that sigmoid-networks are indeed capable of approximating elements of these 3 Figure 2: Left: From top: Example of the piece-wise linear function; Two (out of a continuum of) hypotheses consistent with 3 training points, showing that one needs two points to identify each linear segment; The only hypothesis consistent with 3 training points enriched with derivative information. Right: Logarithm of test error (MSE) for various optimisation benchmarks with varied training set size (20, 100 and 10000 points) sampled uniformly from the problem?s domain. spaces arbitrarily well. However, nowadays we often use activation functions such as ReLU which are neither bounded nor have continuous derivatives. The following theorem shows that for K = 1 we can use ReLU function (or a similar one, like leaky ReLU) to create neural networks that are universal approximators in Sobolev spaces. We will use a standard symbol C 1 (S) (or simply C 1 ) to denote a space of functions which are continuous, differentiable, and have a continuous derivative on a space S [14]. All proofs are given in the Supplementary Materials (SM). Theorem 1. Let f be a C 1 function on a compact set. Then, for every positive ? there exists a single hidden layer neural network with a ReLU (or a leaky ReLU) activation which approximates f in Sobolev space S1 up to  error. This suggests that the Sobolev Training objective is achievable, and that we can seek to encode the values and derivatives of the target function in the values and derivatives of a ReLU neural network model. Interestingly, we can show that if we seek to encode an arbitrary function in the derivatives of the model then this is impossible not only for neural networks but also for any arbitrary differentiable predictor on compact sets. Theorem 2. Let f be a C 1 function. Let g be a continuous function satisfying kg ? ?f ?x k? > 0. Then, there exists an ? > 0 such that for any C 1 function h either kf ? hk? ? ? or g ? ?h ?x ? ? ?. However, when we move to the regime of finite training data, we can encode any arbitrary function in the derivatives (as well as higher order signals if the resulting Sobolev spaces are not degenerate), as shown in the following Proposition. Proposition 1. Given any two functions f : S ? R and g : S ? Rd on S ? Rd and a finite set ? ? S, there exists neural network h with a ReLU (or a leaky ReLU) activation such that ?x ? ? : f (x) = h(x) and g(x) = ?h ?x (x) (it has 0 training loss). Having shown that it is possible to train neural networks to encode both the values and derivatives of a target function, we now formalise one possible way of showing that Sobolev Training has lower sample complexity than regular training. Let F denote the family of functions parametrised by ?. We define Kreg = Kreg (F) to be a measure of the amount of data needed to learn some target function f . That is Kreg is the smallest number for which there holds: for every f? ? F and every set of distinct Kreg points (x1 , ..., xKreg ) such that ?i=1,...,Kreg f (xi ) = f? (xi ) ? f = f? . Ksob is defined analogously, but the final implication is of ?f? form f (xi ) = f? (xi ) ? ?f ?x (xi ) = ?x (xi ) ? f = f? . Straight from the definition there follows: Proposition 2. For any F, there holds Ksob (F) ? Kreg (F). For many families, the above inequality becomes sharp. For example, to determine the coefficients of a polynomial of degree n one needs to compute its values in at least n + 1 distinct points. If we know values and the derivatives at k points, it is a well-known fact that only d n2 e points suffice to determine all the coefficients. We present two more examples in a slightly more formal way. Let FG denote a family of Gaussian PDF-s (parametrised by ?, ?). Let Rd ? D = D1 ? . . . ? Dn and let FPL be a family of functions from D1 ? ... ? Dn (Cartesian product of sets Di ) to Rn of form f (x) = [A1 x1 + b1 , . . . , An xn + bn ] (linear element-wise) (Figure 2 Left). 4 Dataset 20 training samples Regular Sobolev 100 training samples Regular Sobolev Figure 3: Styblinski-Tang function (on the left) and its models using regular neural network training (left part of each plot) and Sobolev Training (right part). We also plot the vector field of the gradients of each predictor underneath the function plot. Proposition 3. There holds Ksob (FG ) < Kreg (FG ) and Ksob (FPL ) < Kreg (FPL ). This result relates to Deep ReLU networks as they build a hyperplanes-based model of the target function. If those were parametrised independently one could expect a reduction of sample complexity by d+1 times, where d is the dimension of the function domain. In practice parameters of hyperplanes in such networks are not independent, furthermore the hinges positions change so the Proposition cannot be directly applied, but it can be seen as an intuitive way to see why the sample complexity drops significantly for Deep ReLU networks too. 4 Experimental Results We consider three domains where information about derivatives is available during training2 . 4.1 Artificial Data First, we consider the task of regression on a set of well known low-dimensional functions used for benchmarking optimisation methods. We train two hidden layer neural networks with 256 hidden units per layer with ReLU activations to regress towards function values, and verify generalisation capabilities by evaluating the mean squared error on a hold-out test set. Since the task is standard regression, we choose all the losses of Sobolev Training to be L2 errors, and use a first order Sobolev method (second order derivatives of ReLU networks with a linear output layer are constant, zero). The optimisation is therefore: min N1 ? N X i=1 kf (xi ) ? m(xi |?)k22 + k?x f (xi ) ? ?x m(xi |?)k22 . Figure 2 right shows the results for the optimisation benchmarks. As expected, Sobolev trained networks perform extremely well ? for six out of seven benchmark problems they significantly reduce the testing error with the obtained errors orders of magnitude smaller than the corresponding errors of the regularly trained networks. The stark difference in approximation error is highlighted in Figure 3, where we show the Styblinski-Tang function and its approximations with both regular and Sobolev Training. It is clear that even in very low data regimes, the Sobolev trained networks can capture the functional shape. Looking at the results, we make two important observations. First, the effect of Sobolev Training is stronger in low-data regimes, however it does not disappear even in the high data regime, when one has 10,000 training examples for training a two-dimensional function. Second, the only case where regular regression performed better is the regression towards Ackley?s function. This particular 2 All experiments were performed using TensorFlow [2] and the Sonnet neural network library [1]. 5 Test DKL Test action prediction error Regular distillation Sobolev distillation Figure 4: Test results of distillation of RL agents on three Atari games. Reported test action prediction error (left) is the error of the most probable action predicted between the distilled policy and target policy, and test DKL (right) is the Kulblack-Leibler divergence between policies. Numbers in the column title represents the percentage of the 100K recorded states used for training (the remaining are used for testing). In all scenarios the Sobolev distilled networks are significantly more similar to the target policy. example was chosen to show that one possible weak point of our approach might be approximating functions with a very high frequency signal component in the relatively low data regime. Ackley?s function is composed of exponents of high frequency cosine waves, thus creating an extremely bumpy surface, consequently a method that tries to match the derivatives can behave badly during testing if one does not have enough data to capture this complexity. However, once we have enough training data points, Sobolev trained networks are able to approximate this function better. 4.2 Distillation Another possible application of Sobolev Training is to perform model distillation. This technique has many applications, such as network compression [21], ensemble merging [9], or more recently policy distillation in reinforcement learning [20]. We focus here on a task of distilling a policy. We aim to distill a target policy ? ? (s) ? a trained neural network which outputs a probability distribution over actions ? into a smaller neural network ?(s|?), such that the two policies ? ? and ? have the same behaviour. In practice this is often done by minimising an expected divergence measure between ? ? and ?, for example, the Kullback?Leibler divergence DKL (?(s)k? ? (s)), over states gathered while following ? ? . Since policies are multivariate functions, direct application of Sobolev Training would mean producing full Jacobian matrices with respect to the s, which for large actions spaces is computationally expensive. To avoid this issue we employ a stochastic approximation described in Section 2, thus resulting in the objective min DKL (?(s|?)k? ? (s)) + ?Ev [k?s hlog ? ? (s), vi ? ?s hlog ?(s|?), vik] , ? where the expectation is taken with respect to v coming from a uniform distribution over the unit sphere, and Monte Carlo sampling is used to approximate it. As target policies ? ? , we use agents playing Atari games [17] that have been trained with A3C [16] on three well known games: Pong, Breakout and Space Invaders. The agent?s policy is a neural network consisting of 3 layers of convolutions followed by two fully-connected layers, which we distill to a smaller network with 2 convolutional layers and a single smaller fully-connected layer (see SM for details). Distillation is treated here as a purely supervised learning problem, as our aim is not to re-evaluate known distillation techniques, but rather to show that if the aim is to minimise a given divergence measure, we can improve distillation using Sobolev Training. Figure 4 shows test error during training with and without Sobolev Training3 . The introduction of Sobolev Training leads to similar effects as in the previous section ? the network generalises much more effectively, and this 3 Testing is performed on a held out set of episodes, thus there are no temporal nor causal relations between training and testing 6 xx x x yxy y y y xx (a)(a) (a)(a)(a) x x yxy Synthetic error gradient error gradient Synthetic error gradient Synthetic error gradient Synthetic error gradient y y ySynthetic (b) (b)(b)(b) (b) Table 1: Various techniques for producing synthetic gradients. Green shaded nodes denote nodes that get supervision from the corresponding object from the main network (gradient or loss value). We report accuracy on the test set ? standard deviation. Backpropagation results are given in parenthesis. 00 SG(h, SG(h, y) y) SG(h, SG(h, y) SG(h, y)SG(h, SG(h, y)SG(h, y) y)y)y) SG(h, y)SG(h, y) SG(h, y) SG(h, SG(h, y) y) y) y) SG(h, y) SG(h, 0 y) SG(h, 0SG(h, 0SG(h, @ @ @h @h @ @ @ @ @h @h @h @h ? ?L L ?L ? L ? ? L L @ @ @@ @ @h@h @h @h @h ? L ? ? L ?L L @ @ @ @h @h @h ? ? L ? L L f (h, y|?) (h, y|?) f (h, f (h, y|?)y|?) f (h,fy|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) p(h|?) h yh y Noprop y hy y hy h hy hy y hy h hy y y y h h h yhhy h y y h hy hhyh y y h Direct SG [12] VFBN [25] Critic Sobolev CIFAR-10 with 3 synthetic gradient modules Top 1 (94.3%) 54.5% ?1.15 79.2% ?0.01 88.5% ?2.70 93.2% ?0.02 93.5% ?0.01 ImageNet with 1 synthetic gradient module Top 1 (75.0%) 54.0% ?0.29 Top 5 (92.3%) 77.3% ?0.06 - 57.9% ?2.03 81.5% ?1.20 71.7% ?0.23 90.5% ?0.15 72.0% ?0.05 90.8% ?0.01 ImageNet with 3 synthetic gradient modules Top 1 (75.0%) 18.7% ?0.18 Top 5 (92.3%) 38.0% ?0.34 - 28.3% ?5.24 52.9% ?6.62 65.7% ?0.56 86.9% ?0.33 66.5% ?0.22 87.4% ?0.11 is especially true in low data regimes. Note the performance gap on Pong is small due to the fact that optimal policy is quite degenerate for this game4 . In all remaining games one can see a significant performance increase from using our proposed method, and as well as minor to no overfitting. Despite looking like a regularisation effect, we stress that Sobolev Training is not trying to find the simplest models for data or suppress the expressivity of the model. This training method aims at matching the original function?s smoothness/complexity and so reduces overfitting by effectively extending the information content of the training set, rather than by imposing a data-independent prior as with regularisation. 4.3 Synthetic Gradients The previous experiments have shown how information about the derivatives can boost approximating function values. However, the core idea of Sobolev Training is broader than that, and can be employed in both directions. Namely, if one ultimately cares about approximating derivatives, then additionally approximating values can help this process too. One recent technique, which requires a model of gradients is Synthetic Gradients (SG) [12] ? a method for training complex neural networks in a decoupled, asynchronous fashion. In this section we show how we can use Sobolev Training for SG. The principle behind SG is that instead of doing full backpropagation using the chain-rule, one splits a network into two (or more) parts, and approximates partial derivatives of the loss L with respect to some hidden layer activations h with a trainable function SG(h, y|?). In other words, given that network parameters up to h are denoted by ? ?L ?h ?h ?L = ? SG(h, y|?) . ?? ?h ?? ?? 2 h ,y) In the original SG paper, this module is trained to minimise LSG (?) = SG(h, y|?) ? ?L(p , ?h 2 where ph is the final prediction of the main network for hidden activations h. For the case of learning a classifier, in order to apply Sobolev Training in this context we construct a loss predictor, composed 4 For majority of the time the policy in Pong is uniform, since actions taken when the ball is far away from the player do not matter at all. Only in crucial situations it peaks so the ball hits the paddle. 7 of a class predictor p(?|?) followed by the log loss, which gets supervision from the true loss, and the gradient of the prediction gets supervision from the true gradient: m(h, y|?) := L(p(h|?), y), SG(h, y|?) := ?m(h, y|?)/?h,   ?m(h,y|?) ?L(ph ,y) Lsob (?) = `(m(h, y|?), L(p , y))) + ` . , h 1 SG ?h ?h In the Sobolev Training framework, the target function is the loss of the main network L(ph , y) for which we train a model m(h, y|?) to approximate, and in addition ensure that the model?s derivatives ?m(h, y|?)/?h are matched to the true derivatives ?L(ph , y)/?h. The model?s derivatives ?m(h, y|?)/?h are used as the synthetic gradient to decouple the main network. This setting closely resembles what is known in reinforcement learning as critic methods [13]. In particular, if we do not provide supervision on the gradient part, we end up with a loss critic. Similarly if we do not provide supervision at the loss level, but only on the gradient component, we end up in a method that resembles VFBN [25]. In light of these connections, our approach in this application setting can be seen as a generalisation and unification of several existing ones (see Table 1 for illustrations of these approaches). One could ask why we need these additional constraints, and what is gained over using a neural network based approximator directly [12]. The answer lies in the fact that gradient vector fields are a tiny subset of all vector fields, and while each neural network produces a valid vector field, almost no (standard) neural network produces valid gradient vector fields. Using non-gradient vector fields as update directions for learning can have catastrophic consequences ? learning divergence, oscillations, chaotic behaviour, etc. The following proposition makes this observation more formal: Proposition 4. If an approximator SG(h, y|?) produces a valid gradient vector field of some scalar function L then the approximator?s Jacobian matrix must be symmetric. It is worth noting that having a symmetric Jacobian is an extremely rare property for a neural network model. For example, a linear model has a symmetric Jacobian if and only if its weight matrix is symmetric. If we sample weights iid from typical distribution (like Gaussian or uniform on an interval), the probability of sampling such a matrix is 0, but it could be easy to learn with strong, symmetric-enforcing updates. On the other hand, for highly non-linear neural networks, it is not only improbable to randomly find such a model, but enforcing this constraint during learning becomes much harder too. This might be one of the reasons why linear SG modules work well in Jaderberg et al. [12], but non-linear convolutional SG struggled to achieve state-of-the-art performance. When using Sobolev-like approach SG always produces a valid gradient vector field by construction, thus avoiding the problem described. We perform experiments on decoupling deep convolutional neural network image classifiers using synthetic gradients produced by loss critics that are trained with Sobolev Training, and compare to regular loss critic training, and regular synthetic gradient training. We report results on CIFAR-10 for three network splits (and therefore three synthetic gradient modules) and on ImageNet with one and three network splits 5 . The results are shown in Table 1. With a naive SG model, we obtain 79.2% test accuracy on CIFAR-10. Using an SG architecture which resembles a small version of the rest of the model makes learning much easier and led to 88.5% accuracy, while Sobolev Training achieves 93.5% final performance. The regular critic also trains well, achieving 93.2%, as the critic forces the lower part of the network to provide a representation which it can use to reduce the classification (and not just prediction) error. Consequently it provides a learning signal which is well aligned with the main optimisation. However, this can lead to building representations which are suboptimal for the rest of the network. Adding additional gradient supervision by constructing our Sobolev SG module avoids this issue by making sure that synthetic gradients are truly aligned and gives an additional boost to the final accuracy. For ImageNet [3] experiments based on ResNet50 [8], we obtain qualitatively similar results. Due to the complexity of the model and an almost 40% gap between no backpropagation and full backpropagation results, the difference between methods with vs without loss supervision grows significantly. This suggests that at least for ResNet-like architectures, loss supervision is a crucial 5 N.b. the experiments presented use learning rates, annealing schedule, etc. optimised to maximise the backpropagation baseline, rather than the synthetic gradient decoupled result (details in the SM). 8 component of a SG module. After splitting ResNet50 into four parts the Sobolev SG achieves 87.4% top 5 accuracy, while the regular critic SG achieves 86.9%, confirming our claim about suboptimal representation being enforced by gradients from a regular critic. Sobolev Training results were also much more reliable in all experiments (significantly smaller standard deviation of the results). 5 Discussion and Conclusion In this paper we have introduced Sobolev Training for neural networks ? a simple and effective way of incorporating knowledge about derivatives of a target function into the training of a neural network function approximator. We provided theoretical justification that encoding both a target function?s value as well as its derivatives within a ReLU neural network is possible, and that this results in more data efficient learning. Additionally, we show that our proposal can be efficiently trained using stochastic approximations if computationally expensive Jacobians or Hessians are encountered. In addition to toy experiments which validate our theoretical claims, we performed experiments to highlight two very promising areas of applications for such models: one being distillation/compression of models; the other being the application to various meta-optimisation techniques that build models of other models dynamics (such as synthetic gradients, learning-to-learn, etc.). In both cases we obtain significant improvement over classical techniques, and we believe there are many other application domains in which our proposal should give a solid performance boost. In this work we focused on encoding true derivatives in the corresponding ones of the neural network. Another possibility for future work is to encode information which one believes to be highly correlated with derivatives. For example curvature [18] is believed to be connected to uncertainty. Therefore, given a problem with known uncertainty at training points, one could use Sobolev Training to match the second order signal to the provided uncertainty signal. Finite differences can also be used to approximate gradients for black box target functions, which could help when, for example, learning a generative temporal model. Another unexplored path would be to apply Sobolev Training to internal derivatives rather than just derivatives with respect to the inputs. References [1] Sonnet. https://github.com/deepmind/sonnet. 2017. [2] Mart?n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. [3] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. 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Neural networks for control. MIT press, 1995. [16] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning, pages 1928?1937, 2016. [17] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013. [18] Razvan Pascanu and Yoshua Bengio. Revisiting natural gradient for deep networks. arXiv preprint arXiv:1301.3584, 2013. [19] Salah Rifai, Gr?goire Mesnil, Pascal Vincent, Xavier Muller, Yoshua Bengio, Yann Dauphin, and Xavier Glorot. Higher order contractive auto-encoder. Machine Learning and Knowledge Discovery in Databases, pages 645?660, 2011. [20] Andrei A Rusu, Sergio Gomez Colmenarejo, Caglar Gulcehre, Guillaume Desjardins, James Kirkpatrick, Razvan Pascanu, Volodymyr Mnih, Koray Kavukcuoglu, and Raia Hadsell. Policy distillation. arXiv preprint arXiv:1511.06295, 2015. [21] Bharat Bhusan Sau and Vineeth N Balasubramanian. Deep model compression: Distilling knowledge from noisy teachers. arXiv preprint arXiv:1610.09650, 2016. [22] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484?489, 2016. [23] Patrice Simard, Bernard Victorri, Yann LeCun, and John S Denker. Tangent prop-a formalism for specifying selected invariances in an adaptive network. In NIPS, volume 91, pages 895?903, 1991. [24] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. [25] Shin-ichi Maeda Koyama Masanori Takeru Miyato, Daisuke Okanohara. Synthetic gradient methods with virtual forward-backward networks. ICLR workshop proceedings, 2017. [26] Yuval Tassa and Tom Erez. Least squares solutions of the hjb equation with neural network value-function approximators. IEEE transactions on neural networks, 18(4):1031?1041, 2007. [27] A?ron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. CoRR abs/1609.03499, 2016. [28] Pascal Vincent. A connection between score matching and denoising autoencoders. Neural computation, 23(7):1661?1674, 2011. [29] Paul J Werbos. Approximate dynamic programming for real-time control and neural modeling. Handbook of intelligent control, 1992. [30] Anqi Wu, Mikio C Aoi, and Jonathan W Pillow. Exploiting gradients and hessians in bayesian optimization and bayesian quadrature. arXiv preprint arXiv:1704.00060, 2017. [31] Sergey Zagoruyko and Nikos Komodakis. Paying more attention to attention: Improving the performance of convolutional neural networks via attention transfer. arXiv preprint arXiv:1612.03928, 2016. 10
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Multi-Information Source Optimization Matthias Poloczek Department of Systems and Industrial Engineering University of Arizona Tucson, AZ 85721 [email protected] Jialei Wang Chief Analytics Office IBM Armonk, NY 10504 [email protected] Peter I. Frazier School of Operations Research and Information Engineering Cornell University Ithaca, NY 14853 [email protected] Abstract We consider Bayesian methods for multi-information source optimization (MISO), in which we seek to optimize an expensive-to-evaluate black-box objective function while also accessing cheaper but biased and noisy approximations (?information sources?). We present a novel algorithm that outperforms the state of the art for this problem by using a Gaussian process covariance kernel better suited to MISO than those used by previous approaches, and an acquisition function based on a one-step optimality analysis supported by efficient parallelization. We also provide a novel technique to guarantee the asymptotic quality of the solution provided by this algorithm. Experimental evaluations demonstrate that this algorithm consistently finds designs of higher value at less cost than previous approaches. 1 Introduction We consider Bayesian multi-information source optimization (MISO), in which we optimize an expensive-to-evaluate black-box objective function while optionally accessing cheaper biased noisy approximations, often referred to as ?information sources (IS)?. This arises when tuning hyperparameters of machine learning algorithms: one may evaluate hyperparameters on a smaller related dataset or subsets of the validation set [34, 15, 17]. We also face this problem in robotics: we can evaluate a parameterized robot control policy in simulation, in a laboratory, or in a field test [15]. Cheap approximations promise a route to tractability, but bias and noise complicate their use. An unknown bias arises whenever a computational model incompletely models a real-world phenomenon, and is pervasive in applications. We present a novel algorithm for this problem, misoKG, that is tolerant to both noise and bias and improves substantially over the state of the art. Specifically, our contributions are: ? The algorithm uses a novel acquisition function that maximizes the incremental gain per unit cost. This acquisition function generalizes and parallelizes previously proposed knowledgegradient methods for single-IS Bayesian optimization [7, 8, 28, 26, 37] to MISO. ? We prove that this algorithm provides an asymptotically near-optimal solution. If the search domain is finite, this result establishes the consistency of misoKG. We present a novel proof technique that yields an elegant, short argument and is thus of independent interest. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Related Work: To our knowledge, MISO was first considered by Swersky, Snoek, and Adams [34], under the the name multi-task Bayesian optimization. This name was used to suggest problems in which the auxiliary tasks could meaningfully be solved on their own, while we use the term MISO to indicate that the IS may be useful only in support of the primary task. Swersky et al. [34] showed that hyperparameter tuning in classification can be accelerated through evaluation on subsets of the validation data. They proposed a GP model to jointly model such ?auxiliary tasks? and the primary task, building on previous work on GP regression for multiple tasks in [3, 10, 35]. They choose points to sample via cost-sensitive entropy search [11, 39], sampling in each iteration a point that maximally reduces uncertainty in the optimum?s location, normalized by the query cost. We demonstrate in experiments that our approach improves over the method of Swersky et al. [34], and we believe this improvement results from two factors: first, our statistical model is more flexible in its ability to model bias that varies across the domain; second, our acquisition function directly and maximally reduces simple regret in one step, unlike predictive entropy search which maximally reduces the maximizer?s entropy in one step and hence only indirectly reduces regret. Lam, Allaire, and Willcox [18] also consider MISO, under the name non-hierarchical multi-fidelity optimization. They propose a statistical model that maintains a separate GP for each IS, and fuse them via the method of Winkler [40]. They apply a modified expected improvement acquisition function on these surrogates to first decide what design x? to evaluate and then select the IS to query; the latter is decided by a heuristic that aims to balance information gain and query cost. We demonstrate in experiments that our approach improves over the method of Lam et al. [18], and we believe this improvement results from two factors: first, their statistical approach assumes an independent prior on each IS, despite their being linked through modeling a common objective; and second their acquisition function selects the point to sample and the IS to query separately via a heuristic rather than jointly using an optimality analysis. Beyond these two works, the most closely related work is in the related problem of multi-fidelity optimization. In this problem, IS are supposed to form a strict hierarchy [16, 14, 6, 24, 20, 19, 15]. In addition, most of these models limit the information that can be obtained from sources of lower fidelity [16, 14, 6, 20, 19]: Given the observation of x at some IS `, one cannot learn more about the value of x at IS with higher fidelity by querying IS ` anywhere else (see Sect. C for details and a proof). Picheny et al. [24] propose a quantile-based criterion for optimization of stochastic simulators, supposing that all simulators provide unbiased approximations of the true objective. From this body of work, we compare against Kandasamy et al. [15], who present an approach for minimizing both simple and cumulative regret, under the assumption that the maximum bias of an information source strictly decreases with its fidelity. Outside of both the MISO and multi-fidelity settings, Klein et al. [17] considered hyperparameter optimization of machine learning algorithms over a large dataset D. Supposing access to subsets of D of arbitrary sizes, they show how to exploit regularity of performance across dataset sizes to significantly speed up the optimization process for support vector machines and neural networks. Our acquisition function is a generalization of the knowledge-gradient policy of Frazier, Powell, and Dayanik [8] to the MISO setting. This generalization requires extending the one-step optimality analysis used to derive the KG policy in the single-IS setting to account for the impact of sampling a cheap approximation on the marginal GP posterior on the primary task. From this literature, we leverage an idea for computing the expectation of the maximum of a collection of linear functions of a normal random variable, and propose a parallel algorithm to identify and compute the required components. The class of GP covariance kernels we propose are a subset of the class of linear models of coregionalization kernels [10, 2], with a restricted form derived from a generative model particular to MISO. Focusing on a restricted class of kernels designed for our application supports accurate inference with less data, which is important when optimizing expensive-to-evaluate functions. Our work also extends the knowledge-gradient acquisition function to the variable cost setting. Similar extensions of expected improvement to the variable cost setting can be found in Snoek et al. [32] (the EI per second criterion) and in Le Gratiet and Cannamela [19]. We now formalize the problem we consider in Sect. 2, describe our statistical analysis in Sect. 3.1, specify our acquisition function and parallel computation method in Sects. 3.2 and 3.3, provide a theoretical guarantee in Sect. 3.4, present numerical experiments in Sect. 4, and conclude in Sect. 5. 2 2 Problem Formulation Given a continuous objective function g : D ? R on a compact set D ? Rd of feasible designs, our goal is to find a design with objective value close to maxx?D g(x). We have access to M possibly biased and/or noisy IS indexed by ` ? [M ]0 . (Here, for any a ? Z+ we use [a] as a shorthand for the set {1, 2, . . . , a}, and further define [a]0 = {0, 1, 2, . . . , a}.) Observing IS ` at design x provides independent, conditional on f (`, x), and normally distributed observations with mean f (`, x) and finite variance ?` (x). In [34], IS ` ? [M ]0 are called ?auxiliary tasks? and g the primary task. These sources are thought of as approximating g, with variable bias. We suppose that g can be observed directly without bias (but possibly with noise) and set f (0, x) = g(x). The bias f (`, x) ? g(x) is also referred to as ?model discrepancy? in the engineering and simulation literature [1, 4]. Each IS ` is also associated with a query cost function c` (x) : D ? R+ . We assume that the cost function c` (x) and the variance function ?` (x) are both known and continuously differentiable (over D). In practice, these functions may either be provided by domain experts or may be estimated along with other model parameters from data (see Sect. 4 and B.2, and [27]). 3 The misoKG Algorithm We now present the misoKG algorithm and describe its two components: a MISO-focused statistical model in Sect. 3.1; and its acquisition function and parallel computation in Sect. 3.2. Sect. 3.3 summarizes the algorithm and Sect. 3.4 provides a theoretical performance guarantee. Extensions of the algorithm are discussed in Sect. D. 3.1 Statistical Model We now describe a generative model for f that results in a Gaussian process prior on f with a parameterized class of mean functions ? : [M ]?D 7? R and covariance kernels ? : ([M ]?D)2 7? R. This allows us to use standard tools for Gaussian process inference ? first finding the MLE estimate of the parameters indexing this class, and then performing Gaussian process regression using the selected mean function and covariance kernel ? while also providing better estimates for MISO than would a generic multi-output GP regression kernel that does not consider the MISO application. We construct our generative model as follows. For each ` > 0 suppose that a function ?` : D 7? R for each ` > 0 was drawn from a separate independent GP, ?` ? GP (?` , ?` ), and let ?0 be identically 0. In our generative model ?` will be the bias f (`, x) ? g(x) for IS `. We additionally set ?` (x) = 0 to encode a lack of a strong belief on the direction of the bias. (If one had a strong belief that an IS is consistently biased in one direction, one may instead set ?` to a constant estimated using maximum a posteriori estimation.) Next, within our generative model, we suppose that g : D 7? R was drawn from its own independent GP, g ? GP (?0 , ?0 ), for some given ?0 and ?0 , and suppose f (`, x) = f (0, x) + ?` (x) for each `. We assume that ?0 and ?` with ` ? 0 belong to one of the standard parameterized classes of mean functions and covariance kernels, e.g., constant ?0 and Mat?rn ?` . With this construction, f is a GP: given any finite collection of points `i ? [M ], xi ? D with i = 1, . . . , I, (f (`i , xi ) : i = 1, . . . , I) is a sum of independent multivariate normal random vectors, and thus is itself multivariate normal. Moreover, we compute the mean function and covariance kernel of f : for `, m ? [M ]0 and x, x0 ? D, ?(`, x) = E [f (`, x)] = E [g(x)] + E [?` (x)] = ?0 (x) ? ((`, x), (m, x0 )) = Cov(g(x) + ?` (x), g(x0 ) + ?m (x0 )) = ?0 (x, x0 ) + 1`,m ? ?` (x, x0 ), where 1`,m denotes Kronecker?s delta, and where we have used independence of ?` , ?m , and g. We refer the reader to https://github.com/misoKG/ for an illustration of the model. This generative model draws model discrepancies ?` independently across IS. This is appropriate when IS are different in kind and share no relationship except that they model a common objective. In the supplement (Sect. B) we generalize this generative model to model correlation between model discrepancies, which is appropriate when IS can be partitioned into groups, such that IS within one group tend to agree more amongst themselves than they do with IS in other groups. Sect. B also discusses the estimation of the hyperparameters in ?0 and ?` . 3 3.2 Acquisition Function Our optimization algorithm proceeds in rounds, selecting a design x ? D and an information source ` ? [M ]0 in each. The value of the information obtained by sampling IS ` at x is the expected gain in the quality of the best design that can be selected using the available information. That is, this value is the difference in the expected quality of the estimated optimum before and after the sample. We then normalize this expected gain by the cost c` (x) associated with the respective query, and sample the IS and design with the largest normalized gain. Without normalization we would always query the true objective, since no other IS provides more information about g than g itself. We formalize this idea. Suppose that we have already sampled n points Xn and made the observations Yn . Denote by En the expected value according to the posterior distribution given Xn , Yn , and let ?(n) (`, x) := En [f (`, x)]. The best expected objective value across the designs, as estimated by our statistical model, is maxx0 ?D ?(n) (0, x0 ). Similarly, if we take an additional sample of IS `(n+1) at design x(n+1) and compute our new posterior mean, the new best expected objective value across the designs is maxx0 ?D ?(n+1) (0, x0 ), whose distribution depends on what IS we sample, and where sample it. Thus, the expected value of sampling at (`, x) normalized by the cost is   maxx0 ?D ?(n+1) (0, x0 ) ? maxx0 ?D ?(n) (0, x0 ) (n+1) n (n+1) MKG (`, x) = En ` = `, x =x , c` (x) (1) which we refer to as the misoKG factor of the pair (`, x). The misoKG policy then samples at the pair (`, x) that maximizes MKGn (`, x), i.e., (`(n+1) , x(n+1) ) ? argmax`?[M ]0 ,x?D MKGn (`, x), which is a nested optimization problem. To make this nested optimization problem tractable, we first replace the search domain D in Eq. (1) by a discrete set A ? D of points, for example selected by a Latin Hypercube design. We may then compute MKGn (`, x) exactly. Toward that end, note that  (n+1) En max ? 0 x ?A  (n+1) (n+1) (0, x ) ` = `, x =x   (n+1) (n) 0 n (n+1) = `, x = x , (2) = En max {? (0, x ) + ? ?x0 (`, x) ? Z} ` 0 0 x ?A 1 where Z ? N (0, 1) and ? ?xn0 (`, x) = ?n ((0, x0 ), (`, x))/ [?` (x) + ?n ((`, x), (`, x))] 2 . Here ?n is the posterior covariance matrix of f given Xn , Yn . We parallelize the computation of MKGn (`, x) for fixed `, x, enabling it to utilize multiple cores. Then (`(n+1) , x(n+1) ) is obtained by computing MKGn (`, x) for all (`, x) ? [M ]0 ? A, a task that can be parallelized over multiple machines in a cluster. We begin by sorting the points in A in parallel by increasing value of ? ?xn0 (`, x) (for fixed `, x). Thereby we remove some points easily identified as dominated. A point xj is dominated if maxi ?(n) (0, xi ) + ? ?xni (`, x)Z is unchanged for all Z if the maximum is taken excluding xj . Note that a point xj is dominated by xk if ? ?xnj (`, x) = ? ?xnk (`, x) and ?(n) (0, xj ) ? ?(n) (0, xk ), since xk has a higher expected value than xj for any realization of Z. Let S be the sorted sequence without such dominated points. We will remove more dominated points later. Since h c` (x) is a constant for i fixed `, x, we may express the conditional expectation in Eq. (1) as maxi {ai +bi Z}?maxi ai i Z}?maxi ai ] En = En [maxi {aic+b , where ai = ?(n) (0, xi ) and bi = ? ?xni (`, x) c` (x) ` (x) for xi ? S. We split S into consecutive sequences S1 , S2 , . . . , SC , where C is the number of cores used for computing MKGn (`, x) and Si , Si+1 overlap in one element: that is, for Sj = {xj1 , . . . , xjk }, x(j?1)k = xj1 and xjk = x(j+1)1 hold. Each xji ? Sj specifies a linear function aji +bji Z (ordered by increasing slopes in S). We are interested in the realizations of Z for which aji +bji Z ? ai0 +bi0 Z for any i0 and hence compute the intersections of these functions. The functions for xji and xji+1 intersect in dji = (aji ?aji+1 )/(bji+1 ?bji ). Observe if dji ? dji?1 , then aji +bji Z ? max{aji?1 +bji?1 Z, aji+1 +bji+1 Z} for all Z: xji is dominated and hence dropped from Sj . In this case we compute the intersection of the affine functions associated with xj?1 and xj+1 and iterate the process. 4 Points in Sj may be dominated by the rightmost (non-dominated) point in Sj?1 . Thus, we compute the intersection of the rightmost point of Sj?1 and the leftmost point of Sj , iteratively dropping all dominated points of Sj . If all points of Sj are dominated, we continue the scan with Sj+1 and so on. Observe that we may stop this scan once there is a point that is not dominated, since the points in any sequence Sj have non-decreasing d-values. If some of the remaining points in Sj are dominated by a point in Sj 0 with j 0 < j ? 1, then this will be determined when the scan initiated by Sj 0 reaches Sj . Subsequently, we check the other direction, i.e. whether xj1 dominates elements of Sj?1 , starting with the rightmost element of Sj?1 . These checks for dominance are performed in parallel for neighboring sequences. [8] showed how to compute sequentially the expected maximum of a collection of affine functions. In particular, their Eq. (14) [8, p. 605] gives En [maxi {ai +bi Z} ? maxi ai ] = PC Pk?1 j=1 h=1 (bjh+1 ?bjh )u(?|djh |), where u is defined as u(z) = z?(z) + ?(z) for the CDF and PDF of the normal distribution. We compute the inner sums simultaneously; the computation of the outer sum could be parallelized by recursively adding pairs of inner sums, although we do not do so to avoid communication overhead. We summarize the parallel algorithm below. The Parallel Algorithm to compute (`(n+1) , x(n+1) ): 1. Scatter the pairs (`, x) ? [M ]0 ? A among the machines. 2. Each computes MKGn (`, x) for its pairs. To compute MKGn (`, x) in parallel: a. Sort the points in A by ascending ? ?xn0 (`, x) in parallel, thereby removing dominated points. Let S be the sorted sequence. b. Split S into sequences S1 , . . . , SC , where P C is the number of cores used to compute MKGn (`, x). Each core computes xi ?SC (bi+1 ? bi )u(?|di |) in parallel, then the partial sums are added to obtain En [maxi {ai + bi Z} ? maxi ai ]. 3. Determine (`(n+1) , x(n+1) ) ? argmax`?[M ]0 ,x?D MKGn (`, x) in parallel. 3.3 Summary of the misoKG Algorithm. 1. Using samples from all information sources, estimate hyperparameters of the Gaussian process prior as described in Sect. B.2. Then calculate the posterior on f based on the prior and samples. 2. Until the budget for samples is exhausted do: Determine the information source `?[M ]0 and the design x?D that maximize the misoKG factor proposed in Eq. (1) and observe IS `(x). Update the posterior distribution with the new observation. 3. Return argmaxx0 ?A ?(n) (0, x0 ). 3.4 Provable Performance Guarantees. The misoKG chooses an IS and an x such that the expected gain normalized by the query cost is maximized. Thus, misoKG is one-step Bayes optimal in this respect, by construction. We establish an additive bound on the difference between misoKG?s solution and the unknown optimum, as the number of queries N ? ?. For this argument we suppose that ?(`, x)=0 ?`, x and ?0 is either the squared exponential kernel or a four times differentiable Mat?rn kernel. Moreover, let xOPT ? argmaxx0 ?D f (0, x0 ), and d = maxx0 ?D minx00 ?A dist(x0 , x00 ). Theorem 1. Let x?N ? A be the point that misoKG recommends in iteration N . For each p ? [0, 1) there is a constant Kp such that with probability p lim f (0, x?N ) ? f (0, xOPT ) ? Kp ? d. N ?? We point out that Frazier, Powell, and Dayanik [8] showed in their seminal work an analogous result for the case of a single information source with uniform query cost (Theorem 4 in [8]). 5 In Sect. A we prove the statement for the MISO setting that allows multiple information sources that each have query costs c` (x) varying over the search domain D. This proof is simple and short. Also note that Theorem 3 establishes consistency of misoKG for the special case that D is finite, since then d = 0. Interestingly, we can compute Kp given ? and p. Therefore, we can control the additive error Kp ? d by increasing the density of A, leveraging the scalability of our parallel algorithm. 4 Numerical Experiments We now compare misoKG to other state-of-the-art MISO algorithms. We implemented misoKG?s statistical model and acquisition function in Python 2.7 and C++ leveraging functionality from the Metrics Optimization Engine [23]. We used a gradient-based optimizer [28] that first finds an optimizer via multiple restarts for each IS ` separately and then picks (`(n+1) , x(n+1) ) with maximum misoKG factor among these. An implementation of our method is available at https://github.com/misoKG/. We compare to misoEI of Lam et al. [18] and to MTBO+, an improved version of Multi-Task Bayesian Optimization proposed by Swersky et al. [34]. Following a recommendation of Snoek 2016, our implementation of MTBO+ uses an improved formulation of the acquisition function given by Hern?ndezLobato et al. [12], Snoek and et al. [31], but otherwise is identical to MTBO; in particular, it uses the statistical model of [34]. Sect. E provides detailed descriptions of these algorithms. Experimental Setup. We conduct experiments on the following test problems: (1) the 2dimensional Rosenbrock function modified to fit the MISO setting by Lam et al. [18]; (2) a MISO benchmark proposed by Swersky et al. [34] in which we optimize the 4 hyperparameters of a machine learning algorithm, using a small, related set of smaller images as cheap IS; (3) an assemble-to-order problem from Hong and Nelson [13] in which we optimize an 8-dimensional target stock vector to maximize the expected daily profit of a company as estimated by a simulator. In MISO settings the amount of initial data that one can use to inform the methods about each information source is typically dictated by the application, in particular by resource constraints and the availability of the respective source. In our experiments all methods were given identical initial datasets for each information source in every replication; these sets were drawn randomly via Latin Hypercube designs. For the sake of simplicity, we provided the same number of points for each IS, set to 2.5 points per dimension of the design space D. Regarding the kernel and mean function, MTBO+ uses the settings provided in [31]. The other algorithms used the squared exponential kernel and a constant mean function set to the average of a random sample. We report the ?gain? over the best initial solution, that is the true objective value of the respective design that a method would return at each iteration minus the best value in the initial data set. If the true objective value is not known for a given design, we report the value obtained from the information source of highest fidelity. This gain is plotted as a function of the total cost, that is the cumulative cost for invoking the information sources plus the fixed cost for the initial data; this metric naturally generalizes the number of function evaluations prevalent in Bayesian optimization. Note that the computational overhead of choosing the next information source and sample is omitted, as it is negligible compared to invoking an information source in real-world applications. Error bars are shown at the mean ? 2 standard errors averaged over at least 100 runs of each algorithm. For deterministic sources a jitter of 10?6 is added to avoid numerical issues during matrix inversion. 4.1 The Rosenbrock Benchmarks We consider the design space D = [?2, 2]2 , and M = 2 information sources. IS 0 is the Rosenbrock function g(x) = (1 ? x1 )2 + 100 ? (x2 ? x21 )2 plus optional Gaussian noise u ? ?. IS 1 returns g(x)+v ?sin(10?x1 +5?x2 ), where the additional oscillatory component serves as model discrepancy. We assume a cost of 1000 for each query to IS 0 and a cost of 1 for IS 1. Since all methods converged to good solutions within few queries, we investigate the ratio of gain to cost: Fig. 1 (l) displays the gain of each method over the best initial solution as a function of the total cost inflicted by querying information sources. The new method misoKG offers a significantly better gain per unit cost and finds an almost optimal solution typically within 5 ? 10 samples. Interestingly, misoKG relies only on cheap samples, proving its ability to successfully handle uncertainty. MTBO+, 6 40 30 35 25 30 misoKG MTBO+ misoEI 15 25 Gain Gain 20 misoKG MTBO+ misoEI 20 15 10 10 5 5 0 5005 5010 5015 5020 Total Cost 5025 0 255 5030 260 265 270 Total Cost 275 280 Figure 1: (l) The Rosenbrock benchmark with the parameter setting of [18]: misoKG offers an excellent gain-to-cost ratio and outperforms its competitors substantially. (r) The Rosenbrock benchmark with the alternative setup. on the other hand, struggles initially but then eventually obtains a near-optimal solution, too. To this end, it makes usually one or two queries of the expensive truth source after about 40 steps. misoEI shows a odd behavior: it takes several queries, one of them to IS 0, before it improves over the best initial design for the first time. Then it jumps to a very good solution and subsequently samples only the cheap IS. For the second setup, we set u = 1, v = 2, and suppose for IS 0 uniform noise of ?0 (x) = 1 and query cost c0 (x) = 50. Now the difference in costs is much smaller, while the variance is considerably bigger. The results are displayed in Fig. 1 (r): as for the first configuration, misoKG outperforms the other methods from the start. Interestingly, misoEI?s performance is drastically decreased compared to the first setup, since it only queries the expensive truth. Looking closer, we see that misoKG initially queries only the cheap information source IS 1 until it comes close to an optimal value after about five samples. It starts to query IS 0 occasionally later. 4.2 The Image Classification Benchmark This classification problem was introduced by Swersky et al. [34] to demonstrate that MTBO can reduce the cost of hyperparameter optimization by leveraging a small dataset as information source. The goal is to optimize four hyperparameters of the logistic regression algorithm [36] using a stochastic gradient method with mini-batches (the learning rate, the L2-regularization parameter, the batch size, and the number of epochs) to minimize the classification error on the MNIST dataset [21]. This dataset contains 70,000 images of handwritten digits: each image has 784 pixels. IS 1 uses the USPS dataset [38] of about 9000 images with 256 pixels each. The query costs are 4.5 for IS 1 and 43.69 for IS 0. A closer examination shows that IS 1 is subject to considerable bias with respect to IS 0, making it a challenge for MISO algorithms. Fig.2 (l) summarizes performance: initially, misoKG and MTBO+ are on par. Both clearly outperform misoEI that therefore was stopped after 50 iterations. misoKG and MTBO+ continued for 150 steps (with a lower number of replications). misoKG usually achieves an optimal test error of about 7.1% on the MNIST testset after about 80 queries, matching the classification performance of the best setting reported by Swersky et al. [34]. Moreover, misoKG achieves better solutions than MTBO+ at the same costs. Note that the results in [34] show that MTBO+ will also converge to the optimum eventually. 4.3 The Assemble-To-Order Benchmark The assemble-to-order (ATO) benchmark is a reinforcement learning problem from a business application where the goal is to optimize an 8-dimensional target level vector over [0, 20]8 (see Sect. G for details). We set up three information sources: IS 0 and 2 use the discrete event simulator of Xie et al. [42], whereas the cheapest source IS 1 invokes the implementation of Hong and Nelson. IS 0 models the truth. 7 40 0.20 misoKG MTBO+ misoEI 0.18 30 25 0.14 Gain Testerror 0.16 35 0.12 20 15 0.10 10 0.08 5 0.06 6.2 6.4 6.6 6.8 7.0 log(Total Cost) 7.2 0 7.4 misoKG MTBO+ misoEI 6.08 6.10 6.12 6.14 6.16 log(Total Cost) 6.18 6.20 6.22 Figure 2: (l) The performance on the image classification benchmark of [34]. misoKG achieves better test errors after about 80 steps and converges to the global optimum. (r) misoKG outperforms the other algorithms on the assemble-to-order benchmark that has significant model discrepancy. The two simulators differ subtly in the model of the inventory system. However, the effect in estimated objective value is significant: on average the outputs of both simulators at the same target vector differ by about 5% of the score of the global optimum, which is about 120, whereas the largest observed bias out of 1000 random samples was 31.8. Thus, we are witnessing a significant model discrepancy. Fig. 2 (r) summarizes the performances. misoKG outperforms the other algorithms from the start: misoKG averages at a gain of 26.1, but inflicts only an average query cost of 54.6 to the information sources. This is only 6.3% of the query cost that misoEI requires to achieve a comparable score. Interestingly, misoKG and MTBO+ utilize mostly the cheap biased IS, and therefore are able to obtain significantly better gain to cost ratios than misoEI. misoKG?s typically first calls IS 2 after about 60 ? 80 steps. In total, misoKG queries IS 2 about ten times within the first 150 steps; in some replications misoKG makes one late call to IS 0 when it has already converged. Our interpretation is that misoKG exploits the cheap, biased IS 1 to zoom in on the global optimum and switches to the unbiased but noisy IS 2 to identify the optimal solution exactly. This is the expected (and desired) behavior for misoKG when the uncertainty of f (0, x? ) is not expected to be reduced sufficiently by queries to IS 1. MTBO+ trades off the gain versus cost differently: it queries IS 0 once or twice after 100 steps and directs all other queries to IS 1, which might explain the observed lower performance. misoEI, which employs a two-step heuristic for trading off predicted gain and query cost, almost always chose to evaluate the most expensive IS. 5 Conclusion We have presented a novel algorithm for MISO that uses a novel mean function and covariance matrix motivated by a MISO-specific generative model. We have also proposed a novel acquisition function that extends the knowledge gradient to the MISO setting and comes with a fast parallel method for computing it. 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An informational approach to the global optimization of expensive-to-evaluate functions. Journal of Global Optimization, 44(4):509?534, 2009. 10 [40] R. L. Winkler. Combining probability distributions from dependent information sources. Management Science, 27(4):479?488, 1981. [41] J. Wu, M. Poloczek, A. G. Wilson, and P. I. Frazier. Bayesian optimization with gradients. In Advances in Neural Information Processing Systems, 2017. Accepted for Publication. Also available at https://arxiv.org/abs/1703.04389. [42] J. Xie, P. I. Frazier, and S. Chick. Assemble to order simulator. http://simopt.org/wiki/ index.php?title=Assemble_to_Order&oldid=447, 2012. Last Accessed on 05/16/2017. 11
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Deep Reinforcement Learning from Human Preferences Paul F Christiano OpenAI [email protected] Miljan Martic DeepMind [email protected] Jan Leike DeepMind [email protected] Shane Legg DeepMind [email protected] Tom B Brown Google Brain? [email protected] Dario Amodei OpenAI [email protected] Abstract For sophisticated reinforcement learning (RL) systems to interact usefully with real-world environments, we need to communicate complex goals to these systems. In this work, we explore goals defined in terms of (non-expert) human preferences between pairs of trajectory segments. We show that this approach can effectively solve complex RL tasks without access to the reward function, including Atari games and simulated robot locomotion, while providing feedback on less than 1% of our agent?s interactions with the environment. This reduces the cost of human oversight far enough that it can be practically applied to state-of-the-art RL systems. To demonstrate the flexibility of our approach, we show that we can successfully train complex novel behaviors with about an hour of human time. These behaviors and environments are considerably more complex than any which have been previously learned from human feedback. 1 Introduction Recent success in scaling reinforcement learning (RL) to large problems has been driven in domains that have a well-specified reward function (Mnih et al., 2015, 2016; Silver et al., 2016). Unfortunately, many tasks involve goals that are complex, poorly-defined, or hard to specify. Overcoming this limitation would greatly expand the possible impact of deep RL and could increase the reach of machine learning more broadly. For example, suppose that we wanted to use reinforcement learning to train a robot to clean a table or scramble an egg. It?s not clear how to construct a suitable reward function, which will need to be a function of the robot?s sensors. We could try to design a simple reward function that approximately captures the intended behavior, but this will often result in behavior that optimizes our reward function without actually satisfying our preferences. This difficulty underlies recent concerns about misalignment between our values and the objectives of our RL systems (Bostrom, 2014; Russell, 2016; Amodei et al., 2016). If we could successfully communicate our actual objectives to our agents, it would be a significant step towards addressing these concerns. If we have demonstrations of the desired task, we can use inverse reinforcement learning (Ng and Russell, 2000) or imitation learning to copy the demonstrated behavior. But these approaches are not directly applicable to behaviors that are difficult for humans to demonstrate (such as controlling a robot with many degrees of freedom but non-human morphology). ? Work done while at OpenAI. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. An alternative approach is to allow a human to provide feedback on our system?s current behavior and to use this feedback to define the task. In principle this fits within the paradigm of reinforcement learning, but using human feedback directly as a reward function is prohibitively expensive for RL systems that require hundreds or thousands of hours of experience. In order to practically train deep RL systems with human feedback, we need to decrease the amount of feedback required by several orders of magnitude. We overcome this difficulty by asking humans to compare possible trajectories of the agent, using that data to learn a reward function, and optimizing the learned reward function with RL. This basic approach has been explored in the past, but we confront the challenges involved in scaling it up to modern deep RL and demonstrate by far the most complex behaviors yet learned from human feedback. Our experiments take place in two domains: Atari games in the Arcade Learning Environment (Bellemare et al., 2013), and robotics tasks in the physics simulator MuJoCo (Todorov et al., 2012). We show that a small amount of feedback from a non-expert human, ranging from fifteen minutes to five hours, suffice to learn both standard RL tasks and novel hard-to-specify behaviors such as performing a backflip or driving with the flow of traffic. 1.1 Related Work A long line of work studies reinforcement learning from human ratings or rankings, including Akrour et al. (2011), Pilarski et al. (2011), Akrour et al. (2012), Wilson et al. (2012), Sugiyama et al. (2012), Wirth and F?rnkranz (2013), Daniel et al. (2015), El Asri et al. (2016), Wang et al. (2016), and Wirth et al. (2016). Other lines of research consider the general problem of reinforcement learning from preferences rather than absolute reward values (F?rnkranz et al., 2012; Akrour et al., 2014; Wirth et al., 2016), and optimizing using human preferences in settings other than reinforcement learning (Machwe and Parmee, 2006; Secretan et al., 2008; Brochu et al., 2010; S?rensen et al., 2016). Our algorithm follows the same basic approach as Akrour et al. (2012) and Akrour et al. (2014), but considers much more complex domains and behaviors. The complexity of our environments force us to use different RL algorithms, reward models, and training strategies. One notable difference is that Akrour et al. (2012) and Akrour et al. (2014) elicit preferences over whole trajectories rather than short clips, and so would require about an order of magnitude more human time per data point. Our approach to feedback elicitation closely follows Wilson et al. (2012). However, Wilson et al. (2012) assumes that the reward function is the distance to some unknown (linear) ?target? policy, and is never tested with real human feedback. TAMER (Knox, 2012; Knox and Stone, 2013) also learns a reward function from human feedback, but learns from ratings rather than comparisons, has the human observe the agent as it behaves, and has been applied to settings where the desired policy can be learned orders of magnitude more quickly. Compared to all prior work, our key contribution is to scale human feedback up to deep reinforcement learning and to learn much more complex behaviors. This fits into a recent trend of scaling reward learning methods to large deep learning systems, for example inverse RL (Finn et al., 2016), imitation learning (Ho and Ermon, 2016; Stadie et al., 2017), semi-supervised skill generalization (Finn et al., 2017), and bootstrapping RL from demonstrations (Silver et al., 2016; Hester et al., 2017). 2 2.1 Preliminaries and Method Setting and Goal We consider an agent interacting with an environment over a sequence of steps; at each time t the agent receives an observation ot 2 O from the environment and then sends an action at 2 A to the environment. In traditional reinforcement learning, the environment would also supply a reward rt 2 R and the agent?s goal would be to maximize the discounted sum of rewards. Instead of assuming that the environment produces a reward signal, we assume that there is a human overseer who can express 2 preferences between trajectory segments. A trajectory segment is a sequence of observations and k 2 actions, = ((o0 , a0 ), (o1 , a1 ), . . . , (ok 1 , ak 1 )) 2 (O ? A) . Write 1 to indicate that the 1 2 human preferred trajectory segment to trajectory segment . Informally, the goal of the agent is to produce trajectories which are preferred by the human, while making as few queries as possible to the human. More precisely, we will evaluate our algorithms? behavior in two ways: Quantitative: We say that preferences o10 , a10 , . . . , o1k are generated by a reward function2 r : O ? A ! R if 1 1 , ak 1 o20 , a20 , . . . , o2k 2 1 , ak 1 whenever r o10 , a10 + ? ? ? + r o1k 1 1 , ak 1 > r o20 , a20 + ? ? ? + r o2k 2 1 , ak 1 . If the human?s preferences are generated by a reward function r, then our agent ought to receive a high total reward according to r. So if we know the reward function r, we can evaluate the agent quantitatively. Ideally the agent will achieve reward nearly as high as if it had been using RL to optimize r. Qualitative: Sometimes we have no reward function by which we can quantitatively evaluate behavior (this is the situation where our approach would be practically useful). In these cases, all we can do is qualitatively evaluate how well the agent satisfies the human?s preferences. In this paper, we will start from a goal expressed in natural language, ask a human to evaluate the agent?s behavior based on how well it fulfills that goal, and then present videos of agents attempting to fulfill that goal. Our model based on trajectory segment comparisons is very similar to the trajectory preference queries used in Wilson et al. (2012), except that we don?t assume that we can reset the system to an arbitrary state3 and so our segments generally begin from different states. This complicates the interpretation of human comparisons, but we show that our algorithm overcomes this difficulty even when the human raters have no understanding of our algorithm. 2.2 Our Method At each point in time our method maintains a policy ? : O ! A and a reward function estimate r? : O ? A ! R, each parametrized by deep neural networks. These networks are updated by three processes: 1. The policy ? interacts with the environment to produce a set of trajectories {? 1 , . . . , ? i }. The parameters of ? are updated by a traditional reinforcement learning algorithm, in order to maximize the sum of the predicted rewards rt = r?(ot , at ). 2. We select pairs of segments 1 , 2 from the trajectories {? 1 , . . . , ? i } produced in step 1, and send them to a human for comparison. 3. The parameters of the mapping r? are optimized via supervised learning to fit the comparisons collected from the human so far. These processes run asynchronously, with trajectories flowing from process (1) to process (2), human comparisons flowing from process (2) to process (3), and parameters for r? flowing from process (3) to process (1). The following subsections provide details on each of these processes. 2 Here we assume here that the reward is a function of the observation and action. In our experiments in Atari environments, we instead assume the reward is a function of the preceding 4 observations. In a general partially observable environment, we could instead consider reward functions that depend on the whole sequence of observations, and model this reward function with a recurrent neural network. 3 Wilson et al. (2012) also assumes the ability to sample reasonable initial states. But we work with high dimensional state spaces for which random states will not be reachable and the intended policy inhabits a low-dimensional manifold. 3 2.2.1 Optimizing the Policy After using r? to compute rewards, we are left with a traditional reinforcement learning problem. We can solve this problem using any RL algorithm that is appropriate for the domain. One subtlety is that the reward function r? may be non-stationary, which leads us to prefer methods which are robust to changes in the reward function. This led us to focus on policy gradient methods, which have been applied successfully for such problems (Ho and Ermon, 2016). In this paper, we use advantage actor-critic (A2C; Mnih et al., 2016) to play Atari games, and trust region policy optimization (TRPO; Schulman et al., 2015) to perform simulated robotics tasks. In each case, we used parameter settings which have been found to work well for traditional RL tasks. The only hyperparameter which we adjusted was the entropy bonus for TRPO. This is because TRPO relies on the trust region to ensure adequate exploration, which can lead to inadequate exploration if the reward function is changing. We normalized the rewards produced by r? to have zero mean and constant standard deviation. This is a typical preprocessing step which is particularly appropriate here since the position of the rewards is underdetermined by our learning problem. 2.2.2 Preference Elicitation The human overseer is given a visualization of two trajectory segments, in the form of short movie clips. In all of our experiments, these clips are between 1 and 2 seconds long. The human then indicates which segment they prefer, that the two segments are equally good, or that they are unable to compare the two segments. The human judgments are recorded in a database D of triples 1 , 2 , ? , where 1 and 2 are the two segments and ? is a distribution over {1, 2} indicating which segment the user preferred. If the human selects one segment as preferable, then ? puts all of its mass on that choice. If the human marks the segments as equally preferable, then ? is uniform. Finally, if the human marks the segments as incomparable, then the comparison is not included in the database. 2.2.3 Fitting the Reward Function We can interpret a reward function estimate r? as a preference-predictor if we view r? as a latent factor explaining the human?s judgments and assume that the human?s probability of preferring a segment i depends exponentially on the value of the latent reward summed over the length of the clip:4 P ? ? exp r? o1t , a1t 2 P P P? 1 = . (1) exp r?(o1t , a1t ) + exp r?(o2t , a2t ) We choose r? to minimize the cross-entropy loss between these predictions and the actual human labels: X ? ? ? ? 2 1 loss(? r) = ?(1) log P? 1 + ?(2) log P? 2 . ( 1 , 2 ,?)2D This follows the Bradley-Terry model (Bradley and Terry, 1952) for estimating score functions from pairwise preferences, and is the specialization of the Luce-Shephard choice rule (Luce, 2005; Shepard, 1957) to preferences over trajectory segments. Our actual algorithm incorporates a number of modifications to this basic approach, which early experiments discovered to be helpful and which are analyzed in Section 3.3: ? We fit an ensemble of predictors, each trained on |D| triples sampled from D with replacement. The estimate r? is defined by independently normalizing each of these predictors and then averaging the results. ? A fraction of 1/e of the data is held out to be used as a validation set for each predictor. We use `2 regularization and adjust the regularization coefficient to keep the validation loss between 1.1 and 1.5 times the training loss. In some domains we also apply dropout for regularization. 4 Equation 1 does not use discounting, which could be interpreted as modeling the human to be indifferent about when things happen in the trajectory segment. Using explicit discounting or inferring the human?s discount function would also be reasonable choices. 4 ? Rather than applying a softmax directly as described in Equation 1, we assume there is a 10% chance that the human responds uniformly at random. Conceptually this adjustment is needed because human raters have a constant probability of making an error, which doesn?t decay to 0 as the difference in reward difference becomes extreme. 2.2.4 Selecting Queries We decide how to query preferences based on an approximation to the uncertainty in the reward function estimator, similar to Daniel et al. (2014): we sample a large number of pairs of trajectory segments of length k from the latest agent-environment interactions, use each reward predictor in our ensemble to predict which segment will be preferred from each pair, and then select those trajectories for which the predictions have the highest variance across ensemble members5 This is a crude approximation and the ablation experiments in Section 3 show that in some tasks it actually impairs performance. Ideally, we would want to query based on the expected value of information of the query (Akrour et al., 2012; Krueger et al., 2016), but we leave it to future work to explore this direction further. 3 Experimental Results We implemented our algorithm in TensorFlow (Abadi et al., 2016). We interface with MuJoCo (Todorov et al., 2012) and the Arcade Learning Environment (Bellemare et al., 2013) through the OpenAI Gym (Brockman et al., 2016). 3.1 Reinforcement Learning Tasks with Unobserved Rewards In our first set of experiments, we attempt to solve a range of benchmark tasks for deep RL without observing the true reward. Instead, the agent learns about the goal of the task only by asking a human which of two trajectory segments is better. Our goal is to solve the task in a reasonable amount of time using as few queries as possible. In our experiments, feedback is provided by contractors who are given a 1-2 sentence description of each task before being asked to compare several hundred to several thousand pairs of trajectory segments for that task (see Appendix B for the exact instructions given to contractors). Each trajectory segment is between 1 and 2 seconds long. Contractors responded to the average query in 3-5 seconds, and so the experiments involving real human feedback required between 30 minutes and 5 hours of human time. For comparison, we also run experiments using a synthetic oracle whose preferences are generated (in the sense of Section 2.1) by the real reward6 . We also compare to the baseline of RL training using the real reward. Our aim here is not to outperform but rather to do nearly as well as RL without access to reward information and instead relying on much scarcer feedback. Nevertheless, note that feedback from real humans does have the potential to outperform RL (and as shown below it actually does so on some tasks), because the human feedback might provide a better-shaped reward. We describe the details of our experiments in Appendix A, including model architectures, modifications to the environment, and the RL algorithms used to optimize the policy. 3.1.1 Simulated Robotics The first tasks we consider are eight simulated robotics tasks, implemented in MuJoCo (Todorov et al., 2012), and included in OpenAI Gym (Brockman et al., 2016). We made small modifications to these tasks in order to avoid encoding information about the task in the environment itself (the modifications are described in detail in Appendix A). The reward functions in these tasks are quadratic functions of distances, positions and velocities, and most are linear. We included a simple cartpole 5 Note that trajectory segments almost never start from the same state. In the case of Atari games with sparse rewards, it is relatively common for two clips to both have zero reward in which case the oracle outputs indifference. Because we considered clips rather than individual states, such ties never made up a large majority of our data. Moreover, ties still provide significant information to the reward predictor as long as they are not too common. 6 5 Figure 1: Results on MuJoCo simulated robotics as measured on the tasks? true reward. We compare our method using real human feedback (purple), our method using synthetic feedback provided by an oracle (shades of blue), and reinforcement learning using the true reward function (orange). All curves are the average of 5 runs, except for the real human feedback, which is a single run, and each point is the average reward over five consecutive batches. For Reacher and Cheetah feedback was provided by an author due to time constraints. For all other tasks, feedback was provided by contractors unfamiliar with the environments and with our algorithm. The irregular progress on Hopper is due to one contractor deviating from the typical labeling schedule. task (?pendulum?) for comparison, since this is representative of the complexity of tasks studied in prior work. Figure 1 shows the results of training our agent with 700 queries to a human rater, compared to learning from 350, 700, or 1400 synthetic queries, as well as to RL learning from the real reward. With 700 labels we are able to nearly match reinforcement learning on all of these tasks. Training with learned reward functions tends to be less stable and higher variance, while having a comparable mean performance. Surprisingly, by 1400 labels our algorithm performs slightly better than if it had simply been given the true reward, perhaps because the learned reward function is slightly better shaped?the reward learning procedure assigns positive rewards to all behaviors that are typically followed by high reward. The difference may also be due to subtle changes in the relative scale of rewards or our use of entropy regularization. Real human feedback is typically only slightly less effective than the synthetic feedback; depending on the task human feedback ranged from being half as efficient as ground truth feedback to being equally efficient. On the Ant task the human feedback significantly outperformed the synthetic feedback, apparently because we asked humans to prefer trajectories where the robot was ?standing upright,? which proved to be useful reward shaping. (There was a similar bonus in the RL reward function to encourage the robot to remain upright, but the simple hand-crafted bonus was not as useful.) 3.1.2 Atari The second set of tasks we consider is a set of seven Atari games in the Arcade Learning Environment (Bellemare et al., 2013), the same games presented in Mnih et al., 2013. Figure 2 shows the results of training our agent with 5,500 queries to a human rater, compared to learning from 350, 700, or 1400 synthetic queries, as well as to RL learning from the real reward. Our method has more difficulty matching RL in these challenging environments, but nevertheless it displays substantial learning on most of them and matches or even exceeds RL on some. Specifically, 6 Figure 2: Results on Atari games as measured on the tasks? true reward. We compare our method using real human feedback (purple), our method using synthetic feedback provided by an oracle (shades of blue), and reinforcement learning using the true reward function (orange). All curves are the average of 3 runs, except for the real human feedback which is a single run, and each point is the average reward over about 150,000 consecutive frames. on BeamRider and Pong, synthetic labels match or come close to RL even with only 3,300 such labels. On Seaquest and Qbert synthetic feedback eventually performs near the level of RL but learns more slowly. On SpaceInvaders and Breakout synthetic feedback never matches RL, but nevertheless the agent improves substantially, often passing the first level in SpaceInvaders and reaching a score of 20 on Breakout, or 50 with enough labels. On most of the games real human feedback performs similar to or slightly worse than synthetic feedback with the same number of labels, and often comparably to synthetic feedback that has 40% fewer labels. On Qbert, our method fails to learn to beat the first level with real human feedback; this may be because short clips in Qbert can be confusing and difficult to evaluate. Finally, Enduro is difficult for A3C to learn due to the difficulty of successfully passing other cars through random exploration, and is correspondingly difficult to learn with synthetic labels, but human labelers tend to reward any progress towards passing cars, essentially shaping the reward and thus outperforming A3C in this game (the results are comparable to those achieved with DQN). 3.2 Novel behaviors Experiments with traditional RL tasks help us understand whether our method is effective, but the ultimate purpose of human interaction is to solve tasks for which no reward function is available. Using the same parameters as in the previous experiments, we show that our algorithm can learn novel complex behaviors. We demonstrate: 1. The Hopper robot performing a sequence of backflips (see Figure 4). This behavior was trained using 900 queries in less than an hour. The agent learns to consistently perform a backflip, land upright, and repeat. 2. The Half-Cheetah robot moving forward while standing on one leg. This behavior was trained using 800 queries in under an hour. 3. Keeping alongside other cars in Enduro. This was trained with roughly 1,300 queries and 4 million frames of interaction with the environment; the agent learns to stay almost exactly even with other moving cars for a substantial fraction of the episode, although it gets confused by changes in background. 7 Figure 3: Performance of our algorithm on MuJoCo tasks after removing various components, as described in Section Section 3.3. All graphs are averaged over 5 runs, using 700 synthetic labels each. Videos of these behaviors can be found at https://goo.gl/MhgvIU. These behaviors were trained using feedback from the authors. 3.3 Ablation Studies In order to better understand the performance of our algorithm, we consider a range of modifications: 1. We pick queries uniformly at random rather than prioritizing queries for which there is disagreement (random queries). 2. We train only one predictor rather than an ensemble (no ensemble). In this setting, we also choose queries at random, since there is no longer an ensemble that we could use to estimate disagreement. 3. We train on queries only gathered at the beginning of training, rather than gathered throughout training (no online queries). 4. We remove the `2 regularization and use only dropout (no regularization). 5. On the robotics tasks only, we use trajectory segments of length 1 (no segments). 6. Rather than fitting r? using comparisons, we consider an oracle which provides the true total reward over a trajectory segment, and fit r? to these total rewards using mean squared error (target). The results are presented in Figure 3 for MuJoCo and Figure 4 for Atari. Training the reward predictor offline can lead to bizarre behavior that is undesirable as measured by the true reward (Amodei et al., 2016). For instance, on Pong offline training sometimes leads our agent to avoid losing points but not to score points; this can result in extremely long volleys (videos at https://goo.gl/L5eAbk). This type of behavior demonstrates that in general human feedback needs to be intertwined with RL rather than provided statically. Our main motivation for eliciting comparisons rather than absolute scores was that we found it much easier for humans to provide consistent comparisons than consistent absolute scores, especially on the continuous control tasks and on the qualitative tasks in Section 3.2; nevertheless it seems important to understand how using comparisons affects performance. For continuous control tasks we found that predicting comparisons worked much better than predicting scores. This is likely because the scale of rewards varies substantially and this complicates the regression problem, which is smoothed significantly when we only need to predict comparisons. In the Atari tasks we clipped rewards 8 Figure 4: Performance of our algorithm on Atari tasks after removing various components, as described in Section 3.3. All curves are an average of 3 runs using 5,500 synthetic labels (see minor exceptions in Section A.2). and effectively only predicted the sign, avoiding these difficulties (this is not a suitable solution for the continuous control tasks because the magnitude of the reward is important to learning). In these tasks comparisons and targets had significantly different performance, but neither consistently outperformed the other. We also observed large performance differences when using single frames rather than clips.7 In order to obtain the same results using single frames we would need to have collected significantly more comparisons. In general we discovered that asking humans to compare longer clips was significantly more helpful per clip, and significantly less helpful per frame. Shrinking the clip length below 1-2 seconds did not significantly decrease the human time required to label each clip in early experiments, and so seems less efficient per second of human time. In the Atari environments we also found that it was often easier to compare longer clips because they provide more context than single frames. 4 Discussion and Conclusions Agent-environment interactions are often radically cheaper than human interaction. We show that by learning a separate reward model using supervised learning, it is possible to reduce the interaction complexity by roughly 3 orders of magnitude. Although there is a large literature on preference elicitation and reinforcement learning from unknown reward functions, we provide the first evidence that these techniques can be economically scaled up to state-of-the-art reinforcement learning systems. This represents a step towards practical applications of deep RL to complex real-world tasks. In the long run it would be desirable to make learning a task from human preferences no more difficult than learning it from a programmatic reward signal, ensuring that powerful RL systems can be applied in the service of complex human values rather than low-complexity goals. Acknowledgments We thank Olivier Pietquin, Bilal Piot, Laurent Orseau, Pedro Ortega, Victoria Krakovna, Owain Evans, Andrej Karpathy, Igor Mordatch, and Jack Clark for reading drafts of the paper. We thank Tyler Adkisson, Mandy Beri, Jessica Richards, Heather Tran, and other contractors for providing the 7 We only ran these tests on continuous control tasks because our Atari reward model depends on a sequence of consecutive frames rather than a single frame, as described in Section A.2 9
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On the Fine-Grained Complexity of Empirical Risk Minimization: Kernel Methods and Neural Networks Arturs Backurs CSAIL MIT [email protected] Piotr Indyk CSAIL MIT [email protected] Ludwig Schmidt CSAIL MIT [email protected] Abstract Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods. While there is a large body of work on algorithms for various ERM problems, the exact computational complexity of ERM is still not understood. We address this issue for multiple popular ERM problems including kernel SVMs, kernel ridge regression, and training the final layer of a neural network. In particular, we give conditional hardness results for these problems based on complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. Under these assumptions, we show that there are no algorithms that solve the aforementioned ERM problems to high accuracy in sub-quadratic time. We also give similar hardness results for computing the gradient of the empirical loss, which is the main computational burden in many non-convex learning tasks. 1 Introduction Empirical risk minimization (ERM) has been highly influential in modern machine learning [37]. ERM underpins many core results in statistical learning theory and is one of the main computational problems in the field. Several important methods such as support vector machines (SVM), boosting, and neural networks follow the ERM paradigm [34]. As a consequence, the algorithmic aspects of ERM have received a vast amount of attention over the past decades. This naturally motivates the following basic question: What are the computational limits for ERM algorithms? In this work, we address this question both in convex and non-convex settings. Convex ERM problems have been highly successful in a wide range of applications, giving rise to popular methods such as SVMs and logistic regression. Using tools from convex optimization, the resulting problems can be solved in polynomial time. However, the exact time complexity of many important ERM problems such as kernel SVMs is not yet well understood. As the size of data sets in machine learning continues to grow, this question is becoming increasingly important. For ERM problems with millions of high-dimensional examples, even quadratic time algorithms can become painfully slow (or expensive) to run. Non-convex ERM problems have also attracted extensive research interest, e.g., in the context of deep neural networks. First order methods that follow the gradient of the empirical loss are not guaranteed to find the global minimizer in this setting. Nevertheless, variants of gradient descent are by far the most common method for training large neural networks. Here, the computational bottleneck is to compute a number of gradients, not necessarily to minimize the empirical loss globally. Although we 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. can compute gradients in polynomial time, the large number of parameters and examples in modern deep learning still makes this a considerable computational challenge. Unfortunately, there are only few existing results concerning the exact time complexity of ERM or gradient computations. Since the problems have polynomial time algorithms, the classical machinery from complexity theory (such as NP hardness) is too coarse to apply. Oracle lower bounds from optimization offer useful guidance for convex ERM problems, but the results only hold for limited classes of algorithms. Moreover, they do not account for the cost of executing the oracle calls, as they simply lower bound their number. Overall, we do not know if common ERM problems allow for algorithms that compute a high-accuracy solution in sub-quadratic or even nearly-linear time for all instances.1 Furthermore, we do not know if there are more efficient techniques for computing (mini-)batch gradients than simply treating each example in the batch independently.2 We address both questions for multiple well-studied ERM problems. Hardness of ERM. First, we give conditional hardness results for minimizing the empirical risk in several settings, including kernel SVMs, kernel ridge regression (KRR), and training the top layer of a neural network. Our results give evidence that no algorithms can solve these problems to high accuracy in strongly sub-quadratic time. Moreover, we provide similar conditional hardness results for kernel PCA. All of these methods are popular learning algorithms due to the expressiveness of the kernel or network embedding. Our results show that this expressiveness also leads to an expensive computational problem. Hardness of gradient computation in neural networks. Second, we address the complexity of computing a gradient for the empirical risk of a neural network. In particular, we give evidence that computing (or even approximating, up to polynomially large factors) the norm of the gradient of the top layer in a neural network takes time that is ?rectangular?. The time complexity cannot be significantly better than O(n ? m), where m is the number of examples and n is the number of units in the network. Hence, there are no algorithms that compute batch gradients faster than handling each example individually, unless common complexity-theoretic assumptions fail. Our hardness results for gradient computation apply to common activation functions such as ReLU or sigmoid units. We remark that for polynomial activation functions (for instance, studied in [24]), significantly faster algorithms do exist. Thus, our results can be seen as mapping the ?efficiency landscape? of basic machine learning sub-routines. They distinguish between what is possible and (likely) impossible, suggesting further opportunities for improvement. Our hardness results are based on recent advances in fine-grained complexity and build on conjectures such as the Strong Exponential Time Hypothesis (SETH) [23, 22, 38]. SETH concerns the classic satisfiability problem for formulas in Conjunctive Normal Form (CNF). Informally, the conjecture states that there is no algorithm for checking satisfiability of a formula with n variables and m clauses in time less than O(cn ? poly(m)) for some c < 2.3 While our results are conditional, SETH has been employed in many recent hardness results. Its plausibility stems from the fact that, despite 60 years of research on satisfiability algorithms, no such improvement has been discovered. Our results hold for a significant range of the accuracy parameter. For kernel methods, our bounds hold for algorithms approximating the empirical risk up to a factor of 1+?, for log(1/?) = ?(log2 n)). Thus, they provide conditional quadratic lower bounds for algorithms with, say, a log 1/? runtime dependence on the approximation error ?. A (doubly) logarithmic dependence on 1/? is generally seen as the ideal rate of convergence in optimization, and algorithms with this property have been studied extensively in the machine learning community (cf. [12].). At the same time, approximate 1 More efficient algorithms exist if the running time is allowed to be polynomial in the accuracy parameter, e.g., [35] give such an algorithm for the kernel SVM problem that we consider as well. See also the discussion at the end of this section. 2 Consider a network with one hidden layer containing n units and a training set with m examples, for simplicity in small dimension d = O(log n). No known results preclude an algorithm that computes a full gradient in time O((n+m) log n). This would be significantly faster than the standard O(n?m?log n) approach of computing the full gradient example by example. 3 Note that SETH can be viewed as a significant strengthening of the P 6= NP conjecture, which only postulates that there is no polynomial time algorithm for CNF satisfiability. The best known algorithms for CNF satisfiability have running times of the form O(2(1?o(1))n ? poly(m)). 2 solutions to ERM problems can be sufficient for good generalization in learning tasks. Indeed, stochastic gradient descent (SGD) is often advocated as an efficient learning algorithm despite its polynomial dependence on 1/? in the optimization error [35, 15]. Our results support this viewpoint since SGD sidesteps the quadratic time complexity of our lower bounds. For other problems, our assumptions about the accuracy parameter are less stringent. In particular, for training the top layer of the neural network, we only need to assume that ? ? 1/n. Finally, our lower bounds for approximating the norm of the gradient in neural networks hold even if ? = nO(1) , i.e., for polynomial approximation factors (or alternatively, a constant additive factor for ReLU and sigmoid activation functions). Finally, we note that our results do not rule out algorithms that achieve a sub-quadratic running time for well-behaved instances, e.g., instances with low-dimensional structure. Indeed, many such approaches have been investigated in the literature, for instance the Nystr?m method or random features for kernel problems [40, 30]. Our results offer an explanation for the wide variety of techniques. The lower bounds are evidence that there is no ?silver bullet? algorithm for solving the aforementioned ERM problems in sub-quadratic time, to high accuracy, and for all instances. 2 Background Fine-grained complexity. We obtain our conditional hardness results via reductions from two well-studied problems: Orthogonal Vectors and Bichromatic Hamming Close Pair. Definition 1 (Orthogonal Vectors problem (OVP)). Given two sets A = {a1 , . . . , an } ? {0, 1}d and B = {b1 , . . . , bn } ? {0, 1}d of n binary vectors, decide if there exists a pair a ? A and b ? B such that aT b = 0. For OVP, we can assume without loss of generality that all vectors in B have the same number of 1s. This can be achieved by appending d entries to every bi and setting the necessary number of them to 1 and the rest to 0. We then append d entries to every ai and set all of them to 0. Definition 2 (Bichromatic Hamming Close Pair (BHCP) problem). Given two sets A = {a1 , . . . , an } ? {0, 1}d and B = {b1 , . . . , bn } ? {0, 1}d of n binary vectors and an integer t ? {2, . . . , d}, decide if there exists a pair a ? A and b ? B such that the number of coordinates in which they differ is less than t (formally, Hamming(a, b) := ||a ? b||1 < t). If there is such a pair (a, b), we call it a close pair. It is known that both OVP and BHCP require almost quadratic time (i.e., n2?o(1) ) for any d = ?(log n) assuming SETH [5].4 Furthermore, if we allow the sizes |A| = n and |B| = m to be different, both problems require (nm)1?o(1) time assuming SETH, as long as m = n? for some constant ? ? (0, 1) [17]. Our proofs will proceed by embedding OVP and BHCP instances into ERM problems. Such a reduction then implies that the ERM problem requires almost quadratic time if the SETH is true. If we could solve the ERM problem faster, we would also obtain a faster algorithm for the satisfiability problem. 3 Our contributions 3.1 Kernel ERM problems We provide hardness results for multiple kernel problems. In the following, let x1 , . . . , xn ? Rd be the n input vectors, where d = ?(log n). We use y1 , . . . , yn ? R as n labels or target values. Finally, let k(x, x0 ) denote a kernel function and let K ? Rn?n be the corresponding kernel matrix, defined as K i,j := k(xi , xj ) [33]. Concretely, we focus on the Gaussian kernel k(x, x0 ) := exp ?Ckx ? x0 k22 for some C > 0. We note that our results can be generalized to any kernel with exponential tail. 4 We use ?(g(n)) to denote any function f such that limn?? f (n)/g(n) = ?. Similarly, we use o(g(n)) to denote any function f such that limn?? f (n)/g(n) = 0. Consequently, we will refer to functions of the form ?(1) as super-constant and to n?(1) as super-polynomial. 3 Kernel SVM. For simplicity, we present our result for hard-margin SVMs without bias terms. This gives the following optimization problem. Definition 3 (Hard-margin SVM). A (primal) hard-margin SVM is an optimization problem of the following form: n 1 X minimize ?i ?j yi yj k(xi , xj ) 2 i,j=1 ?1 ,...,?n ?0 (1) subject to where f (x) := Pn i=1 yi f (xi ) ? 1, i = 1, . . . , n, ?i yi k(xi , x). The following theorem is our main result for SVMs, described in more detail in Section 4. In Sections B, C, and D of the supplementary material we provide similar hardness results for other common SVM variants, including the soft-margin version. Theorem 4. Let k(a, a0 ) be the Gaussian kernel with C = 100 log n and let ? = exp(??(log2 n)). Then approximating the optimal value of Equation (1) within a multiplicative factor 1 + ? requires almost quadratic time assuming SETH. Kernel Ridge Regression. Next we consider Kernel Ridge Regression, which is formally defined as follows. Definition 5 (Kernel ridge regression). Given a real value ? ? 0, the goal of kernel ridge regression is to output ? 1 arg min ||y ? K?||22 + ?T K?. n 2 2 ??R This problem is equivalent to computing the vector (K + ?I)?1 y. We focus on the special case where ? = 0 and the vector y has all equal entries y1 = . . . = yn = 1. In this case, the entrywise sum of K ?1 y is equal to the sum of the entries in K ?1 . Thus, we show hardness for computing the latter quantity (see Section F in the supplementary material for the proof). Theorem 6. Let k(a, a0 ) be the Gaussian kernel for any parameter C = ?(log n) and let ? = exp(??(log2 n)). Then computing the sum of the entries in K ?1 up to a multiplicative factor of 1 + ? requires almost quadratic time assuming SETH. Kernel PCA. Finally, we turn to the Kernel PCA problem, which we define as follows [26]. Definition 7 (Kernel Principal Component Analysis (PCA)). Let 1n be an n ? n matrix where each entry takes value 1/n, and define K 0 := (I ? 1n )K(I ? 1n ). The goal of the kernel PCA problem is to output the n eigenvalues of the matrix K 0 . In the above definition, the output only consists of the eigenvalues, not the eigenvectors. This is because computing all n eigenvectors trivially takes at least quadratic time since the output itself has quadratic size. Our hardness proof applies to the potentially simpler problem where only the eigenvalues are desired. Specifically, we show that computing the sum of the eigenvalues (i.e., the trace of the matrix) is hard. See Section E in the supplementary material for the proof. Theorem 8. Let k(a, a0 ) be the Gaussian kernel with C = 100 log n and let ? = exp(??(log2 n)). Then approximating the sum of the eigenvalues of K 0 = (I ? 1n )K(I ? 1n ) within a multiplicative factor of 1 + ? requires almost quadratic time assuming SETH. We note that the argument in the proof shows that even approximating the sum of the entries of K is hard. This provides an evidence of hardness of the kernel density estimation problem for Gaussian kernels, complementing recent upper bounds of [20]. 3.2 Neural network ERM problems We now consider neural networks. We focus on the problem of optimizing the top layer while keeping lower layers unchanged. An instance of this problem is transfer learning with large networks that would take a long time and many examples to train from scratch [31]. We consider neural networks of depth 2, with the sigmoid or ReLU activation function. Our hardness result holds for a more general class of ?nice? activation functions S as described later (see Definition 12). 4 Given n weight vectors w1 , . . . , wn ? Rd and n weights ?1 , . . . , ?n ? R, consider the function f : Rd ? R using a non-linearity S : R ? R: f (u) := n X ?j ? S(uT wj ) . j=1 This function can be implemented as a neural net that has d inputs, n nonlinear activations (units), and one linear output. To complete the ERM problem, we also require a loss function. Our hardness results hold for a large class of ?nice? loss functions, which includes the hinge loss and the logistic loss.5 Given a nice loss function and m input vectors a1 , . . . , am ? Rd with corresponding labels yi , we consider the following problem: m X minimize loss(yi , f (ui )). (2) ?1 ,...,?n ?R i=1 Our main result is captured by the following theorem (see Section 5 for the proof). For simplicity, we set m = n. Theorem 9. For any d = ?(log n), approximating the optimal value in Equation (2) up to a 1 requires almost quadratic time assuming SETH. multiplicative factor of 1 + 4n 3.3 Hardness of gradient computation Finally, we consider the problem of computing the gradient of the loss function for a given set of examples. We focus on the network architecture from the previous section. Formally, we obtain the following result: Theorem 10. Consider the empirical risk in Equation (2) under the following assumptions: (i) The function f is represented by a neural network with n units, n ? d parameters, and the ReLU activation function. (ii) We have d = ?(log n). (iii) The loss function is the logistic loss or hinge loss. Then approximating the `p -norm (for any p ? 1) of the gradient of the empirical risk for m examples  within a multiplicative factor of nC for any constant C > 0 takes at least O (nm)1?o(1) time assuming SETH. See Section 6 for the proof. We also prove a similar statement for the sigmoid activation function. At the same time, we remark that for polynomial activation functions, significantly faster algorithms do exist, using the polynomial lifting argument. Specifically, for the polynomial activation function of the form xr for some integer r ? 2, all gradients can be computed in O((n + m)dr ) time. Note that the running time of the standard backpropagation algorithm is O(dnm) for networks with this architecture. Thus one can improve over backpropagation for a non-trivial range of parameters, especially for quadratic activation function when r = 2. See Section H in the supplementary material for more details. 3.4 Related work Recent work has demonstrated conditional quadratic hardness results for many combinatorial optimization problems over graphs and sequences. These results include computing diameter in sparse graphs [32, 21], Local Alignment [2], Fr?chet distance [16], Edit Distance [13], Longest Common Subsequence, and Dynamic Time Warping [1, 17]. In the machine learning literature, [14] recently showed a tight lower bound for the problem of inferring the most likely path in a Hidden Markov Model, matching the upper bound achieved by the Viterbi algorithm [39]. As in our paper, the SETH and related assumptions underlie these lower bounds. To the best of our knowledge, our paper is the first application of this methodology to continuous (as opposed to combinatorial) optimization problems. There is a long line of work on the oracle complexity of optimization problems, going back to [28]. We refer the reader to [29] for these classical results. The oracle complexity of ERM problems is still 5 In the binary setting we consider, the logistic loss is equivalent to the softmax loss commonly employed in deep learning. 5 subject of active research, e.g., see [3, 19, 41, 9, 10]. The work closest to ours is [19], which gives quadratic time lower bounds for ERM algorithms that access the kernel matrix through an evaluation oracle or a low-rank approximation. The oracle results are fundamentally different from the lower bounds presented in our paper. Oracle lower bounds are typically unconditional, but inherently apply only to a limited class of algorithms due to their information-theoretic nature. Moreover, they do not account for the cost of executing the oracle calls, as they merely lower bound their number. In contrast, our results are conditional (based on the SETH and related assumptions), but apply to any algorithm and account for the total computational cost. This significantly broadens the reach of our results. We show that the hardness is not due to the oracle abstraction but instead inherent in the computational problem. 4 Overview of the hardness proof for kernel SVMs Let A = {a1 , . . . , an } ? {0, 1}d and B = {b1 , . . . , bn } ? {0, 1}d be the two sets of binary vectors from a BHCP instance with d = ?(log n). Our goal is to determine whether there is a close pair of vectors. We show how to solve this BHCP instance by reducing it to three computations of SVM, defined as follows: 1. We take the first set A of binary vectors, assign label 1 to all vectors, and solve the corresponding SVM on the n vectors: minimize ?1 ,...,?n ?0 n 1 X ?i ?j k(ai , aj ) 2 i,j=1 n X subject to (3) ?j k(ai , aj ) ? 1, i = 1, . . . , n. j=1 Note that we do not have yi in the expressions because all labels are 1. 2. We take the second set B of binary vectors, assign label ?1 to all vectors, and solve the corresponding SVM on the n vectors: minimize ?1 ,...,?n ?0 subject to n 1 X ?i ?j k(bi , bj ) 2 i,j=1 ? n X (4) ?j k(bi , bj ) ? ?1, i = 1, . . . , n. j=1 3. We take both sets A and B of binary vectors, assign label 1 to all vectors from the first set A and label ?1 to all vectors from the second set B. We then solve the corresponding SVM on the 2n vectors: n n n X 1 X 1 X minimize ?i ?j k(ai , aj ) + ?i ?j k(bi , bj ) ? ?i ?j k(ai , bj ) ?1 ,...,?n ?0 2 i,j=1 2 i,j=1 i,j=1 ?1 ,...,?n ?0 subject to n X ?j k(ai , aj ) ? j=1 ? n X j=1 n X ?j k(ai , bj ) j=1 n X ? 1, i = 1, . . . , n , ?j k(bi , aj ) ? ?1, ?j k(bi , bj ) + (5) i = 1, . . . , n . j=1 Intuition behind the construction. To show a reduction from the BHCP problem to SVM computation, we have to consider two cases: ? The YES case of the BHCP problem when there are two vectors that are close in Hamming distance. That is, there exist ai ? A and bj ? B such that Hamming(ai , bj ) < t. ? The NO case of the BHCP problem when there is no close pair of vectors. That is, for all ai ? A and bj ? B, we have Hamming(ai , bj ) ? t. 6 We show that we can distinguish between these two cases by comparing the objective value of the first two SVM instances above to the objective value of the third. Intuition for the NO case. We have Hamming(ai , bj ) ? t for all ai ? A and bj ? B. The Gaussian kernel then gives the inequality k(ai , bj ) = exp(?100 log n ? kai ? bj k22 ) ? exp(?100 log n ? t) for all ai ? A and bj ? B. This means that the value k(ai , bj ) is very small. For simplicity, assume that it is equal to 0, i.e., k(ai , bj ) = 0 for all ai ? A and bj ? B. Consider the third SVM (5). It contains three terms involving k(ai , bj ): the third term in the objective function, the second term in the inequalities of the first type, and the second term in the inequalities of the second type. We assumed that these terms are equal to 0 and we observe that the rest of the third SVM is equal to the sum of the first SVM (3) and the second SVM (4). Thus we expect that the optimal value of the third SVM is approximately equal to the sum of the optimal values of the first and the second SVMs. If we denote the optimal value of the first SVM (3) by value(A), the optimal value of the second SVM (4) by value(B), and the optimal value of the third SVM (5) by value(A, B), then we can express our intuition in terms of the approximate equality value(A, B) ? value(A) + value(B) . Intuition for the YES case. In this case, there is a close pair of vectors ai ? A and bj ? B such that Hamming(ai , bj ) ? t ? 1. Since we are using the Gaussian kernel we have the following inequality for this pair of vectors: k(ai , bj ) = exp(?100 log n ? kai ? bj k22 ) ? exp(?100 log n ? (t ? 1)) . We therefore have a large summand in each of the three terms from the above discussion. Thus the three terms do not (approximately) disappear and there is no reason for us to expect that the approximate equality holds. We can thus expect value(A, B) 6? value(A) + value(B) . Thus, by computing value(A, B) and comparing it to value(A) + value(B) we can distinguish between the YES and NO instances of BHCP. This completes the reduction. The full proofs are given in Section B of the supplementary material. 5 Overview of the hardness proof for training the final layer of a neural network We start by formally defining the class of ?nice? loss functions and ?nice? activation functions. Definition 11. For a label y ? {?1, 1} and a prediction w ? R, we call the loss function loss(y, w) : {?1, 1} ? R ? R?0 nice if the following three properties hold: ? loss(y, w) = l(yw) for some convex function l : R ? R?0 . ? For some sufficiently large constant K > 0, we have that (i) l(x) ? o(1) for all x ? nK , (ii) l(x) ? ?(n) for all x ? ?nK , and (iii) l(x) = l(0) ? o(1/n) for all x ? ?O(n?K ). ? l(0) > 0 is some constant strictly larger than 0. We note that the hinge loss function loss(y, x) = max(0, 1 ? y ? x) and the logistic loss function loss(y, x) = ln12 ln (1 + e?y?x ) are nice loss functions according to the above definition. Definition 12. A non-decreasing activation functions S : R ? R?0 is ?nice? if it satisfies the following property: for all sufficiently large constants T > 0 there exist v0 > v1 > v2 such that S(v0 ) = ?(1), S(v1 ) = 1/nT , S(v2 ) = 1/n?(1) and v1 = (v0 + v2 )/2. The ReLU activation S(z) = max(0, z) satisfies these properties since we can choose v0 = 1, v1 = 1/nT , and v2 = ?1 + 2/nT . For the sigmoid function S(z) = 1+e1?z , we can choose 7 v1 = ? log(nT ? 1), v0 = v1 + C, and v2 = v1 ? C for some C = ?(log n). In the rest of the proof we set T = 1000K, where K is the constant from Definition 11. We now describe the proof of Theorem 9. We use the notation ? := (?1 , . . . , ?n )T . Invoking the first property from Definition 11, we observe that the optimization problem (2) is equivalent to the following optimization problem: minimize n ??R m X l(yi ? (M ?)i ), (6) i=1 where M ? Rm?n is the matrix defined as Mi,j := S(uT i wj ) for i = 1, . . . , m and j = 1, . . . n. For the rest of the section we will use m = ?(n).6 Let A = {a1 , . . . , an } ? {0, 1}d and B = {b1 , . . . , bn } ? {0, 1}d with d = ?(log n) be the input to the Orthogonal Vectors problem. To show hardness we define a matrix M as a vertical concatenation of 3 smaller matrices: M1 , M2 and M2 (repeated). Both matrices M1 , M2 ? Rn?n are of size n ? n. Thus the number of rows of M (equivalently, the number of training examples) is m = 3n. Reduction overview. We select the input examples and weights so that the matrices M1 and M2 , have the following properties: ? M1 : if two vectors ai and bj are orthogonal, then the corresponding entry (M1 )i,j = S(v0 ) = ?(1) and otherwise (M1 )i,j ? 0.7 ? M2 : (M2 )i,i = S(v1 ) = 1/n1000K and (M2 )i,j ? 0 for all i 6= j To complete the description of the optimization problem (6), we assign labels to the inputs corresponding to the rows of the matrix M . We assign label 1 to all inputs corresponding to rows of the matrix M1 and the first copy of the matrix M2 . We assign label ?1 to all remaining rows of the matrix M corresponding to the second copy of matrix M2 . The proof of the theorem is completed by the following two lemmas. See Section G in the supplementary material for the proofs. Lemma 13. If there is a pair of orthogonal vectors, then the optimal value of (6) is upper bounded by (3n ? 1) ? l(0) + o(1). Lemma 14. If there is no pair of orthogonal vectors, then the optimal value of (6) is lower bounded by 3n ? l(0) ? o(1). 6 Hardness proof for gradient computation Finally, we consider the problem of computing the gradient of the loss function for a given set of examples.PWe focus on the network architecture as in the previous section. Specifically, let n F?,B (a) := j=1 ?j S(a, bj ) be the output of a neural net with activation function S, where: (1) a is an input vector from the set A := {a1 , . . . , am } ? {0, 1}d ; (2) B := {b1 , . . . , bn } ? {0, 1}d is a set of binary vectors; (3) ? = {?1 , . . . , ?n }T ? Rn is an n-dimensional real-valued vector. We first prove the following lemma. Lemma 15. For some loss function l : R ? R, let l(F?,B (a)) be the loss for P input a when the label of the input a is +1. Consider the gradient of the total loss l?,A,B := a?A l(F?,B (a)) at ?1 = .P . . = ?n = 0 with respect to ?1 , . . . , ?n . The sum of the entries of the gradient is equal to l0 (0) ? a?A,b?B S(a, b), where l0 (0) is the derivative of the loss function l at 0. For the hinge loss function, we have that the loss function is l(x) = max(0, 1 ? x) if the label is +1. Thus, l0 (0) = ?1. For the logistic loss function, we have that the loss function is l(x) = 1 1 ?x ) if the label is +1. Thus, l0 (0) = ? 2 ln ln 2 ln (1 + e 2 in this case. 6 Note that our reduction does not explicitly construct M . Instead, the values of the matrix are induced by the input examples and weights. 7 We write x ? y if x = y up to an inversely superpolynomial additive factor, i.e., |x ? y| ? n??(1) . 8 Proof of Theorem 10. Since all `p -norms are within a polynomial factor, it suffices to show the statement for `1 -norm. We set S(a, b) := max(0, 1 ? 2aT b). Using P Lemma 15, we get that the `1 -norm of the gradient of the total loss function is equal to |l0 (0)| ? a?A,b?B 1aT b=0 . Since l0 (0) 6= 0, this reduces OV to the gradient computation problem. Note that if there is no orthogonal pair, then the `1 -norm is 0 and otherwise it is a constant strictly greater than 0. Thus approximating the `1 -norm within any finite factor allows us to distinguish the cases. See Section H in the supplementary material for other results. 7 Conclusions We have shown that a range of kernel problems require quadratic time for obtaining a high accuracy solution unless the Strong Exponential Time Hypothesis is false. These problems include variants of kernel SVM, kernel ridge regression, and kernel PCA. We also gave a similar hardness result for training the final layer of a depth-2 neural network. This result is general and applies to multiple loss and activation functions. Finally, we proved that computing the empirical loss gradient for such networks takes time that is essentially ?rectangular?, i.e., proportional to the product of the network size and the number of examples. We note that our quadratic (rectangular) hardness results hold for general inputs. There is a long line of research on algorithms for kernel problems with running times depending on various input parameters, such as its statistical dimension [42], degrees of freedom [11] or effective dimensionality [27]. It would be interesting to establish lower bounds on the complexity of kernel problems as a function of the aforementioned input parameters. Our quadratic hardness results for kernel problems apply to kernels with exponential tails. A natural question is whether similar results can be obtained for ?heavy-tailed? kernels, e.g., the Cauchy kernel. We note that similar results for the linear kernel do not seem achievable using our techniques.8 Several of our results are obtained by a reduction from the (exact) Bichromatic Hamming Closest Pair problem or the Orthogonal Vectors problem. This demonstrates a strong connection between kernel methods and similarity search, and suggests that perhaps a reverse reduction is also possible. Such a reduction could potentially lead to faster approximate algorithms for kernel methods: although the exact closest pair problem has no known sub-quadratic solution, efficient and practical sub-quadratic time algorithms for the approximate version of the problem exist (see e.g., [6, 36, 8, 7, 4]). Acknowledgements Ludwig Schmidt is supported by a Google PhD fellowship. Arturs Backurs is supported by an IBM Research fellowship. This research was supported by grants from NSF and Simons Foundation. References [1] A. Abboud, A. Backurs, and V. V. Williams. Tight hardness results for LCS and other sequence similarity measures. In Symposium on Foundations of Computer Science (FOCS), 2015. [2] A. Abboud, V. V. Williams, and O. Weimann. Consequences of faster alignment of sequences. In International Colloquium on Automata, Languages, and Programming (ICALP), 2014. [3] A. Agarwal and L. Bottou. A lower bound for the optimization of finite sums. In International Conference on Machine Learning (ICML), 2015. 8 In particular, assuming a certain strengthening of SETH, known as the ?non-deterministic SETH? [18], it is provably impossible to prove SETH hardness for any of the linear variants of the studied ERM problems, at least via deterministic reductions. This is due to the fact that these problems have short certificates of optimality via duality arguments. Also, it should be noted that linear analogs of some of the problems considered in this paper (e.g., linear ridge regression) can be solved in O(nd2 ) time using SVD methods. 9 [4] J. Alman, T. M. Chan, and R. Williams. Polynomial Representations of Threshold Functions and Algorithmic Applications. 2016. [5] J. Alman and R. Williams. Probabilistic polynomials and hamming nearest neighbors. 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Policy Gradient With Value Function Approximation For Collective Multiagent Planning Duc Thien Nguyen Akshat Kumar Hoong Chuin Lau School of Information Systems Singapore Management University 80 Stamford Road, Singapore 178902 {dtnguyen.2014,akshatkumar,hclau}@smu.edu.sg Abstract Decentralized (PO)MDPs provide an expressive framework for sequential decision making in a multiagent system. Given their computational complexity, recent research has focused on tractable yet practical subclasses of Dec-POMDPs. We address such a subclass called CDec-POMDP where the collective behavior of a population of agents affects the joint-reward and environment dynamics. Our main contribution is an actor-critic (AC) reinforcement learning method for optimizing CDec-POMDP policies. Vanilla AC has slow convergence for larger problems. To address this, we show how a particular decomposition of the approximate action-value function over agents leads to effective updates, and also derive a new way to train the critic based on local reward signals. Comparisons on a synthetic benchmark and a real world taxi fleet optimization problem show that our new AC approach provides better quality solutions than previous best approaches. 1 Introduction Decentralized partially observable MDPs (Dec-POMDPs) have emerged in recent years as a promising framework for multiagent collaborative sequential decision making (Bernstein et al., 2002). Dec-POMDPs model settings where agents act based on different partial observations about the environment and each other to maximize a global objective. Applications of Dec-POMDPs include coordinating planetary rovers (Becker et al., 2004b), multi-robot coordination (Amato et al., 2015) and throughput optimization in wireless network (Winstein and Balakrishnan, 2013; Pajarinen et al., 2014). However, solving Dec-POMDPs is computationally challenging, being NEXP-Hard even for 2-agent problems (Bernstein et al., 2002). To increase scalability and application to practical problems, past research has explored restricted interactions among agents such as state transition and observation independence (Nair et al., 2005; Kumar et al., 2011, 2015), event driven interactions (Becker et al., 2004a) and weak coupling among agents (Witwicki and Durfee, 2010). Recently, a number of works have focused on settings where agent identities do not affect interactions among agents. Instead, environment dynamics are primarily driven by the collective influence of agents (Varakantham et al., 2014; Sonu et al., 2015; Robbel et al., 2016; Nguyen et al., 2017), similar to well known congestion games (Meyers and Schulz, 2012). Several problems in urban transportation such as taxi supply-demand matching can be modeled using such collective planning models (Varakantham et al., 2012; Nguyen et al., 2017). In this work, we focus on the collective Dec-POMDP framework (CDec-POMDP) that formalizes such a collective multiagent sequential decision making problem under uncertainty (Nguyen et al., 2017). Nguyen et al. present a sampling based approach to optimize policies in the CDec-POMDP model. A key drawback of this previous approach is that policies are represented in a tabular form which scales poorly with the size of observation space of agents. Motivated by the recent suc31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. cess of reinforcement learning (RL) approaches (Mnih et al., 2015; Schulman et al., 2015; Mnih et al., 2016; Foerster et al., 2016; Leibo et al., 2017), our main contribution is a actor-critic (AC) reinforcement learning method (Konda and Tsitsiklis, 2003) for optimizing CDec-POMDP policies. Policies are represented using function approximator such as a neural network, thereby avoiding ns ns ns the scalability issues of a tabular policy. We derive the policy gradient and develop a factored actionom om om 1 2 1 value approximator based on collective agent interactions in CDec-POMDPs. Vanilla AC is slow sm rTm sm sm 1 2 T to converge on large problems due to known issues of learning with global reward in large multiagent am am am 1 2 T systems (Bagnell and Ng, 2005). To address this, m=1:M we also develop a new way to train the critic, our Figure 1: T-step DBN for a CDec-POMDP action-value approximator, that effectively utilizes local value function of agents. 1 2 T We test our approach on a synthetic multirobot grid navigation domain from (Nguyen et al., 2017), and a real world supply-demand taxi matching problem in a large Asian city with up to 8000 taxis (or agents) showing the scalability of our approach to large multiagent systems. Empirically, our new factored actor-critic approach works better than previous best approaches providing much higher solution quality. The factored AC algorithm empirically converges much faster than the vanilla AC validating the effectiveness of our new training approach for the critic. Related work: Our work is based on the framework of policy gradient with approximate value function similar to Sutton et al. (1999). However, as we empirically show, directly applying the original policy gradient from Sutton et al. (1999) into the multi-agent setting and specifically for the CDec-POMDP model results in a high variance solution. In this work, we show a suitable form of compatible value function approximation for CDec-POMDPs that results in an efficient and low variance policy gradient update. Reinforcement learning for decentralized policies has been studied earlier in Peshkin et al. (2000), Aberdeen (2006). Guestrin et al. (2002) also proposed using REINFORCE to train a softmax policy of a factored value function from the coordination graph. However in such previous works, policy gradient is estimated from the global empirical returns instead of a decomposed critic. We show in section 4 that having a decomposed critic along with an individual value function based training of this critic is important for sample-efficient learning. Our empirical results show that our proposed critic training has faster convergence than training with global empirical returns. 2 Collective Decentralized POMDP Model We first describe the CDec-POMDP model introduced in (Nguyen et al., 2017). A T -step Dynamic Bayesian Network (DBN) for this model is shown using the plate notation in figure 1. It consists of the following: ? A finite planning horizon H. ? The number of agents M . An agent m can be in one of the states in the state space S. The joint state space is ?M m=1 S. We denote a single state as i ? S. ? A set of action A for each agent m. We denote an individual action as j ? A. m m m m ? Let (s1:H , a1:H )m = (sm 1 , a1 , s2 . . . , sH , aH ) denote the complete state-action trajectory of an m agent m. We denote the state and action of agent m at time t using random variables sm t , at . Different indicator functions It (?) are defined in table 1. We define the following count given the trajectory of each agent m ? M : nt (i, j, i0 ) = M X 0 0 Im t (i, j, i ) ?i, i ?S, j?A m=1 0 As noted in table 1, count nt (i, j, i ) denotes the number of agents in state i taking action j at time step t and transitioning to next state i0 ; other counts, nt (i) and nt (i, j), are defined analogously. Using these counts, we can define the count tables nst and nst at for the time step t as shown in table 1. 2 Im t (i) ? {0, 1} Im t (i, j) ? {0, 1} 0 Im t (i, j, i ) ? {0, 1} nt (i) ? [0; M ] nt (i, j) ? [0; M ] nt (i, j, i0 ) ? [0; M ] nst nst at nst at st+1 if agent m is at state i at time t or sm t =i m if agent m takes action j in state i at time t or (sm t , at ) = (i, j) m m 0 if agent m takes action j in state i at time t and transitions to state i0 or (sm t , at , st+1 ) = (i, j, i ) Number of agents at state i at time t Number of agents at state i taking action j at time t Number of agents at state i taking action j at time t and transitioning to state i0 at time t + 1 Count table (nt (i) ?i ? S) Count table (nt (i, j) ?i ? S, j ? A) Count table (nt (i, j, i0 ) ?i, i0 ? S, j ? A) Table 1: Summary of notations given the state-action trajectories, (s1:H , a1:H )m ?m, for all the agents ? We assume a general partially observable setting wherein agents can have different observations based on the collective influence of other agents. An agent observes its local state sm t . In m addition, it also observes om t at time t based on its local state st and the count table nst . E.g., an agent m in state i at time t can observe the count of other agents also in state i (=nt (i)) or other agents in some neighborhood of the state i (={nt (j) ?j ? Nb(i)}).  m 0 m ? The transition function is ?t sm t+1 = i |st = i, at = j, nst . The transition function is the same for all the agents. Notice that it is affected by nst , which depends on the collective behavior of the agent population. ? Each agent m has a non-stationary policy ?tm (j|i, om t (i, nst )) denoting the probability of agent m to take action j given its observation (i, om t (i, nst )) at time t. We denote the policy over the m ). planning horizon of an agent m to be ? m = (?1m , . . . , ?H m ? An agent m receives the reward rt = rt (i, j, nst ) dependent on its local state and action, and the counts nst . ? Initial state distribution, bo = (P (i)?i ? S), is the same for all agents. We present here the simplest version where all the agents are of the same type having similar state transition, observation and reward models. The model can handle multiple agent types where agents have different dynamics based on their type. We can also incorporate an external state that is unaffected by agents? actions (such as taxi demand in transportation domain). Our results are extendible to address such settings also. Models such as CDec-POMDPs are useful in settings where agent population is large, and agent identity does not affect the reward or the transition function. A motivating application of this model is for the taxi-fleet optimization where the problem is to compute policies for taxis such that the total profit of the fleet is maximized (Varakantham et al., 2012; Nguyen et al., 2017). The decision making for a taxi is as follows. At time t, each taxi observes its current city zone z (different zones constitute the state-space S), and also the count of other taxis in the current zone and its neighboring zones as well as an estimate of the current local demand. This constitutes the count-based observation o(?) for the taxi. Based on this observation, the taxi must decide whether to stay in the current zone z to look for passengers or move to another zone. These decision choices depend on several factors such as the ratio of demand and the count of other taxis in the current zone. Similarly, the environment is stochastic with variable taxi demand at different times. Such historical demand data is often available using GPS traces of the taxi fleet (Varakantham et al., 2012). Count-Based statistic for planning: A key property in the CDec-POMDP model is that the model dynamics depend on the collective interaction among agents rather than agent identities. In settings such as taxi fleet optimization, the agent population size can be quite large (? 8000 for our real world experiments). Given such a large population, it is not possible to compute unique policy for each agent. Therefore, similar to previous work (Varakantham et al., 2012; Nguyen et al., 2017), our goal is to compute a homogenous policy ? for all the agents. As the policy ? is dependent on counts, it represents an expressive class of policies. For a fixed population M , let {(s1:T , a1:T )m ?m} denote the state-action trajectories of different agents sampled from the DBN in figure 1. Let n1:T ={(nst , nst at , nst at st+1 ) ?t = 1 : T } be the combined vector of the resulting count tables for each time step t. Nguyen et al. show that counts n are the sufficient statistic for planning. That is, the joint-value function of a policy ? over horizon 3 H can be computed by the expectation over counts as (Nguyen et al., 2017): X  M X H H X X X  V (?) = P (n; ?) E[rTm ] = nT (i, j)rT i, j, nT m=1 T =1 (1) T =1 i?S,j?A n??1:H Set ?1:H is the set of all allowed consistent count tables as: X X X nT (i) = M ?T ; nT (i, j) = nT (i) ?j?T ; nT (i, j, i0 ) = nT (i, j) ?i ? S, j ? A, ?T i?S i0 ?S j?A P (n; ?) is the distribution over counts (detailed expression in appendix). A key benefit of this result is that we can evaluate the policy ? by sampling counts n directly from P (n) without sampling individual agent trajectories (s1:H , a1:H )m for different agents, resulting in significant computational savings. Our goal is to compute the optimal policy ? that maximizes V (?). We assume a RL setting with centralized learning and decentralized execution. We assume a simulator is available that can provide count samples from P (n; ?). 3 Policy Gradient for CDec-POMDPs Previous work proposed an expectation-maximization (EM) (Dempster et al., 1977) based sampling approach to optimize the policy ? (Nguyen et al., 2017). The policy is represented as a piecewise linear tabular policy over the space of counts n where each linear piece specifies a distribution over next actions. However, this tabular representation is limited in its expressive power as the number of pieces is fixed apriori, and the range of each piece has to be defined manually which can adversely affect performance. Furthermore, exponentially many pieces are required when the observation o is multidimensional (i.e., an agent observes counts from some local neighborhood of its location). To address such issues, our goal is to optimize policies in a functional form such as a neural network. We first extend the policy gradient theorem of (Sutton et al., 1999) to CDec-POMDPs. Let ? denote the vector of policy parameters. We next show how to compute ?? V (?). Let st , at denote the joint-state and joint-actions of all the agents at time t. The value function of a given policy ? in an expanded form is given as: X Vt (?) = P ? (st , at |bo , ?)Q?t (st , at ) (2) st ,at where P ? (st , at |bo ) = s1:t?1 ,a1:t?1 P ? (s1:t , a1:t |bo ) is the distribution of the joint state-action st , at under the policy ?. The value function Q?t (st , at ) is computed as: X Q?t (st , at ) = rt (st , at ) + P ? (st+1 , at+1 |st , at )Q?t+1 (st+1 , at+1 ) (3) P st+1 ,at+1 We next state the policy gradient theorem for CDec-POMDPs: Theorem 1. For any CDec-POMDP, the policy gradient is given as:   H X X  ? ?? V1 (?) = Est ,at |bo ,? Qt (st , at ) nt (i, j)?? log ?t j|i, o(i, nst ) t=1 (4) i?S,j?A The proofs of this theorem and other subsequent results are provided in the appendix. Notice that computing the policy gradient using the above result is not practical for multiple reasons. The space of join-state action (st , at ) is combinatorial. Given that the agent population size can be large, sampling each agent?s trajectory is not computationally tractable. To remedy this, we later show how to compute the gradient by directly sampling counts n ? P (n; ?) similar to policy evaluation in (1). Similarly, one can estimate the action-value function Q?t (st , at ) using empirical returns as an approximation. This would be the analogue of the standard REINFORCE algorithm (Williams, 1992) for CDec-POMDPs. It is well known that REINFORCE may learn slowly than other methods that use a learned action-value function (Sutton et al., 1999). Therefore, we next present a function approximator for Q?t , and show the computation of policy gradient by directly sampling counts n. 4 3.1 Policy Gradient with Action-Value Approximation One can approximate the action-value function Q?t (st , at ) in several different ways. We consider the following special form of the approximate value function fw : Q?t (st , at ) ? fw (st , at ) = M X m m fwm sm t , o(st , nst ), at  (5) m=1 where each fwm is defined for each agent m and takes as input the agent?s local state, action and the observation. Notice that different components fwm are correlated as they depend on the common count table nst . Such a decomposable form is useful as it leads to efficient policy gradient computation. Furthermore, an important class of approximate value function having this form for CDec-POMDPs is the compatible value function (Sutton et al., 1999) which results in an unbiased policy gradient (details in appendix). Proposition 1. Compatible value function for CDec-POMDPs can be factorized as: X m m fw (st , at ) = fwm (sm t , o(st , nst ), a ) m We can directly replace Q? (?) in policy gradient (4) by the approximate action-value function fw . Empirically, we found that variance using this estimator was high. We exploit the structure of fw and show further factorization of the policy gradient next which works much better empirically. Theorem 2. For any value function having the decomposition as: X  m m fw (st , at ) = fwm sm (6) t , o(st , nst ), at , m the policy gradient can be computed as ?? V1 (?) = H X Est ,at hX  m m i m m m m ?? log ? am t |st , o(st , nst ) fw st , o(st , nst ), at (7) m t=1 The above result shows that if the approximate value function is factored, then the resulting policy gradient also becomes factored. The above result also applies to agents with multiple types as we assumed the function fwm is different for each agent. In the simpler case when all the agents are of same type, then we have the same function fw for each agent, and also deduce the following: X  fw (st , at ) = nt (i, j)fw i, j, o(i, nst ) (8) i,j Using the above result, we simplify the policy gradient as: hX i X  ?? V1 (?) = Est ,at nt (i, j)?? log ? j|i, o(i, nst ) fw (i, j, o(i, nst )) t 3.2 (9) i,j Count-based Policy Gradient Computation Notice that in (9), the expectation is still w.r.t. joint-states and actions (st , at ) which is not efficient in large population sizes. To address this issue, we exploit the insight that the approximate value function in (8) and the inner expression in (9) depends only on the counts generated by the joint-state and action (st , at ).  P Theorem 3. For any value function having the form: fw (st , at ) = i,j nt (i, j)fw i, j, o(i, nst ) , the policy gradient can be computed as: X  H X  En1:H ??1:H nt (i, j)?? log ? j|i, o(i, nt ) fw (i, j, o(i, nt )) (10) t=1 i?S,j?A The above result shows that the policy gradient can be computed by sampling count table vectors n1:H from the underlying distribution P (?) analogous to computing the value function of the policy in (1), which is tractable even for large population sizes. 5 4 Training Action-Value Function In our approach, after count samples n1:H are generated to compute the policy gradient, we also need to adjust the parameters w of our critic fw . Notice that as per (8), the action value function fw (st , at ) depends only on the counts generated by the joint-state and action (st , at ). Training fw can be done by taking a gradient step to minimize the following loss function: min w K X H  X fw (n?t ) ? Rt? 2 (11) ?=1 t=1 where n?1:H is a count sample generated from the distribution P (n; ?); fw (n?t ) is the action value function and Rt? is the total empirical return for time step t computed using (1): fw (n?t ) = X n?t (i, j)fw (i, j, o(i, n?t )); Rt? = i,j H X X n?T (i, j)rT (i, j, n?T ) (12) T =t i?S,j?A However, we found that the loss in (11) did not work well for training the critic fw for larger problems. Several count samples were required to reliably train fw which adversely affects scalability for large problems with many agents. It is already known in multiagent RL that algorithms that solely rely on the global reward signal (e.g. Rt? in our case) may require several more samples than approaches that take advantage of local reward signals (Bagnell and Ng, 2005). Motivated by this observation, we next develop a local reward signal based strategy to train the critic fw . Let n? Individual Value Function: Vt? (i, j) be a count sample. Given the count sample n? , let 1:H 1:H P ? t m m = E[ H t0 =t rt0 |st = i, am = j, n1:H ] denote the total expected reward obtained by an agent that is in state i and takes action j at time t. This individual value function can be computed using dynamic programming as shown in (Nguyen et al., 2017). Based on this value function, we next show an alternative reparameterization of the global empirical reward Rt? in (12): Lemma 1. The empirical return Rt? for the time step t given the count sample n?1:H can be reP parameterized as: Rt? = i?S,j?A n?t (i, j)Vt? (i, j). Individual Value Function Based Loss: Given lemma 1, we next derive an upper bound on the on the true loss (11) which effectively utilizes individual value functions: 2 X X  X 2 XX X ? fw (n? ) ? Rt? = n?t (i, j)fw (i, j, o(i, n?t )) ? nt (i, j)Vt? (i, j) ? t t ? = i,j XXX t ? ?M n?t (i, j) fw (i, j, o(i, n?t )) ? Vt? (i, j) 2 (13) i,j XX ? i,j   2 nt (i, j) fw (i, j, o(i, n?t )) ? Vt? (i, j) (14) t,i,j where the last relation is derived by Cauchy-Schwarz inequality. We train the critic using the modified loss function in (14). Empirically, we observed that for larger problems, this new loss function in (14) resulted in much faster convergence than the original loss function in (13). Intuitively, this is because the new loss (14) tries to adjust each critic component fw (i, j, o(i, n?t )) closer to its counterpart empirical return Vt? (i, j). However, in the original loss function (13), the focus is on minimizing the global loss, rather than adjusting each individual critic factor fw (?) towards the corresponding empirical return. Algorithm 1 shows the outline of our AC approach for CDec-POMDPs. Lines 7 and 8 show two different options to train the critic. Line 7 represents critic update based on local value functions, also referred to as factored critic update (fC). Line 8 shows update based on global reward or global critic update (C). Line 10 shows the policy gradient computed using theorem 2 (fA). Line 11 shows how the gradient is computed by directly using fw from eq. (5) in eq. 4. 6 Algorithm 1: Actor-Critic RL for CDec-POMDPs 1 2 3 4 5 6 7 8 9 10 11 12 13 5 Initialize network parameter ? for actor ? and and w for critic fw ? ? actor learning rate ? ? critic learning rate repeat Sample count vectors n?1:H ? P (n1:H ; ?) ?? = 1 to K Update critic as: hP P  2 i ? ? ? 1 ?w n (i, j) f (i, j, o(i, n )) ? V (i, j) fC : w = w ? ? K w t t t ? t,i,j hP P P 2 i P ? ? ? ? 1 C : w = w ??K ?w ? t i,j nt (i, j)fw (i, j, o(i, nt )) ? i,j nt (i, j)Vt (i, j) Update actor as: i  n?t (i, j) log ? j|i, o(i, n?t ) fw (i, j, o(n?t , i)) i ih P P P hP ? ? ? ? 1 A : ? = ? +?K ?? ? t i,j nt (i, j)fw (i, j, o(nt , i)) i,j nt (i, j) log ? j|i, o(i, nt ) 1 ?? fA : ? = ? + ? K P P hP ? t i,j until convergence return ?, w Experiments This section compares the performance of our AC approach with two other approaches for solving CDec-POMDPs?Soft-Max based flow update (SMFU) (Varakantham et al., 2012), and the Expectation-Maximization (EM) approach (Nguyen et al., 2017). SMFU can only optimize policies m where an agent?s action only depends on its local state, ?(am t |st ), as it approximates the effect of counts n by computing the single most likely count vector during the planning phase. The EM m approach can optimize count-based piecewise linear policies where ?t (am t |st , ?) is a piecewise function over the space of all possible count observations ot . Algorithm 1 shows two ways of updating the critic (in lines 7, 8) and two ways of updating the actor (in lines 10, 11) leading to 4 possible settings for our actor-critic approach?fAfC, AC, AfC, fAC. We also investigate the properties of these different actor-critic approaches. The neural network structure and other experimental settings are provided in the appendix. For fair comparisons with previous approaches, we use three different models for counts-based observation ot . In ?o0? setting, policies depend only on agent?s local state sm t and not on counts. In m ?o1? setting, policies depend on the local state sm t and the single count observation nt (st ). That . In ?oN? setting, is, the agent can only observe the count of other agents in its current state sm t and also the count of other agents from a local neighborhood the agent observes its local state sm t (defined later) of the state sm t . The ?oN? observation model provides the most information to an agent. However, it is also much more difficult to optimize as policies have more parameters. The SMFU only works with ?o0? setting; EM and our actor-critic approach work for all the settings. Taxi Supply-Demand Matching: We test our approach on this real-world domain described in section 2, and introduced in (Varakantham et al., 2012). In this problem, the goal is to compute taxi policies for optimizing the total revenue of the fleet. The data contains GPS traces of taxi movement in a large Asian city over 1 year. We use the observed demand information extracted from this dataset. On an average, there are around 8000 taxis per day (data is not exhaustive over all taxi operators). The city is divided into 81 zones and the plan horizon is 48 half hour intervals over 24 hours. For details about the environment dynamics, we refer to (Varakantham et al., 2012). Figure 2(a) shows the quality comparisons among different approaches with different observation models (?o0?, ?o1? and ?oN?). We test with total number of taxis as 4000 and 8000 to see if taxi population size affects the relative performance of different approaches. The y-axis shows the average per day profit for the entire fleet. For the ?o0? case, all approaches (fAfC-?o0?, SMFU, EM-?o0?) give similar quality with fAfC-?o0? and EM-?o0? performing slightly better than SMFU for the 8000 taxis. For the ?o1? case, there is sharp improvement in quality by fAfC-?o1? over fAfC-?o0? confirming that taking count based observation into account results in better policies. Our approach fAfC-?o1? is also significantly better than the policies optimized by EM-?o1? for both 4000 and 8000 taxi setting. 7 2500000 150 fAfC-o0 fAfC-o1 fAfC-oN SMFU EM-o0 EM-o1 1500000 fAfC-o0 fAfC-o1 fAfC-oN SMFU EM-o0 EM-o1 EM-oN 100 Quality Quality 2000000 1000000 50 0 500000 0 4000 8000 ?50 Taxi Population Grid Navigation (a) Solution quality with varying taxi population (b) Solution quality in grid navigation problem Figure 2: Solution quality comparisons on the taxi problem and the grid navigation fA?fC Quality Quality 500000 1000000 750000 0 Quality 500000 A?fC 1500000 2000000 1000000 1500000 500000 0 0 5000 10000 15000 Iteration 20000 1000000 500000 500000 500000 250000 fA?C Quality 1000000 1250000 A?C 0 5000 10000 15000 Iteration 20000 0 5000 10000 15000 20000 25000 Iteration 0 convergence with ?o0? (a) AC (b) AC convergence with ?o1? (c) AC convergence with ?oN? 250000 Figure 3: Convergence of different actor-critic variants on the taxi problem with 8000 taxis 500000 750000 To further to optimize complex approach in the ?oN? 0 test the scalability 5000 and the ability10000 15000policies by our20000 setting, we define the neighborhood of each Iteration state (which is a zone in the city) to be the set of its geographically connected zones based on the zonal decomposition shown in (Nguyen et al., 2017). On an average, there are about 8 neighboring zones for a given zone, resulting in 9 count based observations available to the agent for taking decisions. Each agent observes both the taxi count and the demand information from such neighboring zones. In figure 2(a), fAfC-?oN? result clearly shows that taking multiple observations into account significantly increases solution quality?fAfC?oN? provides an increase of 64% in quality over fAfC-?o0? and 20% over fAfC-?o1? for the 8000 taxi case. For EM-?oN?, we used a bare minimum of 2 pieces per observation dimension (resulting in 29 pieces per time step). We observed that EM was unable to converge within 30K iterations and provided even worse quality than EM-?o1? at the end. These results show that despite the larger search space, our fAfC approach can effectively optimize complex policies whereas the tabular policy based EM approach was ineffective for this case. Figures 3(a-c) show the quality Vs. iterations for different variations of our actor critic approach? fAfC, AC, AfC, fAC?for the ?o0?, ?o1? and the ?oN? observation model. These figures clearly show that using factored actor and the factored critic update in fAfC is the most reliable strategy over all the other variations and for all the observation models. Variations such as AC and fAC were not able to converge at all despite having exactly the same parameters as fAfC. These results validate different strategies that we have developed in our work to make vanilla AC converge faster for large problems. Robot navigation in a congested environment: We also tested on a synthetic benchmark introduced in (Nguyen et al., 2017). The goal is for a population of robots (= 20) to move from a set of initial locations to a goal state in a 5x5 grid. If there is congestion on an edge, then each agent attempting to cross the edge has higher chance of action failure. Similarly, agents also receive a negative reward if there is edge congestion. On successfully reaching the goal state, agents receive a positive reward and transition back to one of the initial state. We set the horizon to 100 steps. Figure 2(b) shows the solution quality comparisons among different approaches. In the ?oN? observation model, the agent observes its 4 immediate neighbor node?s count information. In this problem, SMFU performed worst, fAfC and EM both performed much better. As expected fAfC-?oN? 8 provides the best solution quality over all the other approaches. In this domain, EM is competitive with fAfC as for this relatively smaller problem with 25 agents, the space of counts is much smaller than in the taxi domain. Therefore, EM?s piecewise policy is able to provide a fine grained approximation over the count range. 6 Summary We addressed the problem of collective multiagent planning where the collective behavior of a population of agents affects the model dynamics. We developed a new actor-critic method for solving such collective planning problems within the CDec-POMDP framework. We derived several new results for CDec-POMDPs such as the policy gradient derivation, and the structure of the compatible value function. To overcome the slow convergence of the vanilla actor-critic method we developed multiple techniques based on value function factorization and training the critic using individual value function of agents. Using such techniques, our approach provided significantly better quality than previous approaches, and proved scalable and effective for optimizing policies in a real world taxi supply-demand problem and a synthetic grid navigation problem. 7 Acknowledgments This research project is supported by National Research Foundation Singapore under its Corp Lab @ University scheme and Fujitsu Limited. First author is also supported by A? STAR graduate scholarship. 9 References Aberdeen, D. (2006). Policy-gradient methods for planning. In Advances in Neural Information Processing Systems, pages 9?16. Amato, C., Konidaris, G., Cruz, G., Maynor, C. A., How, J. P., and Kaelbling, L. P. (2015). Planning for decentralized control of multiple robots under uncertainty. In IEEE International Conference on Robotics and Automation, ICRA, pages 1241?1248. Bagnell, J. A. and Ng, A. Y. (2005). On local rewards and scaling distributed reinforcement learning. In International Conference on Neural Information Processing Systems, pages 91?98. Becker, R., Zilberstein, S., and Lesser, V. (2004a). Decentralized Markov decision processes with event-driven interactions. In Proceedings of the 3rd International Conference on Autonomous Agents and Multiagent Systems, pages 302?309. Becker, R., Zilberstein, S., Lesser, V., and Goldman, C. V. (2004b). Solving transition independent decentralized Markov decision processes. Journal of Artificial Intelligence Research, 22:423? 455. Bernstein, D. S., Givan, R., Immerman, N., and Zilberstein, S. (2002). The complexity of decentralized control of Markov decision processes. Mathematics of Operations Research, 27:819?840. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical society, Series B, 39(1):1?38. Foerster, J. N., Assael, Y. M., de Freitas, N., and Whiteson, S. (2016). Learning to communicate with deep multi-agent reinforcement learning. In Advances in Neural Information Processing Systems, pages 2137?2145. Guestrin, C., Lagoudakis, M., and Parr, R. (2002). Coordinated reinforcement learning. In ICML, volume 2, pages 227?234. Konda, V. R. and Tsitsiklis, J. N. (2003). On actor-critic algorithms. SIAM Journal on Control and Optimization, 42(4):1143?1166. Kumar, A., Zilberstein, S., and Toussaint, M. (2011). Scalable multiagent planning using probabilistic inference. In Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence, pages 2140?2146, Barcelona, Spain. Kumar, A., Zilberstein, S., and Toussaint, M. (2015). Probabilistic inference techniques for scalable multiagent decision making. Journal of Artificial Intelligence Research, 53(1):223?270. Leibo, J. Z., Zambaldi, V. F., Lanctot, M., Marecki, J., and Graepel, T. (2017). Multi-agent reinforcement learning in sequential social dilemmas. In International Conference on Autonomous Agents and Multiagent Systems. Meyers, C. A. and Schulz, A. S. (2012). The complexity of congestion games. Networks, 59:252? 260. Mnih, V., Badia, A. P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., and Kavukcuoglu, K. (2016). Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning, pages 1928?1937. Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., Graves, A., Riedmiller, M. A., Fidjeland, A., Ostrovski, G., Petersen, S., Beattie, C., Sadik, A., Antonoglou, I., King, H., Kumaran, D., Wierstra, D., Legg, S., and Hassabis, D. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540):529?533. Nair, R., Varakantham, P., Tambe, M., and Yokoo, M. (2005). Networked distributed POMDPs: A synthesis of distributed constraint optimization and POMDPs. In AAAI Conference on Artificial Intelligence, pages 133?139. Nguyen, D. T., Kumar, A., and Lau, H. C. (2017). Collective multiagent sequential decision making under uncertainty. In AAAI Conference on Artificial Intelligence, pages 3036?3043. 10 Pajarinen, J., Hottinen, A., and Peltonen, J. (2014). Optimizing spatial and temporal reuse in wireless networks by decentralized partially observable Markov decision processes. IEEE Trans. on Mobile Computing, 13(4):866?879. Peshkin, L., Kim, K.-E., Meuleau, N., and Kaelbling, L. P. (2000). Learning to cooperate via policy search. In Proceedings of the Sixteenth conference on Uncertainty in artificial intelligence, pages 489?496. Morgan Kaufmann Publishers Inc. Robbel, P., Oliehoek, F. A., and Kochenderfer, M. J. (2016). Exploiting anonymity in approximate linear programming: Scaling to large multiagent MDPs. In AAAI Conference on Artificial Intelligence, pages 2537?2543. Schulman, J., Levine, S., Abbeel, P., Jordan, M., and Moritz, P. (2015). Trust region policy optimization. In International Conference on Machine Learning, pages 1889?1897. Sonu, E., Chen, Y., and Doshi, P. (2015). Individual planning in agent populations: Exploiting anonymity and frame-action hypergraphs. In International Conference on Automated Planning and Scheduling, pages 202?210. Sutton, R. S., McAllester, D., Singh, S., and Mansour, Y. (1999). Policy gradient methods for reinforcement learning with function approximation. In International Conference on Neural Information Processing Systems, pages 1057?1063. Varakantham, P., Adulyasak, Y., and Jaillet, P. (2014). Decentralized stochastic planning with anonymity in interactions. In AAAI Conference on Artificial Intelligence, pages 2505?2511. Varakantham, P. R., Cheng, S.-F., Gordon, G., and Ahmed, A. (2012). Decision support for agent populations in uncertain and congested environments. In AAAI Conference on Artificial Intelligence, pages 1471?1477. Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3):229?256. Winstein, K. and Balakrishnan, H. (2013). Tcp ex machina: Computer-generated congestion control. In Proceedings of the ACM SIGCOMM 2013 Conference, SIGCOMM ?13, pages 123?134. Witwicki, S. J. and Durfee, E. H. (2010). Influence-based policy abstraction for weakly-coupled Dec-POMDPs. In International Conference on Automated Planning and Scheduling, pages 185? 192. 11
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Modeling Consistency in a Speaker Independent Continuous Speech Recognition System Yochai Konig, Nelson Morgan, Chuck Wooters International Computer Science Institute 1947 Center Street, Suite 600 Berkeley, CA 94704, USA. Victor Abrash, Michael Cohen, Horacio Franco SRI International 333 Ravenswood Ave. Menlo Park, CA 94025, USA Abstract We would like to incorporate speaker-dependent consistencies, such as gender, in an otherwise speaker-independent speech recognition system. In this paper we discuss a Gender Dependent Neural Network (GDNN) which can be tuned for each gender, while sharing most of the speaker independent parameters. We use a classification network to help generate gender-dependent phonetic probabilities for a statistical (HMM) recognition system. The gender classification net predicts the gender with high accuracy, 98.3% on a Resource Management test set. However, the integration of the GDNN into our hybrid HMM-neural network recognizer provided an improvement in the recognition score that is not statistically significant on a Resource Management test set. 1 INTRODUCTION Earlier work [Bourlard and Morgan, 1991l has shown the ability of Multilayer Perceptrons (MLPs) to estimate emission probabilities for Hidden Markov Models (HMM). As shown in their report, with a few assumptions, an MLP may be viewed as estimating the probability P (qIx ) where q is a sub word model (or a state of a subword model) and x is the input acoustic 682 Modeling Consistency in a Speaker Independent Continuous Speech Recognition System speech data. In this hybrid HMMIMLP recognizer, it was shown that these estimates led to improved performance over standard estimation techniques when a fairly simple HMM was used. More recent results have shown improvements using hybrid HMMIMLP probability estimation over a state-of-the-art pure HMM-based system[Cohen et ai., 1993; Renals et ai., 19921. Some speaker dependencies exist in common parametric representations of speech, and it is possible that making the dependencies explicit may improve performance for a given speaker (essentially enabling the recognizer to soften the influence of the speaker dependency). The basic problem with modeling and estimating explicitly speaker dependent p~ rameters is the lack of training data. In the limit, the only available information about a new speaker is the utterance to be recognized. This limit is our starting point for this study. Even with this limited information, we can incorporate constraints on analysis that rely on the same speaker producing all the frames in an utterance, thus ensuring consistency. As has been observed for some mainstream Hidden Markov Models (HMM) systems [Murveit et ai., 19901, given enough training data, separate phonetic models for male and female speakers can be used to improve performance. Our first attack on consistency, then, is to incorporate gender consistency in the recognition process. In contrast to non-connectionist HMM systems, our proposed architecture attempts to share the gender-independent parameters. Our study had two steps: first we trained an MLP to estimate the probability of gender. Then, we investigated ways to integrate the gender consistency constraint into our existing MLP-HMM hybrid recognizer, resulting in our GDNN architecture. We will give a short description of some related work, followed by an explanation of the two steps described above. We conclude with some discussion and thoughts about future work. 2 RELATED AND PREVIOUS WORK Our previous experiments with the Gender-Dependent Neural Network (GDNN) are described in [Abrash et ai., 1992; Konig and Morgan, 1992]. Other researchers have worked on related problems. For example Hampshire and Waibel presented the "Meta-Pi" architecture [Hampshire and Waibel, 19901. The building blocks for the "Meta-Pi" architecture are multiple TDNN's that are trained to recognize the speech of an individual speaker. These building blocks are integrated by another multiple TDNN trained in a Bayesian MAP scheme to maximize the phoneme recognition rate of the overall architecture. The performance of the "Meta-Pi" architecture on a six speaker /b,d,g/ task was comparable to a speaker dependent system on the same task. Another example of related work is speaker normalization, which attempts to minimize between-speaker variations by transforming the data of a new speaker to that of a reference speaker, and then applying the speaker dependent system for the reference speaker [Huang et ai., 1991]. 3 THE CLASSIFICATION NET In order to classify the gender of a new speaker we need features that distinguish between speakers, in contrast to the features that are used for phoneme recognition that are chosen to suppress speaker variations. Given our constraint that the only available information 683 684 Konig, Morgan, Wooters, Abrash, Cohen, and Franco about the new speaker is the sentence to be recognized, we chose features that are a rough estimate of the vocal tract properties and the fundamental frequency of the new speaker. Furthermore, we tried to suppress the linguistic information in our estimate. More specifically, the goal was to build a net that estimates the probability P(GenderIData). After some experimentation, the first twelve LPC cepstral coefficients were calculated over a 20 msec window every 10 msec (50% overlap) and averaged along each sentence. The sampling rate was 16khz. These features were augmented by an estimate of the fundamental frequency for a total of 13 features per sentence. The MLP had one hidden layer with 24 hidden units. There were two output units, one for each gender. The training set was the 109-speaker DARPA Resource Management corpus. 3500 sentences were used for the training set and 490 in the cross validation set. The size of the test set was 600 sentences, and it was a combination of the DARPA Resource Management speaker-independent Feb89 and Oct89 test sets. The trained MLP predicts the gender for the test set with less than 1.7% error on the sentence level. 4 4.1 INCORPORATING GENDER CONSISTENCY INTO OUR HYBRID HMMIMLP RECOGNIZER DISCUSSION Our goal is to find an architecture that shares the gender independent parameters and models the gender dependent parameters. Given our gender consistency constraint we estimate a probability that is explicitly conditioned on gender, as if the phonetic models were simply doubled to permit male and female forms of each phoneme. We can express P( male, phoneldata) (which is then divided by priors to get the corresponding data likelihood) by expansion to P(phonelmale, data) x P(maleldata). This factorization is realized by two separate MLP's: P(maleldata) is estimated by the classification net described above, and P(phonelmale, data) is realized by our GDNN described below. For further description on how to factorize probabilities by neural networks see [Morgan and Bourlard, 19921. The final likelihood for the male case can be expressed as: P(d I ) _ P(phonelmale, data) x P(maleldata) x P(data) Ih ata pone, ma e P( pone h Ima I) e x P( ma I) e (1) Note that during recognition, P(data) can be ignored. Similarly, a female-assumed probability can be computed for each hypothesized phone. These male and female-assumed probabilities can then be used in separate Viterbi calculations (since we do not permit any hypothesis to switch gender in the midst of an utterance). In other words, dynamic programming is used with the framewise network outputs to evaluate the best hypothesized utterance assuming male gender, and then the same is done for the female case. The case with the lowest cost (highest probability) is then chosen. Note that the output of the classification net only helps in choosing between the sentence recognized according to female gender or male gender. The critical question is how to estimate P(phonelgender, data). A possible answer is to have two separate nets, one trained only on males, and the other trained only on females. This approach has the potential disadvantages of doubling the number of parameters in the system, and of not sharing the gender independent parameters. We have experimented with a such a net [Konig and Morgan, 1992] and it improved our result over the baseline system. Modeling Consistency in a Speaker Independent Continuous Speech Recognition System P(phonelgender,data) t t t t 69 units 1000 units l J ~ t t t f f 9 x (12 mel cepstral + log energy + if(gender == Male) {M=l,F=O} otherwise {M=O,F=l} first derivatives) = 234 Figure 1: A Gender Dependent Neural Network(GDNN) We present here a hybrid GDNN architecture that has the flexibility to tune itself to each gender. The idea is to have extra binary inputs that specify the gender of the speaker and then the probabilities that the network estimates will be conditioned on the gender. The architecture is shown in figure 1. 4.2 EXPERIMENTS AND RESULTS We have compared four different architectures. The first architecture is our baseline system, namely, one large net that was trained on all the sentences in the training set. The second uses two separate nets, for males and females. The third is the hybrid GDNN architecture described in figure 1. The fourth architecture is a variant of the third architecture, the difference being that the binary units are connected to the output units instead of to the hidden units. All the nets have 1000 hidden units and 69 output units, including the totally separate male and female nets. While one might think that the consequent doubling of 685 686 Konig, Morgan, Wooters, Abrash, Cohen, and Franco Table 1: Result Summary Architecture Baseline Two Separate Nets Hybrid Architecture - Variant(Binary to Output) GDNN Binary to hidden Test Set Word Error 10.6% 10.9% 10.9% 10.2% the number of parameters in the system might explain the observed degradation for the perfonnance for the second architecture, we have also experimented with several sizes of male and female separate nets, by changing the number of the hidden units and the number of input frames. None of these experiments resulted in a significant improvement. We used 12 mel-cepstral features and the log-energy along with their first derivatives, so the number of input features per frame was 26. The features were calculated from 20ms of speech, computed every 10 msec (as before). The length of the temporal window (the number of input frames) was 9, so the total number of input features was 234. The training set was the 109-speaker DARPA Resource Management corpus. 3500 sentences were used for the training set and 490 in the cross validation set. The size of the test was the 600 sentences making up the DARPA Feb89 and Oct89 test sets. The results are summarized in table I, and are achieved using the standard Resource Management wordpair grammar(perplexity =60) with a simple context-independent HMM recognizer. We should note here that these results are all somewhat worse than our other results published in [Renals et al., 1992; Cohen et al., 1993], as the latter were achieved using SRI's phonological models, and these were done with a single-pronunciation single-state HMM (with each state repeated for a rough duration model). 5 DISCUSSION AND FUTURE WORK The best results were achieved by the GDNN hybrid architecture that shares the genderindependent parameters while modeling the gender dependent parameters . . However the improvement over our baseline is not statistically significant for this test set, although it is consistent with our experiments with other test sets, not reported here. A possible source for further improvement is using a training set with a more balanced representation of gender. In the DARPA Resource Management speaker independent training set there are 2830 sentences uttered by males versus only 1160 sentences uttered by females. Thus, perfonnance may have suffered from an insufficient number of female training sentences. A reasonable extension to this work would be the modeling of additional speaker dependent parameters such as speech rate, accent, etc. Another direction that might be more fruitful is to combine the gender-dependent models in the local estimation of phonemes, and not to do separate Viterbi recognitions for each gender. We are currently examining this latter alternative. Modeling Consistency in a Speaker Independent Continuous Speech Recognition System Acknowledgements Thanks to Steve Renals for his comments along the way. Computations were done on the RAP machine, with support from software guru Phil Kohn, and hardware wiz Jim Beck. Thanks to Hynek Hermansky for advising us about the features for the gender classification net. Thanks to the other members of the speech group at ICSI for their helpful comments. This work was partially funded by DARPA contract MDA904-90-C-5253. References [Abrash et al., 1992] V. Abrash, H. Franco, M. Cohen, N. Morgan, and Y. Konig. Connectionist gender adaptation in a hybrid neural network / hidden markov model speech recognition system. In Proc. Int'l Conf. on Spoken Lang. Processing, Banff, Canada, October 1992. [Bourlard and Morgan, 19911 H. Bourlard and N. Morgan. Merging multilayer perceptrons & hidden markov models: Some experiments in continuous speech recognition. In E. Gelenbe, editor, Artificial Neural Networks: Advances and Applications. North Holland Press, 1991. [Cohen et al., 1993] M. Cohen, H. Franco, N. Morgan, D. Rumelhart, and V. Abrash. Context-dependent multiple distribution phonetic modeling. In C.L. Giles, Hanson SJ, and J.D. Cowan, editors, Advances in Neural Information Processing Systems, volume 5. Morgan Kaufmann, San Mateo, 1993. [Hampshire and Waibel, 1990] J.B. Hampshire and A. Waibel. Connectionist architectures for multi-speaker phoneme recognition. In D.S. Touretzky, editor, Advances in Neural Information Processing Systems 2, San mateo, CA, 1990. Morgan Kaufman. [Huang et al., 19911 X.D. Huang, K.F. Lee, and A. Waibel. Connectionist speaker normalization and its application to speech recognition. In Neural Networks for Siganl Processing, proc. of 1991 IEEE Workshop" Princeton, New Jersey, October 1991. [Konig and Morgan, 1992] Y. Konig and N. Morgan. Gdnn: A gender -dependent neural network for continuous speech recognition. In Proc. international loint Conference on Neural Networks, Baltimore, Maryland, June 1992. [Morgan and Bourlard, 1992] N. Morgan and H. Bourlard. Factoring neural networks by a statistical method. Neural Computation, (4):835-838,1992. [Murveit et al., 1990] H. Murveit, M. Weintraub, and M. Cohen. Training set issues in sri's decipher speech recognition system. In Proc. speech and Natural Language Workshop. pages 337-340,June 1990. [Renals et al., 1992] S. Renals, N. Morgan, M. Cohen, H. Franco, and H. Bourlard. Connectionist probability estimation in the decipher speech recognition system. In Proceedings IEEE Inti. Conf. on Acoustics, Speech, and Signal Processing, San Francisco, California, March 1992. IEEE. 687
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Adversarial Symmetric Variational Autoencoder Yunchen Pu, Weiyao Wang, Ricardo Henao, Liqun Chen, Zhe Gan, Chunyuan Li and Lawrence Carin Department of Electrical and Computer Engineering, Duke University {yp42, ww109, r.henao, lc267, zg27,cl319, lcarin}@duke.edu Abstract A new form of variational autoencoder (VAE) is developed, in which the joint distribution of data and codes is considered in two (symmetric) forms: (i) from observed data fed through the encoder to yield codes, and (ii) from latent codes drawn from a simple prior and propagated through the decoder to manifest data. Lower bounds are learned for marginal log-likelihood fits observed data and latent codes. When learning with the variational bound, one seeks to minimize the symmetric Kullback-Leibler divergence of joint density functions from (i) and (ii), while simultaneously seeking to maximize the two marginal log-likelihoods. To facilitate learning, a new form of adversarial training is developed. An extensive set of experiments is performed, in which we demonstrate state-of-the-art data reconstruction and generation on several image benchmark datasets. 1 Introduction Recently there has been increasing interest in developing generative models of data, offering the promise of learning based on the often vast quantity of unlabeled data. With such learning, one typically seeks to build rich, hierarchical probabilistic models that are able to fit to the distribution of complex real data, and are also capable of realistic data synthesis. Generative models are often characterized by latent variables (codes), and the variability in the codes encompasses the variation in the data [1, 2]. The generative adversarial network (GAN) [3] employs a generative model in which the code is drawn from a simple distribution (e.g., isotropic Gaussian), and then the code is fed through a sophisticated deep neural network (decoder) to manifest the data. In the context of data synthesis, GANs have shown tremendous capabilities in generating realistic, sharp images from models that learn to mimic the structure of real data [3, 4, 5, 6, 7, 8]. The quality of GAN-generated images has been evaluated by somewhat ad hoc metrics like inception score [9]. However, the original GAN formulation does not allow inference of the underlying code, given observed data. This makes it difficult to quantify the quality of the generative model, as it is not possible to compute the quality of model fit to data. To provide a principled quantitative analysis of model fit, not only should the generative model synthesize realistic-looking data, one also desires the ability to infer the latent code given data (using an encoder). Recent GAN extensions [10, 11] have sought to address this limitation by learning an inverse mapping (encoder) to project data into the latent space, achieving encouraging results on semi-supervised learning. However, these methods still fail to obtain faithful reproductions of the input data, partly due to model underfitting when learning from a fully adversarial objective [10, 11]. Variational autoencoders (VAEs) are designed to learn both an encoder and decoder, leading to excellent data reconstruction and the ability to quantify a bound on the log-likelihood fit of the model to data [12, 13, 14, 15, 16, 17, 18, 19]. In addition, the inferred latent codes can be utilized in downstream applications, including classification [20] and image captioning [21]. However, new images synthesized by VAEs tend to be unspecific and/or blurry, with relatively low resolution. These 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. limitations of VAEs are becoming increasingly understood. Specifically, the traditional VAE seeks to maximize a lower bound on the log-likelihood of the generative model, and therefore VAEs inherit the limitations of maximum-likelihood (ML) learning [22]. Specifically, in ML-based learning one optimizes the (one-way) Kullback-Leibler (KL) divergence between the distribution of the underlying data and the distribution of the model; such learning does not penalize a model that is capable of generating data that are different from that used for training. Based on the above observations, it is desirable to build a generative-model learning framework with which one can compute and assess the log-likelihood fit to real (observed) data, while also being capable of generating synthetic samples of high realism. Since GANs and VAEs have complementary strengths, their integration appears desirable, with this a principal contribution of this paper. While integration seems natural, we make important changes to both the VAE and GAN setups, to leverage the best of both. Specifically, we develop a new form of the variational lower bound, manifested jointly for the expected log-likelihood of the observed data and for the latent codes. Optimizing this variational bound involves maximizing the expected log-likelihood of the data and codes, while simultaneously minimizing a symmetric KL divergence involving the joint distribution of data and codes. To compute parts of this variational lower bound, a new form of adversarial learning is invoked. The proposed framework is termed Adversarial Symmetric VAE (AS-VAE), since within the model (i) the data and codes are treated in a symmetric manner, (ii) a symmetric form of KL divergence is minimized when learning, and (iii) adversarial training is utilized. To illustrate the utility of AS-VAE, we perform an extensive set of experiments, demonstrating state-of-the-art data reconstruction and generation on several benchmarks datasets. 2 Background and Foundations Consider an observed data sample x, modeled as being drawn from p? (x|z), with model parameters ? and latent code z. The prior distribution on the code is denoted p(z), typically a distribution that is easy to draw from, such as isotropic Gaussian. The posterior distribution on the code given data x is p? (z|x), and since this is typically intractable, it is approximated as q? (z|x), parameterized by learned parameters ?. Conditional distributions q? (z|x) and p? (x|z) are typically designed such that they are easily sampled and, for flexibility, modeled in terms of neural networks [12]. Since z is a latent code for x, q? (z|x) is also termed a stochastic encoder, with p? (x|z) a corresponding stochastic decoder. The observed data are assumed drawn from q(x), for which we do not have a explicit form, but from which we have samples, i.e., the ensemble {xi }i=1,N used for learning. R Our goal is to learn the model p? (x) = p? (x|z)p(z)dz such that it synthesizes samples that are well matched to those drawn from q(x). We simultaneously seek to learn a corresponding encoder q? (z|x) that is both accurate and efficient to implement. Samples x are synthesized via x ? p? (x|z) with z ? p(z); z ? q? (z|x) provides an efficient coding of observed x, that may be used for other purposes (e.g., classification or caption generation when x is an image [21]). 2.1 Traditional Variational Autoencoders and Their Limitations Maximum likelihood (ML) learning of ? based on direct evaluation of p? (x) is typically intractable. The VAE [12, 13] seeks to bound p? (x) by maximizing variational expression LVAE (?, ?), with respect to parameters {?, ?}, where   p? (x, z) LVAE (?, ?) = Eq? (x,z) log = Eq(x) [log p? (x) ? KL(q? (z|x)kp? (z|x))] (1) q? (z|x) = ?KL(q? (x, z)kp? (x, z)) + const , (2) with expectations Eq? (x,z) and Eq(x) performed approximately via sampling. Specifically, to evaluate Eq? (x,z) we draw a finite set of samples z i ? q? (z i |xi ), with xi ? q(x) denoting the observed data, and for Eq(x) , we directly use observed data xi ? q(x). When learning {?, ?}, the expectation using samples from z i ? q? (z i |xi ) is implemented via the ?reparametrization trick? [12]. PN Maximizing LVAE (?, ?) wrt {?, ?} provides a lower bound on N1 i=1 log p? (xi ), hence the VAE PN setup is an approximation to ML learning of ?. Learning ? based on N1 i=1 log p? (xi ) is equivalent to learning ? based on minimizing KL(q(x)kp? (x)), again implemented in terms of the N observed samples of q(x). As discussed in [22], such learning does not penalize ? severely for yielding x 2 of relatively high probability in p? (x) while being simultaneously of low probability in q(x). This means that ? seeks to match p? (x) to the properties of the observed data samples, but p? (x) may also have high probability of generating samples that do not look like data drawn from q(x). This is a fundamental limitation of ML-based learning [22], inherited by the traditional VAE in (1). One R reason for the failing R of ML-based learning of ? is that the cumulative posterior on latent codes p? (z|x)q(x)dx ? q? (z|x)q(x)dx = q? (z) is typically different from p(z), which implies that x ? p? (x|z), with z ? p(z) may yield samples x that are different from those generated from q(x). Hence, when learning {?, ?} one may seek to match p? (x) to samples of q(x), as done in (1), while simultaneously matching q? (z) to samples of p(z). The expression in (1) provides a variational bound for matching p? (x) to samples of q(x), thus one may naively think to simultaneously set a similar variational expression for q? (z), with these two variational expressions optimized jointly. However, to compute this additional variational expression we require an analytic expression for q? (x, z) = q? (z|x)q(x), which also means we need an analytic expression for q(x), which we do not have. Examining (2), we also note that LVAE (?, ?) approximates ?KL(q? (x, z)kp? (x, z)), which has limitations aligned with those discussed above for ML-based learning of ?. Analogous to the above discussion, we would also like to consider ?KL(p? (x, z)kq? (x, z)). So motivated, in Section 3 we PN develop a new form of variational lower bound, applicable to maximizing N1 i=1 log p? (xi ) and P M 1 j=1 log q? (z j ), where z j ? p(z) is the j-th of M samples from p(z). We demonstrate that this M new framework leverages both KL(p? (x, z)kq? (x, z)) and KL(q? (x, z)kp? (x, z)), by extending ideas from adversarial networks. 2.2 Adversarial Learning The original idea of GAN [3] was to build an effective generative model p? (x|z), with z ? p(z), as discussed above. There was no desire to simultaneously design an inference network q? (z|x). More recently, authors [10, 11, 23] have devised adversarial networks that seek both p? (x|z) and q? (z|x). As an important example, Adversarial Learned Inference (ALI) [10] considers the following objective function: min max LALI (?, ?, ?) = Eq? (x,z) [log ?(f? (x, z))] + Ep? (x,z) [log(1 ? ?(f? (x, z)))] , ?,? ? (3) where the expectations are approximated with samples, as in (1). The function f? (x, z), termed a discriminator, is typically implemented using a neural network with parameters ? [10, 11]. Note that in (3) we need only sample from p? (x, z) = p? (x|z)p(z) and q? (x, z) = q? (z|x)q(x), avoiding the need for an explicit form for q(x). The framework in (3) can, in theory, match p? (x, z) and q? (x, z), by finding a Nash equilibrium of their respective non-convex objectives [3, 9]. However, training of such adversarial networks is typically based on stochastic gradient descent, which is designed to find a local mode of a cost function, rather than locating an equilibrium [9]. This objective mismatch may lead to the well-known instability issues associated with GAN training [9, 22]. To alleviate this problem, some researchers add a regularization term, such as reconstruction loss [24, 25, 26] or mutual information [4], to the GAN objective, to restrict the space of suitable mapping functions, thus avoiding some of the failure modes of GANs, i.e., mode collapsing. Below we will formally match the joint distributions as in (3), and reconstruction-based regularization will be manifested by generalizing the VAE setup via adversarial learning. Toward this goal we consider the following lemma, which is analogous to Proposition 1 in [3, 23]. Lemma 1 Consider Random Variables (RVs) x and z with joint distributions, p(x, z) and q(x, z). The optimal discriminator D? (x, z) = ?(f ? (x, z)) for the following objective max Ep(x,z) log[?(f (x, z))] + Eq(x,z) [log(1 ? ?(f (x, z)))] , f (4) is f ? (x, z) = log p(x, z) ? log q(x, z). Under Lemma 1, we are able to estimate the log q? (x, z) ? log p? (x)p(z) and log p? (x, z) ? log q(x)q? (z) using the following corollary. 3 Corollary 1.1 For RVs x and z with encoder joint distribution q? (x, z) = q(x)q? (z|x) and decoder joint distribution p? (x, z) = p(z)p? (x|z), consider the following objectives: max LA1 (? 1 ) = Ex?q(x),z?q? (z|x) log[?(f?1 (x, z))] ?1 (5) + Ex?p? (x|z0 ),z0 ?p(z),z?p(z) [log(1 ? ?(f?1 (x, z)))] , max LA2 (? 2 ) = Ez?p(z),x?p? (x|z) log[?(f?2 (x, z))] ?2 (6) + Ez?q? (z|x0 ),x0 ?q(x),x?q(x) [log(1 ? ?(f?2 (x, z)))] , If the parameters ? and ? are fixed, with f??1 the optimal discriminator for (5) and f??2 the optimal discriminator for (6), then f??1 (x, z) = log q? (x, z) ? log p? (x)p(z), f??2 (x, z) = log p? (x, z) ? log q? (z)q(x) . (7) The proof is provided in the Appendix A. We also assume in Corollary 1.1 that f?1 (x, z) and f?2 (x, z) are sufficiently flexible such that there are parameters ? ?1 and ? ?2 capable of achieving the equalities in (7). Toward that end, f?1 and f?2 are implemented as ? 1 - and ? 2 -parameterized neural networks (details below), to encourage universal approximation [27]. 3 Adversarial Symmetric Variational Auto-Encoder (AS-VAE) Consider variational expressions LVAEx (?, ?) = Eq(x) log p? (x) ? KL(q? (x, z)kp? (x, z)) (8) LVAEz (?, ?) = Ep(z) log q? (z) ? KL(p? (x, z)kq? (x, z)) , (9) where all expectations are again performed approximately using samples from q(x) and p(z). Recall that Eq(x) log p? (x) = ?KL(q(x)kp? (x)) + const, and Ep(z) log p? (z) = ?KL(p(z)kq? (z)) + const, thus (8) is maximized when q(x) = p? (x) and q? (x, z) = p? (x, z). Similarly, (9) is maximized when p(z) = q? (z) and q? (x, z) = p? (x, z). Hence, (8) and (9) impose desired constraints on both the marginal and joint distributions. Note that the log-likelihood terms in (8) and (9) are analogous to the data-fit regularizers discussed above in the context of ALI, but here implemented in a generalized form of the VAE. Direct evaluation of (8) and (9) is not possible, as it requires an explicit form for q(x) to evaluate q? (x, z) = q? (z|x)q(x). One may readily demonstrate that LVAEx (?, ?) = Eq? (x,z) [log p? (x)p(z) ? log q? (x, z) + log p? (x|z)] = Eq? (x,z) [log p? (x|z) ? f??1 (x, z)] . A similar expression holds for LVAEz (?, ?), in terms of f??2 (x, z). This naturally suggests the cumulative variational expression LVAExz (?, ?, ? 1 , ? 2 ) = LVAEx (?, ?) + LVAEz (?, ?) (10) = Eq? (x,z) [log p? (x|z) ? f?1 (x, z)] + Ep? (x,z) [log q? (x|z) ? f?2 (x, z)] , where ? 1 and ? 2 are updated using the adversarial objectives in (5) and (6), respectively. Note that to evaluate (10) we must be able to sample from q? (x, z) = q(x)q? (z|x) and p? (x, z) = p(z)p? (x|z), both of which are readily available, as discussed above. Further, we require explicit expressions for q? (z|x) and p? (x|z), which we have. For (5) and (6) we similarly must be able to sample from the distributions involved, and we must be able to evaluate f?1 (x, z) and f?2 (x, z), each of which is implemented via a neural network. Note as well that the bound in (1) for Eq(x) log p? (x) is in terms of the KL distance between conditional distributions q? (z|x) and p? (z|x), while (8) utilizes the KL distance between joint distributions q? (x, z) and p? (x, z) (use of joint distributions is related to ALI). By combining (8) and (9), the complete variational bound LVAExz employs the symmetric KL between these two joint distributions. By contrast, from (2), the original variational lower bound only addresses a one-way KL distance between q? (x, z) and p? (x, z). While [23] had a similar idea of employing adversarial methods in the context variational learning, it was only done within the context of the original form in (1), the limitations of which were discussed in Section 2.1. 4 In the original VAE, in which (1) was optimized, the reparametrization trick [12] was invoked wrt q? (z|x), with samples z ? (x, ) and  ? N (0, I), as the expectation was performed wrt this distribution; this reparametrization is convenient for computing gradients wrt ?. In the AS-VAE in (10), expectations are also needed wrt p? (x|z). Hence, to implement gradients wrt ?, we also constitute a reparametrization of p? (x|z). Specifically, we consider samples x? (z, ?) with ? ? N (0, I). LVAExz (?, ?, ? 1 , ? 2 ) in (10) is re-expressed as   LVAExz (?, ?, ? 1 , ? 2 ) = Ex?q(x),?N (0,I) f?1 (x, z ? (x, )) ? log p? (x|z ? (x, ))   + Ez?p(z),??N (0,I) f?2 (x? (z, ?), z) ? log q? (z|x? (z, ?)) . (11) The expectations in (11) are approximated via samples drawn from q(x) and p(z), as well as samples of  and ?. x? (z, ?) and z ? (x, ) can be implemented with a Gaussian assumption [12] or via density transformation [14, 16], detailed when presenting experiments in Section 5. The complete objective of the proposed Adversarial Symmetric VAE (AS-VAE) requires the cumulative variational in (11), which we maximize wrt ? 1 and ? 1 as in (5) and (6), using the results in (7). Hence, we write min max ?LVAExz (?, ?, ? 1 , ? 2 ) . (12) ?,? ? 1 ,? 2 The following proposition characterizes the solutions of (12) in terms of the joint distributions of x and z. Proposition 1 The equilibrium for the min-max objective in (12) is achieved by specification {? ? , ?? , ? ?1 , ? ?2 } if and only if (7) holds, and p?? (x, z) = q?? (x, z). The proof is provided in the Appendix A. This theoretical result implies that (i) ? ? is an estimator that yields good reconstruction, and (ii) ?? matches the aggregated posterior q? (z) to prior distribution p(z). 4 Related Work VAEs [12, 13] represent one of the most successful deep generative models developed recently. Aided by the reparameterization trick, VAEs can be trained with stochastic gradient descent. The original VAEs implement a Gaussian assumption for the encoder. More recently, there has been a desire to remove this Gaussian assumption. Normalizing flow [14] employs a sequence of invertible transformation to make the distribution of the latent codes arbitrarily flexible. This work was followed by inverse auto-regressive flow [16], which uses recurrent neural networks to make the latent codes more expressive. More recently, SteinVAE [28] applies Stein variational gradient descent [29] to infer the distribution of latent codes, discarding the assumption of a parametric form of posterior distribution for the latent code. However, these methods are not able to address the fundamental limitation of ML-based models, as they are all based on the variational formulation in (1). GANs [3] constitute another recent framework for learning a generative model. Recent extensions of GAN have focused on boosting the performance of image generation by improving the generator [5], discriminator [30] or the training algorithm [9, 22, 31]. More recently, some researchers [10, 11, 33] have employed a bidirectional network structure within the adversarial learning framework, which in theory guarantees the matching of joint distributions over two domains. However, non-identifiability issues are raised in [32]. For example, they have difficulties in providing good reconstruction in latent variable models, or discovering the correct pairing relationship in domain transformation tasks. It was shown that these problems are alleviated in DiscoGAN [24], CycleGAN [26] and ALICE [32] via additional `1 , `2 or adversarial losses. However, these methods lack of explicit probabilistic modeling of observations, thus could not directly evaluate the likelihood of given data samples. A key component of the proposed framework concerns integrating a new VAE formulation with adversarial learning. There are several recent approaches that have tried to combining VAE and GAN [34, 35], Adversarial Variational Bayes (AVB) [23] is the one most closely related to our work. AVB employs adversarial learning to estimate the posterior of the latent codes, which makes the encoder arbitrarily flexible. However, AVB seeks to optimize the original VAE formulation in (1), and hence it inherits the limitations of ML-based learning of ?. Unlike AVB, the proposed use of adversarial learning is based on a new VAE setup, that seeks to minimize the symmetric KL distance between p? (x, z) and q? (x, z), while simultaneously seeking to maximize the marginal expected likelihoods Eq(x) [log p? (x)] and Ep(z) [log p? (z)]. 5 5 Experiments We evaluate our model on three datasets: MNIST, CIFAR-10 and ImageNet. To balance performance and computational cost, p? (x|z) and q? (z|x) are approximated with a normalizing flow [14] of length 80 for the MNIST dataset, and a Gaussian approximation for CIFAR-10 and ImageNet data. All network architectures are provided in the Appendix B. All parameters were initialized with Xavier [36], and optimized via Adam [37] with learning rate 0.0001. We do not perform any dataset-specific tuning or regularization other than dropout [38]. Early stopping is employed based on average reconstruction loss of x and z on validation sets. We show three types of results, using part of or all of our model to illustrate each component. i) AS-VAE-r: This model trained with the first half of the objective in (11) to minimize LVAEx (?, ?) in (8); it is an ML-based method which focuses on reconstruction. ii) AS-VAE-g: This model trained with the second half of the objective in (11) to minimize LVAEz (?, ?) in (9); it can be considered as maximizing the likelihood of q? (z), and designed for generation. iii) AS-VAE This is our proposed model, developed in Section 3. 5.1 Evaluation We evaluate our model on both reconstruction and generation. The performance of the former is evaluated using negative log-likelihood (NLL) estimated via the variational lower bound defined in (1). Images are modeled as continuous. To do this, we add [0, 1]-uniform noise to natural images (one color channel at the time), then divide by 256 to map 8-bit images (256 levels) to the unit interval. This technique is widely used in applications involving natural images [12, 14, 16, 39, 40], since it can be proved that in terms of log-likelihood, modeling in the discrete space is equivalent to modeling in the continuous space (with added noise) [39, 41]. During testing, the likelihood is computed as p(x = i|z) = p? (x ? [i/256, (i + 1)/256]|z) where i = 0, . . . , 255. This is done to guarantee a fair comparison with prior work (that assumed quantization). For the MNIST dataset, we treat the [0, 1]-mapped continuous input as the probability of a binary pixel value (on or off) [12]. The inception score (IS), defined as exp(Eq (x)KL(p(y|x)kp(y))), is employed to quantitatively evaluate the quality of generated natural images, where p(y) is the empirical distribution of labels (we do not leverage any label information during training) and p(y|x) is the output of the Inception model [42] on each generated image. To the authors? knowledge, we are the first to report both inception score (IS) and NLL for natural images from a single model. For comparison, we implemented DCGAN [5] and PixelCNN++ [40] as baselines. The implementation of DCGAN is based on a similar network architectures as our model. Note that for NLL a lower value is better, whereas for IS a higher value is better. 5.2 MNIST We first evaluate our model on the MNIST dataset. The log-likelihood results are summarized in Table 1. Our AS-VAE achieves a negative log-likelihood of 82.51 nats, outperforming normalizing flow (85.1 nats) with a similar architecture. The perfomance of AS-VAE-r (81.14 nats) is competitive to the state-of-the-art (79.2 nats). The generated samples are showed in Figure 1. AS-VAE-g and AS-VAE both generate good samples while the results of AS-VAE-r are slightly more blurry, partly due to the fact that AS-VAE-r is an ML-based model. 5.3 CIFAR Next we evaluate our models on the CIFAR-10 dataset. The quantitative results are listed in Table 2. AS-VAE-r and AS-VAE-g achieve encouraging results on reconstruction and generation, respectively, while our AS-VAE model (leveraging the full objective) achieves a good balance between these two tasks, which demonstrates the benefit of optimizing a symmetric objective. Compared with Table 1: NLL on MNIST. Method NF (k=80) [14] IAF [16] AVB [23] PixelRNN [39] AS-VAE-r AS-VAE-g AS-VAE NLL (nats) 85.1 80.9 79.5 79.2 81.14 146.32 82.51 6 state-of-the-art ML-based models [39, 40], we achieve competitive results on reconstruction but provide a much better performance on generation, also outperforming other adversarially-trained models. Note that our negative ELBO (evidence lower bound) is an upper bound of NLL as reported in [39, 40]. We also achieve a smaller root-mean-square-error (RMSE). Generated samples are shown in Figure 2. Additional results are provided in the Appendix C. ALI [10], which also seeks to match Table 2: Quantitative Results on CIFAR-10; ? 2.96 is based on our the joint encoder and decoder distribu- implementation and 2.92 is reported in [40]. tion, is also implemented as a baseline. Since the decoder in ALI is a deterMethod NLL(bits) RMSE IS ministic network, the NLL of ALI is WGAN [43] 3.82 impractical to compute. Alternatively, MIX+WassersteinGAN [43] 4.05 we report the RMSE of reconstruction DCGAN [5] 4.89 as a reference. Figure 3 qualitatively ALI 14.53 4.79 compares the reconstruction perforPixelRNN [39] 3.06 mance of our model, ALI and VAE. PixelCNN++ [40] 2.96 (2.92)? 3.289 5.51 As can be seen, the reconstruction of AS-VAE-r 3.09 3.17 2.91 ALI is related to but not faithful reproAS-VAE-g 93.12 13.12 6.89 duction of the input data, which eviAS-VAE 3.32 3.36 6.34 dences the limitation in reconstruction ability of adversarial learning. This is also consistent in terms of RMSE. 5.4 ImageNet ImageNet 2012 is used to evaluate the scalability of our model to large datasets. The images are resized to 64?64. The quantitative results are shown in Table 3. Our model significantly improves the performance on generation compared with DCGAN and PixelCNN++, while achieving competitive results on reconstruction compared with PixelRNN and PixelCNN++. Note that the PixelCNN++ takes more than two weeks (44 hours per epoch) for training and 52.0 seconds/image Table 3: Quantitative Results on ImageNet. for generating samples while our model only requires less than 2 days (4 hours per epoch) for training and 0.01 secMethod NLL IS onds/image for generating on a single TITAN X GPU. As a DCGAN [5] 5.965 reference, the true validation set of ImageNet 2012 achieves 3.63 PixelRNN [39] 53.24% accuracy. This is because ImageNet has much 7.65 PixelCNN++ [40] 3.27 greater variety of images than CIFAR-10. Figure 4 shows AS-VAE 3.71 11.14 generated samples based on trained with ImageNet, compared with DCGAN and PixelCNN++. Our model is able to produce sharp images without label information while capturing more local spatial dependencies than PixelCNN++, and without suffering from mode collapse as DCGAN. Additional results are provided in the Appendix C. 6 Conclusions We presented Adversarial Symmetrical Variational Autoencoders, a novel deep generative model for unsupervised learning. The learning objective is to minimizing a symmetric KL divergence between the joint distribution of data and latent codes from encoder and decoder, while simultaneously maximizing the expected marginal likelihood of data and codes. An extensive set of results demonstrated excellent performance on both reconstruction and generation, while scaling to large datasets. A possible direction for future work is to apply AS-VAE to semi-supervised learning tasks. Acknowledgements This research was supported in part by ARO, DARPA, DOE, NGA, ONR and NSF. 7 Figure 1: Generated samples trained on MNIST. (Left) AS-VAE-r; (Middle) AS-VAE-g (Right) AS-VAE. Figure 2: Samples generated by AS-VAE Figure 3: Comparison of reconstruction with ALI [10]. In each block: column one for ground-truth, column two when trained on CIFAR-10. for ALI and column three for AS-VAE. Figure 4: Generated samples trained on ImageNet. (Top) AS-VAE; (Middle) DCGAN [5];(Bottom) PixelCNN++ [40]. 8 References [1] Y. Pu, X. Yuan, A. Stevens, C. Li, and L. Carin. A deep generative deconvolutional image model. Artificial Intelligence and Statistics (AISTATS), 2016. [2] Y. Pu, X. Yuan, and L. Carin. Generative deep deconvolutional learning. In ICLR workshop, 2015. [3] I.. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S.l Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In NIPS, 2014. [4] X. Chen, Y. Duan, R. Houthooft, J. Schulman, I. Sutskever, and P. Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In NIPS, 2016. [5] A. Radford, L. Metz, and S. Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016. 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Unified representation of tractography and diffusion-weighted MRI data using sparse multidimensional arrays Cesar F. Caiafa? Department of Psychological and Brain Sciences Indiana University (47405) Bloomington, IN, USA IAR - CCT La Plata, CONICET / CIC-PBA (1894) V. Elisa, ARGENTINA [email protected] Olaf Sporns Department of Psychological and Brain Sciences Indiana University (47405) Bloomington, IN, USA [email protected] Andrew J. Saykin Department of Radiology - Indiana University School of Medicine. (46202) Indianapolis, IN, USA [email protected] Franco Pestilli? Department of Psychological and Brain Sciences Indiana University (47405) Bloomington, IN, USA [email protected] Abstract Recently, linear formulations and convex optimization methods have been proposed to predict diffusion-weighted Magnetic Resonance Imaging (dMRI) data given estimates of brain connections generated using tractography algorithms. The size of the linear models comprising such methods grows with both dMRI data and connectome resolution, and can become very large when applied to modern data. In this paper, we introduce a method to encode dMRI signals and large connectomes, i.e., those that range from hundreds of thousands to millions of fascicles (bundles of neuronal axons), by using a sparse tensor decomposition. We show that this tensor decomposition accurately approximates the Linear Fascicle Evaluation (LiFE) model, one of the recently developed linear models. We provide a theoretical analysis of the accuracy of the sparse decomposed model, LiFESD , and demonstrate that it can reduce the size of the model significantly. Also, we develop algorithms to implement the optimization solver using the tensor representation in an efficient way. ? ? http://web.fi.uba.ar/~ccaiafa/Cesar.html http://www.brain-life.org/plab/ 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 1 Introduction Multidimensional arrays, hereafter referred to as tensors, are useful mathematical objects to model a variety of problems in machine learning [2, 47] and neuroscience [27, 8, 50, 48, 3, 26, 13]. Tensor decomposition algorithms have a long history of applications in signal processing, however, only recently their relation to sparse representations has started to be explored [35, 11]. In this work, we present a sparse tensor decomposition model and its associated algorithm applied to diffusion-weighted Magnetic Resonance Imaging (dMRI). Diffusion-weighted MRI allows us to estimate structural brain connections in-vivo by measuring the diffusion of water molecules at different spatial directions. Brain connections are comprised of a set of fascicles describing the putative position and orientation of the neuronal axons bundles wrapped by myelin sheaths traveling within the living human brain [25]. The process by which fascicles (the connectome) are identified from dMRI measurements is called tractography. Tractography and dMRI are the primary methods for mapping structural brain networks and white matter tissue properties in living human brains [6, 46, 34]. Despite current limits and criticisms, through these methods we have learned much about the macrostructural organization of the human brain, such that network neuroscience has become one of the fastest-growing scientific fields [38, 43, 44]. In recent years, a large variety of tractography algorithms have been proposed and tested on modern datasets such as the Human Connectome Project (HCP) [45]. However, it has been established that the estimated anatomical properties of the fascicles depend on data type, tractography algorithm and parameters settings [32, 39, 7]. Such variability in estimates makes it difficult to trust a single algorithm for all applications, and calls for routine statistical evaluation methods of brain connectomes [32]. For this reason, linear methods based on convex optimization have been proposed for connectome evaluation [32, 39] and simultaneous connectome and white matter microstructure estimation [15]. However, these methods can require substantial computational resources (memory and computation load) making it prohibitive to apply them to the highest resolution datasets. In this article, we propose a method to encode brain connectomes in multidimensional arrays and perform statistical evaluation efficiently on high-resolution datasets. The article is organized as follows: in section 2, the connectome encoding method is introduced; in section 2.1, a linear formulation of the connectome evaluation problem is described; in section 3, the approximated tensor decomposed model is introduced; in section 3.3, we derive a theoretical bound of the approximation error and compute the theoretical compression factor obtained with the tensor decomposition; in section 4 we develop algorithms to make the operations needed for solving the connectome evaluation optimization problem; in section 5 we present experimental results using high resolution in vivo datasets; finally, in section 6, the main conclusions of our work are outlined. 2 Encoding brain connectomes into multidimensional array structures. We propose a framework to encode brain connectome data (both dMRI and white matter fascicles) into tensors [12, 11, 23] to allow fast and efficient mathematical operations on the structure of the connectome. Here, we introduce the tensor encoding framework and show how it can be used to implement recent methods for statistical evaluation of tractography [32]. More specifically, we demonstrate that the framework can be used to approximate the Linear Fascicle Evaluation model [32] with high accuracy while reducing the size of the model substantially (with measured compression factors up to 40x). Hereafter, we refer to the new tensor encoding method as ENCODE [10]. ENCODE maps fascicles from their natural brain space (Fig. 1(a)) into a three dimensional sparse tensor ? (Fig. 1(b)). The first dimension of ? (1st mode) encodes each individual white matter fascicle?s orientation at each position along their path through the brain. Individual segments (nodes) in a fascicle are coded as non-zero entries in the sparse array (dark-blue cubes in Fig. 1(b)). The second dimension of ? (2nd mode) encodes each fascicle?s spatial position within dMRI data volume (voxels). Slices in this second dimension represent single voxels (cyan lateral slice in Fig. 1(b)). The 2 third dimension (3rd mode) encodes the indices of each fascicle within the connectome. Full fascicles are encoded as ? frontal slices (c.f., yellow and blue in Fig. 1(b)). (a) (b) le cic s Orientation s Fa Voxe ls fascicle fascicle voxel non-zero entry Figure 1: The ENCODE method: mapping structural connectomes from natural brain space to tensor space. (a) Two example white matter fascicles (f1 and f2 ) passing through three voxels (v1 , v2 and v3 ). (b) Encoding of the two fascicles in a three dimensional tensor. The non-zero entries in ? indicate fascicle?s orientation (1st mode), position (voxel, 2nd mode) and identity (3rd mode). Below we demonstrate how to use ENCODE to integrate connectome each fascicle?s structure and measured dMRI signal into a single tensor decomposition model. We then show how to use this decompositon model to implement very efficiently a recent model for tractography evaluation, the linear fascicle evaluation method, also referred to as LiFE [32]. Before introducing the tensor decomposition method, we briefly describe the LiFE model, as this is needed to explain the model decomposition using the ENCODE method. We then calculate the theoretical bounds to accuracy and compression factor that can be achieved using ENCODE and tensor decomposition. Finally, we report the results of experiments on real data and validate the theoretical calculations. 2.1 Statistical evaluation for brain connectomes by convex optimization. The Linear Fascicle Evaluation (LiFE) method was introduced to compute the statistical error of the fascicles comprising a structural brain connectome in predicting the measured diffusion signal [32]. The fundamental idea behind LiFE is that a connectome should contain fascicles whose trajectories represent the measured diffusion signal well. LiFE implements a method for connectome evaluation that can be used, among other things, to eliminate tracked fascicles that do not predict well the diffusion signal. LiFE takes as input the set of fascicles generated by using tractography methods (the candidate connectome) and returns as output the subset of fascicles that best predict the measured dMRI signal (the optimized connectome). Fascicles are scored with respect to how well their trajectories represent the measured diffusion signal in the voxels along the their path. To do so, weights are assigned to each fascicle using convex optimization. Fascicles assigned a weight of zero are removed from the connectome, as their contribution to predicting the diffusion signal is null. The following linear system describes the equation of LiFE (see Fig. 2(a)): y ? Mw, (2.1) N? Nv ? where y ? R is a vector containing the demeaned signal yi = S(?ni , vi ) measured at all white-matter voxels vi ? V = {1, 2, . . . , Nv } and across all diffusion directions ?n ? ? = {?1 , ?2 , . . . , ?N? } ? R3 , and w ? RNf contains the weights for each fascicle in the connectome. Matrix M ? RN? Nv ?Nf contains, at column f , the predicted demeaned signal contributed by fascicle f at all voxels V and across all directions ?: M(i, f ) = S0 (vi )Of (?ni , vf ). (2.2) S0 (v) is defined as the non diffusion-weighted signal and Of (?, vf ) is the orientation distribution function [32] of fascicle f at diffusion direction ?, i.e. T 2 1 X ?b(?nT vf )2 Of (?, vf ) = e?b(? vf ) ? e , (2.3) N? ?n ?? 3 (a) (b) Voxel voxel (c) Empty entries (zero values) Figure 2: The Linear Fascicle Evaluation (LiFE) model. (a) The predicted signal y ? RN? Nv in all voxels and gradient directions is obtained by multiplying matrix M ? RN? Nv ?Nf by the vector of weights w ? RNf (see equation 2.1). (b) A voxel containing two fascicles, f1 and f2 . (c) The predicted diffusion signal yv ? RN? at voxel v is approximated by a nonnegative weighted linear combination of the predicted signals for the fascicles in the voxel. where the simple ?stick? diffusion tensor model [31] was used and vector vf ? R3 is defined as the spatial orientation of the fascicle in that voxel. Whereas vector y and matrix M in equation (2.1) are fully determined by the dMRI measurements and the output of a tractography algorithm, respectively, the vector of weights w needs to be estimated by solving a Non-Negative Least squares (NNLS) optimization problem, which is defined as follows:   1 min ky ? Mwk2 subject to wf ? 0, ?f. (2.4) w 2 As a result, a sparse non-negative vector of weights w is obtained. Whereas nonzero weights correspond to fascicles that contribute to predict the measured dMRI signal, fascicles with zero weight make no contribution to predicting the measurements and can be eliminated. In this way, LiFE identifies the fascicles supported by the data in a candidate connectome providing a principled approach to evaluate connectomes in terms of prediction error as well as the number of non-zero weighted fascicles. A noticeable property of the LiFE method is that the size of matrix M in equation (2.1) can require tens of gigabytes for full-brain connectomes, even when using optimized sparse matrix formats [19]. Below we show how to use ENCODE to implement a sparse tensor decomposition [9, 11] of matrix M. This decomposition allows accurate approximation of the original LiFE model with dramatic reduction in memory requirements. 3 Theoretical results: Tensor decomposition and approximation of the linear model for tractography evaluation. We describe the theoretical approach to factorizing the LiFE model, eq. (2.1). We note that matrix M ? RN? Nv ?Nf (Fig. 2(a)) can be rewritten as a tensor (3D-array) M ? RN? ?Nv ?Nf by decoupling the gradient direction and voxel indices into separate indices, i.e. M(ni , vi , f ) = M(i, f ), where ni = {1, 2, . . . , N? }, vi = {1, 2, . . . , Nv } and f = {1, 2, . . . , Nf }. Thus, equation (2.1) can be rewritten in tensor form as follows: Y ? M ?3 w T , (3.1) where Y ? RN? ?Nv is obtained by converting vector y ? RN? Nv into a matrix (matricization) and ??n ? is the tensor-by-matrix product in mode-n [23], more specifically, the mode-3 4 PNf product in the above equation is defined as follows: Y(n, v) = f =1 M(n, v, f )wf . Below, we show how to approximate the tensor model in equation (3.1) using a sparse Tucker decomposition [9] by first focusing on the dMRI signal in individual voxels and then across voxels. (a) Empty entries (zero values) (b) (c) Max. discretization error, Figure 3: The LiFESD model: (a) Each block Mv of matrix M (a lateral slice in tensor M) is factorized by using a dictionary of diffusion signal predictions D and a sparse matrix of coefficients ?v . (b) LiFESD model is written as a Tucker decomposition model with a sparse core tensor ? and factors D (mode-1) and wT (mode-3). (c). The maximum distance between a fascicle orientation vector v and its approximation va is determined by the discretization of azimuth (?? ) and elevation (?? ) spherical coordinates. More specifically, for ?? = ?? = ?/L, the maximum discretization error is k?v k ? ??2L . 3.1 Approximation of the linear model within individual brain voxels. We focus on writing the linear formulation of the diffusion prediction model (Fig. 2(b)-(c)) by restricting equation (3.1) to individual voxels, v: yv ? Mv w, (3.2) where vector yv = Y(:, v) ? RN? and matrix Mv = M(:, v, :) ? RN? ?Nf , correspond to a column in Y and a lateral slice in tensor M, respectively. We propose to factorize matrix Mv as follows ? v = D?v , Mv ? M (3.3) where matrix D ? RN? ?Na is a dictionary of diffusion predictions whose columns (atoms) correspond to precomputed fascicle orientations, and ?v ? RNa ?Nf is a sparse matrix whose non-zero entries, ?v (a, f ), indicate the orientation of fascicle f in voxel v, which is approximated by atom a (see Fig. 3(a) for an example of a voxel v as shown in Fig. 2(b)-(c)). For computing the diffusion predictions, we use a discrete grid in the sphere by uniformly sampling the spherical coordinates using L points in azimuth and elevation coordinates (Fig. 2(c)). 3.2 Approximation of the linear model across multiple brain voxels. By applying the approximation introduced in equation (3.3) to every slice in tensor M in equation 3.1, we obtain the following tensor Sparse Decomposed LiFE model, hereafter referred to as LiFESD (Fig. 3(b)): Y ? ? ?1 D ?3 w T , (3.4) where D is a common factor in mode-1, i.e., it multiplies all lateral slices. It is noted that, the formula in the above equation (3.4), is a particular case of the Tucker decomposition [42, 16] where the core tensor ? is sparse [9, 11], and only factors in mode-1 (D) and mode-3 (wT ) 5 are present. By comparing equations (3.4) and (3.1) we define the LiFESD approximated tensor model as ? = ? ?1 D M (3.5) 3.3 Theoretical bound for model decomposition accuracy and data compression. In this section, we derive a theoretical bound on the accuracy of LiFESD compared to the original LiFE model (Proposition 3.1) and we theoretically analyze the compression factor associated to the factorized tensor approximation (Proposition 3.2). Hereafter, we assume that, in a given connectome having Nf fascicles, each fascicle has a fixed number of nodes (Nn ), and the diffusion weighted measurements were taken on N? gradient directions with a gradient strength b. The proofs of the propositions can be found in the Supplementary material. Proposition 3.1 (accuracy). For a given connectome, and dictionary D obtained by uniformly sampling the azimuth-elevation (?, ?) space using ?? = ?? = ?/L (see Fig. 3(c)), the following upper bound on the Frobenius norm based model error is verified: p ? F ? 2b? 6Nf Nn N? . kM ? Mk (3.6) L The importance of this theoretical result is that the error is inversely proportional to the discretization parameter L, which allows one to design the decomposed model so that a prescribed accuracy is met. Proposition 3.2 (size reduction). For a given connectome, and a dictionary D ? RN? ?Na containing Na atoms (columns of matrix D), the achieved compression factor is  ?1 4 Na CF = ? , (3.7) 3N? 3Nn Nf ? with C(M) and C(M) ? being the storage costs of LiFE and where CF = C(M)/C(M), LiFESD models, respectively. It is noted that, usually 3Nn Nf  Na , which implies that the compression factor can be approximated by CF ? 3N4 ? , i.e., it is proportional to the number of gradient directions N? . 4 Model optimization using tensor encoding. Once the LiFESD model has been built, the final step to validate a connectome requires finding the non-negative weights that least-squares fit the measured diffusion data. This is a convex optimization problem that can be solved using a variety of NNLS optimization algorithms. We used a NNLS algorithm based on first-order methods specially designed for large scale problems [22]. Next, we show how to exploit the decomposed LiFESD model in the optimization. The gradient of the original objective function for the LiFE model can be written as follows:   1 ?w ky ? Mwk2 = MT Mw ? 2MT y, (4.1) 2 where M ? RN? Nv ?Nf is the original LiFE model, w ? RNf the fascicle weights and y ? RN? Nv the demeaned diffusion signal. Because the decomposed version does not explicitly store M, below we describe how to perform two basic operations (y = Mw and w = MT y) using the sparse decomposition. 4.1 Computing y = Mw Using equation (3.1) we can see that the product Mw can be computed using equation (3.4) and vectorizing the result, i.e. y = vec(Y), where vec() stands for the vectorization 6 operation, i.e., to convert a matrix to a vector by stacking its columns in a long vector. In Algorithm 1, we present the steps for computing y = Mw in an efficient way. Algorithm 1 : y = M_times_w(?,D,w) Require: Decomposition components (?, D and vector w ? RNf ). Ensure: y = Mw 1: Y = ? ?3 wT ; the result is a large but very sparse matrix (Na ? Nv ) 2: Y = DY; the result is a relatively small matrix (N? ? Nv ) 3: y = vec(Y) 4: return y; 4.2 Computing w = MT y The product w = MT y can be computed using LiFESD in the following way: w = MT y = M(3) y = ?(3) (I ? DT )y, (4.2) where M(3) ? RNf ?N? Nv and ?(3) ? RNf ?Na Nv are the unfolding matrices [23] of tensors M ? RN? ?Nv ?Nf and ? ? RNa ?Nv ?Nf , respectively; ? is the Kronecker product and I is the (Nv ? Nv ) identity matrix. Equation (4.2) can be written also as follows [9]: w = ?(3) vec(DT Y). (4.3) Because matrix ?(3) is very sparse, we avoid computing the large and dense matrix DT Y and compute instead only its blocks that are being multiplied by the non-zero entries in ?(3) . This allows maintaining efficient memory usage and limits the number of CPU cycles needed. In Algorithm 2, we present the steps for computing w = MT y in an efficient way. Algorithm 2 : w = Mtransp_times_y(?,D,y) Require: Decomposition components (?, D) and vector y ? RN? Nv . Ensure: w = MT y 1: Y ? RN? ?Nv ? y ? RN? Nv ; reshape vector y into a matrix Y 2: [a, v, f , c] = get_nonzero_entries(?); a(n), v(n), f (n), c(n) indicate the atom, the voxel, the fascicle and the entry in tensor ? associated to node n, respectively, with n = 1, 2, . . . , Nn ; 3: w = 0 ? RNf ; Initialize weights with zeros 4: for n = 1 to Nn do 5: w(f (n)) = w(f (n)) + DT (:, a(n))Y(:, v(n))c(n); 6: end for 7: return w; 5 Experimental results: Validation of the theoretical bounds for model decomposition accuracy and data compression. Here, we validate our theoretical findings by using dMRI data from subjects in a public source (the Stanford dataset [32]). The data were collected using N? = 96 (STN96, five subjects) and N? = 150 (STN150, one subject) directions with b-value b = 2, 000s/mm2 . We performed tractography using these data and both, probabilistic and deterministic methods, in combination with Constrained Spherical Deconvolution (CSD) and the diffusion tensor model (DTI) [41, 17, 5]. We generated candidate connectomes with Nf = 500, 000 fascicles per brain brain. See for [10, 32, 39] for additional details on data preprocessing. We first analyzed the accuracy of the approximated model (LiFESD ) as a function of the parameter, L, which describes the number of fascicles orientations encoded in the dictionary D. In theory, the larger the number of atoms in D the higher the accuracy of the approximation. ? kM?Mk We show that model error (defined as eM = kMkF F ) decreases as a function of the parameter L for all subjects in the dataset Fig. 4(a). This result validates the theoretical upper bound in Proposition 3.1. We also solved the convex optimization problem of equation 7 (2.4) for both, LiFE and LiFESD , and estimated the error in the weights assigned to each ? wk fascicle by the two models (we computed the error in weights as follows ew = kw? kwk ). Fig. 4(b) shows the error ew as a function of the parameter L. It is noted that for L > 180 the error is lower than 0.1% in all subjects. 0.1% 0 23 40 45 90 180 360 720 Matrix based LiFE LiFESD (L=360) 30 (d) 0.1% 0 23 45 40 90 180 360 720 Matrix based LiFE LiFESD (L=360) Model size (GB) 30 20 20 10 10 0 0 50 150 (f) 1,000 Deterministic (64,134 fascicles) 10,000 100,000 1,000,000 (g) 1000 r.m.s.e (det) Probabilistic (121,050 fascicles) 100 500 -3 3x10 0 10 10 500 r.m.s.e (prob) Probability Model size (GB) (e) STN96 (%) 0.5 0.5 0 Subject 1 Subject 2 Subject 3 Subject 4 Subject 5 1 Weights error Model error 1 (c) 1.5 STN150 (b) 1.5 (%) (a) 1000 Figure 4: Experimental results: (a) The model error eM in approximating the matrix M with LiFESD is inversely proportional to the parameter L as predicted by our Proposition 3.1 (eM ? C/L was fitted to the data with C = 27.78 and a fitting error equal to 2.94%). (b) Error in the weights obtained by LiFESD compared with original LiFE?s weights, ew , as a function of parameter L. (c)-(d) Model size (GB) scales linearly with the number of directions N? and the number of fascicles Nf , however it increases much faster in the LiFE model compared to the LiFESD model. LiFESD was computed using L = 360. (e)-(f) Probabilistic and deterministic connectomes validated with LiFESD for a HCP subject. (g) Comparison of the Root-mean-squared-error (r.m.s, as defined in [32]) obtained in all voxels for probabilistic and deterministic connectomes. The averaged r.m.s.e are 361.12 and 423.06 for the probabilistic and deterministic cases, respectively. Having experimentally demonstrated that model approximation error decreases as function of L, we move on to demonstrate the magnitude of model compression achieved by the tensor decomposition approach. To do so, we fixed L = 360 and computed the model size for both, LiFE and LiFESD , as a function of the number of gradient directions N? (Fig. 4(c)) and fascicles Nf (Fig. 4(d)). Results show that, as predicted by our theoretical results in Proposition 3.2, model size scales linearly with the number of directions for both, LiFE and LiFESD , but that the difference in slope is profound. Experimentally measured compression ratios raise up to approximately 40 as it is the case for the subjects in the STN150 dataset (Nf = 500, 000 and N? = 150). 8 Finally, we show an example comparison between two connectomes obtained by applying probabilistic [17] and deterministic [4] tracking algorithms to one brain dataset (a single subject) from the Human Connectome Project dataset [45], with N? = 90, Nv = 267, 306 and Nf = 500, 000. Figs. 4e-f show the detected 20 major tracts in a human brain using only the fascicles with nonzero weigths. In this case, the probabilistic connectome has more fascicles (121, 050) than the deterministic one (64, 134). Moreover, we replicate previous results demonstrating that probabilistic connectomes have lower error than the deterministic one in a majority of the voxels (see Fig. 4(g)). 6 Conclusions We introduced a method to encode brain connectomes in multidimensional arrays and decomposition approach that can accurately approximate the linear model for connectome evaluation used in the LiFE method [32]. We demonstrate that the decomposition approach dramatically reduces the memory requirements of the LiFE model, approximately from 40GB to 1GB, with a small model approximation error of less than 1%. The compactness of the decomposed LIFE model has important implications for other computational problems. For example, model optimization can be implemented by using operations involving tensorial operations avoiding the use of large matrices such as M and using instead the sparse tensor and prediction dictionary (? and D respectively). Multidimensional tensors and decomposition methods have been used to help investigators make sense of large multimodal datasets [27, 11]. Yet to date these methods have found only a few applications in neuroscience, such as performing multi-subject, clustering and electroencephalography analyses [49, 48, 3, 28, 26, 13, 8]. Generally, decomposition methods have been used to find compact representations of complex data by estimating the combination of a limited number of common meaningful factors that best fit the data [24, 27, 23]. We propose a new application that, instead of using the decomposition to estimate latent factors, it encodes the structure of the problem explicitly. The new application of tensor decomposition proposed here has the potential to improve future generations of models of connectomics, tractography evaluation and microstructure [32, 15, 36, 39]. Improving these models will allow going beyond the current limitations of the state of the art methods [14]. Finally, tensorial representations for brain imaging data have the potential to contribute advancing the application of machine learning algorithms to mapping the human connectome [18, 37, 21, 20, 30, 1, 51, 29, 40, 33]. Acknowledgments This research was supported by (NSF IIS-1636893; BCS-1734853; NIH ULTTR001108) to F.P. Data provided by Stanford University (NSF BCS 1228397). F.P. were partially supported by the Indiana University Areas of Emergent Research initiative Learning: Brains, Machines, Children. References [1] Daniel C Alexander, Darko Zikic, Aurobrata Ghosh, Ryutaro Tanno, Viktor Wottschel, Jiaying Zhang, Enrico Kaden, Tim B Dyrby, Stamatios N Sotiropoulos, Hui Zhang, and Antonio Criminisi. Image quality transfer and applications in diffusion MRI. Human Brain Mapping Journal, pages 1?65, March 2017. [2] Animashree Anandkumar, Rong Ge 0001, Daniel J Hsu, and Sham M Kakade. A tensor approach to learning mixed membership community models. 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A Minimax Optimal Algorithm for Crowdsourcing Thomas Bonald Telecom ParisTech [email protected] Richard Combes Centrale-Supelec / L2S [email protected] Abstract We consider the problem of accurately estimating the reliability of workers based on noisy labels they provide, which is a fundamental question in crowdsourcing. We propose a novel lower bound on the minimax estimation error which applies to any estimation procedure. We further propose Triangular Estimation (TE), an algorithm for estimating the reliability of workers. TE has low complexity, may be implemented in a streaming setting when labels are provided by workers in real time, and does not rely on an iterative procedure. We prove that TE is minimax optimal and matches our lower bound. We conclude by assessing the performance of TE and other state-of-the-art algorithms on both synthetic and real-world data. 1 Introduction The performance of many machine learning techniques, and in particular data classification, strongly depends on the quality of the labeled data used in the initial training phase. A common way to label new datasets is through crowdsourcing: many workers are asked to label data, typically texts or images, in exchange of some low payment. Of course, crowdsourcing is prone to errors due to the difficulty of some classification tasks, the low payment per task and the repetitive nature of the job. Some workers may even introduce errors on purpose. Thus it is essential to assign the same classification task to several workers and to learn the reliability of each worker through her past activity so as to minimize the overall error rate and to improve the quality of the labeled dataset. Learning the reliability of each worker is a tough problem because the true label of each task, the so-called ground truth, is unknown; it is precisely the objective of crowdsourcing to guess the true label. Thus the reliability of each worker must be inferred from the comparison of her labels on some set of tasks with those of other workers on the same set of tasks. In this paper, we consider binary labels and study the problem of estimating the workers reliability based on the answers they provide to tasks. We make two novel contributions to that problem: (i) We derive a lower bound on the minimax estimation error which applies to any estimator of the workers reliability. In doing so we identify "hard" instances of the problem, and show that the minimax error depends on two factors: the reliability of the three most informative workers and the mean reliability of all workers. (ii) We propose TE (Triangular Estimation), a novel algorithm for estimating the reliability of each worker based on the correlations between triplets of workers. We analyze the performance of TE and prove that it is minimax optimal in the sense that it matches the lower bound we previously derived. Unlike most prior work, we provide non-asymptotic performance guarantees which hold even for a finite number of workers and tasks. As our analysis reveals, non-asymptotic performance guarantees require to use finer concentration arguments than asymptotic ones. TE has low complexity in terms of memory space and computation time, does not require to store the whole data set in memory and can be easily applied in a setting in which answers to tasks arrive 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. sequentially, i.e., in a streaming setting. Finally, we compare the performance of TE to state-of-theart algorithms through numerical experiments using both synthetic and real datasets. 2 Related Work The first problems of data classification using independent workers appeared in the medical context, where each label refers to the state of a patient (e.g., sick or sane) and the workers are clinicians. [Dawid and Skene, 1979] proposed an expectation-maximization (EM) algorithm, admitting that the accuracy of the estimate was unknown. Several versions and extensions of this algorithm have since been proposed and tested in various settings [Hui and Walter, 1980, Smyth et al., 1995, Albert and Dodd, 2004, Raykar et al., 2010, Liu et al., 2012]. A number of Bayesian techniques have also been proposed and applied to this problem by [Raykar et al., 2010, Welinder and Perona, 2010, Karger et al., 2011, Liu et al., 2012, Karger et al., 2014, 2013] and references therein. Of particular interest is the belief-propagation (BP) algorithm of [Karger et al., 2011], which is provably order-optimal in terms of the number of workers required per task for any given target error rate, in the limit of an infinite number of tasks and an infinite population of workers. Another family of algorithms is based on the spectral analysis of some matrix representing the correlations between tasks or workers. [Ghosh et al., 2011] work on the task-task matrix whose entries correspond to the number of workers having labeled two tasks in the same manner, while [Dalvi et al., 2013] work on the worker-worker matrix whose entries correspond to the number of tasks labeled in the same manner by two workers. Both obtain performance guarantees by the perturbation analysis of the top eigenvector of the corresponding expected matrix. The BP algorithm of Karger, Oh and Shah is in fact closely related to these spectral algorithms: their message-passing scheme is very similar to the power-iteration method applied to the task-worker matrix, as observed in [Karger et al., 2011]. Two notable recent contributions are [Chao and Dengyong, 2015] and [Zhang et al., 2014]. The former provides performance guarantees for two versions of EM, and derives lower bounds on the attainable prediction error (the probability of estimating labels incorrectly). The latter provides lower bounds on the estimation error of the workers? reliability as well as performance guarantees for an improved version of EM relying on spectral methods in the initialization phase. Our lower bound cannot be compared to that of [Chao and Dengyong, 2015] because it applies to the workers? reliability and not the prediction error; and our lower bound is tighter than that of [Zhang et al., 2014]. Our estimator shares some features of the algorithm proposed by [Zhang et al., 2014] to initialize EM, which suggests that the EM phase itself is not essential to attain minimax optimality. All these algorithms require the storage of all labels in memory and, to the best of our knowledge, the only known streaming algorithm is the recursive EM algorithm of [Wang et al., 2013], for which no performance guarantees are available. The remainder of the paper is organized as follows. In section 3 we state the problem and introduce our notations. The important question of identifiability is addressed in section 4. In section 5 we present a lower bound on the minimax error rate of any estimator. In section 6 we present TE, discuss its compexity and prove that it is minimax optimal. In section 7 we present numerical experiments on synthetic and real-world data sets and section 8 concludes the paper. Due to space constraints, we only provide proof outlines for our two main results in this document. Complete proofs are presented in the supplementary material. 3 Model Consider n workers, for some integer n ? 3. Each task consists in determining the answer to a binary question. The answer to task t, the ?ground-truth", is denoted by G(t) ? {+1, ?1}. We assume that the random variables G(1), G(2), . . . are i.i.d. and centered, so that there is no bias towards one of the answers. Each worker provides an answer with probability ? ? (0, 1]. When worker i ? {1, ..., n} provides an answer, this answer is correct with probability 21 (1 + ?i ), independently of the other workers, for some parameter ?i ? [?1, 1] that we refer to as the reliability of worker i. If ?i > 0 then worker 2 i tends to provide correct answers; if ?i < 0 then worker i tends to provide incorrect anwsers; if ?i = 0 then worker i is non-informative. We denote by ? = (?1 , . . . , ?n ) the reliability vector. Both ? and ? are unknown. Let Xi (t) ? {?1, 0, 1} be the output of worker i for task t, where the output 0 corresponds to the absence of an answer. We have: ? i w.p. ? 1+? ? G(t) 2 , 1??i Xi (t) = ?G(t) w.p. ? 2 (1) ? 0 w.p. 1 ? ?. Since the workers are independent, the random variables X1 (t), ..., Xn (t) are independent given G(t), for each task t. We denote by X(t) the corresponding vector. The goal is to estimate the ? that minimizes the error ground-truth G(t) as accurately as possible by designing an estimator G(t) ? ? probability P(G(t) 6= G(t)). The estimator G(t) is adaptive and may be a function of X(1), ..., X(t) but not of the unknown parameters ?, ?. It is well-known that, given ? and ? = 1, an optimal estimator of G(t) is the weighted majority vote [Nitzan and Paroush, 1982, Shapley and Grofman, 1984], namely ? = 1{W (t) > 0} ? 1{W (t) < 0} + Z1{W (t) = 0}, G(t) (2) P n 1+?i 1 where W (t) = n i=1 wi Xi (t), wi = ln( 1??i ) is the weight of worker i (possibly infinite), and Z is a Bernoulli random variable of parameter 21 over {+1, ?1} (for random tie-breaking). We prove this result for any ? ? (0, 1]. Proposition 1 Assuming that ? is known, the estimator (2) is an optimal estimator of G(t). Proof. Finding an optimal estimator of G(t) amounts to finding an optimal statistical test between hypotheses {G(t) = +1} and {G(t) = ?1}, under a symmetry constraint so that type I and type II error probability are equal. For any x ? {?1, 0, 1}n, let L+ (x) and L? (x) be the probabilities that X(t) = x under hypotheses {G(t) = +1} and {G(t) = ?1}, respectively. We have n Y L+ (x) = H(x) (1 + ?i )1{xi =+1} (1 ? ?i )1{xi =?1} , L? (x) = H(x) i=1 n Y i=1 Pn (1 + ?i )1{xi =?1} (1 ? ?i )1{xi =+1} , |xi | is the number of answers and H(x) = 21? ?? (1 ? ?)n?? . We deduce the  +  Pn L (x) log-likelihood ratio ln L = ? (x) i=1 wi xi . By the Neyman-Pearson theorem, for any level of significance, there exists a and b such that the uniformly most powerful test for that level is: 1{wT x > a} ? 1{wT x < a} + Z1{wT x = a}, where Z is a Bernoulli random variable of parameter b over {+1, ?1}. By symmetry, we must have a = 0 and b = 12 , as announced.  where ? = i=1 This result shows that estimating the true answer G(t) reduces to estimating the unknown parameters ? and ?, which is the focus of the paper. Note that the problem of estimating ? is important in itself, due to the presence of "spammers" (i.e., workers with low reliability); a good estimator can be used by the crowdsourcing platform to incentivize good workers. 4 Identifiability Estimating ? and ? from X(1), ..., X(t) is not possible unless we have identifiability, namely there cannot exist two distinct sets of parameters ?, ? and ?? , ?? under which the distribution of X(1), ..., X(t) is the same. Let X ? {?1, 0, 1}n be any sample, for some parameters ? ? (0, 1] and ? ? [?1, 1]n . The parameter ? is clearly identifiable since ? = P(X1 6= 0). The identifiability of ? is less obvious. Assume for instance that ?i = 0 for all i ? 3. It follows from (1) that for any x ? {?1, 0, 1}n, with H(x) defined as in the proof of Proposition 1: ( 1 + ?1 ?2 if x1 x2 = 1, 1 ? ?1 ?2 if x1 x2 = ?1, P(X = x) = H(x) ? 1 if x1 x2 = 0. 3 In particular, two parameters ?, ?? such that ?1 ?2 = ?1? ?2? and ?i = ?i? = 0 for all i ? 3 cannot be distinguished. Similarly, by symmetry, two parameters ?, ?? such that ?? = ?? cannot be distinguished. Let: ) ( n n X X n ?i > 0 . 1{?i 6= 0} ? 3, ? = ? ? [?1, 1] : i=1 i=1 The first condition states that there are at least 3 informative workers, the second that the average reliability is positive. Proposition 2 Any parameter ? ? ? is identifiable. Proof. Any parameter ? ? ? can be expressed as a function of the covariance matrix of X (section 6 below): the absolute value and the sign of ? follow from (4) and (5), respectively.  5 Lower bound on the minimax error The estimation of ? is straightforward and we here focus on the best estimation of ? one can expect, assuming ? is known. Specifically, we derive a lower bound on the minimax error of any estimator ?? of ?. p Define ||?? ? ?||? = maxi=1,...,n |??i ? ?i | and for all ? ? [?1, 1]n , P A(?) = mink maxi,j6=k |?i ?j | and B(?) = ni=1 ?i . Observe that ? = {? ? [?1, 1]n : A(?) > 0, B(?) > 0}. This suggests that the estimation of ? becomes hard when either A(?) or B(?) is small. Define for any a, b ? (0, 1), ?a,b = {? ? [?1, 1]n : A(?) ? a , B(?) ? b}. We have the following lower bound on the minimax error. As the proof reveals, the parameters a and b characterize the difficulty of estimating the absolute value and the sign of ?, respectively. ? of ?. Theorem 1 (Minimax error) Consider any estimator ?(t) For any ? ? (0, min(a, (1 ? a)/2, 1/4)) and ? ? (0, 1/4), we have   ? ? ?||? ? ? ? ? , ?t ? max(T1 , T2 ), min P ||?(t) ???a,b with T1 = c1 ?1?a 2 a4 ?2 ln 1 4?  4 (n?4) ln , T2 = c2 (1?a) ?2 a2 b2 1 4?  and c1 , c2 > 0 two universal constants. Outline of proof. The proof is based on an information theoretic argument. Denote by P? the distribution of X under parameter ? ? ?, and D(.||.) the Kullback-Leibler (KL) divergence. The main element of proof is lemma 1, where we bound D(P?? ||P? ) for two well chosen pairs of parameters. The pair ?, ?? in statement (i) is hard to distinguish when a is small, hence it is hard to estimate the absolute value of ?. The pair ?, ?? of statement (ii) is also hard to distinguish when a or b are small, which shows that it is difficult to estimate the sign of ?. Proving lemma 1 is involved because of the particular form of distribution P? , and requires careful manipulations of the likelihood ratio. We conclude by reduction to a binary hypothesis test between ? and ?? using lemma 2. a a Lemma 1 (i) Let a ? (0, 1), ? = (1, a, a, 0, . . . , 0) and ?? = (1 ? 2?, 1?2? , 1?2? , 0, . . . , 0). Then: D(P?? ||P? ) ? 1 ?2 a4 ?2 c1 1?a ? (ii) Let n > 4, define c = b/(n ? 4), and ? (a, a, ?a, ?a, c, . . . , c), ? = (?a, ?a, a, a, c, . . . , c). Then: D(P?? ||P? ) ? = ?2 a2 b2 1 c2 (n?4)(1?a)4 . ? Lemma 2 [Tsybakov, 2008, Theorem 2.2] Consider any estimator ?(t). For any ?, ?? ? ? with ||? ? ?? ||? ? 2? we have:   ? ? ?? ||? ? ?) ? ? ? ?||? ? ?), P?? (||?(t) min P? (||?(t) 1 . 4 exp(?tD(P?? ||P? )) Relation with prior work. The lower bound derived in [Zhang et al., 2014][Theorem 3] shows 1 that the minimax error of any estimator ?? must be greater than O((?t)? 2 ). Our lower bound is 1 stricter, and shows that the minimax error is in fact greater than O(a?2 ??1 t? 2 ). Another lower bound was derived in [Chao and Dengyong, 2015][Theorems 3.4 and 3.5], but this concerns the ? 6= G), so that it cannot be easily compared to our result. prediction error rate, that is P(G 4 6 Triangular estimation We here present our estimator. The absolute value of the reliability of each worker k is estimated through the correlation of her answers with those of the most informative pair i, j 6= k. We refer to this algorithm as triangular estimation (TE). The sign of the reliability of each worker is estimated in a second step. We use the convention that sign(0) = +. Covariance matrix. Let X ? {?1, 0, 1}n be any sample, for some parameters ? ? (0, 1] and ? ? ?. We shall see that the parameter ? could be recovered exactly if the covariance matrix of X were perfectly known. For any i 6= j, let Cij be the covariance of Xi and Xj given that Xi Xj 6= 0 (that is, both workers i and j provide an answer). In view of (1), Cij = E(Xi Xj ) = ?i ?j . E(|Xi Xj |) (3) In particular, for any distinct indices i, j, k, Cik Cjk = ?i ?j ?k2 = Cij ?k2 . We deduce that, for any k = 1, . . . , n and any pair i, j 6= k such that Cij 6= 0, ?k2 = Cik Cjk . Cij (4) Note exists for each k because ? ? ?. To recover the sign of ?k , we use the fact that Pnthat such a pairP ?k i=1 ?i = ?k2 + i6=k Cik . Since ? ? ?, we get ? ? X sign(?k ) = sign ??k2 + (5) Cik ? . i6=k The TE algorithm consists in estimating the covariance matrix to recover ? from the above expressions. TE algorithm. At any time t, define ?i, j = 1, . . . , n, C?ij = Pt X (s)Xj (s) Ps=1 i . t max |X (s)X (s)|, 1 i j s=1 (6) For all k = 1, . . . , n, find the most informative pair (ik , jk ) ? arg maxi6=j6=k |C?ij | and let ? s ? ? C?ik k C?jk k ? C?ik jk if |Cik jk (t)| > 0, |??k | = ? ? 0 otherwise. P Next, define k ? = arg maxk ??k2 + i6=k C?ik and let sign(??k ) = ( P sign(??k2? + i6=k? C?ik? ) if k = k ? , otherwise, sign(??k? C?kk? ) Complexity. First note that the TE algorithm is a streaming algorithm since C?ij (t) can be written t C?ij = t X X Mij with Mij = Xi (s)Xj (s) and Nij = |Xi (s)Xj (s)|. max(Nij , 1) s=1 s=1 Thus TE requires O(n2 ) memory space (to store the matrices M and N ) and has a time complexity ? O(n2 ln(n)) operations to sort the entries of of O(n2 ln(n)) per task: O(n2 ) operations to update C, ? O(n2 ) operations to compute the sign of ?. ? ? |C(t)|, O(n2 ) operations to compute |?|, 5 Minimax optimality. The following result shows that the proposed estimator is minimax optimal. Namely the sample complexity of our estimator matches the lower bound up to an additive logarithmic term ln(n) and a multiplicative constant. ? the estimator defined above. For any ? ? (0, min( b , 1)) Theorem 2 Let ? ? ?a,b and denote by ?(t) 3 and ? ? (0, 1), we have ? ? ?||? ? ?) ? ? , ?t ? max(T ? , T ? ), P(||?(t) 1 2  2  2 with T1? = c?1 ?2 a14 ?2 ln 6n? , T2? = c?2 ?2 an2 b2 ln 4n? , and c?1 , c?2 > 0 two universal constants. Outline of proof. Define ||C? ? C||? = maxi,j:i6=j |C?ij ? Cij |. The TE estimator is a function of P the empirical pairwise correlations (C?ij )i,j and the sums j6=i C?ij . The main difficulty is to prove P lemma 3, a concentration inequality for j6=i C?ij . Lemma 3 For all i = 1, . . . , n and all ? > 0,   X  ? P | (Cij ? Cij )| ? ? ? 2 exp ? j6=i ?2 ?2 t 30 max(B(?)2 , n)   + 2n exp ? t?2 8(n ? 1)  . Consider i fixed. We dissociate the set of tasks answered by each worker from the actual answers and the truth. Let U = (Uj (t))j,t be i.i.d Bernoulli random variables with E(Uj (t)) = ? and V = (Vj (t))j,t be independent random variables on {?1, 1} with E(Vj (t)) = ?j . One may readily check that (Xj (t))j,t has the same distribution as (G(t)Uj (t)Vj (t))j,t . Hence, in distribution: X C?ij = t XX Ui (s)Uj (s)Vi (s)Vj (s) Nj j6=i s=1 j6=i with Nj = t X Ui (s)Uj (s). s=1 We prove lemma 3 by conditionning with respect to U . Denote by PU the conditional probability with respect to U . Define N = minj6=i Nij . We prove that for all ? ? 0: PU X j6=i t X 2  X ?2 (n ? 1)N + S Ui (s)Uj (s)?j and ? 2 = . (C?ij ? Cij ) ? ? ? e? ?2 with S = N2 s=1 j6=i The quantity ? is an upper bound on the conditional variance of applying Chernoff?s inequality to both N and S. We get: P(N ? ?2 t/2) ? (n ? 1)e? t?2 8 and P j6=i C?ij , which we control by t?2 P(S ? 2t?2 max(Bi (?)2 , n ? 1)) ? e? 3(n?1) . Removing the conditionning on U yields the result. We conclude the proof of theorem 2 by linking the fluctuations of C? to that of ?? in lemma 4. P A(?)B(?) 1 B(?) 2 ? ? Lemma 4 If (a) ||C?C|| , ? ? ? ? A (?) min( 2 , 64 ) and (b) maxi | j6=i Cij ?Cij | ? 8 24? ? then ||? ? ?||? ? 2 . A (?) Relation with prior work. Our upper bound brings improvement over [Zhang et al., 2014] as follows. Two conditions are required for the upper bound of [Zhang et al., 2014][Theorem 4] to hold: (i) it is required that maxi |?i | < 1, and (ii) the number of workers n must grow ? with both ? and t, and in fact must depend on a and b, so that n has to be large if b is smaller than n. Our result does not require condition (i) to hold. Further there are values of a and b such that condition (ii) is ? b b never satisfied, for instance n ? 5, a = 21 , b = n?4 and ? = (a, ?a, a, ?a, n?4 , ..., n?4 ) ? ?a,b . 2 2 For [Zhang et al., 2014][Theorem 4] to hold, n should satisfy n ? c3 nln(t n/?) with c3 a universal constant (see discussion in the supplement) and for t or 1/? large enough no such n exists. It is noted that for such values of a and b, our result remains informative. Our result shows that one can obtain a minimax optimal algorithm for crowdsourcing which does not involve any EM step. The analysis of [Chao and Dengyong, 2015] also imposes n to grow with t and conditions on the minimal value of b. Specifically the first and the last condition of [Chao and Dengyong, 2015][Theorem 6 P 2 3.3], require that n ? ln(t) ? and that i ?i ? 6ln(t). Using the previous example (even for t = 3), this translates to b ? 2 n ? 4. ? In fact, the value b = O( n) seems to mark the transition between "easy" and "hard" ? instances of the crowdsourcing problem. Indeed, when n is large and b is large with respect to n, then the majority vote outputs the truth with high probability by the Central Limit Theorem. 7 Numerical Experiments Synthetic data. We consider three instances: (i) n = 50, t = 103 , ? = 0.25, ?i = a if i ? n/2 and 0 otherwise; (ii) n = 50, t = 104 , ? = 0.25, ? = (1, a, a, 0, ..., 0); (iii) n = 50, t = 104 , ? = 0.25, b b a = 0.9, ? = (a, ?a, a, ?a, n?4 , ..., n?4 ). Instance (i) is an "easy" instance where half of the workers are informative, with A(?) = a and B(?) = na/2. Instance (ii) is a "hard" instance, the difficulty being to estimate the absolute value of ? accurately by identifying the 3 informative workers. Instance (iii) is another "hard" instance, where estimating the sign of the components of ? is difficult. In particular, one must distinguish ? b b from ?? = (?a, a, ?a, a, n?4 , ..., n?4 ), otherwise a large error occurs. Both "hard" instances (ii) and (iii) are inspired by our derivation of the lower bound and constitute the hardest instances in ?a,b . For each instance we average the performance of algorithms on 103 independent runs and apply a random permutation of the components of ? before each run. We consider the following algorithms: KOS (the BP algorithm of [Karger et al., 2011]), Maj (majority voting), Oracle (weighted majority voting with optimal weights, the optimal estimator of the ground truth), RoE (first spectral algorithm of [Dalvi et al., 2013]), EoR (second spectral algorithm of [Dalvi et al., 2013]), GKM (spectral algorithm of [Ghosh et al., 2011]), S-EMk (EM algorithm with spectral initialization of [Zhang et al., 2014] with k iterations of EM) and TE (our algorithm). We do not present the estimation error of KOS, Maj and Oracle since these algorithms only predict the ground truth but do not estimate ? directly. The results are shown in Tables 1 and 2, where the best results are indicated in bold. The spectral algorithms RoE, EoR and GKM tend to be outperformed by the other algorithms. To perform well, GKM needs ?1 to be positive and large (see [Ghosh et al., 2011]); whenever ?1 ? 0 or |?1 | is small, GKN tends to make a sign mistake causing a large error. Also the analysis of RoE and EoR assumes that the task-worker graph is a random D-regular graph (so that the worker-worker matrix has a large spectral gap). Here this assumption is violated and the practical performance suffers noticeably, so that this limitation is not only theoretical. KOS performs consistently well, and seems immune to sign ambiguity, see instance (iii). Further, while the analysis of KOS also assumes that the task-worker graph is random D-regular, its practical performance does not seem sensitive to that assumption. The performance of S-EM is good except when sign estimation is hard (instance (iii), b = 1). This seems due to the fact that the initialization of S-EM ? (see the algorithm description) is not good in this case. Hence the limitation of b being of order n is not only theoretical but practical as well. In fact (combining our results and the ideas of [Zhang et al., 2014]), this suggests a new algorithm where one uses EM with TE as the initial value of ?. Further, the number of iterations of EM brings significant gains in some cases and should affect the universal constants in front of the various error bounds (providing theoretical evidence for this seems non trival). TE performs consistently well except for (i) a = 0.3 (which we believe is due to the fact that t is relatively small in that instance). In particular when sign estimation is hard TE clearly outperforms the competing algorithms. This indeed suggests two regimes for sign estimation: b = ? O(1) (hard regime) and b = O( n) (easy regime). Real-world data. We next consider 6 publicly available data sets (see [Whitehill et al., 2009, Zhou et al., 2015] and summary information in Table 3), each consisting of labels provided by workers and the ground truth. The density is the average number of labels per worker, i.e., ? in our model. The worker degree is the average number of tasks labeled by a worker. First, for data sets with more than 2 possible label values, we split the label values into two groups and associate them with ?1 and +1 respectively. The partition of the labels is given in Table 3. Second, we remove any worker who provides less than 10 labels. Our preliminary numerical experiments (not shown here for concision) show that without this, none of the studied algorithms 7 even match the majority consistently. Workers with low degree create noise and (to the best of our knowledge) any theoretical analysis of crowdsourcing algorithms assumes that the worker degree is sufficiently large. The performance of various algorithms is reported in Table 4. No information ? 6= G). Further, about the workers reliability is available so we only report the prediction error P(G one cannot compare algorithms to the Oracle, so that the main goal is to outperform the majority. Apart from "Bird" and "Web", none of the algorithms seem to be able to significantly outperform the majority and are sometimes noticeably worse. For "Web" which has both the largest number of labels and a high worker degree, there is a significant gain over the majority vote, and TE, despite its low complexity, slightly outperforms S-EM and is competitive with KOS and GKM which both perform best on this dataset. Instance (i) a = 0.3 (i) a = 0.9 (ii) a = 0.55 (ii) a = 0.95 (iii) b = ? 1 (iii) b = n Table 1: Instance (i) a = 0.3 (i) a = 0.9 (ii) a = 0.55 (ii) a = 0.95 (iii) b = ? 1 (iii) b = n Data Set Bird Dog Duchenne RTE Temp Web Oracle 0.227 0.004 0.284 0.219 0.181 0.126 RoE EoR GKM S-EM1 S-EM10 0.200 0.131 0.146 0.100 0.041 0.274 0.265 0.271 0.022 0.022 0.551 0.459 0.479 0.045 0.044 0.528 0.522 0.541 0.034 0.033 0.253 0.222 0.256 0.533 0.389 0.105 0.075 0.085 0.437 0.030 Synthetic data: estimation error E(||?? ? ?||? ). GKM S-EM1 0.374 0.251 0.202 0.004 0.495 0.284 0.483 0.219 0.386 0.388 0.207 0.258 ? Table 2: Synthetic data: prediction error P(G 6= G). # Tasks 108 807 159 800 462 2,653 Data Set Bird Dog Duchenne RTE Temp Web Maj 0.298 0.046 0.441 0.419 0.472 0.315 KOS 0.228 0.004 0.292 0.220 0.185 0.133 RoE 0.402 0.217 0.496 0.495 0.443 0.266 EoR 0.398 0.218 0.497 0.496 0.455 0.284 # Workers # Labels Density Worker Degree 39 4,212 1 108 109 8,070 0.09 74 64 1,221 0.12 19 164 8,000 0.06 49 76 4,620 0.13 61 177 15,539 0.03 88 Table 3: Summary of the real-world datasets. TE 0.134 0.038 0.050 0.039 0.061 0.045 S-EM10 0.228 0.004 0.285 0.219 0.404 0.127 TE 0.250 0.004 0.284 0.219 0.192 0.128 Label Domain {0} vs {1} {0,2} vs {1,3} {0} vs {1} {0} vs {1} {1} vs {2} {1,2,3} vs {4,5} Maj KOS RoE EoR GKM S-EM1 S-EM10 0.24 0.28 0.29 0.29 0.28 0.20 0.28 0.18 0.19 0.18 0.18 0.20 0.24 0.17 0.28 0.30 0.29 0.28 0.29 0.28 0.30 0.10 0.50 0.50 0.89 0.49 0.32 0.16 0.06 0.43 0.24 0.10 0.43 0.06 0.06 0.14 0.02 0.13 0.14 0.02 0.04 0.06 ? Table 4: Real-world data: prediction error P(G 6= G). TE 0.18 0.20 0.26 0.38 0.08 0.03 8 Conclusion We have derived a minimax error lower bound for the crowdsourcing problem and have proposed TE, a low-complexity algorithm which matches this lower bound. Our results open several questions of interest. First, while recent work has shown that one can obtain strong theoretical guarantees by combining one step of EM with a well-chosen initialization, we have shown that, at least in the case of binary labels, one can forgo the EM phase altogether and still obtain both minimax optimality and good numerical performance. It would be interesting to know if this is still possible when there are more than two possible labels, and also if one can do so using a streaming algorithm. 8 References Paul S Albert and Lori E Dodd. A cautionary note on the robustness of latent class models for estimating diagnostic error without a gold standard. Biometrics, 60(2):427?435, 2004. Gao Chao and Zhou Dengyong. 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Estimating Accuracy from Unlabeled Data: A Probabilistic Logic Approach Emmanouil A. Platanios Carnegie Mellon University Pittsburgh, PA [email protected] Hoifung Poon Microsoft Research Redmond, WA [email protected] Tom M. Mitchell Carnegie Mellon University Pittsburgh, PA [email protected] Eric Horvitz Microsoft Research Redmond, WA [email protected] Abstract We propose an efficient method to estimate the accuracy of classifiers using only unlabeled data. We consider a setting with multiple classification problems where the target classes may be tied together through logical constraints. For example, a set of classes may be mutually exclusive, meaning that a data instance can belong to at most one of them. The proposed method is based on the intuition that: (i) when classifiers agree, they are more likely to be correct, and (ii) when the classifiers make a prediction that violates the constraints, at least one classifier must be making an error. Experiments on four real-world data sets produce accuracy estimates within a few percent of the true accuracy, using solely unlabeled data. Our models also outperform existing state-of-the-art solutions in both estimating accuracies, and combining multiple classifier outputs. The results emphasize the utility of logical constraints in estimating accuracy, thus validating our intuition. 1 Introduction Estimating the accuracy of classifiers is central to machine learning and many other fields. Accuracy is defined as the probability of a system?s output agreeing with the true underlying output, and thus is a measure of the system?s performance. Most existing approaches to estimating accuracy are supervised, meaning that a set of labeled examples is required for the estimation. Being able to estimate the accuracies of classifiers using only unlabeled data is important for many applications, including: (i) any autonomous learning system that operates under no supervision, as well as (ii) crowdsourcing applications, where multiple workers provide answers to questions, for which the correct answer is unknown. Furthermore, tasks which involve making several predictions which are tied together by logical constraints are abundant in machine learning. As an example, we may have two classifiers in the Never Ending Language Learning (NELL) project [Mitchell et al., 2015] which predict whether noun phrases represent animals or cities, respectively, and we know that something cannot be both an animal and a city (i.e., the two categories are mutually exclusive). In such cases, it is not hard to observe that if the predictions of the system violate at least one of the constraints, then at least one of the system?s components must be wrong. This paper extends this intuition and presents an unsupervised approach (i.e., only unlabeled data are needed) for estimating accuracies that is able to use information provided by such logical constraints. Furthermore, the proposed approach is also able to use any available labeled data, thus also being applicable to semi-supervised settings. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Classifier Outputs Instance Classifier #1 Classifier #2 Logical Constraints Category animal fish shark bird . animal fish sparrow bird . . . . animal fish shark bird . animal fish sparrow bird . . . . . . . Probability 99% 95% 5% . . 95% 10% 26% . . . . . . 99% 95% 5% 95% 2% 84% animal bird . . . Inputs: Predicted probability for each classifier-object-category. Outputs: Set of object-category classification pairs and categoryclassifier error-rate pairs that are not directly constrained to be 0 or 1 from the logical constraints. Description: Section 3.3.2. animal 1% fish 5% bird 57% . . . animal 1% fish 2% Classifier #2 bird 9% . . . . . . Classifier #1 Unobserved f?1animal (shark) f?fish (shark) = 0.99 eanimal 1 = 0.95 efish 1 f?1bird (shark) = 0.05 ebird 1 f?1animal (sparrow) f?fish (sparrow) = 0.95 f?1bird (sparrow) = 0.26 1 Grounding 1 = 0.10 ... SUB(animal, fish) = 1 SUB(animal, bird) = 1 ME(fish, bird) =1 ... f animal (shark) f fish (shark) f bird (shark) f animal (sparrow) f fish (sparrow) f bird (sparrow) Ground Rules SUB(animal, fish) ^ ?f?1animal (shark) ^ f fish (shark) ! eanimal 1 ... ME(fish, bird) ^ f?1fish (sparrow) ^ f bird (sparrow) Results Error Rates Ground Predicates Observed fish Combined Predictions animal 99% fish 95% bird 8% . . . animal 95% fish 4% sparrow bird 75% . . . . . . ! efish 1 Probabilistic Inference shark Inputs: Ground predicates and rules. Step 1: Create a Markov Random Field (MRF). Step 2: Perform probabilistic inference to obtain the most likely values for the unobserved ground predicates. Inference is performed using a modified version of the Probabilistic Soft Logic (PSL) framework. Outputs: Classifier error rates and underlying function values. Description: Section 3.3. Figure 1: System overview diagram. The classifier outputs (corresponding to the function approximation outputs) and the logical constraints make up the system inputs. The representation of the logical constraints in terms of the function approximation error rates is described in section 3.2. In the logical constraints box, blue arrows represent subsumption constraints, and labels connected by a red dashed line represent a mutually exclusive set. Given the inputs, the first step is grounding (computing all feasible ground predicates and rules that the system will need to perform inference over) and is described in section 3.3.2. In the ground rules box, ?, ?, ? correspond to the logic AND, OR, and IMPLIES. Then, inference is performed in order to infer the most likely truth values of the unobserved ground predicates, given the observed ones and the ground rules (described in detail in section 3.3). The results constitute the outputs of our system and they include: (i) the estimated error rates, and (ii) the most likely target function outputs (i.e., combined predictions). We consider a ?multiple approximations? problem setting in which we have several different apd d proximations, f?1d , . . . , f?N : X 7? {0, 1} for d , to a set of target boolean classification functions, f d = 1, . . . , D, and we wish to know the true accuracies of each of these different approximations, using only unlabeled data, as well as the response of the true underlying functions, f d . Each value of d characterizes a different domain (or problem setting) and each domain can be interpreted as a class or category of objects. Similarly, the function approximations can be interpreted as classifying inputs as belonging or not to these categories. We consider the case where we may have a set of logical constraints defined over the domains. Note that, in contrast with related work, we allow the function approximations to provide soft responses in the interval [0, 1] (as opposed to only allowing binary responses ? i.e., they can now return the probability for the response being 1), thus allowing modeling of their ?certainty?. As an example of this setting, to which we will often refer throughout this paper, let us consider a part of NELL, where the input space of our functions, X , is the space of all possible noun phrases (NPs). Each target function, f d , returns a boolean value indicating whether the input NP belongs to a category, such as ?city? or ?animal?, and these categories correspond to our domains. There also exist logical constraints between these categories that may be hard (i.e., strongly enforced) or soft (i.e., enforced in a probabilistic manner). For example, ?city? and ?animal? may be mutually exclusive (i.e., if an object belongs to ?city?, then it is unlikely that it also belongs to ?animal?). In this case, the function approximations correspond to different classifiers (potentially using a different set of features / different views of the input data), which may return a probability for a NP belonging to a class, instead of a binary value. Our goal is to estimate the accuracies of these classifiers using only unlabeled data. In order to quantify accuracy, we define the error rate of classifier j in domain d as edj , PD [f?jd (X) 6= f d (X)], for the binary case, for j = 1, . . . , N d , where 2 D is the true underlying distribution of the input data. Note that accuracy is equal to one minus error rate. This definition may be relaxed for the case where f?jd (X) ? [0, 1] representing a probability: edj , f?jd (X)PD [f d (X) 6= 1] + (1 ? f?jd (X))PD [f d (X) 6= 0], which resembles an expected probability of error. Even though our work is motivated by the use of logical constraints defined over the domains, we also consider the setting where there are no such constraints. 2 Related Work The literature covers many projects related to estimating accuracy from unlabeled data. The setting we are considering was previously explored by Collins and Singer [1999], Dasgupta et al. [2001], Bengio and Chapados [2003], Madani et al. [2004], Schuurmans et al. [2006], Balcan et al. [2013], and Parisi et al. [2014], among others. Most of their approaches made some strong assumptions, such as assuming independence given the outputs, or assuming knowledge of the true distribution of the outputs. None of the previous approaches incorporated knowledge in the form of logical constraints. Collins and Huynh [2014] review many methods that were proposed for estimating the accuracy of medical tests in the absence of a gold standard. This is effectively the same problem that we are considering, applied to the domains of medicine and biostatistics. They present a method for estimating the accuracy of tests, where these tests are applied in multiple different populations (i.e., different input data), while assuming that the accuracies of the tests are the same across the populations, and that the test results are independent conditional on the true ?output?. These are similar assumptions to the ones made by several of the other papers already mentioned, but the idea of applying the tests to multiple populations is new and interesting. Platanios et al. [2014] proposed a method relaxing some of these assumptions. They formulated the problem of estimating the error rates of several approximations to a function as an optimization problem that uses agreement rates of these approximations over unlabeled data. Dawid and Skene [1979] were the first to formulate the problem in terms of a graphical model and Moreno et al. [2015] proposed a nonparametric extension to that model applied to crowdsourcing. Tian and Zhu [2015] proposed an interesting max-margin majority voting scheme for combining classifier outputs, also applied to crowdsourcing. However, all of these approaches were outperformed by the models of Platanios et al. [2016], which are most similar to the work of Dawid and Skene [1979] and Moreno et al. [2015]. To the best of our knowledge, our work is the first to use logic for estimating accuracy from unlabeled data and, as shown in our experiments, outperforms all competing methods. Logical constraints provide additional information to the estimation method and this partially explains the performance boost. 3 Proposed Method Our method consists of: (i) defining a set of logic rules for modeling the logical constraints between the f d and the f?jd , in terms of the error rates edj and the known logical constraints, and (ii) performing probabilistic inference using these rules as priors, in order to obtain the most likely values of the edj and the f d , which are not observed. The intuition behind the method is that if the constraints are violated for the function approximation outputs, then at least one of these functions has to be making an error. For example, in the NELL case, if two function approximations respond that a NP belongs to the ?city? and the ?animal? categories, respectively, then at least one of them has to be making an error. We define the form of the logic rules in section 3.2 and then describe how to perform probabilistic inference over them in section 3.3. An overview of our system is shown in figure 1. In the next section we introduce the notion of probabilistic logic, which fuses classical logic with probabilistic reasoning and that forms the backbone of our method. 3.1 Probabilistic Logic In classical logic, we have a set of predicates (e.g., mammal(x) indicating whether x is a mammal, where x is a variable) and a set of rules defined in terms of these predicates (e.g., mammal(x) ? animal(x), where ??? can be interpreted as ?implies?). We refer to predicates and rules defined for a particular instantiation of their variables as ground predicates and ground rules, respectively (e.g., mammal(whale) and mammal(whale) ? animal(whale)). These ground predicates and rules take boolean values (i.e., are either true or false ? for rules, the value is true if the rule holds). Our goal 3 is to infer the most likely values for a set of unobserved ground predicates, given a set of observed ground predicate values and logic rules. In probabilistic logic, we are instead interested in inferring the probabilities of these ground predicates and rules being true, given a set of observed ground predicates and rules. Furthermore, the truth values of ground predicates and rules may be continuous and lie in the interval [0, 1], instead of being boolean, representing the probability that the corresponding ground predicate or rule is true. In this case, boolean logic operators, such as AND (?), OR (?), NOT (?), and IMPLIES (?), need to be redefined. For the next section, we will assume their classical logical interpretation. 3.2 Model As described earlier, our goal is to estimate the true accuracies of each of the function approximations, d f?1d , . . . , f?N d for d = 1, . . . , D, using only unlabeled data, as well as the response of the true underlying functions, f d . We now define the logic rules that we perform inference over in order to achieve that goal. The rules are defined in terms of the following predicates, for d = 1, . . . , D: ? Function Approximation Outputs: f?jd (X), defined over all approximations j = 1, . . . , N d , and inputs X ? X , for which the corresponding function approximation has provided a response. Note that the values of these ground predicates lie in [0, 1] due to their probabilistic nature (i.e., they do not have to be binary, as in related work), and some of them are observed. ? Target Function Outputs: f d (X), defined over all inputs X ? X . Note that, in the purely unsupervised setting, none of these ground predicate values are observed, in contrast with the semi-supervised setting. ? Function Approximation Error Rates: edj , defined over all approximations j = 1, . . . , N d . Note that none of these ground predicate values are observed. The primary goal of this paper is to infer their values. The goal of the logic rules we define is two-fold: (i) to combine the function approximation outputs in a single output value, and (ii) to account for the logical constraints between the domains. We aim to achieve both goals while accounting for the error rates of the function approximations. We first define a set of rules that relate the function approximation outputs with the true underlying function output. We call this set of rules the ensemble rules and we describe them in the following section. We then discuss how to account for the logical constraints between the domains. 3.2.1 Ensemble Rules This first set of rules specifies a relation between the target function outputs, f d (X), and the function approximation outputs, f?jd (X), independent of the logical constraints: f?jd (X) ? ?edj ? f d (X), ?f?jd (X) ? ?edj ? ?f d (X), f?d (X) ? ed ? ?f d (X), and ?f?d (X) ? ed ? f d (X), j j d j j (1) (2) for d = 1, . . . , D, j = 1, . . . , N , and X ? X . In words: (i) the first set of rules state that if a function approximation is not making an error, its output should match the output of the target function, and (ii) the second set of rules state that if a function approximation is making an error, its output should not match the output of the target function. An interesting point to make is that the ensemble rules effectively constitute a weighted majority vote for combining the function approximation outputs, where the weights are determined by the error rates of the approximations. These error rates are implicitly computed based on agreement between the function approximations. This is related to the work of Platanios et al. [2014]. There, the authors try to answer the question of whether consistency in the outputs of the approximations implies correctness. They directly use the agreement rates of the approximations in order to estimate their error rates. Thus, there exists an interesting connection in our work in that we also implicitly use agreement rates to estimate error rates, and our results, even though improving upon theirs significantly, reinforce their claim. Identifiability. Let us consider flipping the values of all error rates (i.e., setting them to one minus their value) and the target function responses. Then, the ensemble logic rules would evaluate to the same value as before (e.g., satisfied or unsatisfied). Therefore, the error rates and the target function values are not identifiable when there are no logical constraints. As we will see in the next 4 section, the constraints may sometimes help resolve this issue as, often, the corresponding logic rules do not exhibit that kind of symmetry. However, for cases where that symmetry exists, we can resolve it by assuming that most of the function approximations have error rates better than chance (i.e., < 0.5). This can be done by considering the two rules: (i) f?jd (X) ? f d (X), and ?f?jd (X) ? ?f d (X), for d = 1, . . . , D, j = 1, . . . , N d , and X ? X . Note that all that these rules imply is that f?jd (X) = f d (X) (i.e., they represent the prior belief that function approximations are correct). As will be discussed in section 3.3, in probabilistic frameworks where rules are weighted with a real value in [0, 1], these rules will be given a weight that represents their significance or strength. In such a framework, we can consider using a smaller weight for these prior belief rules, compared to the remainder of the rules, which would simply correspond to a regularization weight. This weight can be a tunable or even learnable parameter. 3.2.2 Constraints The space of possible logical constraints is huge; we do not deal with every possible constraint in this paper. Instead, we focus our attention on two types of constraints that are abundant in structured prediction problems in machine learning, and which are motivated by the use of our method in the context of NELL: ? Mutual Exclusion: If domains d1 and d2 are mutually exclusive, then f d1 = 1 implies that f d2 = 0. For example, in the NELL setting, if a NP belongs to the ?city? category, then it cannot also belong to the ?animal? category. ? Subsumption: If d1 subsumes d2 , then if f d2 = 1, we must have that f d1 = 1. For example, in the NELL setting, if a NP belongs to the ?cat? category, then it must also belong to the ?animal? category. This set of constraints is sufficient to model most ontology constraints between categories in NELL, as well as a big subset of the constraints more generally used in practice. Mutual Exclusion Rule. We first define the predicate ME(d1 , d2 ), indicating that domains d1 and d2 are mutually exclusive1 . This predicate has value 1 if domains d1 and d2 are mutually exclusive, and value 0 otherwise, and its truth value is observed for all values of d1 and d2 . Furthermore, note that it is symmetric, meaning that if ME(d1 , d2 ) is true, then ME(d2 , d1 ) is also true. We define the mutual exclusion logic rule as: ME(d1 , d2 ) ? f?jd1 (X) ? f d2 (X) ? edj 1 , (3) d1 d2 for d1 6= d2 = 1, . . . , D, j = 1, . . . , N , and X ? X . In words, this rule says that if f (X) = 1 and domains d1 and d2 are mutually exclusive, then f?jd1 (X) must be equal to 0, as it is an approximation to f d1 (X) and ideally we want that f?jd1 (X) = f d1 (X). If that is not the case, then f?jd1 must be making an error. Subsumption Rule. We first define the predicate SUB(d1 , d2 ), indicating that domain d1 subsumes domain d2 . This predicate has value 1 if domain d1 subsumes domain d2 , and 0 otherwise, and its truth value is always observed. Note that, unlike mutual exclusion, this predicate is not symmetric. We define the subsumption logic rule as: SUB(d1 , d2 ) ? ?f?jd1 (X) ? f d2 (X) ? edj 1 , d1 (4) d2 for d1 , d2 = 1, . . . , D, j = 1, . . . , N , and X ? X . In words, this rule says that if f (X) = 1 and d1 subsumes d2 , then f?jd1 (X) must be equal to 1, as it is an approximation to f d1 (X) and ideally we want that f?jd1 (X) = f d1 (X). If that is not the case, then f?jd1 must be making an error. Having defined all of the logic rules that comprise our model, we now describe how to perform inference under such a probabilistic logic model, in the next section. Inference in this case comprises determining the most likely truth values of the unobserved ground predicates, given the observed predicates and the set of rules that comprise our model. 1 A set of mutually-exclusive domains can be reduced to pairwise ME constraints for all pairs in that set. 5 3.3 Inference In section 3.1 we introduced the notion of probabilistic logic and we defined our model in terms of probabilistic predicates and rules. In this section we discuss in more detail the implications of using probabilistic logic, and the way in which we perform inference in our model. There exist various probabilistic logic frameworks, each making different assumptions. In what is arguably the most popular such framework, Markov Logic Networks (MLNs) [Richardson and Domingos, 2006], inference is performed over a constructed Markov Random Field (MRF) based on the model logic rules. Each potential function in the MRF corresponds to a ground rule and takes an arbitrary positive value when the ground rule is satisfied and the value 0 otherwise (the positive values are often called rule weights and can be either fixed or learned). Each variable is boolean-valued and corresponds to a ground predicate. MLNs are thus a direct probabilistic extension to boolean logic. It turns out that due to the discrete nature of the variables in MLNs, inference is NP-hard and can thus be very inefficient. Part of our goal in this paper is for our method to be applicable at a very large scale (e.g., for systems like NELL). We thus resorted to Probabilistic Soft Logic (PSL) [Br?cheler et al., 2010], which can be thought of as a convex relaxation of MLNs. Note that the model proposed in the previous section, which is also the primary contribution of this paper, can be used with various probabilistic logic frameworks. Our choice, which is described in this section, was motivated by scalability. One could just as easily perform inference for our model using MLNs, or any other such framework. 3.3.1 Probabilistic Soft Logic (PSL) In PSL, models, which are composed of a set of logic rules, are represented using hinge-loss Markov random fields (HL-MRFs) [Bach et al., 2013]. In this case, inference amounts to solving a convex optimization problem. Variables of the HL-MRF correspond to soft truth values of ground predicates. Specifically, a HL-MRF, f , is a probability density over m random variables, Y = {Y1 , . . . , Ym } with domain D = [0, 1]m , corresponding to the unobserved ground predicate values. Let X = {X1 , . . . , Xn } be an additional set of variables with known values in the domain [0, 1]n , corresponding to observed ground predicate values. Let ? = {?1 , . . . , ?k } be a finite set of k continuous potential functions of the form ?j (X, Y) = (max {`j (X, Y), 0})pj , where `j is a linear function of X and Y, and pj ? {1, 2}. We will soon see how these functions relate to the ground rules of the model. Given the above, for a set of non-negative free parameters ? = {?1 , . . . , ?k } (i.e., the equivalent of MLN rule weights), the HL-MRF density is defined as: f (Y) = k X 1 exp ? ?j ?j (X, Y), Z j=1 (5) where Z is a normalizing constant so that f is a proper probability density function. Our goal is to infer the most probable explanation (MPE), which consists of the values of Y that maximize the likelihood of our data2 . This is equivalent to solving the following convex problem: min m Y?[0,1] k X ?j ?j (X, Y). (6) j=1 Each variable Xi or Yi corresponds to a soft truth value (i.e., Yi ? [0, 1]) of a ground predicate. Each function `j corresponds to a measure of the distance to satisfiability of a logic rule. The set of rules used is what characterizes a particular PSL model. The rules represent prior knowledge we might have about the problem we are trying to solve. For our model, these rules were defined in section 3.2. As mentioned above, variables are allowed to take values in the interval [0, 1]. We thus need to define what we mean by the truth value of a rule and its distance to satisfiability. For the logical operators AND (?), OR (?), NOT (?), and IMPLIES (?), we use the definitions from ?ukasiewicz Logic [Klir and Yuan, 1995]: P ?Q , max {P + Q ? 1, 0}, P ?Q , min {P + Q, 1}, ?P , 1 ? P , and P ? Q , min{1 ? P + Q, 1}. Note that these operators are a simple continuous relaxation of the corresponding boolean operators, in that for boolean-valued variables, with 0 corresponding to FALSE and 1 to TRUE, they are equivalent. By writing all logic rules in the form B1 ? B2 ? ? ? ? ? Bs ? H1 ? H2 ? ? ? ? ? Ht , it is easy to observe that the distance to satisfiability 2 As opposed to performing marginal inference which aims to infer the marginal distribution of these values. 6 Animal Vertebrate Bird Fish Location Invertebrate Mammal Arthropod Mollusk City Country River Lake Figure 2: Illustration of the NELL-11 data set constraints. Each box represents a label, each blue arrow represents a subsumption constraint, and each set of labels connected by a red dashed line represents a mutually exclusive set of labels. For example, Animal subsumes Vertebrate and Bird, Fish, and Mammal are mutually exclusive. Ps Pt (i.e., 1 minus its truth value) of a rule evaluates to max {0, i=1 Bi ? j=1 Ht + 1 ? s}. Note that any set of rules of first-order predicate logic can be represented in this form [Br?cheler et al., 2010], and that minimizing this quantity amounts to making the rule ?more satisfied?. In order to complete our system description we need to describe: (i) how to obtain a set of ground rules and predicates from a set of logic rules of the form presented in section 3.2 and a set of observed ground predicates, and define the objective function of equation 6, and (ii) how to solve the optimization problem of that equation to obtain the most likely truth values for the unobserved ground predicates. These two steps are described in the following two sections. 3.3.2 Grounding Grounding is the process of computing all possible groundings of each logic rule to construct the inference problem variables and the objective function. As already described in section 3.3.1, the variables X and Y correspond to ground predicates and the functions `j correspond to ground rules. The easiest way to ground a set of logic rules would be to go through each one and create a ground rule instance of it, for each possible value of its arguments. However, if a rule depends on n variables and each variable can take m possible values, then mn ground rules would be generated. For example, the mutual exclusion rule of equation 3 depends on d1 , d2 , j, and X, meaning that D2 ?N d1 ?|X| ground rule instances would be generated, where |X| denotes the number of values that X can take. The same applies to predicates; f?jd1 (X) would result in D ? N d1 ? |X| ground instances, which would become variables in our optimization problem. This approach would thus result in a huge optimization problem rendering it impractical when dealing with large scale problems such as NELL. The key to scaling up the grounding procedure is to notice that many of the possible ground rules are always satisfied (i.e., have distance to satisfiability equal to 0), irrespective of the values of the unobserved ground predicates that they depend upon. These ground rules would therefore not influence the optimization problem solution and can be safely ignored. Since in our model we are only dealing with a small set of predefined logic rule forms, we devised a heuristic grounding procedure that only generates those ground rules and predicates that may influence the optimization. Our grounding algorithm is shown in the supplementary material and is based on the idea that a ground rule is only useful if the function approximation predicate that appears in its body is observed. It turns out that this approach is orders of magnitude faster than existing state-of-the-art solutions such as the grounding solution used by Niu et al. [2011]. 3.3.3 Solving the Optimization Problem For large problems, the objective function of equation 6 will be a sum of potentially millions of terms, each one of which only involving a small set of variables. In PSL, the method used to solve this optimization problem is based on the consensus Alternating Directions Method of Multipliers (ADMM). The approach consists of handling each term in that sum as a separate optimization problem using copies of the corresponding variables, while adding the constraint that all copies of each variable must be equal. This allows for solving the subproblems completely in parallel and is thus scalable. The algorithm is summarized in the supplementary material. More details on this algorithm and on its convergence properties can be found in the latest PSL paper [Bach et al., 2015]. We propose a stochastic variation of this consensus ADMM method that is even more scalable. During each iteration, instead of solving all subproblems and aggregating their solutions in the consensus variables, we sample K << k subproblems to solve. The probability of sampling each 7 Table 1: Mean absolute deviation (MAD) of the error rate rankings and the error rate estimates (lower MAD is better), and area under the curve (AUC) of the label estimates (higher AUC is better). The best results for each experiment, across all methods, are shown in bolded text and the results for our proposed method are highlighted in blue. MAJ AR-2 AR BEE CBEE HCBEE LEE ?10?2 MAJ GIBBS-SVM GD-SVM DS AR-2 AR BEE CBEE HCBEE LEE ?10?1 MAJ GIBBS-SVM GD-SVM DS AR-2 BEE CBEE HCBEE LEE MADerror rank 7.71 12.0 11.4 6.00 6.00 5.03 3.71 MADerror rank 23.3 102.0 26.7 170.0 48.3 48.3 40.0 40.0 81.7 30.0 MADerror rank 8.76 7.77 7.60 7.77 16.40 7.98 10.90 28.10 7.60 NELL-7 MADerror 0.238 0.261 0.260 0.231 0.232 0.229 0.152 uNELL-All MADerror 0.47 2.05 0.42 7.08 2.63 2.60 0.60 0.61 2.53 0.37 uBRAIN-All MADerror 0.57 0.43 0.44 0.44 0.87 0.40 0.43 0.85 0.38 AUCtarget 0.372 0.378 0.374 0.314 0.314 0.452 0.508 MADerror rank 7.54 10.8 11.1 5.69 5.69 5.14 4.77 AUCtarget 99.9 28.6 71.3 12.1 96.7 96.7 99.8 99.8 99.4 96.5 MADerror rank 33.3 101.7 93.3 180.0 50.0 48.3 31.7 118.0 81.7 30.0 AUCtarget 8.49 4.65 5.24 8.76 9.71 9.32 9.34 9.20 9.95 MADerror rank 1.52 1.51 1.50 1.32 2.28 1.38 1.77 3.25 1.32 NELL-11 MADerror 0.303 0.350 0.350 0.291 0.291 0.324 0.180 uNELL-10% MADerror 0.54 2.15 1.90 6.96 2.56 2.52 0.64 45.40 2.45 0.43 uBRAIN-10% MADerror 0.68 0.66 0.68 0.63 0.97 0.63 0.89 0.97 0.47 AUCtarget 0.447 0.455 0.477 0.368 0.368 0.462 0.615 AUCtarget 87.7 28.2 67.8 12.3 96.4 96.4 79.5 55.4 84.9 97.3 AUCtarget 7.84 5.28 8.56 4.59 9.89 9.35 9.30 9.37 9.98 subproblem is proportional to the distance of its variable copies from the respective consensus variables. The intuition and motivation behind this approach is that at the solution of the optimization problem, all variable copies should be in agreement with the consensus variables. Therefore, prioritizing subproblems whose variables are in greater disagreement with the consensus variables might facilitate faster convergence. Indeed, this modification to the inference algorithm allowed us to apply our method to the NELL data set and obtain results within minutes instead of hours. 4 Experiments Our implementation as well as the experiment data sets are available at https://github.com/ eaplatanios/makina. Data Sets. First, we considered the following two data sets with logical constraints: ? NELL-7: Classify noun phrases (NPs) as belonging to a category or not (categories correspond to domains in this case). The categories considered for this data set are Bird, Fish, Mammal, City, Country, Lake, and River. The only constraint considered is that all these categories are mutually exclusive. ? NELL-11: Perform the same task, but with the categories and constraints illustrated in figure 2. For both of these data sets, we have a total of 553,940 NPs and 6 classifiers, which act as our function approximations and are described in [Mitchell et al., 2015]. Not all of the classifiers provide a response every input NP. In order to show the applicability of our method in cases where there are no logical constraints between the domains, we also replicated the experiments of Platanios et al. [2014]: ? uNELL: Same task as NELL-7, but without considering the constraints and using 15 categories, 4 classifiers, and about 20,000 NPs per category. 8 ? uBRAIN: Classify which of two 40 second long story passages corresponds to an unlabeled 40 second time series of Functional Magnetic Resonance Imaging (fMRI) neural activity. 11 classifiers were used and the domain in this case is defined by 11 different locations in the brain, for each of which we have 924 examples. Additional details can be found in [Wehbe et al., 2014]. Methods. Some of the methods we compare against do not explicitly estimate error rates. Rather, they combine the classifier outputs to produce a single label. For these methods, we produce an estimate of the error rate using these labels and compare against this estimate. 1. Majority Vote (MV): This is the most intuitive method and it consists of taking the most common output among the provided function approximation responses, as the combined output. 2. GIBBS-SVM/GD-SVM: Methods of Tian and Zhu [2015]. 3. DS: Method of Dawid and Skene [1979]. 4. Agreement Rates (AR): This is the method of Platanios et al. [2014]. It estimates error rates but does not infer the combined label. To that end, we use a weighted majority vote, where the classifiers? predictions are weighted according to their error rates in order to produce a single output label. We also compare against a method denoted by AR-2 in our experiments, which is the same method, except only pairwise function approximation agreements are considered. 5. BEE/CBEE/HCBEE: Methods of Platanios et al. [2016]. In the results, LEE stands for Logic Error Estimation and refers to the proposed method of this paper. Evaluation. We compute the sample error rate estimates using the true target function labels (which are always provided), and we then compute three metrics for each domain and average over domains: ? Error Rank MAD: We rank the function approximations by our estimates and by the sample estimates to produce two vectors with the ranks. We then compute the mean absolute deviation (MAD) between the two vectors, where by MAD we mean the `1 norm of the vectors? difference. ? Error MAD: MAD between the vector of our estimates and the vector of the sample estimates, where each vector is indexed by the function approximation index. ? Target AUC: Area under the precision-recall curve for the inferred target function values, relative to the true function values that are observed. Results. First, note that the largest execution time of our method among all data sets was about 10 minutes, using a 2013 15-inch MacBook Pro. The second best performing method, HCBEE, required about 100 minutes. This highlights the scalability of our approach. Results are shown in table 1. 1. NELL-7 and NELL-11 Data Sets: In this case we have logical constraints and thus, this set of results is most relevant to the central research claims in this paper (our method was motivated by the use of such logical constraints). It is clear that our method outperforms all existing methods, including the state-of-the-art, by a significant margin. Both the MADs of the error rate estimation, and the AUCs of the target function response estimation, are significantly better. 2. uNELL and uBRAIN Data Sets: In this case there exist no logical constraints between the domains. Our method still almost always outperforms the competing methods and, more specifically, it always does so in terms of error rate estimation MAD. This set of results makes it clear that our method can also be used effectively in cases where there are no logical constraints. Acknowledgements We would like to thank Abulhair Saparov and Otilia Stretcu for the useful feedback they provided in early versions of this paper. This research was performed during an internship at Microsoft Research, and was also supported in part by NSF under award IIS1250956, and in part by a Presidential Fellowship from Carnegie Mellon University. References S. H. Bach, B. Huang, B. London, and L. Getoor. Hinge-loss Markov Random Fields: Convex Inference for Structured Prediction. In Conference on Uncertainty in Artificial Intelligence, 2013. 9 S. H. Bach, M. Broecheler, B. Huang, and L. Getoor. Hinge-loss markov random fields and probabilistic soft logic. CoRR, abs/1505.04406, 2015. URL http://dblp.uni-trier.de/ db/journals/corr/corr1505.html#BachBHG15. M.-F. Balcan, A. Blum, and Y. Mansour. Exploiting Ontology Structures and Unlabeled Data for Learning. International Conference on Machine Learning, pages 1112?1120, 2013. Y. Bengio and N. Chapados. Extensions to Metric-Based Model Selection. Journal of Machine Learning Research, 3:1209?1227, 2003. M. Br?cheler, L. Mihalkova, and L. Getoor. Probabilistic Similarity Logic. In Conference on Uncertainty in Artificial Intelligence, pages 73?82, 2010. J. Collins and M. Huynh. Estimation of Diagnostic Test Accuracy Without Full Verification: A Review of Latent Class Methods. Statistics in Medicine, 33(24):4141?4169, June 2014. M. Collins and Y. Singer. Unsupervised Models for Named Entity Classification. In Joint Conference on Empirical Methods in Natural Language Processing and Very Large Corpora, 1999. S. Dasgupta, M. L. Littman, and D. McAllester. PAC Generalization Bounds for Co-training. In Neural Information Processing Systems, pages 375?382, 2001. A. P. Dawid and A. M. Skene. Maximum Likelihood Estimation of Observer Error-Rates Using the EM Algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics), 28(1):20?28, 1979. G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1995. ISBN 0-13-101171-5. O. Madani, D. Pennock, and G. Flake. Co-Validation: Using Model Disagreement on Unlabeled Data to Validate Classification Algorithms. In Neural Information Processing Systems, 2004. T. Mitchell, W. W. Cohen, E. Hruschka Jr, P. Pratim Talukdar, J. Betteridge, A. Carlson, B. Dalvi, M. Gardner, B. Kisiel, J. Krishnamurthy, N. Lao, K. Mazaitis, T. Mohamed, N. Nakashole, E. A. Platanios, A. Ritter, M. Samadi, B. Settles, R. Wang, D. Wijaya, A. Gupta, X. Chen, A. Saparov, M. Greaves, and J. Welling. Never-Ending Learning. In Association for the Advancement of Artificial Intelligence, 2015. P. G. Moreno, A. Art?s-Rodr?guez, Y. W. Teh, and F. Perez-Cruz. Bayesian Nonparametric Crowdsourcing. Journal of Machine Learning Research, 16, 2015. F. Niu, C. R?, A. Doan, and J. Shavlik. Tuffy: Scaling up statistical inference in markov logic networks using an rdbms. Proc. VLDB Endow., 4(6):373?384, Mar. 2011. ISSN 2150-8097. doi: 10.14778/ 1978665.1978669. URL http://dx.doi.org/10.14778/1978665.1978669. F. Parisi, F. Strino, B. Nadler, and Y. Kluger. Ranking and combining multiple predictors without labeled data. Proceedings of the National Academy of Sciences, 2014. E. A. Platanios, A. Blum, and T. M. Mitchell. Estimating Accuracy from Unlabeled Data. In Conference on Uncertainty in Artificial Intelligence, 2014. E. A. Platanios, A. Dubey, and T. M. Mitchell. Estimating Accuracy from Unlabeled Data: A Bayesian Approach. In International Conference on Machine Learning, pages 1416?1425, 2016. M. Richardson and P. Domingos. Markov Logic Networks. Mach. Learn., 62(1-2):107?136, 2006. D. Schuurmans, F. Southey, D. Wilkinson, and Y. Guo. Metric-Based Approaches for SemiSupervised Regression and Classification. In Semi-Supervised Learning. 2006. T. Tian and J. Zhu. Max-Margin Majority Voting for Learning from Crowds. In Neural Information Processing Systems, 2015. L. Wehbe, B. Murphy, P. Talukdar, A. Fyshe, A. Ramdas, and T. Mitchell. Predicting brain activity during story processing. in review, 2014. 10
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A Decomposition of Forecast Error in Prediction Markets Miroslav Dud?k Microsoft Research, New York, NY [email protected] Ryan Rogers University of Pennsylvania, Philadelphia, PA [email protected] S?bastien Lahaie Google, New York, NY [email protected] Jennifer Wortman Vaughan Microsoft Research, New York, NY [email protected] Abstract We analyze sources of error in prediction market forecasts in order to bound the difference between a security?s price and the ground truth it estimates. We consider cost-function-based prediction markets in which an automated market maker adjusts security prices according to the history of trade. We decompose the forecasting error into three components: sampling error, arising because traders only possess noisy estimates of ground truth; market-maker bias, resulting from the use of a particular market maker (i.e., cost function) to facilitate trade; and convergence error, arising because, at any point in time, market prices may still be in flux. Our goal is to make explicit the tradeoffs between these error components, influenced by design decisions such as the functional form of the cost function and the amount of liquidity in the market. We consider a specific model in which traders have exponential utility and exponential-family beliefs representing noisy estimates of ground truth. In this setting, sampling error vanishes as the number of traders grows, but there is a tradeoff between the other two components. We provide both upper and lower bounds on market-maker bias and convergence error, and demonstrate via numerical simulations that these bounds are tight. Our results yield new insights into the question of how to set the market?s liquidity parameter and into the forecasting benefits of enforcing coherent prices across securities. 1 Introduction A prediction market is a marketplace in which participants can trade securities with payoffs that depend on the outcomes of future events [19]. Consider the simple setting in which we are interested in predicting the outcome of a political election: whether the incumbent or challenger will win. A prediction market might issue a security that pays out $1 per share if the incumbent wins, and $0 otherwise. The market price p of this security should always lie between 0 and 1, and can be construed as an event probability. If a trader believes that the likelihood of the incumbent winning is greater than p, she will buy shares with the expectation of making a profit. Market prices increase when there is more interest in buying and decrease when there is more interest in selling. By this process, the market aggregates traders? information into a consensus forecast, represented by the market price. With sufficient activity, prediction markets are competitive with alternative forecasting methods such as polls [4], but while there is a mature literature on sources of error and bias in polls, the impact of prediction market structure on forecast accuracy is still an active area of research [17]. We consider prediction markets in which all trades occur through a centralized entity known as a market maker. Under this market structure, security prices are dictated by a fixed cost function and 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. the current number of outstanding shares [6]. The basic conditions that a cost function should satisfy to correctly elicit beliefs, while bounding the market maker?s loss, are now well-understood, chief among them being convexity [1]. Nonetheless, the class of allowable cost functions remains broad, and the literature so far provides little formal guidance on the specific form of cost function to use in order to achieve good forecast accuracy, including how to set the liquidity parameter which controls price responsiveness to trade. In practice, the impact of the liquidity parameter is difficult to quantify a priori, so implementations typically resort to calibrations based on market simulations [8, 18]. Prior work also suggests that maintaining coherence among prices of logically related securities has informational advantages [8], but there has been little work aimed at understanding why. This paper provides a framework to quantify the impact of the choice of cost function on forecast accuracy. We introduce a decomposition of forecast error, in analogy with the bias-variance decomposition familiar from statistics or the approximation-estimation-optimization decomposition for large-scale machine learning [5]. Our decomposition consists of three components. First, there is the sampling error resulting from the fact that the market consists of a finite population of traders, each holding a noisy estimate of ground truth. Second, there is a market-maker bias which stems from the use of a cost function to provide liquidity and induce trade. Third, there is convergence error due to the fact that the market prices may not have fully converged to their equilibrium point. The central contribution of this paper is a theoretical characterization of the market-maker bias and convergence error, the two components of this decomposition that depend on market structure as defined by the form of the cost function and level of liquidity. We consider a tractable model of agent behavior, originally studied by Abernethy et al. [2], in which traders have exponential utility functions and beliefs drawn from an exponential family. Under this model it is possible to characterize the market?s equilibrium prices in terms of the traders? belief and risk aversion parameters, and thereby quantify the discrepancy between current market prices and ground truth. To analyze market convergence, we consider the trader dynamics introduced by Frongillo and Reid [9], under which trading can be viewed as randomized block-coordinate descent on a suitable potential function. Our analysis is local in that the bounds depend on the market equilibrium prices. This allows us to exactly identify the main asymptotic terms of error. We demonstrate via numerical experiments that these asymptotic bounds are accurate early on and therefore can be used to compare market designs. We make the following specific contributions: 1. We precisely define the three components of the forecasting error. 2. We show that the market-maker bias equals cb ? O(b2 ) as b ! 0, where b is the liquidity parameter, and c is an explicit constant that depends on the cost function and trader beliefs. 3. We show that the convergence error decreases with the number of trades t as t with = 1 ?(b). We provide explicit upper and lower bounds on that depend on the cost function and trader beliefs. In the process, we prove a new local convergence bound for block-coordinate descent. 4. We use our explicit formulas for bias and convergence error to compare two common cost functions: independent markets (IND), under which security prices vary independently, and the logarithmic market scoring rule (LMSR) [10], which enforces logical relationships between security prices. We show that at the same value of the market-maker bias, IND requires at least half-as-many and at most twice-as-many trades as LMSR to achieve the same convergence error. We consider a specific utility model (exponential utility), but our bias and convergence analysis immediately carry over if we assume that each trader is optimizing a risk measure (rather than an exponential utility function) similar to the setup of Frongillo and Reid [9]. Exponential utility was chosen because it was previously well studied and allowed us to focus on the analysis of the cost function and liquidity. The role of the liquidity parameter in trading off the bias and convergence error has been informally recognized in the literature [7, 10, 13], but our precise definition of market-maker bias and explicit formulas for the bias and convergence error are novel. Abernethy et al. [2] provide results that can be used to derive the bias for LMSR, but not for generic cost functions, so they do not enable comparison of biases of different costs. Frongillo and Reid [9] observe that the convergence error can be locally bounded as t , but they only provide an upper bound and do not show how is related to the liquidity or cost function. Our analysis establishes both upper and lower bounds on convergence and relates explicitly to the liquidity and cost function. This is necessary for a 2 meaningful comparison of cost function families. Thus our framework provides the first meaningful way to compare the error tradeoffs inherent in different choices of cost functions and liquidity levels. 2 Preliminaries We use the notation [N ] to denote the set {1, . . . , N }. Given a convex function f : Rd ! R [ {1}, its effective domain, denoted dom f , is the set of points where f is finite. Whenever dom f is non-empty, the conjugate f ? : Rd ! R [ {1} is defined by f ? (v) := supu2Rd [v | u f (u)]. We write k?k for the Euclidean norm. A centralized mathematical reference is provided in Appendix A.1 Cost-function-based market makers We study cost-function-based prediction markets [1]. Let ? be a finite set of mutually exclusive and exhaustive states of the world. A market administrator, known as market maker, wishes to elicit information about the likelihood of various states ! 2 ?, and to that end offers to buy and sell any number of shares of K securities. Each security is associated with a coordinate of a payoff function : ? ! RK , where each share of the k th security is worth k (!) in the event that the true state of the world is ! 2 ?. Traders arrive in the market sequentially and trade with the market maker. The market price is fully determined by a convex potential function C called the cost function. In particular, if the market maker has previously sold sk 2 R shares of each security k and a trader would like to purchase a bundle consisting of k 2 R shares of each, the trader is charged C(s + ) C(s). The instantaneous price of security k is then @C(s)/@sk . Note that negative values of k are allowed and correspond to the trader (short) selling security k. Let M := conv{ (!) : ! 2 ?} be the convex hull of the set of payoff vectors. It is exactly the set of expectations E [ (!)] across all possible probability distributions over ?, which we call beliefs. We refer to elements of M as coherent prices. Abernethy et al. [1] characterize the conditions that a cost function must satisfy in order to guarantee important properties such as bounded loss for the market maker and no possibility of arbitrage. To start, we assume only that C : RK ! R is convex and differentiable and that M ? dom C ? , which corresponds to the bounded loss property. Example 2.1 (Logarithmic Market Scoring Rule: LMSR [10]). Consider a complete market with a single security for each outcome worth $1 if that outcome occurs and $0 otherwise, i.e., ? = [K] and k (!) = 1{k = !} for all k. The LMSR cost function and instantaneous security prices are given by ?P ? @C(s) e sk K sk C(s) = log and = PK , 8k 2 [K]. (1) k=1 e s` @sk `=1 e P Its conjugate is the entropy function, C ? (?) = k ?k log ?k + I{? 2 K }, where K is the simplex in RK and I{?} is the convex indicator, equal to zero if its argument is true and infinity if false. Thus, in this case M = K = dom C ? . Notice that the LMSR security prices are coherent because they always sum to one. This prevents arbitrage opportunities for traders. Our second running example does not have this property. Example 2.2 (Sum of Independent LMSRs: IND). Let ? = [K] and k (!) = 1{k = !} for all k. The cost function and instantaneous security prices for the sum of independent LMSRs are given by C(s) = C ? (?) = P k [?k PK k=1 log (1 + esk ) and @C(s) e sk = , 8k 2 [K], @sk 1 + e sk log ?k +(1 ?k ) log(1 ?k )]+I{? 2 [0, 1]K }, M = K , and dom C (2) ? = [0, 1]K . When choosing a cost function, one important consideration is liquidity, that is, how quickly prices change in response to trades. Any cost function C can be viewed as a member of a parametric family of cost functions of the form Cb (s) := bC(s/b) for all b > 0. With larger values of b, larger trades are required to move market prices by some fixed amount, and the worst-case loss of the market maker is larger; with smaller values, small purchases can result in big changes to the market price. Basic model In our analysis of error we assume that there exists an unknown true probability distribution ptrue 2 |?| over the outcome set ?. The true expected payoffs of the K market securities are then given by the vector ?true := E!?ptrue [ (!)]. 1 A longer version of this paper containing the appendix is available on arXiv and the authors? websites. 3 ? i over We assume that there are N traders and that each trader i 2 [N ] has a private belief p outcomes. We additionally assume that each trader i has a utility function ui : R ! R for wealth and would like to maximize expected utility subject to her beliefs. For now we assume that ui is differentiable and concave, meaning that each trader is risk averse, though later we focus on K exponential utility. The ? expected utility of ? trader i owning a security bundle r i 2 R and cash ci is Ui (r i , ci ) := E!??pi ui ci + (!) ? r i . We assume that each trader begins with zero cash. This is without loss of generality because we could incorporate any initial cash holdings into ui . 3 A Decomposition of Error In this section, we decompose the market?s forecast error into three major components. The first is sampling error, which arises because traders have only noisy observations of the ground truth. The second is market-maker bias, which arises because the shape of the cost function impacts the traders? willingness to invest. Finally, convergence error arises due to the fact that at any particular point in time the market prices may not have fully converged. To formalize our decomposition, we introduce two new notions of equilibrium. Our first notion of equilibrium, called a market-clearing equilibrium, does not assume the existence of a market maker, but rather assumes that traders trade only among themselves, and so no additional securities or cash are available beyond the traders? initial allocations.This equilibrium is described by ? 2 RK and allocations (? security prices ? r i , c?i ) of security bundles and cash to each trader i such that, given her allocation, no trader wants to buy or sell any bundle of securities at the those prices. ? = (? Trader bundles and cash are summarized as r? = (? r i )i2[N ] and c ci )i2[N ] . ?, ?) ? is a market-clearing equilibrium if Definition 3.1 (Market-clearing equilibrium). A triple (? r, c PN PN ? We call r ? = 0, c ? = 0, and for all i 2 [N ], 0 2 argmax r i + , c?i ? ?). i i 2RK Ui (? i=1 i=1 ? market-clearing prices if there exist r? and c ? such that (? ?, ?) ? is a market-clearing equilibrium. ? r, c Similarly, we call r? a market-clearing allocation if there exists a corresponding equilibrium. PN PN The requirements on i=1 r?i and i=1 c?i guarantee that no additional securities or cash have been created. In other words, there exists some set of trades among traders that would lead to the market-clearing allocation, although the definition says nothing about how the equilibrium is reached. Since we rely on a market maker to orchestrate trade, our markets generally do not reach the marketclearing equilibrium. Instead, we introduce the notion of market-maker equilibrium. This equilibrium is again described by a set of security prices ?? and trader allocations (r ?i , c?i ), summarized as (r ? , c? ), such that no trader wants to trade at these prices given her allocation. The difference is that we now require r ? and c? to be reachable via some sequence of trade with the market maker instead of via trade among only the traders, and ?? must be the market prices after such a sequence of trade. Definition 3.2 (Market-maker equilibrium). A triple (r ?, c?, ?? ) is a market-maker equilibrium PN PN for cost function Cb if, for the market state s? = i=1 r ?i , we have i=1 c?i = Cb (0) Cb (s? ), ?? = rCb (s? ), and for all i 2 [N ], 0 2 argmax 2RK Ui r ?i + , c?i Cb (s? + ) + Cb (s? ) . We call ?? market-maker equilibrium prices if there exist r ? and c? such that (r ?, c?, ?? ) is a marketmaker equilibrium. Similarly, we call r ? a market-maker equilibrium allocation if there exists a corresponding equilibrium. We sometimes write ?? (b; C) to show the dependence of ?? on C and b. ? and the market-maker equilibrium prices ?? (b; C) are not unique in The market-clearing prices ? general, but are unique for the specific utility functions that we study in this paper. Using these notions of equilibrium, we can formally define our error components. Sampling error is the difference between the true security values and the market-clearing equilibrium prices. The bias is the difference between the market-clearing equilibrium prices and the market-maker equilibrium prices. Finally, the convergence error is the difference between the market-maker equilibrium prices and the market prices ?t (b; C) at a particular round t. Putting this together, we have that ?true ? +? ? ?t = ?true ? | {z } | Sampling Error ?? (b; C) + ?? (b; C) ?t (b; C) . {z } | {z } Bias 4 Convergence Error (3) 4 The Exponential Trader Model For the remainder of the paper, we work with the exponential trader model introduced by Abernethy et al. [2] in which traders have exponential utility functions and exponential-family beliefs. Under this model, both the market-clearing prices and market-maker equilibrium prices are unique and can be expressed cleanly in terms of potential functions [9], yielding a tractable analysis. The results of this section are immediate consequences of prior work [2, 9], but our equilibrium concepts bring them into a common framework. We consider a specific exponential family [3] of probability distributions over ? defined as p(!; ?) = e (!)?? T (?) , where ? 2PRK is the natural parameter of the distribution, and T is the log partition (!)?? function, T (?) := log . The gradient rT (?) coincides with the expectation of !2? e ? under p(?; ?), and dom T = conv{ (!) : ! 2 ?} = M. Following Abernethy et al. [2], we assume that each trader i has exponential-family beliefs with natural parameter ??i . From the perspective of trader i, the expected payoffs of the K market securities P ? i with ? can then be expressed as the vector ? ?i,k := !2? k (!)p(!; ??i ). As in Abernethy et al. [2], we also assume that traders are risk averse with exponential utility for wealth, so the utility of trader i for wealth W is ui (W ) = (1/ai )e ai W , where ai is the the trader?s risk aversion coefficient. We assume that the traders? risk aversion coefficients are fixed. Using the definitions of the expected utility Ui , the exponential family distribution p(?; ??i ), the log partition function T , and the exponential utility ui , it is straightforward to show [2] that Ui (r i , ci ) = 1 e ai ?i ) ai ci T (? P !2? e ?i ai r i ) (!)?(? = 1 T (??i e ai ?i ) ai ci ai r i ) T (? . (4) Under this trader model, we can use the techniques of Frongillo and Reid [9] to construct potential functions which yield alternative characterizations of the equilibria as solutions of minimization problems. Consider first a market-clearing equilibrium. Define Fi (s) := a1i T (??i + ai s) for each trader i. From Eq. (4) we can observe that Fi ( r i ) + ci is a monotone transformation of trader i?s utility. Since each trader?s utility is locally maximized at a market-clearing equilibrium, the sum PN of traders? utilities is also locally maximized, as is i=1 ( Fi ( r i ) + ci ). Since the equilibrium PN conditions require that i=1 ci = 0, the security allocation associated with any market-clearing PN equilibrium must be a local minimum of i=1 Fi ( r i ). This idea is formalized in the following theorem. The proof follows from an analysis of the KKT conditions of the equilibrium. (See the appendix for all omitted proofs.) Theorem 4.1. Under the exponential trader model, a market-clearing equilibrium always exists and market-clearing prices are unique. Market-clearing allocations and prices are exactly the solutions of the following optimization problems: hP i hP i N N ? ? = argmin r? 2 P argmin ? (5) i=1 Fi ( r i ) , i=1 Fi (?) . r: N i=1 ?2RK r i =0 Using a similar argument, we can show that the allocation associated with any market-maker equilibPN PN rium is a local minimum of the function F (r) := i=1 Fi ( r i ) + Cb i=1 r i . Theorem 4.2. Under the exponential trader model, a market-maker equilibrium always exists and equilibrium prices are unique. Market-maker equilibrium allocations and prices are exactly the solutions of the following optimization problems: hP i N ? ? r ? 2 argmin F (r) , ?? = argmin F (?) + bC (?) . (6) i=1 i r ?2RK Sampling error We finish this section with an analysis of the first component of error identified in Section 3: the sampling error. We begin by deriving a more explicit form of market-clearing prices: Theorem 4.3. Under the exponential trader model, the unique market-clearing equilibrium prices ? := PN ??i /ai / PN 1/ai is the risk-aversion? = E?? [ (!)], where ? can be written as ? i=1 i=1 ? weighted average belief and E?? is the expectation under p(?; ?). 5 The sampling error arises because the beliefs ??i are only noisy signals of the ground truth. From Theorem 4.3 we see that this error may be compounded by the weighting according to risk aversions, ? we need to make which can skew the prices. To obtain a concrete bound on the error term k?true ?k, some assumptions about risk aversion coefficients, the true distribution of the outcome, and how this distribution is related to trader beliefs. For instance, suppose risk aversion coefficients are bounded both below and above, the true outcome is drawn from an exponential-family distribution with natural parameter ? true , and the beliefs ??i are independent samples with mean ? true and a bounded covariance matrix. Under these assumptions, p one can show using standard concentration bounds that with high true ? = O( 1/N ) as N ! 1. In other words, market-clearing prices approach probability, k? ?k the ground truth as the number of traders increases. In Appendix B.4 we make the dependence on risk aversion and belief noise more explicit. The analysis of other information structures (e.g., biased or correlated beliefs) is beyond the scope of this paper; instead, we focus on the two error components that depend on the market design. 5 Market-maker Bias We now analyze the market-maker bias?the difference between the marker-maker equilibrium prices ? We first state a global bound that depends on the liquidity b and cost ?? and market-clearing prices ?. ? with the rate O(b) as b ! 0. The proof function C, but not on trader beliefs, and show that ?? ! ? builds on Theorems 4.1 and 4.2 and uses the facts that C ? is bounded on M (by our assumptions on C), and conjugates Fi? are strongly convex on M (from properties of the log partition function). Theorem 5.1 (Global Bias Bound). Under the exponential trader model, for any C, there exists a ? ? cb for all b 0. constant c such that k?? (b; C) ?k This result makes use of strong convexity constants that are valid over the entire set M, which can ? Furthermore, it gives us only an upper bound, which be overly conservative when ?? is close to ?. cannot be used to compare different cost function families. In the rest of this section we pursue ? Our local analysis requires a tighter local analysis, based on the properties of Fi? and C ? at ?. assumptions that go beyond convexity and differentiability of the cost function. We call the class of functions that satisfy these assumptions convex+ functions. (See Appendix A.3 for their complete treatment and a more general definition than provided here.) These functions are related to functions of Legendre type (see Sec. 26 of Rockafellar [15]). Informally, they are smooth functions that are strictly convex along directions in a certain space (the gradient space) and linear in orthogonal directions. For cost functions, strict convexity means that prices change in response to arbitrarily small trades, while the linear directions correspond to bundles with constant payoffs, whose prices are therefore fixed. Definition 5.2. Let f : Rd ! R be differentiable and convex. Its gradient space is the linear space parallel to the affine hull of its gradients, denoted as G(f ) := span{rf (u) rf (u0 ) : u, u0 2 Rd }. Definition 5.3. We say that a convex function f : Rd ! R is convex+ if it has continuous third derivatives and range(r2 f (u)) = G(f ) for all u 2 Rd . It can be checked that if P is a projection on G(f ) then there exists some a such that f (u) = f (P u) + a| u, so f is up to a linear term fully described by its values on G(f ). The condition on the range of the Hessian ensures that f is strictly convex over G(f ), so its gradient map is invertible over G(f ). This means that the Hessian can be expressed as a function of the gradient, i.e., there exists a matrix-valued function Hf such that r2 f (u) = Hf (rf (u)) (see Proposition A.8). The cost functions C for both the LMSR and the sum of independent LMSRs (IND) are convex+ . Example 5.4 (LMSR as a convex+ function). For LMSR, the gradient space of C is parallel to the simplex: G(C) = {u : 1| u = 0}. The gradients of C are points in the relative interior of the simplex. Given such a point ? = rC(s), the corresponding Hessian is r2 C(s) = HC (?) = (diagk2[K] ?k ) ??| , where diagk2[K] ?k denotes the diagonal matrix with values ?k on the diagonal. The null space of HC (?) is {c1 : c 2 R}, so C is linear in the all-ones direction (buying one share of each security always has cost one), but strictly convex in directions from G(C). Example 5.5 (IND as a convex+ function). For IND, the gradient space is RK and the gradients are the points in (0, 1)K . In this case, HC (?) = diagk [?k (1 ?k )]. This matrix has full rank. ? and C, we have Our next theorem shows that for an appropriate vector u, which depends on ? ? + bu + "b , where k"b k = O(b2 ). Here, the O(?) is taken as b ! 0, so the error term ?? (b; C) = ? 6 "b goes to zero faster than the term bu, which we call the asymptotic bias. Our analysis is local in ? This analysis fully uncovers the the sense that the constants hiding within O(?) may depend on ?. main asymptotic term and therefore allows comparison of cost families. In our experiments, we show that the asymptotic bias is an accurate estimate of the bias even for moderately large values of b. Theorem 5.6 (Local Bias Bound). Assume that the cost function C is convex+ . Then ? ? ? b(? a/N )HT (?)@C (?) + "b , where k"b k = O(b2 ). PN In the expression above, a ? = N/( i=1 1/ai ) is the harmonic mean of risk-aversion coefficients and ? ? ? ? is a set. HT (?)@C (?) is guaranteed to consist of a single point even when @C ? (?) ? ?? (b; C) = ? The theorem is proved by a careful application of Taylor?s Theorem and crucially uses properties of conjugates of convex+ functions, which we derive in Appendix A.3. It gives us a formula to calculate ? or evaluate the worst-case bias the asymptotic bias for any cost function for a particular value of ?, against some set of possible market-clearing prices. It also constitutes an important step in comparing cost function families. To compare the convergence error of two costs C and C 0 in the next section, we require that their liquidities b and b0 be set so that they have (approximately) the same bias, i.e., ? ? k?? (b; C) ?k. ? Theorem 5.6 tells us that this can be achieved by the linear k?? (b0 ; C 0 ) ?k 0 0? ? ? ? ? ? rule b = b/? where ? = kHT (?)@C (?)k / kHT (?)@C (?)k. For C = LMSR and C 0 = IND, we prove that the corresponding ? 2 [1, 2]. Equivalently, this means that for the same value of b the asymptotic bias of IND is at least as large as that of LMSR, but no more than twice as large: ? there exists ? 2 [1, 2] such that for all b, k?? (b/?; IND) ?k ? = Theorem 5.7. For any ? 2 2 ? ? = ?k?? (b; LMSR) ?k?O(b ? k?? (b; LMSR) ?k?O(b ). For this same ?, also k?? (b; IND) ?k ). Theorem 5.6 also captures an intuitive relationship which can guide the market maker in adjusting the market liquidity b as the number of traders N and their risk aversion coefficients ai vary. In particular, ? and the cost function fixed, we can maintain the same amount of bias by setting b / N/? holding ? a. Note that 1/ai plays the role of the budget of trader i in the sense P that at fixed prices, the trader will spend an amount of cash proportional to 1/ai . Thus N/? a = i (1/ai ) corresponds to the total amount of available cash among the traders in the market. Similarly, the marketP maker?s worst-case loss, amounting to the market maker?s cash, is proportional to b, so setting b / i (1/ai ) is natural. 6 Convergence Error We now study the convergence error, namely the difference between the prices ?t at round t and the market-maker equilibrium prices ?? . To do so, we must posit a model of how the traders interact with the market. Following Frongillo and Reid [9], we assume that in each round, a trader i 2 [N ], chosen uniformly at random, buys a bundle 2 RK that optimizes her utility given the current market state s and her existing security and cash allocations, r i and ci . The resulting updates of the allocation vector r = (r i )N i=1 correspond to randomized block-coordinate descent on the potential function F (r) with blocks r i (see Appendix D.1 and Frongillo and Reid [9]). We refer to this model as the all-security (trader) dynamics (ASD).2 We apply and extend the analysis of block-coordinate descent to this setting. We focus on convex+ functions and conduct local convergence analysis around the minimizer of F . Our experiments demonstrate that the local analysis accurately estimates the convergence rate. Let r ? denote an arbitrary minimizer of F and let F ? be the minimum value of F . Also, let r t denote the allocation vector and ?t the market price vector after the tth trade. Instead of directly analyzing the convergence error k?t ?? k, we bound the suboptimality F (r t ) F ? since k?t ?? k2 = ?(F (r t ) F ? ) for convex+ costs C under ASD (see Appendix D.7.1). Convex+ functions are locally strongly convex and have a Lipschitz-continuous gradient, so the standard analysis of block-coordinate descent [9, 11] implies linear convergence, i.e., E [F (r t )] F ? ? O( t ) for some < 1, where the expectation is under the randomness of the algorithm. We refine the standard analysis by (1) proving not only upper, but also lower bounds on the convergence rate, and (2) proving an explicit dependence of on the cost function C and the liquidity b. These two refinements are crucial for comparison of cost families, as we demonstrate with the comparison of LMSR and IND. We begin by formally defining bounds on local convergence of any randomized iterative algorithm that minimizes a function F (r) via a sequence of iterates r t . 2 In Appendix D, we also analyze the single-security (trader) dynamics (SSD), in which a randomly chosen trader randomly picks a single security to trade, corresponding to randomized coordinate descent on F . 7 Definition 6.1. We say that high is an upper bound on the local convergence rate of an algorithm if, with probability 1 under the randomness the algorithm reaches an iteration t0 ? of the algorithm, ? t t0 such that for some c > 0 and all t t0 , E F (r t ) r t0 F ? ? c high . We say that low is a lower bound on the local convergence rate if high holds for all upper bounds low high . To state explicit bounds, we use the notation D := diagi2[N ] ai and P := IN 11| /N , where IN is the N ? N identity matrix and 1 is the all-ones vector. We write M + for the pseudoinverse of a matrix M and min (M ) and max (M ) for its smallest and largest positive eigenvalues. ? and Theorem 6.2 (Local Convergence Bound). Assume that C is convex+ . Let HT := HT (?) ? For the all-securities dynamics, the local convergence rate is bounded between HC := HC (?). ASD high =1 2b N ASD low =1 2b N ? min (P DP ) ? min ? max (P DP ) ? max 1/2 1/2 HT HC+ HT 1/2 1/2 HT HC+ HT + O(b2 ) , O(b2 ) . In our proof, we first establish both lower and upper bounds on convergence of a generic blockcoordinate descent that extend the results of Nesterov [11]. We then analyze the behavior of the algorithm for the specific structure of our objective to obtain explicit lower and upper bounds. Our bounds prove linear convergence with the rate = 1 ?(b). Since the convergence gets worse as b ! 0, there is a trade-off with the bias, which decreases as b ! 0. Theorems 5.6 and 6.2 enable systematic quantitative comparisons of cost families. For simplicity, assume that N 2 and all risk aversions are a, so min (P DP ) = max (P DP ) = a. To compare convergence rates of two costs C and C 0 , we need to control for bias. As discussed after Theorem 5.6, their biases are (asymptotically) equal if their liquidities are linearly related as b0 = b/? for a suitable ?. Theorem 6.2 then states that Cb0 0 requires (asymptotically) at most a factor of ? as many trades as Cb 1/2 1/2 1/2 1/2 to achieve the same convergence error, where ? := ? ? max (HT HC+ HT )/ min (HT HC+0 HT ). 0 0 0 Similarly, Cb requires at most a factor of ? as many trades as Cb0 , with ? defined symmetrically to ?. For C = LMSR and C 0 = IND, we can show that ? ? 2 and ?0 ? 2, yielding the following result: Theorem 6.3. Assume that N 2 and all risk aversions are equal to a. Consider running LMSR with liquidity b and IND with liquidity b0 = b/? such that their asymptotic biases are equal. Denote the iterates of the two runs of the market as ?tLMSR and ?tIND and the respective market-maker equilibria as ??LMSR and ??IND . Then, with probability 1, there exist t0 and t1 t0 such that for all t t1 and sufficiently small b ? 2t(1+") ? ? (t/2)(1 ") 2? 2? 2? Et0 ?LMSR ??LMSR ? Et0 ?tIND ??IND ? Et0 ?LMSR ??LMSR , where " = O(b) and Et0 [?] = E[? | r t0 ] conditions on the t0 th iterate of a given run. This result means that LMSR and IND are roughly equivalent (up to a factor of two) in terms of the number of trades required to achieve a given accuracy. This is somewhat surprising as this implies that maintaining price coherence does not offer strong informational advantages (at least when traders are individually coherent, as assumed here). However, while there is little difference between the two costs in terms of accuracy, there is a difference in terms of the worst-case loss. For K securities, the worst-case loss of LMSR with the liquidity b is b log K, and the worst-case loss of IND with the liquidity b0 is b0 K log 2. If liquidities are chosen as in Theorem 6.3, so that b0 is up to a factor-of-two smaller than b, then the worst-case loss of IND is at least (bK/2) log 2, which is always worse than the LMSR?s loss of b log K, and the ratio of the two losses increases as K grows. When all risk aversion coefficients are equal to some constant a, then the dependence of Theorem 6.2 on the number of traders N and their risk aversion is similar to the dependence in Theorem 5.6. For instance, to guarantee that stays below a certain level for varying N and a requires b = ?(N/a). 7 Numerical Experiments We evaluate the tightness of our theoretical bounds via numerical simulation. We consider a complete market over K = 5 securities and simulate N = 10 traders with risk aversion coefficients equal to 1. These values of N and K are large enough to demonstrate the tightness of our results, but small enough that simulations are tractable. While our theory comprehensively covers heterogeneous risk 8 0.00 0.0 0.2 0.4 0.6 Liquidity Parameter b 0.8 1.0 0.0 0.2 0.4 0.6 Liquidity Parameter b 0.8 1.0 0 ?2 ?4 LMSR IND b 0.01 0.03 0.05 0.07 ?6 Log10 of Suboptimality of Objective Actual Bias ?8 0.08 0.06 LMSR IND 0.04 Market?Maker Bias Asymptotic Bias 0.02 0.12 0.02 0.04 0.06 0.08 0.10 100 200 500 1000 0.00 Bias Plus Convergence Error #Trades 0 200 400 600 800 1000 1200 1400 Number of Trades Figure 1: (Left) The tradeoff between market-maker bias and convergence. Solid lines are for LMSR, dashed for IND, the color indicates the number of trades. (Center) Market-maker bias as a function of b. (Right) Convergence in the objective. Shading indicates 95% confidence based on 20 trading sequences. aversions and the dependence on the number of traders and securities, we have chosen to keep these values fixed and more cleanly explore the impact of liquidity and number of trades. We consider the two most commonly studied cost functions: LMSR and IND. We fix the ground-truth natural parameter ? true and independently sample the belief ??i of each trader from Normal(? true , 2 IK ), with = 5. We consider a single-peaked ground truth distribution with ?1true = log(1 ?(K 1)) and ?ktrue = log ? for k 6= 1, with ? = 0.02. Trading is simulated according to the all-security dynamics (ASD) as described at the start of Section 6. In Appendix E, we show qualitatively similar results using a uniform ground truth distribution and single-security dynamics (SSD). We first examine the tradeoff that arises between market-maker bias and convergence error as the ? liquidity parameter is adjusted. Fig. 1 (left) shows the combined bias and convergence error, k?t ?k, as a function of liquidity and the number of trades t (indicated by the color of the line) for the two cost functions, averaged over twenty random trading sequences. The minimum point on each curve tells us the optimal value of the liquidity parameter b for the particular cost function and particular number of trades. When the market is run for a short time, larger values of b lead to lower error. On the other hand, smaller values of b are preferable as the number of trades grows, with the combined error approaching 0 for small b. ? as a function of b for both LMSR and IND. We In Fig. 1 (center) we plot the bias k?? (b; C) ?k ? ? ? ? b(? ? compare this with the theoretical approximation k?? (b; C) ?k a/N )kHT (?)@C (?)k from Theorem 5.6. Although Theorem 5.6 only gives an asymptotic guarantee as b ! 0, the approximation is fairly accurate even for moderate values of b. In agreement with Theorem 5.7, the bias of IND is higher than that of LMSR at any fixed value of b, but by no more than a factor of two. ? (r t )] F ? as a function of the number of trades t for our two In Fig. 1 (right) we plot the log of E[F cost functions and several liquidity levels. Even for small t the curves are close to linear, showing that the local linear convergence rate kicks in essentially from the start of trade in our simulations. ? (r t )] F ? ? c?? t , or In other words, there exist some c? and ? such that, empirically, we have E[F t ? ? (r )] F ) ? log c? + t log ? . Plugging the belief values into Theorem 6.2, the equivalently, log(E[F slope of the curve for LMSR should be log10 ? ? 0.087b for sufficiently small b, and the slope for IND should be between 0.088b and 0.164b. In Appendix E, we verify that this is the case. 8 Conclusion Our theoretical framework provides a meaningful way to quantitatively evaluate the error tradeoffs inherent in different choices of cost functions and liquidity levels. We find, for example, that to maintain a fixed amount of bias, one should set the liquidity parameter b proportional to a measure of the amount of cash that traders are willing to spend. We also find that, although the LMSR maintains coherent prices while IND does not, the two are equivalent up to a factor of two in terms of the number of trades required to reach any fixed accuracy, though LMSR has lower worst-case loss. We have assumed that traders? beliefs are individually coherent. Experimental evidence suggests that LMSR might have additional informational advantages over IND when traders? beliefs are incoherent or each trader is informed about only a subset of events [12]. We touch on this in Appendix C.2, but leave a full exploration of the impact of different assumptions on trader beliefs to future work. 9 References [1] Jacob Abernethy, Yiling Chen, and Jennifer Wortman Vaughan. Efficient market making via convex optimization, and a connection to online learning. ACM Transactions on Economics and Computation, 1(2):Article 12, 2013. [2] Jacob Abernethy, Sindhu Kutty, S?bastien Lahaie, and Rahul Sami. Information aggregation in exponential family markets. In Proceedings of the 15th ACM Conference on Economics and Computation (EC), 2014. [3] Ole Barndorff-Nielsen. Exponential Families. Wiley Online Library, 1982. [4] Joyce Berg, Robert Forsythe, Forrest Nelson, and Thomas Rietz. Results from a dozen years of election futures markets research. Handbook of experimental economics results, 1:742?751, 2008. [5] Olivier Bousquet and L?on Bottou. The tradeoffs of large scale learning. In Advances in Neural Information Processing Systems (NIPS), 2008. [6] Yiling Chen and David M. Pennock. A utility framework for bounded-loss market makers. In Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence (UAI), 2007. [7] Yiling Chen and Jennifer Wortman Vaughan. A new understanding of prediction markets via no-regret learning. In Proceedings of the 11th ACM Conference on Electronic Commerce (EC), 2010. [8] Miroslav Dud?k, S?bastien Lahaie, David M. Pennock, and David Rothschild. A combinatorial prediction market for the US elections. In Proceedings of the 14th ACM Conference on Electronic Commerce (EC), 2013. [9] Rafael Frongillo and Mark D. Reid. Convergence analysis of prediction markets via randomized subspace descent. In Advances in Neural Information Processing Systems (NIPS), 2015. [10] Robin Hanson. Combinatorial information market design. Information Systems Frontiers, 5(1): 105?119, 2003. [11] Yu. Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM Journal on Optimization, 22(2):341?362, 2012. [12] Kenneth C. Olson, Charles R. Twardy, and Kathryn B. Laskey. Accuracy of simulated flat, combinatorial, and penalized prediction markets. Presented at Collective Intelligence, 2015. [13] Abraham Othman, David M Pennock, Daniel M Reeves, and Tuomas Sandholm. A practical liquidity-sensitive automated market maker. ACM Transactions on Economics and Computation, 1(3):14, 2013. [14] Kaare Brandt Petersen and Michael Syskind Pedersen. The matrix cookbook. Technical Report, Technical University of Denmark, Nov 2012. [15] R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, 1970. [16] R. Tyrrell Rockafellar and Roger J-B Wets. Variational analysis. Springer-Verlag, 2009. [17] David Rothschild. Forecasting elections: comparing prediction markets, polls, and their biases. Public Opinion Quarterly, 73(5):895?916, 2009. [18] Christian Slamka, Bernd Skiera, and Martin Spann. Prediction market performance and market liquidity: A comparison of automated market makers. IEEE Transactions on Engineering Management, 60(1):169?185, 2013. [19] Justin Wolfers and Eric Zitzewitz. Prediction markets. The Journal of Economic Perspectives, 18(2):107?126, 2004. 10
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Safe Adaptive Importance Sampling Sebastian U. Stich EPFL Anant Raj Max Planck Institute for Intelligent Systems [email protected] [email protected] Martin Jaggi EPFL [email protected] Abstract Importance sampling has become an indispensable strategy to speed up optimization algorithms for large-scale applications. Improved adaptive variants?using importance values defined by the complete gradient information which changes during optimization?enjoy favorable theoretical properties, but are typically computationally infeasible. In this paper we propose an efficient approximation of gradient-based sampling, which is based on safe bounds on the gradient. The proposed sampling distribution is (i) provably the best sampling with respect to the given bounds, (ii) always better than uniform sampling and fixed importance sampling and (iii) can efficiently be computed?in many applications at negligible extra cost. The proposed sampling scheme is generic and can easily be integrated into existing algorithms. In particular, we show that coordinate-descent (CD) and stochastic gradient descent (SGD) can enjoy significant a speed-up under the novel scheme. The proven efficiency of the proposed sampling is verified by extensive numerical testing. 1 Introduction Modern machine learning applications operate on massive datasets. The algorithms that are used for data analysis face the difficult challenge to cope with the enormous amount of data or the vast dimensionality of the problems. A simple and well established strategy to reduce the computational costs is to split the data and to operate only on a small part of it, as for instance in coordinate descent (CD) methods and stochastic gradient (SGD) methods. These kind of methods are state of the art for a wide selection of machine learning, deep leaning and signal processing applications [9, 11, 35, 27]. The application of these schemes is not only motivated by their practical preformance, but also well justified by theory [18, 19, 2]. Deterministic strategies are seldom used for the data selection?examples are steepest coordinate descent [4, 34, 20] or screening algorithms [14, 15]. Instead, randomized selection has become ubiquitous, most prominently uniform sampling [27, 29, 7, 8, 28] but also non-uniform sampling based on a fixed distribution, commonly referred to as importance sampling [18, 19, 2, 33, 16, 6, 25, 24]. While these sampling strategies typically depend on the input data, they do not adapt to the information of the current parameters during optimization. In contrast, adaptive importance sampling strategies constantly re-evaluate the relative importance of each data point during training and thereby often surpass the performance of static algorithms [22, 5, 26, 10, 21, 23]. Common strategies are gradientbased sampling [22, 36, 37] (mostly for SGD) and duality gap-based sampling for CD [5, 23]. The drawbacks of adaptive strategies are twofold: often the provable theoretical guarantees can be worse than the complexity estimates for uniform sampling [23, 3] and often it is computationally 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. inadmissible to compute the optimal adaptive sampling distribution. For instance gradient based sampling requires the computation of the full gradient in each iteration [22, 36, 37]. Therefore one has to rely on approximations based on upper bounds [36, 37], or stale values [22, 1]. But in general these approximations can again be worse than uniform sampling. This makes it necessary to develop adaptive strategies that can efficiently be computed in every iteration and that come with theoretical guarantees that show their advantage over fixed sampling. Our contributions. In this paper we propose an efficient approximation of the gradient-based sampling in the sense that (i) it can efficiently be computed in every iteration, (ii) is provably better than uniform or fixed importance sampling and (iii) recovers the gradient-based sampling in the fullinformation setting. The scheme is completely generic and can easily be added as an improvement to both CD and SGD type methods. As our key contributions, we (1) show that gradient-based sampling in CD methods is theoretically better than the classical fixed sampling, the speed-up can reach a factor of the dimension n (Section 2); (2) propose a generic and efficient adaptive importance sampling strategy that can be applied in CD and SGD methods and enjoys favorable properties?such as mentioned above (Section 3); (3) demonstrate how the novel scheme can efficiently be integrated in CD and SGD on an important class of structured optimization problems (Section 4); (4) supply numerical evidence that the novel sampling performs well on real data (Section 5). Notation. For x ? Rn define [x]i := hx, ei i with ei the standard unit vectors in Rn . We abbreviate ?i f := [?f ]i . A convex function f : Rn ? R with L-Lipschitz continuous gradient satisfies f (x + ?u) ? f (x) + ? hu, ?f (x)i + ? 2 Lu 2 2 kuk ?x ? Rn , ?? ? R , (1) for every direction u ? Rn and Lu = L. A function with coordinate-wise Li -Lipschitz continuous gradients1 for constants Li > 0, i ? [n] := {1, . . . , n}, satisfies (1) just along coordinate directions, i.e. u = ei , Lei = Li for every i ? [n]. A function is coordinate-wise L-smooth if Li ? L for i = 1, . . . , n. For convenience we introduce vector l = (L1 , . . . , n )> and matrix L = diag(l). A probability vector p ? ?n := {x ? Rn?0 : kxk1 = 1} defines a probability distribution P over [n] and we denote by i ? p a sample drawn from P. 2 Adaptive Importance Sampling with Full Information In this section we argue that adaptive sampling strategies are theoretically well justified, as they can lead to significant improvements over static strategies. In our exhibition we focus first on CD methods, as we also propose a novel stepsize strategy for CD in this contribution. Then we revisit the results regarding stochastic gradient descent (SGD) already present in the literature. 2.1 Coordinate Descent with Adaptive Importance Sampling We address general minimization problems minx f (x). Let the objective f : Rn ? R be convex with coordinate-wise Li -Lipschitz continuous gradients. Coordinate descent methods generate sequences {xk }k?0 of iterates that satisfy the relation xk+1 = xk ? ?k ?ik f (xk )eik . (2) Here, the direction ik is either chosen deterministically (cyclic descent, steepest descent), or randomly picked according to a probability vector pk ? ?n . In the classical literature, the stepsize is often chosen such as to minimize the quadratic upper bound (1), i.e. ?k = L?1 ik . In this work we propose to set ?k = ?k [pk ]?1 where ? does not depend on the chosen direction ik . This leads to k ik 1 |?i f (x + ?ei ) ? ?i f (x)| ? Li |?| , ?x ? Rn , ?? ? R. 2 directionally-unbiased updates, like it is common among SGD-type methods. It holds   (1) Li ?k2 ?k 2 2 (?ik f (xk )) | xk Eik ?pk [f (xk+1 ) | xk ] ? Eik ?pk f (xk ) ? (?ik f (xk )) + [pk ]ik 2[pk ]2ik n X Li ?k2 2 2 = f (xk ) ? ?k k?f (xk )k2 + (?i f (xk )) . (3) 2[p ] k i i=1 In adaptive strategies we have the freedom to chose both variables ?k and pk as we like. We therefore propose to chose them in such a way that they minimize the upper bound (3) in order to maximize the expected progress. The optimal pk in (3) is independent of ?k , but the optimal ?k depends on pk . We can state the following useful observation. Lemma 2.1. If ?k = ?k (pk ) is the minimizer of (3), then xk+1 := xk ? [p?kk]i ?ik f (xk )eik satisfies k Eik ?pk ?k (pk ) 2 [f (xk+1 ) | xk ] ? f (xk ) ? k?f (xk )k2 . 2 (4) Consider two examples. In the first one we pick a sub-optimal, but very common [18] distribution: Li for i ? [n], where Example 2.2 (Li -based sampling). Let pL ? ?n defined as [pL ]i = Tr[L] 1 L = diag(L1 , . . . , Ln ). Then ?k (pL ) = Tr[L] . The distribution pL is often referred to as (fixed) importance sampling. In the special case when Li = L for all i ? [n], this boils down to uniform sampling. Example 2.3 (Optimal sampling2 ). Equation (3) is minimized for probabilities [p?k ]i = and ?k (p?k ) = k?f (xk )k22 ? 2. k L?f (xk )k ? L |? f (xk )| ?i i k L?f (x)k 1 Observe 1 Tr[L] ? ?k (p?k ) ? 1 Lmin , where Lmin := mini?[n] Li . 1 To prove this result, we rely on the following Lemma?the proof of which, as well as for the claims above, is deferred to Section A.1 of the appendix. Here |?| is applied entry-wise. ? Pn Li [x]2i | Lx| ? Lemma 2.4. Define V (p, x) := i=1 [p] . Then arg min V (p, x) = . n p?? i k Lxk1 The ideal adaptive algorithm. We propose to chose the stepsize and the sampling distribution for CD as in Example 2.3. One iteration of the resulting CD method is illustrated in Algorithm 1. Our bounds on the expected one-step progress can be used to derive convergence rates of this algorithm with the standard techniques. This is exemplified in Appendix A.1. In the next Section 3 we develop a practical variant of the ideal algorithm. Efficiency gain. By comparing the estimates provided in the examples above, we see that the expected progress of the proposed method is always at least as good as for the fixed sampling. For instance in the special case where L = Li for i ? [n], the Li -based sampling is just uniform sampling k?f (x )k2 1 with ?k (punif ) = Ln . On the other hand ?k (p?k ) = Lk?f (xk )k22 , which can be n times larger than k 1 ?k (punif ). The expected one-step progress in this extreme case coincides with the one-step progress of steepest coordinate descent [20]. 2.2 SGD with Adaptive Sampling SGD methods are applicable to objective functions which decompose as a sum Pn f (x) = n1 i=1 fi (x) (5) d with each fi : R ? R convex. In previous work [22, 36, 37] is has been argued that the following k?fi (xk )k2 gradient-based sampling [p??k ]i = Pn k?f is optimal in the sense that it maximizes the i (xk )k2 i=1 expected progress (3). Zhao and Zhang [36] derive complexity estimates for composite functions. For non-composite functions it becomes easier to derive the complexity estimate. For completeness, we add this simpler proof in Appendix A.2. Here ?optimal? refers to the fact that p?k is optimal with respect to the given model (1) of the objective function. If the model is not accurate, there might exist a sampling that yields larger expected progress on f . 2 3 Algorithm 1 Optimal sampling Algorithm 2 Proposed safe sampling Algorithm 3 Fixed sampling (compute full gradient) Compute ?f (xk ) (update l.- and u.-bounds) Update `, u (define optimal sampling) Define (p?k , ?k? ) as in Example 2.3 ik ? p?k xk+1 := xk ? ?? k [p? k ]ik ?ik f (xk ) (compute safe sampling) Define (p?k , ? ? k ) as in (7) ik ? p?k Compute ?ik f (xk ) xk+1 := xk ? ? ?k [p ?k ]ik ?ik f (xk ) (define fixed sampling) Define (pL , ? ? ) as in Example 2.2 ik ? pL Compute ?ik f (xk ) xk+1 := xk ? ? ? [pL ]ik ?ik f (xk ) Figure 1: CD with different sampling strategies. Whilst Alg. 1 requires to compute the full gradient, the compute operation in Alg. 2 is as cheap as for fixed importance sampling, Alg. 3. Defining the safe sampling p?k requires O(n log n) time. 3 Safe Adaptive Importance Sampling with Limited Information In the previous section we have seen that gradient-based sampling (Example 2.3) can yield a massive speed-up compared to a static sampling distribution (Example 2.2). However, sampling according to p?k in CD requires the knowledge of the full gradient ?f (xk ) in each iteration. And likewise, sampling from p??k in SGD requires the knowledge of the gradient norms of all components?both these operations are in general inadmissible, i.e. the compute cost would void all computational benefits of the iterative (stochastic) methods over full gradient methods. However, it is often possible to efficiently compute approximations of p?k or p??k instead. In contrast to previous contributions, we here propose a safe way to compute such approximations. By this we mean that our approximate sampling is provably never worse than static sampling, and moreover, we show that our solution is the best possible with respect to the limited information at hand. 3.1 An Optimization Formulation for Sampling Formally, we assume that we have in each iteration access to two vectors `k , uk ? Rn?0 that provide safe upper and lower bounds on either the absolute values of the gradient entries ([`k ]i ? |?i f (xk )| ? [uk ]i ) for CD, or of the gradient norms in SGD: ([`k ]i ? k?fi (xk )k2 ? [uk ]i ). We postpone the discussion of this assumption to Section 4, where we give concrete examples. The minimization of the upper bound (3) amounts to the equivalent problem3   ?2 V (pk , ck ) 2 min minn ??k kck k2 + k V (pk , ck ) ? min 2 ?k pk ?? pk ??n 2 kck k2 (6) where ck ? Rn represents the unknown true gradient. That is, with respect to the bounds `k , uk , we can write ck ? Ck := {x ? Rn : [`k ]i ? [x]i ? [uk ]i , i ? [n]}. In Example 2.3 we derived the optimal solution for a fixed ck ? Ck . However, this is not sufficient to find the optimal solution for an arbitrary ck ? Ck . Just computing the optimal solution for an arbitrary (but fixed) ck ? Ck is unlikely to yield a good solution. For instance both extreme cases ck = `k and ck = uk (the latter choice is quite common, cf. [36, 23]) might be poor. This is demonstrated in the next example.  ` Example 3.1. Let ` = (1, 2)> , u = (2, 3)> , c = (2, 2)> and L1 = L2 = 1. Then V k`k ,c = 1  25  2 2 2 9 u c 4 kck2 , V kuk , c = 12 kck2 , whereas for uniform sampling V kck , c = 2 kck2 . 1 1 The proposed sampling. As a consequence of these observations, we propose to solve the following optimization problem to find the best sampling distribution with respect to Ck :  V (p, c) vk := minn max , and to set (?k , pk ) := v1k , p?k , (7) 2 p?? c?Ck kck 2 where p?k denotes a solution of (7). The resulting algorithm for CD is summarized in Alg. 2. In the remainder of this section we discuss the properties of the solution p?k (Theorem 3.2) an how such a solution can be efficiently be computed (Theorem 3.4, Algorithm 4). 3 Although only shown here for CD, an equivalent optimization problem arises for SGD methods, cf. [36]. 4 3.2 Proposed Sampling and its Properties Theorem 3.2. Let (p, ? c?) ? ?n ? Rn?0 denote a solution of (7). Then Lmin ? vk ? Tr [L] and (i) max c?Ck V (p, ? c) 2 kck2 ? max V (p, c) 2 kck2 c?Ck , ?p ? ?n ; (p? has the best worst-case guarantee) 2 (p? is always better than Li -based sampling) (ii) V (p, ? c) ? Tr [L] ? kck2 , ?c ? Ck . Remark 3.3. In the special case Li = L for all i ? [n], the Li -based sampling boils down to uniform 2 sampling (Example 2.2) and p? is better than uniform sampling: V (p, ? c) ? Ln kck2 , ?c ? Ck . Proof. Property (i) is an immediate consequence of (7). Moreover, observe that the Li -based L ,c) sampling pL is a feasible solution in (7) with value V (p ? Tr [L] for all c ? Ck . Hence kck22 ? k Lck21 2.4 V (p, c) V (p, ? c) (?) V (p, ? c?) (7) V (pL , c) Lmin ? = minn ? ? ? max = Tr [L] , (8) 2 2 2 2 2 p?? c?C k kck2 kck2 kck2 k? ck2 kck2 for all c ? Ck , thus vk ? [Lmin , Tr [L]] and (ii) follows. We prove inequality (?) in the appendix, by showing that min and max can be interchanged in (7). A geometric interpretation. We show in ?Appendix B that the optimization problem (7) can ? ? k Lck1 h l,ci equivalently be written as vk = maxc?Ck kck = maxc?Ck kck , where [l]i = Li for i ? [n]. 2 2 The maximum is thus attained for vectors c ? Ck that minimize the angle with the vector l. ? ? 2 Lc Theorem 3.4. Let c ? Ck , p = k?Lck and denote m = kck2 ? k Lck?1 1 . If 1 ? ? if [uk ]i ? Li m , ?[uk ]i ? ?i ? [n] , (9) [c]i = [`k ]i if [`k ]i ? Li m , ?? Li m otherwise, then (p, c) is a solution to (7). Moreover, such a solution can be computed in time O(n log n). Proof. This can be proven by examining the optimality conditions of problem (7). This is deferred to Section B.1 of the appendix. A procedure that computes such a solution is depicted in Algorithm 4. The algorithm makes extensive use of (9). For simplicity, assume first L = In for now. In each iteration t , a potential solution vector ct is proposed, and it is verified whether this vector satisfies all optimality conditions. In Algorithm 4, ct is just implicit, with [ct ]i = [c]i for decided indices i ? D ? and [ct ]i = [ Lm]i for undecided ? / D. After at most n iterations a valid solution is found. ? indices i ? By sorting the components of L?1 `k and L?1 uk by their magnitude, at most a linear number of inequality checks in (9) have to be performed in total. Hence the running time is dominated by the O(n log n) complexity of the sorting algorithm. A formal proof is given in the appendix. Algorithm 4 Computing the Safe Sampling for Gradient Information `, u 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: Input: 0n ? ` ? u, ? L, Initialize: c = 0n , u =?1, ` = n, D = ?. `sort := sort_asc( L?1 `), usort := sort_asc( L?1 u), m = max(`sort ) while u ? ` do if [`sort ]` > m then (largest undecided lower bound is violated) ? Set corresponding [c]index := [ L`sort ]` ; ` := ` ? 1; D := D ? {index} else if [usort ]u < m then (smallest undecided upper bound is violated) ? Set corresponding [c]index := [ Lusort ]u ; u := u + 1; D := D ? {index} else break (no constraints are violated) end if ? 2 m := kck2 ? k Lck?1 (update m as in (9)) 1 end while ?   ? ? k Lck2 Lc Set [c]i := Li m for all i ? / D and Return c, p = k?Lck , v = kck2 1 1 5 2 Competitive Ratio. We now compare the proposed sampling distribution p?k with the optimal sampling solution in hindsight. We know that if the true (gradient) vector c? ? C?k would be given to L? c us, then the corresponding optimal probability distribution would be p? (? c) = k?L? (Example 2.3). ck 1 c) k ,? Thus, for this c? we can now analyze the ratio V V(p(?p?(? c),? c) . As we are interested in the worst case ratio among all possible candidates c? ? Ck , we define ?k := max c?Ck Lemma 3.5. Let wk := minc?Ck V (p, ? c) V (p, ? c) = max ? . ? c?C V (p (c), c) k k Lck21 ? k Lck21 . kck22 (10) Then Lmin ? wk ? vk , and ?k ? vk vk wk (? Lmin ). Lemma 3.6. Let ? ? 1. If [Ck ]i ? ?[Ck ]i = ? and ? ?1 [Ck ]i ? [Ck ]i = ? for all i ? [n] (here [Ck ]i denotes the projection on the i-th coordinate), then ?k ? ? 4 . These two lemma provide bounds on the competitive ratio. Whilst Lemma 3.6 relies on a relative accuracy condition, Lemma 3.5 can always be applied. However, the corresponding minimization problem is non-convex. Note that knowledge of ?k is not needed to run the algorithm. 4 Example Safe Gradient Bounds In this section, we argue that for a large class of objective functions of interest in machine learning, suitable safe upper and lower bounds `, u on the gradient along every coordinate direction can be estimated and maintained efficiently during optimization. A similar argument can be given for the efficient approximation of component wise gradient norms in finite sum objective based stochastic gradient optimization. As the guiding example, we will here showcase the training of generalized linear models (GLMs) as e.g. in regression, classification and feature selection. These models are formulated in terms of a given data matrix A ? Rd?n with columns ai ? Rd for i ? [n]. Coordinate Descent - GLMs Pn with Arbitrary Regularizers. Consider general objectives of the form f (x) := h(Ax) + i=1 ?i ([x]i ) with an arbitrary convex separable regularizer term given by the ?i : R ? R for i ? [n]. A key example is when h : Rd ? R describes the least-squares 2 regression objective h(Ax) = 12 kAx ? bk2 for a b ? Rd . Using that this h is twice differentiable 2 with ? h(Ax) = In , it is easy to see that we can track the evolution of all gradient entries, when performing CD steps, as follows: ?i f (xk+1 ) ? ?i f (xk ) = ?k hai , aik i , ?i 6= ik . (11) for ik being the coordinate changed in step k (here we also used the separability of the regularizer). Therefore, all gradient changes can be tracked exactly if the inner products of all datapoints are available, or approximately if those inner products can be upper and lower bounded. For computational efficiency, we in our experiments simply use Cauchy-Schwarz |hai , aik i| ? kai k ? kaik k. This results in safe upper and lower bounds [`k+1 ]i ? ?i f (xk+1 ) ? [uk+1 ]i for all inactive coordinates i 6= ik . (For the active coordinate ik itself one observes the true value without uncertainty). These bounds can be updated in linear time O(n) in every iteration. For general smooth h (again with arbitrary separable regularizers ?i ), (11) can readily be extended to ? ik i instead, when hold [32, Lemma 4.1], the inner product change term becoming hai , ?2 f (Ax)a ? will be an element of the line segment [xk , xk+1 ]. assuming h is twice-differentiable. Here x Stochastic Gradient Descent - GLMs. We now presentP a similar result forPfinite sum problems (5) n n for the use in SGD based optimization, that is f (x) := n1 i=1 fi (x) = n1 i=1 hi (a> i x). Lemma 4.1. Consider f : Rd ? R as above, with twice differentiable hi : R ? R. Let xk , xk+1 ? Rd denote two successive iterates of SGD, i.e. xk+1 := xk ? ?k aik ?hik (a> ik xk ) = xk + ?k aik . d Then there exists x ? ? R on the line segment between xk and xk+1 , x ? ? [xk , xk+1 ] with ?fi (xk+1 ) ? ?fi (xk ) = ?k ?2 hi (a> ?) hai , aik i ai , i x 6 ? i 6= ik . (12) This leads to safe upper and lower bounds for the norms of the partial gradient, [`k ]i ? k?fi (xk )k2 ? [uk ]i , that can be updated in linear time O(n), analogous to the coordinate case discussed above.4 We note that there are many other ways to track safe gradient bounds for relevant machine learning problems, including possibly more tight ones. We here only illustrate the simplest variants, highlighting the fact that our new sampling procedure works for any safe bounds `, u. Computational Complexity. In this section, we have demonstrated how safe upper and lower bounds `, u on the gradient information can be obtained for GLMs, and argued that these bounds can be updated in time O(n) per iteration of CD and SGD. The computation of the proposed sampling takes O(n log n) time (Theorem 3.4). Hence, the introduced overhead in Algorithm 2 compared to fixed sampling (Algorithm 3) is of the order O(n log n) in every iteration. The computation of one coordinate of the gradient, ?ik f (xk ), takes time ?(d) for general data matrices. Hence, when d = ?(n), the introduced overhead reduces to O(log n) per iteration. 5 Empirical Evaluation In this section we evaluate the empirical performance of our proposed adaptive sampling scheme on relevant machine learning tasks. In particular, we illustrate performance on generalized linear models with L1 and L2 regularization, as of the form (5), n 1X hi (a> (13) min i x) + ? ? r(x) x?Rd n i=1 We use square loss, squared hinge loss as well as logistic loss for the data fitting terms hi , and kxk1 and kxk22 for the regularizer r(x). The datasets used in the evaluation are rcv1, real-sim and news20.5 The rcv1 dataset consists of 20,242 samples with 47,236 features, real-sim contains 72,309 datapoints and 20,958 features and news20 contains 19,996 datapoints and 1,355,191 features. For all datasets we set unnormalized features with all the non-zero entries set to 1 (bag-of-words features). By real-sim? and rcv1? we denote a subset of the data chosen by randomly selecting 10,000 features and 10,000 datapoints. By news20? we denote a subset of the data chose by randomly selecting 15% of the features and 15% of the datapoints. A regularization parameter ? = 0.1 is used for all experiments. Our results show the evolution of the optimization objective over time or number of epochs (an epoch corresponding to n individual updates). To compute safe lower and upper bounds we use the methods presented in Section 4 with no special initialization, i.e. `0 = 0n , u0 = ?n . 1 Coordinate Descent. In Figure 2 we compare the effect of the fixed stepsize ?k = Ln (denoted as ?small?) vs. the time varying optimal stepsize (denoted as ?big?) as discussed in Section 2. Results are shown for optimal sampling p?k (with optimal stepsize ?k (p?k ), cf. Example 2.3), our proposed sampling p?k (with optimal stepsize ?k (p?k ) = vk?1 , cf. (7)) and uniform sampling (with 1 optimal stepsize ?k (pL ) = Ln , as here L = LIn , cf. Example 2.2). As the experiment aligns with theory?confirming the advantage of the varying ?big? stepsizes?we only show the results for Algorithms 1?3 in the remaining plots. Performance for squared hinge loss, as well as logistic regression with L1 and L2 regularization is presented in Figure 3 and Figure 4 respectively. In Figures 5 and 6 we report the iteration complexity vs. accuracy as well as timing vs. accuracy results on the full dataset for coordinate descent with square loss and L1 (Lasso) and L2 regularization (Ridge). Theoretical Sampling Quality. As part of the CD performance results in Figures 2?6 we include an additional evolution plot on the bottom of each figure to illustrate the values vk which determine the stepsize (? ?k = vk?1 ) for the proposed Algorithm 2 (blue) and the optimal stepsizes of Algorithm 1 vk (black) which rely on the full gradient information. The plots show the normalized values Tr[L] , i.e. the relative improvement over Li -based importance sampling. The results show that despite only relying on very loose safe gradient bounds, the proposed adaptive sampling is able to strongly benefit from the additional information. 4 5 Here we use the efficient representation ?fi (x) = ?(x) ? ai for ?(x) ? R. All data are available at www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets/ 7 1.00 Uniform Proposed (big step) Proposed (small step) 0.95 1.00 0.99 0.98 0.97 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 f(xk ) Optimal (big step) Optimal (small step) 0.90 0.96 0.95 0.85 0.94 f(x ) k 0 -1 -2 -3 vk -4 10 0 f(xk ) 2 1 Epochs 5 0 -1 -2 -3 vk -4 10 6 0 2 1 (a) rcv1?, L1 reg. 5 Epochs 0 -1 -2 -3 vk -4 10 0 6 Uniform Proposed Optimal 6.90 6.85 0.68 6.80 6.75 f(xk ) 0 -1 -2 -3 vk -4 10 0 2 1 Epochs 5 Uniform Proposed Optimal 0.69 5 Epochs 6 0 -1 -2 -3 vk -4 10 0 Uniform Proposed Optimal 0.69 0.68 0.69 0.67 0.65 0.64 0.64 f(xk ) 0.63 f(xk ) 0 -1 -2 -3 vk -4 10 0 -1 -2 -3 vk -4 10 (a) rcv1?, L1 reg. 5 Epochs 3 0.66 0.65 2 2.5 Uniform Proposed Optimal 0.68 0.66 1 Epochs Figure 3: (CD, squared hinge loss) Function value vs. number of iterations for optimal stepsize ?k = vk?1 . 0.66 0 1 (b) real-sim?, L2 reg. 0.67 f(xk ) Uniform Proposed Optimal 0.5 0.67 0 -1 -2 -3 vk -4 10 6 2 1 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 f(xk ) (a) rcv1?, L1 reg. (b) rcv1?, L2 reg. Figure 2: (CD, square loss) Fixed vs. adaptive sampling strategies, and dependence on stepsizes. 1 . With ?big? ?k = vk?1 and ?small? ?k = Tr[L] 0.1 x Uniform Proposed Optimal 6 0 (b) rcv1?, L2 reg. 1 0.5 2.5 Epochs 3 0 (c) real-sim?, L1 reg. 1 0.5 Epochs 2.5 3 (d) real-sim?, L2 reg. Figure 4: (CD, logistic loss) Function value vs. number of iterations for different sampling strategies. Bottom: Evolution of the value vk which determines the optimal stepsize (? ?k = vk?1 ). The plots vk show the normalized values Tr[L] , i.e. the relative improvement over Li -based importance sampling. 1.00 Uniform Proposed Optimal 0.95 0.90 1.00 Uniform Proposed Optimal 0.95 0.90 0.80 0.75 f(xk ) 0 -1 -2 -3 vk -4 10 0 0.70 0.5 1 1.5 Epochs 3 f(xk ) 1 Epochs (b) real-sim, L1 reg. 90 Uniform Proposed Optimal 60 55 70 45 40 35 0.5 1 Epochs 2 2.5 (a) rcv1?, L1 reg. 0.80 f(xk ) f(xk ) 2 4 6 Time 12 14 16 0.75 0 -1 -2 -3 vk -4 10 0 (a) real-sim, L1 reg. 140 Uniform Proposed Optimal 120 100 60 60 40 40 40 Epochs Time 12 14 16 2 20 0 (b) rcv1?, L2 reg. Uniform Proposed Optimal 80 80 1 6 100 50 0.5 4 (b) real-sim, L2 reg. 60 0 2 Figure 6: (CD, square loss) Function value vs. clock time on the full datasets. (Data for the optimal sampling omitted, as this strategy is not competitive time-wise.) Uniform Proposed Optimal 80 50 0 0.85 0 Figure 5: (CD, square loss) Function value vs. number of iterations on the full datasets. 65 0.85 0 -1 -2 -3 vk -4 10 2 Uniform Proposed 0.95 0.90 0.75 0.5 1.00 0.90 0.80 0 -1 -2 -3 vk -4 3.5 10 0 (a) rcv1, L1 reg. Uniform Proposed 0.95 0.85 0.85 0.80 1.00 0.5 1 Epochs 2 0 2.5 0.5 1 Epochs 2 2.5 (d) real-sim?, L2 reg. (c) real-sim?, L1 reg. Figure 7: (SGD, square loss) Function value vs. number of iterations. 6 5 4 3 2 0 1 2 Epochs Uniform Proposed 40 35 30 25 20 15 10 Uniform Proposed Optimal 7 0 4 5 10 Time 20 25 (a) news20?, L1 reg. (a) news20?, L1 reg. Figure 8: (SGD, square loss) Function value vs. number of iterations. Figure 9: (SGD square loss) Function value vs. clock time. 8 Stochastic Gradient Descent. Finally, we also evaluate the performance of our approach when used within SGD with L1 and L2 regularization and square loss. In Figures 7?8 we report the iteration complexity vs. accuracy results and in Figure 9 the timing vs. accuracy results. The time units in Figures 6 and 9 are not directly comparable, as the experiments were conducted on different machines. We observe that on all three datasets SGD with the optimal sampling performs only slightly better than uniform sampling. This is in contrast with the observations for CD, where the optimal sampling yields a significant improvement. Consequently, the effect of the proposed sampling is less pronounced in the three SGD experiments. Summary. The main findings of our experimental study can be summarized as follows: ? Adaptive importance sampling significantly outperforms fixed importance sampling in iterations and time. The results show that (i) convergence in terms of iterations is almost as good as for the optimal (but not efficiently computable) gradient-based sampling and (ii) the introduced computational overhead is small enough to outperform fixed importance sampling in terms of total computation time. ? Adaptive sampling requires adaptive stepsizes. The adaptive stepsize strategies of Algorithms 1 and 2 allow for much faster convergence than conservative fixed-stepsize strategies. In the experiments, the measured value vk was always significantly below the worst case estimate, in alignment with the observed convergence. ? Very loose safe gradient bounds are sufficient. Even the bounds derived from the the very na?ve gradient information obtained by estimating scalar products resulted in significantly better sampling than using no gradient information at all. Further, no initialization of the gradient estimates is needed (at the beginning of the optimization process the proposed adaptive method performs close to the fixed sampling but accelerates after just one epoch). 6 Conclusion In this paper we propose a safe adaptive importance sampling scheme for CD and SGD algorithms. We argue that optimal gradient-based sampling is theoretically well justified. To make the computation of the adaptive sampling distribution computationally tractable, we rely on safe lower and upper bounds on the gradient. However, in contrast to previous approaches, we use these bounds in a novel way: in each iteration, we formulate the problem of picking the optimal sampling distribution as a convex optimization problem and present an efficient algorithm to compute the solution. The novel sampling provably performs better than any fixed importance sampling?a guarantee which could not be established for previous samplings that were also derived from safe lower and upper bounds. The computational cost of the proposed scheme is of the order O(n log n) per iteration?this is on many problems comparable with the cost to evaluate a single component (coordinate, sum-structure) of the gradient, and the scheme can thus be implemented at no extra computational cost. This is verified by timing experiments on real datasets. We discussed one simple method to track the gradient information in GLMs during optimization. However, we feel that the machine learning community could profit from further research in that direction, for instance by investigating how such safe bounds can efficiently be maintained on more complex models. 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Variational Walkback: Learning a Transition Operator as a Stochastic Recurrent Net Anirudh Goyal MILA, Universit? de Montr?al [email protected] Surya Ganguli Stanford University [email protected] Nan Rosemary Ke MILA, ?cole Polytechnique de Montr?al [email protected] Yoshua Bengio MILA, Universit? de Montr?al [email protected] Abstract We propose a novel method to directly learn a stochastic transition operator whose repeated application provides generated samples. Traditional undirected graphical models approach this problem indirectly by learning a Markov chain model whose stationary distribution obeys detailed balance with respect to a parameterized energy function. The energy function is then modified so the model and data distributions match, with no guarantee on the number of steps required for the Markov chain to converge. Moreover, the detailed balance condition is highly restrictive: energy based models corresponding to neural networks must have symmetric weights, unlike biological neural circuits. In contrast, we develop a method for directly learning arbitrarily parameterized transition operators capable of expressing nonequilibrium stationary distributions that violate detailed balance, thereby enabling us to learn more biologically plausible asymmetric neural networks and more general non-energy based dynamical systems. The proposed training objective, which we derive via principled variational methods, encourages the transition operator to "walk back" (prefer to revert its steps) in multi-step trajectories that start at datapoints, as quickly as possible back to the original data points. We present a series of experimental results illustrating the soundness of the proposed approach, Variational Walkback (VW), on the MNIST, CIFAR-10, SVHN and CelebA datasets, demonstrating superior samples compared to earlier attempts to learn a transition operator. We also show that although each rapid training trajectory is limited to a finite but variable number of steps, our transition operator continues to generate good samples well past the length of such trajectories, thereby demonstrating the match of its non-equilibrium stationary distribution to the data distribution. Source Code: http://github.com/anirudh9119/walkback_nips17 1 Introduction A fundamental goal of unsupervised learning involves training generative models that can understand sensory data and employ this understanding to generate, or sample new data and make new inferences. In machine learning, the vast majority of probabilistic generative models that can learn complex probability distributions over data fall into one of two classes: (1) directed graphical models, corresponding to a finite time feedforward generative process (e.g. variants of the Helmholtz machine (Dayan et al., 1995) like the Variational Auto-Encoder (VAE) (Kingma and Welling, 2013; Rezende et al., 2014)), or (2) energy function based undirected graphical models, corresponding to sampling from a stochastic process whose equilibrium stationary distribution obeys detailed balance with respect to the energy function (e.g. various Boltzmann machines (Salakhutdinov and Hinton, 2009)). This detailed 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. balance condition is highly restrictive: for example, energy-based undirected models corresponding to neural networks require symmetric weight matrices and very specific computations which may not match well with what biological neurons or analog hardware could compute. In contrast, biological neural circuits are capable of powerful generative dynamics enabling us to model the world and imagine new futures. Cortical computation is highly recurrent and therefore its generative dynamics cannot simply map to the purely feed-forward, finite time generative process of a directed model. Moreover, the recurrent connectivity of biological circuits is not symmetric, and so their generative dynamics cannot correspond to sampling from an energy-based undirected model. Thus, the asymmetric biological neural circuits of our brain instantiate a type of stochastic dynamics arising from the repeated application of a transition operator? whose stationary distribution over neural activity patterns is a non-equilibrium distribution that does not obey detailed balance with respect to any energy function. Despite these fundamental properties of brain dynamics, machine learning approaches to training generative models currently lack effective methods to model complex data distributions through the repeated application a transition operator, that is not indirectly specified through an energy function, but rather is directly parameterized in ways that are inconsistent with the existence of any energy function. Indeed the lack of such methods constitutes a glaring gap in the pantheon of machine learning methods for training probabilistic generative models. The fundamental goal of this paper is to provide a step to filling such a gap by proposing a novel method to learn such directly parameterized transition operators, thereby providing an empirical method to control the stationary distributions of non-equilibrium stochastic processes that do not obey detailed balance, and match these distributions to data. The basic idea underlying our training approach is to start from a training example, and iteratively apply the transition operator while gradually increasing the amount of noise being injected (i.e., temperature). This heating process yields a trajectory that starts from the data manifold and walks away from the data due to the heating and to the mismatch between the model and the data distribution. Similarly to the update of a denoising autoencoder, we then modify the parameters of the transition operator so as to make the reverse of this heated trajectory more likely under a reverse cooling schedule. This encourages the transition operator to generate stochastic trajectories that evolve towards the data distribution, by learning to walk back the heated trajectories starting at data points. This walkback idea had been introduced for generative stochastic networks (GSNs) and denoising autoencoders (Bengio et al., 2013b) as a heuristic, and without temperature annealing. Here, we derive the specific objective function for learning the parameters through a principled variational lower bound, hence we call our training method variational walkback (VW). Despite the fact that the training procedure involves walking back a set of trajectories that last a finite, but variable number of time-steps, we find empirically that this yields a transition operator that continues to generate sensible samples for many more time-steps than are used to train, demonstrating that our finite time training procedure can sculpt the non-equilibrium stationary distribution of the transition operator to match the data distribution. We show how VW emerges naturally from a variational derivation, with the need for annealing arising out of the objective of making the variational bound as tight as possible. We then describe experimental results illustrating the soundness of the proposed approach on the MNIST, CIFAR-10, SVHN and CelebA datasets. Intriguingly, we find that our finite time VW training process involves modifications of variational methods for training directed graphical models, while our potentially asymptotically infinite generative sampling process corresponds to non-equilibrium generalizations of energy based undirected models. Thus VW goes beyond the two disparate model classes of undirected and directed graphical models, while simultaneously incorporating good ideas from each. 2 The Variational Walkback Training Process Our goal is to learn a stochastic transition operator pT (s0 |s) such that its repeated application yields samples from the data manifold. Here T reflects an underlying temperature, which we will modify during the training process. The transition operator is further specified by other parameters which must be learned from data. When K steps are chosen to generate a sample, the generative process QK has joint probability p(sK 0 ) = p(sK ) t=1 pTt (st?1 |st ), where Tt is the temperature at step t. We first give an intuitive description of our learning algorithm before deriving it via variational methods in the next section. The basic idea, as illustrated in Fig. 1 and Algorithm 1 is to follow a walkback ? A transition operator maps the previous-state distribution to a next-state distribution, and is implemented by a stochastic transformation which from the previous state of a Markov chain generates the next state 2 Figure 1: Variational WalkBack framework. The generative process is represented in the blue arrows with the sequence of pTt (st?1 |st ) transitions. The destructive forward process starts at a datapoint (from qT0 (s0 )) and gradually heats it through applications of qTt (st |st?1 ). Larger temperatures on the right correspond to a flatter distribution, so the whole destructive forward process maps the data distribution to a Gaussian and the creation process operates in reverse. strategy similar to that introduced in Alain and Bengio (2014). In particular, imagine a destructive process qTt+1 (st+1 |st ) (red arrows in Fig. 1), which starts from a data point s0 = x, and evolves it QK K stochastically to obtain a trajectory s0 , . . . , sK ? sK 0 , i.e., q(s0 ) = q(s0 ) t=1 qTt (st |st?1 ), where q(s0 ) is the data distribution. Note that the p and q chains will share the same parameters for the transition operator (one going backwards and one forward) but they start from different priors for their first step: q(s0 ) is the data distribution while p(s0 ) is a flat factorized prior (e.g. Gaussian). The training procedure trains the transition operator pT to make reverse transitions of the destructive process more likely. For this reason we index time so the destructive process operates forward in time, while the reverse generative process operates backwards in time, with the data distribution occurring at t = 0. In particular, we need only train the transition operator to reverse time by 1-step at each step, making it unnecessary to solve a deep credit assignment problem by performing backpropagation through time across multiple walk-back steps. Overall, the destructive process generates trajectories that walk away from the data manifold, and the transition operator pT learns to walkback these trajectories to sculpt the stationary distribution of pT at T = 1 to match the data distribution. Because we choose qT to have the same parameters as pT , they have the same transition operator but not the same joint over the whole sequence because of differing initial distributions for each trajectory. We also choose to increase temperature with time in the destructive process, following a temperature schedule T1 ? ? ? ? ? TK . Thus the forward destructive (reverse generative) process corresponds to a heating (cooling) protocol. This training procedure is similar in spirit to DAE?s (Vincent et al., 2008) or NET (Sohl-Dickstein et al., 2015) but with one major difference: the destructive process in these works corresponds to the addition of random noise which knows nothing about the current generative process during training. To understand why tying together destruction and creation may be a good idea, consider the special case in which pT corresponds to a stochastic process whose stationary distribution obeys detailed balance with respect to the energy function of an undirected graphical model. Learning any such model involves two fundamental goals: the model must place probability mass (i.e. lower the energy function) where the data is located, and remove probability mass (i.e. raise the energy function) elsewhere. Probability modes where there is no data are known as spurious modes, and a fundamental goal of learning is to hunt down these spurious modes and remove them. Making the destructive process identical to the transition operator to be learned is motivated by the notion that the destructive process should then efficiently explore the spurious modes of the current transition operator. The walkback training will then destroy these modes. In contrast, in DAE?s and NET?s, since the destructive process corresponds to the addition of unstructured noise that knows nothing about the generative process, it is not clear that such an agnostic destructive process will efficiently seek out the spurious modes of the reverse, generative process. We chose the annealing schedule empirically to minimize training time. The generative process starts by sampling a state sK from a broad Gaussian p? (sK ), whose variance is initially equal to 2 the total data variance ?max (but can be later adapted to match the final samples from the inference trajectories). Then we sample from pTmax (sK?1 |sK ), where Tmax is a high enough temperature so that the resultant injected noise can move the state across the whole domain of the data. The injected noise used to simulate the effects of finite temperature has variance linearly proportional to 3 temperature. Thus if ? 2 is the equivalent noise injected by the transition operator pT at T = 1, we ?2 choose Tmax = ?max to achieve the goal of the first sample sK?1 being able to move across the entire 2 range of the data distribution. Then we successively cool the temperature as we sample ?previous? states st?1 according to pT (st?1 |st ), with T reduced by a factor of 2 at each step, followed by n steps at temperature 1. This cooling protocol requires the number of steps to be K = log2 Tmax + n, (1) in order to go from T = Tmax to T = 1 in K steps. We choose K from a random distribution. Thus the training procedure trains pT to rapidly transition from a simple Gaussian distribution to the data distribution in a finite but variable number of steps. Ideally, this training procedure should then indirectly create a transition operator pT at T = 1 whose repeated iteration samples the data distribution with a relatively rapid mixing time. Interestingly, this intuitive learning algorithm for a recurrent dynamical system, formalized in Algorithm 1, can be derived in a principled manner from variational methods that are usually applied to directed graphical models, as we see next. Algorithm 1 VariationalWalkback(?) Train a generative model associated with a transition operator pT (s|s0 ) at temperature T (temperature 1 for sampling from the actual model), parameterized by ?. This transition operator injects noise of variance T ? 2 at each step, where ? 2 is the noise level at temperature 1. Require: Transition operator pT (s|s0 ) from which one can both sample and compute the gradient of log pT (s|s0 ) with respect to parameters ?, given s and s0 . 2 Require: Precomputed ?max , initially data variance (or squared diameter). Require: N1 > 1 the number of initial temperature-1 steps of q trajectory (or ending a p trajectory). repeat Set p? to be a Gaussian with mean and variance of the data. ?2 Tmax ? ?max 2 Sample n as a uniform integer between 0 and N1 K ? ceil(log2 Tmax ) + n Sample x ? data (or equivalently sample a minibatch to parallelize computation and process each element of the minibatch independently) Let s0 = (x) and initial temperature T = 1, initialize L = 0 for t = 1 to K do Sample st ? pT (s|st?1 ) Increment L ? L + log pT (st?1 |st ) (st?1 |st ) Update parameters with log likelihood gradient ? log pT?? If t > n, increase temperature with T ? 2T end for Increment L ? L + log p? (sK ) Update mean and variance of p? to match the accumulated 1st and 2nd moment statistics of the samples of sK until convergence monitoring L on a validation set and doing early stopping =0 3 Variational Derivation of Walkback The marginal probability of a data point s0 at the end of the K-step generative cooling process is ! K X Y p(s0 ) = pT0 (s0 |s1 ) pTt (st?1 |st ) p? (sK ) (2) t=2 sK 1 where sK 1 = (s1 , s2 , . . . , sK ) and v = s0 is a visible variable in our generative process, while the cooling trajectory that lead to it can be thought of as a latent, hidden variable h = sK 1 . Recall the decomposition of the marginal log-likelihood via a variational lower bound, ln p(v) ? ln X p(v|h)p(h) = X h p(v, h) +DKL [q(h|v)||p(h|v)]. q(h|v) {z } q(h|v) ln h | L 4 (3) Here L is the variational lower bound which motivates the proposed training procedure, and q(h|v) is a variational approximation to p(h|v). Applying this decomposition to v = s0 and h = sK 1 , we find ln p(s0 ) = X q(sk1 |s0 ) ln sk 1 p(s0 |sk1 )p(sk1 ) + DKL [q(sk1 |s0 ) || p(sk1 |s0 )]. q(sk1 |s0 ) (4) Similarly to the EM algorithm, we aim to approximately maximize the log-likelihood with a 2-step procedure. Let ?p be the parameters of the generative model p and ?q be the parameters of the approximate inference procedure q. Before seeing the next example we have ?q = ?p . Then in the first step we update ?p towards maximizing the variational bound L, for example by a stochastic gradient descent step. In the second step, we update ?q by setting ?q ? ?p , with the objective to reduce the KL term in the above decomposition. See Sec. 3.1 below regarding conditions for the tightness of the bound, which may not be perfect, yielding a possibly biased gradient when we force the constraint ?p = ?q . We continue iterating this procedure, with training examples s0 . We can obtain an unbiased Monte-Carlo estimator of L as follows from a single trajectory: L(s0 ) ? K X t=1 ln pTt (st?1 |st ) + ln p? (sK ) qTt (st |st?1 ) (5) with respect to p? , where s0 is sampled from the data distribution qT0 (s0 ), and the single sequence sK 1 is sampled from the heating process q(sK 1 |s0 ). We are making the reverse of heated trajectories more likely under the cooling process, leading to Algorithm 1. Such variational bounds have been used successfully in many learning algorithms in the past, such as the VAE (Kingma and Welling, 2013), except that they use an explicitly different set of parameters for p and q. Some VAE variants (S?nderby et al., 2016; Kingma et al., 2016) however mix the p-parameters implicitly in forming q, by using the likelihood gradient to iteratively form the approximate posterior. 3.1 Tightness of the variational lower bound As seen in (4), the gap between L(s0 ) and ln p(s0 ) is controlled by DKL [q(sk1 |s0 )||p(sk1 |s0 )], and is therefore tight when the distribution of the heated trajectory, starting from a point s0 , matches the posterior distribution of the cooled trajectory ending at s0 . Explicitly, this KL divergence is given by K X p(s0 ) Y qTt (st |st?1 ) DKL = q(sk1 |s0 ) ln ? . (6) p (sK ) t=1 pTt (st?1 |st ) k s1 As the heating process q unfolds forward in time, while the cooling process p unfolds backwards in time, we introduce the time reversal of the transition operator pT , denoted by pR T , as follows. Under repeated application of the transition operator pT , state s settles into a stationary distribution ?T (s) at temperature T . The probability of observing a transition st ? st?1 under pT in its stationary state is then pT (st?1 |st )?T (st ). The time-reversal pR T is the transition operator that makes the reverse transition equally likely for all state pairs, and therefore obeys PT (st?1 |st )?T (st ) = PTR (st |st?1 )?T (st?1 ) (7) pR T for all pairs of states st?1 and st . It is well known that is a valid stochastic transition operator and has the same stationary distribution ?T (s) as pT . Furthermore, the process pT obeys detailed balance if and only if it is invariant under time-reversal, so that pT = pR T. To better understand the KL divergence in (6), at each temperature Tt , we use relation (7) to replace the cooling process PTt which occurs backwards in time with its time-reversal, unfolding forward in time, at the expense of introducing ratios of stationary probabilities. We also exploit the fact that q and p are the same transition operator. With these substitutions in (6), we find DKL = X sk 1 q(sk1 |s0 ) ln K K Y pTt (st |st?1 ) X p(s0 ) Y ?Tt (st ) k + q(s |s ) ln . 0 1 p? (sK ) t=1 ?Tt (st?1 ) pR (s |s ) k t=1 Tt t t?1 (8) s1 The first term in (8) is simply the KL divergence between the distribution over heated trajectories, and the time reversal of the cooled trajectories. Since the heating (q) and cooling (p) processes are tied, this KL divergence is 0 if and only if pTt = pR Tt for all t. This time-reversal invariance requirement for vanishing KL divergence is equivalent to the transition operator pT obeying detailed balance at all temperatures. 5 Now intuitively, the second term can be made small in the limit where K is large and the temperature sequence is annealed slowly. To see why, note we can write the ratio of probabilities in this term as, ?T (sK?1 ) ?TK (sK ) p(s0 ) ?T1 (s1 ) ? ? ? K?1 . (9) ?T1 (s0 ) ?T2 (s1 ) ?TK?1 (sK ) p? (sK ) which is similar in shape (but arising in a different context) to the product of probability ratios computed for annealed importance sampling (Neal, 2001) and reverse annealed importance sampling (Burda et al., 2014). Here it is manifest that, under slow incremental annealing schedules, we are comparing probabilities of the same state under slightly different distributions, so all ratios are close to 1. For example, under many steps, with slow annealing, the generative process approximately reaches its stationary distribution, p(s0 ) ? ?T1 (s0 ). This slow annealing to go from p? (sK ) to p(s0 ) corresponds to the quasistatic limit in statistical physics, where the work required to perform the transformation is equal to the free energy difference between states. To go faster, one must perform excess work, above and beyond the free energy difference, and this excess work is dissipated as heat into the surrounding environment. By writing the distributions in terms of energies and free energies: ?Tt (st ) ? e?E(st )/Tt , p? (sK ) = e?[EK (sK )?FK ] , and p(s0 ) = e?[E0 (s0 )?F0 ] , one can see that the second term in the KL divergence is closely related to average heat dissipation in a finite time heating process (see e.g. (Crooks, 2000)). This intriguing connection between the size of the gap in a variational lower bound, and the excess heat dissipation in a finite time heating process opens the door to exploiting a wealth of work in statistical physics for finding optimal thermodynamic paths that minimize heat dissipation (Schmiedl and Seifert, 2007; Sivak and Crooks, 2012; Gingrich et al., 2016), which may provide new ideas to improve variational inference. In summary, tightness of the variational bound can be achieved if: (1) The transition operator of p approximately obeys detailed balance, and (2) the temperature annealing is done slowly over many steps. And intriguingly, the magnitude of the looseness of the bound is related to two physical quantities: (1) the degree of irreversiblity of the transition operator p, as measured by the KL divergence between p and its time reversal pR , and (2) the excess physical work, or equivalently, excess heat dissipated, in performing the heating trajectory. To check, post-hoc, potential looseness of the variational lower bound, we can measure the degree of irreversibility of pT by estimating the KL divergence DKL (pT (s0 |s)?T (s) || pT (s|s0 )?T (s0 )), which is 0 if and only if pT obeys detailed balance and is therefore time-reversal invariant. This quantity PK pT (st+1 |st ) 1 K can be estimated by K t=1 ln pT (st |st+1 ) , where s1 is a long sequence sampled by repeatedly applying transition operator pT from a draw s1 ? ?T . If this quantity is strongly positive (negative) then forward transitions are more (less) likely than reverse transitions, and the process pT is not time-reversal invariant. This estimated KL divergence can be normalized by the corresponding entropy to get a relative value (with 3.6% measured on a trained model, as detailed in Appendix). 3.2 Estimating log likelihood via importance sampling We can derive an importance sampling estimate of the negative log-likelihood by the following procedure. For each training example x, we sample a large number of destructive paths (as in Algorithm 1). We then use the following formulation to estimate the log-likelihood log p(x) via Q  ? ? K ? pT0 (s0 = x|s1 ) t=2 pTt (st?1 |st ) p (sK ) Q ? log Ex?pD ,qT (x)qT (s1 |s0 (x,))(QK qT (st |st?1 )) ? K t=2 t 0 1 q (s |s ) qT0 (x)qT1 (s1 |s0 = x) T t t?1 t t=2 (10) 3.3 VW transition operators and their convergence The VW approach allows considerable freedom in choosing transition operators, obviating the need for specifying them indirectly through an energy function. Here we consider Bernoulli and isotropic Gaussian transition operators for binary and real-valued data respectively. The form of the stochastic state update imitates a discretized version of the Langevin differential equation. The Bernoulli (1??)?st?1 +??F? (st?1 ) transition operator computes the element-wise probability as ? = sigmoid( ). Tt The Gaussian operator computes a conditional mean and standard deviation via ? = (1 ? ?) ? st?1 + ? ? F? (st?1 ) and ? = Tt log(1 + eF? (st?1 ) ). Here the F functions can be arbitrary parametrized functions, such as a neural net and Tt is the temperature at time step t. 6 A natural question is when will the finite time VW training process learn a transition operator whose stationary distribution matches the data distribution, so that repeated sampling far beyond the training time continues to yield data samples. To partially address this, we prove the following theorem: Proposition 1. If p has enough capacity, training data and training time, with slow enough annealing and a small departure from reversibility so p can match q, then at convergence of VW training, the transition operator pT at T = 1 has the data generating distribution as its stationary distribution. A proof can be found in the Appendix, but the essential intuition is that if the finite time generative process converges to the data distribution at multiple different VW walkback time-steps, then it remains on the data distribution for all future time at T = 1. We cannot always guarantee the preconditions of this theorem but we find experimentally that its essential outcome holds in practice. 4 Related Work A variety of learning algorithms can be cast in the framework of Fig. 1. For example, for directed graphical models like VAEs (Kingma and Welling, 2013; Rezende et al., 2014), DBNs (Hinton et al., 2006), and Helmholtz machines in general, q corresponds to a recognition model, transforming data to a latent space, while p corresponds to a generative model that goes from latent to visible data in a finite number of steps. None of these directed models are designed to learn transition operators that can be iterated ad infinitum, as we do. Moreover, learning such models involves a complex, deep credit assignment problem, limiting the number of unobserved latent layers that can be used to generate data. Similar issues of limited trainable depth in a finite time feedforward generative process apply to Generative Adversarial Networks (GANs) (Goodfellow et al., 2014), which also further eschew the goal of specifically assigning probabilities to data points. Our method circumvents this deep credit assignment problem by providing training targets at each time-step; in essence each past time-step of the heated trajectory constitutes a training target for the future output of the generative operator pT , thereby obviating the need for backpropagation across multiple steps. Similarly, unlike VW, Generative Stochastic Networks (GSN) (Bengio et al., 2014) and the DRAW (Gregor et al., 2015) also require training iterative operators by backpropagating across multiple computational steps. VW is similar in spirit to DAE (Bengio et al., 2013b), and NET approaches (Sohl-Dickstein et al., 2015) but it retains two crucial differences. First, in each of these frameworks, q corresponds to a very simple destruction process in which unstructured Gaussian noise is injected into the data. This agnostic destruction process has no knowledge of underlying generative process p that is to be learned, and therefore cannot be expected to efficiently explore spurious modes, or regions of space, unoccupied by data, to which p assigns high probability. VW has the advantage of using a high-temperature version of the model p itself as part of the destructive process, and so should be better than random noise injection at finding these spurious modes. A second crucial difference is that VW ties weights of the transition operator across time-steps, thereby enabling us to learn a bona fide transition operator than can be iterated well beyond the training time, unlike DAEs and NET. There?s also another related recent approach to learning a transition operator with a denoising cost, developed in parallel, called Infusion training (Bordes et al., 2017), which tries to reconstruct the target data in the chain, instead of the previous step in the destructive chain. 5 Experiments VW is evaluated on four datasets: MNIST, CIFAR10 (Krizhevsky and Hinton, 2009), SVHN (Netzer et al., 2011) and CelebA (Liu et al., 2015). The MNIST, SVHN and CIFAR10 datasets were used as is except for uniform noise added to MNIST and CIFAR10, as per Theis et al. (2016), and the aligned and cropped version of CelebA was scaled from 218 x 178 pixels to 78 x 64 pixels and center-cropped at 64 x 64 pixels (Liu et al., 2015). We used the Adam optimizer (Kingma and Ba, 2014) and the Theano framework (Al-Rfou et al., 2016). More details are in Appendix and code for training and generation is at http://github.com/anirudh9119/walkback_nips17. Table 1 compares with published NET results on CIFAR. Image Generation. Figure 3, 5, 6, 7, 8 (see supplementary section) show VW samples on each of the datasets. For MNIST, real-valued views of the data are modeled. Image Inpainting. We clamped the bottom part of CelebA test images (for each step during sampling), and ran it through the model. Figure 1 (see Supplementary section) shows the generated conditional samples. 7 Model bits/dim ? NET (Sohl-Dickstein et al., 2015) 5.40 VW(20 steps) 5.20 Deep VAE < 4.54 VW(30 steps) 4.40 DRAW (Gregor et al., 2015) < 4.13 ResNet VAE with IAF (Kingma et al., 2016) 3.11 Table 1: Comparisons on CIFAR10, test set average number of bits/data dimension(lower is better) 6 6.1 Discussion Summary of results Our main advance involves using variational inference to learn recurrent transition operators that can rapidly approach the data distribution and then be iterated much longer than the training time while still remaining on the data manifold. Our innovations enabling us to achieve this involved: (a) tying weights across time, (b) tying the destruction and generation process together to efficiently destroy spurious modes, (c) using the past of the destructive process to train the future of the creation process, thereby circumventing issues with deep credit assignment (like NET), (d) introducing an aggressive temperature annealing schedule to rapidly approach the data distribution (e.g. NET takes 1000 steps while VWB only takes 30 steps to do so), and (e) introducing variable trajectory lengths during training to encourage the generator to stay on the data manifold for times longer than the training sequence length. Indeed, it is often difficult to sample from recurrent neural networks for many more time steps than the duration of their training sequences, especially non-symmetric networks that could exhibit chaotic activity. Transition operators learned by VW can be stably sampled for exceedingly long times; for example, in experiments (see supplementary section) we trained our model on CelebA for 30 steps, while at test time we sampled for 100000 time-steps. Overall, our method of learning a transition operator outperforms previous attempts at learning transition operators (i.e. VAE, GSN and NET) using a local learning rule. Overall, we introduced a new approach to learning non-energy-based transition operators which inherits advantages from several previous generative models, including a training objective that requires rapidly generating the data in a finite number of steps (as in directed models), re-using the same parameters for each step (as in undirected models), directly parametrizing the generator (as in GANs and DAEs), and using the model itself to quickly find its own spurious modes (the walk-back idea). We also anchor the algorithm in a variational bound and show how its analysis suggests to use the same transition operator for the destruction or inference process, and the creation or generation process, and to use a cooling schedule during generation, and a reverse heating schedule during inference. 6.2 New bridges between variational inference and non-equilibrium statistical physics We connected the variational gap to physical notions like reversibility and heat dissipation. This novel bridge between variational inference and concepts like excess heat dissipation in non-equilbrium statistical physics, could potentially open the door to improving variational inference by exploiting a wealth of work in statistical physics. For example, physical methods for finding optimal thermodynamic paths that minimize heat dissipation (Schmiedl and Seifert, 2007; Sivak and Crooks, 2012; Gingrich et al., 2016), could potentially be exploited to tighten lowerbounds in variational inference. Moreover, motivated by the relation between the variational gap and reversibility, we verified empirically that the model converges towards an approximately reversible chain (see Appendix) making the variational bound tighter. 6.3 Neural weight asymmetry A fundamental aspect of our approach is that we can train stochastic processes that need not exactly 8 obey detailed balance, yielding access to a larger and potentially more powerful space of models. In particular, this enables us to relax the weight symmetry constraint of undirected graphical models corresponding to neural networks, yielding a more brain like iterative computation characteristic of asymmetric biological neural circuits. Our approach thus avoids the biologically implausible requirement of weight transport (Lillicrap et al., 2014) which arises as a consequence of imposing weight symmetry as a hard constraint. With VW, this hard constraint is removed, although the training procedure itself may converge towards more symmetry. Such approach towards symmetry is consistent with both empirical observations (Vincent et al., 2010) and theoretical analysis (Arora et al., 2015) of auto-encoders, for which symmetric weights are associated with minimizing reconstruction error. 6.4 A connection to the neurobiology of dreams The learning rule underlying VW, when applied to an asymmetric stochastic neural network, yields a speculative, but intriguing connection to the neurobiology of dreams. As discussed in Bengio et al. (2015), spike-timing dependent plasticity (STDP), a plasticity rule found in the brain (Markram and Sakmann, 1995), corresponds to increasing the probability of configurations towards which the network intrinsically likes to go (i.e., remembering observed configurations), while reverse-STDP corresponds to forgetting or unlearning the states towards which the network goes (which potentially may occur during sleep). In the VW update applied to a neural network, the resultant learning rule does indeed strengthen synapses for which a presynaptic neuron is active before a postsynaptic neuron in the generative cooling process (STDP), and it weakens synapses in which a postsynaptic neuron is active before a presynaptic neuron in the heated destructive process (reverse STDP). If, as suggested, the neurobiological function of sleep involves re-organizing memories and in particular unlearning spurious modes through reverse-STDP, then the heating destructive process may map to sleep states, in which the brain is hunting down and destroying spurious modes. In contrast, the cooling generative dynamics of VW may map to awake states in which STDP reinforces neural trajectories moving towards observed sensory data. Under this mapping, the relative incoherence of dreams compared to reality is qualitatively consistent with the heated destructive dynamics of VW, compared to the cooled transition operator in place during awake states. 6.5 Future work Many questions remain open in terms of analyzing and extending VW. Of particular interest is the incorporation of latent layers. The state at each step would now include both visible x and latent h components. Essentially the same procedure can be run, except for the chain initialization, with s0 = (x, h0 ) where h0 a sample from the posterior distribution of h given x. Another interesting direction is to replace the log-likelihood objective at each step by a GAN-like objective, thereby avoiding the need to inject noise independently on each of the pixels, during each transition step, and allowing latent variable sampling to inject the required high-level decisions associated with the transition. Based on the earlier results from (Bengio et al., 2013a), sampling in the latent space rather than in the pixel space should allow for better generative models and even better mixing between modes (Bengio et al., 2013a). Overall, our work takes a step to filling a relatively open niche in the machine learning literature on directly training non-energy-based iterative stochastic operators, and we hope that the many possible extensions of this approach could lead to a rich new class of more powerful brain-like machine learning models. Acknowledgments The authors would like to thank Benjamin Scellier, Ben Poole, Tim Cooijmans, Philemon Brakel, Ga?tan Marceau Caron, and Alex Lamb for their helpful feedback and discussions, as well as NSERC, CIFAR, Google, Samsung, Nuance, IBM and Canada Research Chairs for funding, and Compute Canada for computing resources. S.G. would like to thank the Simons, McKnight, James S. McDonnell, and Burroughs Wellcome Foundations and the Office of Naval Research for support. Y.B would also like to thank Geoff Hinton for an analogy which is used in this work, while discussing contrastive divergence (personnal communication). 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Polynomial Codes: an Optimal Design for High-Dimensional Coded Matrix Multiplication ? Qian Yu? , Mohammad Ali Maddah-Ali? , and A. Salman Avestimehr? Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA ? Nokia Bell Labs, Holmdel, NJ, USA Abstract We consider a large-scale matrix multiplication problem where the computation is carried out using a distributed system with a master node and multiple worker nodes, where each worker can store parts of the input matrices. We propose a computation strategy that leverages ideas from coding theory to design intermediate computations at the worker nodes, in order to optimally deal with straggling workers. The proposed strategy, named as polynomial codes, achieves the optimum recovery threshold, defined as the minimum number of workers that the master needs to wait for in order to compute the output. This is the first code that achieves the optimal utilization of redundancy for tolerating stragglers or failures in distributed matrix multiplication. Furthermore, by leveraging the algebraic structure of polynomial codes, we can map the reconstruction problem of the final output to a polynomial interpolation problem, which can be solved efficiently. Polynomial codes provide order-wise improvement over the state of the art in terms of recovery threshold, and are also optimal in terms of several other metrics including computation latency and communication load. Moreover, we extend this code to distributed convolution and show its order-wise optimality. 1 Introduction Matrix multiplication is one of the key building blocks underlying many data analytics and machine learning algorithms. Many such applications require massive computation and storage power to process large-scale datasets. As a result, distributed computing frameworks such as Hadoop MapReduce [1] and Spark [2] have gained significant traction, as they enable processing of data sizes at the order of tens of terabytes and more. As we scale out computations across many distributed nodes, a major performance bottleneck is the latency in waiting for slowest nodes, or ?stragglers? to finish their tasks [3]. The current approaches to mitigate the impact of stragglers involve creation of some form of ?computation redundancy?. For example, replicating the straggling task on another available node is a common approach to deal with stragglers (e.g., [4]). However, there have been recent results demonstrating that coding can play a transformational role for creating and exploiting computation redundancy to effectively alleviate the impact of stragglers [5, 6, 7, 8, 9]. Our main result in this paper is the development of optimal codes, named polynomial codes, to deal with stragglers in distributed high-dimensional matrix multiplication, which also provides order-wise improvement over the state of the art. More specifically, we consider a distributed matrix multiplication problem where we aim to compute C = A| B from input matrices A and B. As shown in Fig. 1, the computation is carried out using 1 a distributed system with a master node and N worker nodes that can each store m fraction of A 1 + and n fraction of B, for some parameters m, n ? N . We denote the stored submtarices at each 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. ?i , which can be designed as arbitrary functions of A and B worker i ? {0, . . . , N ? 1} by A?i and B ?i and returns the result to the master. respectively. Each worker i then computes the product A?|i B ... ... ... Figure 1: Overview of the distributed matrix multiplication framework. Coded data are initially stored distributedly at N workers according to data assignment. Each worker computes the product of the two stored matrices and returns it to the master. By carefully designing the computation strategy, the master can decode given the computing results from a subset of workers, without having to wait for the stragglers (worker 1 in this example). ?i ), the master By carefully designing the computation strategy at each worker (i.e. designing A?i and B only needs to wait for the fastest subset of workers before recovering output C, hence mitigating the impact of stragglers. Given a computation strategy, we define its recovery threshold as the minimum number of workers that the master needs to wait for in order to compute C. In other words, if any subset of the workers with size no smaller than the recovery threshold finish their jobs, the master is able to compute C. Given this formulation, we are interested in the following main problem. What is the minimum possible recovery threshold for distributed matrix multiplication? Can we find an optimal computation strategy that achieves the minimum recovery threshold, while allowing efficient decoding of the final output at the master node? There have been two computing schemes proposed earlier for this problem that leverage ideas from coding theory. The first one, introduced in [5] and extended in [10], injects redundancy in only one of the input matrices using maximum distance separable (MDS) codes [11] 1 . We illustrate this approach, referred to as one dimensional MDS code (1D MDS code), using the example shown in Fig. 2a, where we aim to compute C = A| B using 3 workers that can each store half of A and the entire B. The 1D MDS code evenly divides A along the column into two submatrices denoted by A0 and A1 , encodes them into 3 coded matrices A0 , A1 , and A0 + A1 , and then assigns them to the 3 workers. This design allows the master to recover the final output given the results from any 2 of the 3 workers, hence achieving a recovery threshold of 2. More generally, one can show that the 1D MDS code achieves a recovery threshold of N + m = ?(N ). (1) n An alternative computing scheme was recently proposed in [10] for the case of m = n, referred to as the product code, which instead injects redundancy in both input matrices. This coding technique has also been proposed earlier in the context of?Fault Tolerant ? Computing in [12, 13]. As demonstrated in Fig. 2b, product code aligns workers?in an N ?by? N layout. ? A is divided along the columns into m submatrices, encoded using an ( N , m) MDS code into N coded matrices, and then assigned ? ? to the N columns of workers. Similarly N coded matrices of B are created and assigned to the ? N rows. Given the property of MDS codes, the master can decode an entire row after obtaining any m results in that row; likewise for the columns. Consequently, the master can recover the final output using a peeling algorithm, iteratively decoding the MDS codes on rows and columns until the output C is completely available. For example, if the 5 computing results A|1 B0 , A|1 B1 , (A0 + A1 )| B1 , A|0 (B0 + B1 ), and A|1 (B0 + B1 ) are received as demonstrated in Fig. 2b, the master can recover the K1D-MDS , N ? 1 An (n, k) MDS code is a linear code which transforms k raw inputs to n coded outputs, such that from any subset of size k of the outputs, the original k inputs can be recovered. 2 needed results by computing A|0 B1 = (A0 + A1 )| B1 ? A|1 B1 then A|0 B0 = A|0 (B0 + B1 ) ? A|0 B1 . In general, one can show that the product code achieves a recovery threshold of ? ? Kproduct , 2(m ? 1) N ? (m ? 1)2 + 1 = ?( N ), (2) which significantly improves over K1D-MDS . (a) 1D MDS-code [5] in an example with 3 workers that can each store half of A and the entire B. (b) Product code [10] in an example with 9 workers that can each store half of A and half of B. Figure 2: Illustration of (a) 1D MDS code, and (b) product code. In this paper, we show that quite interestingly, the optimum recovery threshold can be far less than what the above two schemes achieve. In fact, we show that the minimum recovery threshold does not scale with the number of workers (i.e. ?(1)). We prove this fact by designing a novel coded computing strategy, referred to as the polynomial code, which achieves the optimum recovery threshold of mn, and significantly improves the state of the art. Hence, our main result is as follows. For a general matrix multiplication task C = A| B using N workers, where each worker can 1 store m fraction of A and n1 fraction of B, we propose polynomial codes that achieve the optimum recovery threshold of Kpoly , mn = ?(1). (3) Furthermore, polynomial code only requires a decoding complexity that is almost linear to the input size. The main novelty and advantage of the proposed polynomial code is that, by carefully designing the algebraic structure of the encoded submatrices, we ensure that any mn intermediate computations at the workers are sufficient for recovering the final matrix multiplication product at the master. This in a sense creates an MDS structure on the intermediate computations, instead of only the encoded matrices as in prior works. Furthermore, by leveraging the algebraic structure of polynomial codes, we can then map the reconstruction problem of the final output at the master to a polynomial interpolation problem (or equivalently Reed-Solomon decoding [14]), which can be solved efficiently [15]. This mapping also bridges the rich theory of algebraic coding and distributed matrix multiplication. We prove the optimality of polynomial code by showing that it achieves the information theoretic lower bound on the recovery threshold, obtained by cut-set arguments (i.e., we need at least mn matrix blocks returned from workers to recover the final output, which exactly have size mn blocks). Hence, the proposed polynomial code essentially enables a specific computing strategy such that, from any subset of workers that give the minimum amount of information needed to recover C, the master can successfully decode the final output. As a by-product, we also prove the optimality of polynomial code under several other performance metrics considered in previous literature: computation latency [5, 10], probability of failure given a deadline [9], and communication load [16, 17, 18]. We extend the polynomial code to the problem of distributed convolution [9]. We show that by simply reducing the convolution problem to matrix multiplication and applying the polynomial code, we strictly and unboundedly improve the state of the art. Furthermore, by exploiting the computing structure of convolution, we propose a variation of the polynomial code, which strictly reduces the recovery threshold even further, and achieves the optimum recovery threshold within a factor of 2. Finally, we implement and benchmark the polynomial code on an Amazon EC2 cluster. We measure the computation latency and empirically demonstrate its performance gain under straggler effects. 3 2 System Model, Problem Formulation, and Main Result We consider a problem of matrix multiplication with two input matrices A ? Fs?r and B ? Fs?t q q , for some integers r, s, t and a sufficiently large finite field Fq . We are interested in computing the product C , A| B in a distributed computing environment with a master node and N worker nodes, 1 where each worker can store m fraction of A and n1 fraction of B, for some parameters m, n ? N+ (see Fig. 1). We assume at least one of the two input matrices A and B is tall (i.e. s ? r or s ? t), because otherwise the output matrix C would be rank inefficient and the problem is degenerated. r t s? n ?i ? Fs? Specifically, each worker i can store two matrices A?i ? Fq m and B , computed based q ?i , on arbitrary functions of A and B respectively. Each worker can compute the product C?i , A?|i B and return it to the master. The master waits only for the results from a subset of workers, before proceeding to recover the final output C given these products using certain decoding functions.2 2.1 Problem Formulation Given the above system model, we formulate the distributed matrix multiplication problem based on the following terminology: We define the computation strategy as the 2N functions, denoted by f = (f0 , f1 , ..., fN ?1 ), g = (g0 , g1 , ..., gN ?1 ), (4) ?i . Specifically, that are used to compute each A?i and B A?i = fi (A), ?i = gi (B), B ? i ? {0, 1, ..., N ? 1}. (5) For any integer k, we say a computation strategy is k-recoverable if the master can recover C given the computing results from any k workers. We define the recovery threshold of a computation strategy, denoted by k(f , g), as the minimum integer k such that computation strategy (f , g) is k-recoverable. Using the above terminology, we define the following concept: Definition 1. For a distributed matrix multiplication problem of computing A| B using N workers 1 that can each store m fraction of A and n1 fraction of B, we define the optimum recovery threshold, ? denoted by K , as the minimum achievable recovery threshold among all computation strategies, i.e. K ? , min k(f , g). f ,g (6) The goal of this problem is to find the optimum recovery threshold K ? , as well as a computation strategy that achieves such an optimum threshold. 2.2 Main Result Our main result is stated in the following theorem: Theorem 1. For a distributed matrix multiplication problem of computing A| B using N workers 1 that can each store m fraction of A and n1 fraction of B, the minimum recovery threshold K ? is K ? = mn. (7) Furthermore, there is a computation strategy, referred to as the polynomial code, that achieves the above K ? while allowing efficient decoding at the master node, i.e., with complexity equal to that of polynomial interpolation given mn points. Remark 1. Compared to the state of the art [5, 10], the polynomial code provides order-wise improvement in terms of the recovery threshold. Specifically, the?recovery thresholds achieved by 1D MDS code [19] and product code [10] scale linearly with N and N respectively, while the proposed polynomial code actually achieves a recovery threshold that does not scale with N . Furthermore, polynomial code achieves the optimal recovery threshold. To the best of our knowledge, this is the first optimal design proposed for the distributed matrix multiplication problem. 2 Note that we consider the most general model and do not impose any constraints on the decoding functions. However, any good decoding function should have relatively low computation complexity. 4 Remark 2. We prove the optimality of polynomial code using a matching information theoretic lower bound, which is obtained by applying a cut-set type argument around the master node. As a by-product, we can also prove that the polynomial code simultaneously achieves optimality in terms of several other performance metrics, including the computation latency [5, 10], the probability of failure given a deadline [9], and the communication load [16, 17, 18], as discussed in Section 3.4. Remark 3. The polynomial code not only improves the state of the art asymptotically, but also gives strict and significant improvement for any parameter values of N , m, and n (See Fig. 3 for example). Figure 3: Comparison of the recovery thresholds achieved by the proposed polynomial code and the state of the 1 arts (1D MDS code [5] and product code [10]), where each worker can store 10 fraction of each input matrix. ? The polynomial code attains the optimum recovery threshold K , and significantly improves the state of the art. Remark 4. As we will discuss in Section 3.2, decoding polynomial code can be mapped to a polynomial interpolation problem, which can be solved in time almost linear to the input size [15]. This is enabled by carefully designing the computing strategies at the workers, such that the computed products form a Reed-Solomon code [20] , which can be decoded efficiently using any polynomial interpolation algorithm or Reed-Solomon decoding algorithm that provides the best performance depending on the problem scenario (e.g., [21]). Remark 5. Polynomial code can be extended to other distributed computation applications involving linear algebraic operations. In Section 4, we focus on the problem of distributed convolution, and show that we can obtain order-wise improvement over the state of the art (see [9]) by directly applying the polynomial code. Furthermore, by exploiting the computing structure of convolution, we propose a variation of the polynomial code that achieves the optimum recovery threshold within a factor of 2. Remark 6. In this work we focused on designing optimal coding techniques to handle stragglers issues. The same technique can also be applied to the fault tolerance computing setting (e.g., within the algorithmic fault tolerance computing framework of [12, 13], where a module can produce arbitrary error results under failure), to improve robustness to failures in computing. Specifically, given that the polynomial code produces computing results that are coded by Reed-Solomon code, which has the optimum hamming distance, it allows detecting, or correcting the maximum possible number of module errors. This provides the first optimum code for matrix multiplication under fault tolerance computing. 3 Polynomial Code and Its Optimality In this section, we formally describe the polynomial code and its decoding procedure. We then prove its optimality with an information theoretic converse, which completes the proof of Theorem 1. Finally, we conclude this section with the optimality of polynomial code under other settings. 3.1 Motivating Example We first demonstrate the main idea through a motivating example. Consider a distributed matrix multiplication task of computing C = A| B using N = 5 workers that can each store half of the matrices (see Fig. 4). We evenly divide each input matrix along the column side into 2 submatrices: A = [A0 A1 ], B = [B0 B1 ]. Given this notation, we essentially want to compute the following 4 uncoded components:  |  A0 B0 A|0 B1 C = A| B = . | | A1 B 0 A1 B 1 5 (8) (9) Figure 4: Example using polynomial code, with 5 workers that can each store half of each input matrix. (a) Computation strategy: each worker i stores A0 + iA1 and B0 + i2 B1 , and computes their product. (b) Decoding: master waits for results from any 4 workers, and decodes the output using fast polynomial interpolation algorithm. Now we design a computation strategy to achieve the optimum recovery threshold of 4. Suppose elements of A, B are in F7 , let each worker i ? {0, 1, ..., 4} store the following two coded submatrices: A?i = A0 + iA1 , ?i = B0 + i2 B1 . B (10) To prove that this design gives a recovery threshold of 4, we need to design a valid decoding function for any subset of 4 workers. We demonstrate this decodability through a representative scenario, where the master receives the computation results from workers 1, 2, 3, and 4, as shown in Figure 4. The decodability for the other 4 possible scenarios can be proved similarly. According to the designed computation strategy, we have ? ? ? ?? | ? C?1 10 11 12 13 A0 B 0 ?C?2 ? ?20 21 22 23 ? ?A| B0 ? 1 ? ?=? 0 . | ?C?3 ? 3 31 32 33 ? ?A0 B1 ? | 0 1 2 3 A1 B 1 4 4 4 4 C?4 (11) The coefficient matrix in the above equation is a Vandermonde matrix, which is invertible because its parameters 1, 2, 3, 4 are distinct in F7 . So one way to recover C is to directly invert equation (11), which proves the decodability. However, directly computing this inverse using the classical inversion algorithm might be expensive in more general cases. Quite interestingly, because of the algebraic structure we designed for the computation strategy (i.e., equation (10)), the decoding process can be viewed as a polynomial interpolation problem (or equivalently, decoding a Reed-Solomon code). Specifically, in this example each worker i returns ?i = A| B0 + iA| B0 + i2 A| B1 + i3 A| B1 , C?i = A?|i B 0 1 0 1 (12) which is essentially the value of the following polynomial at point x = i: h(x) , A|0 B0 + xA|1 B0 + x2 A|0 B1 + x3 A|1 B1 . (13) Hence, recovering C using computation results from 4 workers is equivalent to interpolating a 3rddegree polynomial given its values at 4 points. Later in this section, we will show that by mapping the decoding process to polynomial interpolation, we can achieve almost-linear decoding complexity. 3.2 General Polynomial Code Now we present the polynomial code in a general setting that achieves the optimum recovery threshold stated in Theorem 1 for any parameter values of N , m, and n. First of all, we evenly divide each input matrix along the column side into m and n submatrices respectively, i.e., A = [A0 A1 ... Am?1 ], B = [B0 B1 ... Bn?1 ], (14) We then assign each worker i ? {0, 1, ..., N ? 1} a number in Fq , denoted by xi , and make sure that all xi ?s are distinct. Under this setting, we define the following class of computation strategies. 6 Definition 2. Given parameters ?, ? ? N, we define the (?, ?)-polynomial code as A?i = m?1 X Aj xj? i , ?i = B j=0 n?1 X Bj xj? i , ? i ? {0, 1, ..., N ? 1}. (15) j=0 In an (?, ?)-polynomial code, each worker i essentially computes ?i = C?i = A?|i B m?1 X n?1 X A|j Bk xj?+k? . i (16) j=0 k=0 In order for the master to recover the output given any mn results (i.e. achieve the optimum recovery threshold), we carefully select the design parameters ? and ?, while making sure that no two terms in the above formula has the same exponent of x. One such choice is (?, ?) = (1, m), i.e, let A?i = m?1 X Aj xji , ?i = B j=0 n?1 X Bj xjm i . (17) j=0 Hence, each worker computes the value of the following degree mn ? 1 polynomial at point x = xi : h(x) , m?1 X n?1 X A|j Bk xj+km , (18) j=0 k=0 where the coefficients are exactly the mn uncoded components of C. Since all xi ?s are selected to be distinct, recovering C given results from any mn workers is essentially interpolating h(x) using mn distinct points. Since h(x) has degree mn ? 1, the output C can always be uniquely decoded. In terms of complexity, this decoding process can be viewed as interpolating degree mn ? 1 polynort mials of Fq for mn times. It is well known that polynomial interpolation of degree k has a complexity 2 of O(k log k log log k) [15]. Therefore, decoding polynomial code also only requires a complexity of O(rt log2 (mn) log log(mn)). Furthermore, this complexity can be reduced by simply swapping in any faster polynomial interpolation algorithm or Reed-Solomon decoding algorithm. Remark 7. We can naturally extend polynomial code to the scenario where input matrix elements are real or complex numbers. In practical implementation, to avoid handling large elements in the coefficient matrix, we can first quantize input values into numbers of finite digits, embed them into a finite field that covers the range of possible values of the output matrix elements, and then directly apply polynomial code. By embedding into finite fields, we avoid large intermediate computing results, which effectively saves storage and computation time, and reduces numerical errors. 3.3 Optimality of Polynomial Code for Recovery Threshold So far we have constructed a computing scheme that achieves a recovery threshold of mn, which upper bounds K ? . To complete the proof of Theorem 1, here we establish a matching lower bound through an information theoretic converse. We need to prove that for any computation strategy, the master needs to wait for at least mn workers in order to recover the output. Recall that at least one of A and B is a tall matrix. Without loss of generality, assume A is tall (i.e. s ? r). Let A be an arbitrary fixed full rank matrix and B be sampled from Fs?t uniformly at random. It is easy to show that C = A| B is uniformly distributed q on Fr?t q . This means that the master essentially needs to recover a random variable with entropy rt of H(C) = rt log2 q bits. Note that each worker returns mn elements of Fq , providing at most rt log q bits of information. Consequently, using a cut-set bound around the master, we can show 2 mn that at least mn results from the workers need to be collected, and thus we have K ? ? mn. Remark 8 (Random Linear Code). We conclude this subsection by noting that, another computation design is to let each worker store two random linear combinations of the input submatrices. Although this design can achieve the optimal recovery threshold with high probability, it creates a large coding overhead and requires high decoding complexity (e.g., O(m3 n3 + mnrt) using the classical inversion decoding algorithm). Compared to random linear code, the proposed polynomial code achieves the optimum recovery threshold deterministically, with a significantly lower decoding complexity. 7 3.4 Optimality of Polynomial Code for Other Performance Metrics In the previous subsection, we proved that polynomial code is optimal in terms of the recovery threshold. As a by-product, we can prove that it is also optimal in terms of some other performance metrics. In particular, we consider the following 3 metrics considered in prior works, and formally establish the optimality of polynomial code for each of them. Proofs can be found in Appendix A. Computation latency is considered in models where the computation time Ti of each worker i is a random variable with a certain probability distribution (e.g, [5, 10]). The computation latency is defined as the amount of time required for the master to collect enough information to decode C. Theorem 2. For any computation strategy, the computation latency T is always no less than the latency achieved by polynomial code, denoted by Tpoly . Namely, T ? Tpoly . (19) Probability of failure given a deadline is defined as the probability that the master does not receive enough information to decode C at any time t [9]. Corollary 1. For any computation strategy, let T denote its computation latency, and let Tpoly denote the computation latency of polynomial code. We have P(T > t) ? P(Tpoly > t) ? t ? 0. (20) Corollary 1 directly follows from Theorem 2 since (19) implies (20) . Communication load is another important metric in distributed computing (e.g. [16, 17, 18]), defined as the minimum number of bits needed to be communicated in order to complete the computation. Theorem 3. Polynomial code achieves the minimum communication load for distributed matrix multiplication, which is given by L? = rt log2 q. 4 (21) Extension to Distributed Convolution We can extend our proposed polynomial code to distributed convolution. Specifically, we consider a convolution task with two input vectors a = [a0 a1 ... am?1 ], b = [b0 b1 ... bn?1 ], (22) where all ai ?s and bi ?s are vectors of length s over a sufficiently large field Fq . We want to compute c , a ? b using a master and N workers. Each worker can store two vectors of length s, which are functions of a and b respectively. We refer to these functions as the computation strategy. Each worker computes the convolution of its stored vectors, and returns it to the master. The master only waits for the fastest subset of workers, before proceeding to decode c. Similar to distributed matrix multiplication, we define the recovery threshold for each computation strategy. We aim to ? characterize the optimum recovery threshold denoted by Kconv , and find computation strategies that closely achieve this optimum threshold, while allowing efficient decoding at the master. Distributed convolution has also been studied in [9], where the coded convolution scheme was proposed. The main idea of the coded convolution scheme is to inject redundancy in only one of the input vectors using MDS codes. The master waits for enough results such that all intermediate values ai ? bj can be recovered, which allows the final output to be computed. One can show that this coded convolution scheme is in fact equivalent to the 1D MDS-coded scheme proposed in [10]. Consequently, it achieves a recovery threshold of K1D-MDS = N ? N n + m. Note that by simply adapting our proposed polynomial code designed for distributed matrix multiplication to distributed convolution, the master can recover all intermediate values ai ? bj after receiving results from any mn workers, to decode the final output. Consequently, this achieves a recovery threshold of Kpoly = mn, which already strictly and significantly improves the state of the art. In this paper, we take one step further and propose an improved computation strategy, strictly reducing the recovery threshold on top of the naive polynomial code. The result is summarized as follows: 8 Theorem 4. For a distributed convolution problem of computing a ? b using N workers that can 1 each store m fraction of a and n1 fraction of b, we can find a computation strategy that achieves a recovery threshold of Kconv-poly , m + n ? 1. (23) Furthermore, this computation strategy allows efficient decoding, i.e., with complexity equal to that of polynomial interpolation given m + n ? 1 points. We prove Theorem 4 by proposing a variation of the polynomial code, which exploits the computation structure of convolution. This new computing scheme is formally demonstrated in Appendix B. Remark 9. Similar to distributed matrix multiplication, our proposed computation strategy provides order-wise improvement compared to the state of the art [9] in many different settings. Furthermore, it achieves almost-linear decoding complexity using the fastest polynomial interpolation algorithm or the Reed-Solomon decoding algorithm. Moreover, we characterize Kconv within a factor of 2, as stated in the following theorem and proved in Appendix C. ? Theorem 5. For a distributed convolution problem, the minimum recovery threshold Kconv can be characterized within a factor of 2, i.e.: 1 ? Kconv-poly < Kconv ? Kconv-poly . (24) 2 5 Experiment Results To examine the efficiency of our proposed polynomial code, we implement the algorithm in Python using the mpi4py library and deploy it on an AWS EC2 cluster of 18 nodes, with the master running on a c1.medium instance, and 17 workers running on t2.micro instances. The input matrices are randomly generated as two numpy matrices of size 4000 by 4000, and then encoded and assigned to the workers in the preprocessing stage. Each worker stores 14 fraction of each input matrix. In the computation stage, each worker computes the product of their assigned matrices, and then returns the result using MPI.Comm.Isend(). The master actively listens to responses from the 17 worker nodes through MPI.Comm.Irecv(), and uses MPI.Request.Waitany() to keep polling for the earliest fulfilled request. Upon receiving 16 responses, the master stops listening and starts decoding the result. To achieve the best performance, we implement an FFT-based algorithm for the Reed-Solomon decoding. Figure 5: Comparison of polynomial code and the uncoded scheme. We implement polynomial code and the uncoded scheme using Python and mpi4py library and deploy them on an Amazon EC2 cluster of 18 instances. We measure the computation latency of both algorithms and plot their CCDF. Polynomial code can reduce the tail latency by 34% even taking into account of the decoding overhead. We compare our results with distributed matrix multiplication without coding.3 The uncoded implementation is similar, except that only 16 out of the 17 workers participate in the computation, each of them storing and processing 14 fraction of uncoded rows from each input matrix. The master waits for all 16 workers to return, and does not need to perform any decoding algorithm to recover the result. To simulate straggler effects in large-scale systems, we compare the computation latency of these two schemes in a setting where a randomly picked worker is running a background thread which approximately doubles the computation time. As shown in Fig. 5, polynomial code can reduce the tail latency by 34% in this setting, even taking into account of the decoding overhead. 3 Due to the EC2 instance request quota limit of 20, 1D MDS code and product code could not be implemented in this setting, which require at least 21 and 26 nodes respectively. 9 6 Acknowledgement This work is in part supported by NSF grants CCF-1408639, NETS-1419632, ONR award N000141612189, NSA grant, and a research gift from Intel. This material is based upon work supported by Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001117C0053. The views, opinions, and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. 10 References [1] J. Dean and S. Ghemawat, ?MapReduce: Simplified data processing on large clusters,? Sixth USENIX Symposium on Operating System Design and Implementation, Dec. 2004. [2] M. Zaharia, M. Chowdhury, M. J. Franklin, S. Shenker, and I. Stoica, ?Spark: cluster computing with working sets,? in Proceedings of the 2nd USENIX HotCloud, vol. 10, p. 10, June 2010. [3] J. Dean and L. A. Barroso, ?The tail at scale,? Communications of the ACM, vol. 56, no. 2, pp. 74?80, 2013. [4] M. Zaharia, A. Konwinski, A. D. Joseph, R. H. Katz, and I. Stoica, ?Improving MapReduce performance in heterogeneous environments,? OSDI, vol. 8, p. 7, Dec. 2008. [5] K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, ?Speeding up distributed machine learning using codes,? e-print arXiv:1512.02673, 2015. [6] S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, ?A unified coding framework for distributed computing with straggling servers,? arXiv preprint arXiv:1609.01690, 2016. [7] A. Reisizadehmobarakeh, S. Prakash, R. Pedarsani, and S. Avestimehr, ?Coded computation over heterogeneous clusters,? arXiv preprint arXiv:1701.05973, 2017. [8] R. Tandon, Q. Lei, A. G. Dimakis, and N. Karampatziakis, ?Gradient coding,? arXiv preprint arXiv:1612.03301, 2016. [9] S. Dutta, V. Cadambe, and P. Grover, ?Coded convolution for parallel and distributed computing within a deadline,? arXiv preprint arXiv:1705.03875, 2017. [10] K. Lee, C. Suh, and K. Ramchandran, ?High-dimensional coded matrix multiplication,? in 2017 IEEE International Symposium on Information Theory (ISIT), pp. 2418?2422, June 2017. [11] R. Singleton, ?Maximum distance q-nary codes,? IEEE Transactions on Information Theory, vol. 10, no. 2, pp. 116?118, 1964. [12] K.-H. Huang and J. A. Abraham, ?Algorithm-based fault tolerance for matrix operations,? IEEE Transactions on Computers, vol. C-33, pp. 518?528, June 1984. [13] J.-Y. Jou and J. A. Abraham, ?Fault-tolerant matrix arithmetic and signal processing on highly concurrent computing structures,? Proceedings of the IEEE, vol. 74, pp. 732?741, May 1986. [14] F. Didier, ?Efficient erasure decoding of reed-solomon codes,? arXiv preprint arXiv:0901.1886, 2009. [15] K. S. Kedlaya and C. Umans, ?Fast polynomial factorization and modular composition,? SIAM Journal on Computing, vol. 40, no. 6, pp. 1767?1802, 2011. [16] S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, ?Coded MapReduce,? 53rd Annual Allerton Conference on Communication, Control, and Computing, Sept. 2015. [17] S. Li, M. A. Maddah-Ali, Q. Yu, and A. S. Avestimehr, ?A fundamental tradeoff between computation and communication in distributed computing,? e-print arXiv:1604.07086. Submitted to IEEE Transactions on Information Theory, 2016. [18] Q. Yu, S. Li, M. A. Maddah-Ali, and A. S. Avestimehr, ?How to optimally allocate resources for coded distributed computing?,? arXiv preprint arXiv:1702.07297, 2017. [19] F. Le Gall, ?Powers of tensors and fast matrix multiplication,? in Proceedings of the 39th international symposium on symbolic and algebraic computation, pp. 296?303, ACM, 2014. [20] R. Roth, Introduction to coding theory. Cambridge University Press, 2006. [21] S. Baktir and B. Sunar, ?Achieving efficient polynomial multiplication in fermat fields using the fast fourier transform,? in Proceedings of the 44th annual Southeast regional conference, pp. 549?554, ACM, 2006. 11
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Unsupervised Learning of Disentangled Representations from Video Emily Denton Department of Computer Science New York University [email protected] Vighnesh Birodkar Department of Computer Science New York University [email protected] Abstract We present a new model D R N ET that learns disentangled image representations from video. Our approach leverages the temporal coherence of video and a novel adversarial loss to learn a representation that factorizes each frame into a stationary part and a temporally varying component. The disentangled representation can be used for a range of tasks. For example, applying a standard LSTM to the time-vary components enables prediction of future frames. We evaluate our approach on a range of synthetic and real videos, demonstrating the ability to coherently generate hundreds of steps into the future. 1 Introduction Unsupervised learning from video is a long-standing problem in computer vision and machine learning. The goal is to learn, without explicit labels, a representation that generalizes effectively to a previously unseen range of tasks, such as semantic classification of the objects present, predicting future frames of the video or classifying the dynamic activity taking place. There are several prevailing paradigms: the first, known as self-supervision, uses domain knowledge to implicitly provide labels (e.g. predicting the relative position of patches on an object [4] or using feature tracks [36]). This allows the problem to be posed as a classification task with self-generated labels. The second general approach relies on auxiliary action labels, available in real or simulated robotic environments. These can either be used to train action-conditional predictive models of future frames [2, 20] or inversekinematics models [1] which attempt to predict actions from current and future frame pairs. The third and most general approaches are predictive auto-encoders (e.g.[11, 12, 18, 31]) which attempt to predict future frames from current ones. To learn effective representations, some kind of constraint on the latent representation is required. In this paper, we introduce a form of predictive auto-encoder which uses a novel adversarial loss to factor the latent representation for each video frame into two components, one that is roughly time-independent (i.e. approximately constant throughout the clip) and another that captures the dynamic aspects of the sequence, thus varying over time. We refer to these as content and pose components, respectively. The adversarial loss relies on the intuition that while the content features should be distinctive of a given clip, individual pose features should not. Thus the loss encourages pose features to carry no information about clip identity. Empirically, we find that training with this loss to be crucial to inducing the desired factorization. We explore the disentangled representation produced by our model, which we call DisentangledRepresentation Net (D R N ET ), on a variety of tasks. The first of these is predicting future video frames, something that is straightforward to do using our representation. We apply a standard LSTM model to the pose features, conditioning on the content features from the last observed frame. Despite the simplicity of our model relative to other video generation techniques, we are able to generate convincing long-range frame predictions, out to hundreds of time steps in some instances. This is significantly further than existing approaches that use real video data. We also show that D R N ET can 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. be used for classification. The content features capture the semantic content of the video thus can be used to predict object identity. Alternately, the pose features can be used for action prediction. 2 Related work On account of its natural invariances, image data naturally lends itself to an explicit ?what? and ?where? representation. The capsule model of Hinton et al. [10] performed this separation via an explicit auto-encoder structure. Zhao et al. [40] proposed a multi-layered version, which has similarities to ladder networks [23]. Several weakly supervised approaches have been proposed to factor images into style and content (e.g. [19, 24]). These methods all operate on static images, whereas our approach uses temporal structure to separate the components. Factoring video into time-varying and time-independent components has been explored in many settings. Classic structure-from-motion methods use an explicit affine projection model to extract a 3D point cloud and camera homography matrices [8]. In contrast, Slow Feature Analysis [38] has no model, instead simply penalizing the rate of change in time-independent components and encouraging their decorrelation. Most closely related to ours is Villegas et al. [33] which uses an unsupervised approach to factoring video into content and motion. Their architecture is also broadly similar to ours, but the loss functions differ in important ways. They rely on pixel/gradient space `p -norm reconstructions, plus a GAN term [6] that encourages the generated frames to be sharp. We also use an `2 pixel-space reconstruction. However, this pixel-space loss is only applied, in combination with a novel adversarial term applied to the pose features, to learn the disentangled representation. In contrast to [33], our forward model acts on latent pose vectors rather than predicting pixels directly. Other approaches explore general methods for learning disentangled representations from video. Kulkarni et al. [14] show how explicit graphics code can be learned from datasets with systematic dimensions of variation. Whitney et al. [37] use a gating principle to encourage each dimension of the latent representation to capture a distinct mode of variation. Grathwohl et al. [7] propose a deep variational model to disentangle space and time in video sequences. A range of generative video models, based on deep nets, have recently been proposed. Ranzato et al. [22] adopt a discrete vector quantization approach inspired by text models. Srivastava et al. [31] use LSTMs to generate entire frames. Video Pixel Networks [12] use these models is a conditional manner, generating one pixel at a time in raster-scan order (similar image models include [27, 32]). Finn et al. [5] use an LSTM framework to model motion via transformations of groups of pixels. Cricri et al. [3] use a ladder of stacked-autoencoders. Other works predict optical flows fields that can be used to extrapolate motion beyond the current frame, e.g. [17, 39, 35]. In contrast, a single pose vector is predicted in our model, rather than a spatial field. Chiappa et al. [2] and Oh et al. [20] focus on prediction in video game environments, where known actions at each frame can be permit action-conditional generative models that can give accurate long-range predictions. In contrast to the above works, whose latent representations combine both content and motion, our approach relies on a factorization of the two, with a predictive model only being applied to the latter. Furthermore, we do not attempt to predict pixels directly, instead applying the forward model in the latent space. Chiappa et al. [2], like our approach, produces convincing long-range generations. However, the video game environment is somewhat more constrained than the real-world video we consider since actions are provided during generation. Several video prediction approaches have been proposed that focus on handling the inherent uncertainty in predicting the future. Mathieu et al. [18] demonstrate that a loss based on GANs can produce sharper generations than traditional `2 -based losses. [34] train a series of models, which aim to span possible outcomes and select the most likely one at any given instant. While we considered GANbased losses, the more constrained nature of our model, and the fact that our forward model does not directly generate in pixel-space, meant that standard deterministic losses worked satisfactorily. 3 Approach In our model, two separate encoders produce distinct feature representations of content and pose for each frame. They are trained by requiring that the content representation of frame xt and the pose representation of future frame xt+k can be combined (via concatenation) and decoded to predict the pixels of future frame xt+k . However, this reconstruction constraint alone is insufficient to induce 2 the desired factorization between the two encoders. We thus introduce a novel adversarial loss on the pose features that prevents them from being discriminable from one video to another, thus ensuring that they cannot contain content information. A further constraint, motivated by the notion that content information should vary slowly over time, encourages temporally close content vectors to be similar to one another. More formally, let xi = (x1i , ..., xTi ) denote a sequence of T images from video i. We subsequently drop index i for brevity. Let Ec denote a neural network that maps an image xt to the content representation htc which captures structure shared across time. Let Ep denote a neural network that maps an image xt to the pose representation htp capturing content that varies over time. Let D denote a decoder network that maps a content representation from a frame, htc , and a pose representation ht+k from future time step t + k to a prediction of the future frame x ?t+k . Finally, C is the scene p discriminator network that takes pairs of pose vectors (h1 , h2 ) and outputs a scalar probability that they came from the same video or not. The loss function used during training has several terms: ?t+k Reconstruction loss: We use a standard per-pixel `2 loss between the predicted future frame x t+k and the actual future frame x for some random frame offset k ? [0, K]: t+k 2 Lreconstruction (D) = ||D(htc , ht+k ||2 p )?x (1) Note that many recent works on video prediction that rely on more complex losses that can capture uncertainty, such as GANs [19, 6]. Similarity loss: To ensure the content encoder extracts mostly time-invariant representations, we penalize the squared error between the content features htc , ht+k of neighboring frames k ? [0, K]: c Lsimilarity (Ec ) = ||Ec (xt ) ? Ec (xt+k )||22 (2) Adversarial loss: We now introduce a novel adversarial loss that exploits the fact that the objects present do not typically change within a video, but they do between different videos. Our desired disenanglement would thus have the content features be (roughly) constant within a clip, but distinct between them. This implies that the pose features should not carry any information about the identity of objects within a clip. We impose this via an adversarial framework between the scene discriminator network C and pose encoder Ep , shown in Fig. 1. The latter provides pairs of pose vectors, either computed from the same t+k t video (htp,i , ht+k p,i ) or from different ones (hp,i , hp,j ), for some other video j. The discriminator then attempts to classify the pair as being from the same/different video using a cross-entropy loss: ?Ladversarial (C) = log(C(Ep (xti ), Ep (xt+k ))) + log(1 ? C(Ep (xti ), Ep (xt+k ))) i j (3) The other half of the adversarial framework imposes a loss function on the pose encoder Ep that tries to maximize the uncertainty (entropy) of the discriminator output on pairs of frames from the same clip: ?Ladversarial (Ep ) = 1 1 log(C(Ep (xti ), Ep (xt+k ))) + log(1 ? C(Ep (xti ), Ep (xt+k ))) i i 2 2 (4) Thus the pose encoder is encouraged to produce features that the discriminator is unable to classify if they come from the same clip or not. In so doing, the pose features cannot carry information about object content, yielding the desired factorization. Note that this does assume that the object?s pose is not distinctive to a particular clip. While adversarial training is also used by GANs, our setup purely considers classification; there is no generator network, for example. Overall training objective: During training we minimize the sum of the above losses, with respect to Ec , Ep , D and C: L = Lreconstruction (Ec , Ep , D)+?Lsimilarity (Ec )+?(Ladversarial (Ep )+Ladversarial (C)) (5) where ? and ? are hyper-parameters. The first three terms can be jointly optimized, but the discriminator C is updated while the other parts of the model (Ec , Ep , D) are held constant. The overall model is shown in Fig. 1. Details of the training procedure and model architectures for Ec , Ep , D and C are given in Section 4.1. 3 Content encoder: Ec(x) xt Scene discriminator: D(Ep(x), Ep(x?)) Pose encoder: Ep(x) xit Target 1 (same scene) Frame decoder: D( Ec(xt), Ep(xt+k) ) Lsimilarity Scene discriminator: C(Ep(x), Ep(x?)) xt+k ~ x t+k LBCE Target 1 (same scene) xit+k Lreconstruction Pose encoder: Ep(x) xit Target 0 (different scenes) Ladversarial xt+k Target 0 (different scenes) Target=0.5 (maximal uncertainty) LBCE xj x t+k t+k? Figure 1: Left: The discriminator C is trained with binary cross entropy (BCE) loss to predict if a pair of pose vectors comes from the same (top portion) or different (lower portion) scenes. xi and xj denote frames from different sequences i and j. The frame offset k is sampled uniformly in the range [0, K]. Note that when C is trained, the pose encoder Ep is fixed. Right: The overall model, showing all terms in the loss function. Note that when the pose encoder Ep is updated, the scene discriminator is held fixed. ~ xt ~t hp hct xt+k Ec Frame decoder: D( Ec(xt), Ep(xt+k) ) xt x t+3 D D D ~ t+1 hp ~ t+2 hp hct LSTM hct hpt-1 Ep x t-1 ~ x t+2 hct LSTM ~ x t+k ~ x t+1 ~ t+3 hp hct LSTM hct hpt hct LSTM ~ t+1 hp ~ t+2 hp hct Ep xt Figure 2: Generating future frames by recurrently predicting hp , the latent pose vector. Ladversary 3.1 Target 1/2 (maximal uncertainty) Scene discriminator not updated, only used for pose encoder loss Forward Prediction After training, the pose and content encoders Ep and Ec provide a representation which enables video prediction in a straightforward manner. Given a frame xt , the encoders produce htp and htc respectively. To generate the next frame, we use these as input to an LSTM model to predict the next pose features ht+1 p . These are then passed (along with the content features) to the decoder, which generates a pixel-space prediction x ?t+1 : ? t+1 = LST M (Ep (xt ), ht ) ? t+1 , ht ) h x ?t+1 = D(h (6) p ? t+2 h p c ? t+1 , ht ) = LST M (h p c p c ? t+2 , ht ) x ?t+2 = D(h p c (7) Note that while pose estimates are generated in a recurrent fashion, the content features htc remain fixed from the last observed real frame. This relies on the nature of Lreconstruction which ensured that content features can be combined with future pose vectors to give valid reconstructions. ? t+1 and The LSTM is trained separately from the main model using a standard `2 loss between h p ht+1 p . Note that this generative model is far simpler than many other recent approaches, e.g. [12]. This largely due to the forward model being applied within our disentangled representation, rather than directly on raw pixels. 3.2 Classification Another application of our disentangled representation is to use it for classification tasks. Content features, which are trained to be invariant to local temporal changes, can be used to classify the semantic content of an image. Conversely, a sequence of pose features can be used to classify actions in a video sequence. In either case, we train a two layer classifier network S on top of either hc or hp , with its output predicting the class label y. 4 4 Experiments We evaluate our model on both synthetic (MNIST, NORB, SUNCG) and real (KTH Actions) video datasets. We explore several tasks with our model: (i) the ability to cleanly factorize into content and pose components; (ii) forward prediction of video frames using the approach from Section 3.1; (iii) using the pose/content features for classification tasks. 4.1 Model details We explored a variety of convolutional architectures for the content encoder Ec , pose encoder Ep and decoder D. For MNIST, Ec , Ep and D all use a DCGAN architecture [21] with |hp | = 5 and |hc | = 128. The encoders consist of 5 convolutional layers with subsampling. Batch normalization and Leaky ReLU?s follow each convolutional layer except the final layer which normalizes the pose/content vectors to have unit norm. The decoder is a mirrored version of the encoder with 5 deconvolutional layers and a sigmoid output layer. For both NORB and SUNCG, D is a DCGAN architecture while Ec and Ep use a ResNet-18 architecture [9] up until the final pooling layer with |hp | = 10 and |hc | = 128. For KTH, Ep uses a ResNet-18 architecture with |hp | = 24. Ec uses the same architecture as VGG16 [29] up until the final pooling layer with |hc | = 128. The decoder is a mirrored version of the content encoder with pooling layers replaced with spatial up-sampling. In the style of U-Net [25], we add skip connections from the content encoder to the decoder, enabling the model to easily generate static background features. In all experiments the scene discriminator C is a fully connected neural network with 2 hidden layers of 100 units. We trained all our models with the ADAM optimizer [13] and learning rate ? = 0.002. We used ? = 0.1 for MNIST, NORB and SUNCG and ? = 0.0001 for KTH experiments. We used ? = 1 for all datasets. For future prediction experiments we train a two layer LSTM with 256 cells using the ADAM optimizer. On MNIST, we train the model by observing 5 frames and predicting 10 frames. On KTH, we train the model by observing 10 frames and predicting 10 frames. 4.2 Synthetic datasets MNIST: We start with a toy dataset consisting of two MNIST digits bouncing around a 64x64 image. Each video sequence consists of a different pair of digits with independent trajectories. Fig. 3(left) shows how the content vector from one frame and the pose vector from another generate new examples that transfer the content and pose from the original frames. This demonstrates the clean disentanglement produced by our model. Interestingly, for this data we found it to be necessary to use a different color for the two digits. Our adversarial term is so aggressive that it prevents the Input frames 1 3 5 Generated frames 6 9 12 15 18 21 50 24 100 200 500 ... actionDim=5-latentDi m=128-maxStep=8-a dvWeight=0-normaliz e=true-ngf=64-ndf=64 -model=basic-output= sigmoid-linWeight=0 ... ... ... ... ... ... Figure 3: Left: Demonstration of content/pose factorization on held out MNIST examples. Each image in the grid is generated using the pose and content vectors hp and hc taken from the corresponding images in the top row and first column respectively. The model has clearly learned to disentangle content and pose. Right: Each row shows forward modeling up to 500 time steps into the future, given 5 initial frames. For each generation, note that only the pose part of the representation is being predicted from the previous time step (using an LSTM), with the content vector being fixed from the 5th frame. The generations remain crisp despite the long-range nature of the predictions. 5 Pose Content Pose Pose Pose Content Content Pose Figure 4: Left: Factorization examples using our D R N ET model on held out NORB images. Each image in the grid is generated using the pose and content vectors hp and hc taken from the corresponding images in the top row and first column respectively. Center: Examples where D R N ET was trained without the adversarial loss term. Note how content and pose are no longer factorized cleanly: the pose vector now contains content information which ends up dominating the generation. Right: factorization examples from Mathieu et al. [19]. x1 Interpolations x2 Content Pose Figure 5: Left: Examples of linear interpolation in pose space between the examples x1 and x2 . Right: Factorization examples on held out images from the SUNCG dataset. Each image in the grid is generated using the pose and content vectors hp and hc taken from the corresponding images in the top row and first column respectively. Note how, even for complex objects, the model is able to rotate them accurately. pose vector from capturing any content information, thus without a color cue the model is unable to determine which pose information to associate with which digit. In Fig. 3(right) we perform forward modeling using our representation, demonstrating the ability to generate crisp digits 500 time steps into the future. NORB: We apply our model to the NORB dataset [16], converted into videos by taking sequences of different azimuths, while holding object identity, lighting and elevation constant. Fig. 4.2(left) shows that our model is able to factor content and pose cleanly on held out data. In Fig. 4.2(center) we train a version of our model without the adversarial loss term, which results in a significant degradation in the model and the pose vectors are no longer isolated from content. For comparison, we also show the factorizations produced by Mathieu et al. [19], which are less clean, both in terms of disentanglement and generation quality than our approach. Table 1 shows classification results on NORB, following the training of a classifier on pose features and also content features. When the adversarial term is used (? = 0.1) the content features perform well. Without the term, content features become less effective for classification. SUNCG: We use the rendering engine from the SUNCG dataset [30] to generate sequences where the camera rotates around a range of 3D chair models. The dataset consists of 324 different chair models of varying size, shape and color. D R N ET learns a clean factorization of content and pose and is able to generate high quality examples of this dataset, as shown in Fig. 4.2(right). 6 4.3 KTH Action Dataset Finally, we apply D R N ET to the KTH dataset [28]. This is a simple dataset of real-world videos of people performing one of six actions (walking, jogging, running, boxing, handwaving, hand-clapping) against fairly uniform backgrounds. In Fig. 4.3 we show forward generations of different held out examples, comparing against two baselines: (i) the MCNet of Villegas et al. [33]which, to the best of our knowledge, produces the current best quality generations of on real-world video and (ii) a baseline auto-encoder LSTM model (AE-LSTM). This is essentially the same as ours, but with a single encoder whose features thus combine content and pose (as opposed to factoring them in D R N ET ). It is also similar to [31]. Fig. 7 shows more examples, with generations out to 100 time steps. For most actions this is sufficient time for the person to have left the frame, thus further generations would be of a fixed background. In Fig. 9 we attempt to quantify the fidelity of the generations by comparing our approach to MCNet [33] using a metric derived from the Inception score [26]. The Inception score is used for assessing generations from GANs and is more appropriate for our scenario that traditional metrics such as PSNR or SSIM (see appendix B for further discussion). The curves show the mean scores of our generations decaying more gracefully than MCNet [33]. Further examples and generated movies may be viewed in appendix A and also at https://sites.google.com/view/drnet-paper//. A natural concern with high capacity models is that they might be memorizing the training examples. We probe this in Fig. 4.3, where we show the nearest neighbors to our generated frames from the training set. Fig. 8 uses the pose representation produced by D R N ET to train an action classifier from very few examples. We extract pose vectors from video sequences of length 24 and train a fully connected classifier on these vectors to predict the action class. We compare against an autoencoder baseline, which is the same as ours but with a single encoder whose features thus combine content and pose. We find the factorization significantly boosts performance. t=1 t=5 t = 10 t = 12 t = 15 t = 17 t = 21 t = 25 t = 27 t = 30 Ground truth future DrNet (ours) AE-LSTM Walking t=1 t=5 MCNet t = 10 t = 12 t = 15 t = 17 t = 21 t = 25 t = 27 t = 30 Ground truth future DrNet (ours) AE-LSTM Running MCNet Figure 6: Qualitative comparison between our D R N ET model, MCNet [33] and the AE-LSTM baseline. All models are conditioned on the first 10 video frames and generate 20 frames. We display predictions of every 3rd frame. Video sequences are taken from held out examples of the KTH dataset for the classes of walking (top) and running (bottom). 7 t=11 t=14 t=17 t=20 t=23 t=26 t=29 t=32 t=35 t=38 t=41 t=44 t=47 t=50 t=60 t=70 t=80 t=90 t=100 DrNet MCNet Figure 7: Four additional examples of generations on held out examples of the KTH dataset, rolled out to 100 timesteps. 1.75 D R N ET ?=0.1 D R N ET ?=0 hc hp hc hp Mathieu et al. [19] Accuracy (%) 1.7 93.3 60.9 72.6 80.8 86.5 1.65 Inception Score Model Table 1: Classification results on t = 15 t = 17 t = 21 t = 25 1.6 1.55 1.5 1.45 1.4 NORB dataset, with/without adversarial loss (? = 0.1/0) using content or pose representations (hc , hp respectively). The adversarial term is crucial for forcing semantic information into the content vectors ? without it performance drops significantly. t = 12 DrNet MCNet 1.35 1.3 0 20 40 60 80 100 Future time step Figure 8: Classification of KTH actions from pose vectors with few labeled examples, with autoencoder baseline. N.B. SOA (fully supervised) is 93.9% [15]. t = 27 t = 30 t = 12 t = 15 t = 17 Figure 9: Comparison of KTH video generation quality using Inception score. X-axis indicated how far from conditioned input the start of the generated sequence is. t = 21 t = 25 t = 27 t = 30 DrNet generations Nearest neighbour in pose space Nearest neighbour in pose+content space DrNet generations Nearest neighbour in pose space Nearest neighbour in pose+content space Figure 10: For each frame generated by D R N ET (top row in each set), we show nearest-neighbor images from the training set, based on pose vectors (middle row) and both content and pose vectors (bottom row). It is evident that our model is not simply copying examples from the training data. Furthermore, the middle row shows that the pose vector generalizes well, and is independent of background and clothing. 8 5 Discussion In this paper we introduced a model based on a pair of encoders that factor video into content and pose. This seperation is achieved during training through novel adversarial loss term. The resulting representation is versatile, in particular allowing for stable and coherent long-range prediction through nothing more than a standard LSTM. Our generations compare favorably with leading approaches, despite being a simple model, e.g. lacking the GAN losses or probabilistic formulations of other video generation approaches. Source code is available at https://github.com/edenton/drnet. Acknowledgments We thank Rob Fergus, Will Whitney and Jordan Ash for helpful comments and advice. Emily Denton is grateful for the support of a Google Fellowship References [1] P. Agrawal, A. 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Federated Multi-Task Learning Virginia Smith Stanford [email protected] Chao-Kai Chiang? USC [email protected] Maziar Sanjabi? USC Ameet Talwalkar CMU [email protected] [email protected] Abstract Federated learning poses new statistical and systems challenges in training machine learning models over distributed networks of devices. In this work, we show that multi-task learning is naturally suited to handle the statistical challenges of this setting, and propose a novel systems-aware optimization method, M OCHA, that is robust to practical systems issues. Our method and theory for the first time consider issues of high communication cost, stragglers, and fault tolerance for distributed multi-task learning. The resulting method achieves significant speedups compared to alternatives in the federated setting, as we demonstrate through simulations on real-world federated datasets. 1 Introduction Mobile phones, wearable devices, and smart homes are just a few of the modern distributed networks generating massive amounts of data each day. Due to the growing storage and computational power of devices in these networks, it is increasingly attractive to store data locally and push more network computation to the edge. The nascent field of federated learning explores training statistical models directly on devices [37]. Examples of potential applications include: learning sentiment, semantic location, or activities of mobile phone users; predicting health events like low blood sugar or heart attack risk from wearable devices; or detecting burglaries within smart homes [3, 39, 42]. Following [25, 36, 26], we summarize the unique challenges of federated learning below. 1. Statistical Challenges: The aim in federated learning is to fit a model to data, {X1 , . . . , Xm }, generated by m distributed nodes. Each node, t ? [m], collects data in a non-IID manner across the network, with data on each node being generated by a distinct distribution Xt ? Pt . The number of data points on each node, nt , may also vary significantly, and there may be an underlying structure present that captures the relationship amongst nodes and their associated distributions. 2. Systems Challenges: There are typically a large number of nodes, m, in the network, and communication is often a significant bottleneck. Additionally, the storage, computational, and communication capacities of each node may differ due to variability in hardware (CPU, memory), network connection (3G, 4G, WiFi), and power (battery level). These systems challenges, compounded with unbalanced data and statistical heterogeneity, make issues such as stragglers and fault tolerance significantly more prevalent than in typical data center environments. In this work, we propose a modeling approach that differs significantly from prior work on federated learning, where the aim thus far has been to train a single global model across the network [25, 36, 26]. Instead, we address statistical challenges in the federated setting by learning separate models for each node, {w1 , . . . , wm }. This can be naturally captured through a multi-task learning (MTL) framework, where the goal is to consider fitting separate but related models simultaneously [14, 2, 58, 28]. Unfortunately, current multi-task learning methods are not suited to handle the systems challenges that arise in federated learning, including high communication cost, stragglers, and fault tolerance. Addressing these challenges is therefore a key component of our work. ? Authors contributed equally. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 1.1 Contributions We make the following contributions. First, we show that MTL is a natural choice to handle statistical challenges in the federated setting. Second, we develop a novel method, M OCHA, to solve a general MTL problem. Our method generalizes the distributed optimization method C O C OA [22, 31] in order to address systems challenges associated with network size and node heterogeneity. Third, we provide convergence guarantees for M OCHA that carefully consider these unique systems challenges and provide insight into practical performance. Finally, we demonstrate the superior empirical performance of M OCHA with a new benchmarking suite of federated datasets. 2 Related Work Learning Beyond the Data Center. Computing SQL-like queries across distributed, low-powered nodes is a decades-long area of research that has been explored under the purview of query processing in sensor networks, computing at the edge, and fog computing [32, 12, 33, 8, 18, 15]. Recent works have also considered training machine learning models centrally but serving and storing them locally, e.g., this is a common approach in mobile user modeling and personalization [27, 43, 44]. However, as the computational power of the nodes within distributed networks grows, it is possible to do even more work locally, which has led to recent interest in federated learning.2 In contrast to our proposed approach, existing federated learning approaches [25, 36, 26, 37] aim to learn a single global model across the data.3 This limits their ability to deal with non-IID data and structure amongst the nodes. These works also come without convergence guarantees, and have not addressed practical issues of stragglers or fault tolerance, which are important characteristics of the federated setting. The work proposed here is, to the best of our knowledge, the first federated learning framework to consider these challenges, theoretically and in practice. Multi-Task Learning. In multi-task learning, the goal is to learn models for multiple related tasks simultaneously. While the MTL literature is extensive, most MTL modeling approaches can be broadly categorized into two groups based on how they capture relationships amongst tasks. The first (e.g., [14, 4, 11, 24]) assumes that a clustered, sparse, or low-rank structure between the tasks is known a priori. A second group instead assumes that the task relationships are not known beforehand and can be learned directly from the data (e.g., [21, 58, 16]). In this work, we focus our attention on this latter group, as task relationships may not be known beforehand in real-world settings. In comparison to learning a single global model, these MTL approaches can directly capture relationships amongst non-IID and unbalanced data, which makes them particularly well-suited for the statistical challenges of federated learning. We demonstrate this empirically on real-world federated datasets in Section 5. However, although MTL is a natural modeling choice to address the statistical challenges of federated learning, currently proposed methods for distributed MTL (discussed below) do not adequately address the systems challenges associated with federated learning. Distributed Multi-Task Learning. Distributed multi-task learning is a relatively new field, in which the aim is to solve an MTL problem when data for each task is distributed over a network. While several recent works [1, 35, 54, 55] have considered the issue of distributed MTL training, the proposed methods do not allow for flexibility of communication versus computation. As a result, they are unable to efficiently handle concerns of fault tolerance and stragglers, the latter of which stems from both data and system heterogeneity. The works of [23] and [7] allow for asynchronous updates to help mitigate stragglers, but do not address fault tolerance. Moreover, [23] provides no convergence guarantees, and the convergence of [7] relies on a bounded delay assumption that is impractical for the federated setting, where delays may be significant and devices may drop out completely. Finally, [30] proposes a method and setup leveraging the distributed framework C O C OA [22, 31], which we show in Section 4 to be a special case of the more general approach in this work. However, the authors in [30] do not explore the federated setting, and their assumption that the same amount of work is done locally on each node is prohibitive in federated settings, where unbalance is common due to data and system variability. 2 The term on-device learning has been used to describe both the task of model training and of model serving. Due to the ambiguity of this phrase, we exclusively use the term federated learning. 3 While not the focus of our work, we note privacy is an important concern in the federated setting, and that the privacy benefits associated with global federated learning (as discussed in [36]) also apply to our approach. 2 3 Federated Multi-Task Learning In federated learning, the aim is to learn a model over data that resides on, and has been generated by, m distributed nodes. As a running example, consider learning the activities of mobile phone users in a cell network based on their individual sensor, text, or image data. Each node (phone), t ? [m], may generate data via a distinct distribution, and so it is natural to fit separate models, {w1 , . . . , wm }, to the distributed data?one for each local dataset. However, structure between models frequently exists (e.g., people may behave similarly when using their phones), and modeling these relationships via multi-task learning is a natural strategy to improve performance and boost the effective sample size for each node [10, 2, 5]. In this section, we suggest a general MTL framework for the federated setting, and propose a novel method, M OCHA, to handle the systems challenges of federated MTL. 3.1 General Multi-Task Learning Setup Given data Xt ? Rd?nt from m nodes, multi-task learning fits separate weight vectors wt ? Rd to the data for each task (node) through arbitrary convex loss functions `t (e.g., the hinge loss for SVM models). Many MTL problems can be captured via the following general formulation: (m n ) t XX T i i min `t (wt xt , yt ) + R(W, ?) , (1) W,? t=1 i=1 where W := [w1 , . . . , wm ] ? Rd?m is a matrix whose t-th column is the weight vector for the t-th task. The matrix ? ? Rm?m models relationships amongst tasks, and is either known a priori or estimated while simultaneously learning task models. MTL problems differ based on their assumptions on R, which takes ? as input and promotes some suitable structure amongst the tasks. As an example, several popular MTL approaches assume that tasks form clusters based on whether or not they are related [14, 21, 58, 59]. This can be expressed via the following bi-convex formulation:  R(W, ?) = ?1 tr W?WT + ?2 kWk2F , (2) with constants ?1 , ?2 > 0, and where the second term performs L2 regularization on each local model. We use a similar formulation with variable clusters (12) in our experiments in Section 5, and provide details on other common classes of MTL models that can be formulated via (1) in Appendix B. 3.2 M OCHA: A Framework for Federated Multi-Task Learning In the federated setting, the aim is to train statistical models directly on the edge, and thus we solve (1) while assuming that the data {X1 , . . . , Xm } is distributed across m nodes or devices. Before proposing our federated method for solving (1), we make the following observations: ? Observation 1: In general, (1) is not jointly convex in W and ?, and even in the cases where (1) is convex, solving for W and ? simultaneously can be difficult [5]. ? Observation 2: When fixing ?, updating W depends on both the data X, which is distributed across the nodes, and the structure ?, which is known centrally. ? Observation 3: When fixing W, optimizing for ? only depends on W and not on the data X. Based on these observations, it is natural to propose an alternating optimization approach to solve problem (1), in which at each iteration we fix either W or ? and optimize over the other, alternating until convergence is reached. Note that solving for ? is not dependent on the data and therefore can be computed centrally; as such, we defer to prior work for this step [59, 21, 58, 16]. In Appendix B, we discuss updates to ? for several common MTL models. In this work, we focus on developing an efficient distributed optimization method for the W step. In traditional data center environments, the task of distributed training is a well-studied problem, and various communication-efficient frameworks have been recently proposed, including the state-of-theart primal-dual C O C OA framework [22, 31]. Although C O C OA can be extended directly to update W in a distributed fashion across the nodes, it cannot handle the unique systems challenges of the federated environment, such as stragglers and fault tolerance, as discussed in Section 3.4. To this end, we extend C O C OA and propose a new method, M OCHA, for federated multi-task learning. Our method is given in Algorithm 1 and described in detail in Sections 3.3 and 3.4. 3 Algorithm 1 M OCHA: Federated Multi-Task Learning Framework 1: Input: Data Xt from t = 1, . . . , m tasks, stored on one of m nodes, and initial matrix ?0 2: Starting point ?(0) := 0 ? Rn , v(0) := 0 ? Rb 3: for iterations i = 0, 1, . . . do 4: Set subproblem parameter ? 0 and number of federated iterations, Hi 5: for iterations h = 0, 1, ? ? ? , Hi do for tasks t ? {1, 2, . . . , m} in parallel over m nodes do 6: 7: call local solver, returning ?th -approximate solution ??t of the local subproblem (4) 8: update local variables ?t ? ?t + ??t return updates ?vt := Xt ??t 9: reduce: vt ? vt + ?vt 10: 11: Update ? centrally based on w(?) for latest ? 12: Central node computes w = w(?) based on the lastest ? 13: return: W := [w1 , . . . , wm ] 3.3 Federated Update of W To update W in the federated setting, we begin by extending works on distributed primal-dual optimization [22, 31, 30] to apply to the generalized multi-task framework (1). This involves deriving the appropriate dual formulation, subproblems, and problem parameters, as we detail below. Dual problem. Considering the dual formulation of (1) will allow us to better separate the Pglobal m problem into distributed subproblems for federated computation across the nodes. Let n := t=1 nt and X := Diag(X1 , ? ? ? , Xm ) ? Rmd?n . With ? fixed, the dual of problem (1), defined with respect to dual variables ? ? Rn , is given by: ( ) nt m X X ? i ? min D(?) := `t (??t ) + R (X?) , (3) ? `?t t=1 i=1 ? where and R are the conjugate dual functions of `t and R, respectively, and ?it is the dual variable for the data point (xit , yti ). Note that R? depends on ?, but for the sake of simplicity, we have removed this in our notation. To derive distributed subproblems from this global dual, we make an assumption described below on the regularizer R. Assumption 1. Given ?, we assume that there exists a symmetric positive definite matrix M ? Rmd?md , depending on ?, for which the function R is strongly convex with respect to M?1 . Note that this corresponds to assuming that R? will be smooth with respect to matrix M. ? = R(W, ?), where ? ?) Remark 1. We can reformulate the MTL regularizer in the form of R(w, ? := ? ? Id?d ? Rmd?md . For example, w ? Rmd is a vector containing the columns of W and ?  ? = tr wT (?1 ? ? + ?2 I)w . Writing the regularizer ? ?) we can rewrite the regularizer in (2) as R(w, ? + ?2 I. in this form, it is clear that it is strongly convex with respect to matrix M?1 = ?1 ? Data-local quadratic subproblems. To solve (1) across distributed nodes, we define the following data-local subproblems, which are formed via a careful quadratic approximation of the dual problem (3) to separate computation across the nodes. These subproblems find updates ??t ? Rnt to the dual variables in ? corresponding to a single node t, and only require accessing data which is available locally, i.e., Xt for node t. The t-th subproblem is given by: nt X 0 ?0 2 min Gt? (??t ; vt , ?t ) := `?t (??it ???it )+hwt (?), Xt ??t i+ kXt ??t kMt +c(?) , (4) ??t 2 i=1 1 where c(?) := m R? (X?), and Mt ? Rd?d is the t-th diagonal block of the symmetric positive definite matrix M. Given dual variables ?, corresponding primal variables can be found via w(?) = ?R? (X?), where wt (?) is the t-th block in the vector w(?). Note that computing w(?) requires the vector v = X?. The t-th block of v, vt ? Rd , is the only information that must be communicated between nodes at each iteration. Finally, ? 0 > 0 measures the difficulty of the data partitioning, and helps to relate progress made to the subproblems to the global dual problem. It can be easily selected based on M for many applications of interest; we provide details in Lemma 9 of the Appendix. 4 3.4 Practical Considerations During M OCHA?s federated update of W, the central node requires a response from all workers before performing a synchronous update. In the federated setting, a naive execution of this communication protocol could introduce dramatic straggler effects due to node heterogeneity. To avoid stragglers, 0 M OCHA provides the t-th node with the flexibility to approximately solve its subproblem Gt? (?), where the quality of the approximation is controled by a per-node parameter ?th . The following factors determine the quality of the t-th node?s solution to its subproblem: 0 1. Statistical challenges, such as the size of Xt and the intrinsic difficulty of subproblem Gt? (?). 2. Systems challenges, such as the node?s storage, computational, and communication capacities due to hardware (CPU, memory), network connection (3G, 4G, WiFi), and power (battery level). 3. A global clock cycle imposed by the central node specifying a deadline for receiving updates. We define ?th as a function of these factors, and assume that each node has a controller that may derive ?th from the current clock cycle and statistical/systems setting. ?th ranges from zero to one, 0 where ?th = 0 indicates an exact solution to Gt? (?) and ?th = 1 indicates that node t made no progress during iteration h (which we refer to as a dropped node). For instance, a node may ?drop? if it runs out of battery, or if its network bandwidth deteriorates during iteration h and it is thus unable to return its update within the current clock cycle. A formal definition of ?th is provided in (5) of Section 4. M OCHA mitigates stragglers by enabling the t-th node to define its own ?th . On every iteration h, the local updates that a node performs and sends in a clock cycle will yield a specific value for ?th . As discussed in Section 4, M OCHA is additionally robust to a small fraction of nodes periodically dropping and performing no local updates (i.e., ?th := 1) under suitable conditions, as defined in Assumption 2. In contrast, prior work of C O C OA may suffer from severe straggler effects in federated settings, as it requires a fixed ?th = ? across all nodes and all iterations while still maintaining synchronous updates, and it does not allow for the case of dropped nodes (? := 1). Finally, we note that asynchronous updating schemes are an alternative approach to mitigate stragglers. We do not consider these approaches in this work, in part due to the fact that the bounded-delay assumptions associated with most asynchronous schemes limits fault tolerance. However, it would be interesting to further explore the differences and connections between asynchronous methods and approximation-based, synchronous methods like M OCHA in future work. 4 Convergence Analysis M OCHA is based on a bi-convex alternating approach, which is guaranteed to converge [17, 45] to a stationary solution of problem (1). In the case where this problem is jointly convex with respect to W and ?, such a solution is also optimal. In the rest of this section, we therefore focus on the convergence of solving the W update of M OCHA in the federated setting. Following the discussion in Section 3.4, we first introduce the following per-node, per-round approximation parameter. Definition 1 (Per-Node-Per-Iteration-Approximation Parameter). At each iteration h, we define the accuracy level of the solution calculated by node t to its subproblem (4) as: 0 ?th := (h) 0 (h) (h) Gt? (??t ; v(h) , ?t ) ? Gt? (???t ; v(h) , ?t ) 0 (h) (h) 0 Gt? (0; v(h) , ?t ) ? Gt? (???t ; v(h) , ?t ) 0 , (5) (h) where ???t is the minimizer of subproblem Gt? (? ; v(h) , ?t ). We allow this value to vary between 0 [0, 1], with ?th := 1 meaning that no updates to subproblem Gt? are made by node t at iteration h. While the flexible per-node, per-iteration approximation parameter ?th in (5) allows the consideration of stragglers and fault tolerance, these additional degrees of freedom also pose new challenges in providing convergence guarantees for M OCHA. We introduce the following assumption on ?th to provide our convergence guarantees. Assumption 2. Let Hh := (?(h) , ?(h?1) , ? ? ? , ?(1) ) be the dual vector history until the beginning of iteration h, and define ?ht := E[?th |Hh ]. For all tasks t and all iterations h, we assume pht := ? ht := E[?th |Hh , ?th < 1] ? ?max < 1. P[?th = 1] ? pmax < 1 and ? 5 This assumption states that at each iteration, the probability of a node sending a result is non-zero, and that the quality of the returned result is, on average, better than the previous iterate. Compared to [49, 30] which assumes ?th = ? < 1, our assumption is significantly less restrictive and better models the federated setting, where nodes are unreliable and may periodically drop out. Using Assumption 2, we derive the following theorem, which characterizes the convergence of the federated update of M OCHA in finite horizon when the losses `t in (1) are smooth. Theorem 1. Assume that the losses `t are (1/?)-smooth. Then, under Assumptions 1 and 2, there exists a constant s ? (0, 1] such that for any given convergence target D , choosing H such that 1 n H? log , (6) ? D (1 ? ?)s will satisfy E[D(?(H) ) ? D(?? )] ? D . ? := pmax + (1 ? pmax )?max < 1. While Theorem 1 is concerned with finite horizon converHere, ? gence, it is possible to get asymptotic convergence results, i.e., H ? ?, with milder assumptions on the stragglers; see Corollary 8 in the Appendix for details. When the loss functions are non-smooth, e.g., the hinge loss for SVM models, we provide the following sub-linear convergence for L-Lipschitz losses. Theorem 2. If the loss functions `t are L-Lipschitz, then there exists a constant ?, defined in (24), such that for any given D > 0, if we choose    2 2L2 ?? 0 H ? H0 + max 1, , (7) ? n 2 D (1 ? ?)  2     2n (D(?? ) ? D(?0 )) 16L2 ?? 0 1 with H0 ? h0 + , ? 2 D , h0 = 1 + (1 ? ?) ? log 4L2 ?? 0 (1 ? ?)n + PH 1 (h) ? := H?H0 ? ? D(?? )] ? D . then ? will satisfy E[D(?) h=H0 +1 ? These theorems guarantee that M OCHA will converge in the federated setting, under mild assumptions on stragglers and capabilities of the nodes. While these results consider convergence in terms of the dual, we show that they hold analogously for the duality gap. We provide all proofs in Appendix C. Remark 2. Following from the discussion in Section 3.4, our method and theory generalize the results in [22, 31]. In the limiting case that all ?th are identical, our results extend the results of C O C OA to the multi-task framework described in (1). Remark 3. Note that the methods in [22, 31] have an aggregation parameter ? ? (0, 1]. Though we prove our results for a general ?, we simplify the method and results here by setting ? := 1, which has been shown to have the best performance, both theoretically and empirically [31]. 5 Simulations In this section we validate the empirical performance of M OCHA. First, we introduce a benchmarking suite of real-world federated datasets and show that multi-task learning is well-suited to handle the statistical challenges of the federated setting. Next, we demonstrate M OCHA?s ability to handle stragglers, both from statistical and systems heterogeneity. Finally, we explore the performance of M OCHA when devices periodically drop out. Our code is available at: github.com/gingsmith/fmtl. 5.1 Federated Datasets In our simulations, we use several real-world datasets that have been generated in federated settings. We provide additional details in the Appendix, including information about data sizes, nt . ? Google Glass (GLEAM)4 : This dataset consists of two hours of high resolution sensor data collected from 38 participants wearing Google Glass for the purpose of activity recognition. Following [41], we featurize the raw accelerometer, gyroscope, and magnetometer data into 180 statistical, spectral, and temporal features. We model each participant as a separate task, and predict between eating and other activities (e.g., walking, talking, drinking). 4 http://www.skleinberg.org/data/GLEAM.tar.gz 6 ? Human Activity Recognition5 : Mobile phone accelerometer and gyroscope data collected from 30 individuals, performing one of six activities: {walking, walking-upstairs, walking-downstairs, sitting, standing, lying-down}. We use the provided 561-length feature vectors of time and frequency domain variables generated for each instance [3]. We model each individual as a separate task and predict between sitting and the other activities. ? Vehicle Sensor6 : Acoustic, seismic, and infrared sensor data collected from a distributed network of 23 sensors, deployed with the aim of classifying vehicles driving by a segment of road [13]. Each instance is described by 50 acoustic and 50 seismic features. We model each sensor as a separate task and predict between AAV-type and DW-type vehicles. 5.2 Multi-Task Learning for the Federated Setting We demonstrate the benefits of multi-task learning for the federated setting by comparing the error rates of a multi-task model to that of a fully local model (i.e., learning a model for each task separately) and a fully global model (i.e., combining the data from all tasks and learning one single model). Work on federated learning thus far has been limited to the study of fully global models [25, 36, 26]. We use a cluster-regularized multi-task model [59, 21], as described in Section 3.1. For each dataset from Section 5.1, we randomly split the data into 75% training and 25% testing, and learn multi-task, local, and global support vector machine models, selecting the best regularization parameter, ? ?{1e5, 1e-4, 1e-3, 1e-2, 0.1, 1, 10}, for each model using 5-fold cross-validation. We repeat this process 10 times and report the average prediction error across tasks, averaged across these 10 trials. Table 1: Average prediction error: Means and standard errors over 10 random shuffles. Model Human Activity Google Glass Vehicle Sensor Global 2.23 (0.30) 5.34 (0.26) 13.4 (0.26) Local 1.34 (0.21) 4.92 (0.26) 7.81 (0.13) MTL 0.46 (0.11) 2.02 (0.15) 6.59 (0.21) In Table 1, we see that for each dataset, multi-task learning significantly outperforms the other models in terms of achieving the lowest average error across tasks. The global model, as proposed in [25, 36, 26] performs the worst, particularly for the Human Activity and Vehicle Sensor datasets. Although the datasets are already somewhat unbalanced, we note that a global modeling approach may benefit tasks with a very small number of instances, as information can be shared across tasks. For this reason, we additionally explore the performance of global, local, and multi-task modeling for highly skewed data in Table 4 of the Appendix. Although the performance of the global model improves slightly relative to local modeling in this setting, the global model still performs the worst for the majority of the datasets, and MTL still significantly outperforms both global and local approaches. 5.3 Straggler Avoidance Two challenges that are prevalent in federated learning are stragglers and high communication. Stragglers can occur when a subset of the devices take much longer than others to perform local updates, which can be caused either by statistical or systems heterogeneity. Communication can also exacerbate poor performance, as it can be slower than computation by many orders of magnitude in typical cellular or wireless networks [52, 20, 48, 9, 38]. In our experiments below, we simulate the time needed to run each method by tracking the operations and communication complexities, and scaling the communication cost relative to computation by one, two, or three orders of magnitude, respectively. These numbers correspond roughly to the clock rate vs. network bandwidth/latency (see, e.g., [52]) for modern cellular and wireless networks. Details are provided in Appendix E. 5 6 https://archive.ics.uci.edu/ml/datasets/Human+Activity+Recognition+Using+Smartphones http://www.ecs.umass.edu/~mduarte/Software.html 7 Human Activity: Statistical Heterogeneity (WiFi) 100 10-1 10-2 10-3 1 2 3 4 Estimated Time 5 6 100 10-1 7 10-3 106 MOCHA CoCoA Mb-SDCA Mb-SGD 101 10-2 0 Human Activity: Statistical Heterogeneity (3G) 102 MOCHA CoCoA Mb-SDCA Mb-SGD 101 Primal Sub-Optimality Primal Sub-Optimality 101 Human Activity: Statistical Heterogeneity (LTE) 102 MOCHA CoCoA Mb-SDCA Mb-SGD Primal Sub-Optimality 102 100 10-1 10-2 0 1 2 3 4 5 6 7 10-3 8 0 0.5 1 106 Estimated Time 1.5 2 107 Estimated Time Figure 1: The performance of M OCHA compared to other distributed methods for the W update of (1). While increasing communication tends to decrease the performance of the mini-batch methods, M OCHA performs well in high communication settings. In all settings, M OCHA with varied approximation values, ?ht , performs better than without (i.e., naively generalizing C O C OA), as it avoids stragglers from statistical heterogeneity. Statistical Heterogeneity. We explore the effect of statistical heterogeneity on stragglers for various methods and communication regimes (3G, LTE, WiFi). For a fixed communication network, we compare M OCHA to C O C OA, which has a single ? parameter, and to mini-batch stochastic gradient descent (Mb-SGD) and mini-batch stochastic dual coordinate ascent (Mb-SDCA), which have limited communication flexibility depending on the batch size. We tune all compared methods for best performance, as we detail in Appendix E. In Figure 1, we see that while the performance degrades for mini-batch methods in high communication regimes, M OCHA and C O C OA are robust to high communication. However, C O C OA is significantly affected by stragglers?because ? is fixed across nodes and rounds, difficult subproblems adversely impact convergence. In contrast, M OCHA performs well regardless of communication cost and is robust to statistical heterogeneity. Systems Heterogeneity. M OCHA is also equipped to handle heterogeneity from changing systems environments, such as battery power, memory, or network connection, as we show in Figure 2. In particular, we simulate systems heterogeneity by randomly choosing the number of local iterations for M OCHA or the mini-batch size for mini-batch methods, between 10% and 100% of the minimum number of local data points for high variability environments, to between 90% and 100% for low variability (see Appendix E for full details). We do not vary the performance of C O C OA, as the impact from statistical heterogeneity alone significantly reduces performance. However, adding systems heterogeneity would reduce performance even further, as the maximum ? value across all nodes would only increase if additional systems challenges were introduced. 5.4 Tolerance to Dropped Nodes Vehicle Sensor: Systems Heterogeneity (Low) 102 Primal Sub-Optimality 10 -1 10 -2 MOCHA CoCoA Mb-CD Mb-SGD 101 100 10-3 100 10-1 10-2 0 1 2 3 4 5 6 7 8 10-3 9 0 1 2 3 106 Estimated Time 4 5 6 7 8 9 106 Estimated Time Figure 2: M OCHA can handle variability from systems heterogeneity. Google Glass: Fault Tolerance, W Step 102 Primal Sub-Optimality 101 100 10 -1 10 -2 10-3 Google Glass: Fault Tolerance, Full Method 102 101 Primal Sub-Optimality Finally, we explore the effect of nodes dropping on the performance of M OCHA. We do not draw comparisons to other methods, as to the best of our knowledge, no other methods for distributed multi-task learning directly address fault tolerance. In M OCHA, we incorporate this setting by allowing ?th := 1, as explored theoretically in Section 4. In Figure 3, we look at the performance of M OCHA, either for one fixed W update, or running the entire M OCHA method, as the probability that nodes drop at each iteration (pht in Assumption 2) increases. We see that the performance of M OCHA is robust to relatively high values of pht , both during a single update of W and in how this affects the performance of the overall method. However, as intuition would suggest, if one of the nodes never sends updates (i.e., ph1 := 1 for all h, green dotted line), the method does not converge to the correct solution. This provides validation for our Assumption 2. Primal Sub-Optimality 101 Vehicle Sensor: Systems Heterogeneity (High) 102 MOCHA CoCoA Mb-SDCA Mb-SGD 100 10-1 10-2 0 2 4 6 Estimated Time 8 10 106 10-3 0 1 2 3 4 5 6 Estimated Time 7 8 107 Figure 3: The performance of M OCHA is robust to nodes periodically dropping out (fault tolerance). 8 6 Discussion To address the statistical and systems challenges of the burgeoning federated learning setting, we have presented M OCHA, a novel systems-aware optimization framework for federated multi-task learning. Our method and theory for the first time consider issues of high communication cost, stragglers, and fault tolerance for multi-task learning in the federated environment. While M OCHA does not apply to non-convex deep learning models in its current form, we note that there may be natural connections between this approach and ?convexified? deep learning models [6, 34, 51, 57] in the context of kernelized federated multi-task learning. Acknowledgements We thank Brendan McMahan, Chlo? 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Learning Cellular Automaton Dynamics with Neural Networks N H Wulff* and J A Hertz t CONNECT, the Niels Bohr Institute and Nordita Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Abstract We have trained networks of E - II units with short-range connections to simulate simple cellular automata that exhibit complex or chaotic behaviour. Three levels of learning are possible (in decreasing order of difficulty): learning the underlying automaton rule, learning asymptotic dynamical behaviour, and learning to extrapolate the training history. The levels of learning achieved with and without weight sharing for different automata provide new insight into their dynamics. 1 INTRODUCTION Neural networks have been shown to be capable of learning the dynamical behaviour exhibited by chaotic time series composed of measurements of a single variable among many in a complex system [1, 2, 3]. In this work we consider instead cellular automaton arrays (CA)[4], a class of many-degree-of-freedom systems which exhibits very complex dynamics, including universal computation. We would like to know whether neural nets can be taught to imitate these dynamics, both locally and globally. One could say we are turning the usual paradigm for studying such systems on its head. Conventionally, one is given the rule by which each automaton updates its state, and the (nontrivial) problem is to find what kind of global dynamical ?Present address: NEuroTech AjS, Copenhagen, Denmark t Address until October 1993: Laboratory of Neuropsychology, NIMH, Bethesda MD 20892. email: [email protected] 631 632 Wulff and Hertz behaviour results. Here we suppose that we are given the history of some CA, and we would like, if possible, to find the rule that generated it. We will see that a network can have different degrees of success in this task, depending on the constraints we place on the learning. Furthermore, we will be able to learn something about the dynamics of the automata themselves from knowing what level of learning is possible under what constraints. This note reports some preliminary investigations of these questions. We study only the simplest automata that produce chaotic or complex dynamic behaviour. Nevertheless, we obtain some nontrivial results which lead to interesting conjectures for future investigation. A CA is a lattice of formal computing units, each of which is characterized by a state variable Si(t), where i labels the site in the lattice and t is the (digital) time. Every such unit updates itself according to a particular rule or function f( ) of its own state and that of the other units in its local neighbourhood. The rule is the same for all units, and the updatings of all units are simultaneous. Different models are characterized by the nature of the state variable (e.g. binary, continuous, vector, etc), the dimensionality of the lattice, and the choice of neighbourhood. In the two cases we study here, the neighbourhoods are of size N = 3, consisting of the unit itself and its two immediate neighbours on a chain, and N = 9, consisting of the unit itself and its 8 nearest neighbours on a square lattice (the 'Moore neighbourhood'). We will consider only binary units, for which we take Si(t) = ?1. Thus, if the neighbourhood (including the unit itself) includes N sites, f( ) is a Boolean function on the N -hypercube. There are 22N such functions. Wolfram [4) has divided the rules for such automata further into three classes: 1. Class 1: rules that lead to a uniform state. 2. Class 2: rules that lead to simple stable or periodic patterns. 3. Class 3: rules that lead to chaotic patterns. 4. Class 4: rules that lead to complex, long-lived transient patterns. Rules in the fourth cla.ss lie near (in a sense not yet fully understood [5)) a critical boundary between classes 2 and 3. They lead eventually to asymptotic behaviour in class 2 (or possibly 3); what distinguishes them is the length of the transient. It is classes 3 and 4 that we are interested in here. More specifically, for class 3 we expect that after the (short) initial transients, the motion is confined to some sort of attractor. Different attractors may be reached for a given rule, depending on initial conditions. For such systems we will focus on the dynamics on these attractors, not on the short transients. We will want to know what we can learn from a given history about the attractor characterizing it, about the asymptotic dynamics of the system generally (i.e. about all attractors), and, if possible, about the underlying rule. For class 4 CA, in contra.st, only the transients are of interest. Different initial conditions will give rise to very different transient histories; indeed, this sensitivity is the dynamical ba.sis for the capability for universal computation that has been Learning Cellular Automaton Dynamics with Neural Networks proved for some of these systems. Here we will want to know what we can learn from a portion of such a history about its future, as well as about the underlying rule. 2 REPRESENTING A CA AS A NETWORK Any Boolean function of N arguments can be implemented by a ~-n unit of order P ::; N with a threshold activation function, i.e. there exist weights wJlh ... jp such that I(SI, S2 ... SN) = sgn [. L . wJd~ ...jp Sjl Sh ... Sjp] . (1) Jl.J~.?"JP The indices ile run over the sites in the neighbourhood (1 to N) and zero, which labels a constant formal bias unit So = 1. Because the updating rule we are looking for is the same for the entire lattice, the weight WJ1 ... jp doesn't depend on i. Furthermore, because of the discrete nature of the outputs, the weights that implement a given rule are not unique; rather, there is a region of weight space for each rule. Although we could work with other architectures, it is natural to study networks with the same structure as the CA to be simulated. We therefore make a lattice of formal 1: - n neurons with short-range connections, which update themselves according to Vi(t+ 1) = 9 r.~ Wit ... jPVjl(t) ... Vjp(t)] , (2) Jt"'Jp In these investigations, we have assumed that we know a priori what the relevant neighbourhood size is, thereby fixing the connectivity of the network. At the end of the day, we will take the limit where the gain of the activation function 9 becomes infinite. However, during learning we use finite gain and continuous-valued units. We know that the order P of our ~ - n units need not be higher than the neighbourhood size N. However, in most cases a smaller P will do. More precisely, a network with any P > ~N can in principle (Le. given the right learning algorithm and sufficient training examples) implement almost all possible rules. This is an asymptotic result for large N but is already quite accurate for N = 3, where only two of the 256 possible rules are not implementable by a second-order unit, and N = 5, where we found from simple learning experiments that 99.87% of 10000 randomly-chosen rules could be implemented by a third-order unit. 3 LEARNING Having chosen a suitable value of P, we can begin our main task: training the network to simulate a CA, with the training examples {Si(t) - t Si(t + I)} taken from a particular known history. The translational invariance of the CA suggests that weight sharing is appropriate in the learning algorithm. On the other hand, we can imagine situations in which we did not possess a priori knowledge that the CA rule was the same for all units, 633 634 Wulff and Hertz or where we only had access to the automaton state in one neighbourhood. This case is analogous to the conventional time series extrapolation paradigm, where we typically only have access to a few variables in a large system. The difference is that here the accessible variables are binary rather than continuous. In these situations we should or are constrained to learn without each unit having access to error information at other units. In what follows we will perform the training both with and without weight sharing. The differences in what can be learned in the two cases will give interesting information about the CA dynamics being simulated. Most of our results are for chaotic (class 3) CA. For these systems, this training history is taken after initial transients have died out. Thus many of the 2N possible examples necessary to specify the rule at each site may be missing from the training set, and it is possible that our training procedure will not result in the network learning the underlying rule of the original system. It might instead learn another rule that coincides with the true one on the training examples. This is even more likely if we are not using weight sharing, because then a unit at one site does not have access to examples from the training history at other sites. However, we may relax our demand on the network, asking only that it evolve exactly like the original system when it is started in a configuration the original system could be in after transients have died out (Le. on an attractor of the original system). Thus we are restricting the test set in a way that is "fairer" to the network, given the instruction it has received. Of course, if the CA has more than one attractor, several rules which yield the same evolution on one attractor need not do so on another one. It is therefore possible that a network can learn the attractor of the training history (Le. will simula.te the original system correctly on a part of the history subsequent to the training sequence) but will not be found to evolve correctly when tested on data from another attractor. For class 4 automata, we cannot formulate the distinctions between different levels of learning meaningfully in terms of attractors, since the object of interest is the transient portion of the history. Nevertheless, we can still ask whether a network trained on part of the transient can learn the full rule, whether it can simulate the dynamics for other initial conditions, or whether it can extrapolate the training history. We therefore distinguish three degrees of successful learning: 1. Learning the rule, where the network evolves exactly like the original system from any initial configuration. 2. Learning the dynamics, the intermediate case where the network can simulate the original system exactly after transients, irrespective of initial conditions, despite not having learned the full rule. 3. Learning to continue the dynamics, where the successful simulation of the original system is only achieved for the particular initial condition used to generate the training history. Our networks are recurrent, but because they have no hidden units, they can be trained by a simple variant of the delta-rule algorithm. It can be obtained formally Learning Cellular Automaton Dynamics with Neural Networks from gradient descent on a modified cross entropy E = ~ '" [(1 + Si(t)) log 1 + Si~t~ + (1 _ L l+v.-t it Si(t)) log 1 - t ~~~t~l 8[-Si(t)Vi(t)] 1- t?t (3) We used the online version: f!lwith ... jp = 7]8[-Si(t+ l)l/i(t+ l)J[Si(t+ 1) - Vi(t+ l)]l';l(t)V}l(t).?. V}p(t) (4) This is like an extension of the Adatron algorithm[6} to E- n units, but with the added feature that we are using a nonlinear activation function. The one-dimensional N = 3 automata we simulated were the 9 legal cha.otic ones identified by Wolfram l4]. Using his system for labeling the rules, these are rules 18, 22, 54, 90, 122, 126, 146, 150, and 182. We used networks of order P = 3 so that all rules were learnable. (Rule 150 would not have been learnable by a second-order net.) Each network was a chain 60 units long, subjected to periodic boundary conditions. The training histories {Si (t)} were 1000 steps long, beginning 100 steps after randomly chosen initial configurations. To test for learning the rules, all neighbourhood configurations were checked at every site. To test for learning the dynamics, the CA were reinitialized with different random starting configurations and run 100 steps to eliminate transients, after which new test histories of length 100 steps were constructed. Networks were then tested on 100 such histories. The test set for continuing the dynamics was made simply by allowing the CA that had generated the training set to continue for 100 more steps. There are no class 4 CA among the one-dimensional N = 3 systems. As an example of such a rule, we chose the Game of Life which is defined on a square lattice with a neighbourhood size N = 9 and has been proved capable of universaJ computation (see, e.g. [7, 8]). We worked with a lattice of 60 x 60 units. The training history for the Game of Life consisted of 200 steps in the transient. The trained networks were tested, as in the case of the chaotic one-dimensional systems, on all possible configurations at every site (learning the rule), on other transient histories generated from different initial conditions (learning the dynamics), and on the evolution of the original system immediately following the training history (learning to continue the dynamics). 4 RESULTS With weight sharing, it proved possible to learn the dynamics for all 9 of the onedimensional chaotic rules very easily. In fact, it took no more than 10 steps of the training history to achieve this. Learning the underlying rules proved harder. After training on the histories of 1000 steps, the networks were able to do so in only 4 of the 9 cases. No qualitative difference in the two groups of patterns is evident to us from looking at their histories (Fig. 1). Nevertheless, we conclude that their ergodic properties must be different, at lea.st quantitatively. Life was also easy with weight sharing. Our network succeed in learning the underlying rule starting almost anywhere in the long transient. 635 636 Wulff and Hertz 22 54 90 126 182 Figure 1: Histories of the 4 one-dimensional rules that could be learned (top) and the 5 that could not (bottom) . (Learning with weight sharing.) Without weight sharing, all learning naturally proved more difficult. While it was possible to learn to continue the dynamics for all the one-dimensional chaotic rules, it proved impossible except in one case (rule 22) to learn the dynamics within the training history of 1000 steps. The networks failed on about 25% of the test histories. It was never possible to learn the underlying rule. Thus, apparently these chaotic states are not as homogeneous as they appear (at least on the time scale of the training period). Life is also difficult without weight sharing. Our network was unable even to continue the dynamics from histories of several hundred steps in the transient (Fig. 2). 5 DISCUSSION In previous studies of learning chaotic behaviour in single-variable time series (e.g. [1, 2, 3]), the test to which networks have been put has been to extrapolate the training series, i.e. to continue the dynamics. We have found that this is also possible in cellular automata for all the chaotic rules we have studied, even when only local information about the training history is available to the units. Thus, the CA evolution history at any site is rich enough to permit error-free extrapolation. However, local training data are not sufficient (except in one system, rule 22) to permit our networks to pass the more stringent test oflearning the dynamics. Thus, viewed from any single site, the different attra.ctors of these systems are dissimilar enough that data from one do not permit generalization to another. Learning Cellular Automaton Dynamics with Neural Networks . . t.=:;"......-0- ~ Ii,. , 0.i!-o.~ -.~ -- , ,, ,-, Q(. -= ,- '\1(0 , oc;> ('v ~ ~ . ... JI~ ~. 0 . _v ~ I (. ,."'~ I , ,- - :~". ...,.~. .~ +.~. -;. .-. Figure 2: The original Game of Life CA (left) and the network (right), both 20 steps a.fter the end of the training history. (Training done without weight sharing.) With the access to training data from other sites implied by weight sharing, the situation changes dramatically. Learning the dynamics is then very easy, implying that all possible asymptotic local dynamics that could occur for any initial condition actually do occur somewhere in the system in any given history. Furthermore, with weight sharing, not only the dynamics but also the underlying rule can be learned for some rules. This suggests that these rules are ergodic, in the sense that all configurations occur somewhere in the system at some time. This division of the chaotic rules into two classes according to this global ergodicity is a new finding . Turning to our class 4 example, Life proves to be impossible without weight sharing, even by OUr most lenient test, continuing the dynamics. Thus, although one might be tempted to think that the transient in Life is so long that it can be treated opera.tionallyas if it were a chaotic attractor, it cannot. For real chaotic attractors, in both in the CA studied here and continuous dynamical systems, networks can learn to continue the dynamics on the basis of local data, while in Life they cannot. On the other hand, the result that the rule of Life is easy to learn with weight sharing implies that looked at globally, the history of the transient is quite rich. Somewhere in the system, it contains sufficient information (together with the a priori knowledge that a second-order network is sufficient) to allow us to predict the evolution from any configuration correctly. This study is a very preliminary one and raises more questions than it answers. We would like to know whether the results we have obtained for these few simple systems are generic to complex and chaotic CA. To answer this question we will have to study systems in higher dimensions and with larger updating neighbourhoods. Perhaps significant universal patterns will only begin to emerge for large neighborhoods (cf [5]). However, we have identified some questions to ask about these problems. 637 638 Wulff and Hertz References [1J A Lapedes and R Farber, Nonlinear Signal Processing Using Neural Networks: Prediction and System Modelling, Tech Rept LA-UR-87 -2662, Los Alamos National Laboratory. Los Alamos NM USA [2] A S Weigend, B A Huberman a.nd D E Rumelhart, Int J Neural Systems 1 193-209 (1990) [3] K Stokbro, D K Umberger and J A Hertz, Complex Systems 4 603-622 (1991) [4] S Wolfram, Theory and Applications of Cellular Automata (World Scientific, 1986) [5] C G Langton, pp 12-37 in Emergent Computation (S Forrest, ed) MIT Press/North Holland, 1991 [6] J K Anlauf and M Biehl, Europhys Letters 10 687 (1989) [7] H V Mcintosh, Physica D 45 105-121 (1990) [8] S Wolfram, Physic a D 10 1-35 (1984)
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Is Input Sparsity Time Possible for Kernel Low-Rank Approximation? Cameron Musco MIT [email protected] David P. Woodruff Carnegie Mellon University [email protected] Abstract Low-rank approximation is a common tool used to accelerate kernel methods: the ? which can be stored n ? n kernel matrix K is approximated via a rank-k matrix K in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error k-rank approximation to K is at least as difficult as multiplying the input data matrix A 2 Rn?d by an arbitrary matrix C 2 Rd?k . Barring a breakthrough in fast matrix multiplication, when k is not too large, this requires ?(nnz(A)k) time where nnz(A) is the number of non-zeros in A. This lower bound matches, in many parameter regimes, recent work on subquadratic time algorithms for low-rank approximation of general kernels [MM16, MW17], demonstrating that these algorithms are unlikely to be significantly improved, in particular to O(nnz(A)) input sparsity runtimes. At the same time there is hope: we show for the first time that O(nnz(A)) time approximation is possible for general radial basis function kernels (e.g., the Gaussian kernel) for the closely related problem of low-rank approximation of the kernelized dataset. 1 Introduction The kernel method is a popular technique used to apply linear learning and classification algorithms to datasets with nonlinear structure. Given training input points a1 , ..., an 2 Rd , the idea is to replace the standard Euclidean dot product hai , aj i = aTi aj with the kernel dot product (ai , aj ), where : Rd ? Rd ! R+ is some positive semidefinite function. Popular kernel functions include 2 e.g., the Gaussian kernel with (ai , aj ) = e kai aj k / for some bandwidth parameter and the polynomial kernel of degree q with (ai , aj ) = (c + aTi aj )q for some parameter c. Throughout this work, we focus on kernels where (ai , aj ) is a function of the dot products aTi ai = kai k2 , aTj aj = kaj k2 , and aTi aj . Such functions encompass many kernels used in practice, including the Gaussian kernel, the Laplace kernel, the polynomial kernel, and the Matern kernels. Letting F be the reproducing kernel Hilbert space associated with (?, ?), we can write (ai , aj ) = h (ai ), (aj )i where : Rd ! F is a typically non-linear feature map. We let = T [ (a1 ), ..., (an )] denote the kernelized dataset, whose ith row is the kernelized datapoint (ai ). There is no requirement that can be efficiently computed or stored ? for example, in the case of the Gaussian kernel, F is an infinite dimensional space. Thus, kernel methods typically work with the kernel matrix K 2 Rn?n with Ki,j = (ai , aj ). We will also sometimes denote K = { (ai , aj )} T to make it clear which kernel function it is generated by. We can equivalently write K = . As long as all operations of an algorithm only access via the dot products between its rows, they can thus be implemented using just K without explicitly computing the feature map. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Unfortunately computing K is expensive, and a bottleneck for scaling kernel methods to large datasets. For the kernels we consider, where depends on dot products between the input points, we must at least compute the Gram matrix AAT , requiring ?(n2 d) time in general. Even if A is sparse, this takes ?(nnz(A)n) time. Storing K then takes ?(n2 ) space, and processing it for downstream applications like kernel ridge regression and kernel SVM can be even more expensive. 1.1 Low-rank kernel approximation For this reason, a vast body of work studies how to efficiently approximate K via a low-rank sur? [SS00, AMS01, WS01, FS02, RR07, ANW14, LSS13, BJ02, DM05, ZTK08, BW09, rogate K ? is rank-k, it can be stored in factored form in O(nk) space and CKS11, WZ13, GM13]. If K operated on quickly ? e.g., it can be inverted in just O(nk 2 ) time to solve kernel ridge regression. ? = Kk where Kk is K?s best k-rank approximation ? the projection onto One possibility is to set K ? the error kK Kk ? F , where kM kF is its top k eigenvectors.PKk minimizes, over all rank-k K, 2 1/2 the Frobenius norm: ( i,j Mi,j ) . It in fact minimizes error under any unitarily invariant norm, e.g., the popular spectral norm. Unfortunately, Kk is prohibitively expensive to compute, requiring ?(n3 ) time in practice, or n! in theory using fast matrix multiplication, where ! ? 2.373 [LG14]. ? which is nearly as good The idea of much prior work on low-rank kernel approximation is to find K ? fulfilling the as Kk , but can be computed much more quickly. Specifically, it is natural to ask for K following relative error guarantee for some parameter ? > 0: kK ? F ? (1 + ?)kK Kk K k kF . (1) Other goals, such as nearly matching the spectral norm error kK Kk k or approximating K entrywise have also been considered [RR07, GM13]. Of particular interest to our results is the closely T related goal of outputting an orthonormal basis Z 2 Rn?k satisfying for any with = K: k ZZ T kF ? (1 + ?)k k kF . (2) (2) can be viewed as a Kernel PCA guarantee ? its asks us to find a low-rank subspace Z such that the projection of our kernelized dataset onto Z nearly optimally approximates this dataset. Given T ? = ZZ T Z, we can approximate K using K ZZ T = ZZ T KZZ T . Alternatively, letting P T ? = P T , which can be computed be the projection onto the row span of ZZ , we can write K efficiently, for example, when P is a projection onto a subset of the kernelized datapoints [MM16]. 1.2 Fast algorithms for relative-error kernel approximation Until recently, all algorithms achieving the guarantees of (1) and (2) were at least as expensive as computing the full matrix K, which was needed to compute the low-rank approximation [GM13]. However, recent work has shown that this is not required. Avron, Nguyen, and Woodruff [ANW14] demonstrate that for the polynomial kernel, Z satisfying (2) can be computed in O(nnz(A)q) + n poly(3q k/?) time for a polynomial kernel with degree q. Musco and Musco [MM16] give a fast algorithm for any kernel, using recursive Nystr?m sampling, ? (in factored form) satisfying kK ? ? , for input parameter . With which computes K Kk the proper setting of , it can output Z satisfying (2) (see Section C.3 of [MM16]). Computing ! 1 ? ? Z requires evaluating O(k/?) columns of the kernel matrix along with O(n(k/?) ) additional time for other computations. Assuming the kernel is a function of the dot products between the ? input points, the kernel evaluations require O(nnz(A)k/?) time. The results of [MM16] can also be ? satisfying (1) with ? = pn in O(nnz(A)k ? used to compute K + nk ! 1 ) time (see Appendix A of [MW17]). Woodruff and Musco [MW17] show that for any kernel, and for any ? > 0, it is p possible to ? ? nk/?2 )? achieve (1) in O(nnz(A)k/?)+n poly(k/?) time plus the time needed to compute an O( p ? nk/?) submatrix of K. If A has uniform row sparsity ? i.e., nnz(ai ) ? c nnz(A)/n for some O( p 2.5 ? constant c and all i, this step can be done in O(nnz(A)k/? ) time. Alternatively, if d ? ( nk/?2 )? 4 4 ? ? for ? < .314 this can be done in O(nk/? ) = O(nnz(A)k/? ) time using fast rectangular matrix multiplication [LG12, GU17] (assuming that there are no all zero data points so n ? nnz(A).) 2 1.3 Our results The algorithms of [MM16, MW17] make significant progress in efficiently solving (1) and (2) for general kernel matrices. They demonstrate that, surprisingly, a relative-error low-rank approximation can be computed significantly faster than the time required to write down all of K. A natural question is if these results can be improved. Even ignoring ? dependencies and typically lower order terms, both algorithms use ?(nnz(A)k) time. One might hope to improve this to input ? sparsity, or near input sparsity time, O(nnz(A)), which is known for computing a low-rank approximation of A itself [CW13]. The work of Avron et al. affirms that this is possible for the kernel PCA guarantee of (2) for degree-q polynomial kernels, for constant q. Can this result be extended to other popular kernels, or even more general classes? 1.3.1 Lower bounds We show that achieving the guarantee of (1) significantly more efficiently than the work of [MM16, MW17] is likely very difficult. Specifically, we prove that for a wide class of kernels, the kernel low-rank approximation problem is as hard as multiplying the input A 2 Rn?d by an arbitrary C 2 Rd?k . We have the following result for some common kernels to which our techniques apply: Theorem 1 (Hardness for low-rank kernel approximation). Consider any polynomial kernel 2 (mi , mj ) = (c + mTi mj )q , Gaussian kernel (mi , mj ) = e kmi mj k / , or the linear kernel (mi , mj ) = mTi mj . Assume there is an algorithm which given M 2 Rn?d with associated kernel matrix K = { (mi , mj )}, returns N 2 Rn?k in o(nnz(M )k) time satisfying: kK N N T k2F ? kK Kk k2F for some approximation factor . Then there is an o(nnz(A)k) + O(nk 2 ) time algorithm for multiplying arbitrary integer matrices A 2 Rn?d , C 2 Rd?k . The above applies for any approximation factor . While we work in the real RAM model, ignoring bit complexity, as long as = poly(n) and A, C have polynomially bounded entries, our reduction from multiplication to low-rank approximation is achieved using matrices that can be represented with just O(log(n + d)) bits per entry. p ? Theorem 1 shows that the runtime of O(nnz(A)k + nk ! 1 ) for = n achieved by [MM16] for general kernels cannot be significantly improved without advancing the state-of-the-art in matrix multiplication. Currently no general algorithm is known for multiplying integer A 2 Rn?d , C 2 Rd?k in o(nnz(A)k) time, except when k n? for ? < .314 and A is dense. In this case, AC can be computed in O(nd) time using fast rectangular matrix multiplication [LG12, GU17]. p As discussed, when A has uniform row sparsity or when d ? ( nk/?2 )? , the runtime of [MW17] ? for = (1 + ?), ignoring ? dependencies and typically lower order terms, is O(nnz(A)k), which is also nearly tight. In recent work, Backurs et al. [BIS17] give lower bounds for a number of kernel learning problems, including kernel PCA for the Gaussian kernel. However, their strong bound, of ?(n2 ) time, requires very small error = exp( !(log2 n), whereas ours applies for any relative error . 1.3.2 Improved algorithm for radial basis function kernels In contrast to the above negative result, we demonstrate that achieving the alternative Kernel PCA guarantee of (2) is possible in input sparsity time for any shift and rotationally invariant kernel ? e.g., any radial basis function kernel where (xi , xj ) = f (kxi xj k). This result significantly extends the progress of Avron et al. [ANW14] on the polynomial kernel. Our algorithm is based off a fast implementation of the random Fourier features method [RR07], which uses the fact that that the Fourier transform of any shift invariant kernel is a probability distribution after appropriate scaling (a consequence of Bochner?s theorem). Sampling frequencies from this distribution gives an approximation to (?, ?) and consequentially the matrix K. 3 ? 2n random We employ a new analysis of this method [AKM+ 17], which shows that sampling O ? ? = ? ? T satisfying the spectral approximation guarantee: Fourier features suffices to give K ? + I) K + I (1 + ?)(K ? + I). (1 ?)(K If we set ? k+1 (K)/k, we can show that ? also gives a projection-cost preserving sketch + [CEM 15] for the kernelized dataset . This ensures that any Z satisfying k ? ZZ T ? k2F ? 2 (1 + ?)k ? ? k k2F also satisfies k ZZ T k2F ? (1 + O(?))k k kF and thus achieves (2). ? ? ? 2n = O ? 2 nk Our algorithm samples s = O ? ? k+1 (K) random Fourier features, which naively requires O(nnz(A)s) time. We show that this can be accelerated to O(nnz(A)) + poly(n, s) time, using a recent result of Kapralov et al. on fast multiplication by random Gaussian matrices [KPW16]. Our technique is analogous to the ?Fastfood? approach to accelerating random Fourier features using fast Hadamard transforms [LSS13]. However, our runtime scales with nnz(A), which can be signif? icantly smaller than the O(nd) runtime given by Fastfood when A is sparse. Our main algorithmic result is: Theorem 2 (Input sparsity time kernel PCA). There is an algorithm that given A 2 Rn?d along with shift and rotation-invariant kernel function : Rd ? Rd ! R+ with (x, x) = 1, outputs, with probability 99/100, Z 2 Rn?k satisfying: 2 k ZZ T k2F ? (1 + ?)k k kF T for any with = K = { (ai , aj )} and any ? > 0. Letting k+1 denote the (k + 1)th largest eigenvalue of K and be the exponent of fast matrix multiplication, the algorithm runs in ? ! < 2.373 ? ?! 1.5 ? k ? n!+1.5 ? O(nnz(A)) + O time. 2 k+1 ? We note that the runtime of our algorithm is O(nnz(A)) whenever n, k, 1/ k+1 , and 1/? are not too large. Due to the relatively poor dependence on n, the algorithm is relevant for very high dimensional datasets with d n. Such datasets are found often, e.g., in genetics applications [HDC+ 01, JDMP11]. While we have dependence on 1/ k+1 , in the natural setting, we only compute a low-rank approximation up to an error threshold, ignoring very small eigenvalues of K, and so k+1 will not be too small. We do note that if we apply Theorem 2 to the low-rank approximation instances given by our lower bound construction, k+1 can be very small, ? 1/ poly(n, d) for matrices with poly(n) bounded entries. Thus, removing this dependence is an important open question in understanding the complexity of low-rank kernel approximation. We leave open the possibility of improving our algorithm, achieving O(nnz(A)) + n ? poly(k, ?) runtime, which would match the state-of-the-art for low-rank approximation of non-kernelized matrices [CW13]. Alternatively, it is possible that a lower bound can be shown, proving the that high n dependence, or the 1/ k+1 term are required even for the Kernel PCA guarantee of (2). 2 Lower bounds Our lower bound proof argues that for a broad class of kernels, given input M , a low-rank approximation of the associated kernel matrix K achieving (1) can be used to obtain a close approximation to the Gram matrix M M T . We write (mi , mj ) as a function of mTi mj (or kmi mj k2 for distance kernels) and expand this function as a power series. We show that the if input points are appropriately rescaled, the contribution of degree-1 term mTi mj dominates, and hence our kernel matrix approximates M M T , up to some easy to compute low-rank components. We then show that such an approximation can be used to give a fast algorithm for multiplying any two integer matrices A 2 Rn?d and C 2 Rd?k . The key idea is to set M = [A, wC] where w is a large weight. We then have: ? AAT wAC MMT = . wC T AT w2 C T C Since w is very large, the AAT block is relatively very small, and so M M T is nearly rank-2k ? it has a ?heavy? strip of elements in its last k rows and columns. Thus, computing a relative-error rank-2k approximation to M M T recovers all entries except those in the AAT block very accurately, and importantly, recovers the wAC block and so the product AC. 4 2.1 Lower bound for low-rank approximation of M M T . We first illustrate our lower bound technique by showing hardness of direct approximation of M M T . Theorem 3 (Hardness of low-rank approximation for M M T ). Assume there is an algorithm A which given any M 2 Rn?d returns N 2 Rn?k such that kM M T N N T k2F ? 1 kM M T (M M T )k k2F in T (M, k) time for some approximation factor 1 . T T For any A 2pRn?d and C 2 Rd?k each with integer entries in [ 2 , 2 ], let B = [A , wC] 2 ! 1 where w = 3 nd. It is possible to compute the product AC in time T (B, 2k) + O(nk ). 1 2 Proof. We can write the (n + k) ? (n + k) matrix BB T as: ? AAT BB T = [AT , wC]T [A, wC] = wC T AT wAC . w2 C T C Let Q 2 Rn?2k be an orthogonal span for the columns of the n ? 2k matrix: ? 0 wAC V w2 C T C where V 2 Rk?k spans the columns of wC T AT 2 Rk?n . The projection QQT BB T gives the best Frobenius norm approximation to BB T in the span of Q. We can see that: ? 2 AAT 0 kBB T (BB T )2k k2F ? kBB T QQT BB T k2F ? ? 42 n2 d2 (3) 0 0 F since each entry of A is bounded in magnitude by 2 and so each entry of AAT is bounded by d 2 2. Let N be the matrix returned by running A on B with rank 2k. In order to achieve the approximation bound of kBB T N N T k2F ? 1 kBB T (BB T )2k k2F we must have, for all i, j: (BB T N N T )2i,j ? kBB T N N T k2F ? 1 4 2 2 2n d p 2 where the last inequality is from (3). This gives |BB T N N T |i,j ? 1 2 nd. Since A and T Cphave integer entries, each entry in the submatrix wAC of BBp is an integer multiple of w = 2 T 2 3 1 2 nd. Since (N N )i,j approximates this entry to error 1 2 nd, by simply rounding T (N N )i,j to the nearest multiple of w, we obtain the entry exactly. Thus, given N , we can exactly recover AC in O(nk ! 1 ) time by computing the n?k submatrix corresponding to AC in BB T . Theorem 3 gives our main bound Theorem 1 for the case of the linear kernel (mi , mj ) = mTi mj . Proof of Theorem 1 ? Linear Kernel. We apply Theorem 3 after noting that for B = [AT , wC]T , nnz(B) ? nnz(A) + nk and so T (B, 2k) = o(nnz(A)k) + O(nk 2 ). We show in Appendix A that there is an algorithm which nearly matches the lower bound of Theorem 1 for any = (1 + ?) for any ? > 0. Further, in Appendix B we show that even just outputting an ? = ZZ T M M T is a relative-error low-rank approximation orthogonal matrix Z 2 Rn?k such that K T ? itself, is enough to give fast multiplication of of M M , but not computing a factorization of K integer matrices A and C. 2.2 Lower bound for dot product kernels We now extend Theorem 3 to general dot product kernels ? where (ai , aj ) = f (aTi aj ) for some function f . This includes, for example, the polynomial kernel. Theorem 4 (Hardness of low-rank approximation for dot product kernels). Consider any kernel : Rd P ? Rd ! R+ with (ai , aj ) = f (aTi aj ) for some function f which can be expanded as 1 f (x) = q=0 cq xq with c1 6= 0 and |cq /c1 | ? Gq 1 and for all q 2 and some G 1. Assume there is an algorithm A which given M 2 Rn?d with kernel matrix K = { (mi , mj )}, returns N 2 Rn?k satisfying kK N N T k2F ? 1 kK Kk k in T (M, k) time. T T For any A 2 Rn?d , C 2 Rd?k with integer entries in [ 2 , 2 ], let B = [w1 A , w2 C] with w1 = 1 p w2 2 p , w = . Then it is possible to compute AC in time T (B, 2k + 1) + O(nk ! 1 ). 2 12 nd 4 Gd 1 2 2 5 Proof. Using our decomposition of (?, ?), we can write the kernel matrix for B and as: ? ? 1 1 w12 AAT w1 w2 AC K = c0 + c1 + c2 K (2) + c3 K (3) + ... 1 1 w1 w2 C T AT w22 C T C (4) (q) where Ki,j = (bTi bj )q and 1 denotes the all ones matrix of appropriate size. The key idea is to show that the contribution of the K (q) terms is small, and so any relative-error rank-(2k+1) approximation to K must recover an approximation to BB T , and thus the product AC as in Theorem 3. By our setting of w2 = we have for all i, j, 1 X q=2 p 1 4 Gd |bTi bj | 2 , the fact that w1 < w2 , and our bound on the entries of A and C, ? w22 d cq Ki,j ? c1 |bTi bj | ? 1 X (q) 1 16G . < Gq q=2 1 Thus, for any i, j, using that |cq /c1 | ? Gq |bTi bj |q 1 ? c1 |bTi bj | 1 X q=2 Gq 1 (16G)q 1 ? 1 : 1 c1 |bTi bj |. 12 (5) ? 1 1 ? just has its last K c0 , with its top right n ? n block set to 0. K 1 1 k columns and rows non-zero, so has rank ? 2k. Let Q 2 Rn?2k+1 be an orthogonal span for the ? along with the all ones vector of length n. Let N be the result of running A on B with columns K rank 2k + 1. Then we have: ? be the matrix Let K kK ? 2 2 ? N N T k2F ? 1 kK K2k+1 k2F ? QQT Kk2F 1 kK ? 1 ? ? (2) + ...) (c1 w12 AAT + c2 K 0 0 0 2 (6) F ? (q) denotes the top left n ? n submatrix of K (q) . By our bound on the entries of A and (5): where K ? ? 13 ? (2) + c3 K ? (3) + ... c1 w12 AAT + c2 K ? c1 w12 AAT i,j ? 2c1 w12 d 22 . 12 i,j Plugging back into (6) and using w1 = (K 12 N N T )i,j ? kK p w2 1 2 nd 2 , this gives for any i, j: N N T kF ? p p 1n 2 ? 2c1 w12 d 2 2 2 1 n ? 2c1 d 2 p ? w1 w2 2 12 1 2 nd w1 w2 c1 ? . 6 ? (7) Since A and C have integer entries, each entry of c1 w1 w2 AC is an integer multiple of c1 w1 w2 . By the decomposition of (4) and the bound of (5), if we subtract c0 from the corresponding entry of K and round it to the nearest multiple of c1 w1 w2 , we will recover the entry of AC. By the bound of (7), we can likewise round the corresponding entry of N N T . Computing all nk of these entries given N takes time O(nk ! 1 ), giving the theorem. Theorem 4 lets us lower bound the time to compute a low-rank kernel approximation for any kernel function expressible as a reasonable power expansion of aTi aj . As a straightforward example, it gives the lower bound for the polynomial kernel of any degree stated in Theorem 1. Proof of Theorem 1 ? Polynomial Kernel. We apply Theorem 4, noting that (mi , mj ) = (c + Pq mTi mj )q can be written as f (mTi mj ) where f (x) = j=0 cj xj with cj = cq j qj . Thus c1 6= 0 and |cj /c1 | ? Gj 1 for G = (q/c). Finally note that nnz(B) ? nnz(A) + nk giving the result. 2.3 Lower bound for distance kernels We finally extend Theorem 4 to handle kernels like the Gaussian kernel whose value depends on the squared distance kai aj k2 rather than just the dot product aTi aj . We prove: 6 Theorem 5 (Hardness of low-rank approximation for distance kernels). Consider any kernel function : RdP ?Rd ! R+ with (ai , aj ) = f (kai aj k2 ) for some function f which can be expanded 1 as f (x) = q=0 cq xq with c1 6= 0 and |cq /c1 | ? Gq 1 and for all q 2 and some G 1. Assume there is an algorithm A which given input M 2 Rn?d with kernel matrix K = { (mi , mj )}, returns N 2 Rn?k satisfying kK N N T k2F ? 1 kK Kk k in T (M, k) time. T T For any A 2 Rn?d , C 2 Rd?k with integer entries in [ 2 , 2 ], let B = [w1 A , w2 C] with 1 p w1 = 36p w2 2 nd , w2 = (16Gd2 4 )(36 . It is possible to compute AC in T (B, 2k + 3) + 2 nd) 1 O(nk ! 1 2 ) time. 2 1 2 The proof of Theorem 5 is similar to that of Theorem 4, and relegated to Appendix C. The key idea is to write K as a polynomial in the distance matrix D with Di,j = kbi bj k22 . Since kbi bj k22 = kbi k22 + kbj k22 2bTi bj , D can be written as 2BB T plus a rank-2 component. By setting w1 , w2 sufficiently small, as in the proof of Theorem 4, we ensure that the higher powers of D are negligible, and thus that our low-rank approximation must accurately recover the submatrix of BB T corresponding to AC. Theorem 5 gives Theorem 1 for the popular Gaussian kernel: Proof of Theorem 1 ? Gaussian Kernel. (mi , mj ) can be written as f (kmi mj k2 ) where f (x) = P1 )q q e x/ = q=0 ( 1/ x . Thus c1 6= 0 and |cq /c1 | ? Gq 1 for G = 1/ . Applying Theorem 5 q! and bounding nnz(B) ? nnz(A) + nk, gives the result. 3 Input sparsity time kernel PCA for radial basis kernels Theorem 1 gives little hope for achieving o(nnz(A)k) time for low-rank kernel approximation. ? Here we show that However, the guarantee of (1) is not the only way of measuring the quality of K. for shift/rotationally invariant kernels, including e.g., radial basis kernels, input sparsity time can be achieved for the kernel PCA goal of (2). 3.1 Basic algorithm Our technique is based on the random Fourier features technique [RR07]. Given any shift-invariant kernel, (x, y) = (x y) with (0) = 1 (we will assume this w.l.o.g. as the function can always be scaled), there is a probability density function p(?) over vectors in Rd such that: Z T (x y) = e 2?i? (x y) p(?)d?. (8) Rd p(?) is just the (inverse) Fourier transform of (?), and is a density function by Bochner?s theorem. Informally, given A 2 Rn?d if we let Z denote the matrix with columns z(?) indexed by ? 2 Rd . T z(?)j = e 2?i? aj . Then (8) gives ZP Z ? = K where P is diagonal with P?,? = p(?), and Z ? denotes the Hermitian transpose. The idea of random Fourier features is to select s frequencies ?1 , ..., ?s according to the density p(?) ? = Z? Z? T is then used to approximate K. and set Z? = p1s [z(?1 ), ...z(?s )]. K In recent work, Avron et al. [AKM+ 17] give a new analysis of random Fourier features. Extending prior work on ridge leverage scores in the discrete setting [AM15, CMM17], they define the ridge leverage function for parameter > 0: ? (?) = p(?)z(?)? (K + I) 1 z(?) (9) ? that spectrally approximates K, they prove the following: As part of their results, which seek K Lemma 6. For all ?, ? (?) ? n/ . While simple, this bound is key to our algorithm. It was shown in [CMM17] that if the columns of a matrix are sampled by over-approximations to their ridge leverage scores (with appropriately set ), the sample is a projection-cost preserving sketch for the original matrix. That is, it can be used as a surrogate in computing a low-rank approximation. The results of [CMM17] carry over to the continuous setting giving, in conjunction with Lemma 6: 7 Lemma 7 (Projection-cost preserving sketch via random Fourier features). Consider any A 2 Rn?d and shift-invariant kernel (?) with (0) = 1, with associated kernel matrix K = { (ai aj )} Pn ) and kernel Fourier transform p(?). For any 0 < ? k1 i=k+1 i (K), let s = cn log(n/ for ?2 1 ? p sufficiently large c and let Z = s [z(?1 ), ..., z(?s )] where ?1 , ..., ?s are sampled independently according to p(?). Then with probability 1 , for any orthonormal Q 2 Rn?k and any with T = K: (1 ?)kQQT Z? ? 2F ? kQQT Zk By (10) if we compute Q satisfying kQQT Z? kQQT k2F ? (1 + ?)2 kZ? k2F ? (1 + ?)kQQT Z? ? 2 ? (1 + ?)kZ? Zk F ? 2F . Zk (10) Z?k k2F then we have: (1 + ?)2 Z?k k2F ? kUk UkT 1 ? = (1 + O(?))k k2F 2 k kF where Uk 2 Rn?k contains the top k column singular vectors of . By adjusting constants on ? by making c large enough, we thus have the relative error low-rank approximation guarantee of (2). It remains to show that this approach can be implemented efficiently. 3.2 Input sparsity time implementation Given Z? sampled as in Lemma 7, we can find a near optimal subspace Q using any input sparsity time low-rank approximation algorithm (e.g., [CW13, NN13]). We have the following Corollary: ? 2 nk Corollary 8. Given Z? sampled as in Lemma 7 with s = ?( ), there is an algorithm running ? ? 2 in time O( ? n2 k ) k+1 (K) k+1 (K) that computes Q satisfying with high probability, for any kQQT k2F ? (1 + ?)k with T = K: 2 k kF . ? With Corollary 8 in place the main bottleneck to our approach becomes computing Z. 3.2.1 Sampling Frequencies ? we first sample ?1 , ..., ?s according to p(?). Here we use the rotational invariance of To compute Z, (?). In this case, p(?) is also rotationally invariant [LSS13] and so, letting p?(?) be the distribution over norms of vectors sampled from p(?) we can sample ?1 , ..., ?n by first selecting s random Gaussian vectors and then rescaling them to have norms distributed according to p?(?). That is, we can write [?1 , ..., ?n ] = GD where G 2 Rd?s is a random Gaussian matrix and D is a diagonal m rescaling matrix with Dii = kG with m ? p?. We will assume that p? can be sampled from in ik O(1) time. This is true for many natural kernels ? e.g., for the Gaussian kernel, p? is just a Gaussian density. 3.2.2 Computing Z? Due to our large sample size, s > n, even writing down G above requires ?(nd) time. However, to form Z? we do not need G itself: it suffices to compute for m = 1, ..., s the column z(?m ) T with z(?m )j = e 2?i?m aj . This requires computing AGD, which contains the appropriate dot products aTj ?m for all m, j. We use a recent result [KPW16] which shows that this can be performed approximately in input sparsity time: Lemma 9 (From Theorem 1 of [KPW16]). There is an algorithm running in O(nnz(A) + log4 dn3 s! 1.5 ) time which outputs random B whose distribution has total variation distance at most from the distribution of AG where G 2 Rd?s is a random Gaussian matrix. Here, ! < 2.373 is the exponent of fast matrix multiplication. Proof. Theorem 1 of [KPW16] shows that for B to have total variation distance from the distribution of AG it suffices to set B = ACG0 where C is a d ? O(log4 dn2 s1/2 / ) CountSketch matrix 8 and G0 is an O(log4 dn2 s1/2 / ) ? s random Gaussian matrix. Computing AC requires O(nnz(A)) 4 3 1.5 time. Multiplying the result by G0 then requires O( log dn s ) time if fast matrix multiplication is not employed. Using fast matrix multiplication, this can be improved to O( log 4 dn3 s! 1.5 ). Applying Lemma 9 with = 1/200 lets us compute random BD with total variation distance 1/200 from AGD. Thus, the distribution of Z? generated from this matrix has total variation distance ? 1/200 from the Z? generated from the true random Fourier features distribution. So, by Corollary 2 8, we can use Z? to compute Q satisfying kQQT k2F ? (1 + ?)k k kF with probability 1/100 accounting for the the total variation difference and the failure probability of Corollary 8. This yields our main algorithmic result, Theorem 2. 3.3 An alternative approach We conclude by noting that near input sparsity time Kernel PCA can also be achieved for a broad class of kernels using a very different approach. We can approximate (?, ?) via an expansion into polynomial kernel matrices as is done in [CKS11] and then apply the sketching algorithms for the polynomial kernel developed in [ANW14]. As long as the expansion achieves high accuracy with low degree, and as long as 1/ k+1 is not too small ? since this will control the necessary approxima? tion factor, this technique can yield runtimes of the form O(nnz(A))+poly(n, k, 1/ k+1 , 1/?), giving improved dependence on n for some kernels over our random Fourier features method. Improving the poly(n, k, 1/ k+1 , 1/?) term in both these methods, and especially removing the 1/ k+1 dependence and achieving linear dependence on n is an interesting open question for future work. 4 Conclusion In this work we have shown that for a broad class of kernels, including the Gaussian, polynomial, and linear kernels, given data matrix A, computing a relative error low-rank approximation to A?s kernel matrix K (i.e., satisfying (1)) requires at least ?(nnz(A)k) time, barring a major breakthrough in the runtime of matrix multiplication. In the constant error regime, this lower bound essentially matches the runtimes given by recent work on subquadratic time kernel and PSD matrix low-rank approximation [MM16, MW17]. We show that for the alternative kernel PCA guarantee of (2), a potentially faster runtime of O(nnz(A)) + poly(n, k, 1/ k+1 , 1/?) can be achieved for general shift and rotation-invariant kernels. Practically, improving the second term in our runtime, especially the poor dependence on n, is an important open question. Generally, computing the kernel matrix K explicitly requires O(n2 d) time, and so our algorithm only gives runtime gains when d is large compared to n ? at least ?(n! .5 ), even ignoring k, k+1 , and ? dependencies. Theoretically, removing the dependence on k+1 would be of interest, as it would give input sparsity runtime without any assumptions on the matrix A (i.e., that k+1 is not too small). Resolving this question has strong connections to finding efficient kernel subspace embeddings, which approximate the full spectrum of K. References [AKM+ 17] Haim Avron, Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker, and Amir Zandieh. Random Fourier features for kernel ridge regression: Approximation bounds and statistical guarantees. In Proceedings of the 34th International Conference on Machine Learning (ICML), 2017. 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The Expxorcist: Nonparametric Graphical Models Via Conditional Exponential Densities Arun Sai Suggala ? Carnegie Mellon University Pittsburgh, PA 15213 Mladen Kolar ? University of Chicago Chicago, IL 60637 Pradeep Ravikumar ? Carnegie Mellon University Pittsburgh, PA 15213 Abstract Non-parametric multivariate density estimation faces strong statistical and computational bottlenecks, and the more practical approaches impose near-parametric assumptions on the form of the density functions. In this paper, we leverage recent developments to propose a class of non-parametric models which have very attractive computational and statistical properties. Our approach relies on the simple function space assumption that the conditional distribution of each variable conditioned on the other variables has a non-parametric exponential family form. 1 Introduction Let X = (X1 , . . . , Xp ) be a p-dimensional random vector. Let G = (V, E) be the graph that encodes conditional independence assumptions underlying the distribution of X, that is, each node of the graph corresponds to a component of vector X and (a, b) ? E if and only if Xa ? 6 ? Xb | X?ab with X?ab := {Xc | c ? V \{a, b}}. The graphical model represented by G is then the set of distributions over X that satisfy the conditional independence assumptions specified by the graph G. There has been a considerable line of work on learning parametric families of such graphical model distributions from data [22, 20, 13, 28], where the distribution is indexed by a finite-dimensional parameter vector. The goal of this paper, however, is on specifying and learning nonparametric families of graphical model distributions, indexed by infinite-dimensional parameters, and for which there has been comparatively limited work. Non-parametric multivariate density estimation broadly, even without the graphical model constraint, has not proved as popular in practical machine learning contexts, for both statistical and computational reasons. Loosely, estimating a non-parametric multivariate density, with mild assumptions, typically requires the number of samples to scale exponentially in the dimension p of the data, which is infeasible even in the big-data era when n is very large. And the resulting estimators are typically computationally expensive or intractable, for instance requiring repeated computations of multivariate integrals. We present a review of multivariate density estimation, that is necessarily incomplete but sets up our proposed approach. A common approach dating back to [15] uses the logistic density transform to satisfy the unity and positivity constraints for densities, and considers densities of the form f (X) = R exp(?(X)) , with some constraints on ? for identifiability such as ?(X0 ) = 0 for some exp(?(x))dx XR X0 ? X or X ?(x)dx = 0. With the logistic density transform, differing approaches for non-parametric density estimation can be contrasted in part by their assumptions on the infinite-dimensional function space domain of ?(?). An early approach [8] considered function spaces of functions with bounded ?roughness? functionals. The predominant line of work however has focused on the setting where ?(?) lies in a Reproducing Kernel Hilbert Space (RKHS), dating back to [21]. Consider the estimation of these logistic density ? [email protected] ? [email protected] ? [email protected] 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. transforms ?(X) given n i.i.d. samples Xn = {X (i) }ni=1 drawn from f? (X). A natural loss functional is penalized log likelihood, with a penalty a smooth fit with respect R P functional that ensures to the function space domain: `(?; Xn ) := ? n1 i?[n] ?(X (i) ) + log exp(?(x))dx + ? pen(?), for functions ?(?) that lie in an RKHS H, and where pen(?) = k?k2H is the squared RKHS norm. This was studied by many [21, 11, 6]. A crucial caveat is that the representer theorem for RKHSs does not hold. Nonetheless, one can consider finite-dimensional function space approximations consisting of the linear span of kernel functions evaluated at the sample points [12]. Computationally this still scales poorly with the dimension due to the need to compute multidimensional integrals of the form R exp(?(x)dx which do not, in general, decompose. These approximations also do not come with strong statistical guarantees. We briefly note that the function space assumption that ?(?) lies in an RKHS could also be viewed from the lens of an infinite-dimensional exponential family [4]. Specifically, let H be a Reproducing Kernel Hilbert Space with reproducing kernel k(?, ?), and inner product h?, ?iH . Then ?(X) = h?(?), k(X, ?)iH , so that the density f (X) can in turn be viewed as a member of an infinite-dimensional exponential family with sufficient statistics k(X, ?) : X 7? H, and natural parameter ?(?) ? H. Following this viewpoint, [4] propose estimators via linear span approximations similar to [11]. Due to the computational caveat with exact likelihood based functionals, a line of approaches have focused on instead. [14] study the following loss functional: R Ppenalized surrogate likelihoods `(?; Xn ) := n1 i?[n] exp(??(X (i) ))+ ?(x)?(x)dx+?pen(?), where ?(X) is some fixed known density with the same support as the unknown density f (X). While this estimation procedure is much more computationally amenable than minimizing the exact penalized likelihood, the caveat, however, is that for a general RKHS this requires solving higher order integrals. The next level of simplification has thus focused on the form of the logistic transform function itself. There has been a line of work on an ANOVA logistic density function into node-wise and Pptype decomposition Pp of Pthe p pairwise terms: ?(X) = s=1 ?s (Xs ) + s=1 t=s+1 ?st (Xs , Xt ). A line of work has coupled such a decomposition with the assumption that each of the terms lie in an RKHS. This does not immediately provide a computational benefit: with penalized likelihood based loss functionals, the loss functional does not necessarily decompose into such node and pairwise terms. [24] thus couple this ANOVA type pairwise decomposition with a score matching based objective. [10] use the above decomposition with the surrogate loss functional of [14] discussed above, but note that this still requires the aforementioned function space approximation as a linear span of kernel evaluations, as well as two-dimensional integrals. A line of recent work has thus focused on further stringent assumptions on the density function space, by assuming some components of the logistic transform to be finite-dimensional. [30] use an ANOVA decomposition but assume the terms belong to finite-dimensional function spaces instead of RKHSs, specified by a pre-defined finite set of basis functions. [29] consider logistic transform functions ?(?) that have the pairwise decomposition above, with a specific class of parametric pairwise functions ?st Xs Xt , and non-parametric node-wise functions. [17, 16] consider the problem of estimating monotonic node-wise functions such that the transformed random vector is multivariate Gaussian; which could also be viewed as estimating a Gaussian copula density. To summarize the (necessarily incomplete) review above, non-parametric density estimation faces strong statistical and computational bottlenecks, and the more practical approaches impose stringent near-parametric assumptions on the form of the (logistic transform of the) density functions. In this paper, we leverage recent developments to propose a very computationally simple non-parametric density estimation algorithm, that still comes with strong statistical guarantees. Moreover, the density could be viewed as a graphical model distribution, with a corresponding sparse conditional independence graph. Our approach relies on the following simple function space assumption: that the conditional distribution of each variable conditioned on the other variables has a non-parametric exponential family form. As we show, for there to exist a consistent joint density, the logistic density transform with respect to a particular necessarily Pp decomposes into the following semi-parametric Pp base measureP p form: ?(X) = s=1 ?s Bs (Xs ) + s=1 t=s+1 ?st Bs (Xs ) Bt (Xt ) in the pairwise case, with both a parametric component {?s : s = 1, . . . , p}, {?st : s < t; s, t = 1, . . . , p}, as well as non-parametric components {Bs : s = 1, . . . , p}. We call this class of models the ?expxorcist?, fol- 2 lowing other ?ghostbusting? semi-parametric models such as the nonparanormal and nonparanormal skeptic [17, 16]. Since the conditional distributions are exponential families, we show that there exist computationally amenable estimators, even in our more general non-parametric setting, where the sufficient statistics have to be estimated as well. The statistical analysis in our non-parametric setting however is more subtle, due in part to non-convexity and in part to the non-parametric setting. We also show how the Expxorcist class of densities is closely related to a semi-parametric exponential family copula density that generalizes the Gaussian copula density of [17, 16]. We corroborate the applicability of our class of models with experiments on synthetic and real data sets. 2 Multivariate Density Specification via Conditional Densities We are interested in the approach of estimating a multivariate density by estimating node-conditional densities. Since node-conditional densities focus on the density of a single variable, though conditioned on the rest of the variables, estimating these is potentially a simpler problem, both statistically and computationally, than estimating the entire joint density itself. Let us consider the general non-parametric conditional density estimation problem. Given the general multivariate density f (X) = R exp(?(X)) , the conditional density of a variable Xs given the rest of the variables X?s exp(?(x))dx X is given by f (Xs | X?s ) = R exp(?((Xs ,X?s ))) , exp(?((x,X?s )))dx X which does not have a multi-dimensional integral, s but otherwise does not have a computationally amenable form. There has been a line of work on such conditional density estimation, mirroring developments in multivariate density estimation [9, 18, 23], but unlike parametric settings, there are no large sample complexity gains with non-parametric conditional density estimation under general settings. There have also been efforts to use ANOVA decompositions in a conditional density context [31, 26]. In addition to computational and sample complexity caveats, recall that in our context, we would like to use conditional density estimates to infer a joint multivariate density. A crucial caveat with using the above estimates to do so is that it is not clear when the estimated node-conditional densities would be consistent with a joint multivariate density. There has been a line of work on this question (of when conditional densities are consistent with a joint density) for parametric densities; see [1] for an overview, with more recent results in [27, 5, 2, 25]. Overall, while estimating node-conditional densities could be viewed as surrogate estimation of a joint density, arbitrary node-conditional distributions need not be consistent in general with any joint density. There has however been a line of work in recent years [3, 28], where it was shown that when the node-conditional distributions belong to an exponential family, then under certain conditions on their parameterization, there do exist multivariate densities consistent with the node-conditional densities. In the next section, we leverage these results towards non-parametric estimation of conditional densities. 3 Conditional Densities of an Exponential Family Form We first recall the definition of an exponential family in the context of a conditional density. Definition 1. A conditional density of a random variable Y ? Y given covariates Z := (Z1 , . . . , Zm ) ? Z is said to have an exponential family form if it can be written as f (Y | Z) = exp(B(Y )T E(Z) + C(Y ) + D(Z)), for some functions B : Y 7? Rk (for some finite integer k > 0), E : Z 7? Rk , C : Y 7? R and D : Z 7? R. Thus, f (Y | Z) belongs to a finite-dimensional exponential family with sufficient statistics B(Y ), base measure exp(C(Y )), and with natural parameter E(Z) and where ?D(Z) is the log-partition function. Contrast this with a general conditional density f (Y | Z) = exp(h(Y, Z) + C(Y ) + D(Z)) with respect to the base measure exp(C(Y )) and ?D(Z) being the log-normalization constant, and it can be seen that a conditional density of the exponential family form has its logistic density transform h(Y, Z) that factorizes as B(Y )T E(Z). Consider the case where the sufficient statistic function is real-valued. The non-parametric estimation problem of a conditional density of exponential form then reduces to the estimation of the sufficient statistics function B(?), the exponential natural parameter function E(?), assuming the base measure C(?) is given. But when would such estimated conditional densities be consistent with a joint density? 3 To answer this question, we draw upon developments in [28]. Suppose that the node-conditional distributions of each random variable Xs conditioned on the rest of random variables have the exponential family form as in Definition 1, so that for each s ? V P(Xs | X?s ) ? exp{Es (X?s )Bs (Xs ) + Cs (Xs )} , (1) for some arbitrary functions Es (?), Bs (?), Cs (?) that specify a valid conditional density. Then [28] show that these node-conditional densities are consistent with a unique joint density over the random vector X, that moreover factors according to a set of cliques C in the graph G, if and only if the functions {Es (?)} P Qs?V specifying the node-conditional distributions have the form Es (X?s ) = ?s + C?C:s?C ?C t?C,t6=s Bt (Xt ), where {?s } ? {?C }C?C is a set of parameters. Moreover, the corresponding consistent joint distribution has the following form nX o X Y X P(X) ? exp ?s Bs (Xs ) + ?C Bs (Xs ) + Cs (Xs ) . (2) s?V C?C s?C s?V In this paper, we are interested in the non-parametric estimation of the Expxorcist class of densities in (2), where we estimate both the finite-dimensional parameters {?s } ? {?C }C?C , as well as the functions {Bs (Xs )}s?V . We assume we are given the base measures {Cs (Xs )}s?V , so that the Q joint density is with respect to a given product base measure s?V exp(Cs (XS )), as is common in the multivariate density estimation literature. Note that this is not a very restrictive assumption. In practice the base measure at each node can be well approximated using the empirical univariate marginal density of that node. We could also extend our algorithm, which we present next, to estimate the base measures along with sufficient statistic functions. 4 Regularized Conditional Likelihood Estimation for Exponential Family Form Densities We consider the nonparametric estimation problem of estimating a joint density of the form in (2), focusing on the pairwise case where the factors have size at most k = 2, so that the joint density takes the form X  X X P(X) ? exp ?s Bs (Xs ) + ?st Bs (Xs ) Bt (Xt ) + Cs (Xs ) . (3) s?V (s,t)?E s?V As detailed in the previous section, estimating this joint density can be reduced to estimating its node-conditional densities, which take the form     X P(Xs | X?s ) ? exp Bs (Xs ) ?s + ?st Bt (Xt ) + Cs (Xs ) . (4) t?NG (s) We now introduce some notation which we use in the sequel. Let ? = {?s }s?V ? {?st }s6=t and ?s = ?s ? {?st }t?V \{s} . Let B = {Bs }s?V be the set of sufficient statistics. Let Xs be the domain of Xs , which we assume is bounded and L2 (Xs ) be the Hilbert space of square integrable functions over Xs with respect to Lebesgue measure. We assume that the sufficient statistics Bs (?) ? L2 (Xs ). Note that the model in Equation (3) is unidentifiable. To overcome this issue we R impose additional constraints on its parameters. Specifically, we require Bs (Xs ) to satisfy Xs Bs (X)dX = 0, R B (X)2 dX = 1 and ?s ? 0, ?s ? V . Xs s Optimization objective: Let Xn = {X (1) , . . . X (n) } be n i.i.d. samples drawn from a joint density of the form in Equation (3), with parameters ?? , B ? . And let Ls (?s , B; Xn ) be the node conditional negative log likelihood at node s     X 1 Xn (i) (i) Ls (?s , B; Xn ) = ?Bs (Xs(i) ) ?s + ?st Bt (Xt ) + A(X?s ; ?s , B) , i=1 t?V \s n where A(X?s ; ?s , B) is the log partition function. To estimate the unknown parameters, we solve the following regularized node conditional log-likelihood estimation problem at each node s ? V min Ls (?s , B; Xn ) + ?n k?s k1 R R s.t. ?s ? 0, Xt Bt (X)dX = 0, Xt Bt (X)2 dX = 1 ?t ? V. ?s ,B 4 (5) The equality constraints on the norm of functions Bt (?) makes the above optimization problem a difficult one to solve. While the norm constraints on Bt (?), ?t ? V \ s can be handled through reparametrization, the constraint on Bs (?) can not be handled efficiently. To make the optimization more amenable for numerical optimization techniques, we solve a closely related optimization problem. At each node s ? V , we consider the following re-parametrization of B: Bs (Xs ) ? ?s Bs (Xs ), Bt (Xt ) ? (?st /?s )Bt (Xt ), ?t ? V \ {s}. With a slight abuse of notation we redefine Ls using this re-parametrization as     X 1 Xn (i) (i) Ls (B; Xn ) = ?Bs (Xs(i) ) 1 + Bt (Xt ) + A(X?s ; B) , (6) i=1 t?V \s n where A(X?s ; B) is the log partition function. We solve the following optimization problem, which is closely related to the original optimization in Equation (5) qR P B (X)2 dX min Ls (B; Xn ) + ?n t?V Xt t B (7) R s.t. Xt Bt (X)dX = 0 ?t ? V. For more details on the relation between (5) and (7), please refer to Appendix. Algorithm: We now present our algorithm for optimization of (7). In the sequel, for simplicity, we assume that the domains Xt of random variables Xt are all the same and equal to X . In order to estimate functions Bt , we expand them over a uniformly bounded, orthonormal basis {?k (?)}? k=0 of L2 (X ) with ?0 (?) ? 1. Expansion of the functions Bt (?) over this basis yields Xm X? Bt (X) = ?t,k ?k (X)+?t,m (X) where ?t,m (X) = ?t,0 ?0 (X)+ ?t,k ?k (X). k=1 k=m+1 R Note that the constraint X Bt (X)dX = 0 in Equation (7), translates to ?t,0 = 0. To convert the infinite dimensional optimization problem in (7) into a finite dimensional problem, we truncate the Pm basis expansion to the top m terms and approximate Bt (?) as k=1 ?t,k ?k (?). The optimization problem in Equation (7) can then be rewritten as X min Ls,m (?m ; Xn ) + ?n k?t,m k2 , (8) ?m t?V {?t,k }m k=1 , ?m = {?t,m }t?V and Ls,m is defined as ? ? ? ? n ? X m m ? X X X 1 (i) (i) Ls,m (?m ; Xn ) = ? ?s,k ?k (Xs(i) ) ?1 + ?t,l ?l (Xt )? + A(X?s ; ?m ) . ? ? n where ?t,m = i=1 t?V \{s} l=1 k=1 Iterative minimization of (8): Note that the objective in (8) is non-convex. In this work, we use a simple alternating minimization technique for its optimization. In this technique, we alternately minimize ?s,m , {?t,m }t?V \s while fixing the other parameters. The resulting optimization problem in each of the alternating steps is convex. We use Proximal Gradient Descent to optimize these sub-problems. To compute the objective and its gradients, we need to numerically evaluate the one-dimensional integrals in the log partition function. To do this, we choose a uniform grid of points over the domain and use quadrature rules to approximate the integrals. Convergence: Although (8) is non-convex, we can show that under certain conditions on the objective function, the alternating minimization procedure converges to the global minimum. In a recent work [32] analyze alternating minimization for low rank matrix factorization problems and show that it converges to a global minimum if the sequence of convex problems are strongly convex and satisfy certain other regularity condition. The analysis of [32] can be extended to show global convergence of alternating minimization for (8). 5 Statistical Properties In this section we provide parameter estimation error rates for the node conditional estimator in Equation (8). Note that these rates are for the re-parameterized model described in Equation (6) and can be easily translated to guarantees on the original model described in Equations (3), (4). 5 Notation: Let B2 (x, r) = {y : ky ? xk2 ? r} be the `2 ball with center x and radius r. Let {Bt? (?)}t?V be the true functions of the re-parametrized model, which we would like to estimate from the data. Denote the basis expansion coefficients of Bt (?) with respect to orthonormal basis ? ? {?k (?)}? k=0 by ?t , which is an infinite dimensional vector and let ?t be the coefficients of Bt (?). And let ?t,m R be the coefficients corresponding to the top m basis in the basis expansion of Bt (?). Note that Bt (X)2 dX = k?t k22 . Let ? = {?t }t?V and ?m = {?t,m }t?V . Let L?s,m (?m ) = E [Ls,m (?m ; Xn )] be the population version of the sample loss defined in Equation (8). We will often omit Xn from Ls,m (?m ; Xn ) when clear from the context. We let (?t ? ?t,m ) be the difference between infinite dimensional vector ?t and the vector obtained by appropriately padding ?t,m with P zeros. Finally, we define the norm R(?) as R(?m ) = t?V k?t,m k2 and its dual as R? (?m ) = supt?V k?t,m k2 . The norms on infinite dimensional vector ? are similarly defined. We now state our key assumption on the loss function Ls,m (?). This assumption imposes strong ? curvature condition on Ls,m along certain directions in a ball around ?m . Assumption 1. There exists rm > 0 and constants c, ? > 0 such that for any ?m ? B2 (0, rm ) the ? ? gradient of the sample loss Ls,m satisfies: h?Ls,m (?m + ?m ) ? ?Ls,m (?m ), ?m i ? ?k?m k22 ? q c m log(p) R(?m ). n Similar assumptions are increasingly common in analysis of non-convex estimators, see [19] and references therein. We are now ready to state our results which give the parameter estimation error rates, the proofs of which can be found in Appendix. We first provide a deterministic bound on ? ? the error k?m ? ?m k2 in terms of the random quantity R? (?Ls,m (?m )). We derive probabilistic results in the subsequent corollaries. Theorem 2. Let Ns be the true neighborhood of node s, with |Ns | = d. Suppose Ls,m satisfies ? Assumption 1. If the regularization parameter ?n is chosen such that ?n ? 2R? (?Ls,m (?m )) + q 2c m log(p) , n ? , rm ) satisfies: then any stationary point ? ? m of (8) in B2 (?m ? 6 2? ? k? ?m ? ?m k2 ? d?n . ? ? We now provide a set of sufficient conditions under which the random quantity R? (?Ls,m (?m )) can be bounded. Assumption 2. There exists a constant L > 0 such that the gradient of the population loss L?s,m at ? ? ? ?m satisfies: R? (?L?s,m (?m )) ? LR? (?? ? ?m ). Corollary 3. Suppose the conditions in Theorem Pm ? 2 are satisfied. Moreover, let ? = supi?N,X?X |?i (X)| and ?m = supt?V,X?X | i=1 ?t,i ?i (X)|. Suppose Ls,m satisfies Assumption q 2 ? 2. If the regularization parameter ?n is chosen such that ?n ? 2LR? (?? ? ?m ) + c??m md nlog(p) , ? then then with probability at least 1 ? 2m/p2 any stationary point ? ? m of (8) in B2 (?m , rm ) satisfies: ? 6 2? ? k? ?m ? ?m k2 ? d?n . ? Theorem 2 and Corollary 3 bound the error of the estimated coefficients in the truncated expansion. The approximation error of the truncated expansion itself depends on the function space assumption, as well as the basis chosen, but can be simply combined with the statement of the above corollary to derive the overall error. As an instance, we present a corollary below for the specific case of Sobolev space of order two, and the trigonometric basis. Corollary 4. Suppose the conditions in Corollary 3 are satisfied. Moreover, suppose the true functions 2 Bt? (?) lie in a Sobolev space of order two. Let {?k }? k=0 be the trigonometric basis of L (X ). If the 2 2/5 2 optimization problem (8) is solved with ?n = c1 (d log(p)/n) and m = c2 (n/d log(p))1/5 , then ? with probability at least 1 ? 2m/p2 any stationary point ? ? m of (8) in B2 (?m , rm ) satisfies:  13/4 2/5 d log(p) ? k? ?m ? ? k2 ? c3 , n where c1 , c2 , c3 depend on L, ?, ?, ?m . 6 Discussion on Assumption 1: We now provide a set of sufficient conditions which ensure the restricted strong convexity (RSC) condition. Suppose the population risk L?s,m (?) is strongly convex ? in a ball of radius rm around ?m ? ? ?L?s,m (?m + ?m ) ? ?L?s,m (?m ), ?m ? ?k?m k22 ??m ? B2 (0, rm ). (9) Moreover, suppose the empirical gradients converge uniformly to the population gradients r  m log p ? ? sup R ?Ls,m (?m ) ? ?Ls,m (?m ) ? c . n ?m ?B2 (?? m ,rm ) (10) For example, this condition holds with high probability when the gradient of Ls,m (?m ) w.r.t ?t,m , for any t ? [p] is a sub-Gaussian process. Equations (9),(10) are easier to check and ensure that Ls,m (?m ) satisfies the RSC property in Assumption 1. 6 Connections to Exponential Family MRF Copulas The Expxorcist class of models could be viewed as being closely related to an exponential family MRF [28] copula density. exponential family MRF joint density o in nP Consider the parametric P P (3): PMRF;? (X) ? exp s?V ?s Bs (Xs ) + (s,t)?E(G) ?st Bs (Xs ) Bt (Xt ) + s?V Cs (Xs ) , where the distribution is indexed by the finite-dimensional parameters {?s }s?V , {?st }(s,t)?E , and where in contrast to the previous sections, we assume we are given the sufficient statistics functions {Bs (?)}s?V as well as the nodewise base measures {Cs (?)}s?V . Now consider the following nonparametric problem. Given a random vector X, suppose we are interested in estimating monotonic node-wise functions {fs (Xs )}s?V such that (f1 (X1 ), . . . , fp (Xp )) follows PMRF;? for some ?. Letting f(X) = (f1 (X1 ), . . . , fp (Xp )), we Q have that P(f(X)) = PMRF;? (f(X)), so that the density of X can be written as P(X) ? P(f(X)) s?V fs0 (Xs ). This is now a semi-parametric estimation problem, where the unknowns are the functions {fs (Xs )}s?V as well as the finite-dimensional parameters ?. To simplify this density, suppose we assume that the given node-wise sufficient statistics are linear, so that Bs (z) = z, for all s ? V , so that density reduces to P(X) ? exp ? ?X ? ?s fs (Xs ) + s?V X ?st fs (Xs ) ft (Xt ) + X (Cs (fs (Xs )) + ? ? log fs0 (Xs )) . (11) ? s?V (s,t)?E(G) In contrast, the Expxorcist nonparametric exponential family graphical model takes the form P(X) ? exp ? ?X ? s?V ?s fs (Xs ) + X (s,t)?E(G) ?st fs (Xs ) ft (Xt ) + X s?V Cs (Xs ) ? ? . (12) ? It can be seen that the two densities have very similar forms, except that the density in (11) has a more complex base measure that depends on the unknown functions {fs }s?V and importantly the functions {fs }s?V in (11) are monotonic. The class of densities in (11) can be cast as an exponential family MRF copula density. Suppose we denote the CDF of the parametric exponential family MRF joint density by FMRF;? (X), with nodewise marginal CDFs FMRF;?,s (Xs ). Then the marginal CDF of the density (11) can be written as Fs (xs ) = P[Xs ? xs ] = P[fs (Xs ) ? fs (xs )] = FMRF;?,s (fs (xs )), so that ?1 fs (xs ) = FMRF;?,s (Fs (xs )). (13)   ?1 ?1 It then follows that: F (X) = FMRF;? FMRF;?,1 (F1 (X1 )), . . . , FMRF;?,p (Fp (Xp )) , where F (X)   ?1 ?1 is the CDF of density (11). By letting FCOP;? (U ) = FMRF;? FMRF;?,1 (U1 ), . . . , FMRF;?,p (Up ) be the exponential family MRF copula density function, we see that the CDF of X is precisely: F (X) = FCOP;? (F1 (X1 ), . . . , Fp (Xp )), which is specified by the marginal CDFs {Fs (Xs )}s?V and the copula density FCOP;? corresponding to the exponential family MRF density. In other words, the non-parametric extension in (11) of the exponential family MRF densities is precisely an exponential family MRF copula density. This development thus generalizes the non-parametric extension of Gaussian MRF densities via the Gaussian copula nonparanormal densities [17]. The caveats with the copula density however are two-fold: the node-wise functions are restricted to be monotonic, but 7 also the estimation of these as in (13) requires the estimation of inverses of marginal CDFs of an exponential family MRF, which is intractable in general. Thus, minor differences in the expressions of the Expxorcist density (12) and an exponential family MRF copula density (11) nonetheless have seemingly large consequences for tractable estimation of these densities from data. 7 Experiments We present experimental results on both synthetic and real datasets. We compare our estimator, Expxorcist, with the Nonparanormal model of [17] and Gaussian Graphical Model (GGM). We use glasso [7] to estimate GGM and the two step estimator of [17] to estimate Nonparanormal model. 7.1 Synthetic Experiments Data: We generated synthetic data from the Expxorcist model with chain and grid graph structures. For both the graph structures, we set ?s = 1, ?s ? V ,?st = 1, ?(s, t) ? E and fix the domain X with two choices  to [?1, 1]. We experimented   for  sufficient statistics Bs (X): sin(4?X) and exp ?20(X ? 0.5)2 + exp ?20(X + 0.5)2 ? 1 and picked the log base measure Cs (X) to be 0. The grid graph we considered has a 10 ? (p/10) structure. We used Gibbs sampling to sample data from these models. We also generated data from Gaussian distribution with chain and grid graph structures. To generate this data we set the off diagonal non-zero entries of inverse covariance matrix to 0.49 for chain graph and 0.25 for grid graph and diagonal entries to 1. Evaluation Metric: We compared the performance of Expxorcist against baselines, on graph structure recovery, using ROC curves. The ROC curve plots the true positive rate (TPR) against false positive rate (FPR) over different choices of regularization parameter, where TPR is the fraction of correctly detected edges and FPR is the fraction of mis-identified non edges. Experiment Settings: For this experiment we set p = 50 and n ? {100, 200, 500} and varied the regularization parameter ? from 10?2 to 1. To fit the data to the non parametric model (3), we used cosine basis and truncated the basis expansion to top 30 terms. In practice, one could choose the number of basis (m) based on domain knowledge (e.g. ?smooth? functions), or in the absence of ? (s), the estimated neighborhood which, one could use hold-out validation/cross validation. Given N for node s, we estimated the overall graph structure as: ?s?V ?t?N? (s) {(s, t)}. To reduce the variance in the ROC plots, we averaged results over 10 repetitions. Results: Figure 1 shows the ROC plots obtained from this experiment. Due to the lack of space, we present more experimental results in Appendix. It can be seen that Expxorcist has much better performance on non-Gaussian data. On these datasets, even at n = 500 the baselines chose edges at random. This suggests that in the presence of multiple modes and fat tails, Expxorcist is a better model. Expxorcist has slightly poor performance than baselines on Gaussian data. However, this is expected because it learns a broader family of distributions than Nonparanormal. 7.2 Futures Intraday Data We now present our analysis on the Futures price returns. This dataset was downloaded from http://www.kibot.com/. We focus on the Top-26 most liquid instruments being traded at the Chicago Mercantile Exchange (CME). The instruments span different sectors like Energy, Agriculture, Currencies, Equity Indices, Metals and Interest Rates. We focus on the hours of maximum liquidity (9am Eastern to 3pm Eastern) and look at the 1 minute price returns. The return distribution is a mixture of 1 minute returns with the overnight return. Since overnight returns tend to be bigger than the 1 minute return within the day, the return distribution is multimodal and fat-tailed. We treat each instrument as a random variable and the 1 minute returns as independent samples drawn from these random variables. We use the data collected in February 2010 as training data and data from March 2010 as held out data for tuning parameter selection. After removing samples with missing entries we are left with 894 training and 650 held out data samples. We fit Expxorcist and baselines on this data with the same parameter settings described above. For each of these models, we select the best tuning parameter through log likelihood on held out data. However, this criteria resulted in complete graphs for Nonparanormal and GGM (325 edges) and a relatively sparser graph for Expxorcist (168 edges). So for a better comparison of these models, we selected tuning parameters for each of the models such that the resulting graphs have almost the same number of edges. Figure 2 shows the 8 Sine(n = 500) TPR 1 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 FPR Expxorcist GGM Nonparanormal 0 0 1 0 Gaussian(n = 200) 1 0.8 0 TPR Exp (n = 500) 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 FPR FPR Figure 1: ROC plots from synthetic experiments. Top and bottom rows show plots for chain and grid graphs respectively. Left column shows plots for data generated from our non-parametric model with Bs (X) = sin(X), n = 500 and center column shows plots for the other choice of sufficient statistic with n = 500. Right column shows plots for Gaussian data with n = 200. (a) Nonparanormal (b) Expxorcist Figure 2: Graph Structures learned for the Futures Intraday Data. The Expxorcist graph shown here was obtained by selecting ? = 0.1. Nodes are colored based on their categories. Edge thickness is proportional to the magnitude of the interaction. learned graphs for one such choice of tuning parameters, which resulted in ? 52 edges in the graphs. Nonparanormal and GGM resulted in very similar graphs, so we only present Nonparanormal here. It can be seen that Expxorcist is able to identify the clusters better than Nonparanormal. More detailed graphs and comparison with GGM can be found in Appendix. 8 Conclusion In this work we considered the problem of non-parametric density estimation and introduced Expxorcist, a new family of non-parametric graphical models. Our approach relies on a simple function space assumption that the conditional distribution of each variable conditioned on the other variables has a non-parametric exponential family form. We proposed an estimator for Expxorcist that is computationally efficient and comes with statistical guarantees. Our empirical results suggest that, in the presence of multiple modes and fat tails in the data, our non-parametric model is a better choice than the Nonparanormal model of [17]. 9 Acknowledgement A.S. and P.R. acknowledge the support of ARO via W911NF-12-1-0390 and NSF via IIS-1149803, IIS-1447574, DMS-1264033, and NIH via R01 GM117594-01 as part of the Joint DMS/NIGMS Initiative to Support Research at the Interface of the Biological and Mathematical Sciences. M. K. acknowledges support by an IBM Corporation Faculty Research Fund at the University of Chicago Booth School of Business. 9 References [1] Barry C. Arnold, Enrique Castillo, and Jos? Mar?a Sarabia. Conditionally specified distributions: an introduction. Stat. Sci., 16(3):249?274, 2001. With comments and a rejoinder by the authors. [2] Patrizia Berti, Emanuela Dreassi, and Pietro Rigo. Compatibility results for conditional distributions. J. Multivar. Anal., 125:190?203, 2014. [3] Julian Besag. 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Improved Graph Laplacian via Geometric Consistency Dominique C. Perrault-Joncas Google, Inc. [email protected] Marina Meil?a Department of Statistics University of Washington [email protected] James McQueen Amazon [email protected] Abstract In all manifold learning algorithms and tasks setting the kernel bandwidth  used construct the graph Laplacian is critical. We address this problem by choosing a quality criterion for the Laplacian, that measures its ability to preserve the geometry of the data. For this, we exploit the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator. Experiments show that this principled approach is effective and robust. 1 Introduction Manifold learning and manifold regularization are popular tools for dimensionality reduction and clustering [1, 2], as well as for semi-supervised learning [3, 4, 5, 6] and modeling with Gaussian Processes [7]. Whatever the task, a manifold learning method requires the user to provide an external parameter, called ?bandwidth? or ?scale? , that defines the size of the local neighborhood. More formally put, a common challenge in semi-supervised and unsupervised manifold learning lies in obtaining a ?good? graph Laplacian estimator L. We focus on the practical problem of optimizing the parameters used to construct L and, in particular, . As we see empirically, since the Laplace-Beltrami operator on a manifold is intimately related to the geometry of the manifold, our estimator for  has advantages even in methods that do not explicitly depend on L. In manifold learning, there has been sustained interest for determining the asymptotic properties of L [8, 9, 10, 11]. The most relevant is [12], which derives the optimal rate for  w.r.t. the sample size N 1 2 = C(M)N ? 3+d/2 , (1) with d denoting the intrinsic dimension of the data manifold M. The problem is that C(M) is a constant that depends on the yet unknown data manifold, so it is rarely known in practice. Considerably fewer studies have focused on the parameters used to construct L in a finite sample problem. A common approach is to ?tune? parameters by cross-validation in the semi-supervised context. However, in an unsurpervised problem like non-linear dimensionality reduction, there is no context in which to apply cross-validation. While several approaches [13, 14, 15, 16] may yield a usable parameter, they generally do not aim to improve L per se and offer no geometry-based justification for its selection. In this paper, we present a new, geometrically inspired approach to selecting the bandwidth parameter  of L for a given data set. Under the data manifold hypothesis, the Laplace-Beltrami operator ?M of the data manifold M contains all the intrinsic geometry of M. We set out to exploit this fact by comparing the geometry induced by the graph Laplacian L with the local data geometry and choose the value of  for which these two are closest. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. 2 Background: Heat Kernel, Laplacian and Geometry Our paper builds on two previous sets of results: 1) the construction of L that is consistent for ?M when the sample size N ? ? under the data manifold hypothesis (see [17]); and 2) the relationship between ?M and the Riemannian metric g on a manifold, as well as the estimation of g (see [18]). Construction of the graph Laplacian. Several methods methods to construct L have been suggested (see [10, 11]). The one we present, due to [17], guarantees that, if the data are sampled from a manifold M, L converges to ?M : Given a set of points D = {x1 , . . . , xN } in high-dimensional Euclidean space Rr , construct a weighted graph G = (D, W ) over them, with W = [wij ]ij=1:N . The weight wij between xi and xj is the heat kernel [1]   2 Wij ? w (xi , xj ) = exp ||xi ? xj ||2 /2 , (2) with  a bandwidth parameter fixed by the user. Next, construct L = [Lij ]ij of G by ti = X Wij , Wij0 = j Wij , ti tj t0i = X j Wij0 , and Lij = X Wij0 j t0j . (3) Equation (3) represents the discrete versions of the renormalized Laplacian construction from [17]. Note that ti , t0i , W 0 , L all depend on the bandwidth  via the heat kernel. Estimation of the Riemannian metric. We follow [18] in this step. A Riemannian manifold (M, g) is a smooth manifold M endowed with a Riemannian metric g; the metric g at point p ? M is a scalar product over the vectors in Tp M, the tangent subspace of M at p. In any coordinate representation of M, gp ? G(p) - the Riemannian metric at p - represents a positive definite matrix1 of dimension d equal to the intrinsic dimension of M. We say that the metric g encodes the geometry of M because p g determines the volume element for any integration over M by det G(x)dx, and the line element q  dx T dx for computing distances along a curve x(t) ? M, by dt G(x) dt . If we assume that the data we observe (in Rr ) lies on a manifold, then under rotation of the original coordinates, the metric G(p) is the unit matrix of dimension d padded with zeros up to dimension r. When the data is mapped to another coordinate system - for instance by a manifold learning algorithm that performs non-linear dimension reduction - the matrix G(p) changes with the coordinates to reflect the distortion induced by the mapping (see [18] for more details). Proposition 2.1 Let x denote local coordinate functions of a smooth Riemannian manifold (M, g) of dimension d and ?M the Laplace-Beltrami operator defined on M. Then, H(p) = (G(p))?1 the (matrix) inverse of the Riemannian metric at point p, is given by   (4) (H(p))kj = 21 ?M xk ? xk (p) xj ? xj (p) |x=x(p) with i, j = 1, . . . , d. Note that the inverse matrices H(p), p ? M, being symmetric and positive definite, also defines a metric h called the cometric on M. Proposition 2.1  says that the  cometric is given by applying the ?M operator to the function ?kj = xk ? xk (p) xj ? xj (p) , where xk , xj denote coordinates k, j seen as functions on M. A converse theorem [19] states that g (or h) uniquely determines ?M . Proposition 2.1 provides a way to estimate h and g from data. Algorithm 1, adapted from [18], implements (4). 3 A Quality Measure for L Our approach can be simply stated: the ?best? value for  is the value for which the corresponding L of (3) best captures the original data geometry. For this we must: (1) estimate the geometry g or h 1 This paper contains mathematical objects like M, g and ?, and computable objects like a data point x, and the graph Laplacian L. The Riemannian metric at a point belongs to both categories, so it will sometimes be denoted gp , gxi and sometimes G(p), G(xi ), depending on whether we refer to its mathematical or algorithmic aspects (or, more formally, whether the expression is coordinate free or in a given set of coordinates). This also holds for the cometric h, defined in Proposition 2.1. 2 Algorithm 1 Riemannian Metric(X, i, L, pow ? {?1, 1}) Input: N ? d design matrix X, i index in data set, Laplacian L, binary variable pow for k = 1 ? d, l = 1 ? d do PN Hk,l ? j=1 Lij (Xjk ? Xik )(Xjl ? Xil ) end for return H pow (i.e. H if pow = 1 and H ?1 if pow = ?1) from L (this is achieved by RiemannianMetric()); (2) find an independent way to estimate the data geometry, locally (this is done in Sections 3.2 and 3.1); (3) define a measure of agreement between the two (Section 3.3). 3.1 The Geometric Consistency Idea and g target There is a natural way to estimate the geometry of the data without the use of L. We consider the canonical embedding of the data in the ambient space Rr for which the geometry is trivially known. This provides a target g target ; we tune the scale of the Laplacian so that the g calculated from Proposition 2.1 matches this target. Hence, we choose  to maximize consistency with the geometry of the data. We denote the inherited metric by gRr |T M , which stands for the restriction of the natural metric of the ambient space Rr to the tangent bundle T M of the manifold M. We tune the parameters of the graph Laplacian L so as to enforce (a coordinate expression of) the identity gp () = g target , with g target = gRr |Tp M ?p ? M . (5) In the above, the l.h.s. will be the metric implied from the Laplacian via Proposition 2.1, and the r.h.s is the metric induced by Rr . Mathematically speaking, (5) is necessary and sufficient for finding the ?correct? Laplacian. The next section describes how to obtain the r.h.s. from a finite sample D. Then, to optimize the graph Laplacian we estimate g from L as prescribed by Proposition 2.1 and compare with gRr |Tp M numerically. We call this approach geometric consistency (GC). The GC method is not limited to the choice of , but can be applied to any other parameter required for the Laplacian. 3.2 Robust Estimation of g target for a finite sample First idea: estimate tangent subspace We use the simple fact, implied by Section 3.1, that projecting the data onto Tp M preserves the metric locally around p. Hence, Gtarget = Id in the projected data. Moreover, projecting on any direction in Tp M does not change the metric in that direction. This remark allows us to work with small matrices (of at most d ? d instead of r ? r) and to avoid the problem of estimating d, the intrinsic dimension of the data manifold. Specifically, we evaluate the tangent subspace around each sampled point xi using weighted (local) Principal Component Analysis (wPCA) and then express gRr |Tp M directly in the resulting lowdimensional subspace as the unit matrix Id . The tangent subspace also serves to define a local coordinate chart, which is passed as input to Algorithm 1 which computes H(xi ), G(xi ) in these coordinates. For computing Txi M, by wPCA, we choose weights defined by the heat kernel (2), centered around xi , with same bandwidth  as for computing L. This approach is similar to samplewise weighted PCA of [20], with one important requirements: the weights must decay rapidly away from xi so that only points close xi are used to estimate Txi M. This is satisfied by the weighted recentered design matrix Z, where Zi: , row i of Z, is given by: ? ? ? ? ? ? N N N X X X Zi: = Wij (xi ? x ?)/ ? Wij 0 ? , with x ? = ? Wij xj ? / ? Wij 0 ? . (6) j 0 =1 j=1 j 0 =1 [21] proves that the wPCA using the heat kernel, and equating the PCA and heat kernel bandwidths as we do, yields a consistent estimator of Txi M. This is implemented in Algorithm 2. In summary, to instantiate equation (5) at point xi ? D, one must (i) construct row i of the graph Laplacian by (3); (ii) perform Algorithm 2 to obtain Y ; (iii) apply Algorithm 1 to Y to obtain G(xi ) ? Rd?d ; (iv) this matrix is then compared with Id , which represents the r.h.s. of (5). 3 Algorithm 2 Tangent Subspace Projection(X, w, d0 ) Input: N ? r design matrix X, weight vector w, working dimension d0 Compute Z using (6) [V, ?] ? eig(Z T Z, d0 ) (i.e.d0 -SVD of Z) Center X around x ? from (6) Y ? XV:,1:d0 (Project X on d0 principal subspace) return Y Second idea: project onto tangent directions We now take this approach a few steps further in terms of improving its robustness with minimal sacrifice to its theoretical grounding. In particular, we perform both Algorithm 2 and Algorithm 1 in d0 dimensions, with d0 < d (and typically d0 = 1). This makes the algorithm faster, and make the computed metrics G(xi ), H(xi ) both more stable numerically and more robust to possible noise in the data2 . Proposition 3.1 shows that the resulting method remains theoretically sound. Proposition 3.1 Let X, Y, Z, V, W:i , H, and d ? 1 represent the quantities in Algorithms 1 and 2; assume that the columns of V are sorted in decreasing order of the singular values, and that the rows and columns of H are sorted according to the same order. Now denote by Y 0 , V 0 , H 0 the quantitities computed by Algorithms 1 and 2 for the same X, W:i but with d ? d0 = 1. Then, V 0 = V:1 ? Rr?1 Y 0 = Y:1 ? RN ?1 H 0 = H11 ? R. (7) The proof of this result is straightforward and omitted for brevity. It is easy to see that Proposition 3.1 generalizes immediately to any 1 ? d0 < d. In other words, by using d0 < d, we will be projecting the data on a proper subspace of Txi M - namely, the subspace of least curvature [22]. The cometric H 0 of this projection is the principal submatrix of order d0 of H, i.e. H11 if d0 = 1. Third idea: use h instead of g Relation (5) is trivially satisfied by the cometrics of g and g target (the latter being H target = Id ). Hence, inverting H in Algorithm 1 is not necessary, and we will use the cometric h in place of g by default. This saves time and increases numerical stability. 3.3 Measuring the Distortion For a finite sample, we cannot expect (5) to hold exactly, and so we need to define a distortion between the two metrics to evaluate how well they agree. We propose the distortion D = 1 N N X ||H(xi ) ? Id || (8) i=1 where ||A|| = ?max (A) is the matrix spectral norm. Thus D measures the average distance of H from the unit matrix over the data set. For a ?good? Laplacian, the distortion D should be minimal: ? = argmin D . (9) The choice of norm in (8) is not arbitrary. Riemannian metrics are R order 2 tensors or T M hence the expression of D is the discrete version of Dg0 (g1 , g2 ) = M ||g1 ? g2 ||g0 dVg0 , with <u,v>gp ||g|| = supu,v?T M\{0} , representing the tensor norm of gp on Tp M with respect to g0 p p <u,v>g0p the Riemannian metric g0p . Now, (8) follows when g0 , g1 , g2 are replaced by I, I and H, respectively. With (9), we have established a principled criterion for selecting the parameter(s) of the graph Laplacian, by minimizing the distortion between the true geometry and the geometry derived from Proposition 2.1. Practically, we have in (9) a 1D optimization problem with no derivatives, and we can use standard algorithms to find its minimum. ?. 4 Related Work We have already mentioned the asymptotic result (1) of [12]. Other work in this area [8, 10, 11, 23] provides the rates of change for  with respect to N to guarantee convergence. These studies are 2 We know from matrix perturbation theory that noise affects the d-th principal vector increasingly with d. 4 Algorithm 3 Compute Distortion(X, , d0 ) Input: N ? r design matrix X, , working dimension d0 , index set I ? {1, . . . , N } Compute the heat kernel W by (2) for each pair of points in X Compute the graph Laplacian L from W by (3) D?0 for i ? I do Y ? TangentSubspaceProjection(X, Wi,: , d0 ) H ? RiemannianMetric(Y, L, pow = 1) D ? D + ||H ? Id0 ||2 /|I| end for return D relevant; but they depend on manifold parameters that are usually not known. Recently, an extremely interesting Laplacian "continuous nearest neighbor? consistent construction method was proposed by [24], from a topological perspective. However, this method depends on a smoothness parameter too, and this is estimated by constructing the persistence diagram of the data. [25] propose a new, statistical approach for estimating , which is very promising, but currently can be applied only to un-normalized Laplacian operators. This approach also depends on unknown pparameters a, b, which are set heuristically. (By contrast, our method depends only weakly on d0 , which can be set to 1.) Among practical methods, the most interesting is that of [14], which estimates k, the number of nearest neighbors to use in the construction of the graph Laplacian. This method optimizes k depending on the embedding algorithm used. By contrast, the selection algorithm we propose estimates an intrinsic quantity, a scale  that depends exclusively on the data. Moreover, it is not known when minimizing reconstruction error for a particular method can be optimal, since [26] even in the limit of infinite data, the most embeddings will distort the original geometry. In semi-supervised learning (SSL), one uses Cross-Validation (CV) [5]. Finally, we mention the algorithm proposed in [27] (CLMR). Its goal is to obtain an estimate of the intrinsic dimension of the data; however, a by-product of the algorithm is a range of scales where the tangent space at a data point is well aligned with the principal subspace obtained by a local singular value decomposition. As these are scales at which the manifold looks locally linear, one can reasonably expect that they are also the correct scales at which to approximate differential operators, such as ?M . Given this, we implement the method and compare it to our own results. From the computational point of view, all methods described above explore exhaustively a range of  values. GC and CLMR only require local PCA at a subset of the data points (with d0 < d components for GC, d0 >> d for CLMR); whereas CV, and [14] require respectively running a SSL algorithm, or an embedding algorithm, for each . In relation to these, GC is by far the most efficient computationally. 3 5 Experimental Results Synthethic Data. We experimented with estimating the bandwidth ? on data sampled from two known manifolds, the two-dimensional hourglass and dome manifolds of Figure 1. We sampled points uniformly from these, adding 10 ?noise? dimensions and Gaussian noise N (0, ? 2 ) resulting in r = 13 dimensions. The range of  values was delimited by min and max . We set max to the average of ||xi ? xj ||2 over all point pairs and min to the limit in Pwhich the heat kernel W becomes approximately equal to the unit matrix; this is tested by maxj ( i Wij ) ? 1 < ? 4 for ? ? 10?4 . This range spans about two orders of magnitude in the data we considered, and was searched by a logarithmic grid with ? approximately 20 points. We saved computatation time by evaluating all pointwise quantities (D, 0 local SVD) on a random sample of size N = 200 of each data set. We replicated each experiment on 10 independent samples. 3 4 In addition, these operations being local, they can be further parallelized or accelerated in the usual ways. Guaranteeing that all eigenvalues of W are less than ? away from 1. 5 ? = 0.001 ? = 0.01 ? = 0.1 Figure 1: Estimates ? (mean and standard deviation over 10 runs) on the dome and hourglass data, vs sample sizes N for various noise levels ?; d0 = 2 is in black and d0 = 1 in blue. In the background, we also show as gray rectangles, for each N, ? the intervals in the  range where the eigengaps of local SVD indicate the true dimension, and, as unfillled rectangles, the estimates proposed by CLMR [27] for these intervals. The variance of ? observed is due to randomness in the subsample N 0 used to evaluate the distortion. Our ? always falls in the true interval (when this exists), and have are less variable and more accurate than the CLMR intervals. Reconstruction of manifold w.r.t. gold standard These results (relegated to the Supplement) are uniformly very positive, and show that GC achieves its most explicit goal, even in the presence of noise. In the remainder, we illustrate the versatility of our method on on other tasks. Effects of d0 , noise and N . The estimated  are presented in Figure 1. Let ?d0 denote the estimate obtained for a given d0 ? d. We note that when d1 < d2 , typically ?d1 > ?d2 , but the values are of the same order (a ratio of about 2 in the synthetic experiments). The explanation is that, chosing d0 < d directions in the tangent subspace will select a subspace aligned with the ?least curvature? directions of the manifold, if any exist, or with the ?least noise? in the random sample. In these directions, the data will tolerate more smoothing, which results in larger ?. The optimal  decreases with N and grows with the noise levels, reflecting the balance it must find between variance and bias. Note that for the hourglass data, the highest noise level of ? = 0.1 is an extreme case, where the original manifold is almost drowned in the 13-dimensional noise. Hence,  is not only commensurately larger, but also stable between the two dimensions and runs. This reflects the fact ?that  captures the noise dimension, and its values are indeed just below the noise amplitude of 0.1 13. The dome data set exhibits the same properties discussed previously, showing that our method is effective even for manifolds with border. Semi-supervised Learning (SSL) with Real Data. In this set of experiments, the task is classification on the benchmark SSL data sets proposed by [28]. This was done by least-square classification, similarly to [5], after choosing the optimal bandwidth by one of the methods below. TE Minimize Test Error, i.e. ?cheat? in an attempt to get an estimate of the ?ground truth?. CV Cross-validation We split the training set (consisting of 100 points in all data sets) into two equal groups;5 we minimize the highly non-smooth CV classification error by simulated annealing. Rec Minimize the reconstruction error We cannot use the method of [14] directly, as it requires an embedding, so we minimize reconstruction error based on the heat kernel weights w.r.t.  2 Pn P W (this is reminiscent of LLE [29]): R() = i=1 xi ? j6=i P ijWij xj l6=i Our method is denoted GC for Geometric Consistency; we evaluate straighforward GC, that uses the cometric H and a variant that includes the matrix inversion in Algorithm 1 denoted GC?1 . 5 In other words, we do 2-fold CV. We also tried 20-fold and 5-fold CV, with no significant difference. 6 Digit1 USPS COIL BCI g241c g241d TE 0.67?0.08 [0.57, 0.78] 1.24?0.15 [1.04, 1.59] 49.79?6.61 [42.82, 60.36] 3.4?3.1 [1.2, 8.9] 8.3? 2.5 [6.3, 14.6] 5.7? 0.24 [5.6, 6.3] CV 0.80?0.45 [0.47, 1.99] 1.25?0.86 [0.50, 3.20] 69.65?31.16 [50.55, 148.96] 3.2?2.5 [1.2, 8.2] 8.8?3.3 [4.4, 14.9] 6.4?1.15 [4.3, 8.2] Rec GC?1 GC 0.64 0.74 0.74 1.68 2.42 1.10 78.37 216.95 116.38 3.31 3.19 5.61 3.79 7.37 7.38 3.77 7.35 7.36 Table 1: Estimates of  by methods presented for the six SSL data sets used, as well as TE. For TE and CV, which depend on the training/test splits, we report the average, its standard error, and range (in brackets below) over the 12 splits. ?1 Digit1 Digit1 USPS COIL BCI g241c g241d CV 3.32 5.18 7.02 49.22 13.31 8.67 Rec 2.16 4.83 8.03 49.17 23.93 18.39 GC?1 2.11 12.00 16.31 50.25 12.77 8.76 GC 2.11 3.89 8.81 48.67 12.77 8.76 USPS COIL BCI g241c g241d GC GC GC?1 GC GC?1 GC GC?1 GC GC?1 GC GC?1 GC d0 =1 0.743 0.744 2.42 1.10 116 187 3.32 5.34 7.38 7.38 7.35 7.35 d0 =2 0.293 0.767 2.31 1.16 87.4 179 3.48 5.34 7.38 9.83 7.35 9.33 d0 =3 0.305 0.781 3.88 1.18 128 187 3.65 5.34 7.38 9.37 7.35 9.78 Table 2: Left panel: Percent classification error for the six SSL data sets using the four  estimation methods described. Right panel:  obtained for the six datasets using various d0 values with GC and GC?1 . ? was computed for d=5 for Digit1, as it is known to have an intrinsic dimension of 5, and found to be 1.162 with GC and 0.797 with GC?1 . Across all methods and data sets, the estimate of  closer to the values determined by TE lead to better classification error, see Table 2. For five of the six data sets6 , GC-based methods outperformed CV, and were 2 to 6 times faster to compute. This is in spite of the fact that GC does not use label information, and is not aimed at reducing the classification error, while CV does. Further, the CV estimates of  are highly variable, suggesting that CV tends to overfit to the training data. Effect of Dimension d0 . Table 2 shows how changing the dimension d0 alters our estimate of . We see that the ? for different d0 values are close, even though we search over a range of two orders of magnitude. Even for g241c and g241d, which were constructed so as to not satisfy the manifold hypothesis, our method does reasonably well at estimating . That is, our method finds the ? for which the Laplacian encodes the geometry of the data set irrespective of whether or not that geometry is lower-dimensional. Overall, we have found that using d0 = 1 is most stable, and that adding more dimensions introduces more numerical problems: it becomes more difficult to optimize the distortion as in (9), as the minimum becomes shallower. In our experience, this is due to the increase in variance associated with adding more dimensions. Using one dimension probably works well because the wPCA selects the dimension that explains the most variance and hence is the closest to linear over the scale considered. Subsequently, the wPCA moves to incrementally ?shorter? or less linear dimensions, leading to more variance in the estimate of the tangent subspace (more evidence for this in the Supplement). 6 In the COIL data set, despite their variability, CV estimates still outperformed the GC-based methods. This is the only data set constructed from a collection of manifolds - in this case, 24 one-dimensional image rotations. As such, one would expect that there would be more than one natural length scale. 7 Figure 2: Bandwidth Estimation For Galaxy Spectra Data. Left: GC results for d0 = 1 (d0 = 2, 3 are also shown); we chose radius = 66 the minimum of D for d = 10 . Right: A log-log plot of radius versus average number of neighbors within this radius. The region in blue includes radius = 66 and indicates dimension d = 3. In the code  = radius/3, hence we use  = 22. Embedding spectra of galaxies (Details of this experiment are in the Supplement.) For these data in r = 3750 dimensions, with N = 650, 000, the goal was to obtain a smooth, low dimensional embedding. The intrinsic dimension d is unknown, CV cannot be applied, and it is impractical to construct multiple embeddings for large N . Hence, we used the GC method with d0 = 1, 2, 3 and N 0 = |I| = 200. We compare the ??s obtained with a heuristic based on the scaling of the neighborhood sizes [30] with the radius, which relates , d and N (Figure 2). Remarkably, both methods yield the same , see the Supplement for evidence that the resulting embedding is smooth. 6 Discussion In manifold learning, supervised and unsupervised, estimating the graph versions of Laplacian-type operators is a fundamental task. We have provided a principled method for selecting the parameters of such operators, and have applied it to the selection of the bandwidth/scale parameter . Moreover, our method can be used to optimize any other parameters used in the graph Laplacian; for example, k in the k-nearest neighbors graph, or - more interestingly - the renormalization parameter ? [17] of the kernel. The latter is theoretically equal to 1, but it is possible that it may differ from 1 in the finite N regime. In general, for finite N , a small departure from the asymptotic prescriptions may be beneficial - and a data-driven method such as ours can deliver this benefit. By imposing geometric self-consistency, our method estimates an intrinsic quantity of the data. GC is also fully unsupervised, aiming to optimize a (lossy) representation of the data, rather than a particular task. This is an efficiency if the data is used in an unsupervised mode, or if it is used in many different subsequent tasks. Of course, one cannot expect an unsupervised method to always be superior to a task-dependent one. Yet, GC has shown to be competitive and sometimes superior in experiments with the widely accepted CV. Besides the experimental validation, there are other reasons to consider an unsupervised method like GC in a supervised task: (1) the labeled data is scarce, so ? will have high variance, (2) the CV cost function is highly non-smooth while D is much smoother, and (3) when there is more than one parameter to optimize, difficulties (1) and (2) become much more severe. Our algorithm requires minimal prior knowledge. In particular, it does not require exact knowledge of the intrinsic dimension d, since it can work satisfactorily with d0 = 1 in many cases. An interesting problem that is outside the scope of our paper is the question of whether  needs to vary over M. This is a question/challenge facing not just GC, but any method for setting the scale, unsupervised or supervised. Asymptotically, a uniform  is sufficient. Practically, however, we believe that allowing  to vary may be beneficial. 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Dual Path Networks Yunpeng Chen1 , Jianan Li1,2 , Huaxin Xiao1,3 , Xiaojie Jin1 , Shuicheng Yan4,1 , Jiashi Feng1 1 National University of Singapore 2 Beijing Institute of Technology 3 National University of Defense Technology 4 Qihoo 360 AI Institute Abstract In this work, we present a simple, highly efficient and modularized Dual Path Network (DPN) for image classification which presents a new topology of connection paths internally. By revealing the equivalence of the state-of-the-art Residual Network (ResNet) and Densely Convolutional Network (DenseNet) within the HORNN framework, we find that ResNet enables feature re-usage while DenseNet enables new features exploration which are both important for learning good representations. To enjoy the benefits from both path topologies, our proposed Dual Path Network shares common features while maintaining the flexibility to explore new features through dual path architectures. Extensive experiments on three benchmark datasets, ImagNet-1k, Places365 and PASCAL VOC, clearly demonstrate superior performance of the proposed DPN over state-of-the-arts. In particular, on the ImagNet-1k dataset, a shallow DPN surpasses the best ResNeXt-101(64 ? 4d) with 26% smaller model size, 25% less computational cost and 8% lower memory consumption, and a deeper DPN (DPN-131) further pushes the state-of-the-art single model performance with about 2 times faster training speed. Experiments on the Places365 large-scale scene dataset, PASCAL VOC detection dataset, and PASCAL VOC segmentation dataset also demonstrate its consistently better performance than DenseNet, ResNet and the latest ResNeXt model over various applications. 1 Introduction ?Network engineering? is increasingly more important for visual recognition research. In this paper, we aim to develop new path topology of deep architectures to further push the frontier of representation learning. In particular, we focus on analyzing and reforming the skip connection, which has been widely used in designing modern deep neural networks and offers remarkable success in many applications [16, 7, 20, 14, 5]. Skip connection creates a path propagating information from a lower layer directly to a higher layer. During the forward propagation, skip connection enables a very top layer to access information from a distant bottom layer; while for the backward propagation, it facilitates gradient back-propagation to the bottom layer without diminishing magnitude, which effectively alleviates the gradient vanishing problem and eases the optimization. Deep Residual Network (ResNet) [5] is one of the first works that successfully adopt skip connections, where each mirco-block, a.k.a. residual function, is associated with a skip connection, called residual path. The residual path element-wisely adds the input features to the output of the same mircoblock, making it a residual unit. Depending on the inner structure design of the mirco-block, the residual network has developed into a family of various architectures, including WRN [22], Inceptionresnet [20], and ResNeXt [21]. More recently, Huang et al. [8] proposed a different network architecture that achieves comparable accuracy with deep ResNet [5], named Dense Convolutional Network (DenseNet). Different from residual networks which add the input features to the output features through the residual path, the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. DenseNet uses a densely connected path to concatenate the input features with the output features, enabling each micro-block to receive raw information from all previous micro-blocks. Similar with residual network family, DenseNet can be categorized to the densely connected network family. Although the width of the densely connected path increases linearly as it goes deeper, causing the number of parameters to grow quadratically, DenseNet provides higher parameter efficiency compared with the ResNet [5]. In this work, we aim to study the advantages and limitations of both topologies and further enrich the path design by proposing a dual path architecture. In particular, we first provide a new understanding of the densely connected networks from the lens of a higher order recurrent neural network (HORNN) [19], and explore the relations between densely connected networks and residual networks. More specifically, we bridge the densely connected networks with the HORNNs, showing that the densely connected networks are HORNNs when the weights are shared across steps. Inspired by [12] which demonstrates the relations between the residual networks and RNNs, we prove that the residual networks are densely connected networks when connections are shared across layers. With this unified view on the state-of-the-art deep architecture, we find that the deep residual networks implicitly reuse the features through the residual path, while densely connected networks keep exploring new features through the densely connected path. Based on this new view, we propose a novel dual path architecture, called the Dual Path Network (DPN). This new architecture inherits both advantages of residual and densely connected paths, enabling effective feature re-usage and re-exploitation. The proposed DPN also enjoys higher parameter efficiency, lower computational cost and lower memory consumption, and being friendly for optimization compared with the state-of-the-art classification networks. Experimental results validate the outstanding high accuracy of DPN compared with other well-established baselines for image classification on both ImageNet-1k dataset and Places365-Standard dataset. Additional experiments on object detection task and semantic segmentation task also demonstrate that the proposed dual path architecture can be broadly applied for various tasks and consistently achieve the best performance. 2 Related work Designing an advanced neural network architecture is one of the most challenging but effective ways for improving the image classification performance, which can also directly benefit a variety of other tasks. AlexNet [10] and VGG [18] are two most important works that show the power of deep convolutional neural networks. They demonstrate that building deeper networks with tiny convolutional kernels is a promising way to increase the learning capacity of the neural network. Residual Network was first proposed by He et al. [5], which greatly alleviates the optimization difficulty and further pushes the depth of deep neural networks to hundreds of layers by using skipping connections. Since then, different kinds of residual networks arose, concentrating on either building a more efficient micro-block inner structure [3, 21] or exploring how to use residual connections [9]. Recently, Huang et al. [8] proposed a different network, called Dense Convolutional Networks, where skip connections are used to concatenate the input to the output instead of adding. However, the width of the densely connected path linearly increases as the depth rises, causing the number of parameters to grow quadratically and costing a large amount of GPU memory compared with the residual networks if the implementation is not specifically optimized. This limits the building of a deeper and wider densenet that may further improve the accuracy. Besides designing new architectures, researchers also try to re-explore the existing state-of-the-art architectures. In [6], the authors showed the importance of the residual path on alleviating the optimization difficulty. In [12], the residual networks are bridged with recurrent neural networks (RNNs), which helps people better understand the deep residual network from the perspective of RNNs. In [3], several different residual functions are unified, trying to provide a better understanding of designing a better mirco structure with higher learning capacity. But still, for the densely connected networks, in addition to several intuitive explanations on better feature reusage and efficient gradient flow introduced, there have been few works that are able to provide a really deeper understanding. In this work, we provide a deeper understanding of the densely connected network, from the lens of Higher Order RNN, and explain how the residual networks are in indeed a special case of densely connected network. Based on these analysis, we then propose a novel Dual Path Network architecture that not only achieves higher accuracy, but also enjoys high parameter and computational efficiency. 2 ? ? + h2 g2(?) + f1(?) h1 h2 Unfold z-1 ?(?)+I(?) Fold g1(?) x0 (a) ResNet with shared weights Output + Output f2(?) + h1 + xt z-1 Unfold Fold hk gk(?) + fk-2k(?) z-1 ... f1k(?) + f0k(?) x0 (b) ResNet in RNN form + fk-1k(?) z-1 x0 (c) DenseNet with shared weights (d) DenseNet in HORNN form Figure 1: The topological relations of different types of neural networks. (a) and (b) show relations between residual networks and RNN, as stated in [12]; (c) and (d) show relations between densely connected networks and higher order recurrent neural network (HORNN), which is explained in this paper. The symbol ?z ?1 ? denotes a time-delay unit; ??? denotes the element-wise summation; ?I(?)? denotes an identity mapping function. 3 Revisiting ResNet, DenseNet and Higher Order RNN In this section, we first bridge the densely connected network [8] with higher order recurrent neural networks [19] to provide a new understanding of the densely connected network. We prove that residual networks [5, 6, 22, 21, 3], essentially belong to the family of densely connected networks except their connections are shared across steps. Then, we present analysis on strengths and weaknesses of each topology architecture, which motivates us to develop the dual path network architecture. For exploring the above relation, we provide a new view on the densely connected networks from the lens of Higher Order RNN, explain their relations and then specialize the analysis to residual networks. Throughout the paper, we formulate the HORNN in a more generalized form. We use ht to denote the hidden state of the recurrent neural network at the t-th step and use k as the index of the current step. Let xt denotes the input at t-th step, h0 = x0 . For each step, ftk (?) refers to the feature extracting function which takes the hidden state as input and outputs the extracted information. The g k (?) denotes a transformation function that transforms the gathered information to current hidden state: "k?1 # X k k k t (1) h =g ft (h ) . t=0 Eqn. (1) encapsulates the update rule of various network architectures in a generalized way. For k HORNNs, weights are shared across steps, i.e. ?t, k, fk?t (?) ? ft (?) and ?k, g k (?) ? g(?). For the densely connected networks, each step (micro-block) has its own parameter, which means ftk (?) and g k (?) are not shared. Such observation shows that the densely connected path of DenseNet is essentially a higher order path which is able to extract new information from previous states. Figure 1(c)(d) graphically shows the relations of densely connected networks and higher order recurrent networks. We then explain that the residual networks are special cases of densely connected networks if taking ?t, k, ftk (?) ? ft (?). Here, for succinctness we introduce rk to denote the intermediate results and let r0 = 0. Then Eqn. (1) can be rewritten as rk , k?1 X t=1 k hk = g ft (ht ) = rk?1 + fk?1 (hk?1 ), (2)  rk . (3) Thus, by substituting Eqn. (3) into Eqn. (2), Eqn. (2) can be simplified as  rk = rk?1 + fk?1 (hk?1 ) = rk?1 + fk?1 (g k?1 rk?1 ) = rk?1 + ?k?1 (rk?1 ), (4) where ?k (?) = fk (g k (?)). Obviously, Eqn. (4) has the same form as the residual network and the recurrent neural network. Specifically, when ?k, ?k (?) ? ?(?), Eqn. (4) degenerates to an RNN; when none of ?k (?) is shared and xk = 0, k > 1, Eqn. (4) produces a residual network. Figure 1(a)(b) 3 graphically shows the relation. Besides, recall that Eqn. (4) is derived under the condition when ?t, k, ftk (?) ? ft (?) from Eqn. (1) and the densely connected networks are in forms of Eqn. (1), meaning that the residual network family essentially belongs to the densely connected network family. Figure 2(a?c) give an example and demonstrate such equivalence, where ft (?) corresponds to the first 1 ? 1 convolutional layer and the g k (?) corresponds to the other layers within a micro-block in Figure 2(b). From the above analysis, we observe: 1) both residual networks and densely connected networks can be seen as a HORNN when ftk (?) and g k (?) are shared for all k; 2) a residual network is a densely connected network if ?t, k, ftk (?) ? ft (?). By sharing the ftk (?) across all steps, g k (?) receives the same feature from a given output state, which encourages the feature reusage and thus reduces the feature redundancy. However, such an information sharing strategy makes it difficult for residual networks to explore new features. Comparatively, the densely connected networks are able to explore new information from previous outputs since the ftk (?) is not shared across steps. However, different ftk (?) may extract the same type of features multiple times, leading to high redundancy. In the following section, we present the dual path networks which can overcome both inherent limitations of these two state-of-the-art network architectures. Their relations with HORNN also imply that our proposed architecture can be used for improving HORNN, which we leave for future works. 4 Dual Path Networks Above we explain the relations between residual networks and densely connected networks, showing that the residual path implicitly reuses features, but it is not good at exploring new features. In contrast the densely connected network keeps exploring new features but suffers from higher redundancy. In this section, we describe the details of our proposed novel dual path architecture, i.e. the Dual Path Network (DPN). In the following, we first introduce and formulate the dual path architecture, and then present the network structure in details with complexity analysis. 4.1 Dual Path Architecture Sec. 3 discusses the advantage and limitations of both residual networks and densely connected networks. Based on the analysis, we propose a simple dual path architecture which shares the ftk (?) across all blocks to enjoy the benefits of reusing common features with low redundancy, while still remaining a densely connected path to give the network more flexibility in learning new features. We formulate such a dual path architecture as follows: xk , k?1 X ftk (ht ), (5) vt (ht ) = y k?1 + ?k?1 (y k?1 ), (6) t=1 yk , k?1 X t=1 k rk , x + yk ,  hk = g k rk , (7) (8) where xk and y k denote the extracted information at k-th step from individual path, vt (?) is a feature learning function as ftk (?). Eqn. (5) refers to the densely connected path that enables exploring new features, Eqn. (6) refers to the residual path that enables common features re-usage, and Eqn. (7) defines the dual path that integrates them and feeds them to the last transformation function in Eqn. (8). The final transformation function g k (?) generates current state, which is used for making next mapping or prediction. Figure 2(d)(e) show an example of the dual path architecture that is being used in our experiments. More generally, the proposed DPN is a family of convolutional neural networks which contains a residual alike path and a densely connected alike path, as explained later. Similar to these networks, one can customize the micro-block function of DPN for task-specific usage or for further overall performance boosting. 4 1?1 3?3 3?3 + 3?3 3?3 1?1 1?1 + + 1?1 3?3 3?3 + 1?1 3?3 1?1 3?3 + 1?1 3?3 + (a) Residual Network (b) Densely Connected Network + 1?1 + (c) Densely Connected Network ( with shared connections ) 1?1 1?1 ~ 1?1 1?1 1?1 1?1 1?1 1?1 + ~ 1?1 + 1?1 ~ 1?1 + 1?1 ~ 3?3 1?1 residual unit 1?1 (d) Dual Path Architecture 1?1 + (e) DPN Figure 2: Architecture comparison of different networks. (a) The residual network. (b) The densely connected network, where each layer can access the outputs of all previous micro-blocks. Here, a 1 ? 1 convolutional layer (underlined) is added for consistency with the micro-block design in (a). (c) By sharing the first 1 ? 1 connection of the same output across micro-blocks in (b), the densely connected network degenerates to a residual network. The dotted rectangular in (c) highlights the residual unit. (d) The proposed dual path architecture, DPN. (e) An equivalent form of (d) from the perspective of implementation, where the symbol ?o? denotes a split operation, and ?+? denotes element-wise addition. 4.2 Dual Path Networks The proposed network is built by stacking multiple modualized mirco-blocks as shown in Figure 2. In this work, the structure of each micro-block is designed with a bottleneck style [5] which starts with a 1 ? 1 convolutional layer followed by a 3 ? 3 convolutional layer, and ends with a 1 ? 1 convolutional layer. The output of the last 1 ? 1 convolutional layer is split into two parts: the first part is element-wisely added to the residual path, and the second part is concatenated with the densly connected path. To enhance the leaning capacity of each micro-block, we use the grouped convolution layer in the second layer as the ResNeXt [21]. Considering that the residual networks are more wildly used than the densely connected networks in practice, we choose the residual network as the backbone and add a thin densely connected path to build the dual path network. Such design also helps slow the width increment of the densely connected path and the cost of GPU memory. Table 1 shows the detailed architecture settings. In the table, G refers to the number of groups, and k refers to the channels increment for the densely connected path. For the new proposed DPNs, we use (+k) to indicate the width increment of the densely connected path. The overall design of DPN inherits backbone architecture of the vanilla ResNet / ResNeXt, making it very easy to implement and apply to other tasks. One can simply implement a DPN by adding one more ?slice layer? and ?concat layer? upon existing residual networks. Under a well optimized deep learning platform, none of these newly added operations requires extra computational cost or extra memory consumption, making the DPNs highly efficient. In order to demonstrate the appealing effectiveness of the dual path architecture, we intentionally design a set of DPNs with a considerably smaller model size and less FLOPs compared with the sate-of-the-art ResNeXts [21], as shown in Table 1. Due to limited computational resources, we set these hyper-parameters based on our previous experience instead of grid search experiments. Model complexity We measure the model complexity by counting the total number of learnable parameters within each neural network. Table 1 shows the results for different models. The DPN-92 costs about 15% fewer parameters than ResNeXt-101 (32 ? 4d), while the DPN-98 costs about 26% fewer parameters than ResNeXt-101 (64 ? 4d). Computational complexity We measure the computational cost of each deep neural network using the floating-point operations (FLOPs) with input size of 224 ? 224, in the number of multiply-adds following [21]. Table 1 shows the theoretical computational cost. Though the actual time cost might be influenced by other factors, e.g. GPU bandwidth and coding quality, the computational cost shows the speed upper bound. As can be see from the results, DPN-92 consumes about 19% less FLOPs than ResNeXt-101(32 ? 4d), and the DPN-98 consumes about 25% less FLOPs than ResNeXt-101(64 ? 4d). 5 Table 1: Architecture and complexity comparison of our proposed Dual Path Networks (DPNs) and other state-of-the-art networks. We compare DPNs with two baseline methods: DenseNet [5] and ResNeXt [21]. The symbol (+k) denotes the width increment on the densely connected path. stage output DenseNet-161 (k=48) ResNeXt-101 (32?4d) ResNeXt-101 (64?4d) DPN-92 (32?3d) DPN-98 (40?4d) conv1 112x112 7 ? 7, 96, stride 2 7 ? 7, 64, stride 2 7 ? 7, 64, stride 2 7 ? 7, 64, stride 2 7 ? 7, 96, stride 2 conv2 56x56 3 ? 3 max pool, stride 2 ? 1?1, 256 ? 3?3, 256, G=64 ? ? 3 1?1, 256 ? ? 1?1, 512 ? 3?3, 512, G=64 ? ? 4 1?1, 512 ? ? 1?1, 1024 ? 3?3, 1024, G=64 ? ? 23 1?1, 1024 ? ? 1?1, 2048 ? 3?3, 2048, G=64 ? ? 3 1?1, 2048 3 ? 3 max pool, stride 2 ? 1?1, 96 ? 3?3, 96, G=32 ? ? 3 1?1, 256 (+16) ? ? 1?1, 192 ? 3?3, 192, G=32 ? ? 4 1?1, 512 (+32) ? ? 1?1, 384 ? 3?3, 384, G=32 ? ? 20 1?1, 1024 (+24) ? ? 1?1, 768 ? 3?3, 768, G=32 ??3 1?1, 2048 (+128) 3 ? 3 max pool, stride 2   conv3 28?28  conv4 14?14 conv5 7?7  1?1, 192 3?3, 48 1?1, 192 3?3, 48 ?  ? 6 ?  ? 12 1?1, 192 3?3, 48  1?1, 192 3?3, 48  ? ? ? ? 36 ? ? ? 24 ? 3 ? 3 max pool, stride 2 ? 1?1, 128 3?3, 128, G=32 ? ? 3 1?1, 256 ? 1?1, 256 3?3, 256, G=32 ? ? 4 1?1, 512 ? 1?1, 512 3?3, 512, G=32 ? ? 23 1?1, 1024 ? 1?1, 1024 3?3, 1024, G=32 ? ? 3 1?1, 2048 ? ? ? ? ? ? ? ? ? ? 3 ? 3 max pool, stride 2 ? 1?1, 160 3?3, 160, G=40 ? ? 3 1?1, 256 (+16) ? 1?1, 320 3?3, 320, G=40 ? ? 6 1?1, 512 (+32) ? 1?1, 640 3?3, 640, G=40 ? ? 20 1?1, 1024 (+32) ? 1?1, 1280 3?3, 1280, G=40 ? ? 3 1?1, 2048 (+128) global average pool 1000-d fc, softmax global average pool 1000-d fc, softmax global average pool 1000-d fc, softmax global average pool 1000-d fc, softmax global average pool 1000-d fc, softmax # params 28.9 ? 106 44.3 ? 106 83.7 ? 106 37.8 ? 106 61.7 ? 106 FLOPs 7.7 ? 109 8.0 ? 109 15.5 ? 109 6.5 ? 109 11.7 ? 109 1?1 5 Experiments Extensive experiments are conducted for evaluating the proposed Dual Path Networks. Specifically, we evaluate the proposed architecture on three tasks: image classification, object detection and semantic segmentation, using three standard benchmark datasets: the ImageNet-1k dataset, Places365Standard dataset and the PASCAL VOC datasets. Key properties of the proposed DPNs are studied on the ImageNet-1k object classification dataset [17] and further verified on the Places365-Standard scene understanding dataset [24]. To verify whether the proposed DPNs can benefit other tasks besides image classification, we further conduct experiments on the PASCAL VOC dataset [4] to evaluate its performance in object detection and semantic segmentation. 5.1 Experiments on image classification task We implement the DPNs using MXNet [2] on a cluster with 40 K80 graphic cards. Following [3], we adopt standard data augmentation methods and train the networks using SGD with a mini-batch size of 32 for each GPU. For the deepest network, i.e. DPN-1311 , the mini-batch ?size is limited to 24 because of the 12GB GPU memory constraint. The learning rate starts from 0.1 for DPN-92 and DPN-131, and from 0.4 for DPN-98. It drops in a ?steps? manner by a factor of 0.1. Following [5], batch normalization layers are refined after training. 5.1.1 ImageNet-1k dataset Firstly, we compare the image classification performance of DPNs with current state-of-the-art models. As can be seen from the first block in Table 2, a shallow DPN with only the depth of 92 reduces the top-1 error rate by an absolute value of 0.5% compared with the ResNeXt-101(32 ? 4d) and an absolute value of 1.5% compared with the DenseNet-161 yet provides with considerably less FLOPs. In the second block of Table 2, a deeper DPN (DPN-98) surpasses the best residual network ? ResNeXt-101 (64 ? 4d), and still enjoys 25% less FLOPs and a much smaller model size (236 MB v.s. 320 MB). In order to further push the state-of-the-art accuracy, we slightly increase the depth of the DPN to 131 (DPN-131). The results are shown in the last block in Table 2. Again, the DPN shows superior accuracy over the best single model ? Very Deep PolyNet [23], with a much smaller model size (304 MB v.s. 365 MB). Note that the Very Deep PolyNet adopts numerous tricks, e.g. initialization by insertion, residual scaling, stochastic paths, to assist the training process. In contrast, our proposed DPN-131 is simple and does not involve these tricks, DPN-131 can be trained using a standard training strategy as shallow DPNs. More importantly, the actual training speed of DPN-131 is about 2 times faster than the Very Deep PolyNet, as discussed in the following paragraph. 1 The DPN-131 has 128 channels at conv1, 4 blocks at conv2, 8 blocks at conv3, 28 blocks at conv4 and 3 blocks at conv5, which has #params=79.5 ? 106 and FLOPs=16.0 ? 109 . 6 Table 2: Comparison with state-of-the-art CNNs on ImageNet-1k dataset. Single crop validation error rate (%) on validation set. *: Performance reported by [21], Table 3: Comparison with state-of-the?: With Mean-Max Pooling (see supplementary material). art CNNs on Places365-Standard dataset. Model x224 x320 / x299 10 crops validation accuracy rate (%) on Method GFLOPs Size top-1 top-5 top-1 top-5 validation set. 111 MB 170 MB 170 MB 145 MB 247 MB 227 MB 320 MB 236 MB 531 MB 365 MB 304 MB 304 MB 7.7 7.8 8.0 6.5 15.0 ? 15.5 11.7 ? ? 16.0 16.0 ? ? ? 4.7 4.8 4.9 4.4 4.4 4.48 4.25 4.23 4.16 20 19.5 ResNeXt-101 (64x4d) 19 DPN-131 (40x4d) DPN-98 (40x4d) 18.5 50 60 70 80 90 100 Training Speed (samples/sec) (a) AlexNet [24] GoogleLeNet [24] VGG-16 [24] ResNet-152 [24] ResNeXt-101 [3] CRU-Net-116 [3] DPN-92 (32 ? 3d) ResNet-200 20 19.5 ResNeXt-101 (64x4d) 19 DPN-98 (40x4d) DPN-131 (40x4d) 18.5 8 9 10 11 Memory Cost (GB), Batch Size = 24 (b) Model Size 223 MB 44 MB 518 MB 226 MB 165 MB 163 MB 138 MB Method 20.5 ResNet-200 Single Crop, Top-1 Error Single Crop, Top-1 Error 20.5 22.2 ? ? 22.0 6.0 ? 21.2 5.6 ? 20.7 5.4 19.3 21.7 5.8 20.1 ? ? 19.9 20.4 5.3 19.1 20.2 5.2 18.9 ? ? 19.10 ? ? 18.71 19.93 5.12 18.62 19.93 5.12 18.55 12 Memory Cost (GB), Batch Size = 24 DenseNet-161(k=48) [8] ResNet-101* [5] ResNeXt-101 (32 ? 4d) [21] DPN-92 (32 ? 3d) ResNet-200 [6] Inception-resnet-v2 [20] ResNeXt-101 (64 ? 4d) [21] DPN-98 (40 ? 4d) Very deep Inception-resnet-v2 [23] Very Deep PolyNet [23] DPN-131 (40 ? 4d) DPN-131 (40 ? 4d) ? top-1 acc. 53.17 53.63 55.24 54.74 56.21 56.60 56.84 top-5 acc. 82.89 83.88 84.91 85.08 86.25 86.55 86.69 12 DPN-131 (40x4d) 11 10 9 ResNet-200 ResNeXt-101 (64x4d) DPN-98 (40x4d) 8 50 60 70 80 90 100 Training Speed (samples/sec) (c) Figure 3: Comparison of total actual cost between different models during training. Evaluations are conducted on a single Node with 4 K80 graphic card with all training samples cached into memory. (For the comparison of Training Speed, we push the mini-batch size to its maximum value given a 12GB GPU memory to test the fastest possible training speed of each model.) Secondly, we compare the training cost between the best performing models. Here, we focus on evaluating two key properties ? the actual GPU memory cost and the actual training speed. Figure 3 shows the results. As can be seen from Figure 3(a)(b), the DPN-98 is 15% faster and uses 9% less memory than the best performing ResNeXt with a considerably lower testing error rate. Note that theoretically the computational cost of DPN-98 shown in Table 2 is 25% less than the best performing ResNeXt, indicating there is still room for code optimization. Figure 3(c) presents the same result in a more clear way. The deeper DPN-131 only costs about 19% more training time compared with the best performing ResNeXt, but achieves the state-of-the-art single model performance. The training speed of the previous state-of-the-art single model, i.e. Very Deep PolyNet (537 layers) [23], is about 31 samples per second based on our implementation using MXNet, showing that DPN-131 runs about 2 times faster than the Very Deep PolyNet during training. 5.1.2 Place365-Standard dataset In this experiment, we further evaluate the accuracy of the proposed DPN on the scene classification task using the Places365-Standard dataset. The Places365-Standard dataset is a high-resolution scene understanding dataset with more than 1.8 million images of 365 scene categories. Different from object images, scene images do not have very clear discriminative patterns and require a higher level context reasoning ability. Table 3 shows the results of different models on this dataset. To make a fair comparison, we perform the DPN-92 on this dataset instead of using deeper DPNs. As can be seen from the results, DPN achieves the best validation accuracy compared with other methods. The DPN-92 requires much less parameters (138 MB v.s. 163 MB), which again demonstrates its high parameter efficiency and high generalization ability. 5.2 Experiments on the object detection task We further evaluate the proposed Dual Path Network on the object detection task. Experiments are performed on the PASCAL VOC 2007 datasets [4]. We train the models on the union set of VOC 2007 trainval and VOC 2012 trainval following [16], and evaluate them on VOC 2007 test set. We use standard evaluation metrics Average Precision (AP) and mean of AP (mAP) following the PASCAL challenge protocols for evaluation. 7 Table 4: Object detection results on PASCAL VOC 2007 test set. The performance is measured by mean of Average Precision (mAP, in %). Method mAP areo bike bird boat bottle bus car cat chair cow table dog horse mbk prsn plant sheep sofa train tv DenseNet-161 (k=48) 79.9 80.4 85.9 81.2 72.8 68.0 87.1 88.0 88.8 64.0 83.3 75.4 87.5 87.6 81.3 84.2 54.6 83.2 80.2 87.4 77.2 ResNet-101 [16] 76.4 79.8 80.7 76.2 68.3 55.9 85.1 85.3 89.8 56.7 87.8 69.4 88.3 88.9 80.9 78.4 41.7 78.6 79.8 85.3 72.0 ResNeXt-101 (32 ? 4d) 80.1 80.2 86.5 79.4 72.5 67.3 86.9 88.6 88.9 64.9 85.0 76.2 87.3 87.8 81.8 84.1 55.5 84.0 79.7 87.9 77.0 DPN-92 (32 ? 3d) 82.5 84.4 88.5 84.6 76.5 70.7 87.9 88.8 89.4 69.7 87.0 76.7 89.5 88.7 86.0 86.1 58.4 85.0 80.4 88.2 83.1 Table 5: Semantic segmentation results on PASCAL VOC 2012 test set. The performance is measured by mean Intersection over Union (mIoU, in %). Method mIoU bkg areo bike bird boat bottle bus car cat chair cow table dog horse mbk prsn plant sheep sofa train tv DenseNet-161 (k=48) 68.7 92.1 77.3 37.1 83.6 54.9 70.0 85.8 82.5 85.9 26.1 73.0 55.1 80.2 74.0 79.1 78.2 51.5 80.0 42.2 75.1 58.6 ResNet-101 73.1 93.1 86.9 39.9 87.6 59.6 74.4 90.1 84.7 87.7 30.0 81.8 56.2 82.7 82.7 80.1 81.1 52.4 86.2 52.5 81.3 63.6 ResNeXt-101 (32 ? 4d) 73.6 93.1 84.9 36.2 80.3 65.0 74.7 90.6 83.9 88.7 31.1 86.3 62.4 84.7 86.1 81.2 80.1 54.0 87.4 54.0 76.3 64.2 DPN-92 (32 ? 3d) 74.8 93.7 88.3 40.3 82.7 64.5 72.0 90.9 85.0 88.8 31.1 87.7 59.8 83.9 86.8 85.1 82.8 60.8 85.3 54.1 82.6 64.6 We perform all experiments based on the ResNet-based Faster R-CNN framework, following [5] and make comparisons by replacing the ResNet, while keeping other parts unchanged. Since our goal is to evaluate DPN, rather than further push the state-of-the-art accuracy on this dataset, we adopt the shallowest DPN-92 and baseline networks at roughly the same complexity level. Table 4 provides the detection performance comparisons of the proposed DPN with several current state-of-the-art models. It can be observed that the DPN obtains the mAP of 82.5%, which makes large improvements, i.e. 6.1% compared with ResNet-101 [16] and 2.4% compared with ResNeXt-101 (32 ? 4d). The better results shown in this experiment demonstrate that the Dual Path Network is also capable of learning better feature representations for detecting objects and benefiting the object detection task. 5.3 Experiments on the semantic segmentation task In this experiment, we evaluate the Dual Path Network for dense prediction, i.e. semantic segmentation, where the training target is to predict the semantic label for each pixel in the input image. We conduct experiments on the PASCAL VOC 2012 segmentation benchmark dataset [4] and use the DeepLab-ASPP-L [1] as the segmentation framework. For each compared method in Table 5, we replace the 3 ? 3 convolutional layers in conv4 and conv5 of Table 1 with atrous convolution [1] and plug in a head of Atrous Spatial Pyramid Pooling (ASPP) [1] in the final feature maps of conv5. We adopt the same training strategy for all networks following [1] for fair comparison. Table 5 shows the results of different convolutional neural networks. It can be observed that the proposed DPN-92 has the highest overall mIoU accuracy. Compared with the ResNet-101 which has a larger model size and higher computational cost, the proposed DPN-92 further improves the IoU for most categories and improves the overall mIoU by an absolute value 1.7%. Considering the ResNeXt-101 (32 ? 4d) only improves the overall mIoU by an absolute value 0.5% compared with the ResNet-101, the proposed DPN-92 gains more than 3 times improvement compared with the ResNeXt-101 (32 ? 4d). The better results once again demonstrate the proposed Dual Path Network is capable of learning better feature representation for dense prediction. 6 Conclusion In this paper, we revisited the densely connected networks, bridged the densely connected networks with Higher Order RNNs and proved the residual networks are essentially densely connected networks with shared connections. Based on this new explanation, we proposed a dual path architecture that enjoys benefits from both sides. The novel network, DPN, is then developed based on this dual path architecture. Experiments on the image classification task demonstrate that the DPN enjoys high accuracy, small model size, low computational cost and low GPU memory consumption, thus is extremely useful for not only research but also real-word application. Experiments on the object detection task and semantic segmentation tasks show that the proposed DPN can also benefit other tasks by simply replacing the base network. Acknowledgments The work of Jiashi Feng was partially supported by National University of Singapore startup grant R-263-000-C08-133, Ministry of Education of Singapore AcRF Tier One grant R-263-000-C21-112 and NUS IDS grant R-263-000-C67-646. 8 References [1] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. arXiv preprint arXiv:1606.00915, 2016. [2] Tianqi Chen, Mu Li, Yutian Li, Min Lin, Naiyan Wang, Minjie Wang, Tianjun Xiao, Bing Xu, Chiyuan Zhang, and Zheng Zhang. Mxnet: A flexible and efficient machine learning library for heterogeneous distributed systems. arXiv preprint arXiv:1512.01274, 2015. [3] Yunpeng Chen, Xiaojie Jin, Bingyi Kang, Jiashi Feng, and Shuicheng Yan. Sharing residual units through collective tensor factorization in deep neural networks. arXiv preprint arXiv:1703.02180, 2017. 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Faster and Non-ergodic O(1/K) Stochastic Alternating Direction Method of Multipliers Cong Fang Feng Cheng Zhouchen Lin? Key Laboratory of Machine Perception (MOE), School of EECS, Peking University, P. R. China Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, P. R. China [email protected] [email protected] [email protected] Abstract We study stochastic convex optimization subjected to linear equality constraints. Traditional Stochastic Alternating Direction Method of Multipliers ? [1] and its Nesterov?s acceleration scheme [2] can only achieve ergodic O(1/ K) convergence rates, where K is the number of iteration. By introducing Variance Reduction (VR) techniques, the convergence rates improve to ergodic O(1/K) [3, 4]. In this paper, we propose a new stochastic ADMM which elaborately integrates Nesterov?s extrapolation and VR techniques. With Nesterov?s extrapolation, our algorithm can achieve a non-ergodic O(1/K) convergence rate which is optimal for separable linearly constrained non-smooth convex problems, ? while the convergence rates of VR based ADMM methods are actually tight O(1/ K) in non-ergodic sense. To the best of our knowledge, this is the first work that achieves a truly accelerated, stochastic convergence rate for constrained convex problems. The experimental results demonstrate that our algorithm is faster than the existing state-of-the-art stochastic ADMM methods. 1 Introduction We consider the following general convex finite-sum problem with linear constraints: n min x1 ,x2 s.t. h1 (x1 ) + f1 (x1 ) + h2 (x2 ) + 1X f2,i (x2 ), n i=1 A1 x1 + A2 x2 = b, (1) where f1 (x1 ) and f2,i (x2 ) with i ? {1, 2, ? ? ? , n} are convex and have Lipschitz continuous gradients, h1 (x1 ) and h2 (x2 ) are also convex, but can be non-smooth. We use the following notations: L1 denotes the Lipschitz constant Pn of f1 (x1 ), L2 is the Lipschitz constant of f2,i (x2 ) with i ? {1, 2, ? ? ? , n}, and f2 (x) = n1 i=1 f2,i (x). And we use ?f to denote the gradient of f . Problem (1) is of great importance in machine learning. The finite-sum functions f2 (x2 ) are typically a loss over training samples, and the remaining functions control the structure or regularize the model to aid generalization [2]. The idea of using linear constraints to decouple the loss and regularization terms enables researchers to consider some more sophisticated regularization terms which might be very complicated to solve through proximity operators for Gradient Descent [5] methods. For example, for multitask learning problems [6, 7], the regularization term is set as ?1 kxk? + ?2 kxk1 , for most graph-guided fused Lasso and overlapping group Lasso problem [8, 4], the regularization term can be written as ?kAxk1 , and for many multi-view learning tasks [9], the regularization terms always involve ?1 kxk2,1 + ?2 kxk? . ? Corresponding author. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Table 1: Convergence rates of ADMM type methods solving Problem (1). Type Batch Stochastic Algorithm ADMM [13] LADM-NE [15] STOC-ADMM [1] OPG-ADMM [16] OPT-ADMM [2] SDCA-ADMM [17] SAG-ADMM [3] SVRG-ADMM [4] ACC-SADMM (ours) Convergence Rate Tight non-ergodic O( ?1K ) 1 ) Optimal non-ergodic O( K 1 ergodic O( ?K ) ergodic O( ?1K ) ergodic O( ?1K ) unknown Tight non-ergodic O( ?1K ) Tight non-ergodic O( ?1K ) 1 ) Optimal non-ergodic O( K Alternating Direction Method of Multipliers (ADMM) is a very popular optimization method to solve Problem (1), with its advantages in speed, easy implementation and good scalability shown in lots of literatures (see survey [10]). A popular criterion of the algorithms? convergence rate is its ergodic convergence. And it is proved in [11, 12] that ADMM converges with an O(1/K) ergodic rate. However, in this paper, it is noteworthy that we consider the convergence in the non-ergodic sense. The reasons are two folded: 1) in real applications, the output of ADMM methods are non-ergodic results (xK ), rather than the ergodic one (convex combination of x1 , x2 , ? ? ? , xK ), as the non-ergodic results are much faster (see detailed discussions in Section 5.3); 2) The ergodic convergence rate is not trivially the same as general-case?s rate. For a sequence {ak } = {1, ?1, 1, ?1, 1, ?1, ? ? ? } (When k is odd, ak is 1, and ?1 when k is even), it is divergent, while in ergodic sense, it converges in O(1/K). So the analysis in the non-ergodic are closer to reality. 2) is especially suit for ADMM methods. In [13], in non? Davis et al. prove that the Douglas-Rachford (DR) splitting converges ? ergodic O(1/ K). They also construct a family of functions showing that non-ergodic O(1/ K) is ? tight. Chen et al. establish O(1/ K) for Linearized ADMM [14]. Then Li et al. accelerate ADMM through Nesterov?s extrapolation and obtain a non-ergodic O(1/K) convergence rate[15]. They also prove that the lower complexity bound of ADMM type methods for the separable linearly constrained nonsmooth convex problems is exactly O(1/K), which demonstrates that their algorithm is optimal. The convergence rates for different ADMM based algorithms are shown in Table 1. On the other hand, to meet the demands of solving large-scale machine learning problems, stochastic algorithms [18] have drawn a lot of interest in recent years. For stochastic ADMM (SADMM), the prior works are from STOC-ADMM [1] and OPG-ADMM ? [16]. Due to the noise of gradient, both of the two algorithms can only achieve an ergodic O(1/ K) convergence rate. There are two lines of research to accelerate SADMM. The first is to introduce the Variance Reduction (VR) [19, 20, 21] techniques into SADMM. VR methods ensure the descent direction to have a bounded variance and so can achieve faster convergence rates. The existing VR based SADMM algorithms include SDCA-ADMM [17], SAG-ADMM [3] and SVRG-ADMM [4]. SAG-ADMM and SVRG-ADMM can provably achieve ergodic O(1/K) rates for Porblem (1). The second way to accelerate SADMM is through the Nesterov?s acceleration [22]. This work is from [2], in which the authors propose Dy +? R2 an ergodic O( K + ??K ) stochastic algorithm (OPT-ADMM). The dependence on the 2 + K smoothness constant of the convergence rate is O(1/K 2 ) and so each term in the convergence?rate seems to have been improved to optimal. However, the worst convergence rate of it is still O(1/ K). In this paper, we propose Accelerated Stochastic ADMM (ACC-SADMM) for large scale general convex finite-sum problems with linear constraints. By elaborately integrating Nesterov?s extrapolation and VR techniques, ACC-SADMM provably achieves a non-ergodic O(1/K) convergence rate which is optimal for non-smooth problems. As in non-ergodic sense, ? the VR based SADMM methods (e.g. SVRG-ADMM, SAG-ADMM) converges in a tight O(1/ K) (please ? see detailed discussions in Section 5.3), ACC-SADMM improve the convergence rates from O(1/ K) to (1/K) in the ergodic sense and fill the theoretical gap between the stochastic and batch (deterministic) ADMM. The original idea to design our ACC-SADMM is by explicitly considering the snapshot vector x ? (approximately the mean value of x in the last epoch) into the extrapolation terms. This is, to some degree, inspired by [23] who proposes an O(1/K 2 ) stochastic gradient algorithm named Katyusha for convex 2 Table 2: Notations and Variables Notation hx, yiG , kxkG Fi (xi ) x y F (x) Meaning ? x Gy, xT Gx hi (xi ) + fi (xi ) (x1 , x2 ) (y1 , y2 ) F1 (x1 ) + F2 (x2 ) Variable k k ys,1 , ys,2 k k xs,1 , xs,2 ? k , ?k ? s s ?s x ?s,1 , x ?s,2 , b x?1 , x?2 , ?? T Meaning extrapolation variables primal variables dual and temp variables snapshot vectors optimal solution of Eq. (1) problems. However, there are many distinctions between the two algorithms (please see detailed discussions in Section 5.1). Our method is also very efficient in practice since we have sufficiently considered the noise of gradient into our acceleration scheme. For example, we adopt extrapolation as ysk = xks + (1 ? ?1,s ? ?2 )(xks ? xk?1 ) in the inner loop, where ?2 is a constant and ?1,s decreases s after every epoch, instead of directly adopting extrapolation as yk = xk + k+1 ?1k (1??1k?1 ) (xk ?1k?1 ? xk?1 ) k 2 ?x k in the original Nesterov?s scheme and adding proximal term kx ?k3/2 as [2] does. There are also variants on updating of multiplier and the snapshot vector. We list the contributions of our work as follows: ? We propose ACC-SADMM for large scale convex finite-sum problems with linear constraints which integrates Nesterov?s extrapolation and VR techniques. We prove that our algorithm converges in non-ergodic O(1/K) which is optimal for separable linearly constrained nonsmooth convex problems. To our best knowledge, this is the first work that achieves a truly accelerated, stochastic convergence rate for constrained convex problems. ? We do experiments on four bench-mark datasets to demonstrate the superiority of our algorithm. We also do experiments on the Multitask Learning [6] problem to demonstrate that our algorithm can be used on very large datasets. 2 Preliminary Most SADMM methods alternately minimize the following variant surrogate of the augmented Lagrangian: L1 kx1 ? xk1 k2G1 (2) 2 ? 2 (x2 ), x2 i + L2 kx2 ? xk2 k2G + ? kA1 x1 + A2 x2 ? b + ? k2 , +h2 (x2 ) + h?f 2 2 2 ? L0 (x1 , x2 , ?, ?) = h1 (x1 ) + h?f1 (x1 ), x1 i + ? 2 (x2 ) is an estimator of ?f2 (x2 ) from one or a mini-batch of training samples. So the where ?f computation cost for each iteration reduces from O(n) to O(b) instead, where b is the mini-batch size. When fi (x) = 0 and Gi = 0, with i = 1, 2, Problem (1) is solved as exact ADMM. When there is no hi (xi ), Gi is set as the identity matrix I, with i = 1, 2, the subproblem in xi can be solved through matrix inversion. This scheme is advocated in many SADMM methods [1, 3]. Another common approach is linearization (also called the inexact Uzawa method) [24, 25], where Gi is set as ?i I ? L?i ATi Ai with ?i ? 1 + L?i kATi Ai k. ? 2 (x2 ) is simply set as: For STOC-ADMM [1], ?f X ? 2 (x2 ) = 1 ?f ?f2,ik (x2 ), b (3) ik ?Ik where Ik is the mini-batch of size b from {1, 2, ? ? ? , n}. For SVRG-ADMM [4], the gradient estimator can be written as: X ? 2 (x2 ) = 1 ?f (?f2,ik (x2 ) ? ?f2,ik (? x2 )) + ?f2 (? x2 ), (4) b ik ?Ik where x ?2 is a snapshot vector (mean value of last epoch). 3 Algorithm 1 Inner loop of ACC-SADMM for k = 0 to m ? 1 do   ?s . ? k + ??2 A1 xk + A2 xk ? b Update dual variable: ?ks = ? s s,1 s,2 ?1,s Update xk+1 s,1 through Eq. (6). Update xk+1 s,2 through Eq. (7).  ? k+1 = ?k + ? A1 xk+1 + A2 xk+1 ? b . Update dual variable: ? s s s,1 s,2 Update ysk+1 through Eq. (5). end for k. 3 3.1 Our Algorithm ACC-SADMM To help readers easier understand our algorithm, we list the notations and the variables in Table 2. Our algorithm has double loops as we use SVRG [19], which also have two layers of nested loops to estimate the gradient. We denote subscript s as the index of the outer loop and superscript k as the index in the inner loops. For example, xks,1 is the value of x1 at the k-th step of the inner iteration and the s-th step of the outer iteration. And we use xks and ysk to denote (xks,1 , xks,2 ), and k k (ys,1 , ys,2 ), respectively. In each inner loop, we update primal variables xks,1 and xks,2 , extrapolation k k terms ys,1 , ys,2 and dual variable ?ks , and s remains unchanged. In the outer loop, we maintain ? s+1 , and then assign the initial value to the extrapolation snapshot vectors x ?s+1,1 , x ?s+1,2 and b 0 0 terms ys+1,1 and ys+1,2 . We directly linearize both the smooth term fi (xi ) and the augmented term ? ? 2 2 kA1 x1 + A2 x2 ? b + ? k . The whole algorithm is shown in Algorithm 2. 3.2 Inner Loop The inner loop of ACC-SAMM is straightforward, shown as Algorithm 1. In each iteration, we do extrapolation, and then update the primal and dual variables. There are two critical steps which ensures us to obtain a non-ergodic results. The first is extrapolation. We do extrapolation as: (5) ysk+1 = xk+1 + (1 ? ?1,s ? ?2 )(xk+1 ? xks ), s s We can find that 1 ? ?1,s ? ?2 ? 1 ? ?1,s . So comparing with original Nesterov?s scheme, our way is more ?mild? to tackle the noise of gradient. The second step is on the updating primal variables. xk+1 s,1 = (6) k argmin h1 (x1 ) + h?f1 (ys,1 ), x1 i x1 +h  ? k k A1 ys,1 + A2 ys,2 ? b + ?ks , A1 x1 i + ?1,s  L1 ?kAT1 A1 k + 2 2?1,s  k kx1 ? ys,1 k2 . And then update x2 with the latest information of x1 , which can be written as: xk+1 s,2 =  ? k A1 xk+1 s,1 + A2 ys,2 ? b ?1,s ! 1 (1 + b?2 )L2 ?kAT2 A2 k k + kx2 ? ys,2 k2 , 2 2?1,s k ? 2 (ys,1 argmin h2 (x2 ) + h?f ), x2 i + h x2 +?ks , A2 x2 i + (7) ? 2 (yk ) is obtained by the technique of SVRG [19] with the form: where ?f s,2 ? 2 (yk ) = 1 ?f s,2 b X  k ?f2,ik,s (ys,2 ) ? ?f2,ik,s (? xs,2 ) + ?f2 (? xs,2 ) . ik,s ?I(k,s) Comparing with unaccelerated SADMM methods, which alternately minimize Eq. (2), our method is k distincted in two ways. The first is that the gradient estimator are computed on the ys,2 . The second ? is that we have chosen a slower increasing penalty factor ?1,s , instead of a fixed one. 4 Algorithm 2 ACC-SADMM ? 0 = 0, x Input: epoch length m > 2, ?, ? = 2, c = 2, x00 = 0, ? ?0 = x00 , y00 = x00 , 0 1 m?? ?1,s = c+? s , ?2 = ? (m?1) . for s = 0 to S ? 1 do Do inner loop, as stated in Algorithm 1. Set primal variables: x0s+1 = xm s . Update snapshot vectors x ?s+1 through Eq. (8). ? 0 = ?m?1 + ?(1 ? ? )(A1 xm + A2 xm ? b). Update dual variable: ? s s+1 s,1 s,2 ? s+1 = A1 x Update dual snapshot variable: b ?s+1,1 + A2 x ?s+1,2 . 0 Update extrapolation terms ys+1 through Eq. (9). end for s. m?1 X 1 ?1,S + ?2 Output: ?S = x xm xkS . + S (m ? 1)(?1,S + ?2 ) + 1 (m ?1)(?1,S + ?2 ) + 1 k=1 3.3 Outer Loop The outer loop of our algorithm is a little complex, in which we preserve snapshot vectors, and then resets the initial value. The main variants we adpot is on the snapshot vector x ?s+1 and the 0 extrapolation term ys+1 . For the snapshot vector x ?s+1 , we update it as:    m?1 !  (? ? 1)?1,s+1 m 1 (? ? 1)?1,s+1 X k 1? xs . (8) x ?s+1 = xs + 1 + m ?2 (m ? 1)?2 k=1 x ?s+1 is not the average of {xks }, different from most SVRG-based methods [19, 4]. The way of 0 generating x ? guarantees a faster convergence rate for the constraints. Then we reset ys+1 as:  ?1,s+1  0 m?1 ys+1 = (1 ? ?2 )xm ?s+1 + (1 ? ?1,s )xm ? ?2 x ?s . (9) s + ?2 x s ? (1 ? ?1,s ? ?2 )xs ?1,s 4 Convergence Analysis In this section, we give the convergence results of ACC-SADMM. The proof and a outline can be found in Supplementary Material. As we have mentioned in Section 3.2, the main strategy that enable us to obtain a non-ergodic results is that we adopt extrapolation as Eq. (5). We first analyze each inner iteration, shown in Lemma 1. We ignore subscript s as s is unchanged in the inner iteration. Lemma 1 Assume that f1 (x1 ) and f2,i (x2 ) with i ? {1, 2, ? ? ? , n} are convex and have Lipschitz continuous gradients. L1 is the Lipschitz constant of f1 (x1 ). L2 is the Lipschitz constant of f2,i (x2 ) with i ? {1, 2, ? ? ? , n} . h1 (x1 ) and h2 (x2 ) is also convex. For Algorithm 2, in any epoch, we have   Eik L(xk+1 , xk+1 , ?? ) ? ?2 L(? x1 , x ?2 , ?? ) ? (1 ? ?2 ? ?1 )L(xk1 , xk2 , ?? ) 1 2  h i ?1 ? k ? ?? k2 ? Ei k? ? k+1 ? ?? k2 + 1 kyk ? (1 ? ?1 ? ?2 )xk ? ?2 x k? ?1 ? ?1 x?1 k2G3 ? 1 k 2? 2 1  1 ? (1 ? ?1 ? ?2 )xk1 ? ?2 x ?1 ? ?1 x?1 k2G3 ? Eik kxk+1 1 2 1 + ky2k ? (1 ? ?1 ? ?2 )xk2 ? ?2 x ?2 ? ?1 x?2 k2G4 2  1 ? Eik kxk+1 ? (1 ? ?1 ? ?2 )xk2 ? ?2 x ?2 ? ?1 x?2 k2G4 , 2 2 where Eik denotes that the expectation is taken over the random samples in the minibatch Ik,s , ?k = ? ? k + ?(1??1 ) (Axk ? b), L(x1 , x2 , ?) = F1 (x1 ) + F2 (x2 ) + h?, A1 x1 + A2 x2 ? bi and ? ?1     T T T ?kA1 A1 k ?A1 A1 ?kA2 A2 k 1 G3 = L1 + I ? , and G = (1 + )L + I. 4 2 ?1 ?1 b?2 ?1 Then Theorem 1 analyses ACC-SADMM in the whole iteration, which is the key convergence result of the paper. 5 Theorem 1 If the conditions in Lemma 1 hold, then we have    1 ?m ?(m?1)?2 0 0 ? 2 ? E k (A? xS ?b) ? Ax0 ? b + ?0 ? ? k (10) 2? ?1,S ?1,0   m (F (? xS ) ? F (x? ) + h?? , A? xS ? bi) +E ?1,S  1 ? 0 ?(1 ? ?1,0 ) ? C3 F (x00 ) ? F (x? ) + h?? , Ax00 ? bi + k? + (Ax00 ? b) ? ?? k2 2? 0 ?1,0 1 1  , + kx00,1 ? x?1 k2(? L +kAT A k)I?AT A + kx00,2 ? x?2 k2 1,0 1 1 1 (1+ b?1 )?1,0 L2 +kAT 1 1 2 2 2 A2 k I 2 where C3 = 1??1,0 +(m?1)?2 . ?1,0 Corollary 1 directly demonstrates that ACC-SADMM have a non-ergodic O(1/K) convergence rate. Corollary 1 If the conditions in Lemma 1 holds, we have E|F (? xS ) ? F (x? )| ? EkA? xS ? bk ? 1 O( ), S 1 O( ). S (11) ? S depends on the latest m information of xkS . So our convergence results is in We can find that x non-ergodic sense, while the analysis for SVRG-ADMM [4] and SAG-ADMM [3] is in ergodic sense, PS Pm 1 k k ? S = mS since they consider the point x s=1 k=1 xs , which is the convex combination of xs over all the iterations. Now we directly use the theoretical results of [15] to demonstrate that our algorithm is optimal when there exists non-smooth term in the objective function. Theorem 2 For the following problem: min F1 (x1 ) + F2 (x2 ), s.t. x1 ? x2 = 0, (12) x1 ,x2 let the ADMM type algorithm to solve it be: ? Generate ?k2 and y2k in any way,   ?k ? xk+1 = ProxF1 /? k y2k ? ? k2 , 1 ? Generate ?k+1 and y1k+1 in any way, 1   ?k+1 ? xk+1 = ProxF2 /? k y1k+1 ? ?1 k . 2 Then there exist convex functions F1 and F2 defined on X = {x ? R6k+5 : kxk ? B} for the above general ADMM method, satsifying ? k1 k + |F1 (? Lk? xk2 ? x xk1 ) ? F1 (x?1 ) + F1 (? xk2 ) ? F2 (x?2 )| ? LB , 8(k + 1) (13) P P ? k2 = ki=1 ?2i xi2 for any ?1i and ?2i with i from 1 to k. ? k1 = ki=1 ?1i xi1 and x where x Theorem 2 is Theorem 11 in [15]. More details can be found in it. Problem (12) is a special case of Problem (1) as we can set each F2,i (x2 ) = F (x2 ) with i = 1, ? ? ? , n or set n = 1. So there is no better ADMM type algorithm which converges faster than O(1/K) for Problem (1). 5 Discussions We discuss some properties of ACC-SADMM and make further comparisons with some related methods. 6 Table 3: Size of datasets and mini-batch size we adopt in the experiments Problem Lasso Multitask 5.1 Dataset a9a covertype mnist dna ImageNet # training 72, 876 290, 506 60, 000 2, 400, 000 1, 281, 167 # testing 72, 875 290, 506 10, 000 600, 000 50, 000 # dimension ? # class 74 ? 2 54 ? 2 784 ? 10 800 ? 2 4, 096 ? 1, 000 # minibatch 100 500 2, 000 Comparison with Katyusha As we have mentioned in Introduction, some intuitions of our algorithm are inspired by Katyusha [23], which obtains an O(1/K 2 ) algorithm for convex finite-sum problems. However, Katyusha cannot solve the problem with linear constraints. Besides, Katyusha uses the Nesterov?s second scheme to accelerate the algorithm while our method conducts acceleration through Nesterov?s extrapolation (Nesterov?s first scheme). And our proof uses the technique of [26], which is different from [23]. Our algorithm can be easily extended to unconstrained convex finite-sum and can also obtain a O(1/K 2 ) rate but belongs to the Nesterov?s first scheme 2 . 5.2 The Growth of Penalty Factor ? ?1,s ? The penalty factor ?1,s increases linearly with the iteration. One might deem that this make our algorithm impractical because after dozens of epoches, the large value of penalty factor might slow down the decrement of function value. However, we have not found any bad influence. There may be two reasons 1. In our algorithm, ?1,s decreases after each epoch (m iterations), which is much slower than LADM-NE [15]. So the growth of penalty factor works as a continuation technique [28], which may help to decrease the function value. 2. From Theorem 1, our algorithm converges in O(1/S) whenever ?1,s is large. So from the theoretical viewpoint, a large ?1,s cannot slow down our algorithm. We find that OPT-ADMM [2] also needs to decrease the step size with the iteration. 3 However, its step size decreasing rate is O(k 2 ) and is faster than ours. 5.3 The Importance of Non-ergodic O(1/K) SAG-ADMM [3] and SVRG-ADMM [4] accelerate SADMM to ergodic O(1/K). In Theorem 9 of [15], the authors ? generate a class of functions showing that the original ADMM has a tight non-ergodic O(1/ K) convergence rate. When n = 1, SAG-ADMM and?SVRG-ADMM are the same as batch ADMM, so their convergence rates are no better than O(1/ K). So in non-ergodic sense, our algorithm does have a faster convergence rate than VR based SADMM methods. Then we are to highlight the importance of our non-ergodic result. As we have mentioned in the Introduction, in practice, the output of ADMM methods is the non-ergodic result xK , not the mean of x1 to xK . For deterministic ADMM, the proof of ergodic O(1/K) rate is proposed in [11], after ADMM had become a prevailing method of solving machine learning problems [29]; for stochastic ADMM, e.g. SVRG-ADMM [4], the authors give an ergodic O(1/K) proof, but in experiment, what they emphasize to use is the mean value of the last epoch as the result. As the non-ergodic results are more close to reality, our algorithm is much faster than VR based SADMM methods, even when its rate is seemingly the same. Actually, though VR based SADMM methods have provably faster rates than STOC-ADMM, the improvement in practice is evident only after numbers of iterations, when point are close to the convergence point, rather than at the early stage. In both [3] and [4], the authors claim that SAG-ADMM and SVRG-ADMM are sensitive to initial points. We also find that if the step sizes are set based on the their theoretical guidances, sometimes they are even slower than STOC-ADMM (see Fig. 1) as the early stage lasts longer when the step size is small. Our algorithm is faster than the two algorithms which demonstrates that Nesterov?s extrapolation has truly accelerated the speed and the integration of extrapolation and VR techniques is harmonious and complementary. 2 We follow [26] to name the extrapolation scheme as Nesterov?s first scheme and the three-step scheme [27] as the Nesterov?s second scheme. 7 10 10 Faster and Non-ergodic O(1/K) Stochastic Alternating Direction Method 10of Multipliers 5 10 15 20 25 30 35 10 -3 40 5 (a) a9a-original 5 10 15 20 25 30 35 20 25 30 35 40 objective gap 5 10 5 10 15 20 25 30 35 10-3 40 25 30 35 40 5 10 15 20 25 30 40 5 10 -2 objective gap test loss 10 -3 0.2 10 -2 20 25 30 35 40 10 -4 3.8 10 15 20 25 30 35 40 35 40 ?10-3 3.6 3.4 3.2 10 -6 0.195 0.37 15 number of effective passes 10 0 0.205-1 10 10 -2 10 -4 10 -3 0.114 10 number of effective passes (d) dna-original 35 10 0 0.21 objective gap 0.372 objective gap test loss objective gap 0.116 0.374 10 -2 5 10-4 10 -1 0.118 -3 number of effective passes 10 0 0.376 20 (c) mnist-original number of effective passes 10 -1 15 number of effective passes (b) covertype-original 10-3 40 number of effective passes 0.12 15 10 -3 10 10 -4 10-2 number of effective passes number of effective passes 10-5 10 -1 10 -2 objective gap 10 -5 10-4 10 -3 test loss 10 -4 -2 objective gap 10 objective gap -3 10 -3 -1 objective gap 10 -2 -2 objective gap objective gap 10 test loss 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 objective gap 10 -2 3 10 -5 0.112 5 10 15 10 20 20 25 30 30 40 35 50 40 60 number of effective passes number of effective passes (a) (e)a9aa9a-group STOC-ADMM STOC-ADMM-ERG 5 1010 15 20 20 3025 40 30 50 35 4060 number of effective passes number of effective passes 5 10 10 20 15 30 20 40 25 50 30 number of effective passes 60 35 40 number of effective passes covertype (f)(b)covertype-group OPT-ADMM 10 -3 0.19 (b) mnist (g) mnist-group SVRG-ADMM SVRG-ADMM-ERG SAG-ADMM 10 -8 2.8 5 10 5 15 20 25 30 35 40 10 15 20 25 30 number of effective passes number of effective passes (d) dna (h) dna-group SAG-ADMM-ERG ACC-SADMM Figure 3. Illustration of the proposed approach. The evolutionary process of our PDE (solid arrow) with respect to the time (t = 0, T /N, ? ? ? , T,) extracts the feature from the image and the gradient descent process (hollow arrow) learns a transform to represent the Figure 1: Experimental results of solving the original Lasso (Top) and Graph-Guided Fused Lasfeature. so (Bottom). The computation time includes the cost of calculating full gradients for SVRG based methods. SVRG-ADMM and SAG-ADMM are initialized by running STOC-ADMM for 3n b iterarepresents the Sham, ergodicand results for the corresponding Frostig,tions. Roy,?-ERG? Ge, Rong, Kakade, Sidford, Lin, Zhouchen,algorithms. Liu, Risheng, and Su, Zhixun. Linearized Aaron. Un-regularizing: approximate proximal point and faster stochastic algorithms for empirical risk minimization. In Proc. Int?l. Conf. on Machine Learning, 6 Experiments 2015. alternating direction method with adaptive penalty for low-rank representation. In Proc. Conf. Advances in Neural Information Processing Systems, 2011. Lin, Zhouchen, Liu, Risheng, and Li, Huan. Linearized We conduct to show the O(1/n) effectiveness method3 .direction We compare ourwith method withsplitting the He, Bingsheng and experiments Yuan, Xiaoming. On the con- of our alternating method parallel and following the-state-of-the-art SADMM algorithms: (1) STOC-ADMM [1],separable (2) SVRG-ADMM [4], in mavergence rate of the douglas?rachford alternating direcadaptive penalty for convex programs OPT-SADMM [2], (4) [3]. We SDCA-ADMM in our Learning, comparison99(2):287?325, since tion (3) method. SIAM Journal on SAG-ADMM Numerical Analysis, 50ignore chine learning. [17] Machine it gives no analysis on general convex problems and it is also not faster than SVRG-ADMM [4]. (2):700?709, 2012. 2015b. Experiments are performed on Intel(R) CPU i7-4770 @ 3.40GHz machine with 16 GB memory. Our Hien, Le Thi Khanh, Lu, Canyi, Huan, and Feng, Canyi,and Li, Huan, Lin,Learning. Zhouchen,Due and to Yan, Shuicheng. experiments focus on two Xu, typical problems [4]:Ji-LassoLu, Problem Multitask space ashi.limited, Accelerated stochastic mirror descentLearning algorithms proximal linearized alternating method of the experiment of Multitask is shownFast in Supplementary Materials. Fordirection the Lasso for composite optimization. arXiv problems,non-strongly we performconvex experiments under the following typical variations. first is P the original multiplier with parallelThe splitting. arXiv preprint arXn 1 preprint arXiv:1605.06892, Lasso problem; and the2016. second is Graph-Guided Fused Lasso model: min iv:1511.05133, 2015. x ?kAxk1 + n i=1 li (x), where li (x) is the logistic loss on sample i, and A = [G; I] is a matrix encoding the feature sparsity Johnson, Rie and Zhang, Tong. pattern Accelerating Nesterov, Yurii. A method for unconstrained pattern. G is the sparsity of the stochastic graph obtained by sparse inverse covariance estimation convex [30]. minigradient descent using predictive variance reduction. In 4 mization problem with the rate of convergence O(1/k 2 ). The experiments are performed on four benchmark data sets: a9a, covertype, mnist and dna . The Proc. Conf. Advances in Neural Information Processing In in Doklady an SSSR, pp. 543?547, details of the dataset and the mini-batch size that we use all SADMM arevolume shown269, in Table 3. And 1983. Systems, 2013. like [3] and [4], we fix ? = 10?5 and report the performance based on (xt , Axt ) to satisfy the Nesterov, Yurii. On an approach to the construction of opconstraints of Kyung-Ah, ADMM. Results areEric averaged over five repetitions. And we set m = 2n for all the Kim, Seyoung, Sohn, and Xing, P. A multimal methods of minimization of bsmooth convex funcalgorithms. For original Lasso problem, the step sizes are set through theoretical guidances for tivariate regression approach to association analysis of a tions. Ekonomika i Mateaticheskie Metody, 24(3):509? each algorithm. For the Graph-Guided Lasso, the best step sizes are obtained through searches on quantitative trait network. Bioinformatics, 25(12):i204? 517, 1988. which give best convergence progress. Except ACC-SADMM, we use the continuation i212,parameters 2009. technique [28] to accelerate algorithms. SAG-ADMM is performed the first three datasets due tooptimizaNesterov, Yurii. on Introductory lectures on convex its and largeLin, memory requirement. Li, Huan Zhouchen. Optimal nonergodic O(1/k) tion: A basic course, volume 87. 2013. convergence rate: When linearized ADM meets nesThe experimental results are shown in Fig. 1. We can find that our algorithm consistently outperforms Nitanda, Atsushi. Stochastic proximal gradient descent terov?s extrapolation. arXiv preprint arXiv:1608.06366, other compared methods in all these datasets for both the two problems, which verifies our theoretical with acceleration techniques. In Proc. Conf. Advances 2016. analysis. The details about parameter setting, experimental results where we set a larger fixed step in Neural Information Processing Systems, 2014. size for theMairal, group Julien, guidedand Lasso problem,Zaid. curvesAof the test error, the memory costs of all algorithms, Lin, Hongzhou, Harchaoui, and Multitask learning experiment are shown in Supplementary Materials. Ouyang, Hua, He, Niao, Tran, Long, and Gray, Alexanuniversal catalyst for first-order optimization. In Proc. der G. Stochastic alternating direction method of multiConf. Advances in Neural Information Processing Sys3 pliers. Proc. Int?l. Conf. on Machine Learning, 2013. tems, 2015a. The code will be available at http://www.cis.pku.edu.cn/faculty/vision/zlin/zlin.htm. 4 a9a, covertype and dna are from: http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/, and mnist is from: http://yann.lecun.com/exdb/mnist/. 8 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 7 Conclusion We propose ACC-SADMM for the general convex finite-sum problems. ACC-SADMM integrates Nesterov?s extrapolation and VR techniques and achieves a non-ergodic O(1/K) convergence rate, which shows theoretical and practical importance. We do experiments to demonstrate that our algorithm is faster than other SADMM methods. Acknowledgment Zhouchen Lin is supported by National Basic Research Program of China (973 Program) (grant no. 2015CB352502) and National Natural Science Foundation (NSF) of China (grant no.s 61625301, 61731018, and 61231002). References [1] Hua Ouyang, Niao He, Long Tran, and Alexander G Gray. Stochastic alternating direction method of multipliers. Proc. Int?l. Conf. on Machine Learning, 2013. [2] Samaneh AzadiSra and Suvrit Sra. Towards an optimal stochastic alternating direction method of multipliers. In Proc. Int?l. Conf. on Machine Learning, 2014. [3] Wenliang Zhong and James Tin-Yau Kwok. Fast stochastic alternating direction method of multipliers. In Proc. Int?l. Conf. on Machine Learning, 2014. [4] Shuai Zheng and James T Kwok. Fast-and-light stochastic admm. In Proc. Int?l. Joint Conf. on Artificial Intelligence, 2016. [5] Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1):183?202, 2009. [6] Andreas Argyriou, Theodoros Evgeniou, and Massimiliano Pontil. Multi-task feature learning. Proc. Conf. Advances in Neural Information Processing Systems, 2007. [7] Li Shen, Gang Sun, Zhouchen Lin, Qingming Huang, and Enhua Wu. Adaptive sharing for image classification. In Proc. Int?l. Joint Conf. on Artificial Intelligence, 2015. [8] Seyoung Kim, Kyung-Ah Sohn, and Eric P Xing. A multivariate regression approach to association analysis of a quantitative trait network. Bioinformatics, 25(12):i204?i212, 2009. [9] Kaiye Wang, Ran He, Liang Wang, Wei Wang, and Tieniu Tan. Joint feature selection and subspace learning for cross-modal retrieval. IEEE Trans. on Pattern Analysis and Machine Intelligence, 38(10):1?1, 2016. [10] Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations R in Machine Learning, 3(1):1?122, 2011. and Trends [11] Bingsheng He and Xiaoming Yuan. On the O(1/n) convergence rate of the Douglas?Rachford alternating direction method. SIAM Journal on Numerical Analysis, 50(2):700?709, 2012. [12] Zhouchen Lin, Risheng Liu, and Huan Li. Linearized alternating direction method with parallel splitting and adaptive penalty for separable convex programs in machine learning. Machine Learning, 99(2):287?325, 2015. [13] Damek Davis and Wotao Yin. Convergence rate analysis of several splitting schemes. In Splitting Methods in Communication, Imaging, Science, and Engineering, pages 115?163. 2016. [14] Caihua Chen, Raymond H Chan, Shiqian Ma, and Junfeng Yang. Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM Journal on Imaging Sciences, 8(4):2239?2267, 2015. [15] Huan Li and Zhouchen Lin. Optimal nonergodic O(1/k) convergence rate: When linearized ADM meets nesterov?s extrapolation. arXiv preprint arXiv:1608.06366, 2016. 9 [16] Taiji Suzuki. Dual averaging and proximal gradient descent for online alternating direction multiplier method. In Proc. Int?l. Conf. on Machine Learning, 2013. [17] Taiji Suzuki. Stochastic dual coordinate ascent with alternating direction method of multipliers. In Proc. Int?l. Conf. on Machine Learning, 2014. [18] L?on Bottou. Stochastic learning. In Advanced lectures on machine learning, pages 146?168. 2004. [19] Rie Johnson and Tong Zhang. Accelerating stochastic gradient descent using predictive variance reduction. In Proc. Conf. Advances in Neural Information Processing Systems, 2013. [20] Aaron Defazio, Francis Bach, and Simon Lacoste-Julien. SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives. In Proc. Conf. Advances in Neural Information Processing Systems, 2014. [21] Mark Schmidt, Nicolas Le Roux, and Francis Bach. Minimizing finite sums with the stochastic average gradient. Mathematical Programming, pages 1?30, 2013. [22] Yurii Nesterov. A method for unconstrained convex minimization problem with the rate of convergence O(1/k 2 ). In Doklady an SSSR, volume 269, pages 543?547, 1983. [23] Zeyuan Allen-Zhu. Katyusha: The first truly accelerated stochastic gradient method. In Annual Symposium on the Theory of Computing, 2017. [24] Zhouchen Lin, Risheng Liu, and Zhixun Su. Linearized alternating direction method with adaptive penalty for low-rank representation. In Proc. Conf. Advances in Neural Information Processing Systems, 2011. [25] Xiaoqun Zhang, Martin Burger, and Stanley Osher. A unified primal-dual algorithm framework based on bregman iteration. Journal of Scientific Computing, 46:20?46, 2011. [26] Paul Tseng. On accelerated proximal gradient methods for convex-concave optimization. In Technical report, 2008. [27] Yurii Nesterov. On an approach to the construction of optimal methods of minimization of smooth convex functions. Ekonomika i Mateaticheskie Metody, 24(3):509?517, 1988. [28] Wangmeng Zuo and Zhouchen Lin. A generalized accelerated proximal gradient approach for total variation-based image restoration. IEEE Trans. on Image Processing, 20(10):2748, 2011. [29] Zhouchen Lin, Minming Chen, and Yi Ma. The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv preprint arXiv:1009.5055, 2010. [30] Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3):432?441, 2008. 10
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A Probabilistic Framework for Nonlinearities in Stochastic Neural Networks Qinliang Su Xuejun Liao Lawrence Carin Department of Electrical and Computer Engineering Duke University, Durham, NC, USA {qs15, xjliao, lcarin}@duke.edu Abstract We present a probabilistic framework for nonlinearities, based on doubly truncated Gaussian distributions. By setting the truncation points appropriately, we are able to generate various types of nonlinearities within a unified framework, including sigmoid, tanh and ReLU, the most commonly used nonlinearities in neural networks. The framework readily integrates into existing stochastic neural networks (with hidden units characterized as random variables), allowing one for the first time to learn the nonlinearities alongside model weights in these networks. Extensive experiments demonstrate the performance improvements brought about by the proposed framework when integrated with the restricted Boltzmann machine (RBM), temporal RBM and the truncated Gaussian graphical model (TGGM). 1 Introduction A typical neural network is composed of nonlinear units connected by linear weights, and such a network is known to have universal approximation ability under mild conditions about the nonlinearity used at each unit [1, 2]. In previous work, the choice of nonlinearity has commonly been taken as a part of network design rather than network learning, and the training algorithms for neural networks have been mostly concerned with learning the linear weights. However, it is becoming increasingly understood that the choice of nonlinearity plays an important role in model performance. For example, [3] showed advantages of rectified linear units (ReLU) over sigmoidal units in using the restricted Boltzmann machine (RBM) [4] to pre-train feedforward ReLU networks. It was further shown in [5] that rectified linear units (ReLU) outperform sigmoidal units in a generative network under the same undirected and bipartite structure as the RBM. A number of recent works have reported benefits of learning nonlinear units along with the inter-unit weights. These methods are based on using parameterized nonlinear functions to activate each unit in a neural network, with the unit-dependent parameters incorporated into the data-driven training algorithms. In particular, [6] considered the adaptive piecewise linear (APL) unit defined by a mixture of hinge-shaped functions, and [7] used nonparametric Fourier basis expansion to construct the activation function of each unit. The maxout network [8] employs piecewise linear convex (PLC) units, where each PLC unit is obtained by max-pooling over multiple linear units. The PLC units were extended to Lp units in [9] where the normalized Lp norm of multiple linear units yields the output of an Lp unit. All these methods have been developed for learning the deterministic characteristics of a unit, lacking a stochastic unit characterization. The deterministic nature limits these methods from being easily applied to stochastic neural networks (for which the hidden units are random variables, rather than being characterized by a deterministic function), such as Boltzmann machines [10], restricted Boltzmann machines [11], and sigmoid belief networks (SBNs) [12]. We propose a probabilistic framework to unify the sigmoid, hyperbolic tangent (tanh) and ReLU nonlinearities, most commonly used in neural networks. The proposed framework represents a 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. unit h probabilistically as p(h|z, ?), where z is the total net contribution that h receives from other units, and ? represents the learnable parameters. By taking the expectation of h, a deterministic R characterization of the unit is obtained as E(h|z, ?) , h p(h|z, ?)dh. We show that the sigmoid, tanh and ReLU are well approximated by E(h|z, ?) under appropriate settings of ?. This is different from [13], in which nonlinearities were induced by the additive noises of different variances, making the model learning much more expensive and nonlinearity producing less flexible. Additionally, more-general nonlinearities may be constituted or learned, with these corresponding to distinct settings of ?. A neural unit represented by the proposed framework is named a truncated Gaussian (TruG) unit because the framework is built upon truncated Gaussian distributions. Because of the inherent stochasticity, TruG is particularly useful in constructing stochastic neural networks. The TruG generalizes the probabilistic ReLU in [14, 5] to a family of stochastic nonlinearities, with which one can perform two tasks that could not be done previously: (i) One can interchangeably use one nonlinearity in place of another under the same network structure, as long as they are both in the TruG family; for example, the ReLU-based stochastic networks in [14, 5] can be extended to new networks based on probabilistic tanh or sigmoid nonlinearities, and the respective algorithms in [14, 5] can be employed to train the associated new models with little modification; (ii) Any stochastic network constructed with the TruG can learn the nonlinearity alongside the network weights, by maximizing the likelihood function of ? given the training data. We can learn the nonlinearity at the unit level, with each TruG unit having its own parameters; or we can learn the nonlinearity at the model level, with the entire network sharing the same parameters for all its TruG units. The different choices entail only minor changes in the update equation of ?, as will be seen subsequently. We integrate the TruG framework into three existing stochastic networks: the RBM, temporal RBM [15] and feedforward TGGM [14], leading to three new models referred to as TruG-RBM, temporal TruG-RBM and TruG-TGGM, respectively. These new models are evaluated against the original models in extensive experiments to assess the performance gains brought about by the TruG. To conserve space, all propositions in this paper are proven in the Supplementary Material. 2 TruG: A Probabilistic Framework for Nonlinearities in Neural Networks For a unit h that receives net contribution z from other units, we propose to relate h to z through the following stochastic nonlinearity,   N h z, ? 2 I(?1 ? h ? ?2 ) (1) , N[?1 ,?2 ] h z, ? 2 , p(h|z, ?) = R ?2 0 2 0 N (h |z, ? ) dh ?1  where I(?) is an indicator function and N ? z, ? 2 is the probability density function (PDF) of a univariate Gaussian distribution with mean z and variance ? 2 ; the shorthand notation N[?1 ,?2 ] indicates the density N is truncated and renormalized such that it is nonzero only in the interval [?1 , ?2 ]; ? , {?1 , ?2 } contains the truncation points and ? 2 is fixed. The units of a stochastic neural network fall into two categories: visible units and hidden units [4]. The network represents a joint distribution over both hidden and visible units and the hidden units are integrated out to yield the marginal distribution of visible units. With a hidden unit expressed in (1), the expectation of h is given by E(h|z, ?) = z + ? ?( ?1??z ) ? ?( ?2??z ) ?( ?2??z ) ? ?( ?1??z ) , (2) where ?(?) and ?(?) are, respectively, the PDF and cumulative distribution function (CDF) of the standard normal distribution [16]. As will become clear below, a weighted sum of these expected hidden units constitutes the net contribution received by each visible unit when the hidden units are marginalized out. Therefore E(h|z, ?) acts as a nonlinear activation function to map the incoming contribution h receives to the outgoing contribution h sends out. The incoming contribution received by h may be a random variable or a function of data such as z = wT x + b; the former case is typically for unsupervised learning and the latter case for supervised learning with x being the predictors. By setting the truncation points to different values, we are able to realize many different kinds of nonlinearities. We plot in Figure 1 three realizations of E(h|z, ?) as a function of z, each with a particular setting of {?1 , ?2 } and ? 2 = 0.2 in all cases. The plots of ReLU, tanh and sigmoid are 2 1 10 5 0 -20 -10 0 10 20 0.5 0.8 0 -0.5 -1 -20 6 sigmoid TruG 0.6 0.4 0.2 -10 0 z 10 20 0 -20 4 3 2 1 -10 0 10 20 0 -20 -10 0 z z (a) sigmoid TruG ReLU 5 activation(z) 15 1 tanh TruG activation(z) ReLU TruG activation(z) activation(z) 20 (b) 10 20 z (c) (d) Figure 1: Illustration of different nonlinearities realized by the TruG with different truncation points. (a) ?1 = 0 and ?2 = +?; (b) ?1 = ?1 and ?2 = 1; (c) ?1 = 0 and ?2 = 1; (d) ?1 = 0 and ?2 = 4. also shown as a comparison. It is seen from Figure 1 that, by choosing appropriate truncation points, E(h|z, ?) is able to approximate ReLU, tanh and sigmoid, the three types of nonlinearities most widely used in neural networks. We can also realize other types of nonlinearities by setting the truncation points to other values, as exemplified in Figure 1(d). The truncation points can be set manually by hand, selected by cross-validation, or learned in the same way as the inter-unit weights. In this paper, we focus on learning them alongside the weights based on training data. The variance of h, given by [16],  Var(h|z, ?) = ? 2 +  ?1 ?z ?1 ?z ? ? ?   ?2 ?2 ?z ? ? ? ?2 ?z ? ? ??   ?1 ?z ? ?2 ?z ?   ? ? ?2 ? ?  ?1 ?z ?  ??  ?  ?2 ?z ?  ??  ?2 ?z ? ?1 ?z ?  ?2  ? , (3) is employed in learning the truncation points and network weights. Direct evaluation of (2) and (3) is prone to the numerical issue of 00 , because both ?(z) and ?(z) are so close to 0 when z < ?38 that they are beyond the maximal accuracy a double float number can represent. We solve this problem by ?(z) using the fact that (2) and (3) can be equivalently expressed in terms of ?(z) by dividing both the numerator and the denominator by ?(?). We make use of the following approximation for the ratio, ? ?(z) z2 + 4 ? z ? , ?(z), for z < ?38, (4) ?(z) 2 the accuracy of which is established in Proposition 1. ? 2 +4?z ?(z) ? 1 < 2 ?zz2 +8?3z ?1; moreover, for all Proposition 1. The relative error is bounded by ?(z)/ ?(z) ?(z) z < ?38, the relative error is guaranteed to be smaller than 4.8 ? 10?7 , that is, ?(z)/ ?(z) ? 1 < 4.8 ? 10?7 for all z < ?38. 3 RBM with TruG Nonlinearity We generalize the ReLU-based RBM in [5] by using the TruG nonlinearity. The resulting TruG-RBM is defined by the following joint distribution over visible units x and hidden units h, p(x, h) = 1 ?E(x,h) e I(x ? {0, 1}n , ?1 ? h ? ?2 ), Z (5) where E(x, h) , 12 hT diag(d)h ? xT Wh ? bT x ? cT h is an energy function and Z is the normalization constant. Proposition 2 shows (5) is a valid probability distribution. Proposition 2. The distribution p(x, h) defined in (5) is normalizable. Qn By (5), the conditional distribution of x given h is still Bernoulli, p(x|h) = i=1 ?([Wh + b]i ), while the conditional p(h|x) is a truncated normal distribution, i.e.,   m Y 1 1 T p(h|x) = N[?1 ,?2 ] hj [W x + c]j , . (6) dj dj j=1 3 By setting ?1 and ?2 to different values, we are able to produce different nonlinearities in (6). We train a TruG-RBM based on maximizing the log-likelihood function `(?, ?) , P ln p(x; ?, ?), where ? , {W, b, c} denotes the network weights, p(x; ?, ?) , R ?2x?X p(x, h)dh is contributed by a single data point x, and X is the training dataset. ?1 3.1 The Gradient w.r.t. Network Weights The gradient w.r.t. ? is known to be ?ln p(x) ?? =E h ?E(x,h) ?? i ?E h i , where E[?] and E[?|x] ?E(x,h) ?? x means the expectation w.r.t. p(x, h) and p(h|x), respectively. If we estimate the gradient using a standard sampling-based method, the variance associated with the estimate is usually very large. To reduce the variance, we follow the traditional RBM in applying the contrastive divergence (CD) to estimate the gradient [4]. Specifically, we approximate the gradient as     ? ln p(x) ?E(x, h) ?E(x, h) (k) x ?E x , (7) ?E ?? ?? ?? where x(k) is the k-th sample of the Gibbs sampler p(h(1) |x(0) ), p(x(1) |h(1) ) ? ? ? p(x(k) |h(k) ), with x(0) being the data x. As shown in (6), p(x|h) and p(h|x) are factorized Bernoulli and univariate truncated normal distributions, for which efficient sampling algorithms exist [17, 18]. = xi , ?E(x,h) = hj and ?E(x,h) = 12 h2j . ?cj    2 (s)  ?dj (s) Thus estimation of the gradient with CD only requires E hj |x and E hj |x , which can be calculated using (2) and (3). Using the estimated gradient, the weights can be updated using the stochastic gradient ascent algorithm or its variants. Furthermore, we can obtain that 3.2 ?E(x,h) ?wij = xi hj , ?E(x,h) ?bi The Gradient w.r.t. Truncation Points The gradient w.r.t. ?1 and ?2 are given by m ? ln p(x) X = (p(hj = ?1 ) ? p(hj = ?1 |x)) , ??1 j=1 (8) m ? ln p(x) X = (p(hj = ?2 |x) ? p(hj = ?2 )) , ??2 j=1 (9) for a single data point, with provided inthe Supplementary Material. It is known that  the derivation 1 T p(hj = ?|x) = N[?1 ,?2 ] hj = ? dj [W x + c]j , d1j , which can be easily calculated. However, if we calculate p(hj = ?) directly, P it would be computationally prohibitive. Fortunately, by noticing the identity p(hj = ?) = x p(hj = ?|x)p(x), are able to estimate it efficiently with CD as we  [WT x(k)+c]j 1 (k) p(hj = ?) ? p(hj = ?|x ) = N[?1 ,?2 ] hj = ? , dj , where x(k) is the k-th sample of dj the Gibbs sampler as described above. Therefore, the gradient w.r.t. the lower and upper truncation  Pm points can be estimated using the equations ? ln??p(x) ? j=1 p(hj = ?2 |x)?p(hj = ?2 |x(k) ) and 2  Pm ? ln p(x) ? ? j=1 p(hj = ?1 |x)?p(hj = ?1 |x(k) ) . After obtaining the gradients, we can update the ??1 truncation points with stochastic gradient ascent methods. It should be emphasized that in the derivation above, we assume a common truncation point pair {?1 , ?2 } shared among all units for the clarity of presentation. The extension to separate truncation points for different units is straightforward, by simply replacing (8) and (9) with ? ln p(x) p(x) = (p(hj = ?2j |x) ? p(hj = ?2j )) and ? ln = (p(hj = ?1j ) ? p(hj = ?1j |x)), where ??2j ??1j ?1j and ?2j are the lower and upper truncation point of j-th unit, respectively. For the models discussed subsequently, one can similarly get the gradient w.r.t. unit-dependent truncations points. After training, due to the conditional independence between x and h and the existence of efficient sampling algorithm for truncated normal, samples can be drawn efficiently from the TruG-RBM using the Gibbs sampler discussed below (7). 4 4 Temporal RBM with TruG Nonlinearity We integrate the TruG framework into the temporal RBM (TRBM) [19] to learn the probabilistic nonlinearity in sequential-data modeling. The resulting temporal TruG-RBM is defined by QT p(X, H) = p(x1 , h1 ) t=2 p(xt , ht |xt?1 , ht?1 ), (10) where p(x1 , h1 ) and p(xt , ht |xt?1 , ht?1 ) are both represented by TruG-RBMs; xt ? Rn and ht ? Rm are the visible and hidden variables at time step t, with X , [x1 , x2 , ? ? ? , xT ] and H , [h1 , h2 , ? ? ? , hT ]. To be specific, the distribution p(xt , ht |xt?1 , ht?1 ) is defined as p(xt , ht |xt?1 , ht?1 ) = Z1t exp?E(xt ,ht ) I(x ? {0, 1}n , ?1 ? ht ? ?2 ),  where the energy function takes the form E(xt , ht ) , 12 xTt diag(a) xt + hTt diag(d) ht ?  T T 2xTt W1 ht ? 2cT ht ? 2 (W2 xt?1 ) ht ? 2bT xt ?2 (W3 xt?1 ) xt ? 2(W4 ht?1 )T ht ; and R +? R +? Zt , ?? 0 e?E(xt ,ht ) dht dxt . Similar to the TRBM, directly optimizing the log-likelihood is difficult. We instead optimize the lower bound L , Eq(H|X) [ln p(X, H; ?, ?) ? ln q(H|X)] , (11) where q(H|X) is an approximating posterior distribution of H. The lower bound is equal to the log-likelihood when q(H|X) is exactly the true posterior p(H|X). We follow [19] to choose the following approximate posterior, q(H|X) = p(h1 |x1 ) ? ? ? p(hT |xT ?1 , hT ?1 , xT ), with which it can be shown that the gradient of the lower bound w.r.t. h the network i  PT ?E(xt ,ht ) ?L weights is given by ?? = ? E E p(h |x ,h ,x ) p(x ,h |x ,h ) t?1 t?2 t?2 t?1 t t t?1 t?1 ?? h i  t=1 ?E(xt ,ht ) Ep(ht |xt?1 ,ht?1 ,xt ) . At any time step t, the outside expectation (which is over ht?1 ) is ?? approximated by sampling from p(ht?1 |xt?2 , ht?2 , xt?1 ); given ht?1 and xt?1 , one can represent p(xt , ht |xt?1 , ht?1 ) as a TruG-RBM and therefore the two inside expectations can be computed in the same way as in Section 3. In particular, the variables in ht are conditionally independent given Qm (xt?1 , ht?1 , xt ), i.e., p(ht |xt?1 , ht?1 , xt ) = j=1 p(hjt |xt?1 , ht?1 , xt ) with each component equal to   [W1T xt+W2 xt?1+W4 ht?1+c]j 1 p(hjt |xt?1 , ht?1 , xt ) =N[?1 ,?2 ] hjt , . (12) dj dj Similarly, the variables in xt are conditionally independent given (xt?1 , ht?1 , ht ). As a result, Ep(ht |xt?1 ,ht?1 ,xt ) [?] can be calculated in closed-form using (2) and (3), and Ep(xt ,ht |xt?1 ,ht?1 ,xt ) [?] can be estimated using the CD algorithm, as in Section Section 3. The gradient of L w.r.t. the upper truncation point is X  T X m T X m X ?L = Eq(H|X) p(hjt = ?2 |xt?1 , ht?1 , xt ) ? p(hjt = ?2 |xt?1 , ht?1 ) , ??2 t=1 j=1 t=1 j=1 ?L with ?? taking a similar form, where the expectations are similarly calculated using the same 1 ?L approach as described above for ?? . 5 TGGM with TruG Nonlinearity We generalize the feedforward TGGM model in [14] by replacing the probabilistic ReLU with the TruG. The resulting TruG-TGGM model is defined by the joint PDF over visible variables y and hidden variables h, p(y, h|x) = N (y|W1 h + b1 , ? 2 I)N[?1 ,?2 ] (h|W0 x + b0 , ? 2 I), 5 (13) given the predictor variables x. After marginalizing out h, we get the expectation of y as E[y|x] = W1 E(h|W0 x + b0 , ?) + b1 , (14) where E(h|W0 x + b0 , ?) is given element-wisely in (2). It is then clear that the expectation of y is related to x through the TruG nonlinearity. Thus E[y|x] yields the same output as a three-layer perceptron that uses (2) to activate its hidden units. Hence, the TruG-TGGM model defined in (13) can be understood as a stochastic perceptron with the TruG nonlinearity. By choosing different values for the truncation points, we are able to realize different kinds of nonlinearities, including ReLU, sigmoid and tanh. ToRtrain the model by maximum likelihood estimation, we need to know the gradient of ln p(y|x) , ln p(y, h|x; ?)dh, where ? , {W1 , W0 , b1 , b0 } represents the model parameters. By rewriting ?E(y,h,x) the joint PDFh as p(y, h|x) to be given by I(? i ? h e i 1 ? h ? ?2 ), the gradient is found ?E(y,h,x) ?E(y,h,x) ||y?W1 h?b1 ||2 +||h?W0 x?b0 ||2 ? ln p(y|x) =E ?? x ?E ?? x, y , where E(y, h, x) , ; ?? 2? 2 E[?|x] is the expectation w.r.t. p(y, h|x); and E[?|x, y] is the expectation w.r.t. p(h|x, y). From (13), we know p(h|x) = N[?1 ,?2 ] (h|W0 x + b0 , ? 2 I) can be factorized into a product of univariate truncated Gaussian PDFs. Thus the expectation E[h|x] can be computed using (2). However, the expectations E[h|x, y] and E[hhT |x, y] involve a multivariate truncated Gaussian PDF and are expensive to calculate directly. Hence mean-field variational Bayesian analysis is used to compute the approximate expectations. The details are similar to those in [14] except that (2) and (3) are used to calculate the expectation and variance of h. p(y|x) = The gradients of the log-likelihood w.r.t. the truncation points ?1 and ?2 are given by ? ln?? 2 PK PK ? ln p(y|x) = ? j=1 (p(hj = ?1 |y, x) ? p(hj = ?1 |x)) j=1 (p(hj = ?2 |y, x) ? p(hj = ?2 |x)) and ??1 for a single data point, with the derivation provided in the Supplementary Material. The probability p(hj = ?1 |x) can be computed directly since it is a univariate truncated Gaussian distribution. For p(hj = ?2 |y, x), we approximate it with the mean-field marginal distributions obtained above. Although TruG-TGGM involves random variables, thanks to the existence of close-form expression ?, for the expectation of univariate truncated normal, the testing is still very easy. Given a predictor x the output can be simply predicted with the conditional expectation E[y|x] in (14). 6 Experimental Results We evaluate the performance benefit brought about by the TruG framework when integrated into the RBM, temporal RBM and TGGM. In each of the three cases, the evaluation is based on comparing the original network to the associated new network with the TruG nonlinearity. For the TruG, we either manually set {?1 , ?2 } to particular values, or learn them automatically from data. We consider both the case of learning a common {?1 , ?2 } shared for all hidden units and the case of learning a separate {?1 , ?2 } for each hidden unit. Results of TruG-RBM The binarized Table 1: Averaged test log-probability on MNIST. (?) MNIST and Caltech101 Silhouettes are con- Results reported in [20]; () Results reported in [21] sidered in this experiment. The MNIST using RMSprop as the optimizer. contains 60,000 training and 10,000 testing Ave. Log-prob Model Trun. Points images of hand-written digits, while CalMNIST Caltech101 tech101 Silhouettes includes 6364 training [0, 1] -97.3 -127.9 and 2307 testing images of objects? silhou[0, +?) -83.2 -105.2 ettes. For both datasets, each image has TruG-RBM [-1, 1] -124.5 -141.5 28 ? 28 pixels [22]. Throughout this experc-Learn -82.9 -104.6 iment, 500 hidden units are used. RMSprop s-Learn -82.5 -104.3 is used to update the parameters, with the RBM ? -86.3? -109.0 delay and mini-batch size set to 0.95 and 100, respectively. The weight parameters are initialized with the Gaussian noise of zero mean and 0.01 variance, while the lower and upper truncation points at all units are initialized to 0 and 1, respectively. The learning rates for weight parameters are fixed to 10?4 . Since truncations points influence the whole networks in a more fundamental way than weight parameters, it is observed that smaller learning rates are often preferred for them. To balance the convergence speed and 6 Sigmoid function <(7) Nonlinearity in Ball Nonlinearity in MNIST Nonlinearity in Motion Nonlinearity in Caltech 0.12 0.1 2.5 2 1.5 Probability 0.1 Probability Output after transform 3 0.12 0.14 4 3.5 0.08 0.06 0.08 0.06 0.04 0.04 1 0.02 0.02 0.5 0 0 -15 -10 -5 0 5 Input before transform: 7 (a) 10 15 0.5 1 1.5 2 2.5 Upper truncation point: (b) 3 2 3.5 2 2.5 3 3.5 Upper truncation point: 4 4.5 2 (c) Figure 2: (a) The learned nonlinearities in TruG models with shared upper truncation point ?; The distribution of unit-level upper truncation points of TruG-RBM for (b) MNIST; (c) Caltech101 Silhouettes. performance, we anneal their learning rates from 10?4 to 10?6 gradually. The evaluation is based on the log-probability averaged over test data points, which are estimated using annealed importance sampling (AIS) [23] with 5 ? 105 inverse temperatures equally spaced in [0, 1]; the reported test log-probability is averaged over 100 independent AIS runs. To investigate the impact of truncation points, we first set the lower and upper truncation points to three fixed pairs: [0, 1], [0, +?) and [?1, 1], which correspond to probabilistic approximations of sigmoid, ReLU and tanh nonlinearities, respectively. From Table 1, we see that the ReLU-type TruG-RBM performs much better than the other two types of TruG-RBM. We also learn the truncation points from data automatically. We can see that the model benefits significantly from nonlinearity learning, and the best performance is achieved when the units learn their own nonlinearities. The learned common nonlinearities (c-Learn) for different datasets are plotted in Figure 2(a), which shows that the model always tends to choose a nonlinearity in between sigmoid and ReLU functions. For the case with separate nonlinearities (s-Learn), the distributions of the upper truncation points in the TruGRBM?s for MNIST and Caltech101 Silhouettes are plotted in Figure 2(b) and (c), respectively. Note that due to the detrimental effect observed for negative truncation points, here the lower truncation points are fixed to zero and only the upper points are learned. To demonstrate the reliability of AIS estimate, the convergence plots of estimated log-probabilities are provided in Supplementary Material. Results of Temporal TruG-RBM The Bouncing Ball and CMU Motion Capture datasets are considered in the experiment with temporal models. Bouncing Ball consists of synthetic binary videos of 3 bouncing balls in a box, with 4000 videos for training and 200 for testing, and each video has 100 frames of size 30 ? 30. CMU Motion Capture is composed of data samples describing the joint angles associated with different motion types. We follow [24] to train a model on 31 sequences and test the model on two testing sequences (one is running and the other is walking). Both the original TRBM and the TruG-TRBM use 400 hidden units for Bouncing Ball and 300 hidden units for CMU Motion Capture. Stochastic gradient descent (SGD) is used to update the parameters, with the momentum set to 0.9. The learning rates are set to be 10?2 and 10?4 for the two datasets, respectively. The learning rate for truncation points is annealed gradually, as done in Section 6. Since calculating the log-probabilities for these temporal models is computationally prohibitive, prediction error is employed here as the performance evaluation criteria, which is widely used [24, 25] in temporal generative models. The performances averaged over 20 independent runs are reported here. Tables 2 and 3 confirm again that models benefit remarkably from nonlinearity learning, especially in the case of learning a separate nonlinearity for each hidden unit. It is noticed that, although the ReLU-type TruG-TRBM performs better the tanh-type TruG-TRBM on Bouncing Ball, the former performs much worse than the latter on CMU Motion Capture. This demonstrates that a fixed nonlinearity cannot perform well on every dataset. However, by learning truncation points automatically, the TruG can adapt the nonlinearity to the data and thus performs the best on every dataset (up to the representational limit of the TruG framework). Video samples drawn from the trained models are provided in the Supplementary Material. Results of TruG-TGGM Ten datasets from the UCI repository are used in this experiment. Following the procedures in [26], datasets are randomly partitioned into training and testing subsets for 7 Table 2: Test prediction error on Bouncing Ball. (?) Taken from [24], in which 2500 hidden units are used. Model TruG-TRBM TRBM RTRBM? Trun. Points [0, 1] [0, +?) [-1, 1] c-Learn s-Learn ? ? Pred. Err. 6.38?0.51 4.16?0.42 6.01?0.52 3.82?0.41 3.66?0.46 4.90?0.47 4.00?0.35 Table 3: Test prediction error on CMU Motion Capture, in which ?w? and ?r? mean walking and running, respectively. (?) Taken from [24]. Model TruG-TRBM TRBM ss-SRTRBM? Trun. Points [0, 1] [0, +?) [-1, 1] c-Learn s-Learn ? ? Err. (w) 8.2?0.18 21.8?0.31 7.3?0.21 6.7?0.29 6.8?0.24 9.6?0.15 8.1?0.06 Err. (r) 6.1?0.22 14.9?0.29 5.9?0.22 5.5?0.22 5.4?0.14 6.8?0.12 5.9?0.05 Table 4: Averaged test RMSEs for multilayer perception (MLP) and TruG-TGGMs under different truncation points. (?) Results reported in [26], where BH, CS, EE, K8 NP, CPP, PS, WQR, YH, YPM are the abbreviations of Boston Housing, Concrete Strength, Kin8nm, Naval Propulsion, Cycle Power Plant, Protein Structure, Wine Quality Red, Yacht Hydrodynamic, Year Prediction MSD, respectively. Dataset MLP (ReLU)? BH CS EE K8 NP CPP PS WQR YH YPM 3.228 ?0.195 5.977?0.093 1.098?0.074 0.091?0.002 0.001?0.000 4.182?0.040 4.539?0.029 0.645?0.010 1.182?0.165 8.932?N/A TruG-TGGM with Different Trun. Points [0, 1] 3.564?0.655 5.210?0.514 1.168?0.130 0.094?0.003 0.002?0.000 4.023?0.128 4.231?0.083 0.662?0.052 0.871?0.367 8.961?N/A [0, +?) 3.214?0.555 5.106?0.573 1.252?0.123 0.086?0.003 0.002?0.000 4.067?0.129 4.387?0.072 0.644?0.048 0.821?0.276 8.985?N/A [-1, 1] 4.003?0.520 4.977?0.482 1.069?0.166 0.091?0.003 0.002? 0.000 3.978?0.132 4.262?0.079 0.659?0.052 0.846?0.310 8.859?N/A c-Learn 3.401?0.375 4.910?0.467 0.881?0.079 0.073?0.002 0.001?0.000 3.952?0.134 4.209?0.073 0.645?0.050 0.803?0.292 8.893?N/A s-Learn 3.622? 0.538 4.743? 0.571 0.913? 0.120 0.075? 0.002 0.001? 0.000 3.951? 0.130 4.206? 0.071 0.643? 0.048 0.793? 0.289 8.965? N/A 10 trials except the largest one (Year Prediction MSD), for which only one partition is conducted due to computational complexity. Table 4 summarizes the root mean square error (RMSE) averaged over the different trials. Throughout the experiment, 100 hidden units are used for the two datasets (Protein Structure and Year Prediction MSD), while 50 units are used for the remaining. RMSprop is used to optimize the parameters, with RMSprop delay set to 0.9. The learning rate is chosen from the set {10?3 , 2 ? 104 , 10?4 }, while the mini-batch size is set to 100 for the two largest datasets and 50 for the others. The number of VB cycles used in the inference is set to 10 for all datasets. The RMSE?s of TGGMs with fixed and learned truncation points are reported in Table 4, along with the RMSE?s of the (deterministic) multilayer perceptron (MLP) using ReLU nonlinearity for comparison. Similar to what we have observed in generative models, the supervised models also benefit significantly from nonlinearity learning. The TruG-TGGM with learned truncation points perform the best for most datasets, with the separate learning performing slightly better than the common learning overall. Due to the limited space, the learned nonlinearities and their corresponding truncation points are provided in Supplementary Material. 7 Conclusions We have presented a probabilistic framework, termed TruG, to unify ReLU, sigmoid and tanh, the most commonly used nonlinearities in neural networks. The TruG is a family of nonlinearities constructed with doubly truncated Gaussian distributions. The ReLU, sigmoid and tanh are three important members of the TruG family, and other members can be obtained easily by adjusting the lower and upper truncation points. A big advantage offered by the TruG is that the nonlinearity is learnable from data, alongside the model weights. Due to its stochastic nature, the TruG can be readily integrated into many stochastic neural networks for which hidden units are random variables. Extensive experiments have demonstrated significant performance gains that the TruG framework can bring about when it is integrated with the RBM, temporal RBM, or TGGM. Acknowledgements The research reported here was supported by the DOE, NGA, NSF, ONR and by Accenture. 8 References [1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097?1105, 2012. [2] Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural networks, 4(2):251? 257, 1991. [3] Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pages 807?814, 2010. [4] Geoffrey E Hinton. Training products of experts by minimizing contrastive divergence. Neural computation, 14(8):1771?1800, 2002. [5] Qinliang Su, Xuejun Liao, Chunyuan Li, Zhe Gan, and Lawrence Carin. Unsupervised learning with truncated gaussian graphical models. In The Thirty-First National Conference on Artificial Intelligence (AAAI), 2016. 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Distral: Robust Multitask Reinforcement Learning Yee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, Razvan Pascanu DeepMind London, UK Abstract Most deep reinforcement learning algorithms are data inefficient in complex and rich environments, limiting their applicability to many scenarios. One direction for improving data efficiency is multitask learning with shared neural network parameters, where efficiency may be improved through transfer across related tasks. In practice, however, this is not usually observed, because gradients from different tasks can interfere negatively, making learning unstable and sometimes even less data efficient. Another issue is the different reward schemes between tasks, which can easily lead to one task dominating the learning of a shared model. We propose a new approach for joint training of multiple tasks, which we refer to as Distral (distill & transfer learning). Instead of sharing parameters between the different workers, we propose to share a ?distilled? policy that captures common behaviour across tasks. Each worker is trained to solve its own task while constrained to stay close to the shared policy, while the shared policy is trained by distillation to be the centroid of all task policies. Both aspects of the learning process are derived by optimizing a joint objective function. We show that our approach supports efficient transfer on complex 3D environments, outperforming several related methods. Moreover, the proposed learning process is more robust to hyperparameter settings and more stable?attributes that are critical in deep reinforcement learning. 1 Introduction Deep Reinforcement Learning is an emerging subfield of Reinforcement Learning (RL) that relies on deep neural networks as function approximators that can scale RL algorithms to complex and rich environments. One key work in this direction was the introduction of DQN [21] which is able to play many games in the ATARI suite of games [1] at above human performance. However the agent requires a fairly large amount of time and data to learn effective policies and the learning process itself can be quite unstable, even with innovations introduced to improve wall clock time, data efficiency, and robustness by changing the learning algorithm [27, 33] or by improving the optimizer [20, 29]. A different approach was introduced by [12, 19, 14], whereby data efficiency is improved by training additional auxiliary tasks jointly with the RL task. With the success of deep RL has come interest in increasingly complex tasks and a shift in focus towards scenarios in which a single agent must solve multiple related problems, either simultaneously or sequentially. Due to the large computational cost, making progress in this direction requires robust algorithms which do not rely on task-specific algorithmic design or extensive hyperparameter tuning. Intuitively, solutions to related tasks should facilitate learning since the tasks share common structure, and thus one would expect that individual tasks should require less data or achieve a higher asymptotic performance. Indeed this intuition has long been pursued in the multitask and transfer-learning literature [2, 31, 34, 5]. Somewhat counter-intuitively, however, the above is often not the result encountered in practice, particularly in the RL domain [26, 23]. Instead, the multitask and transfer learning scenarios are 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. frequently found to pose additional challenges to existing methods. Instead of making learning easier it is often observed that training on multiple tasks can negatively affect performances on the individual tasks, and additional techniques have to be developed to counteract this [26, 23]. It is likely that gradients from other tasks behave as noise, interfering with learning, or, in another extreme, one of the tasks might dominate the others. In this paper we develop an approach for multitask and transfer RL that allows effective sharing of behavioral structure across tasks, giving rise to several algorithmic instantiations. In addition to some instructive illustrations on a grid world domain, we provide a detailed analysis of the resulting algorithms via comparisons to A3C [20] baselines on a variety of tasks in a first-person, visually-rich, 3D environment. We find that the Distral algorithms learn faster and achieve better asymptotic performance, are significantly more robust to hyperparameter settings, and learn more stably than multitask A3C baselines. 2 Distral: Distill and Transfer Learning We propose a framework for simultaneous reindistill regularise ?3 forcement learning of multiple tasks which we ?1 call Distral. Figure 1 provides a high level illustration involving four tasks. The method is distill regularise founded on the notion of a shared policy (shown ? 0 regularise distill in the centre) which distills (in the sense of Bucila and Hinton et al. [4, 11]) common be- ?2 ?4 haviours or representations from task-specific regularise distill policies [26, 23]. Crucially, the distilled policy is then used to guide task-specific policies via regularization using a Kullback-Leibler (KL) di- Figure 1: Illustration of the Distral framework. vergence. The effect is akin to a shaping reward which can, for instance, overcome random walk exploration bottlenecks. In this way, knowledge gained in one task is distilled into the shared policy, then transferred to other tasks. 2.1 Mathematical framework In this section we describe the mathematical framework underlying Distral. A multitask RL setting is considered where there are n tasks, where for simplicity we assume an infinite horizon with discount factor .1 We will assume that the action A and state S spaces are the same across tasks; we use a 2 A to denote actions, s 2 S to denote states. The transition dynamics pi (s0 |s, a) and reward functions Ri (a, s) are different for each task i. Let ?i be task-specific stochastic policies. The dynamics and policies give rise to joint distributions over state and action trajectories starting from some initial state, which we will also denote by ?i by an abuse of notation. Our mechanism for linking the policy learning across tasks is via optimising an objective which consists of expected returns and policy regularizations. We designate ?0 to be the distilled policy which we believe will capture agent behaviour that is common across the tasks. We regularize each task P t |st ) policy ?i towards the distilled policy using -discounted KL divergences E?i [ t 0 t log ??0i (a (at |st ) ]. In addition, we also use a -discounted entropy regularization to further encourage exploration. The resulting objective to be maximized is: 2 3 X X ? (a |s ) i t t t J(?0 , {?i }ni=1 ) = E?i 4 Ri (at , st ) cKL t log cEnt t log ?i (at |st )5 ? (a |s ) 0 t t i t 0 2 3 t t X X ? t = E ?i 4 Ri (at , st ) + log ?0 (at |st ) log ?i (at |st )5 (1) i t 0 where cKL , cEnt 0 are scalar factors which determine the strengths of the KL and entropy regularizations, and ? = cKL /(cKL + cEnt ) and = 1/(cKL + cEnt ). The log ?0 (at |st ) term can be thought 1 The method can be easily generalized to other scenarios like undiscounted finite horizon. 2 of as a reward shaping term which encourages actions which have high probability under the distilled policy, while the entropy term log ?i (at |st ) encourages exploration. In the above we used the same regularization costs cKL , cEnt for all tasks. It is easy to generalize to using task-specific costs; this can be important if tasks differ substantially in their reward scales and amounts of exploration needed, although it does introduce additional hyperparameters that are expensive to optimize. 2.2 Soft Q Learning and Distillation A range of optimization techniques in the literature can be applied to maximize the above objective, which we will expand on below. To build up intuition for how the method operates, we will start in the simple case of a tabular representation and an alternating maximization procedure which optimizes over ?i given ?0 and over ?0 given ?i . With ?0 fixed, (1) decomposes into separate maximization problems for each task, and is an entropy regularized expected return with redefined (regularized) reward Ri0 (a, s) := Ri (a, s) + ? log ?0 (a|s). It can be optimized using soft Q learning [10] aka G learning [7], which are based on deriving the following ?softened? Bellman updates for the state and action values (see also [25, 28, 22]): X 1 Vi (st ) = log ?0? (at |st ) exp [ Qi (at , st )] (2) at Qi (at , st ) = Ri (at , st ) + X st pi (st+1 |st , at )Vi (st+1 ) (3) The Bellman updates are softened in the sense that the usual max operator over actions for the state values Vi is replaced by a soft-max at inverse temperature , which hardens into a max operator as ! 1. The optimal policy ?i is then a Boltzmann policy at inverse temperature : ?i (at |st ) = ?0? (at |st )e Qi (at |st ) Vi (st ) = ?0? (at |st )e Ai (at |st ) (4) where Ai (a, s) = Qi (a, s) Vi (s) is a softened advantage function. Note that the softened state values Vi (s) act as the log normalizers in the above. The distilled policy ?0 can be interpreted as a policy prior, a perspective well-known in the literature on RL as probabilistic inference [32, 13, 25, 7]. However, unlike in past works, it is raised to a power of ? ? 1. This softens the effect of the prior ?0 on ?i , and is the result of the additional entropy regularization beyond the KL divergence. Also unlike past works, we will learn ?0 instead of hand-picking it (typically as a uniform distribution over actions). In particular, notice that the only terms in (1) depending on ?0 are: 2 3 X ?X t E ?i 4 log ?0 (at |st )5 (5) i t 0 which is simply a log likelihood for fitting a model ?0 to a mixture of -discounted state-action distributions, one for each task i under policy ?i . A maximum likelihood (ML) estimator can be derived from state-action visitation frequencies under roll-outs in each task, with the optimal ML solution given by the mixture of state-conditional action distributions. Alternatively, in the non-tabular case, stochastic gradient ascent can be employed, which leads precisely to an update which distills the task policies ?i into ?0 [4, 11, 26, 23]. Note however that in our case the distillation step is derived naturally from a KL regularized objective on the policies. Another difference from [26, 23] and from prior works on the use of distillation in deep learning [4, 11] is that the distilled policy is ?fed back in? to improve the task policies when they are next optimized, and serves as a conduit in which common and transferable knowledge is shared across the task policies. It is worthwhile here to take pause and ponder the effect of the extra entropy regularization. First suppose that there is no extra entropy regularisation, ? = 1, and consider the simple scenario of only n = 1 task.Then (5) is maximized when the distilled policy ?0 and the task policy ?1 are equal, and the KL regularization term is 0. Thus the objective reduces to an unregularized expected return, and so the task policy ?1 converges to a greedy policy which locally maximizes expected returns. Another way to view this line of reasoning is that the alternating maximization scheme is equivalent to trust-region methods like natural gradient or TRPO [24, 29] which use a KL ball centred at the previous policy, and which are understood to converge to greedy policies. If ? < 1, there is an additional entropy term in (1). So even with ?0 = ?1 and KL(?1 k?0 ) = 0, the objective (1) will no longer be maximized by greedy policies. Instead (1) reduces to an entropy 3 regularized expected returns with entropy regularization factor 0 = /(1 ?) = 1/cEnt , so that the optimal policy is of the Boltzmann form with inverse temperature 0 [25, 7, 28, 22]. In conclusion, by including the extra entropy term, we can guarantee that the task policy will not turn greedy, and we can control the amount of exploration by adjusting cEnt appropriately. This additional control over the amount of exploration is essential when there are more than one task. To see this, imagine a scenario where one of the tasks is easier and is solved first, while other tasks are harder with much sparser rewards. Without the entropy term, and before rewards in other tasks are encountered, both the distilled policy and all the task policies can converge to the one that solves the easy task. Further, because this policy is greedy, it can insufficiently explore the other tasks to even encounter rewards, leading to sub-optimal behaviour. For single-task RL, the use of entropy regularization was recently popularized by Mnih et al. [20] to counter premature convergence to greedy policies, which can be particularly severe when doing policy gradient learning. This carries over to our multitask scenario as well, and is the reason for the additional entropy regularization. 2.3 Policy Gradient and a Better Parameterization The above method alternates between maximization of the distilled policy ?0 and the task policies ?i , and is reminiscent of the EM algorithm [6] for learning latent variable models, with ?0 playing the role of parameters, while ?i plays the role of the posterior distributions for the latent variables. Going beyond the tabular case, when both ?0 and ?i are parameterized by, say, deep networks, such an alternating maximization procedure can be slower than simply optimizing (1) with respect to task and distilled policies jointly by stochastic gradient ascent. In this case the gradient update for ?i is simply given by policy gradient with an entropic regularization [20, 28], and can be carried out within a framework like advantage actor-critic [20]. A simple parameterization of policies would be to use a separate network for each task policy ?i , and another one for the distilled policy ?0 . An alternative parameterization, which we argue can result in faster transfer, can be obtained by considering the form of the optimal Boltzmann policy (4). Specifically, consider parameterising the distilled policy using a network with parameters ?0 , ? ?0 (at |st ) = P exp(h?0 (at |st ) 0 a0 exp(h?0 (a |st )) (6) and estimating the soft advantages2 using another network with parameters ?i : A?i (at |st ) = f?i (at |st ) 1 log X ? ?0? (a|st ) exp( f?i (a|st )) (7) a We used hat notation to denote parameterized approximators of the corresponding quantities. The policy for task i then becomes parameterized as, exp(?h?0 (at |st ) + f?i (at |st )) ? ?i (at |st ) = ? ?0? (at |st ) exp( A?i (at |st )) = P 0 0 a0 exp((?h?0 (a |st ) + f?i (a |st )) (8) This can be seen as a two-column architecture for the policy, with one column being the distilled policy, and the other being the adjustment required to specialize to task i. Given the parameterization above, we can now derive the policy gradients. The gradient wrt to the task specific parameters ?i is given by the standard policy gradient theorem [30], h?P ? ?P ?i reg u r?i J =E??i r log ? ? (a |s ) (R (a , s )) ?i i t t u u i t 1 u 1 hP ?P ?i reg u =E??i ?i (at |st ) (Ri (au , su )) (9) t 1 r?i log ? u t where Rireg (a, s) = Ri (a, s) + ? log ? ?0 (a|s) 1 log ? ?i (a|s) is the regularized reward. Note that the partial derivative of the entropy in the integrand has expectation E??i [r?i log ? ?i (at |st )] = 0 because of the log-derivative trick. If a value baseline is estimated, it can be subtracted from the regularized 2 In practice, we do not actually use these as advantage estimates. Instead we use (8) to parameterize a policy which is optimized by policy gradients. 4 DisTra Learning Returns Returns KL ?0 h Baselines Returns KL ?i? ? ?i ? ?0 entropy f h ? ?i i = 1, 2, .. f h i = 1, 2, .. i = 1, 2, .. KL 1col KL+ent 1col KL 2col KL+ent 2col Returns entropy entropy f entropy Returns A3C 2col i = 1, 2, .. ?i entropy f i = 1, 2, .. A3C h ?0 A3C multitask Figure 2: Depiction of the different algorithms and baselines. On the left are two of the Distral algorithms and on the right are the three A3C baselines. Entropy is drawn in brackets as it is optional and only used for KL+ent 2col and KL+ent 1col. returns as a control variate. The gradient wrt ?0 is more interesting: hP ?P ?i X u r ?0 J = E??i ?i (at |st ) (Rireg (au , su ) t 1 r?0 log ? u t i + ?X i E??i hP t 1 t P ?i (a0t |st ) a0t (? ? ?0 (a0t |st ))r?0 h?0 (a0t |st ) (10) i Note that the first term is the same as for the policy gradient of ?i . The second term tries to match the probabilities under the task policy ? ?i and under the distilled policy ? ?0 . The second term would not be present if we simply parameterized ?i using the same architecture ? ?i , but do not use a KL regularization for the policy. The presence of the KL regularization gets the distilled policy to learn to be theP centroid of all task policies, in the sense that the second term would be zero if ? ?0 (a0t |st ) = n1 i ? ?i (a0t |st ), and helps to transfer information quickly across tasks and to new tasks. 2.4 Other Related Works The centroid and star-shaped structure of Distral is reminiscent of ADMM [3], elastic-averaging SGD [35] and hierarchical Bayes [9]. Though a crucial difference is that while ADMM, EASGD and hierarchical Bayes operate in the space of parameters, in Distral the distilled policy learns to be the centroid in the space of policies. We argue that this is semantically more meaningful, and may contribute to the observed robustness of Distral by stabilizing learning. In our experiments we find indeed that absence of the KL regularization significantly affects the stability of the algorithm. Another related line of work is guided policy search [17, 18, 15, 16]. These focus on single tasks, and uses trajectory optimization (corresponding to task policies here) to guide the learning of a policy (corresponding to the distilled policy ?0 here). This contrasts with Distral, which is a multitask setting, where a learnt ?0 is used to facilitate transfer by sharing common task-agnostic behaviours, and the main outcome of the approach are instead the task policies. Our approach is also reminiscent of recent work on option learning [8], but with a few important differences. We focus on using deep neural networks as flexible function approximators, and applied our method to rich 3D visual environments, while Fox et al. [8] considered only the tabular case. We argue for the importance of an additional entropy regularization besides the KL regularization. This lead to an interesting twist in the mathematical framework allowing us to separately control the amounts of transfer and of exploration. On the other hand Fox et al. [8] focused on the interesting problem of learning multiple options (distilled policies here). Their approach treats the assignment of tasks to options as a clustering problem, which is not easily extended beyond the tabular case. 3 Algorithms The framework we just described allows for a number of possible algorithmic instantiations, arising as combinations of objectives, algorithms and architectures, which we describe below and summarize in Table 1 and Figure 2. KL divergence vs entropy regularization: With ? = 0, we get a purely 5 ?=0 ?=1 0<?<1 h?0 (a|s) f?i (a|s) ?h?0 (a|s) + f?i (a|s) A3C multitask A3C KL 1col KL+ent 1col A3C 2col KL 2col KL+ent 2col Table 1: The seven different algorithms evaluated in our experiments. Each column describes a different architecture, with the column headings indicating the logits for the task policies. The rows define the relative amount of KL vs entropy regularization loss, with the first row comprising the A3C baselines (no KL loss). entropy-regularized objective which does not couple and transfer across tasks [20, 28]. With ? = 1, we get a purely KL regularized objective, which does couple and transfer across tasks, but might prematurely stop exploration if the distilled and task policies become similar and greedy. With 0 < ? < 1 we get both terms. Alternating vs joint optimization: We have the option of jointly optimizing both the distilled policy and the task policies, or optimizing one while keeping the other fixed. Alternating optimization leads to algorithms that resemble policy distillation/actor-mimic [23, 26], but are iterative in nature with the distilled policy feeding back into task policy optimization. Also, soft Q learning can be applied to each task, instead of policy gradients. While alternating optimization can be slower, evidence from policy distillation/actor-mimic indicate it might learn more stably, particularly for tasks which differ significantly. Separate vs two-column parameterization: Finally, the task policy can be parameterized to use the distilled policy (8) or not. If using the distilled policy, behaviour distilled into the distilled policy is ?immediately available? to the task policies so transfer can be faster. However if the process of transfer occurs too quickly, it might interfere with effective exploration of individual tasks. From this spectrum of possibilities we consider four concrete instances which differ in the underlying network architecture and distillation loss, identified in Table 1. In addition, we compare against three A3C baselines. In initial experiments we explored two variants of A3C: the original method [20] and the variant of Schulman et al. [28] which uses entropy regularized returns. We did not find significant differences for the two variants in our setting, and chose to report only the original A3C results for clarity in Section 4. Further algorithmic details are provided in the Appendix. 4 Experiments We demonstrate the various algorithms derived from our framework, firstly using alternating optimization with soft Q learning and policy distillation on a set of simple grid world tasks. Then all seven algorithms will be evaluated on three sets of challenging RL tasks in partially observable 3D environments. 4.1 Two room grid world To give better intuition for the role of the distilled behaviour policy, we considered a set of tasks in a grid world domain with two rooms connected by a corridor (see Figure 3) [8]. Each task is distinguished by a different randomly chosen goal location and each MDP state consists of the map location, the previous action and the previous reward. A Distral agent is trained using only the KL regularization and an optimization algorithm which alternates between soft Q learning and policy distillation. Each soft Q learning iteration learns using a rollout of length 10. To determine the benefit of the distilled policy, we compared the Distral agent to one which soft Q learns a separate policy for each task. The learning curves are shown in Figure 3 (left). We see that the Distral agent is able to learn significantly faster than single-task agents. Figure 3 (right) visualizes the distilled policy (probability of next action given position and previous action), demonstrating that the agent has learnt a policy which guides the agent to move consistently in the same direction through the corridor in order to reach the other room. This allows the agent to reach the other room faster and helps exploration, if the agent is shown new test tasks. In Fox et al. [8] two separate options are learnt, while here we learn a single distilled policy which conditions on more past information (previous action and reward). 6 Four di?erent examples of GridWorld tasks A Policy in the corridor if previous action was: left B Policy in the corridor if previous action was: right C D Figure 3: Left: Learning curves on two room grid world. The Distral agent (blue) learns faster, converges towards better policies, and demonstrates more stable learning overall. Center: Example of tasks. Green is goal position which is uniformly sampled for each task. Starting position is uniformly sampled at the beginning of each episode. Right: depiction of learned distilled policy ?0 only in the corridor, conditioned on previous action being left/right and no previous reward. Sizes of arrows depict probabilities of actions. Note that up/down actions have negligible probabilities. The model learns to preserve direction of travel in the corridor. 4.2 Complex Tasks To assess Distral under more challenging conditions, we use a complex first-person partially observed 3D environment with a variety of visually-rich RL tasks. All agents were implemented with a distributed Python/TensorFlow code base, using 32 workers for each task and learnt using asynchronous RMSProp. The network columns contain convolutional layers and an LSTM and are uniform across experiments and algorithms. We tried three values for the entropy costs and three learning rates ?. Four runs for each hyperparameter setting were used. All other hyperparameters were fixed to the single-task A3C defaults and, for the KL+ent 1col and KL+ent 2col algorithms, ? was fixed at 0.5. Mazes In the first experiment, each of n = 8 tasks is a different maze containing randomly placed rewards and a goal object. Figure 4.A1 shows the learning curves for all seven algorithms. Each curve is produced by averaging over all 4 runs and 8 tasks, and selecting the best settings for and ? (as measured by the area under the learning curves). The Distral algorithms learn faster and achieve better final performance than all three A3C baselines. The two-column algorithms learn faster than the corresponding single column ones. The Distral algorithms without entropy learn faster but achieve lower final scores than those with entropy, which we believe is due to insufficient exploration towards the end of learning. We found that both multitask A3C and two-column A3C can learn well on some runs, but are generally unstable?some runs did not learn well, while others may learn initially then suffer degradation later. We believe this is due to negative interference across tasks, which does not happen for Distral algorithms. The stability of Distral algorithms also increases their robustness to hyperparameter selection. Figure 4.A2 shows the final achieved average returns for all 36 runs for each algorithm, sorted in decreasing order. We see that Distral algorithms have a significantly higher proportion of runs achieving good returns, with KL+ent_2col being the most robust. Distral algorithms, along with multitask A3C, use a distilled or common policy which can be applied on all tasks. Panels B1 and B2 in Figure 4 summarize the performances of the distilled policies. Algorithms that use two columns (KL_2col and KL+ent_2col) obtain the best performance, because policy gradients are also directly propagated through the distilled policy in those cases. Moreover, panel B2 reveals that Distral algorithms exhibit greater stability as compared to traditional multitask A3C. We also observe that KL algorithms have better-performing distilled policies than KL+ent ones. We believe this is because the additional entropy regularisation allows task policies to diverge more substantially from the distilled policy. This suggests that annealing the entropy term or increasing the KL term throughout training could improve the distilled policy performance, if that is of interest. Navigation We experimented with n = 4 navigation and memory tasks. In contrast to the previous experiment, these tasks use random maps which are procedurally generated on every episode. The first task features reward objects which are randomly placed in a maze, and the second task requires to return these objects to the agent?s start position. The third task has a single goal object which must be repeatedly found from different start positions, and on the fourth task doors are randomly opened and 7 Figure 4: Panels A1, C1, D1 show task specific policy performance (averaged across all the tasks) for the maze, navigation and laser-tag tasks, respectively. The x-axes are total numbers of training environment steps per task. Panel B1 shows the mean scores obtained with the distilled policies (A3C has no distilled policy, so it is represented by the performance of an untrained network.). For each algorithm, results for the best set of hyperparameters (based on the area under curve) are reported. The bold line is the average over 4 runs, and the colored area the average standard deviation over the tasks. Panels A2, B2, C2, D2 shows the corresponding final performances for the 36 runs of each algorithm ordered by best to worst (9 hyperparameter settings and 4 runs). closed to force novel path-finding. Hence, these tasks are more involved than the previous navigation tasks. The panels C1 and C2 of Figure 4 summarize the results. We observe again that Distral algorithms yield better final results while having greater stability (Figure 4.C2). The top-performing algorithms are, again, the 2 column Distral algorithms (KL_2col and KL+ent_2col). Laser-tag In the final set of experiments, we use n = 8 laser-tag levels. These tasks require the agent to learn to tag bots controlled by a built-in AI, and differ substantially: fixed versus procedurally generated maps, fixed versus procedural bots, and complexity of agent behaviour (e.g. learning to jump in some tasks). Corresponding to this greater diversity, we observe (see panels D1 and D2 of Figure 4) that the best baseline is the A3C algorithm that is trained independently on each task. Among the Distral algorithms, the single column variants perform better, especially initially, as they are able to learn task-specific features separately. We observe again the early plateauing phenomenon for algorithms that do not possess an additional entropy term. While not significantly better than the A3C baseline on these tasks, the Distral algorithms clearly outperform the multitask A3C. Discussion Considering the 3 different sets of complex 3D experiments, we argue that the Distral algorithms are promising solutions to the multitask deep RL problem. Distral can perform significantly better than A3C baselines when tasks have sufficient commonalities for transfer (maze and navigation), while still being competitive with A3C when there is less transfer possible. In terms of specific algorithmic proposals, the additional entropy regularization is important in encouraging continued exploration, while two column architectures generally allow faster transfer (but can affect performance when there is little transfer due to task interference). The computational costs of Distral algorithms are at most twice that of the corresponding A3C algorithms, as each agent need to process two network columns instead of one. However in practice the runtimes are just slightly more than for A3C, because the cost of simulating environments is significant and the same whether single or multitask. 8 5 Conclusion We have proposed Distral, a general framework for distilling and transferring common behaviours in multitask reinforcement learning. In experiments we showed that the resulting algorithms learn quicker, produce better final performances, and are more stable and robust to hyperparameter settings. We have found that Distral significantly outperforms the standard way of using shared neural network parameters for multitask or transfer reinforcement learning. Two ideas in Distral might be worth reemphasizing here. We observe that distillation arises naturally as one half of an optimization procedure when using KL divergences to regularize the output of task models towards a distilled model. The other half corresponds to using the distilled model as a regularizer for training the task models. Another observation is that parameters in deep networks do not typically by themselves have any semantic meaning, so instead of regularizing networks in parameter space, it is worthwhile considering regularizing networks in a more semantically meaningful space, e.g. of policies. We would like to end with a discussion of the various difficulties faced by multitask RL methods. The first is that of positive transfer: when there are commonalities across tasks, how does the method achieve this transfer and lead to better learning speed and better performance on new tasks in the same family? The core aim of Distral is this, where the commonalities are exhibited in terms of shared common behaviours. The second is that of task interference, where the differences among tasks adversely affect agent performance by interfering with exploration and the optimization of network parameters. This is the core aim of the policy distillation and mimic works [26, 23]. As in these works, Distral also learns a distilled policy. But this is further used to regularise the task policies to facilitate transfer. This means that Distral algorithms can be affected by task interference. It would be interesting to explore ways to allow Distral (or other methods) to automatically balance between increasing task transfer and reducing task interference. Other possible directions of future research include: combining Distral with techniques which use auxiliary losses [12, 19, 14], exploring use of multiple distilled policies or latent variables in the distilled policy to allow for more diversity of behaviours, exploring settings for continual learning where tasks are encountered sequentially, and exploring ways to adaptively adjust the KL and entropy costs to better control the amounts of transfer and exploration. Finally, theoretical analyses of Distral and other KL regularization frameworks for deep RL would help better our understanding of these recent methods. 9 References [1] M. G. Bellemare, Y. Naddaf, J. Veness, and M. Bowling. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253?279, june 2013. [2] Yoshua Bengio. Deep learning of representations for unsupervised and transfer learning. In JMLR: Workshop on Unsupervised and Transfer Learning, 2012. 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Online Learning of Optimal Bidding Strategy in Repeated Multi-Commodity Auctions Sevi Baltaoglu Cornell University Ithaca, NY 14850 [email protected] Lang Tong Cornell University Ithaca, NY 14850 [email protected] Qing Zhao Cornell University Ithaca, NY 14850 [email protected] Abstract We study the online learning problem of a bidder who participates in repeated auctions. With the goal of maximizing his T-period payoff, the bidder determines the optimal allocation of his budget among his bids for K goods at each period. As a bidding strategy, we propose a polynomial-time algorithm, inspired by the dynamic programming approach to the knapsack problem. The proposed algorithm, referred to ? as dynamic programming on discrete set (DPDS), achieves ? a regret order of O( T log T ). By showing that the regret is lower bounded by ?( T ) for ? any strategy, we conclude that DPDS is order optimal up to a log T term. We evaluate the performance of DPDS empirically in the context of virtual trading in wholesale electricity markets by using historical data from the New York market. Empirical results show that DPDS consistently outperforms benchmark heuristic methods that are derived from machine learning and online learning approaches. 1 Introduction We consider the problem of optimal bidding in a multi-commodity uniform-price auction (UPA) [1], which promotes the law of one price for identical goods. UPA is widely used in practice. Examples include spectrum auction, the auction of treasury notes, the auction of emission permits (UK), and virtual trading in the wholesale electricity market, which we discuss in detail in Sec. 1.1. A mathematical abstraction of multi-commodity UPA is as follows. A bidder has K goods to bid on at an auction. With the objective to maximize his T-period expected profit, at each period, the bidder determines how much to bid for each good subject to a budget constraint. In the bidding period t, if a bid xt,k for good k is greater than or equal to its auction clearing price ?t,k , then the bid is cleared, and the bidder pays ?t,k . His revenue resulting from the cleared bid will be the good?s spot price (utility) ?t,k . In particular, the payoff obtained from good k at period t is (?t,k ? ?t,k )1{xt,k ? ?t,k } where 1{xt,k ? ?t,k } indicates whether the bid is cleared. Let ?t = [?t,1 , ..., ?t,K ]| and ?t = [?t,1 , ..., ?t,K ]| be the vector of auction clearing and spot market prices at period t, respectively. Similarly, let xt = [xt,1 , ..., xt,K ]| be the vector of bids for period t. We assume that (?t , ?t ) are drawn from an unknown joint distribution and, in our analysis, independent and identically distributed (i.i.d.) over time.1 At the end of each period, the bidder observes the auction clearing and spot prices of all goods. Therefore, before choosing the bid of period t, all the information the bidder has is a vector It?1 containing his observation and decision history {xi , ?i , ?i }t?1 i=1 . Consequently, a bidding policy ? of a bidder is defined as a sequence of decision rules, i.e., ? = (?0 , ?1 ..., ?T ?1 ), such that, at time t ? 1, 1 This implies that the auction clearing price is independent of bid xt , which is a reasonable assumption for any market where an individual?s bid has negligible impact on the market price. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. ?t?1 maps the information history It?1 to the bid xt of period t. The performance of any bidding policy ? is measured by its regret, which is defined by the difference between the total expected payoff of policy ? and that of the optimal bidding strategy under known distribution of (?t , ?t ). 1.1 Motivating applications The mathematical abstraction introduced above applies to virtual trading in the U.S. wholesale electricity markets that are operated under a two-settlement framework. In the day-ahead (DA) market, the independent system operator (ISO) receives offers to sell and bids to buy from generators and retailers for each hour of the next day. To determine the optimal DA dispatch of the next day and DA electricity prices at each location, ISO solves an economic dispatch problem with the objective of maximizing social surplus while taking transmission and operational constraints into account. Due to system congestion and losses, wholesale electricity prices vary from location to location.2 In the real-time (RT) market, ISO adjusts the DA dispatch according to the RT operating conditions, and the RT wholesale price compensates deviations in the actual consumption from the DA schedule. The differences between DA and RT prices occur frequently both as a result of generators and retailers exercising locational market power [2] and as a result of price spikes in the RT due to unplanned outages and unpredictable weather conditions [3]. To promote price convergence between DA and RT markets, in the early 2000s, virtual trading was introduced [4]. Virtual trading is a financial mechanism that allows market participants and external financial entities to arbitrage on the differences between DA and RT prices. Empirical and analytical studies have shown that increased competition in the market due to virtual trading results in price convergence and increased market efficiency [2, 3, 5]. Virtual transactions make up a significant portion of the wholesale electricity markets. For example, the total volume of cleared virtual transactions in five big ISO markets was 13% of the total load in 2013 [4]. In the same year, total payoff resulting from all virtual transactions was around 250 million dollars in the PJM market [2] and 45 million dollars in NYISO market [6]. A bid in virtual trading is a bid to buy (sell) energy in the DA market at a specific location with an obligation to sell (buy) back exactly the same amount in the RT market at the same location if the bid is cleared (accepted). Specifically, a bid to buy in the DA market is cleared if the offered bid price is higher than the DA market price. Similarly, a bid to sell in the DA market is cleared if it is below the DA market price. In this context, different locations and/or different hours of the day are the set of goods to bid on. The DA prices are the auction clearing prices, and the RT prices are the spot prices. The problem studied here may also find applications in other types of repeated auctions where the auction may be of the double, uniform-price, or second-price types. For example, in the case of online advertising auctions [7], different goods can correspond to different types of advertising space an advertiser may consider to bid on. 1.2 Main results and related work We propose an online learning approach to the algorithmic bidding under budget constraints in repeated multi-commodity auctions. The proposed approach falls in the category of empirical risk minimization (ERM) also referred to as the follow the leader approach. The main challenge here is that optimizing the payoff (risk) amounts to solving a multiple choice knapsack problem (MCKP) that is known to be NP hard [8]. The proposed approach, referred to as dynamic programming on discrete set (DPDS), is inspired by a pseudo-polynomial dynamic programming approach to 0-1 Knapsack problems. DPDS allocates the limited budget of the bidder among K goods in polynomial time both in terms of the number of goods K and in terms of the time horizon T . We show that the expected payoffpof DPDS converges to that of the optimal strategy under ? known distribution by a rate no slower than log t/t which results in a regret upper?bound of O( T log T ). By showing that, for any bidding strategy, the regret is lower bounded by ?( T ), we prove that DPDS is order optimal up ? to a log T term. We also evaluate the performance of DPDS empirically in the context of virtual trading by using historical data from the New York energy market. Our empirical results show that 2 For example, transmission congestion may prevent scheduling the least expensive resources at some locations. 2 DPDS consistently outperforms benchmark heuristic methods that are derived from standard machine learning methods. The problem formulated here can be viewed in multiple machine learning perspectives. We highlight below several relevant existing approaches. Since the bidder can calculate the reward that could have been obtained by selecting any given bid value regardless of its own decision, our problem falls into the category of full-feedback version of multi-armed bandit (MAB) problem, referred to as experts problem, where the reward of all arms (actions) are observable at the end of each period regardless of the chosen arm. For the case of finite number of arms, Kleinberg et al. [9] showed that, for stochastic setting, constant regret is achievable by choosing the arm with the highest average reward at each period. A special case of the adversarial setting was studied by Cesa-Bianchi et al. [10] who provided ? matching upper and lower bounds in the order of ?( T ). Later, Freund and Schapire [11] and Auer et al. [12] showed that the Hedge algorithm, a variation of weighted majority algorithm [13], achieves the matching bound for the general setting. These results, however, do not apply to experts problems with continuous action spaces. The stochastic experts problem where the set of arms is an uncountable compact metric space (X , d) rather than finite was studied by Kleinberg and Slivkins [14] (see [15] for an extended version). Since there are uncountable number of arms, it is assumed that, in each period, a payoff function drawn from an i.i.d. distribution is observed rather than the individual payoff of each arm. Under the assumption of Lipschitz expected ? payoff function, they showed that the instance-specific regret of any algorithm is lower bounded by ?( T ). They also showed that their algorithm?NaiveExperts?achieves a regret upper bound of O(T ? ) for any ? > (b + 1)/(b + 2) where b is the isometry invariant of the metric space. However, NaiveExperts is computationally intractable in practice because the computational complexity of its direct implementation grows exponentially with the dimension (number of goods in our case). Furthermore, the lower bound in [14] does not imply a lower bound for our problem with a specific payoff. Krichene et al. [16] studied the adversarial setting and proposed an extension of ? the Hedge algorithm, which achieves O( T log T ) regret under the assumption of Lipschitz payoff functions. For our problem, it is reasonable to assume that the expected payoff function is Lipschitz; yet it is clear that, at each period, the payoff realization is a step function which is not Lipschitz. Hence, Lipschitz assumption of [16] doesn?t hold in our setting. Stochastic gradient descent methods, which have low computational complexity, have been extensively studied in the literature of continuum-armed bandit [17, 18, 19]. However, either the concavity or the unimodality of the expected payoff function is required for regret guarantees of these methods to hold. This may not be the case in our problem depending on the underlying distribution of prices. A relevant work that takes an online learning perspective for the problem of a bidder engaging in repeated auctions is Weed et al. [7]. They are motivated by online advertising auctions and studied the partial information setting of the same problem as ours but without a budget constraint. Under the margin condition, i.e., the probability of auction price occurring in close proximity of mean utility is bounded, they showed that their ? algorithm, inspired by the UCB1 algorithm [20], achieves regret that ranges from O(log T ) to O( T log T ) depending on how tight the margin condition is. They also provided matching lower bounds up to a logarithmic factor. However, their lower bound does not imply a bound for the full information setting we study here. Also, the learning algorithm in [7] does not apply here because the goods are coupled through the budget constraint in our case. Furthermore, we do not have margin condition, and we allow the utility of the good to depend on the auction price. Some other examples of literature on online learning in repeated auctions studied the problem of an advertiser who wants to maximize the number of clicks with a budget constraint [21, 22], or that of a seller who tries to learn the valuation of its buyer in a posted price auction [23, 24]. The settings considered in those problems are considerably different from that studied here in the implementation of budget constraints [21, 22], and in the strategic behavior of the bidder [23, 24]. 2 Problem formulation The total expected payoff at period t given bid xt can be expressed as r(xt ) = E ((?t ? ?t )| 1{xt ? ?t }|xt ) , where the expectation is taken using the joint distribution of (?t , ?t ), and 1{xt ? ?t } is the vector of indicator functions with the k-th entry corresponding to 1{xt,k ? ?t,k }. We assume that the payoff 3 (?t ? ?t )| 1{xt ? ?t } obtained at each period is a bounded random variable with support in [l, u],3 and the auction prices are drawn from a distribution with positive support. Hence, a zero bid for any good is equivalent to not bidding because it will not get cleared. The objective is to determine a bidding policy ? that maximizes the expected T-period payoff subject to a budget constraint for each individual period: ! T X ? maximize E r(xt ) ? t=1 (1) subject to kx?t k1 ? B, for all t = 1, ..., T, x?t ? 0, for all t = 1, ..., T, where B is the auction budget of the bidder, x?t denotes the bid determined by policy ?, and x?t ? 0 is equivalent to x?t,k ? 0 for all k ? {1, 2, ..., K}. 2.1 Optimal solution under known distribution If the joint distribution f (., .) of ?t and ?t is known, the optimization problem (1) decouples to solving for each time instant separately. Since (?t , ?t ) is i.i.d. over t, an optimal solution under known model does not depend on t and is given by x? = arg max r(xt ) (2) xt ?F where F = {x ? <K : x ? 0, kxk1 ? B} is the feasible set of bids. Optimal solution x? may not be unique or it may not have a closed form. The following example illustrates a case where there isn?t a closed form solution and shows that, even in the case of known distribution, the problem is a combinatorial stochastic optimization, and it is not easy to calculate an optimal solution. ? k > 0, and Example. Let ?t and ?t be independent, ?t,k be exponentially distributed with mean ? the mean of ?t,k be ? ?k > 0 for all k ? {1, .., K}. Since not bidding for good k is optimal if ? ?k ? 0, we exclude the case ? ?k ? 0 without loss of generality. For this example, we can use the concavity of r(x) in the interval [0, ? ? ], where ? ? = [? ?1 , ..., ? ?K ]| , to obtain the unique optimal solution x? , which is characterized by ? PK ? ?k if k=1 ? ?k ? B, ?? PK ?k < ? ?, x?k = 0 if k=1 ? ?k > B and ? ? k /? ? P ? ?x satisfying (? ? k = ? ? if K ? ?k ? ? ?, ?k ? xk )e?xk /?k /? ? k /? k k=1 ?k > B and ? where the Lagrange multiplier ? ? > 0 is chosen such that kx? k1 = B is satisfied. This solution takes the form of a "water-filling" strategy. More specifically, if the budget constraint is not binding, then the optimal solution is to bid ? ?k for every good k. However, in the case of a binding budget constraint, the optimal solution is determined by the bid value at which the marginal expected payoff associated ? k ), and this bid value cannot be expressed in closed form. with each good k is equal to min(? ? , ? ?k /? We measure the performance of a bidding policy ? by its regret4 , the difference between the expected T-period payoff of ? and that of x? , i.e., R?T (f ) = T X E(r(x? ) ? r(x?t )), (3) t=1 where the expectation is taken with respect to the randomness induced by ?. The regret of any policy is monotonically increasing. Hence, we are interested in policies with sub-linear regret growth. 3 This is reasonable in the case of virtual trading because DA and RT prices are bounded due to offer/bid caps. The regret definition used here is the same as in [14]. This definition is also known as pseudo-regret in the literature [25]. 4 4 3 Online learning approach to optimal bidding The idea behind our approach is to maximize the sample mean of the expected payoff function, which is an ERM approach [26]. However, we show that a direct implementation of ERM is NP-hard. Hence, we propose a polynomial-time algorithm that is based on dynamic programming on a discretized feasible set. We show that our approach achieves the order optimal regret. 3.1 Approximate expected payoff function and its optimization Regardless of the bidding policy, one can observe the auction and spot prices of past periods. Therefore, the average payoff that could have been obtained by bidding x up to the current period can be calculated for any fixed value of x ? F. Specifically, the average payoff r?t,k (xk ) for a good k as a function of the bid value xk can be calculated at period t + 1 by using observations up to t, i.e., r?t,k (xk ) = (1/t) t X (?i,k ? ?i,k )1{xk ? ?i,k }. i=1 For example, at the end of first period, r?t,k (xk ) = (?1,k ? ?1,k )1{xk ? ?1,k } as illustrated in Fig. 1a. For, t ? 2, this can be expressed recursively;  t?1 ?t?1,k (xk ) if xk < ?t,k , t r r?t,k (xk ) = t?1 (4) 1 r ? (x ) + (? ? ? ) if xk ? ?t,k . t?1,k k t,k t,k t t Since each observation introduces a new breakpoint, and the value of average payoff function is constant between two consecutive breakpoints, we observe that r?t,k (xk ) is a piece-wise constant function with at most t breakpoints. Let  the vector of order | statistics of the observed auction clearing prices {?i,k }ti=1 and zero be ?(k)= 0, ? , ..., ? , and let the vector of associated average (1),k (t),k   (k) (k) (k) payoffs be r , i.e., ri = r?t,k ?i . Then, r?t,k (xk ) can be expressed by the pair ?(k) , r(k) , e.g., see Fig. 1b. r?1,k (xk ) r?4,k (xk ) ?1,k ? ?1,k xk ?1,k 0 (k) r5 (k) r3 (k) r4 (k) r2 0 (k) ?2 (k) ?3 (k) ?4 (k) xk ?5 (b) t = 4 (a) t = 1 Figure 1: Piece-wise constant average payoff function of good k For a vector  y, let ym:n = (ym , ym+1 , ..., yn ) denote the sequence of entries from m to n. Initialize ?(k) , r(k) = (0, 0) at the beginning of first period. Then, at each period t ? 1, the pair ?(k) , r(k) can be updated recursively as follows: |    h i|  t ? 1 1 (k) t ? 1 (k) (k) (k) (k) (k) r , r + (?t,k ? ?t,k ) , (5) ? ,r = ?1:ik , ?t,k , ?ik +1:t , t 1:ik t ik :t t where ik = maxi:?(k) <? i at period t. i t,k Consequently, overall average payoff function r?t (x) can be expressed as a sum of average payoff functions of individual goods. Instead of the unknown expected payoff r(x), let?s consider the maximization of the average payoff function, which corresponds to the ERM approach, i.e., max r?t (x) = max x?F x?F K X r?t,k (xk ). (6) k=1 (k) forhsome i ? {1,  ..., t + 1} contributes (k) (k) the same amount to the overall payoff as choosing any xk ? ?i , ?i+1 if i < t + 1 and any Due to the piece-wise constant structure, choosing xk = ?i 5 (k) (k) xk ? ?i if i = t + 1. However, choosing xk = ?i utilizes a smaller portion of the budget. Hence, an optimal solution to (6) can be obtained by solving the following integer linear program: maximize {zk }K k=1 subject to K  X r(k) | zk k=1 K  X ?(k) | (7) zk ? B, k=1 1| zk ? 1, ?k = 1, ..., K, zk,i ? {0, 1}, ?i = 1, ..., t + 1; ?k = 1, ..., K. | where the bid value xk = ?(k) zk for good k. Observe that (7) is a multiple choice knapsack problem (MCKP) [8], a generalization of 0-1 knapsack. Unfortunately, (7) is NP-hard [8]. If we had a polynomial-time algorithm that finds an optimal solution x ? F to (6), then we could have obtained the solution of (7) in polynomial-time too by setting zk,i = 1 where i = maxi:?(k) ?x i for each k. Therefore, (6) is also NP-hard, and, to the k i best of our knowledge, there isn?t any method in the ERM literature [27], which mostly focuses on classification problems, suitable to implement for the specific problem at hand. 3.2 Dynamic programming on discrete set (DPDS) policy Next, we present an approach that discretizes the feasible set using intervals of equal length and optimizes the average payoff on this new discrete set via a dynamic program. Although this approach doesn?t solve (6), the solution can be arbitrarily close to the optimal depending on the choice of the interval length under the assumption of the Lipschitz continuous expected payoff function. To exploit the smoothness of Lipschitz continuity, discretization approach of the continuous feasible set has been used in the continuous MAB literature previously [17, 14]. However, different than MAB literature, in this paper, discretization approach is utilized to reduce the computational complexity of an NP-hard problem as well. Let ?t be an integer sequence increasing with t and Dt = {0, B/?t , 2B/?t , ..., B} as illustrated in Fig. 2. Then, the new discrete set is given as Ft = {x ? F : xk ? Dt , ?k ? {1, ..., K}}. Our goal is to optimize r?t (.) on the new set Ft rather than F, i.e., max r?t (xt+1 ). (8) xt+1 ?Ft r?4,k (xk ) (k) r5 (k) r3 (k) r4 (k) r2 0 (k) B ?4 ?2 (k) 2B ?4 ?3 (k) 3B ?4 ?4 (k) ?5 4B ?4 xk Figure 2: Example of the discretization of the decision space for good k when t = 4 Now, we use dynamic programming approach that has been used to solve 0-1 Knapsack problems including MCKP given in (7) [28]. However, direct implementation of this approach results in pseudopolynomial computational complexity in the case of 0-1 Knapsack problems. The discretization of the feasible set with equal interval length reduces the computational complexity to polynomial time. We define the maximum payoff one can collect with budget b among goods {1, ..., n} when the bid value xk is restricted to the set Dt for each good k as Vn (b) = Pnmax {xk }n k=1 xk ?b,xk ?Dt ?k k=1 : 6 n X k=1 r?t,k (xk ). Then, the following recursion can be used to solve for VK (B) which gives the optimal solution to (8): ( 0 Vn (jB/?t ) = max (? rt,n (iB/?t ) + Vn?1 ((j ? i)B/?t )) 0?i?j if n = 0, j ? {0, 1, ..., ?t }, if 1 ? n ? K, j ? {0, 1, ..., ?t }. (9) This is the Bellman equation where Vn (b) is the maximum total payoff one can collect using remaining budget b and remaining n goods. Its optimality can be shown via a simple induction argument. Recall that r?t,n (0) = 0 for all (t, n) pairs due to the assumption of positive day-ahead prices. Recursion (9) can be solved starting from n = 1 and proceeding to n = K, where, for each n, Vn (b) is calculated for all b ? Dt . Since the computation of Vn (b) requires at most ?t + 1 comparison for any fixed value of n ? {1, ..., K} and b ? Dt , it has a computational complexity on the order of K?t2 once the average payoff values r?t,n (xn ) for all xn ? Dt and n ? {1, ..., K} are given. For each n ? {1, ..., K}, computation of r?t,n (xn ) for all xn ? Dt introduces an additional computational  complexity of at most on the order of t, which can be observed from the update step of ?(k) , ? (k) , given in (5). Hence, total computational complexity of DPDS is O(K max(t, ?t2 )) at each period t. 3.3 Convergence and regret of DPDS policy Under the assumption of Lipschitz continuity, Theorem 1 shows that the value of DPDS p converges to the value of the optimal policy under known model with a rate faster than or equal to log t/t if the ? DPDS algorithm parameter ? ?t = dt e with ? ? 1/2. Consequently, the regret growth rate of DPDS is upper bounded by O( T log T ). If ? = 1/2, then the computational complexity of the algorithm is bounded by O(Kt) at each period t, and total complexity over the entire horizon is O(KT 2 ). Theorem 1 Let xDPDS t+1 denote the bid of DPDS policy for period t + 1. If r(.) is Lipschitz continuous on F with p-norm and Lipschitz constant L, then, for any ? > 0 and for DPDS parameter choice ?t ? 2, r log t 4 min(u ? l, LK 1/p B)?tK LK 1/p B p ? DPDS + , E(r(x ) ? r(xt+1 )) ? + 2(? + 1)K + 1(u ? l) ?t t t(?+1)K+1/2 (10) and for ?t = max(dt? e, 2) with ? ? 1/2, p p ? RDPDS (f ) ? 2(LK 1/p B +4 min(u?l, LK 1/p B)) T +2 2(? + 1)K + 1(u?l) T log T . (11) T Actually, we can relax the uniform Lipschitz continuity condition. Under the weaker condition of |r(x? ) ? r(x)| ? Lkx? ? xkqp for all x ? F and for some constant L > 0, the incremental regret bound that is given in (10) becomes p p q/p E(r(x? )?r(xDPDS (B/?t )q +(u?l)( 2(? + 1)K + 1 log t/t+4?tK t?(?+1)K?1/2 ). t+1 )) ? LK The proof of Theorem 1 is derived by showing that the value of x?t+1 = arg maxx?Ft r(x) converges ? to the value of x? due to Lipschitz continuity, and the value of xDPDS t+1 converges to the value of xt+1 via the use of concentration inequality inspired by [20, 17]. Even though the upper bound of regret in Theorem 1 depends on the budget B linearly, this dependence can be avoided in the expense of increase in computational complexity. For example, in the literature, the reward is generally assumed to be in the unit interval, i.e., l = 0 and u = 1, and the expected reward is assumed to be Lipschitz continuous with Euclidean norm and constant L = 1. In this case, by following the proof of Theorem 1, we observe ? = 1/2 and ? that assigning ? ?t = max(d?t? e, 2) for some ? > 0 gives a regret upper bound of 2B KT /? +12 KT log T +? ? ? for T > ? + 1. Consequently, if B = O(K), then O(K 3/4 T + KT log T ) regret is achievable by setting ? = K 3/4 . 3.4 Lower bound of regret for any bidding policy We now show that DPDS in fact achieves the slowest possible regret growth. Specifically, Theorem 2 states that, for any bidding policy ? and horizon T , there exists a distribution f for which the regret growth is slower than or equal to the square root of the horizon T . 7 Theorem 2 Consider the case where K = 1, B = 1, and ?t and ?t are independent random variables with distributions f? (?t ) = ?1 1{(1 ? )/2 ? ?t ? (1 + )/2} ? and f? (?t ) = Bernoulli(? ? ), respectively. Let f (?t , ?t ) = f? (?t )f? (?t ) and  = T ?1/2 /2 5. Then, for any bidding policy ?, ? ? RT? (f ) ? (1/16 5) T , either for ? ? = 1/2 +  or for ? ? = 1/2 ? . As seen in Theorem 2, we choose a specific distribution for the auction clearing and spot prices. Observe that, for this distribution, the payoff function is Lipschitz continuous with Lipschitz constant L = 3/2 because the magnitude of the derivative of the payoff function |r0 (x)| ? |? ? ? x|/ ? 3/2 for (1 ? )/2 ? x ? (1 + )/2 and r0 (x) = 0 otherwise. So, it satisfies the condition given in Theorem 1. The proof of Theorem 2 is obtained by showing that, every time the bid is cleared, an incremental regret greater than /2 is incurred under the distribution with ? ? = (1/2?); otherwise, an incremental regret greater than /2 is incurred under the distribution with ? ? = (1/2 + ). However, to distinguish between these two distributions, one needs ?(T ) samples, which results in a regret lower bound ? of ?( T ). The bound is obtained by adapting a similar argument used by [29] in the context of non-stochastic MAB problem. 4 Empirical study New York ISO (NYISO), which consists of 11 zones, allows virtual transactions at zonal nodes only. So, we use historical DA and RT prices of these zones from 2011 to 2016 [30]. Since the price for each hour is different at each zone, there are 11 ? 24 different locations, i.e., zone-hour pairs, to bid on every day. The prices are per unit (MWh) prices. We also consider buy and sell bids simultaneously for all location. As explained in Sec. 1.1, a sell bid is a bid to sell in the DA market with an obligation to buy back in the RT market. Hence, the profit of a sell bid at period t is (?t ? ?t )| 1{xt ? ?t }. Generally, an upper bound p? for the DA prices is known, e.g. p? = $1000 for NYISO. We convert a sell bid to a buy bid by using xsell ? ? xt , ?sell ? ? ?t , and ?tsell = p? ? ?t instead of xt , ?t , t = p t = p and ?t . NYISO DA market for day t closes at 5:00 am on day t ? 1. Hence, the RT prices of all hours of day t ? 1 cannot be observed before the bid submission for day t. Therefore, the most recent information used before the submission for day t was the observations from day t ? 2. (a) y = 2012 (b) y = 2013 (d) y = 2015 (c) y = 2014 (e) y = 2016 Figure 3: Cumulative profit trajectory of year y for B = 100000 We compare DPDS with three algorithms. One of them is UCBID-GR, inspired by UCBID [7]. At each day, UCBID-GR sorts all locations according to their profitabilities, i.e., their price spread (the difference between DA and RT price) sample means. Then, starting from the most profitable location, 8 UCBID-GR sets the bid of a location equal to its RT price sample mean until there isn?t any sufficient budget left. The second algorithm, referred to as SA, is a variant of Kiefer-Wolfowitz stochastic approximation method. SA approximates the gradient of the payoff function by using the current observation and updates the bid of each k as follows; xt,k = xt?1,k + at ((?t?2,k ? ?t?2,k )(1{xt?1,k + ct ? ?t?2,k } ? 1{xt?1,k ? ?t?2,k })) /ct . Then, xt is projected to the feasible set F. The last algorithm is SVM-GR, which is inspired by the use of support vector machines (SVM) by Tang et al. [31] to determine if a buy or a sell bid is profitable at a location, i.e., if the price spread is positive or negative. Due to possible correlation of the price spread at a location on day t with the price spreads observed recently at that and also at other locations, the input of SVM for each location is set as the price spreads of all locations from day t ? 7 to day t ? 2. To test SVM-GR algorithm at a particular year, for each location, the data from the previous year is used to train SVM and to determine the average profit, i.e., average price spread, and the bid level that will be accepted with 95% confidence in the event that a buy or a sell bid is profitable. For the test year, at each period, SVM-GR first determines if a buy or a sell bid is profitable for each location. Then, SVM-GR sorts all locations according to their average profits, and, starting from the most profitable location, it sets the bid of a location equal to the bid level with 95% confidence of acceptance until there isn?t any sufficient budget left. To evaluate the performance of a year, DPDS, UCBID-GR, and SA algorithms have also been trained starting from the beginning of the previous year. The algorithm parameter of DPDS was set as ?t = t; and the step size at and ct of SA were set as 20000/t and 2000/t1/4 , respectively. For B=$100,000, the cumulative profit trajectory of five consecutive years are given in Fig. 3. We observe that DPDS obtains a significant profit in all cases, and it outperforms other algorithms consistently except 2015 where SVM-GR makes approximately 25% more profit. However, in three out of five years, SVM-GR suffers a considerable amount of loss. In general, UCBID-GR performs quite well except 2016, and SA algorithm incurs a loss almost every year. 5 Conclusion By applying general techniques such as ERM, discretization approach, and dynamic programming, we derive a practical and efficient algorithm to the algorithmic bidding problem under budget constraint in repeated multi-commodity auctions. We show that the expected payoff ofp the proposed algorithm, DPDS, converges to that of the optimal strategy by a rate no slower than log ? t/t, which results ? in a O( T log T ) regret. By showing that the regret is lower bounded by ?( T ) for any bidding ? strategy, we prove that DPDS is order optimal up to a log T term. For the motivating application of virtual bidding in electricity markets (see Sec. 1.1), the stochastic setting, studied in this paper, is natural due to the electricity markets being competitive, which implies that the existence of an adversary is very unlikely. However, it is also of interest to study the adversarial setting to extend the results to other applications. For example, the adversarial setting of our problem is a special case of no-regret learning problem of Simultaneous Second Price Auctions (SiSPA), studied by Daskalakis and Syrgkanis [32] and Dudik et al. [33]. In particular, to deal with the adversarial setting, it is possible to use our dynamic programming approach as the offline oracle for the Oracle-Based Generalized FTPL algorithm proposed by Dudik et al. [33] if we fix the discretized action set over the whole time horizon. More specifically, let the interval length of discretization be B/m, i.e., ?t = m. Then, it is possible to show that a 1-admissible translation matrix with Kdlog me columns is implementable with complexity m. Consequently, ? no-regret result of Dudik et al. [33] holds with a regret bound of O(K T log m) if we measure the performance of the algorithm against the best action in hindsight in the discretized finite action set rather than in the original continuous action set considered here. Unfortunately, as shown by Weed et al. [7], it is not possible to achieve sublinear regret with a fixed discretization for the specific problem considered in this paper. Hence, it requires further work to see if this method can be extended to obtain no-regret learning for the adversarial setting under the original continuous action set. 9 Acknowledgments We would like to thank Professor Robert Kleinberg for the insightful discussion. 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Trimmed Density Ratio Estimation Song Liu? University of Bristol [email protected] Taiji Suzuki University of Tokyo, Sakigake (PRESTO), JST, AIP, RIKEN, [email protected] Akiko Takeda The Institute of Statistical Mathematics, AIP, RIKEN, [email protected] Kenji Fukumizu The Institute of Statistical Mathematics, [email protected] Abstract Density ratio estimation is a vital tool in both machine learning and statistical community. However, due to the unbounded nature of density ratio, the estimation proceudre can be vulnerable to corrupted data points, which often pushes the estimated ratio toward infinity. In this paper, we present a robust estimator which automatically identifies and trims outliers. The proposed estimator has a convex formulation, and the global optimum can be obtained via subgradient descent. We analyze the parameter estimation error of this estimator under high-dimensional settings. Experiments are conducted to verify the effectiveness of the estimator. 1 Introduction Density ratio estimation (DRE) [18, 11, 27] is an important tool in various branches of machine learning and statistics. Due to its ability of directly modelling the differences between two probability density functions, DRE finds its applications in change detection [13, 6], two-sample test [32] and outlier detection [1, 26]. In recent years, a sampling framework called Generative Adversarial Network (GAN) (see e.g., [9, 19]) uses the density ratio function to compare artificial samples from a generative distribution and real samples from an unknown distribution. DRE has also been widely discussed in statistical literatures for adjusting non-parametric density estimation [5], stabilizing the estimation of heavy tailed distribution [7] and fitting multiple distributions at once [8]. However, as a density ratio function can grow unbounded, DRE can suffer from robustness and stability issues: a few corrupted points may completely mislead the estimator (see Figure 2 in Section 6 for example). Considering a density ratio p(x)/q(x), a point x that is extremely far away from the high density region of q may have an almost infinite ratio value and DRE results can be dominated by such points. This makes DRE performance very sensitive to rare pathological data or small modifications of the dataset. Here we give two examples: Cyber-attack In change detection applications, a density ratio p(x)/q(x) is used to determine how the data generating model differs between p and q. Consider a ?hacker? who can spy on our data may just inject a few data points in p which are extremely far away from the high-density region of q. This would result excessively large p(x)/q(x) tricking us to believe there is a significant change from q(x) to p(x), even if there is no change at all. If the generated outliers are also far away from the ? This work was done when Song Liu was at The Institute of Statistical Mathematics, Japan 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. high density region of p(x), we end up with a very different density ratio function and the original parametric pattern in the ratio is ruined. We give such an example in Section 6. Volatile Samples The change of external environment may be responded in unpredictable ways. It is possible that a small portion of samples react more ?aggressively? to the change than the others. These samples may be skewed and show very high density ratios, even if the change of distribution is relatively mild when these volatile samples are excluded. For example, when testing a new fertilizer, a small number of plants may fail to adapt, even if the vast majority of crops are healthy. Overly large density ratio values can cause further troubles when the ratio is used to weight samples. For example, in the domain adaptation setting, we may reweight samples from one task and reuse them in another task. Density ratio is a natural choice of such ?importance weighting? scheme [28, 25]. However, if one or a few samples have extremely high ratio, after renormalizing, other samples will have almost zero weights and have little impact to the learning task. Several methods have been proposed to solve this problem. The relative density ratio estimation [33] estimates a ?biased? version of density ratio controlled by a mixture parameter ?. The relative density ratio is always upper-bounded by ?1 , which can give a more robust estimator. However, it is not clear how to de-bias such an estimator to recover the true density ratio function. [26] took a more direct approach. It estimates a thresholded density ratio by setting up a tolerance t to the density ratio value. All likelihood ratio values bigger than t will be clipped to t. The estimator was derived from Fenchel duality for f -divergence [18]. However, the optimization for the estimator is not convex if one uses log-linear models. The formulation also relies on the non-parametric approximation of the density ratio function (or the log ratio function) making the learned model hard to interpret. Moreover, there is no intuitive way to directly control the proportion of ratios that are thresholded. Nonetheless, the concept studied in our paper is inspired by this pioneering work. In this paper, we propose a novel method based on a ?trimmed Maximum Likelihood Estimator? [17, 10]. This idea relies on a specific type of density ratio estimator (called log-linear KLIEP) [30] which can be written as a maximum likelihood formulation. We simply ?ignore? samples that make the empirical likelihood take exceedingly large values. The trimmed density ratio estimator can be formulated as a convex optimization and translated into a weighted M-estimator. This helps us develop a simple subgradient-based algorithm that is guaranteed to reach the global optimum. Moreover, we shall prove that in addition to recovering the correct density ratio under the outlier setting, the estimator can also obtain a ?corrected? density ratio function under a truncation setting. It ignores ?pathological? samples and recovers density ratio only using ?healthy? samples. Although trimming will usually result a more robust estimate of the density ratio function, we also point out that it should not be abused. For example, in the tasks of two-sample test, a diverging density ratio might indicate interesting structural differences between two distributions. In Section 2, we explain some preliminaries on trimmed maximum likelihood estimator. In Section 3, we introduce a trimmed DRE. We solve it using a convex formulation whose optimization procedure is explained in Section 4. In Section 5, we prove the estimation error upper-bound with respect to a sparsity inducing regularizer. Finally, experimental results are shown in Section 6 and we conclude our work in Section 7. 2 Preliminary: Trimmed Maximum Likelihood Estimation Although our main purpose is to estimate the density ratio, we first introduce the basic concept of trimmed estimator using density functions as examples. Given n samples drawn from a distribution  n i.i.d. P , i.e., X := x(i) i=1 ? P, x ? Rd , we want to estimate the density function p(x). Suppose the true density function is a member of exponential family [20], Z p(x; ?) = exp [h?, f (x)i ? log Z(?)] , Z(?) = q(x) exph?, f (x)idx (1) where f (x) is the sufficient statistics, Z(?) is the normalization function and q(x) is the base measure. Maximum Likelihood Estimator (MLE) maximizes the empirical likelihood over the entire dataset. In contrast, a trimmed MLE only maximizes the likelihood over a subset of samples according to 2 their likelihood values (see e.g., [10, 31]). This paradigm can be used to derive a popular outlier detection method, one-class Support Vector Machine (one-SVM) [24]. The derivation is crucial to the development of our trimmed density ratio estimator in later sections. Without loss of generality, we can set the log likelihood function as log p(x(i) ; ?) ? ?0 , where ?0 is a constant. As samples corresponding to high likelihood values are likely to be inliers, we can trim all samples whose likelihood is bigger than ?0 using a clipping function [?]? , i.e., ? = arg max? Pn [log p(x(i) ; ?) ? ?0 ]? , where [`]? returns ` if ` ? 0 and 0 otherwise. This ? i=1 optimization has a convex formulation:   min h, 1i, s.t. ?i, log p x(i) ; ? ? ?0 ? i , (2) ?,?0  where  is the slack variable measuring the difference between log p x(i) ; ? and ?0 . However, formulation (2) is not practical since computing the normalization term Z(?) in (1) is intractable for a general f and it is unclear how to set the trimming level ?0 . Therefore we ignore the normalization term and introduce other control terms: min ?,?0,? ?0 1 1 k?k2 ? ?? + h, 1i s.t. ?i, h?, f (x(i) )i ? ? ? i . 2 n (3) The `2 regularization term is introduced to avoid ? reaching unbounded values. A new hyper parameter ? ? (0, 1] replaces ?0 to control the number of trimmed samples. It can be proven using KKT conditions that at most 1 ? ? fraction of samples are discarded (see e.g., [24], Proposition 1 for details). Now we have reached the standard formulation of one-SVM. This trimmed estimator ignores the large likelihood values and creates a focus only on the low density region. Such a trimming strategy allows us to discover ?novel? points or outliers which are usually far away from the high density area. 3 Trimmed Density Ratio Estimation In this paper, our main focus is to derive a robust density ratio estimator following a similar trimming strategy. First, we briefly review the a density ratio estimator [27] from the perspective of KullbackLeibler divergence minimization. 3.1 Density Ratio Estimation (DRE) (1) i.i.d. (n ) (1) (n ) i.i.d. For two sets of data Xp := {xp , . . . , xp p } ? P, Xq := {xq , . . . , xq q } ? Q, asp(x;? ) sume both the densities p(x) and q(x) are in exponential family (1). We know q(x;?pq ) ? exp [h? p ? ? q , f (x)i] . Observing that the data x only interacts with the parameter ? p ? ? q through f , we can keep using f (x) as our sufficient statistic for the density ratio model, and merge two parameters ? p and ? q into one single parameter ? = ? p ? ? q . Now we can model our density ratio as Z r(x; ?) := exp [h?, f (x)i ? log N (?)] , N (?) := q(x) exph?, f (x)idx, (4) R where N (?) is the normalization term that guarantees q(x)r(x; ?)dx = 1 so that q(x)r(x; ?) is a valid density function and is normalized over its domain. Interestingly, despite the parameterization (changing from ? to ?), (4) is exactly the same as (1) where q(x) appeared as a base measure. The difference is, here, q(x) is a density function from which Xq are drawn so that N (?) can be approximated accurately from samples of Q. Let us define nq h i h i X b (?) , N b (?) := 1 exp h?, f (xq(j) )i . r?(x; ?) := exp h?, f (x)i ? log N nq j=1 (5) Note this model can be computed for any f even if the integral in N (?) does not have a closed form . 3 In order to estimate ?, we minimize the Kullback-Leibler divergence between p and q ? r? : Z Z p(x) min KL [p|q ? r? ] = min p(x) log dx = c ? max p(x) log r(x; ?)dx ? ? ? q(x)r(x; ?) np X 1 log r?(x(i) ? c ? max p ; ?) ? np i=1 (6) where c is a constant irrelevant to ?. It can be seen that the minimization of KL divergence boils down to maximizing log likelihood ratio over dataset Xp . Now we have reached the log-linear Kullback-Leibler Importance Estimation Procedure (log-linear KLIEP) estimator [30, 14]. 3.2 Trimmed Maximum Likelihood Ratio As stated in Section 1, to rule out the influences of large density ratio, we trim samples with large likelihood ratio values from (6). Similarly to one-SVM in (2), we can consider a trimmed MLE Pnp (i) ?? = arg max? i=1 [log r?(xp ; ?) ? t0 ]? where t0 is a threshold above which the likelihood ratios are ignored. It has a convex formulation: min h, 1i, s.t. ?x(i) ?(x(i) p ? Xp , log r p ; ?) ? t0 ? i . ?,?0 (7) (7) is similar to (2) since we have only replaced p(x; ?) with r?(x; ?). However, the ratio model ? while the normalization term Z in p(x; ?) r?(x; ?) in (7) comes with a tractable normalization term N is in general intractable. Similar to (3), we can directly control the trimming quantile via a hyper-parameter ?: 1 h, 1i ? ? ? t + ?R(?), s.t. ?xp(i) ?Xp , log r?(x(i) p ; ?) ? t ? i ?,?0,t?0 np min (8) where R(?) is a convex regularizer. (8) is also convex, but it has np number of non-linear constraints and the search for the global optimal solution can be time-consuming. To avoid such a problem, one could derive and solve the dual problem of (8). In some applications, we rely on the primal parameter structure (such as sparsity) for model interpretation, and feature engineering. In Section 4, we translate (8) into an equivalent form so that its solution is obtained via a subgradient ascent method which is guaranteed to converge to the global optimum. One common way to construct a convex robust estimator is using a Huber loss [12]. Although the proposed trimming technique rises from a different setting, it shares the same guiding principle with Huber loss: avoid assigning dominating values to outlier likelihoods in the objective function. In Section 8.1 in the supplementary material, we show the relationship between trimmed DRE and binary Support Vector Machines [23, 4]. 4 Optimization The key to solving (8) efficiently is reformulating it into an equivalent max min problem. Proposition 1. Assuming ? is chosen such that t? > 0 for all optimal solutions in (8), then ?? is an optimal solution of (8) if and only if it is also the optimal solution of the following max min problem: max ? min h inp w? 0, n1p ,h1,wi=? L(?, w) ? ?R(?), L(?, w) := np X wi ? log r?(x(i) p ; ?). (9) i=1 ? w) ? as a saddle point of (9): The proof is in Section 8.2 in the supplementary material. We define (?, ? w) ? = 0, w ? ? ?? ?R(?) ? ? arg ?? L(?, min w?[0, n1p ]np ,hw,1i=? where the second ?? means the subgradient if R is sub-differentiable. 4 ? w), L(?, (10) Algorithm 1 Gradient Ascent and Trimming max Input: Xp , Xq , ? and step sizes {?it }it it=1 ; Initialize ? 0 , w 0 , Iteration counter: it = 0, Maximum number of iterations: itmax , Best objective, parameter pair (Obest = ??, ? best , wbest ) . while not converged and n ito? itmax do (i) Obtain a sorted set xp 1 np , ?i np i=1 (1) (2) (np ) so that log r?(xp ; ? it ) ? log r?(xp ; ? it ) ? ? ? ? log r?(xp ; ? it ). wit+1,i = ? ?np . wit+1,i = 0, otherwise. Gradient ascent with respect to ?: ? it+1 = ? it + ?it ? ?? [L(? it , wit+1 ) ? ?R(? it )], Obest = max(Obest , L(? it+1 , wit+1 )) and update (? best , wbest ) accordingly. it = it + 1. end while Output: (? best , wbest ) Now the ?trimming? process of our estimator can be clearly seen from (9): The max procedure estimates a density ratio given the currently assigned weights w, and the min procedure trims the large log likelihood ratio values by assigning corresponding wi to 0 (or values smaller than n1p ). For simplicity, we only consider the cases where ? is a multiple of n1p . Intuitively, 1 ? ? is the proportion of likelihood ratios that are trimmed thus ? should not be greater than 1. Note if we set ? = 1, (9) is equivalent to the standard density ratio estimator (6). Downweighting outliers while estimating the model parameter ? is commonly used by robust estimators (See e.g., [3, 29]). ? w) ? is straightforward. It is easy to solve with respect to w or ? while the other The search for (?, is fixed: given a parameter ?, the optimization with respect to w is a linear programming and one of the extreme optimal solutions is attained by assigning weight n1p to the elements that correspond to the ?np -smallest log-likelihood ratio log r?(x(i) , ?). This observation leads to a simple ?gradient ascent and trimming? algorithm (see Algorithm 1). In Algorithm 1, ?? L(?, w) = np nq X 1 X e(j) (i) Pnq (k) f (x(j) := exp(h?, f (x(i) wi ? f (x(i) ) ? ? ? q ), e q )i). p np i=1 e k=1 j=1 In fact, Algorithm 1 is a subgradient method [2, 16], since the optimal value function of the inner problem of (9) is not differentiable at some ? where the inner problem has multiple optimal solutions. The subdifferential of the optimal value of the inner problem with respect to ? can be a set but Algorithm 1 only computes a subgradient obtained using the extreme point solution wit+1 of the inner linear programming. Under mild conditions, this subgradient ascent approach will converge to optimal results with diminishing step size rule and it ? ?. See [2] for details. Algorithm 1 is a simple gradient ascent procedure and can be implemented by deep learning softwares such as Tensorflow2 which benefits from the GPU acceleration. In contrast, the original problem (8), due to its heavily constrained nature, cannot be easily programmed using such a framework. 5 Estimation Consistency in High-dimensional Settings In this section, we show how the estimated parameter ?? in (10) converges to the ?optimal parameters? ? ? as both sample size and dimensionality goes to infinity under the ?outlier? and ?truncation? setting respectively. In the outlier setting (Figure 1a), we assume Xp is contaminated by outliers and all ?inlier? samples in Xp are i.i.d.. The outliers are injected into our dataset Xp after looking at our inliers. For example, hackers can spy on our data and inject fake samples so that our estimator exaggerates the degree of change. In the truncation setting, there are no outliers. Xp and Xq are i.i.d. samples from P and Q respectively. However, we have a subset of ?volatile? samples in Xp (the rightmost mode on histogram in Figure 1b) that are pathological and exhibit large density ratio values. 2 https://www.tensorflow.org/ 5 (a) Outlier Setting. Blue and red points are i.i.d. (b) Truncation Setting. There are no outliers. Figure 1: Two settings of theoretical analysis. In the theoretical results in this section, we focus on analyzing the performance of our estimator for high-dimensional data assuming the number of non-zero elements in the optimal ? ? is k and ? The proofs rely on a recent use the `1 regularizer, i.e., R(?) = k?k1 which induces sparsity on ?. development [35, 34] where a ?weighted? high-dimensional estimator was studied. We also assume the optimization of ? in (9) was conducted within an `1 ball of width ?, i.e., Ball(?), and ? is wisely chosen so that the optimal parameter ? ? ? Ball(?). The same technique was used in previous works [15, 35]. Notations: We denote w? ? Rnp as the ?optimal? weights depending on ? ? and our data. To lighten the notation, we shorten the log density ratio model as z? (x) := log r(x; ?), z?? (x) := log r?(x; ?) The proof of Theorem 1, 2 and 3 can be found in Section 8.4, 8.5 and 8.6 in supplementary materials. 5.1 A Base Theorem Now we provide a base theorem giving an upperbound of k?? ? ? ? k. We state this theorem only with respect to an arbitrary pair (? ? , w? ) and the pair is set properly later in Section 5.2 and 5.3. We make a few regularity conditions on samples from Q. Samples of Xq should be well behaved in terms of log-likelihood ratio values. Assumption 1. ?0 < c1 < 1, 1 < c2 < ? ?xq ? Xq , u ? Ball(?), c1 ? exph? ? + u, xq i ? c2 and collectively c2 /c1 = Cr . We also assume the Restricted Strong Convexity (RSC) condition on the covariance of X q , i.e., cov(X q ) = n1q (X q ? n1q X q 1)(X q ? n1q X q 1)> . Note this property has been verified for various different design matrices X q , such as Gaussian or sub-Gaussian (See, e.g., [21, 22]). Assumption 2. RSC condition of cov(X q ) holds for all u, i.e., there exists ?01 > 0 and c > 0 such that u> cov(X q )u ? ?01 kuk2 ? ?cnq kuk21 with high probability. Theorem 1. In addition to Assumption 1 and 2, there exists coherence between parameter w and ? ? w): ? at a saddle point (?, ? w) ? w? ), u ? ? ?? L(?, ? i ? ??2 k? h?? L(?, uk2 ? ?2 (n, d)k? uk1 , (11) ? := ?? ? ? ? , ?2 > 0 is a constant and ?2 (d, n) > 0. It can be shown that if where u h i ?n ? 2 max k?? L(? ? , w? )k? , 2C??c 2 ?n , ?2 (n, d) q r ??01 ? and k?? ? ? > 2Cr2 ?2 , where c > 0 is a constant determined by RSC condition, ? Cr2 3 k?n k ? (??0 ?2C with probability converging to one. ? 2 2 r ?2 ) 1 we are guaranteed that ? for w? , the change of the gradient ?? L is limited. The condition (11) states that if we swap w Intuitively, it shows that our estimator (9) is not ?picky? on w: even if we cannot have the optimal ? to compute the gradient which is weight assignment w? , we can still use ?the next best thing?, w close enough. We later show how (11) is satisfied. Note if k?? L(? ? , w? )k? , ?2 (n, d) converge to ? In Section zero as np , nq , d ? ?, by taking ?n as such, Theorem 1 guarantees the consistency of ?. ? ? 5.2 and 5.3, we explore two different settings of (? , w? ) that make ||?? ? ? k converges to zero. 6 5.2 Consistency under Outlier Setting Setting: Suppose dataset Xp is the union of two disjoint sets G (Good points) and B (Bad points) i.i.d. (j) (i) i.i.d. such that G ? p(x) and minj?B z?? (xp ) > maxi?G z?? (xp ) (see Figure 1a). Dataset Xq ? ? q(x) does not contain any outlier. We set ? = |G| np . The optimal parameter ? is set such that p(x) = q(x)r(x; ? ? ). We set w?i = (i) 1 np , ?xp ?G and 0 otherwise. Remark: Knowing the inlier proportion |G|/np is a strong assumption. However it is only imposed for theoretical analysis. As we show in Section 6, our method works well even if ? is only a rough guess (like 90%). Loosening this assumption will be an important future work. Assumption 3. ?u ? Ball(?), supx |? z ?? +u (x) ? z??? (x)| ? Clip kuk1 . This assumption says that the log density ratio model is Lipschitz continuous around its optimal parameter ? ? and hence there is a limit how much a log ratio model can deviate from the optimal model under a small perturbation u. As our estimated weights w ?i depends on the relative ranking of (i) z??? (xp ), this assumption implies that the relative ranking between two points will remain unchanged under a small perturbation u if they are far apart. The following theorem shows that if we have enough clearance between ?good?and ?bad samples?, ?? converges to the optimal parameter ? ? . Theorem 2. In addition to Assumption 1, 2 and a few mild technical conditions (see Section 8.5 in the (j) (i) ? ? supplementary material), Assumptions 3 holds. Suppose q minj?B z? (x p ) ? maxi?G z? (xp ) ? 3Clip ?, ? = |G| np , nq constants, we are guaranteed that ||?? ? ? ? k ? It can be seen that k?? ? ? ? k = O 5.3 K1 log d ??c ? |G| , 2Cr2 nq = ?(|G|2 ). If ?n ? 2 ? max p Cr2 ??01 , where K1 > 0, c > 0 are ? ? 3 k?n with probability converging to 1.  log d/min(|G|, nq ) if d is reasonably large. Consistency under Truncation Setting In this setting, we do not assume there are outliers in the observed data. Instead, we examine the ability of our estimator recovering the density ratio up to a certain quantile of our data. This ability is especially useful when the behavior of the tail quantile is more volatile and makes the standard estimator (6) output unpredictable results. Notations: Given ? ? (0, 1], we call t? (?) is the ?-th quantile of z? if P [z? < t? (?))] ? ? and P [z? ? t? (?))] ? ?. In this setting, we consider ? is fixed by a user thuswe drop the subscript ? from all subsequent discussions. Let?s define a truncated domain: X(?) = x ? Rd |z? (x) < t(?) , p q X (?) = Xp ? X(?) and X (?) = Xq ? X(?). See Figure 1b for a visualization of t(?) and X(?) (the dark shaded region). i.i.d. i.i.d. Setting: Suppose dataset Xp ? P and Xq ? Q. Truncated densities p? and q ? are the unbounded densities p and q restricted only on the truncated domain X(?). Note that the truncated densities are dependent on the parameter ? and ?. We show that under some assumptions, the parameter ?? obtained from (9) using a fixed hyperparameter ? will converge to the ? ? such that (i) q ?? (x)r(x; ? ? ) = p?? (x). We also define the ?optimal? weight assignment wi? = n1p , ?i, xp ? X(? ? ) and 0 otherwise. Interestingly, the constraint in (9), hw? , 1i = ? may not hold, but our ? w) ? in the feasible region so that analysis in this section suggests we can always find a pair (?, ? ? k? ? ? k converges to 0 under mild conditions. We first assume the log density ratio model and its CDF is Lipschitz continuous. Assumption 4. ?u ? Ball(?), sup |? z ?? +u (x) ? z??? (x)| ? Clip kuk. x 7 (12)  Define T (u, ) := x ? Rd | |z?? (x) ? t(? ? )| ? 2Clip kuk +  where 0 <  ? 1. We assume ?u ? Ball(?), 0 <  ? 1 P [xp ? T (u, )] ? CCDF ? kuk + . In this assumption, we define a ?zone? T (u, ) near the ?-th quantile t(? ? ) and assume the CDF of our ratio model is upper-bounded over this region. Different from Assumption 3, the RHS of (12) is with respect to `2 norm of u. In the following assumption, we assume regularity on P and Q. Assumption 5. ?xq ? Rd , kf (xq )k? ? Cq and ?u ? Ball(?), ?xp ? T (u, 1), kf (xp )k? ? Cp . (see Section 8.6 in the Theorem 3. In addition Assumption 1 and 2 and other mild assumptions ? 8CCDF kCp Cr2 supplementary material), Assumption 4 and 5 hold. If 1 ? ? ? , nq = ?(|X p (? ? )|2 ), ?01 i hq 0 q K1 log d 2Cr2 Cq |Xq \X (? ? )| 2L?Cp ??c ? , , , ?n ? 2 max + 2 ?n ? p n n 2C q p q |X (? )| r ? 4C 2 where K10 > 0, c > 0 are constants, we are guaranteed that ||?? ? ? ? k ? ??0r ? 3 k?n with high 1 probability. q  p It can be seen that k?? ? ? ? k = O log d/min(|X (? ? )|, nq ) if d is reasonably large and q |Xq \X (? ? )|/nq decays fast. 6 Experiments 6.1 Detecting Sparse Structural Changes between Two Markov Networks (MNs) [14] In the first experiment3 , we learn changes between two Gaussian MNs under the outlier setting. The P ratio between two Gaussian MNs can be parametrized as p(x)/q(x) ? exp(? i,j?d ?i,j xi xj ), where ?i,j := ?pi,j ? ?qi,j is the difference between precision matrices. We generate 500 samples as Xp and Xq using two randomly structured Gaussian MNs. One point [10, . . . , 10] is added as an Pd outlier to Xp . To induce sparsity, we set R(?) = i,j=1,i?j |?i,j | and fix ? = 0.0938. Then run DRE and TRimmed-DRE to learn the sparse differential precision matrix ? and results are plotted on Figure 2a and 2b4 where the ground truth (the position i, j, ??i,j 6= 0) is marked by red boxes. It can be seen that the outlier completely misleads DRE while TR-DRE performs reasonably well. We also run experiments with two different settings (d = 25, d = 36) and plot True Negative Rate (TNR) - True Positive Rate (TPR) curves. We fix ? in TR-DRE to 90% and compare the performance of DRE and TR-DRE using DRE without any outliers as gold standard (see Figure 2c). It can be seen that the added outlier makes the DRE fail completely while TR-DRE can almost reach the gold standard. It also shows the price we pay: TR-DRE does lose some power for discarding samples. However, the loss of performance is still acceptable. 6.2 Relative Novelty Detection from Images In the second experiment, we collect four images (see Figure 3a) containing three objects with a textured background: a pencil, an earphone and an earphone case. We create data points from these four images using sliding windows of 48 ? 48 pixels (the green box on the lower right picture on Figure 3a). We extract 899 features using MATLAB HOG method on each window and construct an 899-dimensional sample. Although our theorems in Section 5 are proved for linear models, here f (x) is an RBF kernel using all samples in Xp as kernel basis. We pick the top left image as Xp and using all three other images as Xq , then run TR-DRE, THresholded-DRE [26], and one-SVM. In this task, we select high density ratio super pixels on image Xp . It can be expected that the super pixels containing the pencil will exhibit high density ratio values as they did not appear in the reference dataset Xq while super pixels containing the earphone case, the earphones and the background, repeats similar patches in Xq will have lower density ratio values. This is different from 3 4 Code can be found at http://allmodelsarewrong.org/software.html Figures are best viewed in color. 8 1 0.8 TNR 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 TPR ? obtained by DRE, d = 20, with(b) ? ? obtained by TR-DRE, ? = (a) ? one outlier. 90%, with one outlier. (c) TNR-TPR plot, ? = 90% Figure 2: Using DRE to learn changes between two MNs. We set R(?) = k ? k1 and f (xi , xj ) = xi xj . (a) Dataset (b) ? = 97% (c) ? = 90% (d) ? = 85% (e) TH-DRE (f) one-SVM 2 Figure 3: Relative object detection using super pixels. We set R(?) = k ? k , f (x) is an RBF kernel. a conventional novelty detection, as a density ratio function help us capture only the relative novelty. For TR-DRE, we use the trimming threshold t? as the threshold for selecting high density ratio points. It can be seen on Figure 3b, 3c and 3d, as we tune ? to allow more and more high density ratio windows to be selected, more relative novelties are detected: First the pen, then the case, and finally the earphones, as the lack of appearance in the reference dataset Xq elevates the density ratio value by different degrees. In comparison, we run TH-DRE with top 3% highest density ratio values thresholded, which corresponds to ? = 97% in our method. The pattern of the thresholded windows (shaded in red) in Figure 3e is similar to Figure 3b though some parts of the case are mistakenly shaded. Finally, one-SVM with 3% support vectors (see Figure 3f) does not utilize the knowledge of a reference dataset Xq and labels all salient objects in Xp as they corresponds to the ?outliers? in Xp . 7 Conclusion We presents a robust density ratio estimator based on the idea of trimmed MLE. It has a convex formulation and the optimization can be easily conducted using a subgradient ascent method. We also investigate its theoretical property through an equivalent weighted M-estimator whose `2 estimation error bound was provable under two high-dimensional, robust settings. Experiments confirm the effectiveness and robustness of the our trimmed estimator. Acknowledgments We thank three anonymous reviewers for their detailed and helpful comments. Akiko Takeda thanks Grant-in-Aid for Scientific Research (C), 15K00031. Taiji Suzuki was partially supported by MEXT KAKENHI (25730013, 25120012, 26280009 and 15H05707), JST-PRESTO and JST-CREST. Song Liu and Kenji Fukumizu have been supported in part by MEXT Grant-in-Aid for Scientific Research on Innovative Areas (25120012). 9 References [1] F. Azmandian, J. G. Dy, J. A. Aslam, and D. R. Kaeli. Local kernel density ratio-based feature selection for outlier detection. In Proceedings of 8th Asian Conference on Machine Learning (ACML2012), JMLR Workshop and Conference Proceedings, pages 49?64, 2012. [2] S. Boyd. Subgradient methods. Technical report, Stanford University, 2014. Notes for EE364b, Stanford University, Spring 2013?14. [3] W. S. Cleveland. Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 74(368):829?836, 1979. [4] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, 2000. [5] B. Efron and R. Tibshirani. Using specially designed exponential families for density estimation. The Annals of Statistics, 24(6):2431?2461, 1996. [6] F. Fazayeli and A. Banerjee. Generalized direct change estimation in ising model structure. In Proceedings of The 33rd International Conference on Machine Learning (ICML2016), page 2281?2290, 2016. [7] W. Fithian and S. Wager. 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C. Lozano. Robust gaussian graphical modeling with the trimmed graphical lasso. In Advances in Neural Information Processing Systems, pages 2602?2610, 2015. 11
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Training recurrent networks to generate hypotheses about how the brain solves hard navigation problems Ingmar Kanitscheider & Ila Fiete Department of Neuroscience The University of Texas Austin, TX 78712 ikanitscheider, ilafiete @mail.clm.utexas.edu Abstract Self-localization during navigation with noisy sensors in an ambiguous world is computationally challenging, yet animals and humans excel at it. In robotics, Simultaneous Location and Mapping (SLAM) algorithms solve this problem through joint sequential probabilistic inference of their own coordinates and those of external spatial landmarks. We generate the first neural solution to the SLAM problem by training recurrent LSTM networks to perform a set of hard 2D navigation tasks that require generalization to completely novel trajectories and environments. Our goal is to make sense of how the diverse phenomenology in the brain?s spatial navigation circuits is related to their function. We show that the hidden unit representations exhibit several key properties of hippocampal place cells, including stable tuning curves that remap between environments. Our result is also a proof of concept for end-to-end-learning of a SLAM algorithm using recurrent networks, and a demonstration of why this approach may have some advantages for robotic SLAM. 1 Introduction Sensory noise and ambiguous spatial cues make self-localization during navigation computationally challenging. Errors in self-motion estimation cause rapid deterioration in localization performance, if localization is based simply on path integration (PI), the integration of self-motion signals. Spatial features in the world are often spatially extended (e.g. walls) or similar landmarks are found at multiple locations, and thus provide only partial position information. Worse, localizing in novel environments requires solving a chicken-or-egg problem: Since landmarks are not yet associated with coordinates, agents must learn landmark positions from PI (known as mapping), but PI location estimates drift rapidly and require correction from landmark coordinates. Despite the computational difficulties, animals exhibit stable neural tuning in familiar and novel environments over several 10s of minutes [1, 2], even though the PI estimates in the same animals is estimated to deteriorate within a few minutes [3]. These experimental and computational findings suggest that the brain is solving some version of the simultaneous localization and mapping (SLAM) problem. In robotics, the SLAM problem is solved by algorithms that approximate Bayes-optimal sequential probabilistic inference: at each step, a probability distribution over possible current locations and over the locations of all the landmarks is updated based on noisy motion and noisy, ambiguous landmark inputs [4]. These algorithms simultaneously update location and map estimates, effectively bootstrapping their way to better estimates of both. Quantitative studies of neural responses in rodents suggest that their brains might also perform high-quality sequential probabilistic fusion of motion and landmark cues during navigation [3]. The required probabilistic computations are difficult to 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. translate by hand into forms amenable to neural circuit dynamics, and it is entirely unknown how the brain might perform them. We ask here how the brain might solve the SLAM problem. Instead of imposing heavy prior assumptions on the form a neural solution might take, we espouse a relatively model-free approach [5, 6, 7]: supervised training of recurrent neural networks to solve spatial localization in familiar and novel environments. A recurrent architecture is necessary because self-localization from motion inputs and different landmark encounters involves integration over time, which requires memory. We expect that the network will form representations of the latent variables essential to solving the task . Unlike robotic SLAM algorithms that simultaneously acquire a representation of the agent?s location and a detailed metric map of a novel environment, we primarily train the network to perform accurate localization; the map representation is only explicitly probed by asking the network to extract features to correctly classify the environment it is currently in. However, even if the goal is to merely localize in one of several environments, the network must have created and used a map of the environment to enable accurate localization with noisy PI. In turn, an algorithm that successfully solves the problem of accurate localization in novel environments can automatically solve the SLAM problem, as mapping a space then simply involves assigning correct coordinates to landmarks, walls, and other features in the space [4]. Our network solution exploits the fact that the SLAM problem can be considered as one of mapping sequences of ambiguous motion and landmark observations to locations, in a way that generalizes across trajectories and environments. Our goal is to better understand how the brain solves such problems, by relating emergent responses in the trained network to those observed in the brain, and through this process to synthesize, from a function-driven perspective, the large body of phenomenology on the brain?s spatial navigation circuits. Because we have access to all hidden units and control over test environments and trajectories, this approach allows us to predict the effective dimensionality of the dynamics required to solve the 2D SLAM task and make novel predictions about the representations the brain might construct to solve hard inference problems. Even from the perspective of well-studied robotic SLAM, this approach could allow for the learning and use of rich environment structure priors from past experience, which can enable faster map building in novel environments. 2 2.1 Methods Environments and trajectories We study the task of a simulated rat that must estimate its position (i.e., localize itself) while moving along a random trajectory in two-dimensional enclosure, similar to a typical task in which rats chase randomly scattered food pellets [8]. The enclosure is polygon-shaped and the rat does not have access to any local or distal spatial cues other than touch-based information upon contact with the boundaries of the environment (Figure 1A-B; for details see SI Text, section 1-4). We assume that the rat has access to noisy estimates of self-motion speed and direction, as might be derived from proprioceptive and vestibular cues (Figure 1A), and to boundary-contact information derived from its rare encounters with a boundary whose only feature is its geometry. On boundary contact, the rat receives information only about its distance and angle relative to the boundary (Figure 1B). This information is degenerate: it depends simply on the pose of the rat with respect to the boundary, and the same signal could arise at various locations along the boundary. Self-motion and boundary contact estimates are realistically noisy, with magnitudes based on work in [3]. 2.2 Navigation tasks We study the following navigation tasks: ? Localization only: Localization in a single familiar environment. The rat is familiar with the geometry of the environment but starts each trial at a random unknown location. To successfully solve the task, the rat must infer its location relative to a fixed point in the interior on the basis of successive boundary contacts and its knowledge of the environment?s geometry, and be able to generalize this computation across novel random trajectories. ? Generalized SLAM: Localization in novel environments. Each trial takes place in a novel environment, sampled from a distribution of random polygons (Figure 1C; SI Text, section 2 A B C D localization or classification ? ? recurrent layer (LSTM) input: motion and boundary Figure 1: Task setup. Self-localization in 2D enclosures. A Noisy heading direction and speed inputs allow the simulated rat to update its location in the interior. B Occasional boundary contacts provide noisy estimates of the its relative angle (?) and distance (d) from the wall. C Samples from the distribution of random environments. D Architecture of the recurrent neural network. 1); the rat must accurately infer its location relative to the starting point by exploiting boundary inputs despite not knowing the geometry of its enclosure. To solve the task, the rat must be able to generalize its localization computations to trials with both novel trajectories and novel environments. ? Specialized task: Localization in and classification of any of 100 familiar environments. Each trial takes place in one of 100 known environments, sampled from a distribution of random polygons (Figure 1C; SI Text, section 1), but the rat does not know which one. The trial starts at a fixed point inside the polygon (known to rat through training), and the ongoing trajectory is random. In addition to the challenges of the localization tasks above, the rat must correctly classify the environment. The environments are random polygons with 10 vertices. The center-to vertex lengths are drawn randomly from a distribution with mean 1m in the localization-only task or 0.33m in the specialized and generalized SLAM tasks. 2.3 Recurrent network architecture and training The network has three layers: input, recurrent hidden and output layer (Figure 1D). The input layer encodes noisy self-motion cues like velocity and head direction change, as well as noisy boundarycontact information like relative angle and distance to boundary (SI Text, section 9). The recurrent layer contains 256 Long Short-Term Memory (LSTM) units with peepholes and forget gates [9], an architecture demonstrated to be able to learn dependencies across many timesteps [10]. We adapt the nonlinearity of the LSTM units to produce non-negative hidden activations in order to facilitate the comparison with neural firing rates1 . Two self-localization units in the output perform a linear readout; their activations correspond to the estimated location coordinates. The cost function for localization is mean squared error. The classification output is implemented by a softmax layer with 100 neurons (1 per environment); the cost function is cross-entropy. When the network is trained to both localize and classify, the relative weight is tuned such that the classification cost is half of the localization cost. Independent trials used for training: 5000 trials in the localization-only task, 250,000 trials in the specialized task, and 300,000 trials in the generalized task. The network is trained using the Adam algorithm [11], a form of stochastic gradient descent. Gradients are clipped to 1. During training performance is monitored on a validation set of 1000 independent trials, and 1 The LSTM equations are implemented by the equations: it ft ct ot ht = = = = = ?(Wxi xt + Whi ht?1 + wci ct?1 + bi ) ?(Wxf xt + Whf ht?1 + wcf ct?1 + bf ) ft ct?1 + it tanh(Wxc xt + Whc ht?1 + bc ) ?(Wxo xt + Who ht?1 + wco ct + bo ) ot tanh([ct ]+ ) where ? is the logistic sigmoid function, h is the hidden activation vector, i, f , o and c are respectively the input gate, forget gate, output gate and cell activation vectors, a b denotes point-wise multiplication and [x]+ denotes rectification. 3 network parameters with the smallest validation error are selected. All results are cross-validated on a separate set of 1000 test trials to ensure the network indeed generalizes across new random trajectories and/or environments. 3 Results 3.1 Network performance on spatial tasks rivals optimal performance 3.1.1 Localization in a familiar environment The trained network, starting a trial from an unknown random initial position and running along a new random trajectory, quickly localizes itself within the space (Figure 2, red curve). The mean location error (averaged over new test trials) drops as a function of time in each trial, as the rat encounters more boundaries in the environment. After about 5 boundary contacts, the initial error has sharply declined. 0.5 0 0 500 1000 timestep 1500 0 0 PF ang error [rads] 1 2 1 NN PF SH PI radial error [m] mean abs err [m] 1.5 1 500 1000 1500 timestep 0 0 SH 500 1000 timestep 1500 Figure 2: Localization in a single familiar environment. Mean absolute error on the localization-only task (left), radial error measured from origin (middle) and angular error (right). One time step corresponds to 0.77 seconds. Network performance (red, NN) is compared to that of the particle filter (black, PF). Also shown: single hypothesis filter (light red, SH) and simple path integration (gray, PI) estimates as controls. The drop in error over time and the final error of the network match that of the optimal Bayesian estimator with access to the same noisy sensory data but perfect knowledge of the boundary coordinates (Figure 2, black). The optimal Bayesian estimator is implemented as a particle filter (PF) with 1000 particles and performs fully probabilistic sequential inference about position, using the environment coordinates and the noisy sensory data. The posterior location distributions are frequently elongated in an angular arc and multimodal (thus far from Gaussian). Both network and PF vastly outperform pure PI. First, since the PI estimate does not have access to boundary information, it cannot overcome initial localization uncertainty due to the unknown starting point. Second, the error in the PI estimate of location grows unbounded with time, as expected due to the accumulating effects of noise in the motion estimates (Figure 2, gray). In contrast, the errors in the network and PF ? which make use of the same motion estimates ? remain bounded. Finally we contrast the performance of the network and PF with the single hypothesis (SH) algorithm, which updates a single location estimate (rather than a probability distribution) by taking into account motion, contact, and arena shape. The SH algorithm can be thought of as an abstraction of neural bump attractor models [12, 13], in which an activity bump is updated using PI and corrected when a landmark or boundary with known spatial coordinates is observed. The SH algorithm overcomes, to a certain degree, the initial localization uncertainty due to the unknown starting position, but the error steadily increases thereafter. It still vastly underperforms the network and PF, since it is not able to efficiently resolve the complex-shaped uncertainties induced by featureless boundaries. 3.1.2 Localization in novel environments The network is trained to localize within a different environment in each trial, then tested on a set of trials in different novel environments. 4 Strikingly, the network localizes well in the novel environments, despite its ignorance about their specific geometry (Figure 3A, red). While the network (unsurprisingly) does not match the performance of an oracular PF that is supplied with the arena geometry at the beginning of the trial (Figure 3A, black), its error exceeds the oracular PF by only ? 50%, and it vastly outperforms PI-based estimation (Figure 3A, gray) and a naive Bayesian (NB) approach that takes into account the distribution of locations across the ensemble of environments (Figure 3A, reddish-gray; SI section 8). Compared to robotic SLAM in open-field environments, this task setting is especially difficult since distant boundary information is gathered only from sparse contacts, rather than spatially extended and continuous measurements with laser or radar scanners. 3.1.3 Localization in and classification of 100 familiar environments The network is trained on 100 environments then tested in an arbitrary environment from that set. The goal is to identify the environment and localize within it, from a known starting location. Localization initially deteriorates because of PI errors (Figure 3B, red). After a few boundary encounters, the network correctly identifies the environment (Figure 3C), and simultaneously, localization error drops as the network now associates the boundary with coordinates for the appropriate environment. The network?s localization error post-classification matches that of an oracular PF with full knowledge about the environment geometry. Within 200s of exploration within the environment, classification performance is close to 100%. As a measure of the efficacy of the neural network in solving the specialized task, we compare its performance to PFs that do not know the identity of the environment at the outset of the trial (PF SLAM) and that perform both localization and classification, with varying numbers of particles, Figure 3D-E. For classification, the asymptotic network performance with 256 recurrent units is comparable to a 10,000 particle PF SLAM, while for localization, the asymptotic network performance is comparable to a 4,000 particle PF SLAM, suggesting that the network is extremely efficient. Even the 10,000 particle PF SLAM classification estimate sometimes prematurely collapses to not always the correct value. The network is slower to select a classification, and is more accurate, improving on a common problem with particle-filter based SLAM caused by particle depletion. generalized network B 0.4 0 0.1 500 PF SLAM 1000 4000 10000 oracular PF F 0.05 0.2 0 D specialized network NN oracular PF NB PI 500 1000 timestep 0 100 0 1500 C 100 E 0 500 1000 timestep 1500 0 generalized network, specialized task class (%) 0.6 class (%) mean abs err [m] A 500 1000 timestep 0 0 500 1000 timestep 1500 1500 Figure 3: Localization and classification in the generalized and specialized SLAM tasks. A Localization performance of the generalized network (red, NN) tested in novel environments, compared to a PF that knows the environment identity (black, oracular PF). Controls: PI only (gray, PI) and a naive Bayes filter (see text and SI; reddish-gray, NB). B Same as (A), but for the specialized network tested in 100 familiar environments. C Classification performance of the specialized network in 100 familiar environments. D-E Localization and classification by a SLAM PF with different number of particles, compared to the specialized network in 100 familiar environments. F Classification performance of the general network after retraining of the readout weights on the specialized task. 3.1.4 Spontaneous classification of novel environments In robotic SLAM, algorithms that self-localize accurately in novel environments in the presence of noise must simultaneously build a map of the environments. Since the network in the general task in Figure 3A successfully localizes in novel environments, we conjecture that it must entertain a 5 spontaneous representation of the environment, even though the environments are quite similar to each other. To test this hypothesis we fix the input and recurrent weights of the network trained on the generalized task (completely novel environments) and retrain it on the specialized task (one out of hundred familiar environments), whereby only the readout weights are trained for classification. We find that the classification performance late in each trial is close to 80%, much higher than chance (1%), Figure 3F. This implies that the hidden neurons spontaneously build a representation that separates novel environments so they can be linearly classified. This separation can be interpreted as a simple form of spontaneous map-building. However, this spontaneous map-building is done with fixed weights this is different than standard Hopfield-type network models that require synaptic plasticity to learn a new environment. 3.2 Comparison with and predictions for neural representation Neural activity in the hippocampus and entorhinal cortex ? areas involved in spatial navigation ? has been extensively catalogued, usually while animals chase randomly dropped food pellets in open field environments. It is not always clear what function the observed responses play in solving hard navigation problems, or why certain responses exist. Here we compare the responses of our network, which is trained to solve such tasks, with the experimental phenomenology. Hidden units in our network exhibit stable place tuning, similar to place cells in CA1/CA3 of the hippocampus [14, 15, 16], Figure 4A,B (left two columns). Stable place fields are observed across tasks ? the network trained to localize in a single familiar environment exhibits stable fields there, while the networks trained on the specialized and generalized tasks exhibit repeatedly stable fields in all tested environments. generalized network env A env A env B C D E 1 weight B 30 cumul freq 0 1 head direction frequency specialized network env A env A env B activity distribution A 0 0.05 0.1 0.15 normalized similarity 0.2 20 SS 10 0 0 0.5 1 spatial selectivity (SS) Figure 4: Neuron-like representations. A Spatial tuning of four typical hidden units from the specialized network, measured twice with different trajectories in the same environment (columns 1-2, blue box). The same cells are measured in a second environment (column 3, red box). B Same as A but for the generalized network; both environments were not in the training set. C Hidden units (representative sample of 20) are not tuned to head direction. D Cumulative distribution of similarity of hidden unit states in the specialized (top) and generalized (bottom) networks, for trials in the same environment (blue) versus trials in different environments (purple). Control: similarity after randomizing over environments (gray). E Spatial selectivities of hidden units in the specialized network. Inset: spatial selectivity (averaged across environments) versus effective projection strength to classifier neurons, per hidden unit. The hidden units, all of which receive head direction inputs and use this data to compute location estimates, nevertheless exhibit weak to nil head direction tuning, Figure 4C, again similar to observations in rodent place cells [17] (but see [18] for a report of head direction tuning in bat place cells). Between different environments, the network trained on the specialized task exhibits clear global remapping [19, 20]: cells fire in some environments and not others, and cells that were co-active in one environment are not in another, Figure 4A,B (third column). Strikingly, the network trained on the generalized task exhibits stable and reproducible maps of different novel environments with remapping, even though the input and recurrent connections were never readjusted for these novel environments, Figure 4B. The similarity and dissimilarity of the representations within the same environment and across environments, in the specialized and generalized tasks are quantified in Figure 4D: the representations are randomized across environments but stable within an environment. 6 For networks trained on the specialized or generalized tasks, the spatial selectivity of hidden units in an environment - measured as the fraction of the variance of each hidden neuron?s activation that can be explained by location - is broad and long-tailed or sparse, Figure 4E: a few cells exhibit high selectivity, many have low selectivity. Interestingly, cells with low spatial selectivity in one environment also tend to have low selectivity across environments (in other words, the distribution in selectivity per cell across environments is narrower than the distribution of selectivity across cells per environment). Indeed, spatial information in hippocampal neurons seems to be concentrated in a small set of neurons [21], an experimental observation that seemed to run counter to the informationtheoretic view that whitened representations are most efficient. However, our 256-neuron recurrent network, which efficiently solves a hard task that requires 104 particles, seems to do the same. There is a negative correlation between spatial selectivity and the strength of feedforward connections to the classification units: Hidden units that more strongly drive classification also tend to be less spatially selective, Figure 4E (inset). In other words, some low spatial selectivity cells correspond to what are termed context cells [22]. It remains unclear and the focus of future work to understand the role of the remaining cells with low spatial selectivity. Inner workings of the network r = 0.45 Cyy (PF) Cxx (PF) 0 0.1 r = 0.49 0 0.02 Cxx (net prediction) 0 0.05 C r = 0.45 15 5 -0.05 0 0.02 -0.01 0.01 Cyy (net prediction) Cxy (net prediction) B 0 10 8.6 s 10.1 s 30.3 s d = 5.6 +/- 0.03 0 15 10 5 3.9 s d = 5.0 +/- 0.04 10 log(#elements within) A 0.1 Cxy (PF) 3.3 d = 8.6 +/- 0.1 0 -1.5 -1 -0.5 0 log(radius) 0.5 Figure 5: Inner workings of the network A Hidden units in the localization-only network predict the covariances (Cxx , Cyy , Cxy ) of the posterior location (x, y) distributions in the particle filter. B Light red: snapshots of the narrowing set of potential environment classifications by the specialized neural network at different early times in a trajectory, as determined by the activation of classifier neurons in the output layer. C Dimensionality of the hidden representations: localization network (top), specialized network (middle), generalized network (bottom). Dimensionality estimated from across-environment pooled responses for the latter two networks. Beyond the similarities between representations in our hidden units and neural representations, what can we learn about how the network solves the SLAM problem? The performance of the network compared to the particle filter (and its superiority to simpler strategies used as controls) already implies that the network is performing sophisticated probabilistic computations about location. If it is indeed tracking probabilities, it should be possible to predict the uncertainties in location estimation from the hidden units. Indeed, all three covariance components related to the location estimate of the particle filter can be predicted by cross-validated linear regression from the hidden units in the localization-only network (Figure 5A). When first placed into one of 100 familiar environments, the specialized network simultaneously entertains multiple possibilities for environment identity, Figure 5B. The activations of neurons in the soft-max classification layer may be viewed as a posterior distribution over environment identity. 7 With continued exploration and boundary encounters, the represented possibilities shrink until the network has identified the correct environment. Unlike the particle filter and contrary to neural models that implement probabilistic inference by stochastic sampling of the underlying distribution [23], this network implements ongoing near-optimal probabilistic location estimation through a fully deterministic dynamics. Location in 2D spaces is a continuous 2D metric variable, so one might expect location representations to lie on a low-dimensional manifold. On the other hand, SLAM also involves the representation of landmark and boundary coordinates and the capability to classify environments, which may greatly expand the effective dimension of a system solving the problem. We analyze the fractal manifold dimension of the hidden layer activities in the three networks, Figure 5C2 . The localization-only network has a dimension D = 5.0. Surprisingly, the specialized network states (pooled across all 100 environments) are equally low-dimensional: D = 5.6. The generalized network states, pooled across environments, have dimension D = 8.6. (The dimensionality of activity in the latter two networks, considered in single environments only, remains the same as when pooled across environments.) This implies that the network extracts and representing only the most relevant summary statistics required to solve the 2D localization tasks, and that these statistics have fairly low dimension. These dimension estimates could serve as a prediction for hippocampal dynamics in the brain. 4 Discussion By training a recurrent network on a range of challenging navigation tasks, we have generated ? to our knowledge ? the first fully neural SLAM solution that is as effective as particle filter-based implementations. Existing neurally-inspired SLAM algorithms such as RatSLAM [24] have combined attractor models with semi-metric topological maps, but only the former was neurally implemented. [25] trained a bidirectional LSTM network to transform laser range sensor data into location estimates, but the network was not shown to generalize across environments. In contrast, our recurrent network implementation is fully neural and generalizes successfully across environments with very different shapes. (Note that since this paper was under review, a new paper implementing SLAM using a network with recurrent components has appeared [26].) Previous hand-designed models such as the multichart attractor model of Samsonovich & McNaughton [12] could path integrate and use landmark information to correct the network?s PI estimate in many different environments. Yet our model substantially transcends those computational capabilities: First, our model performs sequential probabilistic inference, not simply a hard resetting of the PI estimate according to external cues. Second, our network reliably localizes in 100 environments with 256 LSTM units (which corresponds to 512 dynamical units); the low capacity of the multichart attractor model would require about 175,000 neurons for the same number of environments. This comparison suggests that the special network architecture of the LSTM not only affects learnability, but also capacity. Finally, unlike the multichart attractor model, our model is able to linearly separate completely novel environments without changing its weights, as shown in section 3.1.4. Despite its success in reproducing key elements of the phenomenology of the hippocampus, our network model does not incorporate many biological constraints. This is in itself interesting, since it suggests that observed phenomena like stable place fields and remapping may emerge from the computational demands of hard navigation tasks rather than from detailed biological constraints. It will be interesting to see whether incorporating constraints like Dale?s law and the known gross architecture of the hippocampal circuit results in the emergence of additional features associated with the brain?s navigation circuits, such as sparse population activity, directionality in place representations in 1D environments, and grid cell-like responses. The choice of an LSTM architecture for the hidden layer units, involving multiplicative input, output and forget gates and persistent cells, was primarily motivated by its ability to learn long timedependencies. One might wonder whether such multiplicative interactions could be implemented in biological neurons. A model by [27] proposed that dendrites of granule cells in the dental gyrus contextually gate projections from grid cells in the entorhinal cortex to place cells. Similarly, granule 2 To estimate the fractal dimension, we use ?correlation dimension?: measure the number of states across trials that fall into a ball of radius r around a point in state space. The slope of log(#states) versus log(r) is the fractal dimension at that point. 8 cells could implement LSTM gates by modulating recurrent connections between pyramidal neurons in hippocampal area CA3. LSTM cells might be interpreted as neural activity or as synaptic weights updated by a form of synaptic plasticity. The learning of synaptic weights by gradient descent does not map well to biologically plausible synaptic plasticity rules, and such learning is slow, requiring a vast number of supervised training examples. Our present results offer a hint that, through extensive learning, the generalized network acquires useful general prior knowledge about the structure of natural navigation tasks, which it then uses to map and localize in novel environments with minimal further learning. One could thus argue that the slow phase of learning is evolutionary, while learning during a lifetime can be brief and driven by relatively little experience in new environments. At the same time, progress in biologically plausible learning may one day bridge the efficiency gap to gradient descent [28]. Finally, although our work is focused on understanding the phenomenology of navigation circuits in the brain, it might also be of some interest for robotic SLAM. 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Biologically Plausible Local Learning Rules for the Adaptation of the Vestibulo-Ocular Reflex Olivier Coenen* Terrence J. Sejnowski Computational Neurobiology Laboratory Howard Hughes Medical Institute The Salk Institute P.O.Box 85800 San Diego, CA 92186-5800 Stephen G. Lisberger Department of Physiology W.M. Keck Foundation Center for Integrative Neuroscience University of California, San Fransisco, CA, 94143 Abstract The vestibulo-ocular reflex (VOR) is a compensatory eye movement that stabilizes images on the retina during head turns. Its magnitude, or gain, can be modified by visual experience during head movements. Possible learning mechanisms for this adaptation have been explored in a model of the oculomotor system based on anatomical and physiological constraints. The local correlational learning rules in our model reproduce the adaptation and behavior of the VOR under certain parameter conditions. From these conditions, predictions for the time course of adaptation at the learning sites are made. 1 INTRODUCTION The primate oculomotor system is capable of maintaining the image of an object on the fovea even when the head and object are moving simultaneously. The vestibular organs provide information about the head velocity with a short delay of 14 ms but visual Signals from the moving object are relatively slow and can take 100 ms to affect eye movemen.ts. The gain, a, of the VOR, defined as minus the eye velocity over the head velocity (-if h), can be modified by wearing magnifying or diminishing glasses (figure 1). VOR adaptation, absent in the dark, is driven by the combination of image slip on the retina and head turns. ?University of California, San Diego. Dept. of Physics. La Jolla, CA, 92037. Email address: oli [email protected] 961 962 Coenen, Sejnowski, and Lisberger During head turns on the first day of wearing magnifying glasses, the magnified image of an object slips on the retina. After a few days of adaptation, the eye velocity and hence the gain of the VOR increases to compensate for the image magnification. We have constructed a model of the VOR and smooth pursuit systems that uses biologically plausible local learning rules that are consistent with anatomical path ways and physiological recordings. The learning rules in the model are local in the sense that the adaptation of a synapse depends solely on signals that are locally available. A similar model with different local learning rules has been recently proposed (Quinn et at., Neuroscience 1992). xl .O xl.O Spectacles off Spectacles on 1B 9 1 I; , 1.4 1.2 z < <.:) ~ ~ ,, . o Gain = 1.01 + 0 .68(1 . e-<1020 Gain = 1.01 + 0.68 Ie t) l? , ~-------- 1.0 ? :: r~--:--:---:----:----- m._____ 0.6 [ + Gain" = O.~ + 027 Ie" -0 1J t) " II xO.5 SpectaCles Spectacles , o tl ? f xO.5 on " -- " -------? -0._ off , I ! ! I I ? _...L'_..L.'_.I..-' ,L-....L.---.J'_....l,_....L, 23456780 234567 TIME IDaysl Figure 1: Tune course of the adapting VOR and its recovery of gain in monkeys exposed to the longterm influence of magnifying (upper curves) and diminishing (lower curves) spectacles. Different symbols obtained from different animals, demonstrating the consistency of the adaptive change. From Melvill Jones (1991), selected from Miles and Eighmy (1980). 2 THEMODEL Feedforward and recurrent models of the VOR have been proposed (Fujita, 1982; Galiana, 1986; Kawato and Gomi, 1992; Quinn et al., 1992; Arnold and Robinson, 1992; Lisberger and Sejnowski, 1992). In this paper we study a static and linear version of a previously studied recurrent network model of the VOR and smooth pursuit system (Lisberger, 1992; Lisberger and Sejnowski, 1992; Viola, Lisberger and Sejnowski, 1992). The time delays and time constants associated with nodes in the network were eliminated so that the time course of the VOR plasticity could be more easily analyzed (figure 2). The model describes the system ipsilateral to one eye. The visual error, which carries the image retinal slip velocity signal, is a measure of the performance of both the VOR and smooth pursuit system as well as the main error signal for learning. The value at each node represents changes in its firing rate from its resting firing rate. The transformation from the rate of firing of premotor signal (N) to eye velocity is represented in the model by a gain Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex o: Visual error: mossy fibers Gains P : Purkinje Cell N : Vestibular Nucleus g : Desired gain ? h -(g Inhibitory h + e) e? eye velocity Visual error: climbing fibers Figure 2: Diagram of the VOR and smooth pursuit model. The input and output of the model are, respectively, head velocity and eye velocity. The model has three main parts: the node P represents an ensemble of Purkinje cells from the ventral paraflocculus of the cerebellum, the node N represents an ensemble of flocculus-target neurons in the vestibular nucleus, and the visual inputs which provide the visual error signals in the mossy and climbing fibers. The capital letter gains A and D, multiplying the input signals to the nodes, are modified according to their learning rules. The lower case letters b, v, and 9 are also multiplicative gains, but remain constant during adaptation. The traces represent head and eye velocity modulation in time. The visual error signal in the climbing fibers drives learning in node N but does not constitute one of its inputs in the present model. of -1. The gain of the VOR in this model is given by ~=:. We have not modeled the neural integrator that converts eye velocity commands to eye position signals that drive the motoneurons. 3 LEARNING RULES We have adopted the learning rules proposed by Marr (1969), Albus (1971) and Ito (1970) for adaptation in the cerebellum and by Lisberger (1988), Miles and Lisberger (1981) for plasticity in the brain stem (figure 3). These are variations of the delta rule and depend on an explicit representation of the error signal at the synapses. Long term depression at mossy fiber synapses on Purkinje cells has been observed in vitro under simultaneous stimulation of climbing fibers and mossy fibers (Ito, Sakurai and Tongroach, 1982). In addition, we have included a learning mechanism for potentiation of mossy fiber head velocity inputs under concurrent mossy fiber visual and head velocity inputs. Although the climbing fiber inputs to the cerebellum were not directly represented in this model (figure 2), the image velocity signal carried by the mossy fibers to P was used in the model to achieve the same result. There is good indirect evidence that learning also occurs in the vestibular nucleus. We have adopted the suggestion of Lisberger (1988) that the effectiveness of the head velocity input to some neurons in the vestibular nucleus may be modified by head velocity input in 963 964 Coenen, Sejnowski, and Lisberger ~ - Learning Rate x ( InPut) (Error) Signal x Signal Cerebellum (P): A qA X ( - qA x - qA X ex: h2 Head Velocity ) x ( Mossy fiber ) Visual signal h x -v(gh + e) h x -v[(g - D)h + P] Vestibular nucleus (N): b qD X ( Head ) (Climbing fiber x Visual signal Velocity x h x [(1 - q)(gh - qD - qD X oc h2 Purkinje Signal ) + e) - qP] h x [(1 - q)(g - D)h + (1 - 2q)P] where P A - bD - (g - D)v . h I-b+v Figure 3: Learning rules for the cerebellum and vestibular nucleus. The gains A and D change according to the correlation of their input signal and the error signal to the node, as shown for ~ at the top. The parameter q determines the proportion of learning from Purkinje cell inputs compared to learning from climbing fiber inputs. When q = I, only Purkinje cell inputs drive the adaptation at node N; if q = 0, learning occurs solely from climbing fiber inputs. association with Purkinje cells firing. We have also added adaptation from pairing the head velocity input with climbing fiber firing. The relative effectiveness of these two learning mechanisms is controlled by the parameter q (figure 3). Learning for gain D depends on the interplay between several signals. If the VOR gain is too small. a rightward head turn P (positive value for head velocity) results in too small a leftward eye turn (a negative value for eye velocity). Consequently, the visual scene appears to move to the left (negative image slip). P then fires below its resting level (negative) and its inhibitory influence on N decreases so that N increases its firing rate (figure 4 bottom left). This corrects the VOR gain and increases gain D according to figure 3. Concurrently, the climbing fiber visual signal is above resting firing rate (positive) which also leads to an increase in gain D. Since the signal passing through gain A has an inhibitory influence via Ponto N, decreasing gain A has the opposite effect on the eye velocity as decreasing gain D. Hence, if the VOR is too small we expect gain A to decrease. This is what happens during the early phase of learning (figure 4 top left). 4 RESULTS Finite difference equations of the learning rules were used to calculate changes in gains A and D at the end of each cycle during our simulations. A cycle was defined as one biphasic Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex Desired gain 9 = 1.6 Magnitude Magnitude 2 1.75 1.5 t.25 1 0.7!1 2 G / t.75 1.5 1.25 D 1 0.5 0.25 0.75 0.5 0.25 A 0 G D 20 40 60 10lime 80 0 A 2000 4000 6000 10(J;ime 8000 A, D & VOR gain G vs time Amplitude Amplitude 2 2 1.5 1.5 1 / N 0.5 ~.5 N 1 0.5 ~ 40 60 100 Time 80 P ?1 2000 4000 6000 ~.5 P ?1 P & N responses to a head turn during learning vs time Figure 4: Simulation of change in gain from 1.0 to 1.6. Top: Short-term (left) and long-term (right) adaptation of the gains A, D and G. Bottom: Changes on two time scales of P and N responses to a .!lA = 10. , head turn of amplitude 1 during learning. The parameters were v = 1.0, b .88, T f1D and q .01. = = = head velocity input as shown in figure 2. We assumed that the learning rates were so small that the changes in gains, and hence in the node responses, were negligibly small during each iteration. This allowed the replacement ~f A(t) .and D(t) by their values obtained on the previous iteration for the calculations of A and D. The period of the iteration as well as the amplitude of the head velocity input were chosen so that the integral of the head velocity squared over one iteration equaled l. For the simulations shown in figure 4 the gain G of the VOR increased monotonically from 1 to reach the desired value 1.6 within 60 time steps. This rapid adaptation was mainly due to a rapid decrease in A, as expected from the local learning rule (figure 3), since the learning rate 'f/A was greater than the learning rate 'f/D. Over a longer time period, learning was transferred from A to D: D increased from 1 to reach its final value 1.6 while the VOR gain stayed constant. Transfer of learning occurs when P fires in conjunction with a head turn. P can have an elevated firing rate even though the visual error signal is zero (that is, even if the VOR gain G has reached the desired gain g) because of the difference between its two other inputs: the head velocity input through A and the eye velocity feedback input through b. It is only when these two inputs become equal in.amplit~lde that P firing goes to zero. It can be shown that when learning settles (when D and A equal zero) D = g, A = bg, and P = O. With these values for A and D, the two other inputs to P are indeed equal in amplitude: one equals Ah, while the other equals b( -1 )Dh. During the later part of learning, gain A is driven in the opposite direction (increase) than during the earlier 965 966 Coenen, Sejnowski, and Lisberger part (decrease). This comes from a sign reversal of the visual error input to P. After the first 60 time steps, the gain has reached the desired gain due to a rapid decrease in A, this means that any subsequent increase in D, due to transfer of learning as explained above, will cause the gain of the VOR G to become larger than the desired gain g, hence the visual error changes sign. In order to compensate for this small error, gain A increases promptly, keeping G very close to the desired gain. This process goes on until A and D reach their equilibrium values stated above. The short and long-term changes in P and N responses to a velocity step are also shown. As the firing of P decreased with the adaptation of A, the firing rate of N increased to the right level. 5 OVERSHOOT OF THE VOR GAIN G In this section we show that for some ranges of the learning parameters, the gain G in the model overshoots the desired value g. Since an overshoot is not observed in animals (figure I), this provides constraints on the parameters. The parameter q in the learning rule for the vestibular nucleus (node N, gain D), determines the proportion of learning from Purkinje cell inputs compared to learning from climbing fiber inputs. When q = 1, only Purkinje cell inputs drive the adaptation at node N; if q = 0, learning at N occurs solely from climbing fiber inputs. These two inputs have quite different effects on learning as shown in figure 5. Asymptotically, P goes to 0, and D goes to 9 if q = 1; and P can only 0. The gain has an overshoot for any value of q different than 0, as differ from if q shown in figure 6. Nevertheless, its amplitude is only significant for a limited extent in the parameter space of q and r (graph of figure 6). The overshoot is reduced with a smaller q and a larger r. One possibility is that q is chosen close to and r > I, that is TJA > 7JD. These conditions were used to choose parameter values in the simulations (figure 4). ? = ? 6 DISCUSSION AND CONCLUSION The VOR model analyzed here is a static model without time delays and multiple time scales. We are currently studying how these factors affect the time course of learning in a dynamical model of the VOR and smooth pursuit. In our model, learning occurs in the dark if P #- 0, which has not been observed in animals. One way to avoid learning in the dark when P is firing would be to gate the learning by a visual input, such as that provided by climbing fibers. The responses of vestibular afferents to head motion can be classified into two categories: phase-tonic and tonic. In this model, only the tonic afferents were represented. Both afferent types encode head velocity, while the phasic-tonic responds to head acceleration as well. The steady state VOR gain can also be changed by altering the relative proportions of phasic and tonic afferents to the Purkinje cells (Lisberger and Sejnowski, 1992). We are currently investigating learning rules for which this occurs. The model predicts that adaptation in the cerebellum is faster than in the vestibular nucleus, and that learning in the vestibular nucleus is mostly driven by the climbing fiber error signals. The model shows how the dynamics of the whole system can lead to long-term adaptation Biologically Plausible Local Learning Rules for Adaptation of Vestibulo-Ocular Reflex Desired gain g = 1. 6 q=1 q=O Magnibtde Magnibtde 1 2.5 1.75 G G 1.5 0 A 0.5 1.15 1 0 0.75 0.5 A 0.15 0 100 200 300 400 0 sooTime 100 200 300 400 50lime A, D & VOR gain G vs time Amplitude Amplitude 1.5 2.5 1 N 300 N 500 Time 400 ?1 100 P 200 300 400 500 Time ?1 P & N responses to a head turn during learning vs time Figure 5: Effect of q on learning curves for gain increase. Left: q = 1 leads to an (wershoot in the VOR gain G above the desired gain. D increases up to the desired gain, P starts from 0 and asymptotically goes back to 0; both indicate that learning is totally transferred from P to N. Right: With q = 0, there is no overshoot in the VOR gain, but since A decreases to a constant value and D only increases very slightly, learning is not transfered. Consequently, P firing rate stays constant after an initial drop. E (I-b+v) (I-b) (D - ) q 9 (2q-I)-rv 10 Figure 6: Overshoot f. of the VOR gain G as a function of q and r. The parameter q is the proportion of learning to node N (vestibular nucleus), coming from the P node (cerebellum) compared to learning from climbing fibers. The parameter T is the ratio of the learning rates TJA and TJD. No overshoot is seen in animals, which restricts the parameters space of q and r for the model to be valid. Note that the overshoot diverges for some parameter values.' which differs from what may be expected from the local learning rules at the synapses because of differences in time scales and shifts of activity in the system during learning. This may reconcile apparently contradictory evidence between local learning rules observed in vitro (Ito, 1970) and the long-term adaptation seen in vivo in animals (Miles and Lisberger, 1981). 967 968 Coenen, Sejnowski, and Lisberger Acknowledgments O.c. was supported by NSERC during this research. References Albus, J. S. (1971). A theory of cerebellar function. Math. Biosci., 10:25-61. Arnold, D. B. and Robinson, D. A. (1992). A neural network model of the vestibulo-ocular reflex using a local synaptic learning rule. Phil. Trans. R. Soc. Lond. B, 337:327-330. Fujita, M. (1982). Simulations of adaptive modification of the vestibulo-ocular reflex with an adaptive filter model of the cerebellum. Biological Cybernetics, 45:207-214. Galiana, H. L. (1986). A new approach to understanding adaptive visual-vestibular interactions in the central nervous system. Journal of Neurophysiology, 55:349-374. Ito, M. (1970). Neurophysiological aspects of the cerebellar motor control system. Int.J.Neurol., 7:162-176. Ito, M., Sakurai, M., and Tongroach, P. (1982). Climbing fibre induced depression of both mossy fibre responsiveness and glutamate sensitivity of cerebellar purkinje cells. J. Physiol. Lond., 324:113-134. Kawato, M. and Gomi, H. (1992). The cerebellum and VORlOKR learning models. Trends in Neuroscience, 15 :445-453. Lisberger, S. G. (1988). The neural basis for learning of simple motor skills. Science, 242:728-735. Lisberger, S. G. (1992). Neural basis for motor learning in the vestibulo-ocularreflex ofprimates:IV. The sites of learning. In preparation. Lisberger, S. G. and Sejnowski, T. J. (1992). Computational analysis suggests a new hypothesis for motor learning in the vestibulo-ocular reflex. Technical Report 9201, INC, Univ. of California, San Diego. Marr, D. (1969). A theory of cerebellar cortex. J. Physiol., 202:437-470. MelviIl Jones, G. M. (1991). The Vestibular Contribution, volume 8 of Vision and Visual Dysfunction, chapter 2, pages 293-303. CRC Press, Inc., Boston. General Editor: J. R. Cronly-Dillon. Miles, E A. and Eighmy, B. B. (1980). Long-term adaptive changes in primate vestibulo-ocular reflex.l. Behavioural observations. Journal of Neurophysiology, 43:140&-1425. Miles, F. A. and Lisberger, S. G. (1981). Plasticity in the vestibulo-ocular reflex: A new hypothesis. Ann. Rev. Neurosci., 4:273-299. Quinn, K. J., Baker, J., and Peterson, B. (1992). Simulation of cerebellar-vestibular interactions during VOR adaptation. In Program 22nd Annual Meeting. Society for Neuroscience. Quinn, K. J., Schmajuk, N., Jain, A., Baker, J. E, and Peterson, B. W. (1992). Vestibuloocular reflex arc analysis using an experimentally constrained network. Biologtcal Cybernetics, 67: 113-122. Viola, P. A., Lisberger, S. G., and Sejnowski, T. J. (1992). Recurrent eye tracking network using a distributed representation of image motion. In Moody, 1. E., Hansen, S. J., and Lippman, R. P., editors, Advances in Neural Information Processing Systems 4, San Mateo. IEEE, Morgan Kaufmann Publishers.
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Visual Interaction Networks: Learning a Physics Simulator from Video Nicholas Watters, Andrea Tacchetti, Th?ophane Weber Razvan Pascanu, Peter Battaglia, Daniel Zoran DeepMind London, United Kingdom {nwatters, atacchet, theophane, razp, peterbattaglia, danielzoran}@google.com Abstract From just a glance, humans can make rich predictions about the future of a wide range of physical systems. On the other hand, modern approaches from engineering, robotics, and graphics are often restricted to narrow domains or require information about the underlying state. We introduce the Visual Interaction Network, a generalpurpose model for learning the dynamics of a physical system from raw visual observations. Our model consists of a perceptual front-end based on convolutional neural networks and a dynamics predictor based on interaction networks. Through joint training, the perceptual front-end learns to parse a dynamic visual scene into a set of factored latent object representations. The dynamics predictor learns to roll these states forward in time by computing their interactions, producing a predicted physical trajectory of arbitrary length. We found that from just six input video frames the Visual Interaction Network can generate accurate future trajectories of hundreds of time steps on a wide range of physical systems. Our model can also be applied to scenes with invisible objects, inferring their future states from their effects on the visible objects, and can implicitly infer the unknown mass of objects. This work opens new opportunities for model-based decision-making and planning from raw sensory observations in complex physical environments. 1 Introduction Physical reasoning is a core domain of human knowledge [22] and among the earliest topics in AI [24, 25]. However, we still do not have a system for physical reasoning that can approach the abilities of even a young child. A key obstacle is that we lack a general-purpose mechanism for making physical predictions about the future from sensory observations of the present. Overcoming this challenge will help close the gap between human and machine performance on important classes of behavior that depend on physical reasoning, such as model-based decision-making [3], physical inference [13], and counterfactual reasoning [10, 11]. We introduce the Visual Interaction Network (VIN), a general-purpose model for predicting future physical states from video data. The VIN is learnable and can be trained from supervised data sequences which consist of input image frames and target object state values. It can learn to approximate a range of different physical systems which involve interacting entities by implicitly internalizing the rules necessary for simulating their dynamics and interactions. The VIN model is comprised of two main components: a visual encoder based on convolutional neural networks (CNNs) [17], and a recurrent neural network (RNN) with an interaction network (IN) [2] as its core, for making iterated physical predictions. Using this architecture we are able to learn a model which infers object states and can make accurate predictions about these states in future time 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. steps. We show that this model outperforms various baselines and can generate compelling future rollout trajectories. 1.1 Related work One approach to learning physical reasoning is to train models to make state-to-state predictions. One early algorithm using this approach was the ?NeuroAnimator? [12], which was able to simulate articulated bodies. Ladicky et al. [16] proposed a learned model for simulating fluid dynamics based on regression forests. Battaglia et al. [2] introduced a general-purpose learnable physics engine, termed an Interaction Network (IN), which could learn to predict gravitational systems, rigid body dynamics, and mass-spring systems. Chang et al. [7] introduced a similar model in parallel that could likewise predict rigid body dynamics. Another class of approaches learn to predict summary physical judgments and produce simple actions from images. There have been several efforts [18, 19] which used CNN-based models to predict whether a stack of blocks would fall. Mottaghi et al. [20, 21] predicted coarse, image-space motion trajectories of objects in real images. Several efforts [4, 6, 26, 27] have fit the parameters of Newtonian mechanics equations to systems depicted in images and videos, though the dynamic equations themselves were not learned. Agrawal et al. [1] trained a system that learns to move objects by poking. A third class of methods [5, 8, 9, 23], like our Visual Interaction Network, have been used to predict future state descriptions from pixels. However, in contrast to the Visual Interaction Network, these models have to be tailored to the particular physical domain of interest, are only effective over a few time steps, or use side information such as object locations and physical constraints at test time. 2 Model The Visual Interaction Network (VIN) learns to produce future trajectories of objects in a physical system from video frames of that system. The VIN is depicted in Figure 1, and consists of the following components: ? The visual encoder takes a triplet of frames as input and outputs a state code. A state code is a list of vectors, one for each object in the scene. Each of these vectors is a distributed representation of the position and velocity of its corresponding object. We apply the encoder in a sliding window over a sequence of frames, producing a sequence of state codes. See Section 2.1 and Figure 2a for details. ? The dynamics predictor takes a sequence of state codes (output from a visual encoder applied in a sliding-window manner to a sequence of frames) and predicts a candidate state code for the next frame. The dynamics predictor is comprised of several interaction-net cores, each taking input at a different temporal offset and producing candidate state codes. These candidates are aggregated by an MLP to produce a predicted state code for the next frame. See Section 2.2 and Figure 2b for details. ? The state decoder converts a state code to a state. A state is a list of each object?s position/velocity vector. The training targets for the system are ground truth states. See Section 2.3 for details. 2.1 Visual Encoder The visual encoder is a CNN that produces a state code from a sequence of 3 images. It has a frame pair encoder Epair shown in Figure 2a which takes a pair of consecutive frames and outputs a candidate state code. This frame pair encoder is applied to both consecutive pairs of frames in a sequence of 3 frames. The two resulting candidate state codes are aggregated by a shared MLP applied to the concatenation of each pair of slots. The result is an encoded state code. Epair itself applies a CNN with two different kernel sizes to a channel-stacked pair of frames, appends constant x, y coordinate channels, and applies a CNN with alternating convolutional and max-pooling layers until unit width and height. The resulting matrix of shape 1 ? 1 ? (Nobject ? Lcode ) is reshaped into a state code of shape Nobject ? Lcode , where Nobject is the number of objects in the scene and Lcode is the length of each state code slot. The two state codes are fed into an MLP to produce the final 2 Figure 1: Visual Interaction Network: The general architecture is depicted here (see legend on the bottom right). The visual encoder takes triplets of consecutive frames and produces a state code for the third frame in each triplet. The visual encoder is applied in a sliding window over the input sequence to produce a sequence of state codes. Auxiliary losses applied to the decoded output of the encoder help in training. The state code sequence is then fed into the dynamics predictor which has several Interaction Net cores (2 in this example) working on different temporal offsets. The outputs of these Interaction Nets are then fed into an aggregator to produce the prediction for the next time step. The core is applied in a sliding window manner as depicted in the figure. The predicted state codes are linearly decoded and are used in the prediction loss when training. encoder output from the triplet. See the Supplementary Material for further details of the visual encoder model. One important feature of this visual encoder architecture is its weight sharing given by applying the same Epair on both pairs of frames, which approximates a temporal convolution over the input sequence. Another important feature is the inclusion of constant coordinate channels (an x- and y-coordinate meshgrid over the image), which allows positions to be incorporated throughout much of the processing. Without the coordinate channels, such a convolutional architecture would have to infer position from the boundaries of the image, a more challenging task. 2.2 Dynamics Predictor The dynamics predictor is a variant of an Interaction Network (IN) [2]. An IN, summarized in Figure 2b, is a state-to-state physical predictor model that uses a shared relation net on pairs of objects as well as shared self-dynamics and global affector nets to predict per-object dynamics. The main difference between our predictor and a vanilla IN is aggregation over multiple temporal offsets. Our predictor has a set of temporal offsets (in practice we use {1, 2, 4}), with one IN core for each. Given an input state code sequence, for each offset t a separate IN core computes a candidate predicted state code from the input state code at index t. An MLP aggregator transforms the list of candidate state codes into a predicted state code. This aggregator is applied independently to the concatenation over candidate state codes of each slot and is shared across slots to enforce some consistency of object representations. See the Supplementary Material for further details of the dynamics predictor model. As with the visual encoder, we explored many dynamics predictor architectures (some of which we compare as baselines below). The temporal offset aggregation of this architecture enhances its power by allowing it to accommodate both fast and slow movements by different objects within a sequence of frames. See the Supplementary Material for an exploration of the importance of temporal offset aggregation. The factorized representation of INs, which allows efficient learning of interactions even in scenes with many objects, is an important contributor to our predictor architecture?s performance. 3 (b) Interaction Net (a) Frame Pair Encoder Figure 2: Frame Pair Encoder and Interaction Net. (a) The frame pair encoder is a CNN which transforms two consecutive input frame into a state code. Important features are the concatenation of coordinate channels before pooling to unit width and height. The pooled output is reshaped into a state code. (b) An Interaction Net (IN) is used for each temporal offset by the dynamics predictor. For each slot, a relation net is applied to the slot?s concatenation with each other slot. A self-dynamics net is applied to the slot itself. Both of these results are summed and post-processed by the affector to produce the predicted slot. 2.3 State Decoder The state decoder is simply a linear layer with input size Lcode and output size 4 (for a position/velocity vector). This linear layer is applied independently to each slot of the state code. We explored more complicated architectures, but this yielded the best performance. The state decoder is applied to both encoded state codes (for auxiliary encoding loss) and predicted state codes (for prediction loss). 3 3.1 Experiments Physical Systems Simulations We focused on five types of physical systems with high dynamic complexity but low visual complexity, namely 2-dimensional simulations of colored objects on natural-image backgrounds interacting with a variety of forces (see the Supplementary Material for details). In each system the force law is applied pair-wise to all objects and all objects have the same mass and density unless otherwise stated. ? Spring Each pair of objects has an invisible spring connection with non-zero equilibrium. All springs share the same equilibrium and Hooke?s constant. ? Gravity Objects are massive and obey Newton?s Law of gravity. ? Billiards No long-distance forces are present, but the billiards bounce off each other and off the boundaries of the field of vision. ? Magnetic Billiards All billiards are positively charged, so instead of bouncing, they repel each other according to Coulomb?s Law. They still bounce off the boundaries. ? Drift No forces of any kind are present. Objects drift with their initial velocities. These systems include previously studied gravitational and billiards systems [3, 1] with the added challenge of natural image backgrounds. For example videos of these systems, see the Supplementary Material or visit (https://goo.gl/yVQbUa). One limitation of the above systems is that the positions, masses, and radii of all objects are either visually observable in every frame or global constants. Furthermore, while occlusion is allowed, the objects have the same radius so total occlusion never occurs. In contrast, systems with hidden quantities that influence dynamics abound in the real world. To mimic this, we explored a few challenging additional systems: ? Springs with Invisibility. In each simulation a random object is not rendered. In this way a model must infer the location of the invisible object from its effects on the other objects. ? Springs and Billiards with Variable Mass. In each simulation, each object?s radius is randomly generated. This not only causes total occlusion (in the Spring system), but density is held constant, so a model must determine each object?s mass from its radius. 4 To simulate each system, we initialized the position and velocity of each ball randomly and used a physics engine to simulate the resulting dynamics. See the Supplementary Material for more details. To generate video data, we rendered the system state on top of a CIFAR-10 natural image background. The background was randomized between simulations. Importantly, we rendered the objects with 15-fold anti-aliasing so the visual encoder could learn to distinguish object positions much more finely than pixel resolution, as evident by the visual encoder accuracy described in Section 4.1. For each system we generated a dataset with 3 objects and a dataset with 6 objects. Each dataset had a training set of 2.5 ? 105 simulations and a test set of 2.5 ? 104 simulations, with each simulation 64 frames long. Since we trained on sequences of 14 frames, this ensures we had more than 1 ? 107 training samples with distinct dynamics. We rendered natural image backgrounds online from separate training and testing CIFAR-10 sets. 3.2 Baseline Models We compared the VIN to a suite of baseline and competitor models, including ablation experiments. For each model, we performed hyperparameter sweeps across all datasets and choose the hyperparameter set with the lowest average test loss. The Visual RNN has the same visual encoder as the VIN, but the core of its dynamics predictor core is an MLP instead of an IN. Each state code is flattened before being passed to the dynamics predictor. The dynamics predictor is still treated as a recurrent network with temporal offset aggregation, but the dynamics predictor no longer supports the factorized representation of the IN core. Without the weight-sharing of the IN, this model is forced to learn the same force law for each pair of objects, which is not scalable as the object number increases. The Visual LSTM has the same visual encoder as the VIN, but its dynamics predictor is an LSTM [14] with MLP pre- and post-processors. It has no temporal offset aggregation, since the LSTM implicitly integrates temporal information through state updates. During rollouts, the output state code from the post-processor MLP is fed into the pre-processor MLP. The VIN Without Relations is an ablation modification of the VIN. The only difference between this and the VIN is an omitted relation network in the dynamics predictor cores. Note that there is still ample opportunity to compute relations between objects (both in the visual encoder and the dynamics predictor?s temporal offset aggregator), just not specifically through the relation network. Note that we performed a second ablation experiment to isolate the effect of temporal offset aggregation. See the Supplementary Material for details. The Vision With Ground-Truth Dynamics model uses a visual encoder and a miniature version of the dynamics predictor to predict not the next-step state but the current-step state (i.e. the state corresponding to the last observed frame). Since this predicts static dynamics, we did not train it on rollouts. However, when testing, we fed the static state estimation into a ground-truth physics engine to generate rollouts. This model is not a fair comparison to the other models because it does not learn dynamics. It serves instead as a performance bound imposed by the visual encoder. We normalized our results by the performance of this model, as described in Section 4. All models described above learn state from pixels. However, we also trained two baselines with privileged information: IN from State and LSTM from State models, which have the IN and LSTM dynamics predictors, but make their predictions directly from state to state. Hence, they do not have a visual encoder but instead have access to the ground truth states for observed frames. These, in combination with the Vision with Ground Truth Dynamics, allowed us to comprehensively test our model in part and in full. 3.3 Training procedure Our goal was for the models to accurately predict physical dynamics into the future. As shown in Figure 1, the VIN lends itself well to long-term predictions because the dynamics predictor can be treated as a recurrent net and rolled out on state codes. We trained the model to predict a sequence of 8 consecutive unseen future states from 6 frames of input video. Our prediction loss was a normalized weighted sum of the corresponding 8 error terms. The sum was weighted by a discount factor that started at 0.0 and approached 1.0 throughout training, so at the start of training the model must only predict the first unseen state and at the end it must predict an average of all 8 future states. Our 5 training loss was the sum of this prediction loss and an auxiliary encoding loss, as indicated in Figure 1. The model was trained by backpropagation with an Adam optimizer [15]. See the Supplementary Material for full training parameters. 4 Results Our results show that the VIN predicts dynamics accurately, outperforming baselines on all datasets (see Figures 3 and 4). It is scalable, can accommodate forces with a variety of strengths and distance ranges, and can infer visually unobservable quantities (invisible object location) from dynamics. Our model also generates long rollout sequences that are both visually plausible and similar to a ground-truth physics, even outperforming state-of-the-art state-to-state models on this measure. 4.1 Inverse Normalized Loss We evaluated the performance of each model with the Inverse Normalized Loss, defined as Lbound /Lmodel . Here Lbound is the test loss of the Vision with Ground Truth Dynamics and Lmodel is the test loss of the model in question (See Section 3.3). We used this metric because it is much more interpretable than Lmodel itself. The Vision with Ground Truth Dynamics produces the best possible predictions given the visual encoder?s error, so the Inverse Normalized Loss always lies in [0, 1], where a value closer to 1.0 indicates better performance. The visual encoder learned position predictions accurate to within 0.15% of the framewidth (0.048 times the pixel width), so we have no concerns about the accuracy of the Vision with Ground Truth Dynamics. Figure 3 shows the Inverse Normalized Loss on all test datasets after 3 ? 105 training steps. The VIN out-performs all baselines on nearly all systems. The only baseline with comparable performance is the VIN Without Relations on Drift, which matches the VIN?s performance. This makes sense, because the objects do not interact in the Drift system, so the relation net should be unnecessary. Of particular note is the performance of the VIN on the invisible dataset (spring system with random invisible object), where its performance is comparable to the fully visible 3-object Spring system. It can locate the invisible object?s position to within 4% of the frame width (1.3 times the pixel width) for the first 8 rollout steps. Figure 3: Performance. We compare our model?s Inverse Normalized Loss to that of the baselines on all test datasets. 3-object dataset are on the upper row, and 6-object datasets are on the lower row. By definition of the Inverse Normalized Loss, all values are in [0, 1] with 1.0 being the performance of a ground-truth simulator given the visual encoder. The VIN (red) outperforms every baseline on every dataset (except the VIN Without Relations on Drift, the system with no object interactions). 4.2 Euclidean Prediction Error of Rollout Positions One important desirable feature of a physical predictor is the ability to extrapolate from a short input video. We addressed this by comparing performance of all models on long rollout sequences and observing the Euclidean Prediction Error. To compute the Euclidean Prediction Error from a 6 predicted state and ground-truth state, we calculated the mean over objects of the Euclidean norm between the predicted and true position vectors. We computed the Euclidean Prediction Error at each step over a 50-timestep rollout sequence. Figure 4 shows the average of this quantity over all 3-object test datasets with respect to both timestep and object distance traveled. The VIN out-performs all other models, including the IN from state and LSTM from state even though they have access to privileged information. This demonstrates the remarkable robustness and generalization power of the VIN. We hypothesize that it outperforms state-to-state models in part because its dynamics predictor must tolerate visual encoder noise during training. This noise-robustness translates to rollouts, where the dynamics predictor remains accurate even as its predictions deviate from true physical dynamics. The state-to-state methods are not trained on noisy state inputs, so they exhibit poorer generalization. See the Supplementary Material for a dataset-specific quantification of these results. (a) Distance Comparison (b) Time Comparison Figure 4: Euclidean Prediction Error on 3-object datasets. We compute the mean over all test datasets of the Euclidean Prediction Error for 50-timestep rollouts. The VIN outperforms all other pixel-to-state models (solid lines) and state-to-state models (dashed lines). Errorbars show 95% confidence intervals. (a) Mean Euclidean Prediction Error with respect to object distance traveled (measured as a fraction of the frame-width). The VIN is accurate to within 6% after objects have traversed 0.72 times the framewidth. (b) Mean Euclidean Prediction Error with respect to timestep. The VIN is accurate to within 7.5% after 50 timesteps. The optimal information-less predictor (predicting all objects to be at the frame?s center) has an error of 37%, higher than all models. 4.3 Visualized Rollouts To qualitatively evaluate the plausibility of the VIN?s rollout predictions, we generated videos by rendering the rollout predictions. These are best seen in video format, though we show them in trajectory-trail images here as well. The backgrounds made trajectory-trails difficult to see, so we masked the background (only for rendering purposes). Trajectory trails are shown for rollouts between 40 and 60 time steps, depending on the dataset. We encourage the reader to view the videos at (https://goo.gl/RjE3ey). Those include the CIFAR backgrounds and show very long rollouts of up to 200 timesteps, which demonstrate the VIN?s extremely realistic predictions. We find no reason to doubt that the predictions would continue to be visually realistic (if not exactly tracking the ground-truth simulator) ad infinitum. 5 Conclusion Here we introduced the Visual Interaction Network and showed that it can infer the physical states of multiple objects from video input and make accurate predictions about their future trajectories. The model uses a CNN-based visual encoder to obtain accurate measurements of object states in the scene. The model also harnesses the prediction abilities and relational computation of Interaction Networks, providing accurate predictions far into the future. We have demonstrated that our model performs well on a variety of physical systems and is robust to visual complexity and partially observable data. 7 Truth Prediction Sample Frame Truth Prediction Drift Billiards Magnetic Billiards Gravity Spring Sample Frame Table 1: Rollout Trajectories. For each of our datasets, we show a sample frame, an example true future trajectory, and a corresponding predicted rollout trajectory (for 40-60 frames, depending on the dataset). The left half shows the 3-object regime and the right half shows the 6-object regime. For visual clarity, all objects are rendered at a higher resolution here than in the training input. One property of our model is the inherent presence of noise from the visual encoder. In contrast to state-to-state models such as the Interaction Net, here the dynamic predictor?s input is inherently noisy due to the discretization of our synthetic dataset rendering. Surprisingly, this noise seemed to confer an advantage because it helped the model learn to overcome temporally compounding errors generated by inaccurate predictions. This is especially notable when doing long term roll outs where we achieve performance which surpasses even a pure state-to-state Interaction Net. Since this dependence on noise would be inherent in any model operating on visual input, we postulate that this is an important feature of any prediction model and warrants further research. While experimentation with variable number of objects falls outside the scope of the material presented here, this is an important direction that could be explored in further work. Importantly, INs generalize out of the box to scenes with a variable number of objects. Should the present form of the perceptual encoder be insufficient to support this type of generalization, this could be addressed by using an attentional encoder and order-agnostic loss function. Our Visual Interaction Network provides a step toward understanding how representations of objects, relations, and physics can be learned from raw data. This is part of a broader effort toward understanding how perceptual models support physical predictions and how the structure of the physical world influences our representations of sensory input, which will help AI research better capture the powerful object- and relation-based system of reasoning that supports humans? powerful and flexible general intelligence. Acknowledgments We thank Max Jaderberg, David Reichert, Daan Wierstra, and Koray Kavukcuoglu for helpful discussions and insights. 8 References [1] Pulkit Agrawal, Ashvin Nair, Pieter Abbeel, Jitendra Malik, and Sergey Levine. Learning to poke by poking: Experiential learning of intuitive physics. arXiv preprint arXiv:1606.07419, 2016. [2] Peter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al. Interaction networks for learning about objects, relations and physics. In Advances in Neural Information Processing Systems, pages 4502?4510, 2016. [3] Peter W Battaglia, Jessica B Hamrick, and Joshua B Tenenbaum. Simulation as an engine of physical scene understanding. Proceedings of the National Academy of Sciences, 110(45):18327?18332, 2013. [4] Kiran Bhat, Steven Seitz, Jovan Popovi?c, and Pradeep Khosla. Computing the physical parameters of rigid-body motion from video. Computer Vision?ECCV 2002, pages 551?565, 2002. [5] Apratim Bhattacharyya, Mateusz Malinowski, Bernt Schiele, and Mario Fritz. Long-term image boundary extrapolation. arXiv preprint arXiv:1611.08841, 2016. [6] Marcus A Brubaker, Leonid Sigal, and David J Fleet. Estimating contact dynamics. In Computer Vision, 2009 IEEE 12th International Conference on, pages 2389?2396. IEEE, 2009. [7] Michael B Chang, Tomer Ullman, Antonio Torralba, and Joshua B Tenenbaum. A compositional objectbased approach to learning physical dynamics. arXiv preprint arXiv:1612.00341, 2016. [8] Sebastien Ehrhardt, Aron Monszpart, Niloy J Mitra, and Andrea Vedaldi. Learning a physical long-term predictor. arXiv preprint arXiv:1703.00247, 2017. [9] Katerina Fragkiadaki, Pulkit Agrawal, Sergey Levine, and Jitendra Malik. Learning visual predictive models of physics for playing billiards. arXiv preprint arXiv:1511.07404, 2015. [10] Tobias Gerstenberg, Noah Goodman, David A Lagnado, and Joshua B Tenenbaum. Noisy newtons: Unifying process and dependency accounts of causal attribution. In In proceedings of the 34th. Citeseer, 2012. [11] Tobias Gerstenberg, Noah Goodman, David A Lagnado, and Joshua B Tenenbaum. From counterfactual simulation to causal judgment. In CogSci, 2014. [12] Radek Grzeszczuk, Demetri Terzopoulos, and Geoffrey Hinton. Neuroanimator: Fast neural network emulation and control of physics-based models. In Proceedings of the 25th annual conference on Computer graphics and interactive techniques, pages 9?20. ACM, 1998. [13] Jessica B Hamrick, Peter W Battaglia, Thomas L Griffiths, and Joshua B Tenenbaum. Inferring mass in complex scenes by mental simulation. Cognition, 157:61?76, 2016. [14] Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735?1780, 1997. [15] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [16] Lubor Ladicky, SoHyeon Jeong, Barbara Solenthaler, Marc Pollefeys, Markus Gross, et al. Data-driven fluid simulations using regression forests. ACM Transactions on Graphics (TOG), 34(6):199, 2015. [17] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436?444, 2015. [18] Adam Lerer, Sam Gross, and Rob Fergus. Learning physical intuition of block towers by example. arXiv preprint arXiv:1603.01312, 2016. [19] Wenbin Li, Seyedmajid Azimi, Ale? Leonardis, and Mario Fritz. To fall or not to fall: A visual approach to physical stability prediction. arXiv preprint arXiv:1604.00066, 2016. [20] Roozbeh Mottaghi, Hessam Bagherinezhad, Mohammad Rastegari, and Ali Farhadi. Newtonian scene understanding: Unfolding the dynamics of objects in static images. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3521?3529, 2016. [21] Roozbeh Mottaghi, Mohammad Rastegari, Abhinav Gupta, and Ali Farhadi. ?what happens if...? learning to predict the effect of forces in images. In European Conference on Computer Vision, pages 269?285. Springer, 2016. [22] Elizabeth S Spelke and Katherine D Kinzler. Core knowledge. Developmental science, 10(1):89?96, 2007. [23] Russell Stewart and Stefano Ermon. Label-free supervision of neural networks with physics and domain knowledge. arXiv preprint arXiv:1609.05566, 2016. [24] Terry Winograd. Procedures as a representation for data in a computer program for understanding natural language. Technical report, DTIC Document, 1971. [25] Patrick H Winston. Learning structural descriptions from examples. 1970. [26] Jiajun Wu, Joseph J Lim, Hongyi Zhang, Joshua B Tenenbaum, and William T Freeman13. Physics 101: Learning physical object properties from unlabeled videos. psychological science, 13(3):89?94, 2016. [27] Jiajun Wu, Ilker Yildirim, Joseph J Lim, Bill Freeman, and Josh Tenenbaum. Galileo: Perceiving physical object properties by integrating a physics engine with deep learning. In Advances in neural information processing systems, pages 127?135, 2015. 9
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Reconstruct & Crush Network Erin? Merdivan1,2 , Mohammad Reza Loghmani3 and Matthieu Geist4 1 AIT Austrian Institute of Technology GmbH, Vienna, Austria 2 LORIA (Univ. Lorraine & CNRS), CentraleSup?lec, Univ. Paris-Saclay, 57070 Metz, France 3 Vision4Robotics lab, ACIN, TU Wien, Vienna, Austria 4 Universit? de Lorraine & CNRS, LIEC, UMR 7360, Metz, F-57070 France [email protected], [email protected] [email protected] Abstract This article introduces an energy-based model that is adversarial regarding data: it minimizes the energy for a given data distribution (the positive samples) while maximizing the energy for another given data distribution (the negative or unlabeled samples). The model is especially instantiated with autoencoders where the energy, represented by the reconstruction error, provides a general distance measure for unknown data. The resulting neural network thus learns to reconstruct data from the first distribution while crushing data from the second distribution. This solution can handle different problems such as Positive and Unlabeled (PU) learning or covariate shift, especially with imbalanced data. Using autoencoders allows handling a large variety of data, such as images, text or even dialogues. Our experiments show the flexibility of the proposed approach in dealing with different types of data in different settings: images with CIFAR-10 and CIFAR-100 (not-in-training setting), text with Amazon reviews (PU learning) and dialogues with Facebook bAbI (next response classification and dialogue completion). 1 Introduction The main purpose of machine learning is to make inferences about unknown data based on encoded dependencies between variables learned from known data. Energy-based learning [16] is a framework that achieves this goal by using an energy function that maps each point of an input space to a single scalar, called energy. The fact that energy-based models are not subject to the normalizability condition of probabilistic models makes them a flexible framework for dealing with tasks such as prediction or classification. In the recent years, with the advancement of deep learning, astonishing results have been achieved in classification [15, 25, 8, 26]. These solutions focus on the standard setting, in which the classifier learns to discriminate between K classes, based on the underlying assumption that the training and test samples belong to the same distribution. This assumption is violated in many applications in which the dynamic nature [6] or the high cardinality [19] of the problem prevent the collection of a representative training set. In the literature, this problem is referred to as covariate shift [7, 24]. In this work, we address the covariate shift problem by explicitly learning features that define the intrinsic characteristics of a given class of data rather than features that discriminate between different classes. The aim is to distinguish between samples of a positive class (A) and samples that do not belong to this class (?A), even when test samples are not drawn from the same distribution as the training samples. We achieve this goal by introducing an energy-based model that is adversarial regarding data: it minimizes the energy for a given data distribution (the positive samples) while maximizing the energy for another given data distribution (the negative or unlabeled samples). The model is instantiated with autoencoders because of their ability to learn data manifolds. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. In summary, our contributions are the following: ? a simple energy-based model dealing with the A/?A classification problem by providing a distance measure of unknown data as the energy value; ? a general framework that can deal with a large variety of data (images, text and sequential data) by using features extracted from an autoencoder architecture; ? a model that implicitly addresses the imbalanced classification problem; ? state-of-the-art results for the dialogue completion task on the Facebook bAbI dataset and competitive results for the general A/?A classification problem using different datasets such as CIFAR-10, CIFAR-100 and Amazon Reviews. The next section introduces the proposed ?reconstruct & crush? network, section 3 positions our approach compared to related works, section 4 presents the experimental results on the aforementioned problems and section 5 draws the conclusions. 2 Model Let define ppos as the probability distribution producing positive samples, xpos ? ppos . Similarly, write pneg the distribution of negative samples, xneg ? pneg . More generally, these negative samples can be unlabeled samples (possibly containing positive samples). This case will be considered empirically, but we keep this notation for now. Let N denote a neural network that takes as input a sample x and outputs a (positive) energy value E: N (x) = E ? R+ . The proposed approach aims at learning a network N that assign low energy values to positive samples (N (xpos ) small for xpos ? ppos ) and high energy values for negative samples (N (xneg ) high for xneg ? pneg ). Let m > 0 be a user-defined margin, we propose to use the following loss LN and associated risk RN : L(xpos , xneg ; N ) = N (xpos ) + max(0, m ? N (xneg )) R(N ) = Expos ?ppos ,xneg ?pneg L(xpos , xneg ) = Expos ?ppos [N (xpos )] + Exneg ?pneg [max(0, m ? N (xneg ))]. (1) Ideally, minimizing this risk amounts to have no reconstruction error over positive samples and a reconstruction error greater than m (in expectation) over negative samples. The second term of the risk acts as a regularizer that enforces the network to assign a low energy only to positive samples. The choice of the margin m will affect the behavior of the network: if m is too small a low energy will be assigned to all inputs (both positive and negative), while if m is too large assigning a large energy to negative samples can prevent from reconstructing the positive ones. We specialize our model with autoencoders, that are a natural choice to represent energy-based models. An autoencoder is composed of two parts, the encoder (Enc) that projects the data into an encoding space, and the decoder (Dec) that reconstructs the data from this projection: Enc :X ? Z Dec :Z ? X argmin kx? Dec(Enc(x))k2 . Enc,Dec Here, X is the space of the input data (either positive or negative) and Z is the space of encoded data. In this setting, the reconstruction error of a sample x can be interpreted as the energy value associated to that sample: N (x) = kx ? Dec(Enc(x))k2 = E. Our resulting reconstruct & crush network (RCN) is thus trained to assign a low reconstruction error to xpos (reconstruct) and an high reconstruction error to xneg (crush). Any stochastic gradient descent method can be used to optimize the risk of Eq. (1), the mini-batches of positive and negative samples being sampled independently from the corresponding distributions. 2 3 Related work With the diffusion of deep neural networks, autoencoders have received a new wave of attention due to their use for layer-wise pretraining [1]. Although the concept of autoencoders goes back to the 80s [23, 3, 10], many variations have been proposed more recently, such as denoising autoencoder [27], stacked autoencoders [9] or variational autoencoders [13]. Despite the use of autoencoders for pretraining is not a common practice anymore, various researches still take advantage of their properties. In energy-based generative adversarial networks (EBGAN) [30], an autoencoder architecture is used to discriminate between real samples and "fake" ones produced by the generator. Despite not being a generative model, our method shares with EBGAN the interpretation of the reconstruction error provided by the autoencoder as energy value and the fundamentals of the discriminator loss. However, instead of the samples produced by the generator network, we use negative or unlabeled samples to push the autoencoder to discover the data manifold during training. In other words, EBGAN searches for a generative model by training adversarial networks, while in our framework the network tries to make two distributions adversarial. The use of unlabeled data (that could contain both positive and negative samples) together with positive samples during training is referred to as PU (Positive and Unlabeled) learning [5, 17]. In the literature, works in the PU learning setting [29, 18] focus on text-based applications. Instead, we show in the experiments that our work can be applied to different type of data such as images, text and sequential data. Similarly to our work, [11] uses the reconstruction error as a measure to differentiate between positive and negative samples. However they train their network with either positive or negative data only. In addition, instead of end-to-end training, they provide a two-stage process in which a classifier is trained to discriminate between positive and negative samples based on the reconstruction error. In the context of dialogue management systems, the score proposed in [21] has been used as a quality measure of the response. Nevertheless, [19] shows that this score fails when a correct response, that largely diverges from the ground truth, is given. The energy value of the RCN is a valid score to discriminate between good and bad responses, as we show in section 4.4. 4 Experimental results In this section, we experiment the proposed RCN on various tasks with various kind of data. We consider a not-in-training setting for CIFAR-10 and CIFAR-100 (sections 4.1 and 4.2), a PU learning setting for the amazon reviews dataset (section 4.3) and a dialogue completion setting for the Facebook bAbI dataset (section 4.4). For an illustrative purpose, we also provide examples of reconstructed and crushed images from CIFAR-10 and CIFAR-100 in figure 1, corresponding to experiments of sections 4.1 and 4.2. 4.1 CIFAR-10 CIFAR-10 consists of 60k 32x32 color images in 10 classes, with 6k images per class. There are 50k training images and 10k test images [14]. We converted the images to gray-scale and used 5k images per class. This set of experiments belong to the not-in-training setting [6]: the training set contains positive and negative samples and the test set belongs to a different distribution than the training set. The ?automobile? class is used as the positive class (A) and the rest of the classes are considered to be the negative class (?A) (binary classification problem). All the training samples are used for training, except for those belonging to the ?ship? class. Test samples of ?automobile? and ?ship? are used for testing. It is worth noticing that the size of positive and negative training sets is highly imbalanced: 5k positive samples and 40k negative samples. In this experiment, we show the superior performances of our network with respect to standard classifiers in dealing with images of an unseen class. Since we are dealing with a binary classification problem, we define a threshold T for the energy value. This threshold is used in RCN to distinguish between the positive and the negative class. For our autoencoder, we used a convolutional network defined as: (32)3c1s-(32)3c1s-(64)3c2s-(64)3c2-(32)3c1s-512f-1024f, where ?(32)3c1s? denotes 3 Figure 1: Illustrations of Reconstructed and Crushed images by RCN from CIFAR10 and CIFAR100. a convolution layer with 32 output feature maps, kernel size 3 and stride 1, and ?512f? denotes a fully-connected layer with 512 hidden units. The size of the last layer corresponds to the size of the images (32x32=1024). For standard classification we add on top of the last layer another fully-connected layer with 2 output neurons (A/?A). The choice of the architectures for standard classifier and autoencoder is driven by necessity of fair comparison. ReLU activation functions are used for all the layers except for the last fully-connected layer of the standard classifier in which a Softmax function is used. These models are implemented in Tensorflow and trained with the adam optimizer [12] (learning rate of 0.0004) and a mini-batch size of 100 samples. The margin m was set to 1.0 and the threshold T to 0.5. Table 1 shows the true positive rate (TPR=#(correctly classified cars)/#cars) and the true negative rate (TNR=#(correctly classified ships)/#ships) obtained by the standard classifier (CNN / CNN-reduced) and our network (RCN). CNN-reduced shows the performance of the standard classifier when using the same amount of positive and negative samples. It can be noticed that RCN presents the best TNR and a TPR comparable to the one of CNN-reduced. These results shows that RCN is a better solution when dealing with not-in-training data. In addition, the TPR and TNR of our method is comparable despite the imbalanced training set. Figure 2 clearly shows that not-in-training samples (ship images) are positioned between positive in-training samples (automobile images) and in-training-negative samples (images from all classes except automobile and ship). It can be noticed that negative in-training samples have a reconstruction loss close to margin value 1.0. Table 1: Performances of standard classifier (CNN / CNN-reduced) and our method (RCN) on CIFAR-10. The positive class corresponds to "automobile" and the negative class corresponds to "ship" (unseen during the training phase). Method True Positive Rate True Negative Rate CNN-reduced CNN RCN 0.82 0.74 0.81 0.638 0.755 0.793 4 Figure 2: Mean reconstruction error over the epochs of positive in-training, negative in-training and negative not-in-training samples of CIFAR-10. 4.2 CIFAR-100 CIFAR-100 is similar to CIFAR-10, except it has 100 classes containing 600 images each (500 for training and 100 for testing) [14]. The 100 classes in the CIFAR-100 are grouped into 20 super-classes with 5 classes each. Each image comes with a pair of labels: the class and the super-class. In this set of experiments, the ?food containers? super-class is used as the positive class (A) and the all the other super-classes are considered to be the negative class (?A) (binary classification problem). During training, 4 out of 5 classes belonging to the ?food containers? super-class (?bottles?, ?bowls?, ?cans?, ?cups?) are used as the positive training set and 4 out of 5 classes belonging to the ?flowers? super-class (?orchids?, ?poppies?, ?roses?, ?sunflowers?) are used as the negative training set. At test time, two in-training classes (?cups? and ?sunflowers?), two not-in-training classes belonging to ?food containers? (?plates?) and ?flowers? (?tulips?) and two not-in-training classes belonging to external super-classes (?keyboard? and ?chair?) are used. In this experiment, we show the superior performances of our network with respect to standard classifiers in dealing with data coming from unknown distributions and from unseen modes of the same distributions as the training data. The same networks and parameters of section 4.1 are used here. Table 2 shows the true positive rate (TPR=#(correctly classified plates)/#plates) and the true negative rate (TNR=#(correctly classified tulips)/#tulips) obtained by the standard classifier (CNN) and our network (RCN). It can be noticed that RCN presents the best results both for TNR and for TPR. These results shows that RCN is a better solution when dealing with not-in-training data coming from unseen modes of the data distribution. It is worth noticing that only 4k samples (2k positive and 2k negative) have been used during training. Figure 3 clearly shows the effectiveness of the learning procedure of our framework: the networks assigns low energy value (close to 0) to positive samples, high energy value (close to m) to negative samples related to the negative training set and medium energy value (close to m/2) to negative samples unrelated to the negative training set. Table 2: Performances of the standard classifier (CNN) and our method (RCN) on CIFAR-100. The positive class corresponds to "plates" and the negative class corresponds to "tulips". Method True Positive Rate True Negative Rate CNN RCN 0.718 0.861 0.81 0.853 5 Figure 3: Mean reconstruction error over the epochs of positive in-training and not-in-training (blue), negative in-training and not-in-training (red) and not-in-training unrelated (green,black) of CIFAR-100. 4.3 Amazon review Amazon reviews is a dataset containing product reviews (ratings, text, helpfulness votes) and metadata (descriptions, category information, price, brand, and image features) from Amazon, including 142.8 million reviews spanning [20]. Here, we only use the ratings and text features. This set of experiments belong to the PU learning setting: the training set contains positive and unlabeled data. The positive training set contains 10k "5-star" reviews and the unlabeled training set contains 10k unlabeled review (containing both positive and negative review). The test set is composed of 10k samples: 5k "5-star" (positive) reviews and 5k "1-star" (negative) reviews. The aim here is to show that RCN performs well in the PU learning setting with unlabeled sets with different positive/negative samples ratio. For handling the text data, we used the pretrained Glove word-embedding [22] with 100 feature dimensions. We set the maximum number of words in a sentence to 40 and zero-padded shorter sentences. For our autoencoder, we used a 1-dimensional (1D) convolutional network defined as: (128)7c1s(128)7c1s-(128)3c1s-(128)3c1-(128)3c1s-2048f-4000f, where ?(128)7c1s? denotes a 1D convolution layer with 128 output feature maps, kernel size 7 and stride 1. ReLU activation functions are used for all the layers. These models are implemented in Tensorflow and trained with the adam optimizer (learning rate of 0.0004) and a mini-batch size of 100 samples. The margin m was set to 0.85 and the threshold T to 0.425. Table 3 shows the results of different well-established PU learning methods, together with ours (RCN), on the Amazon review dataset. In can be noticed that, despite the fact that the architecture of our method is not specifically designed for handling the PU learning setting, it shows comparable results to the other methods, even when unlabeled training data with a considerable amount of positive samples (50%) are used. Table 4 presents some examples from the test set. It can be noticed that positive comments are assigned a low reconstruction error (energy) and vice-versa. 4.4 Facebook bAbI dialogue Facebook bAbI dialogue is a dataset containing dialogues related to 6 different tasks in which the user books a table in a restaurant with the help of a bot [2]. For each task 1k training and 1k test dialogues are provided. Each dialogue has 4 to 11 turns between the user and the bot for a total of 6 Table 3: F-measure of positive samples obtained with Roc-SVM [28], Roc-EM [18], Spy-SVM [18], NB-SVM [18], NB-EM [18] and RCN (ours). The scores are obtained on two different configuration of the unlabeled training set: one containing 5% of positive samples and one containing 50% of positive samples. Method F-measure for pos. samples (%5-%95) F-measure for pos. samples (%50-%50) Roc-SVM [28] Roc-EM [18] Spy-SVM [18] NB-SVM [18] NB-EM [18] RCN 0.92 0.91 0.92 0.92 0.91 0.90 0.89 0.90 0.89 0.86 0.86 0.83 Table 4: Examples of positive (5/5 score) and negative (1/5 score) reviews from Amazon review with the corresponding reconstruction error assigned from RCN. Review Score Error excellent funny fast reading i would recommend to all my friends 5/5 0.00054 this is easily one of my favorite books in the series i highly recommend it 5/5 0.00055 super book liked the sequence and am looking forward to a sequel keeping the s and characters would be nice 5/5 0.00060 i truly enjoyed all the action and the characters in this book the interactions between all the characters keep you drawn in to the book 5/5 0.00066 this book was the worst zombie book ever not even worth the review 1/5 1.00627 way too much sex and i am not a prude i did not finish and then deleted the book 1/5 1.00635 in reality it rates no stars it had a political agenda in my mind it was a waste my money 1/5 1.00742 fortunately this book did not cost much in time or money it was very poorly written an ok idea poorly executed and poorly developed 1/5 1.00812 ?6k turns in each set (training and test) for task 1 and ?9.5k turns in each set for task 2. Here, we consider the training and test data associated to tasks 1 and 2 because the other tasks require querying Knowledge Base (KB) upon user request: this is out of the scope of the paper. In task 1, the user requests to make a new reservation in a restaurant by defining a query that can contain from 0 to 4 required fields (cuisine type, location, number of people and price range) and the bot asks questions for filling the missing fields. In task 2, the user requests to update a reservation in a restaurant between 1 and 4 times. The training set is built in such a way that, for each turn in a dialogue, together with the positive (correct) response, 100 possible negative responses are selected from the candidate set (set of all bot responses in the Facebook bAbI dialogue dataset with a total of 4212 samples). The test set is built in such a way that, for each turn in a dialogue, all possible negative responses are selected from the candidate set. More precisely, for task 1, the test set contains approximately 6k positive and 25 million negative dialogue history-reply pairs, while for task 2, it contains approximately 9k positive and 38 million negative pairs. For our autoencoder, we use a gated recurrent unit (GRU) [4] with 1024 hidden units and a projection layer on top of it in order to replicate the input sequence in output. An upper limit of 100 was set for 7 the sequence length and a feature size of 50 was selected for word embeddings. The GRU uses ReLU activation and a dropout of 0.1. This model is implemented in Tensorflow and trained with the adam optimizer (learning rate of 0.0004) and a mini-batch size of 100 samples. In this experiments, our network equals the state-of-the-art performance of memory networks presented in [2] by achieving 100% accuracy both for next response classification and for dialogue completion where dialogue is considered as completed if all responses within the dialogue are correctly chosen. 5 Conclusions We have introduced a simple energy-based model, adversarial regarding data by minimizing the energy of positive data and maximizing the energy of negative data. The model is instantiated with autoencoders where the specific architecture depends on the considered task, thus providing a family of RCNs. Such an approach can address various covariate shift problems, such as not-in-training and positive and unlabeled learning and various types of data. The efficiency of our approach has been studied with exhaustive experiments on CIFAR-10, CIFAR100, the Amazon reviews dataset and the Facebook bAbI dialogue dataset. These experiments showed that RCN can obtain state-of-the art results for the dialogue completion task and competitive results for the general A/?A classification problem. These outcomes suggest that the energy value provided by RCN can be used to asses the quality of response given the dialogue history. Future works will extend the RCN to the multi-class classification setting. These results suggest that the energy value provided by RCN can be used to assess the quality of the response given the dialogue history. We plan to study further this aspect in the near future, in order to provide an alternative metric for dialogue systems evaluation. Acknowledgments This work has been funded by the European Union Horizon2020 MSCA ITN ACROSSING project (GA no. 616757). The authors would like to thank the members of the project?s consortium for their valuable inputs. References [1] Y. Bengio. Learning deep architectures for ai. Foundations and trends in Machine Learning, 2(1):1?127, 2009. [2] A. Bordes and J. Weston. Learning end-to-end goal-oriented dialog. arXiv:1605.07683, 2016. [3] H. Bourlard and Y. Kamp. Auto-association by multilayer perceptrons and singular value decomposition. Biological Cybernetics, 59(4):291?294, 1988. [4] K. Cho, B. van Merrienboer, D. Bahdanau, and Y. Bengio. On the properties of neural machine translation: Encoder-decoder approaches. arXiv preprint arXiv:1409.1259, 2014. [5] F. Denis. Pac learning from positive statistical queries. Algorithmic Learning Theory,112?126, 1998. [6] F. Geli and L. Bing. Social media text classification under negative covariate shift. EMNLP, 2015. [7] W.H. Greene. Sample selection bias as a specification error: A comment. Econometrica: Journal of the Econometric Society, pages 795?798, 1981. [8] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. CVPR, 2016. [9] G. E. Hinton and R. R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504?507, 2006. 8 [10] G.E. Hinton and R.S. Zemel. Autoencoders, minimum description length and helmholtz free energy. NIPS, 1994. [11] N. Japkowicz, C. Myers, and M. Gluck. A novelty detection approach to classification. IJCAI, 1995. [12] D. Kingma and J. Ba. Adam: A method for stochastic optimization. ICLR, 2015. [13] D. Kingma and M. Welling. Auto-encoding variational bayes. ICLR, 2013. [14] A. Krizhevsky. Learning multiple layers of features from tiny images. Technical report, 2009. [15] A. Krizhevsky, I. Sutskever, and G. E Hinton. Imagenet classification with deep convolutional neural networks. NIPS, 2012. [16] Y. LeCun, S. Chopra, R. Hadsell, M. Ranzato, and F.J. Huang. A tutorial on energy-based learning. Technical report, MIT Press, 2006. [17] X. Li and L. Bing. Learning from positive and unlabeled examples with different data distributions. ECML, 2005. [18] B Liu, Y. Dai, X. Li, W-S. Lee, and P. Yu. Building text classifiers using positive and unlabeled examples. ICDM, 2003. [19] C. Liu, R. Lowe, I.V. Serban, M. Noseworthy, L. Charlin, and J. Pineau. How not to your dialogue system: An empirical study of unsupervised evaluation metrics for dialogue response generation. EMNLP, 2016. [20] J. McAuley and J. Leskovec. Hidden factors and hidden topics: understanding rating dimensions with review text. RecSys, 2013. [21] K. Papineni, S. Roukos, T. Ward, and W. Zhu. Bleu: a method for automatic evaluation of machine translation. ACL, 2002. [22] J. Pennington, R. Socher, and C. D. Manning. Glove: Global vectors for word representation. EMNLP, 2014. [23] D.E. Rumelhart, G.E. Hinton, and R.J. Williams. Learning representations by back-propagating errors. Cognitive Modeling, 5(3):1, 1988. [24] H. Shimodaira. Improving predictive inference under covariate shift by weighting the loglikelihood function. Journal of Statistical Planning and Inference, 90(2):227?244, 2000. [25] K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. [26] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. CVPR, 2015. [27] P. Vincent, H. Larochelle, Y. Bengio, and P. Manzagol. Extracting and composing robust features with denoising autoencoders. ACM, 2008. [28] Li X. and Liu B. Learning to classify text using positive and unlabeled data. IJCAI, 2003. [29] H. Yu, J. Han, and K. Chang. Pebl: Positive example based learning for web page classification using svm. KDD, 2002. [30] J. Zhao, M. Mathieu, and Y. LeCun. Energy-based generative adversarial networks. ICLR, 2017. 9
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Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach Slobodan Mitrovi?c? EPFL Ilija Bogunovic? EPFL Ashkan Norouzi-Fard? EPFL Jakub Tarnawski? EPFL Volkan Cevher? EPFL Abstract We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm?s memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-G REEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance. 1 Introduction A central challenge in many large-scale machine learning tasks is data summarization ? the extraction of a small representative subset out of a large dataset. Applications include image and document summarization [1, 2], influence maximization [3], facility location [4], exemplar-based clustering [5], recommender systems [6], and many more. Data summarization can often be formulated as the problem of maximizing a submodular set function subject to a cardinality constraint. On small datasets, a popular algorithm is the simple greedy method [7], which produces solutions provably close to optimal. Unfortunately, it requires repeated access to all elements, which makes it infeasible for large-scale scenarios, where the entire dataset does not fit in the main memory. In this setting, streaming algorithms prove to be useful, as they make only a small number of passes over the data and use sublinear space. In many settings, the extracted representative set is also required to be robust. That is, the objective value should degrade as little as possible when some elements of the set are removed. Such removals may arise for any number of reasons, such as failures of nodes in a network, or user preferences which the model failed to account for; they could even be adversarial in nature. ? e-mail: e-mail: ? e-mail: ? e-mail: ? e-mail: ? [email protected] [email protected] [email protected] [email protected] [email protected] 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. A robustness requirement is especially challenging for large datasets, where it is prohibitively expensive to reoptimize over the entire data collection in order to find replacements for the removed elements. In some applications, where data is produced so rapidly that most of it is not being stored, such a search for replacements may not be possible at all. These requirements lead to the following two-stage setting. In the first stage, we wish to solve the robust streaming submodular maximization problem ? one of finding a small representative subset of elements that is robust against any possible removal of up to m elements. In the second, query stage, after an arbitrary removal of m elements from the summary obtained in the first stage, the goal is to return a representative subset, of size at most k, using only the precomputed summary rather than the entire dataset. For example, (i) in dominating set problem (also studied under influence maximization) we want to efficiently (in a single pass) compute a compressed but robust set of influential users in a social network (whom we will present with free copies of a new product), (ii) in personalized movie recommendation we want to efficiently precompute a robust set of user-preferred movies. Once we discard those users who will not spread the word about our product, we should find a new set of influential users in the precomputed robust summary. Similarly, if some movies turn out not to be interesting for the user, we should still be able to provide good recommendations by only looking into our robust movie summary. Contributions. In this paper, we propose a two-stage procedure for robust submodular maximization. For the first stage, we design a streaming algorithm which makes one pass over the data and finds a summary that  is robust against removal of up to m elements, while containing at most O (m log k + k) log2 k elements. In the second (query) stage, given any set of size m that has been removed from the obtained summary, we use a simple greedy algorithm that runs on the remaining elements and produces a solution of size at most k (without needing to access the entire dataset). We prove that this solution satisfies a constant-factor approximation guarantee. Achieving this result requires novelty in the algorithm design as well as the analysis. Our streaming algorithm uses a structure where the constructed summary is arranged into partitions consisting of buckets whose sizes increase exponentially with the partition index. Moreover, buckets in different partitions are associated with greedy thresholds, which decrease exponentially with the partition index. Our analysis exploits and combines the properties of the described robust structure and decreasing greedy thresholding rule. In addition to algorithmic and theoretical contributions, we also demonstrate in several practical scenarios that our procedure matches (and in some cases outperforms) the S IEVE -S TREAMING algorithm [8] (see Section 5) ? even though we allow the latter to know in advance which elements will be removed from the dataset. 2 Problem Statement We consider a potentially large universe of elements V of size n equipped with a normalized monotone submodular set function f : 2V ? R?0 defined on V . We say that f is monotone if for any two sets X ? Y ? V we have f (X) ? f (Y ). The set function f is said to be submodular if for any two sets X ? Y ? V and any element e ? V \ Y it holds that f (X ? {e}) ? f (X) ? f (Y ? {e}) ? f (Y ). We use f (Y | X) to denote the marginal gain in the function value due to adding the elements of set Y to set X, i.e. f (Y | X) := f (X ? Y ) ? f (X). We say that f is normalized if f (?) = 0. The problem of maximizing a monotone submodular function subject to a cardinality constraint, i.e., max f (Z), Z?V,|Z|?k (1) has been studied extensively. It is well-known that a simple greedy algorithm (henceforth refered to as G REEDY) [7], which starts from an empty set and then iteratively adds the element with highest marginal gain, provides a (1 ? e?1 )-approximation. However, it requires repeated access to all elements of the dataset, which precludes it from use in large-scale machine learning applications. 2 We say that a set S is robust for a parameter m if, for any set E ? V such that |E| ? m, there is a subset Z ? S \ E of size at most k such that f (Z) ? cf (OPT(k, V \ E)), where c > 0 is an approximation ratio. We use OPT(k, V \ E) to denote the optimal subset of size k of V \ E (i.e., after the removal of elements in E): OPT(k, V \ E) ? argmax f (Z). Z?V \E,|Z|?k In this work, we are interested in solving a robust version of Problem (1) in the setting that consists of the following two stages: (i) streaming and (ii) query stage. In the streaming stage, elements from the ground set V arrive in a streaming fashion in an arbitrary order. Our goal is to design a one-pass streaming algorithm that has oracle access to f and retains a small set S of elements in memory. In addition, we want S to be a robust summary, i.e., S should both contain elements that maximize the objective value, and be robust against the removal of prespecified number of elements m. In the query stage, after any set E of size at most m is removed from V , the goal is to return a set Z ? S \ E of size at most k such that f (Z) is maximized. Related work. A robust, non-streaming version of Problem (1) was first introduced in [9]. In that setting, the algorithm must output a set Z of size k which maximizes the smallest objective value guaranteed to be obtained after a set of size m is removed, that is, max min f (Z \ E). Z?V,|Z|?k E?Z,|E|?m The work [10] ? provides the first constant (0.387) factor approximation result to this problem, valid for m = o( k). Their solution consists of buckets of size O(m2 log k) that are constructed greedily, one after another. Recently, in [11], a centralized algorithm PRO has been proposed that achieves the same approximation result and allows for a greater robustness m = o(k). PRO constructs a set that is arranged into partitions consisting of buckets whose sizes increase exponentially with the partition index. In this work, we use a similar structure for the robust set but, instead of filling the buckets greedily one after another, we place an element in the first bucket for which the gain of adding the element is above the corresponding threshold. Moreover, we introduce a novel analysis that allows us to be robust to any number of removals m as long as we are allowed to use O(m log2 k) memory. Recently, submodular streaming algorithms (e.g. [5], [12] and [13]) have become a prominent option for scaling submodular optimization to large-scale machine learning applications. A popular submodular streaming algorithm S IEVE -S TREAMING [8] solves Problem (1) byperforming  one pass k over the data, and achieves a (0.5 ? )-approximation while storing at most O k log elements.  Our algorithm extends the algorithmic ideas of S IEVE -S TREAMING, such as greedy thresholding, to the robust setting. In particular, we introduce a new exponentially decreasing thresholding scheme that, together with an innovative analysis, allows us to obtain a constant-factor approximation for the robust streaming problem. Recently, robust versions of submodular maximization have been considered in the problems of influence maximization (e.g, [3], [14]) and budget allocation ([15]). Increased interest in interactive machine learning methods has also led to the development of interactive and adaptive submodular optimization (see e.g. [16], [17]). Our procedure also contains the interactive component, as we can compute the robust summary only once and then provide different sub-summaries that correspond to multiple different removals (see Section 5.2). Independently and concurrently with our work, [18] gave a streaming algorithm for robust submodular maximization under the cardinality constraint. Their approach provides a 1/2 ? ? approximation guarantee. However, their algorithm uses O(mk log k/?) memory. While the memory requirement of their method increases linearly with k, in the case of our algorithm this dependence is logarithmic. 3 Data Stream decreasing thresholds ? k ? buckets ?/2 (k / 2) ? buckets partitions 2? ?/k 1? Set S Figure 1: Illustration of the set S returned by STAR-T. It consists of dlog ke + 1 partitions such that each partition i contains wdk/2i e buckets of size 2i (up to rounding). Moreover, each partition i has its corresponding threshold ? /2i . 3 A Robust Two-stage Procedure Our approach consists of the streaming Algorithm 1, which we call Streaming Robust submodular algorithm with Partitioned Thresholding (STAR-T). This algorithm is used in the streaming stage, while Algorithm 2, which we call STAR-T-G REEDY, is used in the query stage. As the input, STAR-T requires a non-negative monotone submodular function f , cardinality constraint k, robustness parameter m and thresholding parameter ? . The parameter ? is an ?approximation to f (OPT(k, V \ E)), for some ? ? (0, 1] to be specified later. Hence, it depends on f (OPT(k, V \ E)), which is not known a priori. For the sake of clarity, we present the algorithm as if f (OPT(k, V \ E)) were known, and in Section 4.1 we show how f (OPT(k, V \ E)) can be approximated. The algorithm makes one pass over the data and outputs a set of elements S that is later used in the query stage in STAR-T-G REEDY. The set S (see Figure 1 for an illustration) is divided into dlog ke + 1 partitions, where every partition i ? {0, . . . , dlog ke} consists of wdk/2i e buckets Bi,jl, j ? {1, . m. . , wdk/2i e}. Here, w ? N+ is a memory parameter that depends on m; we use w ? 4dlogkkem in our asymptotic theory, while our numerical results show that w = 1 works well in practice. Every bucket Bi,j stores at most min{k, 2i } elements. If |Bi,j | = min{2i , k}, then we say that Bi,j is full. Every partition has a corresponding threshold that is exponentially decreasing with the partition index i as ? /2i . For example, the buckets in the first partition will only store elements that have marginal value at least ? . Every element e ? V arriving on the stream is assigned to the first non-full bucket Bi,j for which the marginal value f (e | Bi,j ) is at least ? /2i . If there is no such bucket, the element will not be stored. Hence, the buckets are disjoint sets that in the end (after one pass over the data) can have a smaller number of elements than specified by their corresponding cardinality constraints, and some of them might even be empty. The set S returned by STAR-T is the union of all the buckets. In the second stage, STAR-T-G REEDY receives as input the set S constructed in the streaming stage, a set E ? S that we think of as removed elements, and the cardinality constraint k. The algorithm then returns a set Z, of size at most k, that is obtained by running the simple greedy algorithm G REEDY on the set S \ E. Note that STAR-T-G REEDY can be invoked for different sets E. 4 Theoretical Bounds In this section we discuss our main theoretical results. We initially assume that the value f (OPT(k, V \ E)) is known; later, in Section 4.1, we remove this assumption. The more detailed versions of our proofs are given in the supplementary material. We begin by stating the main result. 4 Algorithm 1 STreAming Robust - Thresholding submodular algorithm (STAR-T) Input: Set V , k, ? , w ? N+ 1: Bi,j ? ? for all 0 ? i ? dlog ke and 1 ? j ? wdk/2i e 2: for each element e in the stream do 3: for i ? 0 to dlog ke do . loop over partitions 4: for j ? 1 to wdk/2i e do . loop over buckets 5: if |Bi,j | < min{2i , k} and f (e | Bi,j ) ? ? / min{2i , k} then 6: Bi,j ? Bi,j ? {e} 7: break: proceed to the next element in the stream S 8: S ? i,j Bi,j 9: return S Algorithm 2 STAR-T- G REEDY Input: Set S, query set E and k 1: Z ? G REEDY(k, S \ E) 2: return Z Theorem 4.1 Let f be a normalized monotone submodular function defined over the set V . l ground m 4dlog kem Given a cardinality constraint k and parameter m, for a setting of parameters w ? and k ?= 2+ 1  f (OPT(k, V (1?e?1 )  1 1? dlog ke ?1/3 (1?e ) \ E)), STAR-T performs a single pass over the data set and constructs a set S of size at most O((k + m log k) log k) elements. For such a set S and any set E ? V such that |E| ? m, STAR-T-G REEDY yields a set Z ? S \ E of size at most k with f (Z) ? c ? f (OPT(k, V \ E)),   1 for c = 0.149 1 ? dlog ke . Therefore, as k ? ?, the value of c approaches 0.149. Proof sketch. We first consider the case when there is a partition i? in S such that at least half of its buckets are full. We show that there is at least one full bucket Bi? ,j such that f (Bi? ,j \ E) is only a constant factor smaller than f (OPT(k, V \ E)), as long as the threshold ? is set close to f (OPT(k, V \ E)). We make this statement precise in the following lemma: Lemma 4.2 If there exists a partition in S such that at least half of its buckets are full, then for the set Z produced by STAR-T-G REEDY we have    4m f (Z) ? 1 ? e?1 1 ? ?. (2) wk To prove this lemma, we first observe that from the properties of G REEDY it follows that  f (Z) = f (G REEDY(k, S \ E)) ? 1 ? e?1 f (Bi? ,j \ E) . Now it remains to show that f (Bi? ,j \ E) is close to ? . We observe that for any full bucket Bi? ,j , we have |Bi? ,j | = min{2i , k}, so its objective value f (Bi? ,j ) is at least ? (every element added to this bucket increases its objective value by at least ? / min{2i , k}). On average, |Bi? ,j ? E| is relatively small, and hence we can show that there exists some full bucket Bi? ,j such that f (Bi? ,j \ E) is close to f (Bi? ,j ). Next, we consider the other case, i.e., when for every partition, more than half of its buckets are not full after the execution of STAR-T. For every partition i, we let Bi denote a bucket that is not fully populated and for which |Bi ? E| is minimized over all the buckets of that partition. Then, we look at such a bucket in the last partition: Bdlog ke . We provide two lemmas that depend on f (Bdlog ke ). If ? is set to be small compared to f (OPT(k, V \ E)): 5 ? Lemma 4.3 shows that if f (Bdlog ke ) is close to f (OPT(k, V \ E)), then our solution is within a constant factor of f (OPT(k, V \ E)); ? Lemma 4.4 shows that if f (Bdlog ke ) is small compared to f (OPT(k, V \ E)), then our solution is again within a constant factor of f (OPT(k, V \ E)). Lemma 4.3 If there does not exist a partition of S such that at least half of its buckets are full, then for the set Z produced by STAR-T-G REEDY we have     4m ?1/3 f (Z) ? 1 ? e f Bdlog ke ? ? , wk where Bdlog ke is a not-fully-populated bucket in the last partition that minimizes Bdlog ke ? E and |E| ? m. Using standard properties of submodular functions and the G REEDY algorithm we can show that     4m ?1/3 f (Z) = f (G REEDY(k, S \ E)) ? 1 ? e f Bdlog ke ? ? . wk The complete proof of this result can be found in Lemma B.2, in the supplementary material. Lemma 4.4 If there does not exist a partition of S such that at least half of its buckets are full, then for the set Z produced by STAR-T-G REEDY,  f (Z) ? (1 ? e?1 ) f (OP T (k, V \ E)) ? f (Bdlog ke ) ? ? , where Bdlog ke is any not-fully-populated bucket in the last partition. To prove this lemma, we look at two sets X and Y , where Y contains all the elements from OPT(k, V \ E) that are placed in the buckets that precede bucket Bdlog ke in S, and set X := OPT(k, V \ E) \ Y . By monotonicity and submodularity of f , we bound f (Y ) by:  X  f (Y ) ? f (OPT(k, V \ E)) ? f (X) ? f (OPT(k, V \ E)) ? f Bdlog ke ? f e Bdlog ke . e?X  To bound the sum on the right hand side we use that for every e ? X we have f e Bdlog ke < ?k , which holds due to the fact that Bdlog ke is a bucket in the last partition and is not fully populated.  We conclude the proof by showing that f (Z) = f (G REEDY(k, S \ E)) ? 1 ? e?1 f (Y ). Equipped with the above results, we proceed to prove our main result. Proof of Theorem 4.1. First, we prove the bound on the size of S: dlog ke |S| = X dlog ke wdk/2i e min{2i , k} ? i=0 By setting w ? l 4dlog kem k X w(k/2i + 1)2i ? (log k + 5)wk. (3) i=0 m we obtain S = O((k + m log k) log k).  ?1/3 Next, we show the approximation guarantee. We first define ? := 4m , and wk , ?1 := 1 ? e  ?2 := 1 ? e?1 . Lemma 4.3 and 4.4 provide two bounds on f (Z), one increasing and one decreasing in f (Bdlog ke ). By balancing out the two bounds, we derive   ?1 ?2 f (Z) ? (f (OPT(k, V \ E)) ? (1 + ?)? ), (4) ?1 + ?2 with equality for f (Bdlog ke ) = ?2 f (OPT(k,V \E))?(?2 ???1 )? . ?2 +?1 Next, as ? ? 0, we can observe that Eq. (4) is decreasing, while the bound on f (Z) given by Lemma 4.2 is increasing in ? for ? < 1. Hence, by balancing out the two inequalities, we obtain our final bound 1 f (Z) ? (5) 2 1 f (OPT(k, V \ E)). ?2 (1??) + ?1 6 l m For w ? 4dlogkkem we have ? ? 1/dlog ke, and hence, by substituting ?1 and ?2 in Eq. (5), we prove our main result:    1 ? e?1/3 1 ? e?1 1 ? dlog1 ke  f (OPT(k, V \ E)) f (Z) ? 2 1 ? e?1/3 + (1 ? e?1 )   1 ? 0.149 1 ? f (OPT(k, V \ E)). dlog ke 2 4.1 Algorithm without access to f (OPT(k, V \ E)) Algorithm STAR-T requires in its input a parameter ? which is a function of an unknown value f (OPT(k, V \ E)). To deal with this shortcoming, we show how to extend the idea of [8] of maintaining multiple parallel instances of our algorithm in order to approximate f (OPT(k, V \ E)). For a given constant  > 0, this approach increases the space by a factor of log1+ k and provides a (1 + )-approximation compared to the value obtained in Theorem 4.1. More precisely, we prove the following theorem. Theorem 4.5 For any given constant  > 0 there exists a parallel variant of STAR-T that makes one pass over the stream and outputs a collection of sets S of total size O (k + m log k) log k log1+ k with the following property: There exists a set S ? S such that applying STAR-T-G REEDY on S yields a set Z ? S \ E of size at most k with   0.149 1 f (Z) ? 1? f (OPT(k, V \ E)). 1+ dlog ke The proof of this theorem, along with a description of the corresponding algorithm, is provided in Appendix E. 5 Experiments In this section, we numerically validate the claims outlined in the previous section. Namely, we test the robustness and compare the performance of our algorithm against the S IEVE -S TREAMING algorithm that knows in advance which elements will be removed. We demonstrate improved or matching performance in two different data summarization applications: (i) the dominating set problem, and (ii) personalized movie recommendation. We illustrate how a single robust summary can be used to regenerate recommendations corresponding to multiple different removals. 5.1 Dominating Set In the dominating set problem, given a graph G = (V, M ), where V represents the set of nodes and M stands for edges, the objective function is given by f (Z) = |N (Z) ? Z|, where N (Z) denotes the neighborhood of Z (all nodes adjacent to any node of Z). This objective function is monotone and submodular. We consider two datasets: (i) ego-Twitter [19], consisting of 973 social circles from Twitter, which form a directed graph with 81306 nodes and 1768149 edges; (ii) Amazon product co-purchasing network [20]: a directed graph with 317914 nodes and 1745870 edges. Given the dominating set objective function, we run STAR-T to obtain the robust summary S. Then we compare the performance of STAR-T-G REEDY, which runs on S, against the performance of S IEVE -S TREAMING, which we allow to know in advance which elements will be removed. We also compare against a method that chooses the same number of elements as STAR-T, but does so uniformly at random from the set of all elements that will not be removed (V \ E); we refer to it as R ANDOM. Finally, we also demonstrate the peformance of STAR-T-S IEVE, a variant of our algorithm that uses the same robust summary S, but instead of running G REEDY in the second stage, it runs S IEVE -S TREAMING on S \ E. 7 8000 6000 4000 2000 0 10 20 30 40 50 60 70 80 90 ?104 2 0.5 Obj. value Obj. value 4000 3000 2000 20 30 40 50 60 70 20 10 20 30 40 50 60 70 80 90 0 100 10 30 80 90 100 90 60 Star-T-Greedy Star-T-Sieve Sieve-Str Random 50 1 0.5 0 10 70 (f) Movies, by genre ?104 1.5 50 Cardinality k Star-T-Greedy Star-T-Sieve Sieve-Str Greedy 40 30 20 1000 0 10 30 (d) ego-Twitter,|E| = 2k 2 Star-T-Greedy Star-T-Sieve Sieve-Str Random 5000 40 Cardinality k (b) Amazon communities,|E| = 2k 6000 Star-T-Greedy Star-T-Sieve Sieve-Str Greedy 50 1 Cardinality k 7000 (e) Movies, already-seen 60 1.5 0 10 100 (c) ego-Twitter,|E| = k Star-T-Greedy Star-T-Sieve Sieve-Str Random Obj. value 10000 Avg. obj. value Avg. obj. value 2.5 Star-T-Greedy Star-T-Sieve Sieve-Str Random Obj. value (a) Amazon communities,|E| = k 12000 20 30 Cardinality k 40 50 60 70 80 90 100 Cardinality k 10 10 30 50 70 90 110 130 150 170 190 Cardinality k Figure 2: Numerical comparisons of the algorithms STAR-T-G REEDY, STAR-T-S IEVE and S IEVE S TREAMING. Figures 2(a,c) show the objective value after the random removal of k elements from the set S, for different values of k. Note that E is sampled as a subset of the summary of our algorithm, which hurts the performance of our algorithm more than the baselines. The reported numbers are averaged over 100 iterations. STAR-T-G REEDY, STAR-T-S IEVE and S IEVE -S TREAMING perform comparably (STAR-T-G REEDY slightly outperforms the other two), while R ANDOM is significantly worse. In Figures 2(b,d) we plot the objective value for different values of k after the removal of 2k elements from the set S, chosen greedily (i.e., by iteratively removing the element that reduces the objective value the most). Again, STAR-T-G REEDY, STAR-T-S IEVE and S IEVE -S TREAMING perform comparably, but this time S IEVE -S TREAMING slightly outperforms the other two for some values of k. We observe that even when we remove more than k elements from S, the performance of our algorithm is still comparable to the performance of S IEVE -S TREAMING (which knows in advance which elements will be removed). We provide additional results in the supplementary material. 5.2 Interactive Personalized Movie Recommendation The next application we consider is personalized movie recommendation. We use the MovieLens 1M database [21], which contains 1000209 ratings for 3900 movies by 6040 users. Based on these ratings, we obtain feature vectors for each movie and each user by using standard low-rank matrix completion techniques [22]; we choose the number of features to be 30. For a user u, we use the following monotone submodular function to recommend a set of movies Z: X X fu (Z) = (1 ? ?) ? hvu , vz i + ? ? max hvm , vz i . z?Z m?M z?Z The first term aggregates the predicted scores of the chosen movies z ? Z for the user u (here vu and vz are non-normalized feature vectors of user u and movie z, respectively). The second term corresponds to a facility-location objective that measures how well the set Z covers the set of all movies M [4]. Finally, ? is a user-dependent parameter that specifies the importance of global movie coverage versus high scores of individual movies. Here, the robust setting arises naturally since we do not have complete information about the user: when shown a collection of top movies, it will likely turn out that they have watched (but not rated) many of them, rendering these recommendations moot. In such an interactive setting, the user may also require (or exclude) movies of a specific genre, or similar to some favorite movie. We compare the performance of our algorithms STAR-T-G REEDY and STAR-T-S IEVE in such scenarios against two baselines: G REEDY and S IEVE -S TREAMING (both being run on the set V \ E, i.e., knowing the removed elements in advance). Note that in this case we are able to afford running 8 G REEDY, which may be infeasible when working with larger datasets. Below we discuss two concrete practical scenarios featured in our experiments. Movies by genre. After we have built our summary S, the user decides to watch a drama today; we retrieve only movies of this genre from S. This corresponds to removing 59% of the universe V . In Figure 2(f) we report the quality of our output compared to the baselines (for user ID 445 and ? = 0.95) for different values of k. The performance of STAR-T-G REEDY is within several percent of the performance of G REEDY (which we can consider as a tractable optimum), and the two sieve-based methods STAR-T-S IEVE and S IEVE -S TREAMING display similar objective values. Already-seen movies. We randomly sample a set E of movies already watched by the user (500 out of all 3900 movies). To obtain a realistic subset, each movie is sampled proportionally to its popularity (number of ratings). Figure 2(e) shows the performance of our algorithm faced with the removal of E (user ID = 445, ? = 0.9) for a range of settings of k. Again, our algorithm is able to almost match the objective values of G REEDY (which is aware of E in advance). Recall that we are able to use the same precomputed summary S for different removed sets E. This summary was built for parameter w = 1, which theoretically allows for up to k removals. However, despite having |E|  k in the above scenarios, our performance remains robust; this indicates that our method is more resilient in practice than what the proved bound alone would guarantee. 6 Conclusion We have presented a new robust submodular streaming algorithm STAR-T based on a novel partitioning structure and an exponentially decreasing thresholding rule. It makes one pass over the data and retains a set of size O (k + m log k) log2 k . We have further shown that after the removal of any m elements, a simple greedy algorithm that runs on the obtained set achieves a constant-factor approximation guarantee for robust submodular function maximization. In addition, we have presented two numerical studies where our method compares favorably against the S IEVE -S TREAMING algorithm that knows in advance which elements will be removed. Acknowledgment. IB and VC?s work was supported in part by the European Research Council (ERC) under the European Union?s Horizon 2020 research and innovation program (grant agreement number 725594), in part by the Swiss National Science Foundation (SNF), project 407540_167319/1, in part by the NCCR MARVEL, funded by the Swiss National Science Foundation, in part by Hasler Foundation Switzerland under grant agreement number 16066 and in part by Office of Naval Research (ONR) under grant agreement number N00014-16-R-BA01. 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Simple Strategies for Recovering Inner Products from Coarsely Quantized Random Projections Ping Li Baidu Research, and Rutgers University [email protected] Martin Slawski Department of Statistics George Mason University [email protected] Abstract Random projections have been increasingly adopted for a diverse set of tasks in machine learning involving dimensionality reduction. One specific line of research on this topic has investigated the use of quantization subsequent to projection with the aim of additional data compression. Motivated by applications in nearest neighbor search and linear learning, we revisit the problem of recovering inner products (respectively cosine similarities) in such setting. We show that even under coarse scalar quantization with 3 to 5 bits per projection, the loss in accuracy tends to range from ?negligible? to ?moderate?. One implication is that in most scenarios of practical interest, there is no need for a sophisticated recovery approach like maximum likelihood estimation as considered in previous work on the subject. What we propose herein also yields considerable improvements in terms of accuracy over the Hamming distance-based approach in Li et al. (ICML 2014) which is comparable in terms of simplicity. 1 Introduction The method of random projections (RPs) for linear dimensionality reduction has become more and more popular over the years after the basic theoretical foundation, the celebrated JohnsonLindenstrauss (JL) Lemma [12, 20, 33], had been laid out. In a nutshell, it states that it is possible to considerably lower the dimension of a set of data points by means of a linear map in such a way that squared Euclidean distances and inner products are roughly preserved in the low-dimensional representation. Conveniently, a linear map of this sort can be realized by a variety of random matrices [1, 2, 18]. The scope of applications of RPs has expanded dramatically in the course of time, and includes dimension reduction in linear classification and regression [14, 30], similarity search [5, 17], compressed sensing [8], clustering [7, 11], randomized numerical linear algebra and matrix sketching [29], and differential privacy [21], among others. The idea of achieving further data compression by means of subsequent scalar quantization of the projected data has been considered for a while. Such setting can be motivated from constraints concerning data storage and communication, locality-sensitive hashing [13, 27], or the enhancement of privacy [31]. The extreme case of one-bit quantization can be associated with two seminal works in computer science, the SDP relaxation of the MAXCUT problem [16] and the simhash [10]. One-bit compressed sensing is introduced in [6], and along with its numerous extensions, has meanwhile developed into a subfield within the compressed sensing literature. A series of recent papers discuss quantized RPs with a focus on similarity estimation and search. The papers [25, 32] discuss quantized RPs with a focus on image retrieval based on nearest neighbor search. Independent of the specific application, [25, 32] provide JL-type statements for quantized RPs, and consider the trade-off between the number of projections and the number of bits per projection under a given budget of bits as it also appears in the compressed sensing literature [24]. The paper [19] studies approximate JL-type results for quantized RPs in detail. The approach to quantized RPs taken in the present paper follows [27, 28] 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. in which the problem of recovering distances and inner products is recast within the framework of classical statistical point estimation theory. The paper [28] discusses maximum likelihood estimation in this context, with an emphasis on the aforementioned trade-off between the number of RPs and the bit depth per projection. In the present paper we focus on the much simpler and computationally much more convenient approach in which the presence of the quantizer is ignored, i.e., quantized data are treated in the same way as full-precision data. We herein quantify the loss of accuracy of this approach relative to the full-precision case, which turns out to be insignificant in many scenarios of practical interest even under coarse quantization with 3 to 5 bits per projection. Moreover, we show that the approach compares favorably to the Hamming distance-based (or equivalently collision-based) scheme in [27] which is of similar simplicity. We argue that both approaches have their merits: the collision-based scheme performs better in preserving local geometry (the distances of nearby points), whereas the one studied in more detail herein yields better preservation globally. Notation. For a positive integer m, we let [m] = {1, . . . , m}. For l ? [m], v(l) denotes the l-th component of a vector v ? Rm ; if there is no danger of confusion with another index, the brackets in the subscript are omitted. I(P ) denotes the indicator function of expression P . Supplement: Proofs and additional experimental results can be found in the supplement. Basic setup. Let X = {x1 , . . . , xn } ? Rd be a set of input data with squared Euclidean norms ?2i := kxi k22 , i ? [n]. We think of d being large. RPs reduce the dimensionality of the input data by means of a linear map A : Rd ? Rk , k  d. We assume throughout the paper that the map A is realized by a random matrix with i.i.d. entries from the standard Gaussian distribution, i.e., Alj ? N (0, 1), l ? [k], j ? [d]. One standard goal of RPs is to approximately preserve distances in X while lowering the dimension, i.e., kAxi ? Axj k22 /k ? kxi ? xj k22 for all (i, j). This is implied by approximate inner product preservation hxi , xj i ? hAxi , Axj i /k for all (i, j). For the time being, we assume that it is possible to compute and store the squared norms {?2i }ni=1 , and to rescale the input data to unit norm, i.e., one first forms x ei ? xi /?i , i ? [n], before applying hx ,x i A. In this case, it suffices to recover the (cosine) similarities ?ij := ?ii ?jj = he xi , x ej i, i, j ? [n], of the input data X from their compressed representation Z = {z1 , . . . , zn }, zi := Ae xi , i ? [n]. 2 Estimation of cosine similarity based on full-precision RPs As preparation for later sections, we start by providing background concerning the usual setting without quantization. Let (Z, Z 0 )r be random variables having a bivariate Gaussian distribution with zero mean, unit variance, and correlation r ? (?1, 1):     0 1 r 0 (Z, Z )r ? N2 , . (1) 0 r 1 Let further x, x0 be a generic pair of points from X , and let z := Ae x, z 0 := Ae x0 be the counterpart in 0 k 0 Z. Then the components {(z(l) , z(l) )}l=1 of (z, z ) are distributed i.i.d. as in (1) with r = ? =: he x, x e0 i. 0 Hence the problem of recovering the cosine similarity of x and x can be re-cast as estimating the correlation from an i.i.d. sample of k bivariate Gaussian random variables. To simplify our exposition, we henceforth assume that 0 ? ? < 1 as this can easily be achieved by flipping the sign of one of x or x0 . The standard estimator of ? is what is called the ?linear estimator? herein: ?blin = k 1 1X 0 hz, z 0 i = . z(l) z(l) k k (2) l=1 As pointed out in [26] this estimator can be considerably improved upon by the maximum likelihood estimator (MLE) given (1):    1 1 1 1 1 0 2 1 2 2 0 ?bMLE = argmax ? log(1 ? r ) ? kzk2 + kz k2 ? hz, z i 2r . (3) 2 2 1 ? r2 k k k r The estimator ?bMLE is not available in closed form, which is potentially a serious concern since it needs to be evaluated for numerous different pairs of data points. However, this can be addressed 2 n  o 2 2 by tabulation of the two statistics kzk2 + kz 0 k2 /k, hz, z 0 i /k and the corresponding solutions ?bMLE over a sufficiently fine grid. At processing time, computation of ?bMLE can then be reduced to a look-up in a pre-computed table. One obvious issue of ?blin is that it does not respect the range of the underlying parameter. A natural fix is the use of the ?normalized linear estimator? ?bnorm = hz, z 0 i /(kzk2 kz 0 k2 ). (4) When comparing different estimators of ? in terms of statistical accuracy, we evaluate the mean squared error (MSE), possibly asymptotically as the number of RPs k ? ?. Specifically, we consider MSE? (b ?) = E? [(? ? ?b)2 ] = Bias2? (b ?) + Var? (b ?), Bias? (b ?) := E? [b ?] ? ?, (5) where ?b is some estimator, and the subscript ? indicates that expectations are taken with respect to a sample (z, z 0 ) following the bivariate normal distribution in (1) with r = ?. It turns out that ?bnorm and ?bMLE can have dramatically lower (asymptotic) MSEs than ?blin for large values of ?, i.e., for points of high cosine similarity. It can be shown that (cf. [4], p.132, and [26]) Var? (b ?lin ) = (1 + ?2 )/k, Bias? (b ?lin ) = 0, Bias2? (b ?norm ) Bias2? (b ?MLE ) 2 2 = O(1/k ), = O(1/k ), (6) 2 2 2 Var? (b ?norm ) = (1 ? ? ) /k + O(1/k ), (7) (1??2 )2 1+?2 /k + O(1/k ). (8) Var? (b ?MLE ) = 2 While for ? = 0, the (asymptotic) MSEs are the same, we note that the leading terms of the MSEs of ?bnorm and ?bMLE decay at rate ?((1 ? ?)2 ) as ? ? 1, whereas the MSE of ?blin grows with ?. The following table provides the asymptotic MSE ratios of ?blin and ?bnorm for selected values of ?. ? 0.5 0.6 0.7 0.8 0.9 0.95 0.99 MSE? (b ?lin ) MSE? (b ?norm ) 2.2 3.3 5.7 12.6 50 200 5000 In conclusion, if it is possible to pre-compute and store the norms of the data prior to dimensionality reduction, a simple form of normalization can yield important benefits with regard to the recovery of inner products and distances for pairs of points having high cosine similarity. The MLE can provide a further refinement, but the improvement over ?bnorm can be at most by a factor of 2. 3 Estimation of cosine similarity based on quantized RPs The following section contains our main results. After introducing preliminaries regarding quantization, we review previous approaches to the problem, before analyzing estimators following a different paradigm. We conclude with a comparison and some recommendations about what to use in practice. Quantization. After obtaining the projected data Z, the next step is scalar quantization. Let t = (t1 , . . . , tK?1 ) with 0 = t0 < t1 < . . . < tK?1 < tK = +? be a set of thresholds inducing a partitioning of the positive real line into K intervals {[ts?1 , ts ), s ? [K]}, and let M = {?1 , . . . , ?K } be a set of codes with ?s representing interval [ts?1 , ts ), s ? [K]. Given t and M, the scalar quantizer (or quantization map) is defined by PK Q : R ? M? := ?M ? M, z 7? Q(z) = sign(z) s=1 ?s I(|z| ? [ts?1 , ts )). (9)  k The projected and quantized data result as Q = {qi }ni=1 ? (M? )k , qi = Q(zi(l) ) l=1 , where zi(l) denotes the l-th component of zi ? Z, l ? [k], i ? [n]. The bit depth b of the quantizer is given by b := 1 + log2 (K). For simplicity, we only consider the case where b is an integer. The case b = 1 is well-studied [10, 27] and is hence disregarded in our analysis to keep our exposition compact. Bin-based vs. code-based approaches. Let q = Q(z) and q 0 = Q(z 0 ) be the points resulting from quantization of the generic pair z, z 0 in the previous section. In this paper, we distinguish between two basic paradigms for estimating the cosine similarity of the underlying pair x, x0 from q, q 0 . The first paradigm, which we refer to as bin-based estimation, does not make use of the specific values of 3 the codes M? , but only of the intervals (?bins?) associated with each code. This is opposite to the second paradigm, referred to as code-based estimation which only makes use of the values of the codes. As we elaborate below, an advantage of the bin-based approach is that working with intervals reflects the process of quantization more faithfully and hence can be statistically more accurate; on the other hand, a code-based approach tends to be more convenient from the point of view computation. In this paper, we make a case for the code-based approach by showing that the loss in statistical accuracy can be fairly minor in several regimes of practical interest. Lloyd-Max (LM) quantizer. With b respectively K being fixed, one needs to choose the thresholds t and the codes M of the quantizer (the second is crucial only for a code-based approach). In our setting, with zi(l) ? N (0, 1), i ? [n], l ? [k], which is inherited from the distribution of the entries of A, a standard choice is LM quantization [15] which minimizes the squared distortion error: (t? , ?? ) = argmin Eg?N (0,1) [{g ? Q(g; t, ?)}2 ]. (10) t,? Problem (10) can be solved by an iterative scheme that alternates between optimization of t for fixed ? and vice versa. That scheme can be shown to deliver the global optimum [22]. In the absence of any prior information about the cosine similarities that we would like to recover, (10) appears as a reasonable default whose use for bin-based estimation has been justified in [28]. In the limit of cosine similarity ? ? 1, it may seem more plausible to use (10) with g replaced by its square, and taking the root of the resulting optimal thresholds and codes. However, it turns out that empirically this yields reduced performance more often than improvements, hence we stick to (10) in the sequel. 3.1 Bin-based approaches 0 k MLE. Given a pair q = (q(l) )kl=1 and q 0 = (q(l) )l=1 of projected and quantized points, maximum likelihood estimation of the underlying cosine similarity ? is studied in depth in [28]. The associated likelihood function L(r) is based on bivariate normal probabilities of the form Pr (Z ? [ts?1 , ts ), Z 0 ? [tu?1 , tu )), P?r (Z ? [ts?1 , ts ), Z 0 ? [tu?1 , tu )) with (Z, Z 0 )r as in (1). It is shown in [28] that the MLE with b ? 2 can be more efficient at the bit level than common single-bit quantization [10, 16]; the optimal choice of b increases with ?. While statistically optimal in the given setting, the MLE remains computationally cumbersome even when using the approximation in [28] because it requires cross-tabulation of the empirical frequencies corresponding to the bivariate normal probabilities above. This makes the use of the MLE unattractive particularly in situations in which it is not feasible to materialize all O(n2 ) pairwise similarities estimable from (qi , qj )i<j so that they would need to be re-evaluated frequently. approach Collision-based estimator. The collision-based estimatorproposed in [27] is a bin-based  Pk 0 as the MLE. The similarity ? is estimated as ?bcol = ??1 I(q = q )/k , where the map (l) l=1 (l) ? : [0, 1] ? [0, 1] is defined by r 7? ?(r) = Pr (Q(Z) = Q(Z 0 )), shown to be monotonically increasing in [27]. Compared to the MLE, ?bcol uses less information ? it only counts ?collisions?, 0 i.e., events {q(l) = q(l) }. The loss in statistical efficiency is moderate for b = 2, in particular for ? close to 1. However, as b increases that loss becomes more and more substantial; cf. Figure 1. On the positive side, ?bcol is convenient to compute given that the evaluation of the function ??1 can be approximated by employing a look-up table after tabulating ? on a fine grid. 1.5 0 -0.5 -1 -1.5 -2 -2.5 -3 0 0.2 0.4 0.6 0.8 1 5 Relative Efficiency log10(MSE) 0.5 30 b=4 b=3 b=2 Relative Efficiency 1 4 b=2 3 2 1 0.2 0.4 0.6 0.8 1 20 b=4 10 1 0 0.2 0.4 0.6 0.8 1 Figure 1: (L): Asymptotic MSEs [27] of ?bcol (to be divided by k) for 2 ? b ? 4. (M,R): Asymptotic relative efficiencies MSE? (b ?col )/MSE? (b ?MLE ) for b ? {2, 4}, where ?bMLE is the MLE in [28]. 4 2 -2 b=2 -3 b=3 -4 b=4 -5 1.8 b=5 -6 b=6 -7 -8 -9 -10 -11 0 0.2 0.4 0.6 0.8 b bound on bias2 2 3 4 5 6 1.2 ? 10?1 7.2 ? 10?3 4.5 ? 10?4 2.8 ? 10?5 1.8 ? 10?6 1 1.6 variance log10(squared Bias) -1 1.4 1.2 1 0.8 0 0.2 0.4 Bias2? (b ?lin ) 0.6 Figure 2: (L): and the bound of Theorem 1. (M): uniform upper bounds on obtained from Theorem 1 by setting ? = 1. (R): Var? (b ?lin ) (to be divided by k). 3.2 0.8 1 Bias2? (b ?lin ) Code-based approaches In the code-based approach, we simply ignore the fact that the quantized data actually represent intervals and treat them precisely in the same way as full-precision data. Recovery of cosine similarity is performed by means of the estimator in ?2 with z, z 0 replaced by q, q 0 . Perhaps surprisingly, it turns out that depending on ? the loss of information incurred by this rather crude approach can be small already for bit depths between b = 3 and b = 5. That loss increases with ?, with a fundamental gap compared to bin-based approaches and to the full precision case in the limit ? ? 1. Linear estimator. We first consider ?blin = hq, q 0 i /k. We note that ?blin = ?blin,b depends on b; b = ? corresponds to the estimator ?blin = ?blin,? in ?2 denoted by the same symbol. A crucial difference between the code-based and the bin-based approaches discussed above is that the latter have vanishing asymptotic squared bias of the order O(k ?2 ) for any b [27, 28]. This is not the case for code-based approaches whose bias needs to be analyzed carefully. The exact bias of ?blin in dependence of ? and b can be evaluated exactly numerically. Numerical evaluations of bias and variance of estimators discussed in the present section only rely on the computation of coefficients ??,? defined by ??,? := E? [Q(Z)? Q(Z 0 )? ] = X K X  ? 0 0 ? ? (? 0 )? ?? s ?u P? Z ? ?(ts?1 , ts ), Z ? ? (tu?1 , tu ) , ?,? 0 ?{?1,1} s,u=1 (11) where ?, ? are non-negative integers and (Z, Z 0 ) are bivariate normal (1) with r = ?. Specifically, 2 we have E? [b ?lin ] = ?1,1 , Var? (b ?lin ) = (?2,2 ? ?1,1 )/k. In addition to exact numerical evaluation, we provide a bound on the bias of ?blin which quantifies explicitly the rate of decay in dependence b. Theorem 1. We have Bias2? (b ?lin ) ? 4?2 Db2 , where Db = 33/2 2? ?2b 12 2 ? 2.72 ? 2?2b . As shown in Figure 2 (L), the bound on the squared bias in Theorem 1 constitutes a reasonable proxy of the exact squared bias. The rate of decay is O(2?4b ). Moreover, it can be verified numerically that the variance in the full precision case upper bounds the variance for finite b, i.e., Var? (b ?lin,b ) ? Var? (b ?lin,? ), ? ? [0, 1). Combining bias and variance, we may conclude that depending on k, the MSE of ?blin based on coarsely quantized data does not tend to be far from what is achieved with full precision data. The following two examples illustrate this point. (i) Suppose k = 100 and b = 3. With full precision, we have MSE? (b ?lin,? ) = (1+?2 )/k ? [.01, .02]. From Figure 2 (M) and the observation that Var? (b ?lin,3 ) ? Var? (b ?lin,? ), we find that the MSE can go up by at most 7.2 ? 10?3 , i.e., it can at most double relative to the full precision case. (ii) Suppose k = 1000 and b = 4. With the same reasoning as in (i), the MSE under quantization can increase at most by a factor of 1.45 as compared to full precision data. Figure 3 shows that these numbers still tend to be conservative. In general, the difference of the MSEs for b = ? on the one hand and b ? {3, 4, 5} on the other hand gets more pronounced for large values of the similarity ? and large values of k. This is attributed to the (squared) bias of ?blin . In particular, it does not pay off to choose k significantly larger than the order of the squared bias. 5 k = 500 k = 1000 -3 k = 2000 log10(MSE) log10(MSE) k = 200 -2 k = 5000 -3.5 k = 10000 -4 0.4 0.6 k = 50 -2 k = 200 k = 500 k = 1000 -2.5 k = 2000 -3 k = 5000 -3.5 0.8 1 k = 10000 0.2 0.4 0.6 k = 50 k = 100 -2 k = 200 k = 500 -2.5 k = 1000 -3 k = 2000 k = 5000 -3.5 k = 10000 -4 b=4 0 k = 20 -1.5 k = 100 -4 b=3 0.2 -1 k = 20 -1.5 k = 100 -2.5 0 -1 k = 20 k = 50 log10(MSE) -1 -1.5 0.8 b=5 0 1 0.2 0.4 0.6 0.8 1 Figure 3: MSEs of ?blin for various k and b ? {3, 4, 5} (dotted). The solid (red) lines indicate the corresponding MSEs for ?blin in the full-precision case (b = ?). Normalized estimator. In the full precision case we have seen that simple normalization of the form ?bnorm = hz, z 0 i /(kzk2 kz 0 k2 ) can yield substantial benefits. Interestingly, it turns out that the counterpart ?bnorm = hq, q 0 i /(kqk2 kq 0 k2 ) for quantized data is even more valuable as it helps reducing the bias of ?blin = hq, q 0 i /k. This effect can be seen easily in the limit ? ? 1 in which case Bias? (b ?norm ) ? 0 by construction. In general, bias and variance can be evaluated as follows. Proposition 1. In terms of the coefficients ??,? defined in (11), as k ? ?, we have ? ? ? + O(k ?1 ) | Bias? [b ?norm ]| = ?1,1 2,0   2 ?1,1 (?4,0 +?2,2 ) ? 2? ?3,1 Var(b ?norm ) = k1 ?2,2 ? 1,1 + + O(k ?2 ). 2 3 4 ? 2? 2,0 2,0 2,0 Figure 4 (L,M) graphs the above two expressions. In particular, the plots highlight the reduction in bias compared to ?blin and the fact that the variance is decreasing in ? as for b = ?. While Proposition 1 is asymptotic, we verify a tight agreement in simulations for reasonably small k (cf. supplement). -1 1 -3 b=3 -4 b=4 -5 0.8 b=5 -6 b=6 -7 -2 k = 100 b=2 0.6 b=3 0.4 -8 -9 k = 200 -2.5 k = 500 -3 k = 1000 k = 2000 -3.5 k = 5000 -4k = 10000 0.2 -4.5 -10 -11 0 k = 20 -1.5 k = 50 log10(MSE) b=2 variance -2 b= log10(squared Bias) -1 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 -5 0 0.2 0.4 0.6 0.8 1 Figure 4: (L): Asymptotic Bias2? (b ?norm ) relative to Bias2? (b ?lin ). (M): Var? (b ?norm ) (asymptotic, to be divided by k). (R): MSEs of ?blin,4 vs. the MSEs of ?bcoll,2 using twice the number of RPs (comparison at the bit level). The stars indicate the values of ? at which the MSEs of the two estimators are equal. 3.3 Coding-based estimation vs. Collision-based estimation Both schemes are comparable in terms of simplicity, but at the level of statistical performance none of the two dominates the other. The collision-based approach behaves favorably in a high similarity regime as shows a comparison of MSE? (b ?col ) (b = 2) and MSE? (b ?norm ) (b = 4) at the bit level (Figure 4 (R)): since ?bcol uses only two bits for each of the k RPs, while ?bnorm uses twice as many bits, we have doubled the number of RPs for ?bcol . The values of ? for which the curves of the two approaches (for fixed k) intersect are indicated by stars. As k decreases from 104 to 102 , these values increase from about ? = 0.55 to ? = 0.95. In conclusion, ?bcol is preferable in applications in which high similarities prevail, e.g., in duplicate detection. On the other hand, for generic high-dimensional data, one would rather not expect ? to take high values given that two points drawn uniformly at random from the sphere are close to orthogonal with high probability. Figure 1 (L) shows that as b is raised, ?bcol requires ? to be increasingly closer to one to achieve lower MSE. By contrast, increasing b for the coding-based schemes yields improvements essentially for the 6 whole range of ?. An interesting phenomenon occurs in the limit ? ? 1. It turns out that the rate of decay of Var? (b ?norm ) is considerably slower than the rate of decay of Var? (b ?col ). Theorem 2. For any finite b, we have Var? (b ?norm ) = ?((1 ? ?)1/2 ), Var? (b ?col ) = ?((1 ? ?)3/2 ) as ? ? 1. The rate ?((1 ? ?)3/2 ) is the same as the MLE [28] which is slower than the rate ?((1 ? ?)2 ) in the full precision case (cf. ?2). We conjecture that the rate ?((1 ? ?)1/2 ) is intrinsic to code-based estimation as this rate is also obtained when computing the full precision MLE (3) with quantized data (i.e., z, z 0 gets replaced by q, q 0 ). 3.4 Quantization of norms Let us recall that according to our basic setup in ?1, we have assumed so far that it is possible to compute the norms ?i = kxi k22 , i ? [n], of the original data prior to projection and quantization, and store them in full precision to approximately recover inner products and squared distances via hxi , xj i ? ?i ?j ?bij , kxi ? xj k22 ? ?2i + ?2j ? 2?i ?j ?bij , where ?bij is an estimate of the cosine similarity of xi and xj . Depending on the setting, it may be required to quantize the {?i }ni=1 as well. It turns out that the MSE for estimating distances can be bi ? ?i |, where tightly bounded in terms of the MSE for estimating cosine similarities and max1?i?n |? n n b {?i }i=1 denote the quantized versions of {?i }i=1 ; the precise bound is stated in the supplement. 4 Empirical results: linear classification using quantized RPs One traditional application of RPs is dimension reduction in linear regression or classification with high-dimensional predictors [14, 30]. The results of ?3.2 suggest that as long as the number of RPs k are no more than a few thousand, subsequent scalar quantization to four bits is not expected to have much of a negative effect relative to using full precision data. In this section, we verify this hypothesis for four high-dimensional data sets from the UCI repository: arcene (d = 104 ), Dexter (d = 2 ? 104 ), farm (d = 5.5 ? 104 ) and PEMS (d = 1.4 ? 105 ). Setup. All data points are scaled to unit Euclidean norm before dimension reduction and scalar quantization based on the Lloyd-Max quantizer (10). The number of RPs k is varied according to {26 , 27 , . . . , 212 }. For each of these values for k, we consider 20 independent realizations of the random projection matrix A. Given projected and quantized data {q1 , . . . , qn }, we estimate the underlying cosine similarities ?ij as ?bij = ?b(qi , qj ), i, j ? [n], where ?b(qi , qj ) is a placeholder for either the collision-based estimator ?bcoll based on b = 2 bits or the normalized estimator ?bnorm for b ? {1, 2, 4, ?} using data {qi(l) , qj (l) }kl=1 ; one-bit quantization (b = 1) is here included as a reference. The {b ?ij }1?i,j?n are then used as a kernel matrix fed into LIBSVM [9] to train a binary classifier. Prediction on test sets is performed accordingly. LIBSVM is run with 30 different values of its tuning parameter C ranging from 10?3 to 104 . Results. A subset of the results is depicted in Figure 5 which is composed of three columns (one for each type of plot) and four rows (one for each data set). All results are averages over 20 independent sets of random projections. The plots in the left column show the minimum test errors over all 30 choices of the tuning parameter C under consideration in dependency of the number of RPs k. The plots in the middle column show the test errors in dependency of C for a selected value of k (the full set of plots can be found in the supplement). The plots in the right column provide a comparison of the minimum (w.r.t. C) test errors of ?bcoll,2 and ?bnorm,4 at the bit level, i.e., with k doubled for ?bcoll,2 . In all plots, classification performance improves as b increases. What is more notable though is that the gap between b = 4 and b = ? is indeed minor as anticipated. Regarding the performance of ?bcoll,2 and ?bnorm,4 , the latter consistently achieves better performance. 5 Conclusion In this paper, we have presented theoretical and empirical evidence that it is possible to achieve additional data compression in the use of random projections by means of coarse scalar quantization. 7 0.7 0.78 0.75 0.72 0.69 0.66 0.5 0.4 0.3 7 8 9 log2(k) 10 11 -4 12 accuracy on test set 0.8 0.75 0.7 -2 0 log10(C) 2 0.4 6 7 8 9 log2(k) 10 11 12 0.65 0.6 0.75 0.7 0.65 -2 0 log10(C) 2 0.6 6 4 0.75 7 8 9 log2(k) 10 11 0.65 0.6 0.55 -2 0 log10(C) 2 0.75 0.7 0.65 6 0.75 0.7 0.65 0.6 7 8 9 log2(k) 10 11 12 0.55 -4 -2 0 log10(C) 0.8 0.75 7 8 9 log2(k) 10 11 arcene 0.8 accuracy on test set accuracy on test set accuracy on test set 0.8 11 0.85 arcene, k = 512 arcene 10 0.85 0.7 6 4 0.85 0.85 8 9 log2(k) farm 0.7 0.5 -4 12 7 0.9 accuracy on test set 0.8 11 0.8 farm, k = 64 0.85 10 0.85 0.75 farm 8 9 log2(k) Dexter 0.7 0.55 -4 7 0.9 0.75 accuracy on test set accuracy on test set 0.5 4 0.65 0.7 6 0.6 0.55 Dexter, k = 512 0.85 0.9 0.7 0.65 0.8 Dexter 0.6 6 0.75 0.45 0.2 0.9 accuracy on test set 0.6 accuracy on test set 0.63 6 0.8 PEMS PEMS, k = 64 accuracy on test set PEMS accuracy on test set accuracy on test set 0.81 2 4 0.8 0.75 0.7 0.65 0.6 6 7 8 9 log2(k) 10 11 Figure 5: Results of the classification experiments. Each row corresponds to one data set. (L): Accuracy on the test set (optimized over C) in dependence of the number of RPs k (log2 scale). (M): Accuracy on the test set for a selected value of k in dependence of log10 (C). (R): Comparison of the test accuracies when using the estimators ?bnorm,4 respectively ?bcoll,2 with twice the number of RPs. The loss of information incurred at this step tends to be mild even with the naive approach in which quantized data are treated in the same way as their full precision counterparts. An exception only arises for cosine similarities close to 1 (Theorem 2). We have also shown that the simple form of normalization employed in the construction of the estimator ?bnorm can be extremely beneficial, even more so for coarsely quantized data because of a crucial bias reduction. Regarding future work, it is worthwhile to consider the extension to the case in which the random projections are not Gaussian but arise from one of the various structured Johnson-Lindenstrauss transforms, e.g., those in [2, 3, 23]. A second direction of interest is to analyze the optimal trade-off between the number of RPs k and the bit depth b in dependence of the similarity ?; in the present work, the choice of b has been driven with the goal of roughly matching the full precision case. 8 Acknowledgments The work was partially supported by NSF-Bigdata-1419210, NSF-III-1360971. Ping Li also thanks Michael Mitzenmacher for helpful discussions. References [1] D. Achlioptas. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. Journal of Computer and System Sciences, 66:671?687, 2003. [2] N. Ailon and B. Chazelle. Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In Proceedings of the Symposium on Theory of Computing (STOC), pages 557?563, 2006. [3] N. Ailon and E. Liberty. Almost optimal unrestricted fast Johnson?Lindenstrauss transform. ACM Transactions on Algorithms, 9:21, 2013. [4] T. Anderson. An Introduction to Multivariate Statistical Analysis. Wiley, 2003. [5] E. Bingham and H. Mannila. Random projection in dimensionality reduction: applications to image and text data. In Conference on Knowledge Discovery and Data Mining (KDD), pages 245?250, 2001. [6] P. Boufounos and R. Baraniuk. 1-bit compressive sensing. In Information Science and Systems, 2008. [7] C. Boutsidis, A. Zouzias, and P. Drineas. Random Projections for k-means Clustering. In Advances in Neural Information Processing Systems (NIPS), pages 298?306. 2010. [8] E. Candes and T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 52:5406?5425, 2006. [9] C-C. Chang and C-J. Lin. LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2:27:1?27:27, 2011. http://www.csie.ntu.edu.tw/~cjlin/libsvm. [10] M. Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the Symposium on Theory of Computing (STOC), pages 380?388, 2002. [11] S. Dasgupta. Learning mixtures of Gaussians. In Symposium on Foundations of Computer Science (FOCS), pages 634?644, 1999. [12] S. Dasgupta. An elementary proof of a theorem of Johnson and Lindenstrauss. Random Structures and Algorithms, 22:60?65, 2003. [13] M. Datar, N. Immorlica, P. Indyk, and V. Mirrokni. Locality-Sensitive Hashing Scheme Based on p-Stable Distributions. In Symposium on Computational Geometry (SCG), pages 253?262, 2004. [14] D. Fradkin and D. Madigan. Experiments with random projections for machine learning. In Conference on Knowledge Discovery and Data Mining (KDD), pages 517?522, 2003. [15] A. Gersho and R. Gray. Vector Quantization and Signal Compression. Springer, 1991. [16] M. Goemans and D. Williamson. Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the ACM, 42:1115?1145, 1995. [17] P. Indyk and R. Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the Symposium on Theory of Computing (STOC), pages 604?613, 1998. [18] J. Matousek. On variants of the Johnson-Lindenstrauss lemma. Random Structures and Algorithms, 33:142?156, 2008. [19] L. Jacques. A Quantized Johnson-Lindenstrauss Lemma: The Finding of Buffon?s needle. IEEE Transactions on Information Theory, 61:5012?5027, 2015. [20] W. Johnson and J. Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, pages 189?206, 1984. [21] K. Kenthapadi, A. Korolova, I. Mironov, and N. Mishra. Privacy via the Johnson-Lindenstrauss Transform. Journal of Privacy and Confidentiality, 5, 2013. [22] J. Kieffer. Uniqueness of locally optimal quantizer for log-concave density and convex error weighting function. IEEE Transactions on Information Theory, 29:42?47, 1983. 9 [23] F. Krahmer and R. Ward. New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property. SIAM Journal on Mathematical Analysis, 43:1269?1281, 2011. [24] J. Laska and R. Baraniuk. Regime change: Bit-depth versus measurement-rate in compressive sensing. IEEE Transactions on Signal Processing, 60:3496?3505, 2012. [25] M. Li, S. Rane, and P. Boufounos. Quantized embeddings of scale-invariant image features for mobile augmented reality. In International Workshop on Multimedia Signal Processing (MMSP), pages 1?6, 2012. [26] P. Li, T. Hastie, and K. Church. Improving Random Projections Using Marginal Information. In Annual Conference on Learning Theory (COLT), pages 635?649, 2006. [27] P. Li, M. Mitzenmacher, and A. Shrivastava. Coding for Random Projections. In Proceedings of the International Conference on Machine Learning (ICML), pages 676?678, 2014. [28] P. Li, M. Mitzenmacher, and M. Slawski. Quantized Random Projections and Non-Linear Estimation of Cosine Similarity. In Advances in Neural Information Processing Systems (NIPS), pages 2756?2764. 2016. [29] M. Mahoney. Randomized Algorithms for Matrices and Data. Foundations and Trends in Machine Learning, 3:123?224, 2011. [30] O. Maillard and R. Munos. Compressed least-squares regression. In Advances in Neural Information Processing Systems (NIPS), pages 1213?1221. 2009. [31] S. Rane and P. Boufounos. Privacy-preserving nearest neighbor methods: Comparing signals wihtout revealing them. IEEE Signal Processing Magazine, 30:18?28, 2013. [32] S. Rane, P. Boufounos, and A. Vetro. Quantized embeddings: An efficient and universal nearest neighbor method for cloud-based image retrieval. In SPIE Optical Engineering and Applications, pages 885609? 885609. International Society for Optics and Photonics, 2013. [33] S. Vempala. The Random Projection Method. American Mathematical Society, 2005. 10
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Discovering Potential Correlations via Hypercontractivity Hyeji Kim1? Weihao Gao1? Sreeram Kannan2? Sewoong Oh1? Pramod Viswanath1? University of Illinois at Urbana Champaign1 and University of Washington2 {hyejikim,wgao9}@illinois.edu,[email protected],{swoh,pramodv}@illinois.edu Abstract Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations. 1 Introduction Measuring the strength of an association between two random variables is a fundamental topic of broad scientific interest. Pearson?s correlation coefficient [1] dates from over a century ago and has been generalized seven decades ago as maximal correlation (mCor) to handle nonlinear dependencies [2?4]. Novel correlation measures to identify different kinds of associations continue to be proposed in the literature; these include maximal information coefficient (MIC) [5] and distance correlation (dCor) [6]. Despite the differences, a common theme of measurement of the empirical average dependence unites the different dependence measures. Alternatively, these are factual measures of dependence and their relevance is restricted when we seek a potential dependence of one random variable on another. For instance, consider a hypothetical city with very few smokers. A standard measure of correlation on the historical data in this town on smoking and lung cancer will fail to discover the fact that smoking causes cancer, since the average correlation is very small. On the other hand, clearly, there is a potential correlation between smoking and lung cancer; indeed applications of this nature abound in several scenarios in modern data science, including a recent one on genetic pathway discovery [7]. Discovery of a potential correlation naturally leads one to ask for a measure of potential correlation that is statistically well-founded and addresses practical needs. Such is the focus of this work, where our proposed measure of potential correlation is based on a novel interpretation of the Information Bottleneck (IB) principle [8]. The IB principle has been used to address one of the fundamental tasks in supervised learning: given samples {Xi , Yi }ni=1 , how do we find a compact summary of a variable ? Coordinated Science Lab and and Department of Electrical and Computer Engineering Department of Electrical Engineering ? Coordinated Science Lab and Department of Industrial and Enterprise Systems Engineering ? 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. X that is most informative in explaining another variable Y . The output of the IB principle is a compact summary of X that is most relevant to Y and has a wide range of applications [9, 10]. We use this IB principle to create a measure of correlation based on the following intuition: if X is (potentially) correlated with Y , then a relatively compact summary of X can still be very informative about Y . In other words, the maximal ratio of how informative a summary can be in explaining Y to how compact a summary is with respect to X is, conceptually speaking, an indicator of potential correlation from X to Y . Quantifying the compactness by I(U ; X) and the information by I(U ; Y ) we consider the rate of information bottleneck as a measure of potential correlation: s(X; Y ) ? sup U ?X?Y I(U ; Y ) , I(U ; X) (1) where U X Y forms a Markov chain and the supremum is over all summaries U of X. This intuition is made precise in Section 2, where we formally define a natural notion of potential correlation (Axiom 6), and show that the rate of information bottleneck s(X; Y ) captures this potential correlation (Theorem 1) while other standard measures of correlation fail (Theorem 2). This ratio has only recently been identified as the hypercontractivity coefficient [11]. Hypercontractivity has a distinguished and central role in a large number of technical arenas including quantum physics [12, 13], theoretical computer science [14, 15], mathematics [16?18] and probability theory [19, 20]. In this paper, we provide a novel interpretation to the hypercontractivity coefficient as a measure of potential correlation by demonstrating that it satisfies a natural set of axioms such a measure is expected to obey. For practical use in discovering correlations, the standard correlation coefficients are equipped with corresponding natural sample-based estimators. However, for hypercontractivity coefficient, estimating it from samples is widely acknowledged to be challenging, especially for continuous random variables [21?23]. There is no existing algorithm to estimate the hypercontractivity coefficient in general [21], and there is no existing algorithm for solving IB from samples either [22, 23]. We provide a novel estimator of the hypercontractivity coefficient ? the first of its kind ? by bringing together the recent theoretical discoveries in [11, 24] of an alternate definition of hypercontractivity coefficient as ratio of Kullback-Leibler divergences defined in (5), and recent advances in joint optimization (the max step in Equation 1) and estimating information measures from samples using importance sampling [25]. Our main contributions are the following: ? We postulate a set of natural axioms that a measure of potential correlation from X to Y should satisfy (Section 2). p ? We show that s(X; Y ), our proposed measure of potential correlation, satisfies all the axioms we postulate. In comparison, we prove that existing standard measures of correlation not only fail to satisfy thepproposed axioms, but also fail to capture canonical potential correlations captured by s(X; Y ) (Section 2). Another natural candidate is mutual information, but it is not clear how to interpret the value of mutual information as it is unnormalized, unlike all other measures of correlation which are between zero and one. ? Computation of the hypercontractivity coefficient from samples is known to be a challenging open problem. We introduce a novel estimator to compute hypercontractivity coefficient from i.i.d. samples in a statistically consistent manner for continuous random variables, using ideas from importance sampling and kernel density estimation (Section 3). ? In a series of synthetic experiments, we show empirically that our estimator for the hypercontractivity coefficient is statistically more powerful in discovering a potential correlation than existing correlation estimators; a larger power means a larger successful detection rate for a fixed false alarm rate (Section 4.1). ? We show applications of our estimator of hypercontractivity coefficient in two important datasets: In Section 4.2, we demonstrate that it discovers hidden potential correlations among various national indicators in WHO datasets, including how aid is potentially correlated with the income growth. In Section 4.3, we consider the following gene pathway recovery problem: we are given samples of four gene expressions time series. Assuming we know that gene A causes B, that B causes C, and that C causes D, the problem is to discover that 2 these causations occur in the sequential order: A to B, and then B to C, and then C to D. We show empirically that the estimator of the hypercontractivity coefficient recovers this order accurately from a vastly smaller number of samples compared to other state-of-the art causal influence estimators. 2 Axiomatic approach to measure potential correlations We propose a set of axioms that a measure of potential correlation should satisfy and propose a new measure of correlation that satisfies all the proposed axioms. Axioms for potential correlation. We postulate that a measure of potential correlation ?? : X ?Y ! [0, 1] between two random variables X 2 X and Y 2 Y should satisfy: 1. ?? (X, Y ) is defined for any pair of non-constant random variables X and Y . 2. 0 ? ?? (X, Y ) ? 1. 3. ?? (X, Y ) = 0 iff X and Y are statistically independent. 4. For bijective Borel-measurable functions f, g : R ! R, ?? (X, Y ) = ?? (f (X), g(Y )). 5. If (X, Y ) ? N (?, ?), then ?? (X, Y ) = |?|, where ? is the Pearson correlation coefficient. 6. ?? (X, Y ) = 1 if there exists a subset Xr ? X such that for a pair of continuous random variables (X, Y ) 2 Xr ?Y, Y = f (X) for a Borel-measurable and non-constant continuous function f . ? ? ? ? ? 0.2 ?? 0.0 ?? ? ?? ? ? 0.2 0.4 0.6 0.8 0.8 Y 0.6 ? ? ? ? ? ? ? 0.4 0.6 ?? ? ? ? ? ? ? ? 0.2 ?? ? ? ? 0.4 Y ? ? ? ? ? ? ? ? ? ? ?? ? ? 0.0 0.8 ? ? ? ? ? ? ?? ?? ? ?? ?? ? ?? ? ? ?? ? ? ? ??? ?? ? ? ??? ??? ? ?? ? ??? ??? ?? ????? ? ? ?? ?? ? ?? ? ?? ? ? ?? ? ? ? ?? ? ???? ? ?? ? ? ????? ? ?? ??? ?? ? ?? ?? ? ? ? ? ??? ? ? ?? ? ? ??? ? ? ? ? ? ?? ?? ? ? ? ?? ?? ? ? ? ? ?? ? ?? ? ?? ?? ? ?? ? ? ? ??? ?? ??? ? ? ?? ? ?? ? ? ? ?? ? ?? ?? ?? ? ? ? ? ? ???? ? ? ? ? ?? ? ?????? ? ? ?? ? ??? ? ? ? ?? ? ? ? ? ?? ?? ??? ? ? ? ?? ? ? ?? ? ??? ? ??? ? ? ? ?? ?? ? ?? ?? ?? ???? ? ? ? ? ? ? ? ? ??? ?? 1.0 Quadratic 1.0 Linear 1.0 0.2 X 0.4 ? ? ? ? ? 0.6 0.8 ??? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ?? ?? ? ? ? ?? ? ? ?? ? ??? ? ?? ?? ? ? ?? ?? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ?? ?? ? ??? ?? ? ? ? ?? ? ? ? ?? ?? ? ? ?? ? ??? ? ?? ??? ? ? ? ?? ? ?? ? ? ? ?? ?? ?? ? ? ? ? ??? ?? ? ? ? ?? ??? ?? ? ? ? ? ?? ? ? ? ??? ??? ? ? ? ? ?? ? ?? ? ???? ?? ? ? ?? ? ? ? ? ?? ?? ? ???? ? ? ?? ? ??? ? ??? ? ? ? ?? ??? ? ? ?? ?? ? ??? ? ??? ? ? ?? ? ? ? ? ??? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? 1.0 X Figure 1: A measure of potential correlation should capture the rare correlation in X 2 [0, 1] in these examples which satisfy Axiom 6 for a linear and a quadratic function, respectively. Axioms 1-5 are identical to a subset of the celebrated axioms of R?nyi in [4], which ensure that the measure is properly normalized and invariant under bijective transformations, and recovers the Pearson correlation for jointly Gaussian random variables. R?nyi?s original axioms for a measure of correlation in [4] included Axioms 1-5 and also that the measure ?? of correlation should satisfy 6?. ?? (X, Y ) = 1 if for Borel-measurable functions f or g, Y = f (X) or X = g(Y ). 7?. ?? (X; Y ) = ?? (Y ; X). The Pearson correlation violates a subset (3, 4, and 6?) of R?nyi?s axioms. Together with recent empirical successes in multimodal deep learning (e.g. [26?28]), R?nyi?s axiomatic approach has been a major justification of Hirschfeld-Gebelein-R?nyi (HGR) maximum correlation coefficient defined as mCor(X, Y ) := supf,g E[f (X)g(Y )], which satisfies all R?nyi?s axioms [2]. Here, the supremum is over all measurable functions with E[f (X)] = E[g(Y )] = 0 and E[f 2 (X)] = E[g 2 (Y )] = 1. However, maximum correlation is not the only measure satisfying all of R?nyi?s axioms, as we show in the following. Proposition 1. For any function F : [0, 1]?[0, 1] ! [0,p 1] satisfyingp F (x, y) = F (y, x), F (x, x) = x, and F (x, y) = 0 only if xy = 0, the symmetrized F ( s(X; Y ), s(Y ; X)) satisfies all R?nyi?s axioms. p This follows from the fact that the hypercontractivity coefficient s(X; Y ) satisfies all but the symmetry in Axiom 7 (Theorem 1), and it follows that a symmetrized version satisfies all axioms, 3 p p e.g. (1/2)( s(X; Y ) + s(Y ; X)) and (s(X; Y )s(Y ; X))1/4 . A formal proof is provided in Appendix A.1. From the original R?nyi?s axioms, for potential correlation measure, we remove Axiom 7? that ensures symmetry, as directionality is fundamental in measuring the potential correlation from X to Y . We further replace Axiom 6? by Axiom 6, as a variable X has a full potential to be correlated with Y if there exists a domain Xr such that X and Y are deterministically dependent and non-degenerate (i.e. not a constant function), as illustrated in Figure 1 for a linear function and a quadratic function. The hypercontractivity coefficient satisfies all axioms. We propose the hypercontractivity coefficient s(X; Y ), first introduced in [19], as the measure of potential correlation satisfying all Axioms 1-6. Intuitively, s(X; Y ) measures how much potential correlation X has with Y . For example, if X and Y are independent, then s(X; Y ) = 0 as X has no correlation with Y (Axiom 3). By data processing inequality, it follows that it is a measure between zero and one (Axiom 2) and also invariant under bijective transformations (Axiom 4). For jointly Gaussian variables X and Y with 2 the Pearson correlation ?, we can show p that s(X; Y ) = s(Y ; X) = ? . Hence, the squared-root of s(X; Y ) satisfies Axiom 5. In fact, s(X; Y ) satisfies all desired axioms for potential correlation, and we make this precise in the following theorem whose proof is provided in Appendix A.2. p Theorem 1. Hypercontractivity coefficient s(X; Y ) satisfies Axioms 1-6. In particular, the hypercontractivity coefficient satisfies Axiom 6 for potential correlation, unlike other measures of correlation (see Theorem 2 for examples). If there is a potential for X in a possibly rare regime in X to be fully correlated with Y such that Y = f (X), then the hypercontractivity coefficient is maximum: s(X; Y ) = 1. However, just as HGR correlation is not the only one satisfying R?nyi?s original axioms, the hypercontractivity coefficient is not the only one satisfying our axioms. There is a family of measures known as hypercontractivity ribbon that includes the hypercontractivity coefficient as a special case, all of which satisfy the axioms. However, a few properties of the hypercontractivity coefficient makes it more attractive for practical use; it can be efficiently estimated from samples (see Section 3) and is a natural extension of the popular HGR maximal correlation coefficient. Axiom 5p is restricted to univariate X and Y , and it can be naturally extended to multivariate variables where s(X; Y ) is a multivariate measure that satisfies all the axioms. For the discussion of hypercontractivity ribbon, connection between hypercontractivity coefficient and HGR maximal correlation, and extension of axioms to multivariate variables, see the journal version [29]. Beside standard correlation measures, another measure widely used to quantify the strength of dependence is mutual information. We can show that mutual information satisfies Axiom 6 if we replace 1 by 1. However there are two key problems: (a) Practically, mutual information is unnormalized, i.e., I(X; Y ) 2 [0, 1). Hence, it provides no absolute indication of the strength of the dependence. (b) Mathematically, we are looking for a quantity that tensorizes, i.e., doesn?t change when there are many i.i.d. copies of the same pair of random variables. Hypercontractivity coefficient tensorizes, i.e, s(X1 , ..., Xn ; Y1 , .., Yn ) = s(X1 , Y1 ), for i.i.d. (Xi , Yi ), i = 1, ? ? ? , n. On the other hand, mutual information is additive, i.e., I(X1 , ? ? ? , Xn ; Y1 , ? ? ? , Yn ) = nI(X1 ; Y1 ), for i.i.d. (Xi , Yi ), i = 1, ? ? ? , n. Tensorizing quantities capture the strongest relationship among independent copies while additive quantities capture the sum. For instance, mutual information could be large because a small amount of information accumulates over many of the independent components of X and Y (when X and Y are high dimensional) while tensorizing quantities would rule out this scenario, where there is no strong dependence. When the components are not independent, hypercontractivity indeed pools information from different components to find the strongest direction of dependence, which is a desirable property. One natural way to normalize mutual information is by the log of the cardinality of the input/output alphabets [30]. One can interpret a popular correlation measure MIC as a similar effort for normalizing mutual information and is one of our baselines. Standard correlation coefficients violate the Axioms. We next analyze existing measures of correlations under the scenario with potential correlation (Axiom 6), where we find that none of the 4 existing correlation measures satisfy Axiom 6. Suppose X and Y are independent (i.e. no correlation) in a subset Xd of the domain X , and allow X and Y to be arbitrarily correlated in the rest Xr of the domain, such that X = Xd [ Xr . We further assume that the independent part is dominant and the correlated part is rare; let ? := P(X 2 Xr ) and we consider the scenario when ? is small. A good measure of potential correlation is expected to capture the correlation in Xr even if it is rare (i.e., ? is small). To make this task more challenging, we assume that the conditional distribution of Y |{X 2 Xr } is the same as Y |{X 2 / Xr }. Figure 1 (of this section) illustrates sampled points for two examples from such a scenario and more examples are in Figure 5 in Appendix B. Our main result is the analysis of HGR maximal correlation (mCor) [2], distance correlation (dCor) [6], maximal information coefficients (MIC) [5], which shows that these measures are vanishing with ? even if the dependence in the rare regime is very high. Suppose Y |(X 2 Xr ) = f (X), then all three correlation coefficients are vanishing as ? gets small. This in particular violates Axiom 6. The reason is that standard correlation coefficients measure the average correlation whereas the hypercontractivity coefficient measures the potential correlation. The experimental comparisons on the power of these measures confirm our analytical predictions in Figure 2. The formal statement is below and the proof is provided in Appendix A.3. Theorem 2. Consider a pair of continuous random variables (X, Y ) 2 X ? Y. Suppose X is partitioned as Xr [ Xd = X such that PY |X (S|X 2 Xr ) = PY |X (S|X 2 Xd ) for all S ? Y, and Y is independent of X for X 2 Xd . Let ? = P{X 2 Xr }. The HGR maximal correlation coefficient is p mCor(X, Y ) = ? mCor(Xr , Y ) , (2) the distance correlation coefficient is dCor(X, Y ) = ? dCor(Xr , Y ) , (3) the maximal information coefficient is upper bounded by MIC(X, Y ) ? ? MIC(Xr , Y ) , (4) where Xr is the random variable X conditioned on the rare domain X 2 Xr . 3 Estimator of the hypercontractivity coefficient from samples In this section, we present an algorithm1 to compute the hypercontractivity coefficient s(X; Y ) from i.i.d. samples {Xi , Yi }ni=1 . The computation of the hypercontractivity coefficient from samples is known to be challenging for continuous random variables [22, 23], and to the best of our knowledge, there is no known efficient algorithm to compute the hypercontractivity coefficient from samples. Our estimator is the first efficient algorithm to compute the hypercontractivity coefficient, based on the following equivalent definition of the hypercontractivity coefficient, shown recently in [11]: s(X; Y ) ? sup rx 6=px D(ry ||py ) . D(rx ||px ) (5) There are two main challenges for computing s(X; Y ). The first challenge is ? given a marginal distribution rx and samples from pxy , how do we estimate the KL divergences D(ry ||py ) and D(rx ||px ). The second challenge is the optimization over the infinite dimensional simplex. We need to combine estimation and optimization together in order to compute s(X; Y ). Our approach is to combine ideas from traditional kernel density estimates and from importance sampling. Let wi = rx (Xi )/px (Xi ) be the likelihood ratio evaluated at sample i. We propose the estimation and optimization be solved jointly as follows: Estimation: To estimate KL divergence D(rx ||px ), notice that ? rx (X) rx (X) D(rx | |px ) = EX?px log . px (X) px (X) Using empirical average to replace the expectation over px , we propose n 1 n X rx (Xi ) rx (Xi ) 1X b x | |px ) = 1 D(r log = wi log wi . n i=1 px (Xi ) px (Xi ) n i=1 Code is available at https://github.com/wgao9/hypercontractivity 5 For D(ry ||py ), we follow the similar idea, but the challenge is in computing vj = ry (Yj )/py (Yj ). To do this, notice that rxy = rx py|x , so ? ? ? rx (X) . ry (Yj ) = EX?rx py|x (Yj |X) = EX?px py|x (Yj |X) px (X) Replacing the expectation by empirical average again, we get the following estimator of vj : n vbj = n 1 X py|x (Yj |Xi ) rx (Xi ) 1 X pxy (Xi , Yj ) = wi . n i=1 py (Yj ) px (Xi ) n i=1 px (Xi )py (Yj ) | {z } Aji b = AT w. We use a kernel density estimator We can write this expression in matrix form as v from [31] to estimate the matrix A, but our approach is compatible with any density estimator of choice. Optimization: Given the estimators of the KL divergences, we are able to convert the problem of computing s(X; Y ) into an optimization problem over the vector w. Here a constraint of Pn (1/n) i=1 wi = 1 is needed to satisfy Epx [rx /px ] = 1. To improve numerical stability, we use log s(X; Y ) as the objective function. Then the optimization problem has the following form: maxw subject to Pn log (wT A log(AT w) n 1X wi = 1 n i=1 wi log wT log w 0, 8 i where w log w = i=1 wi log wi for short. Although this problem is not convex, we apply gradient descent to maximize the objective. In practice, we initialize wi = 1 + N (0, 2 ) for 2 = 0.01. Hence, the initial rx is perturbed mildly from px . Although we are not guaranteed to achieve the global maximum, we consistently observe in extensive numerical experiments that we have 50%-60% probability of achieving the same maximum value, which we believed to be the global maximum. A theoretical analysis of the landscape of local and global optima and their regions of attraction with respect to gradient descent is an interesting and challenging open question, outside the scope of this paper. A theoretical understanding of the performance of gradient descent on the optimization step (where the number of samples is fixed) above is technically very challenging and is left to future work. T 4 Experimental results We present experimental results on synthetic and real datasets showing that the hypercontractivity coefficient (a) is more powerful in detecting potential correlation compared to existing measures; (b) discovers hidden potential correlations among various national indicators in WHO datasets; and (c) is more robust in discovering pathways of gene interactions from gene expression time series data. 4.1 Synthetic data: power test on potential correlation As our estimator (and the measure itself) involves a maximization, it is possible that we are sensitive to outliers and may capture spurious noise. A formal statistical approach to test the robustness as well as accuracy is to run power tests: testing for the power of the estimator in binary hypothesis tests. Via a series of experiments we show that the hypercontractivity coefficient and our estimator are capturing the true potential correlation. We compare the power of the hypercontractivity coefficient and other correlation coefficients in the binary hypothesis testing scenario of Theorem 2. As shown in Figure 5 in Appendix B, we generate pairs of datasets ? one where X and Y are independent and one where there is a potential correlation as per our scenario. We experiment with eight types of functional associations, following the examples from [5, 32, 33]. For the correlated datasets, out of n samples {(xi , yi )}ni=1 , ?n rare but correlated samples are in X = [0, 1] and (1 ?)n dominant but independent samples are in X 2 [1, 1.1]. 6 The rare but correlated samples are generated as xi ? Unif[0, 1], yi ? f (xi ) + N (0, 2 ) for i 2 [1 : ?n]. The dominant samples are generated as xi ? Unif[1, 1.1], yi ? f (Unif[0, 1])+N (0, 2 ) for i 2 [?n + 1, n]. A formal comparison is done via testing their powers: comparing the false negative rate at a fixed false positive rate of, say, 5%. We show empirically that for linear, quadratic, sine with period 1/2, and the step function, the hypercontractivity coefficient is more powerful as compared to other measures. For a given setting, a larger power means a larger successful detection rate for a fixed false alarm rate. Figure 2 shows the power of correlation estimators as a function of the additive noise level, 2 , for ? = 0.05 and n = 320. The hypercontractivity coefficient is more powerful than other correlation estimators for most functions. The power of all the estimators are very small for sine (period 1/8) and circle functions. This is not surprising given that it is very hard to discern the correlated and independent cases even visually, as shown in Figure 5. We give extensive experimental results in the journal version [29]. 0.01 0.03 0.1 0.3 1 3 0.3 1 3 1 3 0.1 0.3 1 3 0 Noise level 0.03 0.1 0.3 1 3 Cor dCor MIC mCor HC 0.4 0.6 0.4 0.03 0.01 Step function Cor dCor MIC mCor HC 0.2 0.01 0 Noise level 0.0 0 1.0 0.6 0.4 0.2 0.3 0.6 1.0 0.6 0.4 0.2 0.1 Noise level 0.1 Circle Cor dCor MIC mCor HC 0.0 0.03 0.03 Noise level 0.8 1.0 0.8 0.6 0.4 0.2 0.0 0.01 0.01 X^(1/4) Cor dCor MIC mCor HC 0 0.0 0 Noise level Sine: period 1/8 0.8 1.0 0.8 0.4 0.2 0.0 0 1.0 3 0.8 1 0.2 0.3 Cor dCor MIC mCor HC 0.0 0.1 Noise level 1.0 0.03 0.8 0.01 Sine: period 1/2 Cor dCor MIC mCor HC 0.6 0.8 0.6 0.2 0.0 0 Power Cubic Cor dCor MIC mCor HC 0.4 0.6 0.4 0.0 0.2 Power 0.8 Cor dCor MIC mCor HC 1.0 Quadratic 1.0 Linear 0.01 0.03 0.1 0.3 1 3 0 0.01 Noise level 0.03 0.1 0.3 1 3 Noise level Figure 2: Power vs. noise level for ? = 0.05, n = 320 4.2 Real data: correlation between indicators of WHO datasets We compute the hypercontractivity coefficient, MIC, and Pearson correlation of 1600 pairs of indicators for 202 countries in the World Health Organization (WHO) dataset [5]. Figure 3 illustrates that the hypercontractivity coefficient discovers hidden potential correlation (e.g. in (E) and (F)), whereas other measures fail. Scatter plots of Pearson correlation vs. the hypercontractivity coefficient and MIC vs. the hypercontractivity coefficient for all pairs are presented in Figure 3 (A) and (D). The samples for pairs of indicators corresponding to B,C,E,F in Figure 3 (A) and (D) are shown in Figure 3 (B),(C),(E),(F), respectively. In (B), it is reasonable to assume that the number of bad teeth per child is uncorrelated with the democracy score. The hypercontractivity coefficient, MIC, and Pearson correlation are all small, as expected. In (C), the correlation between CO2 emissions and energy use is clearly visible, and all three correlation estimates are close to one. However, only the hypercontractivity coefficient discovers the hidden potential correlation in (E) and (F). In (E), the data is a mixture of two types of countries ? one with small amount of aid received (less than $5 ? 108 ), and the other with large amount of aid received (larger than $5 ? 108 ). Dominantly many countries (104 out of 146) belong to the first type (small aid), and for those countries, the amount of aid received and the income growth are independent. For the remaining countries with larger aid received, although those are rare, there is a clear correlation between the amount of aid received and the income growth. Similarly in (F), there are two types of countries ? one with small arms exports (less than $2 ? 108 ) and the other with large arms exports (larger than $2 ? 108 ). Dominantly many countries (71 out of 82) belong to the first type, for which the amount of arms exports and the health expenditure are independent. For the remaining countries that belong to the second type, on the other hand, there is a visible correlation between the arms exports and the health expenditure. This is expected as for those countries that export arms the GDP is positively correlated 7 ? ? ? ? ? ? ? F ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ?? ? ? ? ? ??? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ?? ? ?? ? ? ? ? ? 2 3 4 5 ? 6 ? ?? ? ? ?? ? ??? ? ? ? ? ? ? ? ? ?? ? ? ?? ?? ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ?? ? ? ? ? ??? ?? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? 0 10 20 ? ? ? ? ? ? ? ? ? ? ? ? ??? ?? ? ? ? ? ? ? ? ?? ? ? ? ?? ?? ? ?? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ?? ? ????? ? ? ?? ? ? ? ???? ?? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ?? ? ? ??? ? ? ? ? ? ?????? ? ? ? ? ? ?? ? ? ? ??? ??? ? ?? ? ?? ?? ? ? ? ?? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ?? ? ? ?? ? ? ??? ? ? ?? ? ?? ? ? ? ? ? ? ??? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ?? ?? ?? ?? ? ?? ? ? ? ? ???? ? ? ? ??? ? ?? ? ? ?? ?? ? ???? ? ?? ??? ? ?? ? ? ?? ? ?? ? ? ? ? ? ? ?? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ??? ? ? ?? ? ?? ? ? ?? ? ? ? ???? ? ? ?? ?? ? ? ? ? ? ? ?? ?? ? ? ?? ? ? ?? ? ? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ??? ??? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ?? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?????????? ? ? ? ?? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ? ? ? ? ?? ?? ? ? ? ?? ? ?? ?? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ??? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? F ? ? E ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ??? ? ? ?? ? 0.25 0.50 0.75 Hypercontractivity 15 ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? 1.00 ? ? ?? ? ? ? ?? ? ?? ? ? ? ?? ? ??? ?? ? ?? ? ? ??? ? ? ? ? ? ?? ? ????? ? ? ? ? ? ? ?? ? ? ? ? ? ??? ?? ? ? ? ? ? ? ? ?? ? ? ? ??? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ?? ? ? ? ? ? ? ? ??? ? ? ?? ? ? ?? ?? ? 50 60 16 ? ? ? 0.0e+00 5.0e+09 1.0e+10 1.5e+10 Aid_received_total (E) 2.0e+10 14 ? ? ? 4 ? ? ? ? ? ? ? ? ? 2 ? Health_expenditure_total ?? ?? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? 20 ? ? ? ? 40 (C) 12 ? ? ? ?? ? ? ? ? 10 ? ? ? 30 CO2_emissions 10 ? 5 ?? ? ? ? ? ?? Income_growth ? ? ? ? ? ? ? ? ? ? 0 ? (D) ? ? ? ?? ? ? ? 0.00 ? ? ? 1 ? ? ? 0.00 ?? ? ? ? ? ? 25 C ? ? ?? B ? ? ?5 Pearson correlation 0.75 0.25 ? Bad_teeth_per_child ? ? ?? ? ? (B) ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ?? ? ? 1.00 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 15000 5 ?? ? ? ? ? 1.00 ? ??? ?? Energy_use ? ? (A) ? ? ?? ? 5000 ? ? ? ? ? ? ? ? ? ? ? ?? 0.25 0.50 0.75 Hypercontractivity 0.50 ?? ? 8 0.00 ? ? ? E B ?? 0 ? ? ? ? ? ? ? 0.00 ? ? ?? 6 0.25 ? ?? ?? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? 0 MIC 0.50 ? ? ??? ?? ? ? Democracy_score ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ?? ?? ?? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ?? ? ? ? ?? ? ? ?? ??? ? ? ?? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ? ? ?? ? ? ? ?? ? ?? ? ? ? ? ?? ?? ? ??? ? ? ? ?? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ??? ?? ? ?? ? ? ??? ? ? ?? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?????? ? ? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ?? ? ? ? ??? ? ? ?? ? ?? ? ? ? ???? ?? ?? ? ? ? ? ?? ? ? ? ? ? ?? ? ?? ? ? ? ?? ? ?? ? ? ? ???? ? ? ? ? ? ?? ?? ? ?? ? ? ? ? ? ? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? ? ??? ? ?? ? ? ? ?? ??? ?? ? ? ? ? ?? ?? ? ?? ? ?? ? ?? ? ??? ? ? ??? ? ?? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ?? ?? ? ??? ? ? ? ?? ?? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ?? ?? ??? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ?? ????? ??? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ??? ?? ? ?? ? ? ? ?? ?? ? ? ?? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ??? ?? ? ?? ?? ? ? ? ? ??? ? ?? ?? ? ? ?? ?? ? ? ?? ? ? ?? ? ? ? ? ? ?? ?? ?? ?? ?? ? ?? ? ?? ? ? ??? ? ??? ? ? ?? ?? ?? ??? ?? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ?? ? ? ?? ? ?? ? ????? ? ?? ?? ? ? ?? ?? ?? ?? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ????? ?? ? ?? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ???? ? ? ? ? ? ? ? ? ? ??? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ??? ? ?? ? ? ? ? ? ? ? ? ? ? ??? ? ??? ? ? ? ?? ? ?? ? ? ?? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ??? ? ? ? ? ?? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ???? ??? ???? ? ? ? ? ?? ? ? ? ? ? ? ? ??? ? ? ? ? ? ?? ? ? ?? ??? ?? ? ? ? ?? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ??? ? ?? ? ? ? ? ?? ?? ? ? ?? ??? ?? ? ? ??? ? ? ? ? ? ? ? ? ? ??? ? C ? ? ? ?10 0.75 ? ? ? ? ?5 ? ? ? ? 20000 ? ? ? ? ? ? ? 10000 ? ? ? 10 1.00 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? 0e+00 2e+09 4e+09 6e+09 Arms_exports (F) Figure 3: (A) and (D): Scatter plot of correlation measures. (B): Correlations are small. (C): Correlations are large. (E) and (F): Only the hypercontractivity coefficient discovers potential correlation. with both arms exports and health expenditure, whereas for those do not have arms industry, these two will be independent. We give extensive numerical analyses of the WHO dataset in the journal version [29]. 4.3 Gene pathway recovery from single cell data We replicate the genetic pathway detection experiment from [7], and show that hypercontractivity correctly discovers the genetic pathways from smaller number of samples. A genetic pathway is a series of genes interacting with each other as a chain. Consider the following setup where four genes whose expression values in a single cell are modeled by random processes Xt , Yt , Zt and Wt respectively. These 4 genes interact with each other following a pathway Xt ! Yt ! Zt ! Wt ; it is biologically known that Xt causes Yt with a negligible delay, and later at time t0 , Yt0 causes Zt0 , and so on. Our goal is to recover this known gene pathway from sampled data points. For a sequence of (j) (j) (j) (j) ni time points {ti }m i=0 , we observe ni i.i.d. samples {Xti , Yti , Zti , Wti }j=1 generated from the random process P (Xti , Yti , Zti , Wti ). We use the real data obtained by the single-cell mass flow cytometry technique [7]. Given these samples from time series, the goal of [7] is to recover the direction of the interaction along the known pathway using correlation measures as follows, where they proposed a new measure called DREMI. The DREMI correlation measure is evaluated on each pairs on the pathway, ? (Xti , Yti ), ? (Yti , Zti ) and ? (Zti , Wti ), at each time points ti . It is declared that a genetic pathway is correctly recovered if the peak of correlation follows the expected trend: arg maxti ? (Xti , Yti ) ? arg maxti ? (Yti , Zti ) ? arg maxti ? (Zti , Wti ). In [25], the same experiment has been done with ? evaluated by UMI and CMI estimators. In this paper, we evaluate ? using our proposed estimator of hypercontractivity. We subsample the raw data from [7] to evaluate the ability to find the trend from smaller samples. Precisely, given a resampling rate 2 (0, 1], we randomly select a subset of indices Si ? [ni ] with card(Si ) = d ni e, compute ? (Xti , Yti ), ? (Yti , Zti ) and ? (Zti , Wti ) from sub8 (j) (j) (j) (j) probability of success samples {Xti , Yti , Zti , Wti }j2Si , and determine whether we can recover the trend successfully, i.e., whether arg maxti ? (Xti , Yti ) ? arg maxti ? (Yti , Zti ) ? arg maxti ? (Zti , Wti ). We repeat the experiment several times with independent subsamples and compute the probability of successfully recovering the trend. Figure 4 illustrates that when the entire dataset is available, all methods are able to recover the trend correctly. When only fewer samples are available, hypercontractivity improves upon other competing measures in recovering the hidden chronological order of interactions of the pathway. For completeness, we run datasets for both regular T-cells (shown in left figure) and T-cells exposed with an antigen (shown right figure), for which we expect distinct biological trends. Hypercontractivity method can capture the trend for both datasets correctly and sample-efficiently. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 HyperContractivity CMI UMI DREMI 0.5 0.4 0.02 0.05 0.1 0.25 resampling rate 0.5 HyperContractivity CMI UMI DREMI 0.5 0.4 1.0 0.2 0.4 0.6 resampling rate 0.8 1.0 Figure 4: Accuracy vs. subsampling rate. Hypercontractivity method has higher probability to recover the trend when data size is smaller compared to other methods. Left: regular T-cells. Right: T-cells exposed with an antigen [7]. Acknowledgments This work was partially supported by NSF grants CNS-1527754, CNS-1718270, CCF-1553452, CCF-1617745, CCF-1651236, CCF-1705007, and GOOGLE Faculty Research Award. References [1] K. Pearson, ?Note on regression and inheritance in the case of two parents,? Proceedings of the Royal Society of London, vol. 58, pp. 240?242, 1895. [2] H. Hirschfeld, ?A connection between correlation and contingency,? Mathematical Proceedings of the Cambridge Philosophical Society, pp. 31(4), pp. 520?524., 1935. [3] H. Gebelein, ?Das statistische problem der korrelation als variations-und eigenwertproblem und sein zusammenhang mit der ausgleichsrechnung,? ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f?r Angewandte Mathematik und Mechanik, vol. 21, no. 6, pp. 364?379, 1941. [4] A. 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Nair and S. Kamath, Personal communication, 2016. [22] A. A. Alemi, I. Fischer, J. V. Dillon, and K. Murphy, ?Deep variational information bottleneck,? ICLR, 2017. [23] A. Achille and S. Soatto, ?Information dropout: Learning optimal representations through noisy computation,? ArXiv e-prints. 1611.01353, 2016. [24] C. Nair, ?An extremal inequality related to hypercontractivity of Gaussian random variables,? in Information Theory and Applications Workshop, 2014. [25] W. Gao, S. Kannan, S. Oh, and P. Viswanath, ?Conditional dependence via shannon capacity: Axioms, estimators and applications,? in Proceedings of The 33rd International Conference on Machine Learning, 2016, pp. 2780?2789. [26] J. Ngiam, A. Khosla, M. Kim, J. Nam, H. Lee, and A. Y. Ng, ?Multimodal deep learning,? in Proceedings of the 28th international conference on machine learning (ICML-11), 2011, pp. 689?696. [27] N. Srivastava and R. R. Salakhutdinov, ?Multimodal learning with deep boltzmann machines,? in Advances in neural information processing systems, 2012, pp. 2222?2230. [28] G. Andrew, R. Arora, J. Bilmes, and K. Livescu, ?Deep canonical correlation analysis,? in International Conference on Machine Learning, 2013, pp. 1247?1255. [29] H. Kim, W. Gao, S. Kannan, S. Oh, and P. Viswanath, ?Discovering potential correlations via hypercontractivity,? in preparation. 10 [30] C. Bell, ?Mutual information and maximal correlation as measures of dependence,? The Annals of Mathematical Statistics, vol. 33, no. 2, pp. 587?595, 1962. [31] T. Michaeli, W. Wang, and K. Livescu, ?Nonparametric canonical correlation analysis,? in Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, ser. ICML?16, 2016, pp. 1967?1976. [32] N. Simon and R. Tibshirani, ?Comment on ?Detecting Novel Associations In Large Data Sets? by Reshef Et Al, Science Dec 16, 2011,? 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Doubly Stochastic Variational Inference for Deep Gaussian Processes Hugh Salimbeni Imperial College London and PROWLER.io [email protected] Marc Peter Deisenroth Imperial College London and PROWLER.io [email protected] Abstract Gaussian processes (GPs) are a good choice for function approximation as they are flexible, robust to overfitting, and provide well-calibrated predictive uncertainty. Deep Gaussian processes (DGPs) are multi-layer generalizations of GPs, but inference in these models has proved challenging. Existing approaches to inference in DGP models assume approximate posteriors that force independence between the layers, and do not work well in practice. We present a doubly stochastic variational inference algorithm that does not force independence between layers. With our method of inference we demonstrate that a DGP model can be used effectively on data ranging in size from hundreds to a billion points. We provide strong empirical evidence that our inference scheme for DGPs works well in practice in both classification and regression. 1 Introduction Gaussian processes (GPs) achieve state-of-the-art performance in a range of applications including robotics (Ko and Fox, 2008; Deisenroth and Rasmussen, 2011), geostatistics (Diggle and Ribeiro, 2007), numerics (Briol et al., 2015), active sensing (Guestrin et al., 2005) and optimization (Snoek et al., 2012). A Gaussian process is defined by its mean and covariance function. In some situations prior knowledge can be readily incorporated into these functions. Examples include periodicities in climate modelling (Rasmussen and Williams, 2006), change-points in time series data (Garnett et al., 2009) and simulator priors for robotics (Cutler and How, 2015). In other settings, GPs are used successfully as black-box function approximators. There are compelling reasons to use GPs, even when little is known about the data: a GP grows in complexity to suit the data; a GP is robust to overfitting while providing reasonable error bars on predictions; a GP can model a rich class of functions with few hyperparameters. Single-layer GP models are limited by the expressiveness of the kernel/covariance function. To some extent kernels can be learned from data, but inference over a large and richly parameterized space of kernels is expensive, and approximate methods may be at risk of overfitting. Optimization of the marginal likelihood with respect to hyperparameters approximates Bayesian inference only if the number of hyperparameters is small (Mackay, 1999). Attempts to use, for example, a highly parameterized neural network as a kernel function (Calandra et al., 2016; Wilson et al., 2016) incur the downsides of deep learning, such as the need for application-specific architectures and regularization techniques. Kernels can be combined through sums and products (Duvenaud et al., 2013) to create more expressive compositional kernels, but this approach is limited to simple base kernels, and their optimization is expensive. A Deep Gaussian Process (DGP) is a hierarchical composition of GPs that can overcome the limitations of standard (single-layer) GPs while retaining the advantages. DGPs are richer models than standard GPs, just as deep networks are richer than generalized linear models. In contrast to models with highly parameterized kernels, DGPs learn a representation hierarchy non-parametrically with very few hyperparmeters to optimize. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Unlike their single-layer counterparts, DGPs have proved difficult to train. The mean-field variational approaches used in previous work (Damianou and Lawrence, 2013; Mattos et al., 2016; Dai et al., 2016) make strong independence and Gaussianity assumptions. The true posterior is likely to exhibit high correlations between layers, but mean-field variational approaches are known to severely underestimate the variance in these situations (Turner and Sahani, 2011). In this paper, we present a variational algorithm for inference in DGP models that does not force independence or Gaussianity between the layers. In common with many state-of-the-art GP approximation schemes we start from a sparse inducing point variational framework (Matthews et al., 2016) to achieve computational tractability within each layer, but we do not force independence between the layers. Instead, we use the exact model conditioned on the inducing points as a variational posterior. This posterior has the same structure as the full model, and in particular it maintains the correlations between layers. Since we preserve the non-linearity of the full model in our variational posterior we lose analytic tractability. We overcome this difficulty by sampling from the variational posterior, introducing the first source of stochasticity. This is computationally straightforward due to an important property of the sparse variational posterior marginals: the marginals conditioned on the layer below depend only on the corresponding inputs. It follows that samples from the marginals at the top layer can be obtained without computing the full covariance within the layers. We are primarily interested in large data applications, so we further subsample the data in minibatches. This second source of stochasticity allows us to scale to arbitrarily large data. We demonstrate through extensive experiments that our approach works well in practice. We provide results on benchmark regression and classification data problems, and also demonstrate the first DGP application to a dataset with a billion points. Our experiments confirm that DGP models are never worse than single-layer GPs, and in many cases significantly better. Crucially, we show that additional layers do not incur overfitting, even with small data. 2 Background In this section, we present necessary background on single-layer Gaussian processes and sparse variational inference, followed by the definition of the deep Gaussian process model. Throughout we emphasize a particular property of sparse approximations: the sparse variational posterior is itself a Gaussian process, so the marginals depend only on the corresponding inputs. 2.1 Single-layer Gaussian Processes We consider the task of inferring a stochastic function f : RD ? R, given a likelihood p(y|f ) and a set of N observations y = (y1 , . . . , yN )> at design locations X = (x1 , . . . , xN )> . We place a GP prior on the function f that models all function values as jointly Gaussian, with a covariance function k : RD ? RD ? R and a mean function m : RD ? R. We further define an additional set of M inducing locations Z = (z1 , . . . , zM )> . We use the notation f = f (X) and u = f (Z) for the function values at the design and inducing points, respectively. We define also [m(X)]i = m(xi ) and [k(X, Z)]ij = k(xi , zj ). By the definition of a GP, the joint density p(f , u) is a Gaussian whose mean is given by the mean function evaluated at every input (X, Z)> , and the corresponding covariance is given by the covariance function evaluated at every pair of inputs. The joint density of y, f and u is YN p(y, f , u) = p(f |u; X, Z)p(u; Z) p(yi |fi ) . (1) | {z } | i=1{z } GP prior likelihood In (1) we factorized the joint GP prior p(f , u; X, Z) 1 into the prior p(u) = N (u|m(Z), k(Z, Z)) and the conditional p(f |u; X, Z) = N (f |?, ?), where for i, j = 1, . . . , N [?]i = m(xi ) + ?(xi )> (u ? m(Z)) , > [?]ij = k(xi , xj ) ? ?(xi ) k(Z, Z)?(xj ) , (2) (3) 1 Throughout this paper we use the semi-colon notation to clarify the input locations of the corresponding function values, which will become important later when we discuss multi-layer GP models. For example, p(f |u; X, Z) indicates that the input locations for f and u are X and Z, respectively. 2 with ?(xi ) = k(Z, Z)?1 k(Z, xi ). Note that the conditional mean ? and covariance ? defined via (2) and (3), respectively, take the form of mean and covariance functions of the inputs xi . Inference in the model (1) is possible in closed form when the likelihood p(y|f ) is Gaussian, but the computation scales cubically with N . We are interested in large datasets with non-Gaussian likelihoods. Therefore, we seek a variational posterior to overcome both these difficulties simultaneously. Variational inference seeks an approximate posterior q(f , u) by minimizing the Kullback-Leibler divergence KL[q||p] between the variational posterior q and the true posterior p. Equivalently, we maximize the lower bound on the marginal likelihood (evidence)   p(y, f , u) L = Eq(f ,u) log , (4) q(f , u) where p(y, f , u) is given in (1). We follow Hensman et al. (2013) and choose a variational posterior q(f , u) = p(f |u; X, Z)q(u) , (5) where q(u) = N (u|m, S). Since both terms in the variational posterior are Gaussian, we can analytically marginalize u, which yields Z ? . ? ?) q(f |m, S; X, Z) = p(f |u; X, Z)q(u)du = N (f |?, (6) ? can be written as mean and covariance functions ? and ? Similar to (2) and (3), the expressions for ? of the inputs. To emphasize this point we define ?m,Z (xi ) = m(xi ) + ?(xi )> (m ? m(Z)) , > ?S,Z (xi , xj ) = k(xi , xj ) ? ?(xi ) (k(Z, Z) ? S)?(xj ) . (7) (8) ? ij = ?S,Z (xi , xj ). We have written the ? i = ?m,Z (xi ) and [?] With these functions we define [?] mean and covariance in this way to make the following observation clear. Remark 1. The fi marginals of the variational posterior (6) depend only on the corresponding inputs xi . Therefore, we can write the ith marginal of q(f |m, S; X, Z) as q(fi |m, S; X, Z) = q(fi |m, S; xi , Z) = N (fi |?m,Z (xi ), ?S,Z (xi , xi )) . (9) Using our variational posterior (5) the lower bound (4) simplifies considerably since (a) the conditionals p(f |u; X, Z) inside the logarithm cancel and (b) the likelihood expectation requires only the variational marginals. We obtain L= XN i=1 Eq(fi |m,S;xi ,Z) [log p(yi |fi )] ? KL[q(u)||p(u)] . (10) The final (univariate) expectation of the log-likelihood can be computed analytically in some cases, with quadrature (Hensman et al., 2015) or through Monte Carlo sampling (Bonilla et al., 2016; Gal et al., 2015). Since the bound is a sum over the data, an unbiased estimator can be obtained through minibatch subsampling. This permits inference on large datasets. In this work we refer to a GP with this method of inference as a sparse GP (SGP). The variational parameters (Z, m and S) are found by maximizing the lower bound (10). This maximization is guaranteed to converge since L is a lower bound to the marginal likelihood p(y|X). We can also learn model parameters (hyperparameters of the kernel or likelihood) through the maximization of this bound, though we should exercise caution as this introduces bias because the bound is not uniformly tight for all settings of hyperparameters (Turner and Sahani, 2011) So far we have considered scalar outputs yi ? R. In the case of D-dimensional outputs yi ? RD we define Y as the matrix with ith row containing the ith observation yi . Similarly, we define F and U. QD If each output is an independent GP we have the GP prior d=1 p(Fd |Ud ; X, Z)p(Ud ; Z), which we abbreviate as p(F|U; X, Z)p(U; Z) to lighten the notation. 3 2.2 Deep Gaussian Processes A DGP (Damianou and Lawrence, 2013) defines a prior recursively on vector-valued stochastic functions F 1 , . . . , F L . The prior on each function F l is an independent GP in each dimension, with input locations given by the noisy corruptions of the function values at the next layer: the outputs of the GPs at layer l are Fdl , and the corresponding inputs are F l?1 . The noise between layers is assumed i.i.d. Gaussian. Most presentations of DGPs (see, e.g. Damianou and Lawrence, 2013; Bui et al., 2016) explicitly parameterize the noisy corruptions separately from the outputs of each GP. Our method of inference does not require us to parameterize these variables separately. For notational convenience, we therefore absorb the noise into the kernel knoisy (xi , xj ) = k(xi , xj ) + ?l2 ?ij , where ?ij is the Kronecker delta, and ?l2 is the noise variance between layers. We use Dl for the dimension of the outputs at layer l. As with the single-layer case, we have inducing locations Zl?1 at each layer and inducing function values Ul for each dimension. An instantiation of the process has the joint density YN YL p(Y, {Fl , Ul }L p(yi |fiL ) p(Fl |Ul ; Fl?1 , Zl?1 )p(Ul ; Zl?1 ) , l=1 ) = i=1 l=1 | {z }| {z } likelihood (11) DGP prior where we define F0 = X. Inference in this model is intractable, so approximations must be used. The original DGP presentation (Damianou and Lawrence, 2013) uses a variational posterior that maintains the exact model conditioned on Ul , but further forces the inputs to each layer to be independent from the outputs of the previous layer. The noisy corruptions are parameterized separately, and the variational distribution over these variables is a fully factorized Gaussian. This approach requires 2N (D1 + ? ? ? + DL?1 ) variational parameters but admits a tractable lower bound on the log marginal likelihood if the kernel is of a particular form. A further problem of this bound is that the density over the outputs is simply a single layer GP with independent Gaussian inputs. Since the posterior loses all the correlations between layers it cannot express the complexity of the full model and so is likely to underestimate the variance. In practice, we found that optimizing the objective in Damianou and Lawrence (2013) results in layers being ?turned off? (the signal to noise ratio tends to zero). In contrast, our posterior retains the full conditional structure of the true model. We sacrifice analytical tractability, but due to the sparse posterior within each layer we can sample the bound using univariate Gaussians. 3 Doubly Stochastic Variational Inference In this section, we propose a novel variational posterior and demonstrate a method to obtain unbiased samples from the resulting lower bound. The difficulty with inferring the DGP model is that there are complex correlations both within and between layers. Our approach is straightforward: we use sparse variational inference to simplify the correlations within layers, but we maintain the correlations between layers. The resulting variational lower bound cannot be evaluated analytically, but we can draw unbiased samples efficiently using univariate Gaussians. We optimize our bound stochastically. We propose a posterior with three properties. Firstly, the posterior maintains the exact model, conditioned on Ul . Secondly, we assume that the posterior distribution of {Ul }L l=1 is factorized between layers (and dimension, but we suppress this from the notation). Therefore, our posterior takes the simple factorized form YL q({Fl , Ul }L p(Fl |Ul ; Fl?1 , Zl?1 )q(Ul ) . (12) l=1 ) = l=1 Thirdly, and to complete specification of the posterior, we take q(Ul ) to be a Gaussian with mean ml and variance Sl . A similar posterior was used in Hensman and Lawrence (2014) and Dai et al. (2016), but each of these works contained additional terms for the noisy corruptions at each layer. As in the single layer SGP, we can marginalize the inducing variables from each layer analytically. After this marginalization we obtain following distribution, which is fully coupled within and between layers: YL YL ? l) . ? l, ? q(Fl |ml , Sl ; Fl?1 , Zl?1 ) = N (Fl |? (13) q({Fl }L l=1 ) = l=1 l=1 4 ?l Here, q(Fl |ml , Sl ; Fl?1 , Zl?1 ) is as in (6). Specifically, it is a Gaussian with mean and variance ? l l ? ]ij = ?Sl ,Zl?1 (f l , f l ) (recall that f l is the ith row of ? , where [? ? l ]i = ?ml ,Zl?1 (f l ) and [? and ? i i j i Fl ). Since (12) is a product of terms that each take the form of the SGP variational posterior (5), we have again the property that within each layer the marginals depend on only the corresponding inputs. In particular, fiL depends only on fiL?1 , which in turn depends only on fiL?2 , and so on. Therefore, we have the following property: Remark 2. The ith marginal of the final layer of the variational DGP posterior (12) depends only on the ith marginals of all the other layers. That is, Z Y L?1 q(fiL ) = q(fil |ml , Sl ; fil?1 , Zl?1 )dfil . (14) l=1 The consequence of this property is that taking a sample from q(fiL ) is straightforward, and furthermore we can perform the sampling using only univariate unit Gaussians using the ?re-parameterization trick? (Rezende et al., 2014; Kingma et al., 2015). Specifically, we first sample li ? N (0, IDl ) and then recursively draw the sampled variables ? fil ? q(fil |ml , Sl ; ? fil?1 , Zl?1 ) for l = 1, . . . , L ? 1 as q ? fil = ?ml ,Zl?1 (? fil?1 ) + li ?Sl ,Zl?1 (? fil?1 , ? fil?1 ) , (15) where the terms in (15) are Dl -dimensional and the square root is element-wise. For the first layer we define ? fi0 := xi . Efficient computation of the evidence lower bound The evidence lower bound of the DGP is   p(Y, {Fl , Ul }L l=1 ) LDGP = Eq({Fl ,Ul }Ll=1 ) . (16) q({Fl , Ul }L l=1 ) Using (11) and (12) for the corresponding expressions in (16), we obtain after some re-arranging XN XL LDGP = Eq(fiL ) [log p(yn |fnL )] ? KL[q(Ul )||p(Ul ; Zl?1 )] , (17) i=1 l=1 where we exploited the exact marginalization of the inducing variables (13) and the property of the marginals of the final layer (14). A detailed derivation is provided in the supplementary material. This bound has complexity O(N M 2 (D1 + ? ? ? + DL )) to evaluate. We evaluate the bound (17) approximately using two sources of stochasticity. Firstly, we approximate the expectation with a Monte Carlo sample from the variational posterior (14), which we compute according to (15). Since we have parameterized this sampling procedure in terms of isotropic Gaussians, we can compute unbiased gradients of the bound (17). Secondly, since the bound factorizes over the data we achieve scalability through sub-sampling the data. Both stochastic approximations are unbiased. Predictions To predict we sample from the variational posterior changing the input locations to the test location x? . We denote the function values at the test location as f?l . To obtain the density over f?L we use the Gaussian mixture 1 XS (s) L?1 q(f?L ) ? q(f?L |mL , SL ; f? , ZL?1 ) , (18) s=1 S (s) L?1 where we draw S samples f? using (15), but replacing the inputs xi with the test location x? . Further Model Details While GPs are often used with a zero mean function, we consider such a choice inappropriate for the inner layers of a DGP. Using a zero mean function causes difficulties with the DGP prior as each GP mapping is highly non-injective. This effect was analyzed in Duvenaud et al. (2014) where the authors suggest adding the original input X to each layer. Instead, we consider an alternative approach and include a linear mean function m(X) = XW for all the inner layers. If the input and output dimension are the same we use the identity matrix for W, otherwise we compute the SVD of the data and use the top Dl left eigenvectors sorted by singular value (i.e. the PCA mapping). With these choices it is effective to initialize all inducing mean values ml = 0. This choice of mean function is partly inspired by the ?skip layer? approach of the ResNet (He et al., 2016) architecture. 5 boston N=506, D=13 concrete N=1030, D=8 energy N=768, D=8 kin8nm N=8192, D=8 PBP DGP 5 DGP 4 DGP 3 DGP 2 AEDGP 2 SGP 500 SGP Linear -2.89 PBP DGP 5 DGP 4 DGP 3 DGP 2 AEDGP 2 SGP 500 SGP Linear -2.63 -2.37 naval N=11934, D=26 3.92 -3.75 -3.43 -3.11 -2.39 power N=9568, D=4 -1.55 -0.71 protein N=45730, D=9 5.39 6.86 -2.92 -2.83 -2.73 -3.05 Bayesian NN Single layer benchmarks 0.25 0.78 1.31 wine_red N=1599, D=22 -2.89 -2.73 -1.01 DGP with approx EP PBP DGP 5 DGP 4 DGP 3 DGP 2 AEDGP 2 SGP 500 SGP Linear PBP DGP 5 DGP 4 DGP 3 DGP 2 AEDGP 2 SGP 500 SGP Linear -0.97 -0.93 This work Figure 1: Regression test log-likelihood results on benchmark datasets. Higher (to the right) is better. The sparse GP with the same number of inducing points is highlighted as a baseline. 4 Results We evaluate our inference method on a number of benchmark regression and classification datasets. We stress that we are interested in models that can operate in both the small and large data regimes, with little or no hand tuning. All our experiments were run with exactly the same hyperparameters and initializations. See the supplementary material for details. We use min(30, D0 ) for all the inner layers of our DGP models, where D0 is the input dimension, and the RBF kernel for all layers. Regression Benchmarks We compare our approach to other state-of-the-art methods on 8 standard small to medium-sized UCI benchmark datasets. Following common practice (e.g. Hern?ndez-Lobato and Adams, 2015) we use 20-fold cross validation with a 10% randomly selected held out test set and scale the inputs and outputs to zero mean and unit standard deviation within the training set (we restore the output scaling for evaluation). While we could use any kernel, we choose the RBF kernel with a lengthscale for each dimension for direct comparison with Bui et al. (2016). The test log-likelihood results are shown in Fig. 1. We compare our models of 2, 3, 4 and 5 layers (DGP 2?5), each with 100 inducing points, with (stochastically optimized) sparse GPs (Hensman et al., 2013) with 100 and 500 inducing points points (SGP, SGP 500). We compare also to a two-layer Bayesian neural network with ReLu activations, 50 hidden units (100 for protein and year), with inference by probabilistic backpropagation (Hern?ndez-Lobato and Adams, 2015) (PBP). The results are taken from Hern?ndez-Lobato and Adams (2015) and were found to be the most effective of several other methods for inferring Bayesian neural networks. We compare also with a DGP model with approximate expectation propagation (EP) for inference (Bui et al., 2016). Using the authors? code 2 we ran a DGP model with 1 hidden layer using approximate expectation propagation (Bui et al., 2016) (AEPDGP 2). We used the input dimension for the hidden layer for a fair comparison with our models3 . We found the time requirements to train a 3-layer model with this inference prohibitive. Plots for test RMSE and further results tables can be found in the supplementary material. On five of the eight datasets, the deepest DGP model is the best. On ?wine?, ?naval? and ?boston? our DGP recovers the single-layer GP, which is not surprising: ?boston? is very small, ?wine? is 2 https://github.com/thangbui/deepGP_approxEP We note however that in Bui et al. (2016) the inner layers were 2D, so the results we obtained are not directly comparable to those reported in Bui et al. (2016) 3 6 near-linear (note the proximity of the linear model and the scale) and ?naval? is characterized by extremely high test likelihoods (the RMSE on this dataset is less than 0.001 for all SGP and DGP models), i.e. it is a very ?easy? dataset for a GP. The Bayesian network is not better than the sparse GP for any dataset and significantly worse for six. The Approximate EP inference for the DGP models is also not competitive with the sparse GP for many of the datasets, but this may be because the initializations were designed for lower dimensional hidden layers than we used. Our results on these small and medium sized datasets confirm that overfitting is not observed with the DGP model, and that the DGP is never worse and often better than the single layer GP. We note in particular that on the ?power?, ?protein? and ?kin8nm? datasets all the DGP models outperform the SGP with five times the number of inducing points. Rectangles Benchmark We use the Rectangle-Images dataset4 , which is specifically designed to distinguish deep and shallow architectures. The dataset consists of 12,000 training and 50,000 testing examples of size 28 ? 28, where each image consists of a (non-square) rectangular image against a different background image. The task is to determine which of the height and width is greatest. We run 2, 3 and 4 layer DGP models, and observe increasing performance with each layer. Table 1 contains the results. Note that the 500 inducing point single-layer GP is significantly less effective than any of the deep models. Our 4-layer model achieves 77.9% classification accuracy, exceeding the best result of 77.5% reported in Larochelle et al. (2007) with a three-layer deep belief network. We also exceed the best result of 76.4% reported in Krauth et al. (2016) using a sparse GP with an Arcsine kernel, a leave-one-out objective, and 1000 inducing points. Table 1: Results on Rectangles-Images dataset (N = 12000, D = 784) Single layer GP Accuracy (%) Likelihood Ours Larochelle [2007] Krauth [2016] SGP SGP 500 DGP 2 DGP 3 DGP 4 DBN-3 SVM SGP 1000 76.1 ?0.493 76.4 ?0.485 77.3 0.475 77.8 ?0.460 77.9 ?0.460 77.5 - 76.96 - 76.4 ?0.478 Large-Scale Regression To demonstrate our method on a large scale regression problem we use the UCI ?year? dataset and the ?airline? dataset, which has been commonly used by the large-scale GP community. For the ?airline? dataset we take the first 700K points for training and next 100K for testing. We use a random 10% split for the ?year? dataset. Results are shown in Table 2, with the log-likelihood reported in the supplementary material. In both datasets we see that the DGP models perform better with increased depth, significantly improving in both log likelihood and RMSE over the single-layer model, even with 500 inducing points. Table 2: Regression test RMSE results for large datasets year airline taxi N D SGP SGP 500 DGP 2 DGP 3 DGP 4 DGP 5 463810 700K 1B 90 8 9 10.67 25.6 337.5 9.89 25.1 330.7 9.58 24.6 281.4 8.98 24.3 270.4 8.93 24.2 268.0 8.87 24.1 266.4 MNIST Multiclass Classification We apply the DGP with 2 and 3 layers to the MNIST multiclass classification problem. We use the robust-max multiclass likelihood (Hern?ndez-Lobato et al., 2011) and use full unprocessed data with the standard training/test split of 60K/10K. The single-layer GP with 100 inducing points achieves a test accuracy of 97.48% and this is increased to 98.06% and 98.11% with two and three layer DGPs, respectively. The 500 inducing point single layer model achieved 97.9% in our implementation, though a slightly higher result for this model has previously been reported of 98.1% (Hensman et al., 2013) and 98.4% (Krauth et al., 2016) for the same model with 1000 inducing points. We attribute this difference to different hyperparameter initialization and training schedules, and stress that we use exactly the same initialization and learning schedule for all our models. The only other DGP result in the literature on this dataset is 94.24% (Wang et al., 2016) for a two layer model with a two dimensional latent space. 4 http://www.iro.umontreal.ca/~lisa/twiki/bin/view.cgi/Public/RectanglesData 7 Large-Scale Classification We use the HIGGS (N = 11M, D = 28) and SUSY (N = 5.5M, D = 18) datasets for large-scale binary classification. These datasets have been constructed from Monte Carlo physics simulations to detect the presence of the Higgs boson and super-symmetry (Baldi et al., 2014). We take a 10% random sample for testing and use the rest for training. We use the AUC metric for comparison with Baldi et al. (2014). Our DGP models are the highest performing on the SUSY dataset (AUC of 0.877 for all the DGP models) compared to shallow neural networks (NN, 0.875), deep neural networks (DNN, 0.876) and boosted decision trees (BDT, 0.863). On the HIGGS dataset we see a steady improvement in additional layers (0.830, 0.837, 0.841 and 0.846 for DGP 2?4 respectively). On this dataset the DGP models exceed the performance of BDT (0.810) and NN (0.816) and both single layer GP models SGP (0.785) and SGP 500 (0.794). The best performing model on this dataset is a 5 layer DNN (0.885). Full results are reported in the supplementary material. Massive-Scale Regression To demonstrate the efficacy of our Table 3: Typical computation model on massive data we use the New York city yellow taxi trip time in seconds for a single dataset of 1.21 billion journeys 5 . Following Peng et al. (2017) we use gradient step. 9 features: time of day; day of the week; day of the month; month; CPU GPU pick-up latitude and longitude; drop-off latitude and longitude; travel SGP 0.14 0.018 distance. The target is to predict the journey time. We randomly select SGP 500 1.71 0.11 1B (109 ) examples for training and use 1M examples for testing, and 0.36 0.030 we scale both inputs and outputs to zero mean and unit standard de- DGP 2 DGP 3 0.49 0.045 viation in the training data. We discard journeys that are less than 10 s DGP 4 0.65 0.056 or greater than 5 h, or start/end outside the New York region, which DGP 5 0.87 0.069 we estimate to have squared distance less than 5o from the center of New York. The test RMSE results are the bottom row of Table 2 and test log likelihoods are in the supplementary material. We note the significant jump in performance from the single layer models to the DGP. As with all the large-scale experiments, we see a consistent improvement extra layers, but on this dataset the improvement is particularly striking (DGP 5 achieves a 21% reduction in RMSE compared to SGP) 5 Related Work The first example of the outputs of a GP used as the inputs to another GP can be found in Lawrence and Moore (2007). MAP approximation was used for inference. The seminal work of Titsias and Lawrence (2010) demonstrated how sparse variational inference could be used to propagate Gaussian inputs through a GP with a Gaussian likelihood. This approach was extended in Damianou et al. (2011) to perform approximate inference in the model of Lawrence and Moore (2007), and shortly afterwards in a similar model L?zaro-Gredilla (2012), which also included a linear mean function. The key idea of both these approaches is the factorization of the variational posterior between layers. A more general model (flexible in depth and dimensions of hidden layers) introduced the term ?DGP? and used a posterior that also factorized between layers. These approaches require a linearly increasing number of variational parameters in the number of data. For high-dimensional observations, it is possible to amortize the cost of this optimization with an auxiliary model. This approach is pursued in Dai et al. (2016), and with a recurrent architecture in Mattos et al. (2016). Another approach to inference in the exact model was presented in Hensman and Lawrence (2014), where a sparse approximation was used within layers for the GP outputs, similar to Damianou and Lawrence (2013), but with a projected distribution over the inputs to the next layer. The particular form of the variational distribution was chosen to admit a tractable bound, but imposes a constraint on the flexibility. An alternative approach is to modify the DGP prior directly and perform inference in a parametric model. This is achieved in Bui et al. (2016) with an inducing point approximation within each layer, and in Cutajar et al. (2017) with an approximation to the spectral density of the kernel. Both approaches then apply additional approximations to achieve tractable inference. In Bui et al. (2016), an approximation to expectation propagation is used, with additional Gaussian approximations to the log partition function to propagate uncertainly through the non-linear GP mapping. In Cutajar et al. (2017) a fully factorized variational approximation is used for the spectral components. Both these 5 http://www.nyc.gov/html/tlc/html/about/trip_record_data.shtml 8 approaches require specific kernels: in Bui et al. (2016) the kernel must have analytic expectations under a Gaussian, and in Cutajar et al. (2017) the kernel must have an analytic spectral density. Vafa (2016) also uses the same initial approximation as Bui et al. (2016) but applies MAP inference for the inducing points, such that the uncertainty propagated through the layers only represents the quality of the approximation. In the limit of infinitely many inducing points this approach recovers a deterministic radial basis function network. A particle method is used in Wang et al. (2016), again employing an online version of the sparse approximation used by Bui et al. (2016) within each layer. Similarly to our approach, in Wang et al. (2016) samples are taken through the conditional model, but differently from us they then use a point estimate for the latent variables. It is not clear how this approach propagates uncertainty through the layers, since the GPs at each layer have point-estimate inputs and outputs. A pathology with the DGP with zero mean function for the inner layers was identified in Duvenaud et al. (2014). In Duvenaud et al. (2014) a suggestion was made to concatenate the original inputs at each layer. This approach is followed in Dai et al. (2016) and Cutajar et al. (2017). The linear mean function was original used by L?zaro-Gredilla (2012), though in the special case of a two layer DGP with a 1D hidden layer. To the best of our knowledge there has been no previous attempt to use a linear mean function for all inner layers. 6 Discussion Our experiments show that on a wide range of tasks the DGP model with our doubly stochastic inference is both effective and scalable. Crucially, we observe that on the small datasets the DGP does not overfit, while on the large datasets additional layers generally increase performance and never deteriorate it. In particular, we note that the largest gain with increasing layers is achieved on the largest dataset (the taxi dataset, with 1B points). We note also that on all the large scale experiments the SGP 500 model is outperformed by the all the DGP models. Therefore, for the same computational budget increasing the number of layers can be significantly more effective than increasing the accuracy of approximate inference in the single-layer model. Other than the additional computation time, which is fairly modest (see Table 3), we do not see downsides to using a DGP over a single-layer GP, but substantial advantages. While we have considered simple kernels and black-box applications, any domain-specific kernel could be used in any layer. This is in contrast to other methods (Damianou and Lawrence, 2013; Bui et al., 2016; Cutajar et al., 2017) that require specific kernels and intricate implementations. Our implementation is simple (< 200 lines), publicly available 6 , and is integrated with GPflow (Matthews et al., 2017), an open-source GP framework built on top of Tensorflow (Abadi et al., 2015). 7 Conclusion We have presented a new method for inference in Deep Gaussian Process (DGP) models. With our inference we have shown that the DGP can be used on a range of regression and classification tasks with no hand-tuning. Our results show that in practice the DGP always exceeds or matches the performance of a single layer GP. Further, we have shown that the DGP often exceeds the single layer significantly, even when the quality of the approximation to the single layer is improved. Our approach is highly scalable and benefits from GPU acceleration. The most significant limitation of our approach is the dealing with high dimensional inner layers. We used a linear mean function for the high dimensional datasets but left this mean function fixed, as to optimize the parameters would go against our non-parametric paradigm. It would be possible to treat this mapping probabilistically, following the work of Titsias and L?zaro-Gredilla (2013). Acknowledgments We have greatly appreciated valuable discussions with James Hensman and Steindor Saemundsson in the preparation of this work. We thank Vincent Dutordoir and anonymous reviewers for helpful feedback on the manuscript. We are grateful for a Microsoft Azure Scholarship and support through a Google Faculty Research Award to Marc Deisenroth. 6 https://github.com/ICL-SML/Doubly-Stochastic-DGP 9 References M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, L. Kaiser, M. Kudlur, J. Levenberg, D. Man, R. Monga, S. Moore, D. Murray, J. Shlens, B. Steiner, I. Sutskever, P. Tucker, V. Vanhoucke, V. Vasudevan, O. Vinyals, P. Warden, M. Wicke, Y. Yu, and X. Zheng. TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems. arXiv preprint:1603.04467, 2015. P. Baldi, P. Sadowski, and D. Whiteson. Searching for Exotic Particles in High-Energy Physics with Deep Learning. Nature Communications, 2014. E. V. Bonilla, K. Krauth, and A. Dezfouli. 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MCMC for Variationally Sparse Gaussian Processes. Advances in Neural Information Processing Systems, 2015. D. Hern?ndez-Lobato, H. Lobato, J. Miguel, and P. Dupont. Robust Multi-class Gaussian Process Classification. Advances in Neural Information Processing Systems, 2011. J. M. Hern?ndez-Lobato and R. Adams. Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks. International Conference on Machine Learning, 2015. D. P. Kingma, T. Salimans, and M. Welling. Variational Dropout and the Local Reparameterization Trick. 2015. J. Ko and D. Fox. GP-BayesFilters: Bayesian Filtering using Gaussian Process Prediction and Observation Models. IEEE Intelligent Robots and Systems, 2008. K. Krauth, E. V. Bonilla, K. Cutajar, and M. Filippone. AutoGP: Exploring the Capabilities and Limitations of Gaussian Process Models. arXiv preprint:1610.05392, 2016. H. Larochelle, D. Erhan, A. Courville, J. Bergstra, and Y. Bengio. An Empirical Evaluation of Deep Architectures on Problems with Many Factors of Variation. International Conference on Machine Learning, 2007. N. D. Lawrence and A. J. Moore. Hierarchical Gaussian Process Latent Variable Models. International Conference on Machine Learning, 2007. M. L?zaro-Gredilla. Bayesian Warped Gaussian Processes. Advances in Neural Information Processing Systems, 2012. D. J. C. Mackay. Comparison of Approximate Methods for Handling Hyperparameters. Neural computation, 1999. A. G. Matthews, M. Van Der Wilk, T. Nickson, K. Fujii, A. Boukouvalas, P. Le?n-Villagr?, Z. Ghahramani, and J. Hensman. GPflow: A Gaussian process library using TensorFlow. Journal of Machine Learning Research, 2017. A. G. d. G. Matthews, J. Hensman, R. E. Turner, and Z. Ghahramani. On Sparse Variational Methods and The Kullback-Leibler Divergence Between Stochastic Processes. Artificial Intelligence and Statistics, 2016. C. L. C. Mattos, Z. Dai, A. Damianou, J. Forth, G. A. Barreto, and N. D. 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Turner and M. Sahani. Two Problems with Variational Expectation Maximisation for Time-Series Models. Bayesian Time Series Models, 2011. 11 K. Vafa. Training Deep Gaussian Processes with Sampling. Advances in Approximate Bayesian Inference Workshop, Neural Information Processing Systems, 2016. Y. Wang, M. Brubaker, B. Chaib-Draa, and R. Urtasun. Sequential Inference for Deep Gaussian Process. Artificial Intelligence and Statistics, 2016. A. G. Wilson, Z. Hu, R. Salakhutdinov, and E. P. Xing. Deep Kernel Learning. Artificial Intelligence and Statistics, 2016. 12
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Ranking Data with Continuous Labels through Oriented Recursive Partitions Stephan Cl?emenc?on Mastane Achab LTCI, T?el?ecom ParisTech, Universit?e Paris-Saclay 75013 Paris, France [email protected] Abstract We formulate a supervised learning problem, referred to as continuous ranking, where a continuous real-valued label Y is assigned to an observable r.v. X taking its values in a feature space X and the goal is to order all possible observations x in X by means of a scoring function s : X ? R so that s(X) and Y tend to increase or decrease together with highest probability. This problem generalizes bi/multi-partite ranking to a certain extent and the task of finding optimal scoring functions s(x) can be naturally cast as optimization of a dedicated functional criterion, called the IROC curve here, or as maximization of the Kendall ? related to the pair (s(X), Y ). From the theoretical side, we describe the optimal elements of this problem and provide statistical guarantees for empirical Kendall ? maximization under appropriate conditions for the class of scoring function candidates. We also propose a recursive statistical learning algorithm tailored to empirical IROC curve optimization and producing a piecewise constant scoring function that is fully described by an oriented binary tree. Preliminary numerical experiments highlight the difference in nature between regression and continuous ranking and provide strong empirical evidence of the performance of empirical optimizers of the criteria proposed. 1 Introduction The predictive learning problem considered in this paper can be easily stated in an informal fashion, as follows. Given a collection of objects of arbitrary cardinality, N ? 1 say, respectively described by characteristics x1 , . . . , xN in a feature space X , the goal is to learn how to order them by increasing order of magnitude of a certain unknown continuous variable y. To fix ideas, the attribute y can represent the ?size? of the object and be difficult to measure, as for the physical measurement of microscopic bodies in chemistry and biology or the cash flow of companies in quantitative finance and the features x may then correspond to indirect measurements. The most convenient way to define a preorder on a feature space X is to transport the natural order on the real line onto it by means of a (measurable) scoring function s : X ? R: an object with charcateristics x is then said to be ?larger? (?strictly larger?, respectively) than an object described by x0 according to the scoring rule s when s(x0 ) ? s(x) (when s(x) < s(x0 )). Statistical learning boils down here to build a scoring function s(x), based on a training data set Dn = {(X1 , Y1 ), . . . , (Xn , Yn )} of objects for which the values of all variables (direct and indirect measurements) have been jointly observed, such that s(X) and Y tend to increase or decrease together with highest probability or, in other words, such that the ordering of new objects induced by s(x) matches that defined by their true measures as well as possible. This problem, that shall be referred to as continuous ranking throughout the article can be viewed as an extension of bipartite ranking, where the output variable Y is assumed to be binary and the objective can be naturally formulated as a functional M -estimation problem by means of the concept of ROC curve, see [7]. Refer also to [4], [11], [1] for approaches based on the optimization 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. of summary performance measures such as the AUC criterion in the binary context. Generalization to the situation where the random label is ordinal and may take a finite number K ? 3 of values is referred to as multipartite ranking and has been recently investigated in [16] (see also e.g. [14]), where distributional conditions guaranteeing that ROC surface and the VUS criterion can be used to determine optimal scoring functions are exhibited in particular. It is the major purpose of this paper to formulate the continuous ranking problem in a quantitative manner and explore the connection between the latter and bi/multi-partite ranking. Intuitively, optimal scoring rules would be also optimal for any bipartite subproblem defined by thresholding the continuous variable Y with cut-off t > 0, separating the observations X such that Y < t from those such that Y > t. Viewing this way continuous ranking as a continuum of nested bipartite ranking problems, we provide here sufficient conditions for the existence of such (optimal) scoring rules and we introduce a concept of integrated ROC curve (IROC curve in abbreviated form) that may serve as a natural performance measure for continuous ranking, as well as the related notion of integrated AUC criterion, a summary scalar criterion, akin to Kendall tau. Generalization properties of empirical Kendall tau maximizers are discussed in the Supplementary Material. The paper also introduces a novel recursive algorithm that solves a discretized version of the empirical integrated ROC curve optimization problem, producing a scoring function that can be computed by means of a hierarchical combination of binary classification rules. Numerical experiments providing strong empirical evidence of the relevance of the approach promoted in this paper are also presented. The paper is structured as follows. The probabilistic framework we consider is described and key concepts of bi/multi-partite ranking are briefly recalled in section 2. Conditions under which optimal solutions of the problem of ranking data with continuous labels exist are next investigated in section 3, while section 4 introduces a dedicated quantitative (functional) performance measure, the IROC curve. The algorithmic approach we propose in order to learn scoring functions with nearly optimal IROC curves is presented at length in section 5. Numerical results are displayed in section 6. Some technical proofs are deferred to the Supplementary Material. 2 Notation and Preliminaries Throughout the paper, the indicator function of any event E is denoted by I{E}. The pseudo-inverse of any cdf F (t) on R is denoted by F ?1 (u) = inf{s ? R : F (s) ? u}, while U([0, 1]) denotes the uniform distribution on the unit interval [0, 1]. 2.1 The probabilistic framework Given a continuous real valued r.v. Y representing an attribute of an object, its ?size? say, and a random vector X taking its values in a (typically high dimensional euclidian) feature space X modelling other observable characteristics of the object (e.g. ?indirect measurements? of the size of the object), hopefully useful for predicting Y , the statistical learning problem considered here is to learn from n ? 1 training independent observations Dn = {(X1 , Y1 ), . . . , (Xn , Yn )}, drawn as the pair (X, Y ), a measurable mapping s : X ? R, that shall be referred to as a scoring function throughout the paper, so that the variables s(X) and Y tend to increase or decrease together: ideally, the larger the score s(X), the higher the size Y . For simplicity, we assume throughout the article that X = Rd with d ? 1 and that the support of Y ?s distribution is compact, equal to [0, 1] say. For any q ? 1, we denote by ?q the Lebesgue measure on Rq equipped with its Borelian ?-algebra and suppose that the joint distribution FX,Y (dxdy) of the pair (X, Y ) has a density fX,Y (x, y) w.r.t. the tensor product measure ?d ? ?1 . We also introduces the Rmarginal distributions FY (dy) = fY (y)? R 1 (dy) and FX (dx) = fX (x)?d (dx), where fY (y) = x?X fX,Y (x, y)?d (dx) and fX (x) = y?[0,1] fX,Y (x, y)?1 (dy) as well as the conditional densities fX|Y =y (x) = fX,Y (x, y)/fY (y) and fY |X=x (y) = fX,Y (x, y)/fX (x). Observe incidentally that the probabilistic framework of the continuous ranking problem is quite similar to that of distribution-free regression. However, as shall be seen in the subsequent analysis, even if the regression function m(x) = E[Y | X = x] can be optimal under appropriate conditions, just like for regression, measuring ranking performance involves criteria that are of different nature than the expected least square error and plug-in rules may not be relevant for the goal pursued here, as depicted by Fig. 2 in the Supplementary Material. 2 Scoring functions. The set of all scoring functions is denoted by S here. Any scoring function s ? S defines a total preorder on the space X : ?(x, x0 ) ? X 2 , x s x0 ? s(x) ? s(x0 ). We also set x ?s x0 when s(x) < s(x0 ) and x =s x0 when s(x) = s(x0 ) for (x, x0 ) ? X 2 . 2.2 Bi/multi-partite ranking Suppose that Z is a binary label, taking its values in {?1, +1} say, assigned to the r.v. X. In bipartite ranking, the goal is to pick s in S so that the larger s(X), the greater the probability that Y is equal to 1 ideally. In other words, the objective is to learn s(x) such that the r.v. s(X) given Y = +1 is as stochastically larger1 as possible than the r.v. s(X) given Y = ?1: the difference between ? s (t) = P{s(X) ? t | Y = +1} and H ? s (t) = P{s(X) ? t | Y = ?1} should be thus maximal G for all t ? R. This can be naturally quantified by means of the notion of ROC curve of a candidate ? s (t), G ? s (t)), which can be viewed as the graph s ? S, i.e. the parametrized curve t ? R 7? (H of a mapping ROCs : ? ? (0, 1) 7? ROCs (?), connecting possible discontinuity points by linear ?1 ? s ? (1 ? H?1 segments (so that ROCs (?) = G s )(1 ? ?) when Hs has no flat part in Hs (1 ? ?), ? where Hs = 1 ? Hs ). A basic Neyman Pearson?s theory argument shows that the optimal elements s? (x) related to this natural (functional) bipartite ranking criterion (i.e. scoring functions whose ROC curve dominates any other ROC curve everywhere on (0, 1)) are transforms (T ? ?)(x) of the posterior probability ?(x) = P{Z = +1 | X = x}, where T : SUPP(?(X)) ? R is any strictly increasing borelian mapping. Optimization of the curve in sup norm has been considered in [7] or in [8] for instance. However, given its functional nature, in practice the ROC curve of any s ? S is often summarized by the area under it, which performance measure can be interpreted in a probabilistic manner, as the theoretical rate of concording pairs 1 AUC(s) = P {s(X) < s(X0 ) | Z = ?1, Z0 = +1} + P {s(X) = s(X0 ) | Z = ?1, Z0 = +1} , 2 (1) where (X 0 , Z 0 ) denoted an independent copy of (X, Z). A variety of algorithms aiming at maximizing the AUC criterion or surrogate pairwise criteria have been proposed and studied in the literature, among which [11], [15] or [3], whereas generalization properties of empirical AUC maximizers have been studied in [5], [1] and [12]. An analysis of the relationship between the AUC and the error rate is given in [9]. Extension to the situation where the label Y takes at least three ordinal values (i.e. multipartite ranking) has been also investigated, see e.g. [14] or [6]. In [16], it is shown that, in contrast to the bipartite setup, the existence of optimal solutions cannot be guaranteed in general and conditions on (X, Y )?s distribution ensuring that optimal solutions do exist and that extensions of bipartite ranking criteria such as the ROC manifold and the volume under it can be used for learning optimal scoring rules have been exhibited. An analogous analysis in the context of continuous ranking is carried out in the next section. 3 Optimal elements in ranking data with continuous labels In this section, a natural definition of the set of optimal elements for continuous ranking is first proposed. Existence and characterization of such optimal scoring functions are next discussed. 3.1 Optimal scoring rules for continuous ranking Considering a threshold value y ? [0, 1], a considerably weakened (and discretized) version of the problem stated informally above would consist in finding s so that the r.v. s(X) given Y > y is as stochastically larger than s(X) given Y < y as possible. This subproblem coincides with the bipartite ranking problem related to the pair (X, Zy ), where Zy = 2I{Y > y} ? 1. As briefly recalled in subsection 2.2, the optimal set Sy? is composed of the scoring functions that induce the same ordering as ?y (X) = P{Y > y | X} = 1 ? (1 ? py )/(1 ? py + py ?y (X)), where py = 1 ? FY (y) = P{Y > y} and ?y (X) = (dFX|Y >y /dFX|Y <y )(X). Given two real-valued r.v.?s U and U 0 , recall that U is said to be stochastically larger than U 0 when P{U ? t} ? P{U 0 ? t} for all t ? R. 1 3 A continuum of bipartite ranking problems. The rationale behind the definition of the set S ? of optimal scoring rules for continuous ranking is that any element s? should score observations x in the same order as ?y (or equivalently as ?y ). Definition 1. (O PTIMAL SCORING RULE ) An optimal scoring rule for the continuous ranking problem related to the random pair (X, Y ) is any element s? that fulfills: ?y ? (0, 1), ?(x, x0 ) ? X 2 , ?y (x) < ?y (x0 ) ? s? (x) < s? (x0 ). T In other words, the set of optimal rules is defined as S ? = y?(0,1) Sy? . (2) It is noteworthy that, although the definition above is natural, the set S ? can be empty in absence of any distributional assumption, as shown by the following example. Example 1. As a counter-example, consider the distributions FX,Y such that FY = U([0, 1]) and d FX|Y =y = N (|2y ? 1|, (2y ? 1)2 ). Observe that (X, 1 ? Y )=(X, Y ), so that ?1?t = ??1 t for all t ? (0, 1) and there exists t 6= 0 s.t. ?t is not constant. Hence, there exists no s? in S such that (2) holds true for all t ? (0, 1). Remark 1. (I NVARIANCE ) We point out that the class S ? of optimal elements for continuous ranking thus defined is invariant by strictly increasing transform of the ?size? variable Y (in particular, a change of unit has no impact on the definition of S ? ): for any borelian and strictly increasing mapping H : (0, 1) ? (0, 1), any scoring function s? (x) that is optimal for the continuous ranking problem related to the pair (X, Y ) is still optimal for that related to (X, H(Y )) (since, under these hypotheses, for any y ? (0, 1): Y > y ? H(Y ) > H(y)). 3.2 Existence and characterization of optimal scoring rules We now investigate conditions guaranteeing the existence of optimal scoring functions for the continuous ranking problem. Proposition 1. The following assertions are equivalent. 1. For all 0 < y < y 0 < 1, for all (x, x0 ) ? X 2 : ?y (x) < ?y (x0 ) ? ?y0 (x) ? ?y0 (x0 ). 2. There exists an optimal scoring rule s? (i.e. S ? 6= ?). 3. The regression function m(x) = E[Y | X = x] is an optimal scoring rule. 4. The collection of probability distributions FX|Y =y (dx) = fX|Y =y (x)?d (dx), y ? (0, 1) satisfies the monotone likelihood ratio property: there exist s? ? S and, for all 0 < y < y 0 < 1, an increasing function ?y,y0 : R ? R+ such that: ?x ? Rd , fX|Y =y0 (x) = ?y,y0 (s? (x)). fX|Y =y Refer to the Appendix section for the technical proof. Truth should be said, assessing that Assertion 1. is a very challenging statistical task. However, through important examples, we now describe (not uncommon) situations where the conditions stated in Proposition 1 are fulfilled. Example 2. We give a few important examples of probabilistic models fulfilling the properties listed in Proposition 1. ? Regression model. Suppose that Y = m(X) + , where m : X ? R is a borelian function and  is a centered r.v. independent from X. One may easily check that m ? S ? . ? Exponential families. Suppose that fX|Y =y (x) = exp(?(y)T (x) ? ?(y))f (x) for all x ? Rd , where f : Rd ? R+ is borelian, ? : [0, 1] ? R is a Rborelian strictly increasing function and T : Rd ? R is a borelian mapping such that ?(y) = log x?Rd exp(?(y)T (x))f (x)dx < +?. We point out that, although the regression function m(x) is an optimal scoring function when S ? 6= ?, the continuous ranking problem does not coincide with distribution-free regression (notice incidentally that, in this case, any strictly increasing transform of m(x) belongs to S ? as well). As depicted by Fig. 2 the least-squares criterion is not relevant to evaluate continuous ranking performance and naive plug-in strategies should be avoided, see Remark 3 below. Dedicated performance criteria are proposed in the next section. 4 4 Performance measures for continuous ranking We now investigate quantitative criteria for assessing the performance in the continuous ranking problem, which practical machine-learning algorithms may rely on. We place ourselves in the situation where the set S ? is not empty, see Proposition 1 above. A functional performance measure. It follows from the view developped in the previous section that, for any (s, s? ) ? S ? S ? and for all y ? (0, 1), we have: ?? ? (0, 1), ROCs,y (?) ? ROCs? ,y (?) = ROC?y (?), (3) denoting by ROCs,y the ROC curve of any s ? S related to the bipartite ranking subproblem (X, Zy ) and by ROC?y the corresponding optimal ROC curve, i.e. the ROC curve of strictly increasing transforms of ?y (x). Based on this observation, it is natural to design a dedicated performance measure by aggregating these ?sub-criteria?. Integrating over y w.r.t. a ?-finite measure ? with supR port equal to [0, 1], this leads to the following definition IROC?,s (?) = ROCs,y (?)?(dy). The functional criterion thus defined inherits properties from the ROCs,y ?s (e.g. monotonicity, concavity). In addition, the curve IROC?,s? with s? ? S ? dominates everywhere on (0, 1) any other curve IROC?,s for s ? S. However, except in pathologic situations (e.g. when s(x) is constant), the curve IROC?,s is not invariant when replacing Y ?s distribution by that of a strictly increasing transform H(Y ). In order to guarantee that this desirable property is fulfilled (see Remark 1), one should integrate w.r.t. Y ?s distribution (which boils down to replacing Y by the uniformly distributed r.v. FY (Y )). Definition 2. (I NTEGRATED ROC/AUC CRITERIA ) The integrated ROC curve of any scoring rule s ? S is defined as: ?? ? (0, 1), Z 1 IROCs (?) = ROCs,y (?)FY (dy) = E [ROCs,Y (?)] . (4) y=0 The integrated AUC criterion is defined as the area under the integrated ROC curve: ?s ? S, Z 1 IAUC(s) = IROCs (?)d?. (5) ?=0 The following result reveals the relevance of the functional/summary criteria defined above for the continuous ranking problem. Additional properties of IROC curves are listed in the Supplementary Material. Theorem 1. Let s? ? S. The following assertions are equivalent. 1. The assertions of Proposition 1 are fulfilled and s? is an optimal scoring function in the sense given by Definition 1. 2. For all ? ? (0, 1), IROCs? (?) = E [ROC?Y (?)]. 3. We have IAUCs? = E [AUC?Y ], where AUC?y = R1 ?=0 ROC?y (?)d? for all y ? (0, 1). If S ? 6= ?, then we have: ?s ? S, def IROCs (?) ? IROC? (?) = E [ROC?Y (?)] , for any ? ? (0, 1, ) def IAUC(s) ? IAUC? = E [AUC?Y ] . In addition, for any borelian and strictly increasing mapping H : (0, 1) ? (0, 1), replacing Y by H(Y ) leaves the curves IROCs , s ? S, unchanged. Equipped with the notion defined above, a scoring rule s1 is said to be more accurate than another one s2 if IROCs2 (?) ? IROCs1 (?) for all ? ? (0, 1).The IROC curve criterion thus provides a partial preorder on S. Observe also that, by virtue of Fubini?s theorem, we have R IAUC(s) = AUCy (s)FY (dy) for all s ? S, denoting by AUCy (s) the AUC of s related to the bipartite ranking subproblem (X, Zy ). Just like the AUC for bipartite ranking, the scalar IAUC criterion defines a full preorder on S for continuous ranking. Based on a training dataset Dn of independent copies of (X, Y ), statistical versions of the IROC/IAUC criteria can be straightforwardly computed by replacing the distributions FY , FX|Y >t and FX|Y <t by their empirical counterparts in (3)-(5), see the Supplementary Material for further details. The lemma below provides a probabilistic interpretation of the IAUC criterion. 5 Lemma 1. Let (X 0 , Y 0 ) be a copy of the random pair (X, Y ) and Y 00 a copy of the r.v. Y . Suppose that (X, Y ), (X 0 , Y 0 ) and Y 00 are defined on the same probability space and are independent. For all s ? S, we have: 1 IAUC(s) = P {s(X) < s(X0 ) | Y < Y00 < Y0 } + P {s(X) = s(X0 ) | Y < Y00 < Y0 } . (6) 2 This result shows in particular that a natural statistical estimate of IAUC(s) based on Dn involves U -statistics of degree 3. Its proof is given in the Supplementary Material for completeness. The Kendall ? statistic. The quantity (6) is akin to another popular way to measure the tendency to define the same ordering on the statistical population in a summary fashion: 1 def d? (s) = P {(s(X) ? s(X 0 )) ? (Y ? Y 0 ) > 0} + P {s(X) = s(X 0 )} (7) 2 1 = P{s(X) < s(X 0 ) | Y < Y 0 } + P {X =s X 0 } , 2 where (X 0 , Y 0 ) denotes an independent copy of (X, Y ), observing that P{Y < Y 0 } = 1/2. The empirical counterpart of (7) based on the sample Dn , given by X X 1 2 I {(s(Xi ) ? s(Xj )) ? (Yi ? Yj ) > 0} + I {s(Xi ) = s(Xj )} dbn (s) = n(n ? 1) i<j n(n ? 1) i<j (8) is known as the Kendall ? statistic and is widely used in the context of statistical hypothesis testing. The quantity (7) shall be thus referred to as the (theoretical or true) Kendall ? . Notice that d? (s) is invariant by strictly increasing transformation of s(x) and thus describes properties of the order it defines. The following result reveals that the class S ? , when non empty, is the set of maximizers of the theoretical Kendall ? . Refer to the Supplementary Material for the technical proof. Proposition 2. Suppose that S ? 6= ?. For any (s, s? ) ? S ? S ? , we have: d? (s) ? d? (s? ). Equipped with these criteria, the objective expressed above in an informal manner can be now formulated in a quantitative manner as a (possibly functional) M -estimation problem. In practice, the goal pursued is to find a reasonable approximation of a solution to the optimization problem maxs?S d? (s) (respectively maxs?S IAUC(s)), where the supremum is taken over the set of all scoring functions s : X ? R. Of course, these criteria are unknown in general, just like (X, Y )?s probability distribution, and the empirical risk minimization (ERM in abbreviated form) paradigm (see [10]) invites for maximizing the statistical version (8) over a class S0 ? S of controlled complexity when considering the criterion d? (s) for instance. The generalization capacity of empirical maximizers of the Kendall ? can be straightforwardly established using results in [5]. More details are given in the Supplementary Material. Before describing a practical algorithm for recursive maximization of the IROC curve, a few remarks are in order. Remark 2. (O N K ENDALL ? AND AUC) We point out that, in the bipartite ranking problem (i.e. when the output variable Z takes its values in {?1, +1}, see subsection 2.2) as well, the AUC criterion can be expressed as a function of the Kendall ? related to the pair (s(X), Z) when the r.v. s(X) is continuous. Indeed, we have in this case 2p(1?p)AUC(s) = d? (s), where p = P{Z = +1} and d? (s) = P{(s(X) ? s(X 0 )) ? (Z ? Z 0 ) > 0}, denoting by (X 0 , Z 0 ) an independent copy of (X, Z). Remark 3. (C ONNECTION TO DISTRIBUTION - FREE REGRESSION ) Consider the nonparametric regression model Y = m(X) + , where  is a centered r.v. independent from X. In this case, it is well-known that the regression function m(X) = E[Y | X] is the (unique) solution of the expected least squares minimization. However, although m ? S ? , the least squares criterion is far from appropriate to evaluate ranking performance, as depicted by Fig. 2. Observe additionally that, in contrast to the criteria introduced above, increasing transformation of the output variable Y may have a strong impact on the least squares minimizer: except for linear stransforms, E[H(Y ) | X] is not an increasing transform of m(X). Remark 4. (O N DISCRETIZATION ) Bi/multi-partite algorithms are not directly applicable to the continuous ranking problem. Indeed a discretization of the interval [0, 1] would be first required but this would raise a difficult question outside our scope: how to choose this discretization based on the training data? We believe that this approach is less efficient than ours which reveals problemspecific criteria, namely IROC and IAUC. 6 Figure 1: A scoring function described by an oriented binary subtree T . For any element x ? X , one may compute the quantity sT (x) very fast in a top-down fashion by means of the heap structure: starting from the initial value 2J at the root node, at each internal node Cj,k , the score remains unchanged if x moves down to the left sibling, whereas one subtracts 2J?(j+1) from it if x moves down to the right. 5 Continuous Ranking through Oriented Recursive Partitioning It is the purpose of this section to introduce the algorithm CR ANK, a specific tree-structured learning algorithm for continuous ranking. 5.1 Ranking trees and Oriented Recursive Partitions Decision trees undeniably figure among the most popular techniques, in supervised and unsupervised settings, refer to [2] or [13] for instance. This is essentially due to the visual model summary they provide, in the form of a binary tree graphic that permits to describe predictions by means of a hierachichal combination of elementary rules of the type ?X (j) ? ?? or ?X (j) > ??, comparing the value taken by a (quantitative) component of the input vector X (the split variable) to a certain threshold (the split value). In contrast to local learning problems such as classification or regression, predictive rules for a global problem such as ranking cannot be described by a (tree-structured) partition of the feature space: cells (corresponding to the terminal leaves of the binary decision tree) must be ordered so as to define a scoring function. This leads to the definition of ranking trees as binary trees equipped with a ?left-to-right? orientation, defining a tree-structured collection of anomaly scoring functions, as depicted by Fig. 1. Binary ranking trees have been in the context of bipartite ranking in [7] or in [3] and in [16] in the context of multipartite ranking. The root node of a ranking tree TJ of depth J ? 0 represents the whole feature space X : C0,0 = X , while each internal node (j, k) with j < J and k ? {0, . . . , 2j ? 1} corresponds to a subset Cj,k ? X , whose left and right siblings respectively correspond to disjoint subsets Cj+1,2k and Cj+1,2k+1 such that Cj,k = Cj+1,2k ? Cj+1,2k+1 . Equipped with the left-to-right orientation, any subtree T ? TJ defines a preorder on X : elements lying in the same terminal cell of T being equally ranked. The scoring function related to the oriented tree T can be written as: sT (x) = X J 2 Cj,k : terminal leaf of T 5.2  k 1? j 2  ? I{x ? Cj,k }. (9) The CR ANK algorithm Based on Proposition 2, as mentioned in the Supplementary Material, one can try to build from the training dataset Dn a ranking tree by recursive empirical Kendall ? maximization. We propose below an alternative tree-structured recursive algorithm, relying on a (dyadic) discretization of the ?size? variable Y . At each iteration, the local sample (i.e. the data lying in the cell described by the current node) is split into two halves (the highest/smallest halves, depending on Y ) and the algorithm calls a binary classification algorithm A to learn how to divide the node into right/left children. The theoretical analysis of this algorithm and its connection with approximation of IROC? are difficult questions that will be adressed in future work. Indeed we found out that the IROC cannot be 7 represented as a parametric curve contrary to the ROC, which renders proofs much more difficult than in the bipartite case. T HE CR ANK A LGORITHM 1. Input. Training data Dn , depth J ? 1, binary classification algorithm A. 2. Initialization. Set C0,0 = X . 3. Iterations. For j = 0, . . . , J ? 1 and k = 0, . . . , 2J ? 1, (a) Compute a median yj,k of the dataset {Y1 , . . . , , Yn } ? Cj,k and assign the binary label Zi = 2I{Yi > yj,k } ? 1 to any data point i lying in Cj,k , i.e. such that Xi ? Cj,k . (b) Solve the binary classification problem related to the input space Cj,k and the training set {(Xi , Yi ) : 1 ? i ? n, Xi ? Cj,k }, producing a classifier gj,k : Cj,k ? {?1, +1}. (c) Set Cj+1,2k = {x ? Cj,k , gj,k = +1} = Cj,k \ Cj+1,2k+1 . 4. Output. Ranking tree TJ = {Cj,k : 0 ? j ? J, 0 ? k < D}. Of course, the depth J should be chosen such that 2J ? n. One may also consider continuing to split the nodes until the number of data points within a cell has reached a minimum specified in advance. In addition, it is well known that recursive partitioning methods fragment the data and the unstability of splits increases with the depth. For this reason, a ranking subtree must be selected. The growing procedure above should be classically followed by a pruning stage, where children of a same parent are progressively merged until the root T0 is reached and a subtree among the sequence T0 ? . . . ? TJ with nearly maximal IAUC should be chosen using cross-validation. Issues related to the implementation of the CR ANK algorithm and variants (e.g. exploiting randomization/aggregation) will be investigated in a forthcoming paper. 6 Numerical Experiments In order to illustrate the idea conveyed by Fig. 2 that the least squares criterion is not appropriate for the continuous ranking problem we compared on a toy example CR ANK with CART. Recall that the latter is a regression decision tree algorithm which minimizes the MSE (Mean Squared Error). We also runned an alternative version of CR ANK which maximizes the empirical Kendall ? instead of the empirical IAUC: this method is refered to as K ENDALL from now on. The experimental setting is composed of a unidimensional feature space X = [0, 1] (for visualization reasons) and a simple regression model without any noise: Y = m(X). Intuitively, a least squares strategy can miss slight oscillations of the regression function, which are critical in ranking when they occur in high probability regions as they affect the order among the feature space. The results are presented in Table 1. See Supplementary Material for further details. CR ANK K ENDALL CART IAUC 0.95 0.94 0.61 Kendall ? 0.92 0.93 0.58 MSE 0.10 0.10 7.4 ? 10?4 Table 1: IAUC, Kendall ? and MSE empirical measures 7 Conclusion This paper considers the problem of learning how to order objects by increasing ?size?, modeled as a continuous r.v. Y , based on indirect measurements X. We provided a rigorous mathematical formulation of this problem that finds many applications (e.g. quality control, chemistry) and is referred to as continuous ranking. In particular, necessary and sufficient conditions on (X, Y )?s distribution for the existence of optimal solutions are exhibited and appropriate criteria have been proposed for evaluating the performance of scoring rules in these situations. In contrast to distribution-free regression where the goal is to recover the local values taken by the regression function, continuous 8 ranking aims at reproducing the preorder it defines on the feature space as accurately as possible. The numerical results obtained via the algorithmic approaches we proposed for optimizing the criteria aforementioned highlight the difference in nature between these two statistical learning tasks. Acknowledgments This work was supported by the industrial chair Machine Learning for Big Data from T?el?ecom ParisTech and by a public grant (Investissement d?avenir project, reference ANR-11-LABX-0056LMH, LabEx LMH). References [1] S. Agarwal, T. Graepel, R. Herbrich, S. Har-Peled, and D. Roth. Generalization bounds for the area under the ROC curve. J. Mach. Learn. Res., 6:393?425, 2005. [2] L. Breiman, J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Wadsworth and Brooks, 1984. [3] G. Cl?emenc?on, M. Depecker, and N. Vayatis. Ranking Forests. J. Mach. Learn. Res., 14:39?73, 2013. [4] S. Cl?emenc?on, G. Lugosi, and N.Vayatis. Ranking and scoring using empirical risk minimization. In Proceedings of COLT 2005, volume 3559, pages 1?15. Springer., 2005. [5] S. Cl?emenc?on, G. Lugosi, and N. Vayatis. Ranking and empirical risk minimization of ustatistics. The Annals of Statistics, 36:844?874, 2008. [6] S. Cl?emenc?on and S. Robbiano. The TreeRank Tournament algorithm for multipartite ranking. Journal of Nonparametric Statistics, 25(1):107?126, 2014. [7] S. Cl?emenc?on and N. Vayatis. Tree-based ranking methods. IEEE Transactions on Information Theory, 55(9):4316?4336, 2009. [8] S. Cl?emenc?on and N. Vayatis. The RankOver algorithm: overlaid classification rules for optimal ranking. Constructive Approximation, 32:619?648, 2010. [9] Corinna Cortes and Mehryar Mohri. Auc optimization vs. error rate minimization. In Advances in neural information processing systems, pages 313?320, 2004. [10] L. Devroye, L. Gy?orfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer, 1996. [11] Y. Freund, R. D. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. Journal of Machine Learning Research, 4:933?969, 2003. [12] Aditya Krishna Menon and Robert C Williamson. Bipartite ranking: a risk-theoretic perspective. Journal of Machine Learning Research, 17(195):1?102, 2016. [13] J.R. Quinlan. Induction of Decision Trees. Machine Learning, 1(1):1?81, 1986. [14] S. Rajaram and S. Agarwal. Generalization bounds for k-partite ranking. In NIPS 2005 Workshop on Learn to rank, 2005. [15] A. Rakotomamonjy. Optimizing Area Under Roc Curve with SVMs. In Proceedings of the First Workshop on ROC Analysis in AI, 2004. [16] S. Robbiano S. Cl?emenc?on and N. Vayatis. Ranking data with ordinal labels: optimality and pairwise aggregation. Machine Learning, 91(1):67?104, 2013. 9
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Scalable Model Selection for Belief Networks Zhao Song? , Yusuke Muraoka? , Ryohei Fujimaki? , Lawrence Carin? ? Department of ECE, Duke University Durham, NC 27708, USA {zhao.song, lcarin}@duke.edu ? NEC Data Science Research Laboratories Cupertino, CA 95014, USA {ymuraoka, rfujimaki}@nec-labs.com Abstract We propose a scalable algorithm for model selection in sigmoid belief networks (SBNs), based on the factorized asymptotic Bayesian (FAB) framework. We derive the corresponding generalized factorized information criterion (gFIC) for the SBN, which is proven to be statistically consistent with the marginal log-likelihood. To capture the dependencies within hidden variables in SBNs, a recognition network is employed to model the variational distribution. The resulting algorithm, which we call FABIA, can simultaneously execute both model selection and inference by maximizing the lower bound of gFIC. On both synthetic and real data, our experiments suggest that FABIA, when compared to state-of-the-art algorithms for learning SBNs, (i) produces a more concise model, thus enabling faster testing; (ii) improves predictive performance; (iii) accelerates convergence; and (iv) prevents overfitting. 1 Introduction The past decade has witnessed a dramatic increase in popularity of deep learning [20], stemming from its state-of-the-art performance across many domains, including computer vision [19], reinforcement learning [27], and speech recognition [15]. However, one important issue in deep learning is that its performance is largely determined by the underlying model: a larger and deeper network tends to possess more representational power, but at the cost of being more prone to overfitting [32], and increased computation. The latter issue presents a challenge for deployment to devices with constrained resources [2]. Inevitably, an appropriate model-selection method is required to achieve good performance. Model selection is here the task of selecting the number of layers and the number of nodes in each layer. Despite the rapid advancement in performance of deep models, little work has been done to address the problem of model selection. As a basic approach, cross-validation selects a model according to a validation score. However, this is not scalable, as its complexity is exponential with respect to LMAX the number of layers in the network: O(JMAX ), where JMAX and LMAX represent the maximum allowed numbers of nodes in each layer and number of layers, respectively. In Alvarez and Salzmann [2], a constrained optimization approach was proposed to infer the number of nodes in convolutional neural networks (CNNs); the key idea is to incorporate a sparse group Lasso penalty term to shrink all edges flowing into a node. Based on the shrinkage mechanism of the truncated gamma-negative binomial process, Zhou et al. [36] showed that the number of nodes in Poisson gamma belief networks (PGBNs) can be learned. Furthermore, we empirically observe that the shrinkage priors employed in Gan et al. [11], Henao et al. [14], Song et al. [31] can potentially perform model selection in certain tasks, even though this was not explicitly discussed in those works. One common problem for these approaches, however, is that the hyperparameters need to be tuned in order to achieve good performance, which may be time-consuming for some applications involving deep networks. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. The factorized asymptotic Bayesian (FAB) approach has recently been shown as a scalable modelselection framework for latent variable models. Originally proposed for mixture models [9], it was later extended to the hidden Markov model (HMM) [8], latent feature model (LFM) [12], and relational model [22]. By maximizing the approximate marginal log-likelihood, FAB introduces an `0 regularization term on latent variables, which can automatically estimate the model structure by eliminating irrelevant latent features through an expectation maximization [7] (EM)-like alternating optimization, with low computational cost. We develop here a scalable model selection algorithm within the FAB framework to infer the size of SBNs [28], a popular component of deep models, e.g., deep belief networks (DBN) [16] and deep Poisson factor analysis (DPFA) [10], and we assume here the depth of the SBN is fixed. Since the mean-field assumption used in FAB does not hold in SBNs, we employ a recognition network [18, 29, 25, 26] to represent the variational distribution. As our method combines Figure 1: Requirement for removal of nodes in (Left) the advantages of FAB Inference and Auto- SBN and (Right) FNN (dashed circles denote nodes encoding variational Bayesian (VB) frame- that can be removed). Note that a node in the SBN works, we term it as FABIA. To handle large can be removed only if all of its connected edges datasets, we also derive a scalable version shrink. For FNN, shrinkage of all incoming edges of FABIA with mini-batches. As opposed to eliminates a node. previous works, which predefine the SBN size [28, 30, 25, 5, 11, 6, 31, 26], FABIA determines it automatically. It should be noted that model selection in SBNs is more challenging than CNNs and feedforward neural networks (FNNs). As shown in Figure 1, simply imposing a sparsity prior or a group sparsity prior as employed in CNNs [2] and SBNs [11, 14, 31] does not necessarily shrink a node in SBN, since such approaches cannot guarantee to shrink all edges connected to a node. FABIA possesses the following distinguishing features: (i) a theoretical guarantee that its objective function, the generalized factorized information criterion (gFIC), is statistically consistent with the model?s marginal log-likelihood; and (ii) prevention of overfitting in large networks when the amount of training data is not sufficiently large, thanks to an intrinsic shrinkage mechanism. We also detail that FABIA has important connections with previous work on model regularization, such as Dropout [32], Dropconnect [35], shrinkage priors [11, 36, 14, 31], and automatic relevance determination (ARD) [34]. 2 Background An SBN is a directed graphical model for which the distribution of each layer is determined by the preceding layer via the sigmoid function, defined as ?(x) , 1/[1 + exp(?x)]. Let h(l) denote the lth hidden layer with Jl units, and v represent the visible layer with M units. The generative model of the SBN, with L hidden layers, is represented as p(h(L) |b) = JL Y (L) [?(bi )]hi (L) [?(?bi )]1?hi , p(h(l) |h(l+1) ) = i=1 Jl Y (l) (l) (l) (l) [?(?i )]hi [?(??i )]1?hi i=1 (l) ?i (l) (l) Wi? h(l+1) +ci , where l = 1, . . . , L?1, = and b corresponds to prior parameters; the notation i? means the ith row of a matrix. For the link function of the visible layer, i.e., p(v|h(1) ), we use the sigmoid function for binary data and the multinomial function for count data, as in Mnih and Gregor [25], Carlson et al. [6]. One difficulty of learning SBNs is the evaluation of the expectation with respect to the posterior distribution of hidden variables [31]. In Mnih and Gregor [25], a recognition network under the variational auto-encoding (VAE) framework [18] was proposed to approximate this intractable expectation. Compared with the Gibbs sampler employed in Gan et al. [11], Carlson et al. [6], Song et al. [31], the recognition network enables fast sampling of hidden variables in blocks. The variational parameters in 2 the recognition network can be learned via stochastic gradient descent (SGD), as shown in the neural variational inference and learning (NVIL) algorithm [25], for which multiple variance reduction techniques have been proposed to obtain better gradient estimates. Note that all previous work on learning SBNs assumes that a model with a fixed number of nodes in each layer has been provided. To select a model for an SBN, we follow the FAB framework [9], which infers the structure of a latent variable model by Bayesian inference. Let ? = {W, b, c} denote the model parameters and M be the model, with the goal in the FAB framework being to obtain the following maximum-likelihood (ML) estimate: N N X X XZ cML = arg max M ln p(v n |M) = arg max ln p(v n , hn |?)p(?|M) d? (1) M M n=1 n=1 hn As a key feature of the FAB framework, the `0 penalty term on hn induced by approximating (1) can remove irrelevant latent variables from the model (?shrinkage mechanism"). In practice, we can start from a large model and gradually reduce its size through this ?shrinkage mechanism" until convergence. Although a larger model has more representational capacity, a smaller model with similar predictive performance is preferred in practice, given a computational budget. A smaller model also enables faster testing, a desirable property in many machine learning tasks. Furthermore, a smaller model implies more robustness to overfitting, a common danger in deeper and larger models with insufficient training data. Since the integration in (1) is in general intractable, Laplace?s method [23] is employed in FAB inference for approximation. Consequently, gFIC can be derived as a surrogate function of the marginal log-likelihood. By maximizing the variational lower bound of gFIC, one obtains estimates of both parameters and the underlying model size. Note that while FAB inference uses the mean-field approximation for the variational distribution [9, 8, 22, 21], the same does not hold for SBNs, due to the correlation within hidden variables given the data. In contrast, the recognition network has been designed to approximate the posterior distribution of hidden variables with more fidelity [18, 29, 25]. Therefore, it can be a better candidate for the variational distribution in our task. 3 The FABIA Algorithm 3.1 gFIC for SBN Following the FAB inference approach, we first lower bound the marginal log-likelihood in (1) via a variational distribution q(h|?) as 1 R  XZ X p(v n , hn |?) p(?|M) d? ln p(v n , hn |?)p(?|M) d? ? q(hn |?) ln . q(hn |?) hn hn By applying Laplace?s method [23], we obtain ln p(v, h|M) = N M X 2? 1 X D? b + ln p(?|M) b ln( ) + ln p(v n , hn |?) ? ln |?m | + O(1) (2) 2 N 2 n=1 m=1 b represents the ML estimate of ?, and ?m represents the where D? refers to the dimension of ?, ? negative Hessian of the log-likelihood with respect to Wm? . Since ln |?m | in (2) cannot be represented with an analytical form, we must approximate it first, for the purpose of efficient optimization of the marginal log-likelihood. Following the gFIC [13] approach, we propose performing model selection in SBNs by introducing the shrinkage mechanism from this approximation. We start by providing the following assumptions, which are useful in the proof of our main theoretical results in Theorem 1. PN Assumption 1. The matrix n=1 ?n hTn hn has full rank with probability 1 as N ? ?, where ?n ? (0, 1). 1 For derivation clarity, we assume only one hidden layer and drop the bias term in the SBN 3 Note that this full-rank assumption implies that the SBN can preserve information in the large-sample limit, based on the degeneration analysis of gFIC [13]. Assumption 2. hn,j , ?j is generated from a Bernoulli distribution as hn,j ? Ber(?j ), where ?j > 0. Theorem 1. As N ? ?, ln |?m | can be represented with the following equality:  X X ln |?m | = ln hn,j ? ln N + O(1) j (3) n Proof. We first compute the negative Hessian as X 1 ? 1 X ?m = ? ln p(v n , hn |?) = ?(Wm? hn ) ?(?Wm? hn ) hTn hn . T N ?Wm? ?Wm? n N n From Assumption 1, ?m has full rank, since ?(x) ? (0, 1), ?x ? R. Furthermore, the determinant of ?m is bounded, since ?m ij ? (0, 1), ?i, j. Next, we define the following diagonal matrix  P   P  ( n hn,1 ) ( n hn,J ) ? , diag ,..., . N N P From Assumption 2, limN ?? P r[ n hn,j = 0] = 0, ?j. Therefore, ? is full-rank and its determinant is bounded, when N ? ?. Subsequently, we can decompose it as ?m = ? F (4) where F also has full rank and bounded determinant. Finally, applying the log determinant operator to the right side of (4) leads to our conclusion. To obtain the gFIC for SBN, we first follow the previous FAB approaches [9, 12, 22] to assume ? |M) = 0. We then apply the log-prior of ? to be constant with respect to N , i.e., limN ?? ln p(N Theorem 1 to (2) and have   N  X M X X b + M J ? D? ln N + H(q) gFIC SBN = max Eq ? ln hn,j + ln p(v n , hn |?) q 2 j 2 n n=1 (5) where H(q) is the entropy for the variational distribution q(h). P P As a key quantity in (5), M j (ln n hn,j ) can be viewed as a regularizer over the model to 2 execute model selection. This term directly operates on hidden nodes to perform shrinkage, which distinguishes our approach from previous work [11, 14, 31], where sparsity priors are assigned over edges. As illustrated in Figure 1, these earlier approaches do not necessarily shrink hidden nodes, as setting up a prior or a penalty term to shrink all edges connected to a node is very challenging in SBNs. Furthermore, the introduction of this quantity does not bring any cost of tuning parameters with crossvalidation. In contrast, the Lagrange parameter in Alvarez and Salzmann [2] and hyperparameters for priors in Gan et al. [11], Henao et al. [14], Zhou et al. [36], Song et al. [31] all need to be properly set, which may be time-consuming in certain applications involving deep and large networks. Under the same regularity conditions as Hayashi and Fujimaki [12], gFIC SBN is statistically consistent with the marginal log-likelihood, an important property of the FAB framework. Corollary 1. As N ? ?, ln p(v|M) = gFIC SBN + O(1). Proof. The conclusion holds as a direct extension of the consistency results in Hayashi and Fujimaki [12]. 3.2 Optimization of gFIC b is in general not The gFIC SBN in (5) cannot be directly optimized, because (i) the ML estimator ? available, and (ii) evaluation of the expectation over hidden variables is computationally expensive. Instead, the proposed FABIA algorithm optimizes the lower bound as gFIC SBN ? ? N  X   M X X ln Eq (hn,j ) + Eq ln p(v n , hn |?) + H(q) 2 j n n=1 4 (6) b ? p(v n , hn |?), ??; (ii) where we use the following facts to get the lower bound: (i) p(v n , hn |?) the concavity of the logarithm function; (iii) D? ? M J; and (iv) the maximum of all possible variational distributions q in (5). This leaves the choice of the form of the variational distribution. We could use the mean-field approximation as in previous FAB approaches [9, 8, 12, 13, 22, 21]. However, this approximation fails to capture the dependencies between hidden variables in SBN, as discussed in Song et al. [31]. Instead, we follow the recent auto-encoding VB approach [18, 29, 25, 26] to model the variational distribution with a recognition network, which maps v n to q(hn |v n , ?). Specifically, q(hn |v n , ?) = QJ QJ J?M parameterizes the recognition j=1 q(hn,j |v n , ?) = j=1 Ber[?(?j? v n )], where ? ? R network. Not only does using a recognition network allow us to more accurately model the variational distribution, it also enables faster sampling of hidden variables. The optimization of the lower bound in (6) can be executed via SGD; we use the Adam algorithm [17] as our optimizer. To reduce gradient variance, we employ the NVIL algorithm to estimate gradients in both generative and recognition networks. We also note that other methods, such as the importancesampled objectives method [5, 26, 24], can be used and such an extension is left for future work.  P  P Since M j ln n Eq (hn,j ) in (6) is only dependent on q, gradients of the generative model in 2 our FABIA algorithm and NVIL should be the same. However, gradients of the recognition network in FABIA are regularized to shrink the model, which is lacking in the standard VAE framework. We note that FABIA is a flexible framework, as its shrinkage term can be combined with any gradientbased variational auto-encoding methods to perform model selection, where only minimal changes to the gradients of the recognition network of the original methods are necessary. PN (l) A node j at level l will be removed from the model if it satisfies N1 n=1 Eq (hn,j ) ? (l) , where (l) is a threshold parameter to control the model size. This criterion has an intuitive interpretation that a node should be removed if the proportion of its samples equaling 1 is small. When the expectation is not exact, such as in the top layers, we use samples drawn from the recognition network to approximate it. 3.3 Minibatch gFIC To handle large datasets, we adapt the gFIC SBN developed in (5) to use minibatches (which is also appropriate for online learning). Suppose that each mini-batch contains Nmini data points, and currently we have seen T mini-batches, an unbiased estimator for (5) (up to constant terms) is then  NX Nmini mini  b p(v i+NT , hi+NT |?) MX  X ^ ln gFIC = max E ? ln h + T SBN q i+NT ,j q 2 j q(hi+NT |?) i=1 i=1  M J ? D? ln NT +1 + 2 (7) where NT = (T ? 1)Nmini . Derivation details are provided in Supplemental Materials. ^ An interesting observation in (7) is that gFIC SBN can automatically adjust shrinkage over time: At the P PNmini beginning of the optimization, i.e., when T is small, the shrinkage term M hi+NT ,j ) j ln( i=1 2 is more dominant in (7). As T becomes larger, the model is more stable and shrinkage gradually disappears. This phenomenon is also observed in our experiments in Section 5. 3.4 Computational complexity The NVIL algorithm has complexity O(M JNtrain ) for computing gradients in both the generative model and recognition network. FABIA needs an extra model selection step, also with complexity O(M JNtrain ) per step. As the number of training iteration increases, the additional cost to perform model selection is offset by the reduction of time when computing gradients, as observed in Figure 3. In test, the complexity is O(M JNtest K) per step, with K being the number of samples taken to compute the variational lower bound. Therefore, shrinkage of nodes can linearly reduce the testing time. 5 4 Related Work Dropout As a standard approach to regularize deep models, Dropout [32] randomly removes a certain number of hidden units during training. Note that FABIA shares this important characteristic by directly operating on nodes, instead of edges, to regularize the model, which has a more direct connection with model selection. One important difference is that in each training iteration, Dropout updates only a subset of the model; in contrast, FABIA updates every parameter in the model, which enables faster convergence. Shrinkage prior The shrinkage sparsity-inducing approach aims to shrink edges in a model, by employing either shrinkage priors [11, 14, 36, 31] or a random mask [35] on the weight matrix. In FABIA, the penalty term derived in gFIC of (5) also has the shrinkage property, but the shrinkage effect is instead imposed on the nodes. Furthermore, shrinkage priors are usually approached from the Bayesian framework, where Markov chain Monte Carlo (MCMC) is often needed for inference. In contrast, FABIA integrates the shrinkage mechanism from gFIC into the auto-encoding VB approach and thus is scalable to large deep models. Group Sparsity Application of group sparsity can be viewed as an extension of the shrinkage prior, with the key idea being to enforce sparsity on entire rows (columns) of the weight matrix [2]. This corresponds to the ARD prior [34] where each row (column) has an individual hyperparameter. In FNNs and CNNs, this is equivalent to node shrinkage in FABIA for SBNs. The structure of SBNs precludes a direct application of the group sparsity approach for model selection, but there exists an interesting opportunity for future work to extend FABIA to FNNs and CNNs. Nonparametric Prior In Adams et al. [1], a cascading Indian buffet process (IBP) based approach was proposed to infer the structure of the Gaussian belief network with continuous hidden units, for which the inference was performed via MCMC. By employing the nonparametric properties of the IBP prior, this approach can adjust the model size with observations. Due to the high computational cost of MCMC, however, it may not be scalable to large problems. 5 Experiments We test the proposed FABIA algorithm on synthetic data, as well as real image and count data. For comparison, we use the NVIL algorithm [25] as a baseline method, which does not have the model selection procedure. Both FABIA and NVIL are implemented in Theano [4] and tested on a machine with 3.0GHz CPU and 64GB RAM. The learning rate in Adam is set to be 0.001 and we follow the default settings of other parameters in all of our experiments. We set the threshold parameter (l) to be 0.001, ?l unless otherwise stated. We also tested Dropout but did not notice any clear improvement. The purpose of these experiments is to show that FABIA can automatically learn the model size, and achieve better or competitive performance with a more compact model. 5.1 Synthetic Dataset The synthetic data are generated from a one-layer SBN and a two-layer SBN, with M = 30 visible units in both cases. We simulate 1250 data points, and then follow an 80/20% split to obtain the training and test sets. For the one-layer case, we employ a true model with 5 nodes and initialize FABIA and NVIL with 25 nodes. For the two-layer case, the true network has the structure of 10-5 2 , and we initialize FABIA and NVIL with a network of 25-15. We compare the inferred SBN structure and test log-likelihood for FABIA, the NVIL algorithm initialized with the same model size as FABIA (denoted as ?NVIL"), and the NVIL algorithm initialized with the true model size (denoted as ?NVIL (True)?). One hundred independent random trials are conducted to report statistics. Figure 2(a) shows the mean and standard deviation of the number of nodes inferred by FABIA, as a function of iteration number. In both one- and two-layer cases, the mean of the inferred model size is very close to the ground truth. In Figure 2(b), we compare the convergence in terms of the test log-likelihood for different algorithms: FABIA has almost the same convergence speed as NVIL with 2 We list the number of nodes in the deeper layer first in all of our experiments. 6 ?16.0 25 Level 1 Level 1 10 5 0 20 Test log-likelihood Number of nodes Number of nodes 15 15 10 5 1 2 3 4 5 Iteration 6 7 1e2 ?17.0 ?17.5 ?18.0 ?18.5 FABIA NVIL (True) NVIL ?19.0 ?19.5 0 0 ?16.5 ?16.5 Level 2 20 0 1 2 3 4 5 Iteration 6 7 1e2 (a) Test log-likelihood 25 0 50 100 150 Time (seconds) 200 ?17.0 ?17.5 ?18.0 FABIA NVIL (True) NVIL ?18.5 ?19.0 0 50 100 150 200 250 Time (seconds) (b) Figure 2: (a) Inferred number of nodes from FABIA in (Left) one- and (Right) two-layer cases; (b) Test log-likelihood for different methods in (Left) one- and (Right) two-layer cases. the true model, both of which have remarkable gaps over the NVIL variant initialized with the same model size as FABIA. 5.2 Image Modeling We use the publicly available MNIST dataset, which contains 60, 000 training and 10, 000 test images of size 28 ? 28. Our performance metric is the variational lower bound of the test log-likelihood. The mini-batches for FABIA and NVIL are set to 100. For this dataset we compared FABIA with the VB approach in Gan et al. [11] and Rec-MCEM in Song et al. [31]. The VB approach in Gan et al. [11] can potentially shrink nodes, due to the three parameter beta-normal (TPBN) prior [3]. We claim a node P (l) (l) hj can be removed from the model, if its adjacent weight matrices satisfy k [Wk,j ]2 /J (l?1) < P (l+1) 10?8 and k [Wj,k ]2 /J (l+1) < 10?8 . We run the code provided in https://github.com/ zhegan27/dsbn_aistats2015 and use default parameter settings to report the VB results. We also implemented the Rec-MCEM approach but only observed shrinkage of edges, not nodes. Table 1 shows the variational lower bound of Table 1: Model size, test variational lower bound the test log-likelihood, model size, and test (VLB) (in nats), and test time (in seconds) on the time for different algorithms. FABIA achieves MNIST dataset. Note that FABIA and VB start from the highest test log-likelihood in all cases and the same model size as NVIL and Rec-MCEM. converges to smaller models, compared to Method Size VLB Time NVIL. FABIA also benefits from its more compact model to have the smallest test time. VB 81 -117.04 8.94 Furthermore, we observe that VB always overRec-MCEM 200 -116.70 8.52 shrinks nodes in the top layer, which might NVIL 200 -115.63 8.47 be related to the settings of hyperparameters. FABIA 107 ?114.96 6.88 Unlike VB, FABIA avoids the difficult task of VB 200-11 -113.69 22.37 tuning hyperparameters to balance predictive Rec-MCEM 200-200 -106.54 12.25 performance and model size. We also notice NVIL 200-200 -105.62 12.34 that the deeper layer in the two-layer model FABIA 135-93 ?104.92 9.18 did not shrink in VB, as our experiments suggest that all nodes in the deeper layer still have NVIL 200-200-200 -101.99 15.66 connections with nodes in adjacent layers. FABIA 136-77-72 ?101.14 10.97 Figure 3 shows the variational lower bound of the test log-likelihood and number of nodes in FABIA, as a function of CPU time, for different initial model sizes. Additional plots as a function of the number of iterations are provided in Supplemental Materials, which are similar to Figure 3. We note that FABIA initially has a similar log-likelihood that gradually outperforms NVIL, which can be explained by the fact that FABIA initially needs additional time to perform the shrinkage step but later converges to a smaller and better model. This gap becomes more obvious when we increase the number of hidden units from 200 to 500. The deteriorating performance of NVIL is most likely due to overfitting. In contrast, FABIA is robust to the change of the initial model size. 7 200 Number of nodes Test log-likelihood ?110 ?115 ?120 160 Level 1 Level 2 Level 3 140 120 100 ?125 ?130 Test log-likelihood 180 ?105 Fabia NVIL 0.0 0.5 1.0 1.5 2.0 500 ?105 450 ?110 400 ?115 ?120 ?125 ?130 ?135 80 0.0 Time (seconds) 1e5 ?100 0.5 1.0 1.5 ?140 2.0 Time (seconds) 1e5 (a) Number of nodes ?100 Fabia NVIL 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (seconds) 1e5 Level 1 Level 2 Level 3 350 300 250 200 150 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (seconds) 1e5 (b) Figure 3: Test log-likelihood and the number of nodes in FABIA, as a function of CPU time on the MNIST dataset, for an SBN with initial size as (a) 200-200-200 (b) 500-500-500. 5.3 Topic Modeling The two benchmarks we used for topic modeling are Reuters Corpus Volume I (RCV1) and Wikipedia, as in Gan et al. [10], Henao et al. [14]. RCV1 contains 794,414 training and 10,000 testing documents, with a vocabulary size of 10,000. Wikipedia is composed of 9,986,051 training documents, 1,000 test documents, and 7,702 words. The performance metric we use is the predictive perplexity on the test set, which cannot be directly evaluated. Instead, we follow the approach of 80/20% split on the test set, with details provided in Gan et al. [10]. We compare FABIA against DPFA [10], deep Poisson factor modeling (DPFM) [14], MCEM [31], Over-RSM [33], and NVIL. For both FABIA and NVIL, we use a mini-batch of 200 documents. The results for other methods are cited from corresponding references. We test DPFA and DPFM with the publicly available code provided by the authors; however, no shrinkage of nodes are observed in our experiments. Table 2 shows the perplexities of different algorithms on the RCV1 and Wikipedia datasets, respectively. Both FABIA and NVIL outperform other methods with marked margins. Test Perplexity Test Perplexity Interestingly, we note that FABIA does not shrink any nodes in the first layer, which is likely RCV1 Wikipedia 1000 1000 due to the fact that these two datasets have a FABIA FABIA 950 large number of visible units and thus a suffiNVIL NVIL 950 900 ciently large first hidden layer is necessary. This requirement of a large first hidden layer to prop850 erly model the data may also explain why NVIL 900 800 does not overfit on these datasets as much as it 750 does on MNIST; the training set of these datasets 850 700 being sufficiently large is another possible ex650 planation. We also computed test time but did 800 600 not observe any clear improvement of FABIA 100 400 1000 2000 100 400 1000 2000 # of nodes in the 1st layer # of nodes in the 1st layer over NVIL, which may be explained by the fact that most of the computation is spent on the first Figure 4: Test perplexities as a function of number layer in these two benchmarks. of nodes in the first layer, in the two-layer case. In Figure 4, we vary the number of hidden units in the first layer and fix the number of nodes in other layers to be 400. We use early stopping for NVIL to prevent it from overfitting with larger networks. For the networks with 100 and 400 nodes in the first layer, FABIA and NVIL have roughly the same perplexities. Once the number of nodes is increased to 1000, FABIA starts to outperform NVIL with remarkable gaps, which implies that FABIA can handle the overfitting problem, as a consequence of its shrinkage mechanism for model selection. We also observed that setting a larger (1) for the first layer in the 2000 units case for FABIA can stabilize its performance; 8 Table 2: Test perplexities and model size on the benchmarks. FABIA starts from a model initialized with 400 hidden units in each layer. RCV1 Over-RSM MCEM DPFA-SBN DPFA-RBM DPFM NVIL FABIA Wikipedia Perplexity Size Perplexity Size 128 128 1024-512-256 128-64-32 128-64 400-400 400-156 1060 1023 964 920 908 857 856 1024-512-256 128-64-32 128-64 400-400 400-151 770 942 783 735 730 we choose this value by cross-validation. The results for three layers are similar and are included in Supplemental Materials. 6 Conclusion and Future Work We develop an automatic method to select the number of hidden units in SBNs. The proposed gFIC criterion is proven to be statistically consistent with the model?s marginal log-likelihood. By maximizing gFIC, the FABIA algorithm can simultaneously execute model selection and inference tasks. Furthermore, we show that FABIA is a flexible framework that can be combined with autoencoding VB approaches. Our experiments on various datasets suggest that FABIA can effectively select a more-compact model and achieve better held-out performance. Our future work will be to extend FABIA to importance-sampling-based VAEs [5, 26, 24]. We also aim to explicitly select the number of layers in SBNs, and to tackle other popular deep models, such as CNNs and FNNs. Finally, investigating the effect of FABIA?s shrinkage mechanism on the gradient noise is another interesting direction. Acknowledgements The authors would like to thank Ricardo Henao for helpful discussions, and the anonymous reviewers for their insightful comments and suggestions. Part of this work was done during the internship of the first author at NEC Laboratories America, Cupertino, CA. This research was supported in part by ARO, DARPA, DOE, NGA, ONR, NSF, and the NEC Fellowship. 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Targeting EEG/LFP Synchrony with Neural Nets Yitong Li1 , Michael Murias2 , Samantha Major2 , Geraldine Dawson2 , Kafui Dzirasa2 , Lawrence Carin1 and David E. Carlson3,4 1 Department of Electrical and Computer Engineering, Duke University Departments of Psychiatry and Behavioral Sciences, Duke University 3 Department of Civil and Environmental Engineering, Duke University 4 Department of Biostatistics and Bioinformatics, Duke University {yitong.li,michael.murias,samantha.major,geraldine.dawson, kafui.dzirasa,lcarin,david.carlson}@duke.edu 2 Abstract We consider the analysis of Electroencephalography (EEG) and Local Field Potential (LFP) datasets, which are ?big? in terms of the size of recorded data but rarely have sufficient labels required to train complex models (e.g., conventional deep learning methods). Furthermore, in many scientific applications, the goal is to be able to understand the underlying features related to the classification, which prohibits the blind application of deep networks. This motivates the development of a new model based on parameterized convolutional filters guided by previous neuroscience research; the filters learn relevant frequency bands while targeting synchrony, which are frequency-specific power and phase correlations between electrodes. This results in a highly expressive convolutional neural network with only a few hundred parameters, applicable to smaller datasets. The proposed approach is demonstrated to yield competitive (often state-of-the-art) predictive performance during our empirical tests while yielding interpretable features. Furthermore, a Gaussian process adapter is developed to combine analysis over distinct electrode layouts, allowing the joint processing of multiple datasets to address overfitting and improve generalizability. Finally, it is demonstrated that the proposed framework effectively tracks neural dynamics on children in a clinical trial on Autism Spectrum Disorder. 1 Introduction There is significant current research on methods for Electroencephalography (EEG) and Local Field Potential (LFP) data in a variety of applications, such as Brain-Machine Interfaces (BCIs) [21], seizure detection [24, 26], and fundamental research in fields such as psychiatry [11]. The wide variety of applications has resulted in many analysis approaches and packages, such as Independent Component Analysis in EEGLAB [8], and a variety of standard machine learning approaches in FieldTrip [22]. While in many applications prediction is key, such as for BCIs [18, 19], in applications such as emotion processing and psychiatric disorders, clinicians are ultimately interested in the dynamics of underlying neural signals to help elucidate understanding and design future experiments. This goal necessitates development of interpretable models, such that a practitioner may understand the features and their relationships to outcomes. Thus, the focus here is on developing an interpretable and predictive approach to understanding spontaneous neural activity. A popular feature in these analyses is based on spectral coherence, where a specific frequency band is compared between pairwise channels, to analyze both amplitude and phase coherence. When two regions have a high power (amplitude) coherence in a spectral band, it implies that these areas are 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. coordinating in a functional network to perform a task [3]. Spectral coherence has been previously used to design classification algorithms on EEG [20] and LFP [30] data. Furthermore, these features have underlying neural relationships that can be used to design causal studies using neurostimulation [11]. However, fully pairwise approaches face significant challenges with limited data because of the proliferation of features when considering pairwise properties. Recent approaches to this problem include first partitioning the data to spatial areas and considering only broad relationships between spatial regions [33], or enforcing a low-rank structure on the pairwise relationships [30]. To analyze both LFP and EEG data, we follow [30] to focus on low-rank properties; however, this previous approach focused on a Gaussian process implementation for LFPs, that does not scale to the greater number of electrodes used in EEG. We therefore develop a new framework whereby the low-rank spectral patterns are approximated by parameterized linear projections, with the parametrization guided by neuroscience insights from [30]. Critically, these linear projections can be included in a convolutional neural network (CNN) architecture to facilitate end-to-end learning with interpretable convolutional filters and fast test-time performance. In addition to being interpretable, the parameterization dramatically reduces the total number of parameters to fit, yielding a CNN with only hundreds of parameters. By comparison, conventional deep models require learning millions of parameters. Even special-purpose networks such as EEGNet [15], a recently proposed CNN model for EEG data, still require learning thousands of parameters. The parameterized convolutional layer in the proposed model is followed by max-pooling, a single fully-connected layer, and a cross-entropy classification loss; this leads to a clear relationship between the proposed targeted features and outcomes. When presenting the model, interpretation of the filters and the classification algorithms are discussed in detail. We also discuss how deeper structures can be developed on top of this approach. We demonstrate in the experiments that the proposed framework mitigates overfitting and yields improved predictive performance on several publicly available datasets. In addition to developing a new neuroscience-motivated parametric CNN, there are several other contributions of this manuscript. First, a Gaussian Process (GP) adapter [16] within the proposed framework is developed. The idea is that the input electrodes are first mapped to pseudo-inputs by using a GP, which allows straightforward handling of missing (dropped or otherwise noise-corrupted) electrodes common in real datasets. In addition, this allows the same convolutional neural network to be applied to datasets recorded on distinct electrode layouts. By combining data sources, the result can better generalize to a population, which we demonstrate in the results by combining two datasets based on emotion recognition. We also developed an autoencoder version of the network to address overfitting concerns that are relevant when the total amount of labeled data is limited, while also improving model generalizability. The autoencoder can lead to minor improvements in performance, which is included in the Supplementary Material. 2 Basic Model Setup: Parametric CNN The following notation is employed: scalars are lowercase italicized letters, e.g. x, vectors are bolded lowercase letters, e.g. p x, and matrices are bolded uppercase letters, e.g. X. The convolution operator is denoted ?, and | = 1. ? denotes the Kronecker product. denotes an element-wise product. The input data are Xi 2 RC?T , where C is the number of simultaneously recorded electrodes/channels, and T is given by the sampling rate and time length; i = 1, . . . , N , where N | is the total number of trials. The data can also be represented as Xi = [xi1 , ? ? ? , xiC ] , where xic 2 RT is the data restricted to the cth channel. The associated labels are denoted yi , which is an integer corresponding to a label. The trial index i is added only when necessary for clarity. An example signal is presented in Figure 1 (Left). The data are often windowed, the ith of which yields Xi and the associated label yi . Clear identification of phase and power relationships among channels motivates the development of a structured neural network model for which the convolutional filters target this synchrony, or frequency-specific power and phase correlations. 2.1 SyncNet Inspired both by the success of deep learning and spectral coherence as a predictive feature [12, 30], a CNN is developed to target these properties. The proposed model, termed SyncNet, performs a structured 1D convolution to jointly model the power, frequency and phase relationships between channels. 2 Figure 1: (Left) Visualization of EEG dataset on 8 electrodes split into windows. The markers (e.g., ?FP1?) denote electrode names, which have corresponding spatial locations. (Right) 8 channels of synthetic data. Refer to Section 2.2 for more detail. Figure 2: SyncNet follows a convolutional neural network structure. The right side is the SyncNet (Section 2.1), which is parameterized to target relevant quantities. The left side is the GP adapter, which aims at unifying different electrode layout and reducing overfitting (Section 3). This goal is achieved by using parameterized 1-dimensional convolutional filters. Specifically, the kth of K filters for channel c is (k) (k) fc (? ) = bc cos(! (k) ? + (k) c ) exp( (k) 2 ? ). (1) The frequency ! (k) 2 R+ and decay (k) 2 R+ parameters are shared across channels, and they define the real part of a (scaled) Morlet wavelet1 . These two parameters define the spectral properties targeted by the kth filter, where ! (k) controls the center of the frequency spectrum and (k) controls (k) (k) the frequency-time precision trade-off. The amplitude bc 2 R+ and phase shift c 2 [0, 2?] are channel-specific. Thus, the convolutional filter in each channel will be a discretized version of a scaled and rotated Morlet wavelet. By parameterizing the model in this way, all channels are targeted collectively. The form in (1) is motivated by the work in [30], but the resulting model we develop is far more computationally efficient. A fuller discussion of the motivation for (1) is detailed in Section 2.2. For practical reasons,?the filters are restricted to have finite length each ?time step ? takes ? ? NN? , and N? N? ? 1 N? 1 an integer value from , 1 when N is even and from , when N? is odd. ? 2 2 2 2 (k) For typical learned ?s, the convolutional filter vanishes by the edges of the window. Succinctly, PC (k) the output of the k convolutional filter bank is given by h(k) = c=1 fc (? ) ? xc . The simplest form of SyncNet contains only one convolution layer, as in Figure 2. The output from each filter bank h(k) is passed through a Rectified Linear Unit (ReLU), followed by max pooling ? (k) for each filter. The filter outputs h ? (k) for k = 1, . . . , K are over the entire window, to return h concatenated and used as input to a softmax classifier with the cross-entropy loss to predict y?. Because of the temporal and spatial redundancies in EEG, dropout is instituted at the channel level, with ? xc /p, with probability p dropout(xc ) = . (2) 0, with probability 1 p p determines the typical percentage of channels included, and was set as p = 0.75. It is straightforward to create deeper variants of the model by augmenting SyncNet with additional standard convolutional 1 It is straightforward to use the Morlet wavelet directly and define the outputs as complex variables and define the neural network to target the same properties, but this leads to both computational and coding overhead. 3 layers. However, in our experiments, adding more layers typically resulted in over-fitting due to the limited numbers of training samples, but will likely be beneficial in larger datasets. 2.2 SyncNet Targets Class Differences in Cross-Spectral Densities The cross-spectral density [3] is a widely used metric for understanding the synchronous nature of signal in frequency bands. The cross-spectral density is typically constructed by converting a time-series into a frequency representation, and then calculating the complex covariance matrix in each frequency band. In this section we sketch how the SyncNet filter bank targets cross-spectral densities to make optimal classifications. The discussion will be in the complex domain first, and then it will be demonstrated why the same result occurs in the real domain. In the time-domain, it is possible to understand the cross-spectral density of a single frequency band by using a cross-spectral kernel [30] to define the covariance function of a Gaussian process. Letting ? = t t0 , the cross-spectral kernel is defined CSD Kcc 0 tt0 = cov(xct , xc0 t0 ) = Acc0 ?(? ), ?(? ) = exp 1 2 ? 2 ? + |! ? ? . (3) Here, ! ? and ? control the frequency band. c and c0 are channel indexes. A 2 CC?C is a positive semi-definite matrix that defines the cross-spectral density for that frequency band controlled by ?(? ). Each entry Acc0 is made of of a magnitude |Acc0 | that controls the power (amplitude) coherence between electrodes in that frequency band and a complex phase that determines the optimal time offset between the signals. The covariance over the complete multi-channel times series is given by K CSD = A ? ?(? ). The power (magnitude) coherence is given by the absolute value of the entry, and the phase offset can be determined by the rotation in the complex space. A generative model for oscillatory neural signals is given by a Gaussian process with this kernel [30], where vec(X) ? CN (0, K CSD + 2 IC?T ). The entries of K CSD are given from (3). CN denotes the circularly symmetric complex normal. The additive noise term 2 IC?T is excluded in the following for clarity. Note that the complex form of (1) in SyncNet across channels is given as f (? ) = f! (? )s, where f! (? ) = exp( 12 ? 2 + |!? ) is the filter over time and s = b exp(| ) are the weights and rotations of a single SyncNet filter. Suppose that each channel was filtered independently by the filter ? c = f! ? xc = F!? xc , f! = f! (? ) with a vector input ? . Writing the convolution in matrix form as x T ?T where F! 2 C is a matrix formulation of the convolution operator, results in a filtered signal ? c ? CN 0, Acc F!? ?(? )F! . For a filtered version over all channels, X T = [xT1 , ? ? ? , xTC ], the x distribution would be given by ? ? ? = vec(F ? X T ) ? CN 0, A ? F ? ?(? )F! , x ? t ? CN (0, A F!? ?(? )F! tt ). (4) vec(X) ! ! ? ? ? t 2 RC is defined as the observation at time t for all C channels. The diagonal of F!? ?(? )F! will x ? ? ? reach a steady-state quickly away from the edge effects, so we state this as const = F! ?(? )F! tt . ?t ? The output from the SyncNet filter bank prior to the pooling stage is then given by ht = s? x CN (0, const ? s? As). We note that the signal-to-noise ratio would be maximized by matching the filter?s (f! ) frequency properties to the generated frequency properties; i.e. and ! from (1) should match ? and ! ? from (3). We next focus on the properties of an optimal s. Suppose that two classes are generated from (3) with cross-spectral densities of A0 and A1 for classes 0 and 1, respectively. Thus, the signals are drawn from CN (0, Ay ? ?(? )) for y = {0, 1}. The optimal projection s? would maximize the differences in the distribution ht depending on the class, which is equivalent to maximizing the ratio between the variances of the two cases. Mathematically, this is equivalent to finding n ? o ? 1 s s A0 s s? = arg maxs max ss? A , = arg maxs | log(s? A1 s) log(s? A0 s)|. (5) A 0 s s? A 1 s Note that the constant dropped out due to the ratio. Because the SyncNet filter is attempting to classify the two conditions, it should learn to best differentiate the classes and match the optimal s? . We demonstrate in Section 5.1 on synthetic data that SyncNet filters do in fact align with this optimal direction and is therefore targeting properties of the cross-spectral densities. In the above discussion, the argument was made with respect to complex signals and models; however, a similar result holds when only the real domain is used. Note that if the signals are oscillatory, then 4 the result after the filtering of the domain and the max-pooling will be essentially the same as using a max-pooling on the absolute value of the complex filters. This is because the filtered signal is rotated through the complex domain, and will align with the real domain within the max-pooling period for standard signals. This is shown visually in Supplemental Figure 9. 3 Gaussian Process Adapter A practical issue in EEG datasets is that electrode layouts are not constant, either due to inconsistent device design or electrode failure. Secondly, nearby electrodes are highly correlated and contain redundant information, so fitting parameters to all electrodes results in overfitting. These issues are addressed by developing a Gaussian Process (GP) adapter, in the spirit of [16], trained with SyncNet as shown in the left side of Figure 2. Regardless of the electrode layout, the observed signal X at electrode locations p = {p1 , ? ? ? , pC } are mapped to a shared number of pseudo-inputs at locations p? = {p?1 , ? ? ? , p?L } before being input to SyncNet. In contrast to prior work, the proposed GP adapter is formulated as a multi-task GP [4] and the pseudoinput locations p? are learned. A GP is used to map X 2 RC?T at locations p to the pseudo-signals X ? 2 RL?T at locations p? , where L < C is the number of pseudo-inputs. Distances are constructed by projecting each electrode into a 2D representation by the Azimuthal Equidistant Projection. When evaluated at a finite set of points, the multi-task GP [4] can be written as a multivariate normal vec(X) ? N f , 2 IC?T , f ? N (0, K) . (6) K is constructed by a kernel function K(?, c, c0 ) that encodes separable relationships through time and through space. The full covariance matrix can be calculated as K = Kpp ? Ktt , where Kpc pc0 = ?1 exp( ?2 ||pc pc0 ||1 ) and Ktt is set to identity matrix IT . Kpp 2 RC?C targets the spatial relationship across channels using the exponential kernel. Note that this kernel K is distinct from K CSD used in section 2.2. Let the pseudo-inputs locations be defined as p?l for l = 1, ? ? ? , L. Using the GP formulation, the signal can be inferred at the L pseudo-input locations from the original signal. Following [16], only the expectation of the signal is used (to facilitate fast computation), which is given by X ? = E(X ? |X) = Kp? p (Kpp + 2 IC ) 1 X. An illustration of the learned new locations is shown under X ? in Figure 2. The derivation of this mathematical form and additional details on the GP adapter are included in Supplemental Section A. The GP adapter parameters p? , ?1 , ?2 are optimized jointly with SyncNet. The input signal Xi is mapped to Xi? , which is then input to SyncNet. The predicted label y?i is given by y?i = Sync(Xi? ; ?), where Sync()? is the prediction function of SyncNet. Given the SyncNet loss function PN PN yi , yi ) = i=1 ` (Sync(Xi? ; ?), yi ), the overall training loss function i=1 ` (? PN PN L = i=1 ` (Sync(E[Xi? |Xi ]; ?), yi ) = i=1 ` Sync(Kp? p (Kpp + 2 IC ) 1 Xi ; ?), yi , (7) is jointly minimized over the SyncNet parameters ? and the GP adapter parameters {p? , ?1 , ?2 }. The GP uncertainty can be included in the loss at the expense of significantly increased optimization cost, but does not result in performance improvements to justify the increased cost [16]. 4 Related Work Frequency-spectrum features are widely used for processing EEG/LFP signals. Often this requires calculating synchrony- or entropy-based features within predefined frequency bands, such as [20, 5, 9, 14]. There are many hand-crafted features and classifiers for a BCI task [18]; however, in our experiments, these hand-crafted features did not perform well on long oscillatory signals. The EEG signal is modeled in [1] as a matrix-variate model with spatial and spectral smoothing. However, the number of parameters scales with time length, rendering the approach ineffective for longer time series. A range-EEG feature has been proposed [23], which measures the peak-to-peak amplitude. In contrast, our approach learns frequency bands of interest and we can deal with long time series evaluated in our experiments. Deep learning has been a popular recent area of research in EEG analysis. This includes Restricted Boltzmann Machines and Deep Belief Networks [17, 36], CNNs [32, 29], and RNNs [2, 34]. These 5 approaches focus on learning both spatial and temporal relationships. In contrast to hand-crafted features and SyncNet, these deep learning methods are typically used as a black box classifier. EEGNET [15] considered a four-layer CNN to classify event-related potentials and oscillatory EEG signals, demonstrating improved performance over low-level feature extraction. This network was designed to have limited parameters, requiring 2200 for their smallest model. In contrast, the SyncNet filters are simple to interpret and require learning only a few hundred parameters. An alternative approach is to design GP kernels to target synchrony properties and learn appropriate frequency bands. The phase/amplitude synchrony of LFP signals has been modeled [30, 10] with the cross-spectral mixture (CSM) kernel. This approach was used to define a generative model over differing classes and may be used to learn an unsupervised clustering model. A key issue with the CSM approach is the computational complexity, where gradients cost O(N T C 3 ) (using approximations), and is infeasible with the larger number of electrodes in EEG data. In contrast, the proposed GP adapter requires only a single matrix inversion shared by most data points, which is O(C 3 ). The use of wavelets has previously been considered in scattering networks [6]. Scattering networks used Morlet wavelets for image classification, but did not consider the complex rotation of wavelets over channels nor the learning of the wavelet widths and frequencies considered here. 5 Experiments To demonstrate that SyncNet is targeting synchrony information, we first apply it to synthetic data in Section 5.1. Notably, the learned filter bank recovers the optimal separating filter. Empirical performance is given for several EEG datasets in Section 5.2, where SyncNet often has the highest hold-out accuracy while maintaining interpretable features. The usefulness of the GP adapter to combine datasets is demonstrated in Section 5.3, where classification performance is dramatically improved via data augmentation. Empirical performance on an LFP dataset is shown in Section 5.4. Both the LFP signals and the EEG signals measure broad voltage fluctuations from the brain, but the LFP has a significantly cleaner signal because it is measured inside the cortical tissue. In all tested cases, SyncNet methods have essentially state-of-the-art prediction while maintaining interpretable features. The code is written in Python and Tensorflow. The experiments were run on a 6-core i7 machine with a Nvidia Titan X Pascal GPU. Details on training are given in Supplemental Section C. 5.1 2 Synthetic Dataset Optimal Learned Synthetic data are generated for two classes by drawing data from a circularly symmetric normal matching the synchrony assumptions discussed in Section 2.2. The frequency band is pre-defined as ! ? = 10Hz and ? is defined as 40 (frequency variance of 2.5Hz) in (3). The number of channels is set to C = 8. Example data generated by this procedure is shown in Figure 1 (Right), where only the real part of the signal is kept. 1 0 -1 -2 -2 -1 0 1 2 A1 and A0 are set such that the optimal vector Figure 3: Each dot represents one of 8 electrodes. from solving (5) is given by the shape visual- The dots give complex directions for optimal and ized in Figure 3. This is accomplished by set- learned filters, demonstrating that SyncNet approxting A0 = IC and A1 = I + s? (s? )? . Data imately recovers optimal filters. is then simulated by drawing from vec(X) ? CN (0, K CSD + 2 IC?T ) and keeping only the real part of the signal. K CSD is defined in equation (3) with A set to A0 or A1 depending on the class. In this experiment, the goal is to relate the filter learned in SyncNet and to this optimal separating plane s? . To show that SyncNet is targeting synchrony, it is trained on this synthetic data using only one single convolutional filter. The learned filter parameters are projected to the complex space by s = b exp(| ), and are shown overlaid (rotated and rescaled to handle degeneracies) with the 6 optimal rotations in Figure 3. As the amount of data increases, the SyncNet filter recovers the expected relationship between channels and the predefined frequency band. In addition, the learned ! is centered at 11Hz, which is close to the generated feature band ! ? of 10Hz. These synthetic data results demonstrate that SyncNet is able to recover frequency bands of interest and target synchrony properties. 5.2 Performance on EEG Datasets We consider three publicly available datasets for EEG classification, described below. After the validation on the publicly available data, we then apply the method to a new clinical-trial data, to demonstrate that the approach can learn interpretable features that track the brain dynamics as a result of treatment. UCI EEG: This dataset2 has a total of 122 subjects with 77 diagnosed with alcoholism and 45 control subjects. Each subject undergoes 120 separate trials. The stimuli are pictures selected from 1980 Snodgrass and Vanderwart picture set. The EEG signal is of length one second and is sampled at 256Hz with 64 electrodes. We evaluate the data both within subject, which is randomly split as 7 : 1 : 2 for training, validation and testing, and using 11 subjects rotating test set. The classification task is to recover whether the subject has been diagnosed with alcoholism or is a control subject. DEAP dataset: The ?Database for Emotion Analysis using Physiological signals? [14] has a total of 32 participants. Each subject has EEG recorded from 32 electrodes while they are shown a total of 40 one-minute long music videos with strong emotional score. After watching each video, each subject gave an integer score from one to nine to evaluate their feelings in four different categories. The self-assessment standards are valence (happy/unhappy), arousal (bored/excited), dominance (submissive/empowered) and personal liking of the video. Following [14], this is treated as a binary classification with a threshold at a score of 4.5. The performance is evaluated with leave-one-out testing, and the remaining subjects are split to use 22 for training and 9 for validation. SEED dataset: This dataset [35] involves repeated tests on 15 subjects. Each subject watches 15 movie clips 3 times. It clip is designated with a negative/neutral/positive emotion label, while the EEG signal is recorded at 1000Hz from 62 electrodes. For this dataset, leave-one-out cross-validation is used, and the remaining 14 subjects are split with 10 for training and 4 for validation. ASD dataset: The Autism Spectral Disorder (ASD) dataset involves 22 children from ages 3 to 7 years undergoing treatment for ASD with EEG measurements at baseline, 6 months post treatment, and 12 months post treatment. Each recording session involves 3 one-minute videos designed to measure responses to social stimuli and controls, measured with a 121 electrode array. The trial was approved by the Duke Hospital Institutional Review Board and conducted under IND #15949. Full details on the experiments and initial clinical results are available [7]. The classification task is to predict the time relative to treatment to track the change in neural signatures post-treatment. The cross-patient predictive ability is estimated with leave-one-out cross-validation, where 17 patients are used to train the model and 4 patients are used as a validation set. Dataset DE [35] PSD [35] rEEG [23] Spectral [14] EEGNET [15] MC-DCNN [37] SyncNet GP-SyncNet UCI DEAP [14] Within Cross Arousal Valence Domin. Liking 0.821 0.622 0.529 0.517 0.528 0.577 0.816 0.605 0.584 0.559 0.595 0.644 0.702 0.614 0.549 0.538 0.557 0.585 * * 0.620 0.576 * 0.554 0.878 0.672 0.536 0.572 0.589 0.594 0.840 0.300 0.593 0.604 0.635 0.621 0.918 0.705 0.611 0.608 0.651 0.679 0.923 0.723 0.592 0.611 0.621 0.659 Table 1: Classification accuracy on EEG datasets. SEED [35] Emotion 0.491 0.352 0.468 * 0.533 0.527 0.558 0.516 ASD Stage 0.504 0.499 0.361 * 0.363 0.584 0.630 0.637 The accuracy of predictions on these EEG datasets, from a variety of methods, is given in Table 1. We also implemented other hand-crafted spatial features, such as the brain symmetric index [31]; however, their performance was not competitive with the results here. EEGNET is an EEG-specific convolutional network proposed in [15]. The ?Spectral? method from [14] uses an SVM on extracted 2 https://kdd.ics.uci.edu/databases/eeg/eeg.html 7 (a) Spatial pattern of learned amplitude b. (b) Spatial pattern of learned phase . Figure 4: Learned filter centered at 14Hz on the ASD dataset. Figures made with FieldTrip [22]. spectral power features from each electrode in different frequency bands. MC-DCNN [37] denotes a 1D CNN where the filters are learned without the constraints of the parameterized structure. The SyncNet used 10 filter sets both with (GP-SyncNet) and without the GP adapter. Remarkably, the basic SyncNet already delivers state-of-the-art performance on most tasks. In contrast, the handcrafted features did not effectively cannot capture available information and the alternative CNN based methods severely overfit the training data due to the large number of free parameters. In addition to state-of-the-art classification performance, a key component of SyncNet is that the features extracted and used in the classification are interpretable. Specifically, on the ASD dataset, the proposed method significantly improves the state-of-the-art. However, the end goal of this experiment is to understand how the neural activity is changing in response to the treatment. On this task, the ability of SyncNet to visualize features is important for dissemination to medical practitioners. To demonstrate how the filters can be visualized and communicated, we show one of the filters learned in SyncNet on the ASD dataset in Figure 4. This filter, centered at 14Hz, is highly associated with the session at 6 months post-treatment. Notably, this filter bank is dominantly using the signals measured at the forward part of the scalp (Figure 4, Left). Intriguingly, the phase relationships are primarily in phase for the frontal regions, but note that there are off-phase relationships between the midfrontal and the frontal part of the scale (Figure 4, Right). Additional visualizations of the results are given in Supplemental Section E. 5.3 Experiments on GP adapter In the previous section, it was noted that the GP adapter can improve performance within an existing dataset, demonstrating that the GP adapter is useful to reduce the number of parameters. However, our primary designed use of the GP Adapter is to unify different electrode layouts. This is explored further by applying the GP-SyncNet to the UCI EEG dataset and changing the number of pseudo-inputs. Notably, a mild reduction in the number of pseudo-inputs improves performance over directly using the measured data (Supplemental Figure 6(a)) by reducing the total number of parameters. This is especially true when comparing the GP adapter to using a random subset of channels to reduce dimensionality. SyncNet GP-SyncNet GP-SyncNet Joint DEAP [14] dataset 0.521 ? 0.026 0.557 ? 0.025 0.603 ? 0.020 SEED [35] dataset 0.771 ? 0.009 0.762 ? 0.015 0.779 ? 0.009 Table 2: Accuracy mean and standard errors for training two datasets separately and jointly. To demonstrate that the GP adapter can be used to combine datasets, the DEAP and SEED datasets were trained jointly using a GP adapter. The SEED data was downsampled to 128Hz to match the frequency of DEAP dataset, and the data was separated into 4 second windows due to their different lengths. The label for the trial is attached for each window. To combine the labeling space, only the negative and positive emotion labels were kept in SEED and valence was used in the DEAP dataset. The number of pseudo-inputs is set to L = 26. The results are given in Table 2, which demonstrates that combining datasets can lead to dramatically improved generalization ability due to the data 8 augmentation. Note that the basic SyncNet performances in Table 2 differ from the results in Table 1. Specifically, the DEAP dataset performance is worse; this is due to significantly reduced information when considering a 4 second window instead of a 60 second window. Second, the performance on SEED has improved; this is due to considering only 2 classes instead of 3. 5.4 Performance on an LFP Dataset Due to the limited publicly available multi-region LFP datasets, only a single LFP data was included in the experiments. The intention of this experiment is to show that the method is broadly applicable in neural measurements, and will be useful with the increasing availability of multi-region datasets. An LFP dataset is recorded from 26 mice from two genetic backgrounds (14 wild-type and 12 CLOCK 19). CLOCK 19 mice are an animal model of a psychiatric disorder. The data are sampled at 200 Hz for 11 channels. The data recording from each mouse has five minutes in its home cage, five minutes from an open field test, and ten minutes from a tail-suspension test. The data are split into temporal windows of five seconds. SyncNet is evaluated by two distinct prediction tasks. The first task is to predict the genotype (wild-type or CLOCK 19) and the second task is to predict the current behavior condition (home cage, open field, or tail-suspension test). We separate the data randomly as 7 : 1 : 2 for training, validation and testing Behavior Genotype PCA + SVM 0.911 0.724 DE [35] 0.874 0.771 PSD [35] 0.858 0.761 rEEG [23] 0.353 0.449 EEGNET [15] 0.439 0.689 SyncNet 0.946 0.926 Table 3: Comparison between different methods on an LFP dataset. Results from these two predictive tasks are shown in Table 3. SyncNet used K = 20 filters with filter length 40. These results demonstrate that SyncNet straightforwardly adapts to both EEG and LFP data. These data will be released with publication of the paper. 6 Conclusion We have proposed SyncNet, a new framework for EEG and LFP data classification that learns interpretable features. In addition to our original architecture, we have proposed a GP adapter to unify electrode layouts. Experimental results on both LFP and EEG data show that SyncNet outperforms conventional CNN architectures and all compared classification approaches. Importantly, the features from SyncNet can be clearly visualized and described, allowing them to be used to understand the dynamics of neural activity. Acknowledgements In working on this project L.C. received funding from the DARPA HIST program; K.D., L.C., and D.C. received funding from the National Institutes of Health by grant R01MH099192-05S2; K.D received funding from the W.M. 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Near-Optimal Edge Evaluation in Explicit Generalized Binomial Graphs Sanjiban Choudhury The Robotics Institute Carnegie Mellon University [email protected] Shervin Javdani The Robotics Institute Carnegie Mellon University [email protected] Siddhartha Srinivasa The Robotics Institute Carnegie Mellon University [email protected] Sebastian Scherer The Robotics Institute Carnegie Mellon University [email protected] Abstract Robotic motion-planning problems, such as a UAV flying fast in a partially-known environment or a robot arm moving around cluttered objects, require finding collision-free paths quickly. Typically, this is solved by constructing a graph, where vertices represent robot configurations and edges represent potentially valid movements of the robot between these configurations. The main computational bottlenecks are expensive edge evaluations to check for collisions. State of the art planning methods do not reason about the optimal sequence of edges to evaluate in order to find a collision free path quickly. In this paper, we do so by drawing a novel equivalence between motion planning and the Bayesian active learning paradigm of decision region determination (DRD). Unfortunately, a straight application of existing methods requires computation exponential in the number of edges in a graph. We present B I SEC T, an efficient and near-optimal algorithm to solve the DRD problem when edges are independent Bernoulli random variables. By leveraging this property, we are able to significantly reduce computational complexity from exponential to linear in the number of edges. We show that B I SEC T outperforms several state of the art algorithms on a spectrum of planning problems for mobile robots, manipulators, and real flight data collected from a full scale helicopter. Open-source code and details can be found here: https://github.com/sanjibac/matlab_learning_collision_checking 1 Introduction Motion planning, the task of computing collision-free motions for a robotic system from a start to a goal configuration, has a rich and varied history [23]. Up until now, the bulk of the prominent research has focused on the development of tractable planning algorithms with provable worst-case performance guarantees such as computational complexity [3], probabilistic completeness [24] or asymptotic optimality [20]. In contrast, analysis of the expected performance of these algorithms on the real world planning problems a robot encounters has received considerably less attention, primarily due to the lack of standardized datasets or robotic platforms. However, recent advances in affordable sensors and actuators have enabled mass deployment of robots that navigate, interact and collect real data. This motivates us to examine the following question: ?How can we design planning algorithms that, subject to on-board computation constraints, maximize their expected performance on the actual distribution of problems that a robot encounters?? This paper addresses a class of robotic motion planning problems where path evaluation is expensive. For example, in robot arm planning [12], evaluation requires expensive geometric intersection 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. || || || ? ?|| && && ! ! ?? && ! && ? ? X X X !X tcas eq enabled && tcas eq && X X ? ? enabled X X ? && tcas enabled ? && tcas eq eq X X enabled tcas eq intent known X not intent known ? ? tcas eq Xnot (a) (b) tcas intent known tcas eq eq intent notnot known Figure 1: The feasible path identification problem (a) The explicit graph contains dynamically feasible maneuvers [27] for a UAV flying fast, with a set candidate paths. The map shows the distribution of edge validity for the graph. (b) Given a distribution over edges, our algorithm checks an edge, marks it as invalid (red) or valid (green), and updates its belief. We continue until a feasible path is identified as free. We aim to minimize the number of expensive edge evaluations. computations. In UAV path planning [9], evaluation must be done online with limited computational resources (Fig. 1). State of the art planning algorithms [11] first compute a set of unevaluated paths quickly, and then evaluate them sequentially to find a valid path. Oftentimes, candidate paths share common edges. Hence, evaluation of a small number of edges can provide information about the validity of many candidate paths simultaneously. Methods that check paths sequentially, however, do not reason about these common edges. This leads us naturally to the feasible path identification problem - given a library of candidate paths, identify a valid path while minimizing the cost of edge evaluations. We assume access to a prior distribution over edge validity, which encodes how obstacles are distributed in the environment (Fig. 1(a)). As we evaluate edges and observe outcomes, the uncertainty of a candidate path collapses. Our first key insight is that this problem is equivalent to decision region determination (DRD) [19, 5]) - given a set of tests (edges), hypotheses (validity of edges), and regions (paths), the objective is to drive uncertainty into a single decision region. This linking enables us to leverage existing methods in Bayesian active learning for robotic motion planning. Chen et al. [5] provide a method to solve this problem by maximizing an objective function that satisfies adaptive submodularity [15] - a natural diminishing returns property that endows greedy policies  with near-optimality guarantees. Unfortunately, naively applying this algorithm requires O 2E computation to select an edge to evaluate, where E is the number of edges in all paths. We define the Bern-DRD problem, which leverages additional structure in robotic motion planning by assuming edges are independent Bernoulli random variables 1 , and regions correspond to sets of edges evaluating to true. We propose Bernoulli Subregion Edge Cutting (B I SEC T), which provides a greedy policy to select candidate edges in O (E). We prove our surrogate objective also satisfies adaptive submodularity [15], and provides the same bounds as Chen et al. [5] while being more efficient to compute. We make the following contributions: 1. We show a novel equivalence between feasible path identification and the DRD problem, linking motion planning to Bayesian active learning. 2. We develop B I SEC T, a near-optimal algorithm for the special case of Bernoulli tests, which  selects tests in O (E) instead of O 2E . 3. We demonstrate the efficacy of our algorithm on a spectrum of planning problems for mobile robots, manipulators, and real flight data collected from a full scale helicopter. 1 1 1 1 Generally, edges in this graph are correlated, as edges in collision are likely to have neighbours in collision. Unfortunately, even measuring this correlation is challenging, especially in the high-dimensional non-linear configuration space of robot arms. Assuming independent edges is a common simplification [23, 25, 7, 2, 11] 2 1 2 Problem Formulation 2.1 Planning as Feasible Path Identification on Explicit Graphs Let G = (V, E) be an explicit graph that consists of a set of vertices V and edges E. Given a pair of start and goal vertices, (vs , vg ) ? V , a search algorithm computes a path ? ? E - a connected sequence of valid edges. To ascertain the validity of an edge, it invokes an evaluation function Eval : E ? {0, 1}. We address applications where edge evaluation is expensive, i.e., the computational cost c(e) of computing Eval(e) is significantly higher than regular search operations2 . We define a world as an outcome vector o ? {0, 1}|E| which assigns to each edge a boolean validity when evaluated, i.e. Eval(e) = o(e). We assume that the outcome vector is sampled from an independent Bernoulli distribution P (o), giving rise to a Generalized Binomial Graph (GBG) [13]. We make a second simplification to the problem - from that of search to that of identification. Instead of searching G online for a path, we frame the problem as identifying a valid path from a library of ?good? candidate paths ? = (?1 , ?2 , . . . , ?m ). The candidate set of paths ? is constructed offline, while being cognizant of P (o), and can be verified to ensure that all paths have acceptable solution quality when valid. 3 Hence we care about completeness with respect to ? instead of G. We wish to design an adaptive edge selector Select(o) which is a decision tree that operates on a world o, selects an edge for evaluation and branches on its outcome. The total cost of edge evaluation is c(Select(o)). Our objective is to minimize the cost required to find a valid path: Y min Eo?P (o) [c(Select(o))] s.t ?o, ?? : o(e) = 1 , ? ? Select(o) (1) e?? 2.2 Decision Region Determination with Independent Bernoulli Tests We now define an equivalent problem - decision region determination with independent Bernoulli tests (Bern-DRD). Define a set of tests T = {1, . . . , n}, where the outcome of each test is a Bernoulli random variable Xt ? {0, 1}, P (Xt = xt ) = ?txt (1 ? ?t )1?xt . We define a set of hypotheses h ? H, where each is an outcome vector h ? {0, 1}T mapping all tests t ? T to outcomes h(t). We define a m set of regions {Ri }i=1 , each of which is a subset of tests R ? T . A regionQis determined to be valid if all tests in that region evaluate to true, which has probability P (R) = P (Xt = 1). t?R If a set of tests A ? T are performed, let the observed outcome vector be denoted by xA ? {0, 1}|A| . Let the version space H(xA ) be the set of hypotheses consistent with observation vector xA , i.e. H(xA ) = {h ? H | ?t ? A, h(t) = xA (t)}. We define a policy ? as a mapping from observation vector xA to tests. A policy terminates when it shows that at least one region is valid, or all regions are invalid. Let xT ? {0, 1}T be the ground truth - the outcome vector for all tests. Denote the observation vector of a policy ? given ground truth xT as xA (?, xT ). The expected cost of a policy ? is c(?) = ExT [c(xA (?, xT )] where c(xA ) is the cost of all tests t ? A. The objective is to compute a policy ? ? with minimum cost that ensures at least one region is valid, i.e. ? ? ? arg min c(?) s.t ?xT , ?Rd : P (Rd | xA (?, xT )) = 1 (2) ? m Note that we can cast problem (1) to (2) by setting E = T and ? = {Ri }i=1 . That is, driving uncertainty into a region is equivalent to identification of a valid path (Fig. 2). This casting enables us to leverage efficient algorithms with near-optimality guarantees for motion planning. 3 Related Work The computational bottleneck in motion planning varies with problem domain and that has led to a plethora of planning techniques ([23]). When vertex expansions are a bottleneck, A* [17] is optimally efficient while techniques such as partial expansions [28] address graph searches with large branching factors. The problem class we examine, that of expensive edge evaluation, has inspired a variety of 2 3 It is assumed that c(e) is modular and non-zero. It can scale with edge length. Refer to supplementary on various methods to construct a library of good candidate paths 3 ?1 R1 R2 ?2 R1 ?1 R2 ?2 ?3 (a) R3 ?1 R1 R2 ?2 ?3 (b) R3 ?3 (c) R3 Figure 2: Equivalence between the feasible path identification problem and Bern-DRD. A path ?i is equivalent to a region Ri over valid hypotheses (blue dots). Tests eliminate hypotheses and the algorithm terminates when uncertainty is pushed into a region (R1 ) and the corresponding path (?1 ) is determined to be valid. ?lazy? approaches. The Lazy Probabilistic Roadmap (PRM) algorithm [1] only evaluates edges on the shortest path while Fuzzy PRM [26] evaluates paths that minimize probability of collision. The Lazy Weighted A* (LWA*) algorithm [8] delays edge evaluation in A* search and is reflected in similar techniques for randomized search [14, 6]. An approach most similar in style to ours is the LazyShortestPath (LazySP) framework [11] which examines the problem of which edges to evaluate on the shortest path. Instead of the finding the shortest path, our framework aims to efficiently identify a feasible path in a library of ?good? paths. Our framework is also similar to the Anytime Edge Evaluation (AEE*) framework [25] which deals with edge evaluation on a GBG. However, our framework terminates once a single feasible path is found while AEE* continues to evaluation in order to minimize expected cumulative sub-optimality bound. Similar to Choudhury et al. [7] and Burns and Brock [2], we leverage priors on the distribution of obstacles to make informed planning decisions. We draw a novel connection between motion planning and optimal test selection which has a wide-spread application in medical diagnosis [21] and experiment design [4]. Optimizing the ideal metric, decision theoretic value of information [18], is known to be NPPP complete [22]. For hypothesis identification (known as the Optimal Decision Tree (ODT) problem), Generalized Binary Search (GBS) [10] provides a near-optimal policy. For disjoint region identification (known as the Equivalence Class Determination (ECD) problem), EC2 [16] provides a near-optimal policy. When regions overlap (known as the Decision Region Determination (DRD) problem), HEC [19] provides a near-optimal policy. The D I REC T algorithm [5], a computationally more efficient alternative to HEC, forms the basis of our approach. 4 The Bernoulli Subregion Edge Cutting Algorithm The DRD problem in general is addressed by the Decision Region Edge Cutting (D I REC T) [5] algorithm. The intuition behind the method is as follows - as tests are performed, hypotheses inconsistent with test outcomes are pruned away. Hence, tests should be incentivized to push the probability mass over hypotheses into any region as fast as possible. Chen et al. [5] derive a surrogate objective function that provides such an incentive by creating separate sub-problems for each region and combining them in a Noisy-OR fashion such that quickly solving any one sub-problem suffices. Importantly, this objective is adaptive submodular [15] - greedily maximizing such an objective results in a near-optimal policy. We adapt the framework of D I REC T to address the Bern-DRD problem. We first provide a modification to the EC2 sub-problem objective which is simpler to compute when the distribution over hypotheses is non-uniform, while providing the same guarantees. Unfortunately, naively apply ing D I REC T requires O 2T computation per sub-problem. For the special case of independent Bernoulli tests, we present a more efficient Bernoulli Subregion Edge Cutting (B I SEC T) algorithm, which computes each subproblem in O (T ) time. We provide a brief exposition deferring to the supplementary for detailed derivations. 4.1 A simple subproblem: One region versus all Following Chen et al. [5], we define a ?one region versus all? subproblem, the solution of which helps address the Bern-DRD. Given a single region, the objective is to either push the version space to that region, or collapse it to a single hypothesis. We view a region R as a version space RH ? H 4 consistent with its constituent tests. We define this subproblem over a set of disjoint subregions Si . Let the hypotheses in the target region RH be S1 . Every other hypothesis h ? RH is defined as its own subregion Si , i > 1, where RH is a set of hypothesis where a region is not valid. Determining which subregion is valid falls under the framework of Equivalence Class Determination (ECD), (a special case of the DRD problem) and can be solved efficiently by the EC2 algorithm (Golovin et al. [16]). This objective defines a graph with nodes as subregions and edges between distinct subregions, where the weight of an edge is the product of probabilities of subregions. As tests are performed and outcomes are received, the version space shrinks, and probabilities of different subregions are driven to 0. This has the effect of decreasing the total weight of edges. Importantly, the problem is solved i.f.f. the weight of all edges is zero. The weight over the set of subregions is: X w[16] ({Si }) = P (Sj )P (Sk ) (3) j6=k When hypotheses have uniformP weight, this can be computed efficiently for the ?one region versus all? subproblem. Let P (S1 ) = P (Si ): i>1  1 w[16] ({Si }) = P (S1 )P (S1 ) + P (S1 ) P (S1 ) ? |H|  (4) For non-uniform prior however, this quantity is more difficult to compute. We modify this objective slightly, adding self-edges on subregions Si , i > 1, enabling more efficient computation while still maintaining the same guarantees: X X X wEC ({Si }) = P (S1 )( P (Si )) + ( P (Si ))( P (Sj )) j?1 i6=1 i6=1 (5) = P (S1 )P (S1 ) + P (S1 )2 = P (RH )(P (RH ) + P (RH )) For region R, let the relevant version space be HR (xA ) = {h ? H | ?t ? A ? R, h(t) = xA (t)}. The set of all hypotheses in RH consistent with relevant outcomes in xA is given by RH ? HR (xA ). The terms P (RH ? HR (xA )) and P (RH ? HR (xA )) allows us to quantify the progress made on determining region validity. Naively computing these terms would require computing all hypotheses  and assigning them to correct subregions, thus requiring a runtime of O 2T . However, for the special case of Bernoulli tests, we can reduce this to O (T ) as we can see from the expression ? ? !2 Y Y Y x (k) A R 1?x (k) A wEC ({Si }?H (xA )) = ?1 ? I(Xi = 1) ?j ? ?k (1 ? ?k ) i?(R?A) k?R?A j?(R\A) (6) We can further reduce this to O (1) when iteratively updated (see supplementary for derivations). We now define a criterion that incentivizes removing edges quickly and has theoretical guarantees. Let fEC (xA ) be the weight of edges removed on observing outcome vector xA . This is evaluated as fEC (xA ) = 1 ? =1? wEC ({Si } ? HR (xA )) wEC ({Si }) 1? Q i?(R?A) I(Xi = 1) ! Q Q ?j k?R?A j?(R\A) 1? Q x (k) ?k A (1 1?xA (k) 2 ? ?k ) (7) ?i i?R Lemma 1. The expression fEC (xA ) is strongly adaptive monotone and adaptive submodular. 4.2 Solving the Bern-DRD problem using B I SEC T We now return to Bern-DRD problem (2) where we have multiple regions {R1 , . . . , Rm } that r overlap. Each region Rr is associated with an objective fEC (xA ) for solving the ?one region versus all? problem. Since solving any one such subproblem suffices, we combine them in a Noisy-OR 5 m Algorithm 1: Decision Region Determination with Independent Bernoulli Test({Ri }i=1 , ?, xT ) 1 2 3 4 5 6 A??; while (@Ri , P (Ri |xA ) = 1) and (?Ri , P (Ri |xA ) > 0) do Tcand ? SelectCandTestSet(xA ) ; . Using either (10) or (12) t? ? SelectTest(Tcand , ?, xA ) ; . Using either (11),(13),(14),(15) or (16) A ? A ? t? ; xt? ? xT (t? ) ; . Observe outcome for selected test formulation by defining an objective fDRD (xA ) = 1 ? ? m Q r (1 ? fEC (xA )) [5] which evaluates to r=1 !2 ? ! x (k) ? 1? I(Xi = 1) ?j ?k A (1 ? ?k )1?xA (k) m ? Y k?R ?A i?(R ?A) j?(R \A) r r r ? Q 1? ? ? 1? ?i r=1 ? i?Rr Q Q Q ? ? ? ? ? ? (8) r Since fDRD (xA ) = 1 iff fEC (xA ) = 1 for at least one r, we define the following surrogate problem to Bern-DRD ? ? ? arg min c(?) s.t ?xT : fDRD (xA (?, xT )) ? 1 (9) ? The surrogate problem has a structure that allows greedy policies to have near-optimality guarantees Lemma 2. The expression fDRD (xA ) is strongly adaptive monotone and adaptive submodular. Theorem 1. Let m be the number of regions, phmin the minimum prior probability of any hypothesis, ?DRD be the greedy policy and ? ? with the optimal policy. Then c(?DRD ) ? c(? ? )(2m log ph1 +1). min We now describe the B I SEC T algorithm. Algorithm 1 shows the framework for a general decision region determination algorithm. In order to specify B I SEC T, we need to define two options - a candidate test set selection function SelectCandTestSet(xA ) and a test selection function SelectTest(Tcand , ?, xA ). The unconstrained version of B I SEC T implements SelectCandTestSet(xA ) to return the set of all tests Tcand that contains only unevaluated tests belonging to active regions (m ) [ {Ri | P (Ri |xA ) > 0} \ A (10) Tcand = i=1 We now examine the B I SEC T test selection rule SelectTest(Tcand , ?, xA ) ? ? ? m Y Y Y 1 ?1 ? I(Xi = 1) ?j ? Ext ? t? ? arg max t?Tcand c(t) r=1 i?(Rr ?A) j?(Rr \A) ? ? ?? ? (11) m P m Y Y Y 2 I(t?Rk ) ?1 ? ? ?? I(Xi = 1) ?j ?? (?txt (1 ? ?t )1?xt ) k=1 r=1 i?(Rr ?A?t) j?(Rr \A?t) The intuition behind this update is that tests are selected to squash the probability of regions not being valid. It also additionally incentivizes selection of tests on which multiple regions overlap. 4.3 Adaptively constraining test selection to most likely region We observe in our experiments that the surrogate (8) suffers from a slow convergence problem fDRD (xA ) takes a long time to converge to 1 when greedily optimized. To alleviate the convergence problem, we introduce an alternate candidate selection function SelectCandTestSet(xA ) that assigns to Tcand the set of all tests that belong to the most likely region TmaxP which is evaluated as follows (we will refer to this variant as M AX P ROB R EG) ( ) TmaxP = arg max Ri =(R1 ,R2 ,...,Rm ) 6 P (Ri |xA ) \A (12) Applying the constraint in (12) leads to a dramatic improvement for any test selection policy as we will show in Sec. 5.2. The following theorem offers a partial explanation Theorem 2. A policy that greedily latches to a region according the the posterior conditioned on the region outcomes has a near-optimality guarantee of 4 w.r.t the optimal region evaluation sequence. Applying the constraint in (12) implies we are no longer greedily optimizing fDRD (xA ). However, the following theorem bounds the sub-optimality of this policy. Theorem 3. Let pmin = mini P (Ri ), phmin = minh?H P (h) and l = maxi |Ri |. The policy using   ?1     2 l + 1 where ? ? 1 ? max (1 ? pmin )2 , pmin (12) has a suboptimality of ? 2m log ph1 . min 5 Experiments We evaluate B I SEC T on a collection of datasets spanning across a spectrum of synthetic problems and real-world planning applications. The synthetic problems are created by randomly selecting problem parameters to test the general applicability of B I SEC T. The motion planning datasets range from simplistic yet insightful 2D problems to more realistic high dimension problems as encountered by an UAV or a robot arm. The 7D arm planning dataset is obtained from a high fidelity simulation as shown in Fig. 4(a). Finally, we test B I SEC T on experimental data collected from a full scale helicopter flying that has to avoid unmapped wires at high speed as it comes into land as shown in Fig. 4(b). Refer to supplementary for exhaustive details on experiments and additional results. Open-source code and details can be found here: https://github.com/sanjibac/matlab_learning_collision_checking 5.1 Heuristic approaches to solving the Bern-DRD problem We propose a collection of competitive heuristics that can also be used to solve the Bern-DRD problem. These heuristics are various SelectTest(Tcand , ?, xA ) policies in the framework of Alg. 1. To simplify the setting, we assume unit cost c(t) = 1 although it would be possible to extend these to nonuniform setting. The first heuristic R ANDOM selects a test by sampling uniform randomly t? ? Tcand (13) We adopt our next heuristic M AX TALLY from Dellin and Srinivasa [11] where the test belonging to most regions is selected. It uses the following criteria, which exhibits a ?fail-fast? characteristic t? ? arg max t?Tcand m X I (t ? Ri , P (Ri |xA ) > 0) (14) i=1 The next policy S ET C OVER selects tests that maximize the expected number of ?covered? tests, i.e. if a selected test is in collision, how many other tests does it remove from consideration. ( m ) m [ [  x , ? A t ? arg max (1 ? ?t ) {Ri | P (Ri |xA ) > 0} ? Rj P (Rj |, Xt =0) > 0 \ {A ? {t}} t?Tcand i=1 j=1 (15) Theorem 4. S ET C OVER is a near-optimal policy for the problem of optimally checking all regions. The last heuristic is derived from a classic heuristic in decision theory: myopic value of information (Howard [18]). MVO I greedily chooses the test that maximizes the change in the probability mass of the most likely region. This test selection works only with SelectCandTestSet(xA ) = TmaxP . t? ? arg max (1 ? ?t ) max P (Ri | xA , Xt = 0) t?TmaxP i=1,...,m (16) We also evaluate against state of the art L AZY SP [11] planner which explicitly minimizes collision checking effort while trying to guarantee optimality. We ran two variants of LazySP. The first variant is the vanilla unconstrained algorithm that searches for the shortest path on the entire graph, collision checks the path and repeats. The second variant is constrained to the library of paths used by all other baselines. 5.2 Analysis of results Table 1 shows the evaluation cost of all algorithms on various datasets normalized w.r.t B I SEC T. The two numbers are lower and upper 95% confidence intervals - hence it conveys how much fractionally poorer are algorithms w.r.t B I SEC T. The best performance on each dataset is highlighted. We present a set of observations to interpret these results. O 1. B I SEC T has a consistently competitive performance across all datasets. 7 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 0 0.9 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.1 0 0 0.4 0.3 0.2 0.1 0 0 MaxTally (|A| : 29) SetCover (|A| : 30) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 MVoI (|A| : 28) 0.9 1 0 0.7 0.8 0.9 1 BiSECt (|A| : 20) Figure 3: Performance (number of evaluated edges) of all algorithms on 2D geometric planning. Snapshots, at start, interim and final stages respectively, show evaluated valid edges (green), invalid edges (red) and the final path (magenta). The utility of edges as computed by algorithms is shown varying from low (black) to high (cream). ? Wires in Region 2: Many edges with high probability real flight Region 1: Single edge with low probability (a) (b) (c) Figure 4: (a) A 7D arm has to perform pick and place tasks at high speed in a table with clutter. (b) Experimental data from a full-scale helicopter that has to react quickly to avoid unmapped wires detected by the sensor. B I SEC T (given an informative prior) checks a small number of edges around the detected wire and identifies a path. (c) Scenario where regions have size disparity. Unconstrained B I SEC T significantly outperforms other algorithms on such a scenario. Table 1 shows that on 13 out of the 14 datasets, B I SEC T is at par with the best. On 7 of those it is exclusively the best. O 2. The M AX P ROB R EG variant improves the performance of all algorithms on most datasets Table 1 shows that this is true on 12 datasets. The impact is greatest on R ANDOM on the 2D Forest dataset performance improves from (19.45, 27.66) to (0.13, 0.30). However, this is not true in general. On datasets with large disparity in region sizes as illustrated in Fig. 4(c), unconstrained B I SEC T significantly outperforms other algorithms. In such scenarios, M AX P ROB R EG latches on to the most probable path which also happens to have a large number of edges. It performs poorly on instances where this region is invalid, while the other region containing a single edge is valid. Unconstrained B I SEC T prefers to evaluate the single edge belonging to region 1 before proceeding to evaluate region 2, performing optimally on those instances. Hence, the myopic nature of M AX P ROB R EG is the reason behind its poor performance. O 3. On planning problems, B I SEC T strikes a trade-off between the complimentary natures of M AX TALLY and MVO I. 8 Table 1: Normalized evaluation cost - (lower, upper) bound of 95% confidence interval L AZY SP Small m : 100 Medium m : 500 Large m : 1e3 Forest OneWall TwoWall OneWall m : 300 OneWall m : 858 Forest OneWall Table Clutter Synth. (T : 10) 2D Plan (m : 2) R ANDOM M AX TALLY S ET C OVER B I SEC T Unconstrained MVO I Unconstrained Unconstrained Unconstrained Unconstrained Constrained MaxProbReg MaxProbReg MaxProbReg MaxProbReg Synthetic Bernoulli Test: Variation across region overlap (4.18, 6.67) (3.49, 5.23) (1.77, 3.01) (0.00, 0.08) (0.12, 0.29) (0.12, 0.25) (0.18, 0.40) (3.27, 4.40) (3.04, 4.30) (3.55, 4.67) (0.00, 0.00) (0.05, 0.25) (0.14, 0.24) (0.14, 0.33) (2.86, 4.26) (2.62, 3.85) (2.94, 3.71) (?0.11, 0.00) (0.00, 0.28) (0.06, 0.26) (0.09, 0.22) 2D Geometric Planning: Variation across environments (10.8, 14.3) (19.5, 27.7) (4.68, 6.55) (3.53, 5.07) (1.38, 2.51) (0.03, 0.18) (0.13, 0.30) (0.09, 0.18) (0.00, 0.09) (6.96, 11.3) (13.4, 17.8) (4.12, 4.89) (1.36, 2.11) (0.16, 0.55) (0.045, 0.21) (0.11, 0.42) (0.00, 0.12) (0.14, 0.29) (18.9, 25.6) (13.8, 16.6) (2.76, 3.93) (2.07, 2.94) (?0.17, 0.01) (0.00, 0.09) (0.33, 0.51) (0.10, 0.20) (0.00, 0.00) 2D Geometric Planning: Variation across region size (5.82, 12.1) (12.1, 16.0) (4.47, 5.13) (2.00, 3.41) (0.00, 0.57) (0.00, 0.17) (0.12, 0.42) (0.06, 0.24) (0.00, 0.38) (5.43, 10.02) (13.3, 16.8) (2.18, 3.77) (1.04, 1.62) (?0.03, 0.45) (0.00, 0.14) (0.09, 0.27) (?0.04, 0.08) (0.00, 0.14) Non-holonomic Path Planning: Variation across environments (1.97, 3.81) (22.4, 29.7) (9.79, 11.14) (2.63, 5.28) (0.15, 0.47) (0.09, 0.18) (0.46, 0.79) (0.25, 0.38) (0.00, 0.00) (0.97, 2.45) (13.0, 15.8) (8.40, 11.47) (3.72, 4.54) (0.02, 0.51) (?0.11, 0.11) (0.00, 0.12) (0.21, 0.28) (?0.11, 0.11) 7D Arm Planning: Variation across environments (0.97, 1.59) (15.1, 19.4) (4.80, 6.98) (1.36, 2.17) (0.24, 0.72) (0.28, 0.54) (0.13, 0.31) (0.00, 0.04) (0.00, 0.11) (0.28, 1.19) (7.92, 9.85) (3.96, 6.44) (1.42, 2.07) (0.00, 0.38) (0.02, 0.20) (0.14, 0.36) (0.00, 0.00) (0.00, 0.11) Datasets with large disparity in region sizes (6.50, 8.00) (5.50, 6.50) (3.00, 3.50) (3.00, 3.50) (3.00, 4.50) (5.00, 7.50) (3.00, 3.50) (9.50, 11.3) (2.80, 6.10) (6.60, 10.5) (6.60, 10.5) (6.90, 10.8) (6.80, 8.30) (6.60, 10.5) (1.42, 2.36) (0.00, 0.00) (1.77, 2.64) (0.00, 0.00) (1.33, 1.81) (0.00, 0.00) (1.90, 2.46) (0.00, 0.00) (0.76, 1.20) (0.00, 0.00) (0.91, 1.44) (0.00, 0.00) (0.94, 1.42) (0.00, 0.00) (0.41, 0.91) (0.00, 0.00) (1.54, 2.46) (0.00, 0.00) (3.28, 3.78) (0.00, 0.00) (0.32, 0.67) (0.00, 0.00) (1.23, 1.75) (0.00, 0.00) (0.00, 0.00) (3.00, 3.50) (0.00, 0.00) (7.30, 11.2) We examine this in the context of 2D planning as shown in Fig. 3. M AX TALLY selects edges belonging to many paths which is useful for path elimination but does not reason about the event when the edge is not in collision. MVO I selects edges to eliminate the most probable path but does not reason about how many paths a single edge can eliminate. B I SEC T switches between these behaviors thus achieving greater efficiency than both heuristics. O 4. B I SEC T checks informative edges in collision avoidance problems encountered a helicopter Fig. 4(b) shows the efficacy of B I SEC T on experimental flight data from a helicopter avoiding wire. 6 Conclusion In this paper, we addressed the problem of identification of a feasible path from a library while minimizing the expected cost of edge evaluation given priors on the likelihood of edge validity. We showed that this problem is equivalent to a decision region determination problem where the goal is to select tests (edges) that drive uncertainty into a single decision region (a valid path). 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A Recurrent Neural Network for Generation of Ocular Saccades Lina L.E. Massone Department of Physiology Department of Electrical Engineering and Computer Scienc~ Northwestern University 303 E. Chicago Avenue, Chicago, 1160611 Abstract This paper presents a neural network able to control saccadic movements. The input to the network is a specification of a stimulation site on the collicular motor map. The output is the time course of the eye position in the orbit (horizontal and vertical angles). The units in the network exhibit a one-to-one correspondance with neurons in the intermediate layer of the superior colliculus (collicular motor map), in the brainstem and with oculomotor neurons. Simulations carried out with this network demonstrate its ability to reproduce in a straightforward fashion many experimental observations. 1. INTRODUCTION It is known that the superior colliculus (SC) plays an important role in the control of eye movements (Schiller et a1. 1980). Electrophysiological studies (Cynader and Berman 1972, Robinson 1972) showed that the intermediate layer of SC is topographically organized into a motor map. The location of active neurons in this area was found to be related to the oculomotor error (Le. how far the eyes are from the target) and their firing rate to saccade velocity (Roher et al. 1987, Berthoz et al. 1987). Neurons in the rostral area of the motor map, the so-called fixation neurons, tend to become active when the eyes are on target (Munoz and Wurtz 1992) and they can provide a gating mechanism to 1014 A Recurrent Neural Network for Generation of Ocular Saccades arrest the movement (Guitton 1992). SC sends signals to the brainstem whose circuitry translates them into commands to the oculomotor neurons that innervate the eye muscles (Robinson 1981). This paper presents a recurrent neural network that performs a spatio-temporal transformation from a stimulation site on the collicular motor map and an eye movement. The units in the network correspond to neurons in the intermediate layer of the colliculus, neurons in the brainstem and to oculomotor neurons. Medial (up) : : . . Caudal (right) Caudal (left) Lateral (down) Figure 1: An array of units that represents the collicular motor map. The dark square represents the fixation area. The units in the array project to four units that represent burst cells devoted to process rightward, leftward, upward and downward saccades. The network was built entirely on anatomical and physiological observations. Specifically, the following assumptions were used: (1) The activity on the collicular motor map shifts towards the fixation area during movement (Munoz et aI. 1991, Droulez and Berthoz 1991). (2) The output of the superior colliculus is a vectorial velocity signal 1015 1016 Massone that is the sum of the contributions from each active collicular neuron. (3) Such signal is decomposed into horizontal velocity and vertical velocity by a topographic and graded connectivity pattern from SC to the burst cells in the brainstem. (4) The computation performed from the burst-cells level down to the actual eye movement is carried out according to the push-pull arrangement proposed by Robinson (1981). (5) The activity on the collicular motor map is shifted by signals that represent the eye velocity. Efferent copies of the horizontal and vertical eye velocities are fed back onto the collicular map in order to implement the activity shift. (a) (c) (b) (d) Figure 2: The topographic and graded pattern of connectivity from the collicular array to the four burst cells. Black means no connection, brighter colors represent larger weight values. (a) To the right cell. (b) To the left cell. (c) To the up cell. (d) To the down cell. Simulations conducted with such a system (Massone submitted) demonstrated the network's ability to reproduce a number of experimental observations. Namely the network can: (1) Spontaneously produce oblique saccades whose curvature varies with the ratio between the horizontal and vertical components of the motor error.(2) Automatically hold the eye position in the orbit at the end of a saccade by exploiting the internal dynamic of the network. (3) Continuously produce efferent copies of the movements A Recurrent Neural Network for Generation of Ocular Saccades without the need for reset signals. (4) Account for the outcome of the lidocaine experiment (Lee et al. 1988) without assuming a population averaging mechanism. Section 2 describes the network architecture. A more detailed description of the network, it mechanisms and physiological ground as well as a number of simulation results can be found in Massone (submitted). 2. THE NETWORK The network input layer is a bidimensional array of linear units that represent neurons in the collicular motor map. The array is topographically arranged as shown in Figure 1. Activity along the caudal axis produces horizontal saccades in a contralateral fashion, activity along the medio-Iateral axis produces vertical saccades, activity in the rest of the array produces oblique saccades. The dark square in the center (rostral area) represents the fixation area. The units in this array project to four logistic units that represent two pairs of burst cells, one pair devoted to control horizontal movements, one pair devoted to control vertical movements. The pattern of connectivity between the collicular array and the units that represent the burst cells is qualitatively shown in Figure 2. The value of the weights of such connections increases exponentially when one moves from the center towards the periphery of the array. The fixation area projects to four other units that represent the so-called omnipause neurons. These units send a gating signal to the burstcells units and are responsible for arresting the movement when the eyes are on target. i.e. when the activity in the input array reaches the center. Each pair of burst-cells units project to the network shown in Figure 3. This network is a computational version of the push-pull arrangement proposed by Robinson (1981). The bottom part of the network represents the oculomotor plant, the top part represents the brainstem circuitry and the oculomotor neurons. The weights in the bottom part of the network were derived by splitting into two equations the differential equation proposed by Robinson (1981) to describe the behavior of the oculomotor plant under a combined motorneuron input R. d91 R1 =k91 + r <l R 1 and R2 are the firing rates of the agonist and antagonist motorneurons, 91 and 92 are the components of the eye position due to motions in opposite directions (e.g. left and right), k is the eye stiffness and r is the eye viscosity. The weights in the top part of the network were analytically computed from the weights in the bottom part of the network by imposing the following constraints: (1) The difference between 91 and 92 must produce the correct 9. (2) The output of the neural integrators must be an efferent copy of the eye movement. (3) The output of the motorneurons must hold the eye at the current orbital position when the burst-cells units are shut off by the gating action of the omnipause cells. Efferent copies of the horizontal and vertical eye velocities were computed by differentiating the output of the neural 1017 1018 Massone from fixation neurons from fixation neurons from collicular array \1/ \1/ a ,-- Rl .1t/r .1t1r R2 -, l-k.1t1r + 1 -1 a1 a J--------- , "":YE_ I Figure 3: The recurrent network used to control eye movements in one direction, e.g. horizontal. An identical network is required to control vertical movements. OPN: omnipause neurons. Bel, BC2: burst cells. NIl, NI2: neural integrators. MNI, MN2: motor neurons. The architecture is based on Robinson'S push-pull arrangement. k=4.0, r=O.95, a=O.5, L\t=l msec. A Recurrent Neural Network for Generation of Ocular Saccades integrators. These signals were recurrently fed back onto the input array and made the activity in the array shift towards the fixation area. This architecture assumes that the output of the collicular array represents saccade velocity. The network is started by selecting one unit in the input array, i.e. a "stimulation" site. When the unit is selected, a square area centered at that unit becomes active with a gaussian activity profile (Ottes et a1. 1986. Munoz and Guitton 1991). At the time the input units are activated the eye starts moving and, as a consequence of the velocity feedback the activity on the input array starts shifting. The movement is arrested when the fixation area becomes activated. The activity of all units i~l the network represents neurons firing rates and is expressed in spikes/second. Figure 4 shows the response of the network when the collicular array is stimulated at two sites sequentially. Each site causes an oblique saccade with unequal components. Stimulation number 1 brings the eye up and to the right. stimulation number 2 brings the eye back to the initial position. Fixation is maintained for a while inbetween stimulations and at the end of the two movements. The reSUlting trajectories in the movement plane (vertical angle versus horizontal angle) demonstrate the ability of the network to (i) maintain the eye position in the orbit when the burst cells activation is set to zero by the gating action of the omnipause neurons. (ii) produce curved trajectories with opposite curvatures when the eye moves back and forth between the same two angular positions. None of the units in the network is ever reset between saccades; because of the push-pull arrangement, when the activity of one neural integrator increases, the activity of the antagonist integrator decreases. This mechanism ensures that their activity does not grow indefinetely. 3. CONCLUSIONS In this paper I presented an anatomically and physiologically inspired network able to control saccadic movements and to reproduce the outcome of some experimental observations. The results of simulations carried out with this network can be found in Massone (submitted). This work is currently being extended to (i) modeling the activity shift phenomenon as the relaxation of a dynamical system to its equilibrium configuration rather than as a feedback-driven mechanism, (ii) studying the role of the collicular output signals in the calibration and accuracy of arm movements (Massone 1992). Acknowledgements This work was supported by the National Science Foundation, grant BCS-9113455 to the author. References Berthoz A., Grantyn A., Droulez J. (1987) Some collicular neurons code saccadic eye velocity, Neuroscience Letters, 72.289-294. 1019 "'No"' o ~ I\) VI VI o ::3 ~ BC_JtP& BC}eft BC_.. ~.~. ~. [A. TJwta BC_..... C. E=. L.~. M PhI c ~~ ........... Figure 4: The response of the network to two sequential stimulations that produce two oblique saccades with unequal components. A Recurrent Neural Network for Generation of Ocular Saccades Cynader M., Berman N. (1972) Receptive-field organization of monkey superior colliculus, Journal of Neurophysiology, 35, 187-201. Droulez J., Berthoz A. (1991) The concept of dynamic memory in sensorimotor control, in Motor Control Concepts and Issues, Humphrey D.R. and Freund H.J. Eds., 1. Whiley and Sons, 137-161. GuiUon D. (1992) Control of eye-head coordination during orienting gaze shifts, Trends i1l Neuroscience, 15(5),174-179. Lee C., Roher W.H., Sparks D.L. (1988) Population coding of saccadic eye movements by neurons in the superior colliculus. Nature, 332, 357-360. Massone L. E. (1992) A biologically-inspired architecture for reactive motor control, in Neural Networks for Control, G. Beckey and K. Goldberg Eds., Kluwer Academic Publishers, 1992. Massone L.E. (submitted) A velocity-based model for control of ocular saccades, Neural Computation. Munoz D.P., Pellisson D., Guitton D. (1991) Movement of Neural Activity on the Superior Colliculus Motor Map during Gaze Shifts, Science, 251. 1358-1360. Munoz D.P., Guitton D. (1991) Gaze control by the tecto-reticulo-spinal system in the head-free cat. II. Sustained discharges coding gaze position error, Journal of Neurophysiology, 66, 1624-1641. Munoz D.P., Wurtz R.H. (1992) Role of the rostral superior colliculus in active visual fixation and execution of express saccades, Journal of Neurophysiology, 67, 1000-1002. Ottes F.P., Van Gisbergen J.A.M., Eggermont J.J. (1986) Visuomotor fields of the superior colliculus: a quantitative model, Vision Research, 26, 857-873. Robinson D.A. (1972) Eye movements evoked by collicular stimulation in the alert monkey, Vision Research, 12, 1795-1808. Robinson D.A. (1981) Control of eye movements, in Handbook of Physiology - The Nervous System fl, V.B. Brooks Ed., 1275-1320. Roher W.H., White J.M., Sparks D.L. (1987) Saccade-related burst cells in the superior colliculus: relationship of activity with saccade velocity. Society of Neuroscience Abstracts, 13, 1092. 1021
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Non-Stationary Spectral Kernels Sami Remes Markus Heinonen Samuel Kaski [email protected] [email protected] [email protected] Helsinki Institute for Information Technology HIIT Department of Computer Science, Aalto University Abstract We propose non-stationary spectral kernels for Gaussian process regression by modelling the spectral density of a non-stationary kernel function as a mixture of input-dependent Gaussian process frequency density surfaces. We solve the generalised Fourier transform with such a model, and present a family of non-stationary and non-monotonic kernels that can learn input-dependent and potentially longrange, non-monotonic covariances between inputs. We derive efficient inference using model whitening and marginalized posterior, and show with case studies that these kernels are necessary when modelling even rather simple time series, image or geospatial data with non-stationary characteristics. 1 Introduction Gaussian processes are a flexible method for non-linear regression [18]. They define a distribution over functions, and their performance depends heavily on the covariance function that constrains the function values. Gaussian processes interpolate function values by considering the value of functions at other similar points, as defined by the kernel function. Standard kernels, such as the Gaussian kernel, lead to smooth neighborhood-dominated interpolation that is oblivious of any periodic or long-range connections within the input space, and can not adapt the similarity metric to different parts of the input space. Two key properties of covariance functions are stationarity and monotony. A stationary kernel K(x, x0 ) = K(x + a, x0 + a) is a function only of the distance x ? x0 and not directly the value of x. Hence it encodes an identical similarity notion across the input space, while a monotonic kernel decreases over distance. Kernels that are both stationary and monotonic, such as the Gaussian and Mat?rn kernels, can encode neither input-dependent function dynamics nor long-range correlations within the input space. Non-monotonic and non-stationary functions are commonly encountered in realistic signal processing [19], time series analysis [9], bioinformatics [5, 20], and in geostatistics applications [7, 8]. Recently, several authors have explored kernels that are either non-monotonic or non-stationary. A non-monotonic kernel can reveal informative manifolds over the input space by coupling distant points due to periodic or other effects. Non-monotonic kernels have been derived from the Fourier decomposition of kernels [13, 24, 30], which renders them inherently stationary. Non-stationary kernels, on the other hand, are based on generalising monotonic base kernels, such as the Mat?rn family of kernels [6, 15], by partitioning the input space [4], or by input transformations [25]. We propose an expressive and efficient kernel family that is ? in contrast to earlier methods ? both non-stationary and non-monotonic, and hence can infer long-range or periodic relations in an input-dependent manner. We derive the kernel from first principles by solving the more expressive generalised Fourier decomposition of non-stationary functions, than the more limited standard Fourier decomposition exploited by earlier works. We propose and solve the generalised spectral density as a mixture of Gaussian process density surfaces that model flexible input-dependent frequency patterns. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. The kernel reduces to a stationary kernel with appropriate parameterisation. We show the expressivity of the kernel with experiments on time series data, image-based pattern recognition and extrapolation, and on climate data modelling. 2 Related Work Bochner?s theorem for stationary signals, whose covariance can be written as k(? ) = k(x ? x0 ) = k(x, x0 ), implies a Fourier dual [30] Z k(? ) = S(s)e2?is? ds Z S(s) = k(? )e?2?is? d?. The dual is a special case of the more general Fourier transform (1), and has been exploited to design rich, yet stationary kernel representations [24, 32] and used for large-scale inference [17]. Lazaro-Gredilla et al. [13] proposed to directly learn the spectral density as a mixture of Dirac delta PQ 1 T functions leading to a sparse spectrum (SS) kernel kSS (? ) = Q i=1 cos(2?si ? ). Wilson et al. [30] derived a stationary spectral mixture P (SM) kernel by modelling the univariate 2 2 spectral density using a mixture of normals SSM (s) = i wi [N (s|?i , ?i ) + N (s| ? ?i , ?i )]/2, P 2 2 corresponding to the kernel function kSM (? ) = i wi exp(?2? ?i ? ) cos(2??i ? ), which we generalize to the non-stationary case. The SM kernel was also extended for multidimensional inputs using Kronecker structure for scalability [27]. Kernels derived from the spectral representation are particularly well suited to encoding long-range, non-monotonic or periodic kernels; however, they have so far been unable to handle non-stationarity, although [29] presented a partly non-stationary SM kernel that has input-dependent mixture weights. Kom Samo and Roberts also derived a kernel similar to our bivariate spectral mixture kernel in a recent technical report [11]. Non-stationary kernels, on the other hand, have been constructed by non-stationary extensions of Mat?rn and Gaussian kernels with input-dependent length-scales [3, 6, 15, 16], input space warpings [22, 25], and with local stationarity with products of stationary and non-stationary kernels [2, 23]. The simplest non-stationary kernel is arguably the dot product kernel [18], which has been used as a way to assign input-dependent signal variances [26]. Non-stationary kernels are a good match for functions with transitions in their dynamics, yet are unsuitable for modelling non-monotonic properties. Our work can also be seen as a generalisation of wavelets, or time-dependent frequency components, into general and smooth input-dependent components. In signal processing, Hilbert-Huang transforms and Hilbert spectral analysis explore input-dependent frequencies, but with deterministic transform functions on the inputs [8, 9]. 3 Non-stationary spectral mixture kernels This section introduces the main contributions. We employ the generalised spectral decomposition of non-stationary functions and derive a practical and efficient family of kernels based on non-stationary spectral components. Our approach relies on associating input-dependent frequencies for data inputs, and solving a kernel through the generalised spectral transform. The most general family of kernels is the non-stationary kernels, which include stationary kernels as special cases [2]. A non-stationary kernel k(x, x0 ) ? R for scalar inputs x, x0 ? R can be characterized by its spectral density S(s, s0 ) over frequencies s, s0 ? R, and the two are related via a generalised Fourier inverse transform1 Z Z 0 0 0 k(x, x ) = e2?i(xs?x s ) ?S (ds, ds0 ) , (1) R R 1 We focus on scalar inputs and frequencies for simplicity. An extension based on vector-valued inputs and frequencies [2, 10] is straightforward. 2 (a) (b) Figure 1: (a): Spectral density surface of a single component bivariate spectral mixture kernel with 8 permuted peaks. (b): The corresponding kernel on inputs x ? [?1, 1]. where ?S is a Lebesgue-Stieltjes measure associated to some positive semi-definite (PSD) spectral density function S(s, s0 ) with bounded variations [2, 14, 31], which we denote as the spectral surface since it considers the amplitude of frequency pairs (See Figure 1a). The generalised Fourier transform (1) specifies that a spectral surface S(s, s0 ) generates a PSD kernel K(x, x0 ) that is non-stationary unless the spectral measure mass is concentrated only on the diagonal s = s0 . We design a practical, efficient and flexible parameterisation of spectral surfaces that, in turn, specifies novel non-stationary kernels with input-dependent characteristics and potentially long-range non-monotonic correlation structures. 3.1 Bivariate Spectral Mixture kernel Next, we introduce spectral kernels that remove the restriction of stationarity of earlier works. We start by modeling the spectral density as a mixture of Q bivariate Gaussian components     2  X ?i ?i ?i ?i0 s 0 Si (s, s ) = N |?i , ?i , ?i = , (2) 2 s0 ?i ?i ?i0 ?i0 0 2 ?i ??{?i ,?i } 2 with parameterisation using the correlation ?i , means ?i , ?0i and variances ?i2 , ?i0 . To produce a PSD spectral density Si as required by equation (1) we need to include symmetries Si (s, s0 ) = Si (s0 , s) and sufficient diagonal components Si (s, s), Si (s0 , s0 ). To additionally result in a real-valued kernel, symmetryP is required with respect to the negative frequencies as well, i.e., Si (s, s0 ) = Si (?s, ?s0 ). The sum ?i ??{?i ,?0 }2 satisfies all three requirements by iterating over the four permutations of i {?i , ?0i }2 and the opposite signs (??i , ??0i ), resulting in eight components (see Figure 1a). The generalised Fourier inverse transform (1) can be solved in closed form for a weighted spectral PQ surface mixture S(s, s0 ) = i=1 wi2 Si (s, s0 ) using Gaussian integral identities (see the Supplement): k(x, x0 ) = Q X ? T ?i x ? )??i ,?0i (x)T ??i ,?0i (x0 ) wi2 exp(?2? 2 x (3) i=1 where  ??i ,?0i (x) =  cos 2??i x + cos 2??0i x , sin 2??i x + sin 2??0i x ? = (x, ?x0 )T and introduce mixture weights wi for each component. We and where we define x denote the proposed kernel as the bivariate spectral mixture (BSM) kernel (see Figure 1b). The positive definiteness of the kernel is guaranteed by the spectral transform, and is also easily verified since the sinusoidal components form an inner product and the exponential component resembles an unscaled Gaussian density. A similar formulation for non-stationary spectral kernels was presented also in a technical report [11]. 3 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 2: (a)-(d): Examples of kernel matrices on inputs x ? [?1, 1] for a Gaussian kernel (a), sparse spectrum kernel [13] (b), spectral mixture kernel [30] (c), and for the GSM kernel (d). (e)-(h): The corresponding generalised spectral density surfaces of the four kernels. (i)-(l): The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of Q = 2 Gaussian process surface functions. We immediately notice that the BSM kernel vanishes rapidly outside the origin (x, x0 ) = (0, 0). We would require a huge number of components centered at different points xi to cover a reasonably-sized input space. 3.2 Generalised Spectral Mixture (GSM) kernel We extend the kernel derived in Section 3.1 further by parameterising the frequencies, length-scales and mixture weights as a Gaussian processes2 , that form a smooth spectrogram (See Figure 2(l)): log wi (x) ? GP(0, kw (x, x0 )), (4) log `i (x) ? GP(0, k` (x, x0 )), (5) logit ?i (x) ? GP(0, k? (x, x0 )). (6) Here the log transform is used to ensure the weights w(x) and lengthscales `(x) are non-negative, and the logit transform logit ?(x) = log FN??? limits the learned frequencies between zero and the Nyquist frequency FN , which is defined as half of the sampling rate of the signal. A GP prior f (x) ? GP(0, k(x, x0 )) defines a distribution over zero-mean functions, and denotes the covariance between function values cov[f (x), f (x0 )] = k(x, x0 ) equals their prior kernel. For any collection of inputs, x1 , . . . , xN , the function values follow a multivariate normal distribution (f (x1 ), . . . , f (xN ))T ? N (0, K), where Kij = k(xi , xj ). The key property of Gaussian processes is that they can encode smooth functions by correlating function values of input points that are similar according to the kernel k(x, x0 ). We use standard Gaussian kernels kw , k` and k? . 2 See the Supplement for a tutorial on Gaussian processes. 4 We accommodate the input-dependent lengthscale by replacing the exponential part of (3) by the Gibbs kernel s   (x ? x0 )2 2`i (x)`i (x0 ) 0 kGibbs,i (x, x ) = exp ? , `i (x)2 + `i (x0 )2 `i (x)2 + `i (x0 )2 which is a non-stationary generalisation of the Gaussian kernel [3, 6, 15]. We propose a non-stationary generalised spectral mixture (GSM) kernel with a simple closed form (see the Supplement): kGSM (x, x0 ) = Q X wi (x)wi (x0 )kgibbs,i (x, x0 ) cos(2?(?i (x)x ? ?i (x0 )x0 )) . (7) i=1 The kernel is a product of three PSD terms. The GSM kernel encodes the similarity between two data points based on their combined signal variance w(x)w(x0 ), and the frequency surface based on the frequencies ?(x), ?(x0 ) and frequency lengthscales `(x), `(x0 ) associated with both inputs. The GSM kernel encodes the spectrogram surface mixture into a relatively simple kernel. The kernel reduces to the stationary Spectral Mixture (SM) kernel [30] with constant functions wi (x) = wi , ?i (x) = ?i and `i (x) = 1/(2??i ) (see the Supplement). We have presented the proposed kernel (7) for univariate inputs for simplicity. The kernel can be extended to multivariate inputs in a straightforward manner using the generalised Fourier transform with vector-valued inputs [2, 10]. However, in many applications multivariate inputs have a gridlike structure, for instance in geostatistics, image analysis and temporal models. We exploit this assumption and propose a multivariate extension that assumes the inputs to decompose across input dimensions [1, 27]: kGSM (x, x0 |?) = P Y kGSM (xp , x0p |? p ) . (8) p=1 Here x, x0 ? RP , ? = (? 1 , . . . , ? P ) collects the dimension-wise kernel parameters ? p = n (wip , `ip , ?ip )Q i=1 of the n-dimensional realisations wip , `ip , ?ip ? R per dimension p. Then, the kernel matrix can be expressed using Kronecker products as K? = K?1 ? ? ? ? ? K?P , while missing values and data not on a regular grid can be handled with standard techniques [1, 21, 28, 27]. 4 Inference We use the Gaussian process regression framework and assume a Gaussian likelihood over N = nP N data points3 (xj , yj )N j=1 with all outputs collected into a vector y ? R , yj = f (xj ) + ?j , ?j ? N (0, ?n2 ) f (x) ? GP(0, kGSM (x, x0 |?)), (9) with a standard predictive GP posterior f (x? |y) for a new input point x? [18]. The posterior can be efficiently computed using Kronecker identities [21] (see the Supplement). We aim to infer the noise variance ?n2 and the kernel parameters ? = (wip , `ip , ?ip )Q,P i=1,p=1 that reveal the input-dependent frequency-based correlation structures in the data, while regularising the learned kernel to penalise overfitting. We perform MAP inference over the log marginalized posterior log p(?|y) ? log p(y|?)p(?) = L(?), where the functions f (x) have been marginalised out, ? ? Q,P Y ? ip |0, Kwp )N (? ? ip |0, K?p )N (?`ip |0, K`p )? , (10) L(?) = log ?N (y|0, K? + ?n2 I) N (w i,p=1 ? ? ? and ?` represent the log or where Kwp , K?p , K`p are n ? n prior matrices per dimensions p, and w, logit transformed variables. The marginalized posterior automatically balances between parameters ? that fit the data and a model that is not overly complex [18]. We can efficiently evaluate both 3 Assuming that we have equal number of points n in all dimensions. 5 the marginalized posterior and its gradients in O(P N [21, 27] (see the Supplement). P +1 P ) instead of the usual O(N 3 ) complexity ? ip , ? ? ip , ?`ip Gradient-based optimisation of (10) is likely to converge very slowly due to parameters w ? ? being highly self-correlated. We remove the correlations by whitening the variables as ? = L?1 ? where L is the Cholesky decomposition of the prior covariances. We maximize L using gradient ? by evaluating L(L?) ? and the gradient as [6, 12] ascent with respect to the whitened variables ? ?L ?L ?? ? ?? ?L = = LT . ? ? ? ?? ?? ?? ? ?? ?? 5 (11) Experiments We apply our proposed kernel first on simple simulated time series, then on texture images and lastly on a land surface temperature dataset. With the image data, we compare our method to two stationary mixture kernels, specifically the spectral mixture (SM) [30] and sparse spectrum (SS) kernels [13], and the standard squared exponential (SE) kernel. We employ the GPML Matlab toolbox, which directly implements the SM and SE kernels, and the SS kernel as a meta kernel combining simple cosine kernels. The GPML toolbox also implements Kronecker inference automatically for these kernels. We implemented the proposed GSM kernel and inference in Matlab4 . For optimising the log posterior (10) we employ the L-BFGS algorithm. For both our method and the comparisons, we restart the optimisation from 10 different initialisations, each of which is chosen as the best among 100 randomly sampled hyperparameter values as evaluating the log posterior is cheap compared to evaluating gradients or running the full optimisation. 5.1 Simulated time series with a decreasing frequency component First we experiment whether the GSM kernel can find a simulated time-varying frequency pattern. We simulated a dataset where the frequency of the signal changes deterministically as ?(x) = 1+(1?x)2 on the interval x ? [?1, 1]. We built a single-component GSM kernel K using the specified functions ?(x), `(x) = ` = exp(?1) and w(x) = w = 1. We sampled a noisy function y ? N (0, K + ?n2 I) with a noise variance ?n2 = 0.1. The example in Figure 3 shows the learned GSM kernel, as well as the data and the function posterior f (x). For this 1D case, we also employed the empirical spectrogram for initialising the hyperparameter values. The kernel correctly captures the increasing frequency towards negative values (towards left in Figure 3a). 5.2 Image data We applied our kernel to two texture images. The first image of a sheet of metal represents a mostly stationary periodic pattern. The second, a wood texture, represents an example of a very non-stationary pattern, especially on the horizontal axis. We use majority of the image as training data (the non-masked regions of Figure 3a and 3f) , and use the compared kernels to predict a missing cross-section in the middle, and also to extrapolate outside the borders of the original image. Figure 4 shows the two texture images, and extrapolation predictions given by the proposed GSM kernel, with a comparison to the spectral mixture (SM), sparse spectrum (SS) and standard squared exponential (SE) kernels. For GSM, SM and SS we used Q = 5 mixture components for the metal texture, and Q = 10 components for the more complex wood texture. The GSM kernel gives the most pleasing result visually, and fills in both patterns well with consistent external extrapolation as well. The stationary SM kernel does capture the cross-section, but has trouble extrapolation outside the borders. The SS kernel fails to represent even the training data, it lacks any smoothness in the frequency space. The gaussian kernel extrapolates poorly. 4 Implementation available at https://github.com/sremes/nonstationary-spectral-kernels 6 3.5 1.5 3 1 2.5 0.5 2 0 1.5 -0.5 1 -1 0.5 -1.5 (a) (b) (c) (d) Figure 3: (a) A simulated time series with a single decreasing frequency component and a GP fitted using a GSM kernel. (b) The learned kernel shows that close to x = ?1 the signal is highly correlated and anti-correlated with close time points, while these periodic dependencies vanish when moving p towards x = 1. For visualisation, the values are scaled as K = sgn(K) |K|. (c) The spectrogram shows the decreasing frequency. (d) The learned latent frequency function ?(x) correctly finds the decreasing trend. The length-scale `(x) is almost a constant, and weights w(x) slightly decrease in time. 5.3 Spatio-Temporal Analysis of Land Surface Temperatures NASA5 provides a land surface temperature dataset that we used to demonstrate our kernel in analysis of spatio-temporal data. Our primary objective is to demonstrate the capability of the kernel in inferring long-range, non-stationary spatial and temporal covariances. We took a subset of four years (February 2000 to February 2004) of North American land temperatures for training data. In total we get 407,232 data points, constituting 48 monthly temperature measurements on a 84 ? 101 map grid. The grid also contains water regions, which we imputed with the mean temperature of each month. We experimented with the data by learning a generalized spectral mixture kernel using Q = 5 components. Figure 5 presents our results. Figure 5b highlights the training data and model fits for a winter and summer month, respectively. Figure 5a shows the non-stationary kernel slices at two locations across both latitude and longitude, as well as indicating that the spatial covariances are remarkably non-symmetric. Figure 5c indicates five months of successive training data followed by three months of test data predictions. 6 Discussion In this paper we have introduced non-stationary spectral mixture kernels, with treatment based on the generalised Fourier transform of non-stationary functions. We first derived the bivariate spectral mixture (BSM) kernel as a mixture of non-stationary spectral components. However, we argue it has only limited practical use due to requiring an impractical amount of components to cover any sufficiently sized input space. The main contribution of the paper is the generalised spectral mixture (GSM) kernel with input-dependent Gaussian process frequency surfaces. The Gaussian process components can cover non-trivial input spaces with just a few interpretable components. The GSM kernel is a flexible, practical and efficient kernel that can learn both local and global correlations 5 https://neo.sci.gsfc.nasa.gov/view.php?datasetId=MOD11C1_M_LSTDA 7 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 4: A metal texture data with Q = 5 components used for GSM, SM and SS kernels shown in (a)-(e) and a wood texture in (f)-(j) (with Q = 10 components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in (b) and (g). The SM kernel fills in the missing cross pattern in (c) but does not extrapolate well. In (h) the SM kernel fills in the vertical middle block only with the mean value while GSM in (g) is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture (d) or (i), while the SE kernel overfits by using a too short length-scale in (e) and (j). across the input domains in an input-dependent manner. We highlighted the capability of the kernel to find interesting patterns in the data by applying it on climate data where it is highly unrealistic to assume the same (stationary) covariance pattern for every spatial location irrespective of spatial structures. Even though the proposed kernel is motivated by the generalised Fourier transform, the solution to its spectral surface ZZ 0 0 0 SGSM (s, s ) = kGSM (x, x0 )e?2?i(xs?x s ) dxdx0 (12) remains unknown due to having multiple GP functions inside the integral. Figure 2h highlights a numerical integration of the surface equation (12) on an example GP frequency surface. Furthermore, the theoretical work of Kom Samo and Roberts [11] on generalised spectral transforms suggests that the GSM kernel may also be dense in the family of non-stationary kernels, that is, to reproduce arbitrary non-stationary kernels. Acknowledgments This work has been partly supported by the Finnish Funding Agency for Innovation (project Re:Know) and Academy of Finland (COIN CoE, and grants 299915, 294238 and 292334). We acknowledge the computational resources provided by the Aalto Science-IT project. References [1] S. Flaxman, A. G. Wilson, D. Neill, H. Nickisch, and A. Smola. Fast kronecker inference in Gaussian processes with non-Gaussian likelihoods. In ICML, volume 2015, 2015. [2] M. Genton. Classes of kernels for machine learning: A statistics perspective. Journal of Machine Learning Research, 2:299?312, 2001. [3] M. Gibbs. Bayesian Gaussian Processes for Regression and Classification. PhD thesis, University of Cambridge, 1997. 8 (b) (a) (c) Figure 5: (a) Demonstrates the non-stationary spatial covariances in the land surface data. The vertical black lines denote the point x0 at which the kernel function k(?, x0 ) is centered. (b) Sample reconstructions. In all plots, only the land area temperatures are shown. (c) Posterior for five last training months (until Jan 2004) and prediction for the three next months (February 2004 to April 2004), which the model is able to to construct reasonably accurately. [4] R. Gramacy and H. Lee. Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103:1119?1130, 2008. [5] M. Grzegorczyk, D. Husmeier, K. Edwards, P. Ghazal, and A. Millar. Modelling non-stationary gene regulatory processes with a non-homogeneous bayesian network and the allocation sampler. Bioinformatics, 24:2071?2078, 2008. [6] M. Heinonen, H. Mannerstr?m, J. Rousu, S. Kaski, and H. L?hdesm?ki. Non-stationary Gaussian process regression with Hamiltonian Monte Carlo. In AISTATS, volume 51, pages 732?740, 2016. [7] D. Higdon, J. Swall, and J. Kern. Non-stationary spatial modeling. Bayesian statistics, 6:761? 768, 1999. [8] N. Huang. A review on hilbert-huang transform: Method and its applications to geophysical studies. Reviews of Geophysics, 46, 2008. [9] N. Huang, S. Zheng, S. Long, M. Wu, H. Shih, Q. Zheng, N.-Q. Yen, C. Tung, and H. Liu. The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 454:903?995, 1998. [10] Y. Kakihara. A note on harmonizable and v-bounded processes. Journal of Multivariate Analysis, 16:140?156, 1985. 9 [11] Y.-L. Kom Samo and S. Roberts. Generalized spectral kernels. Technical report, University of Oxford, 2015. arXiv:1506.02236. [12] M. Kuss and C. E. Rasmussen. Assessing approximate inference for binary Gaussian process classification. Journal of Machine Learning Research, 6:1679?1704, 2005. [13] M. L?zaro-Gredilla, J. Qui?onero-Candela, C. E. Rasmussen, and A. R. Figueiras-Vidal. Sparse spectrum Gaussian process regression. Journal of Machine Learning Research, 11:1865?1881, 2010. [14] M. Loeve. Probability Theory II, volume 46 of Graduate Texts in Mathematics. Springer, 1978. [15] C. Paciorek and M. Schervish. Nonstationary covariance functions for Gaussian process regression. In NIPS, pages 273?280, 2004. [16] C. Paciorek and M. Schervish. Spatial modelling using a new class of nonstationary covariance functions. Environmetrics, 17(5):483?506, 2006. [17] A. Rahimi and B. Recht. Random features for large-scale kernel machines. In Advances in neural information processing systems, pages 1177?1184, 2008. [18] C. E. Rasmussen and C. Williams. Gaussian processes for machine learning. MIT Press, 2006. [19] O. Rioul and V. Martin. 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Expectation propagation for nonstationary heteroscedastic Gaussian process regression. In Machine Learning for Signal Processing (MLSP), 2014 IEEE International Workshop on, pages 1?6. IEEE, 2014. [27] A. Wilson, E. Gilboa, J. P. Cunningham, and A. Nehorai. Fast kernel learning for multidimensional pattern extrapolation. In NIPS, 2014. [28] A. Wilson and H. Nickisch. Kernel interpolation for scalable structured gaussian processes (KISS-GP). In International Conference on Machine Learning, pages 1775?1784, 2015. [29] A. G. Wilson. Covariance kernels for fast automatic pattern discovery and extrapolation with Gaussian processes. PhD thesis, University of Cambridge, 2014. [30] A. G. Wilson and R. Adams. Gaussian process kernels for pattern discovery and extrapolation. In ICML, 2013. [31] A. M. Yaglom. Correlation theory of stationary and related random functions: Volume I: Basic results. Springer Series in Statistics. Springer, 1987. [32] Z. Yang, A. Smola, L. Song, and A. Wilson. A la carte: Learning fast kernels. In AISTATS, 2015. 10
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Overcoming Catastrophic Forgetting by Incremental Moment Matching Sang-Woo Lee1 , Jin-Hwa Kim1 , Jaehyun Jun1 , Jung-Woo Ha2 , and Byoung-Tak Zhang1,3 Seoul National University1 Clova AI Research, NAVER Corp2 Surromind Robotics3 {slee,jhkim,jhjun}@bi.snu.ac.kr [email protected] [email protected] Abstract Catastrophic forgetting is a problem of neural networks that loses the information of the first task after training the second task. Here, we propose a method, i.e. incremental moment matching (IMM), to resolve this problem. IMM incrementally matches the moment of the posterior distribution of the neural network which is trained on the first and the second task, respectively. To make the search space of posterior parameter smooth, the IMM procedure is complemented by various transfer learning techniques including weight transfer, L2-norm of the old and the new parameter, and a variant of dropout with the old parameter. We analyze our approach on a variety of datasets including the MNIST, CIFAR-10, Caltech-UCSDBirds, and Lifelog datasets. The experimental results show that IMM achieves state-of-the-art performance by balancing the information between an old and a new network. 1 Introduction Catastrophic forgetting is a fundamental challenge for artificial general intelligence based on neural networks. The models that use stochastic gradient descent often forget the information of previous tasks after being trained on a new task [1, 2]. Online multi-task learning that handles such problems is described as continual learning. This classic problem has resurfaced with the renaissance of deep learning research [3, 4]. Recently, the concept of applying a regularization function to a network trained by the old task to learning a new task has received much attention. This approach can be interpreted as an approximation of sequential Bayesian [5, 6]. Representative examples of this regularization approach include learning without forgetting [7] and elastic weight consolidation [8]. These algorithms succeeded in some experiments where their own assumption of the regularization function fits the problem. Here, we propose incremental moment matching (IMM) to resolve the catastrophic forgetting problem. IMM uses the framework of Bayesian neural networks, which implies that uncertainty is introduced on the parameters in neural networks, and that the posterior distribution is calculated [9, 10]. The dimension of the random variable in the posterior distribution is the number of the parameters in the neural networks. IMM approximates the mixture of Gaussian posterior with each component representing parameters for a single task to one Gaussian distribution for a combined task. To merge the posteriors, we introduce two novel methods of moment matching. One is mean-IMM, which simply averages the parameters of two networks for old and new tasks as the minimization of the average of KL-divergence between one approximated posterior distribution for the combined task 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. ?1:2Mode = (S1-1 + S 2-1 )-1 (S1-1?1 + S -21?2 ) ?1:2Mode ?1 ?1:2Mean ?1:2Mean = ( ?1 + ?2 ) / 2 ?2 weight-transfer ?1 ? ?2 L2-transfer ?2 - ?1 2 2 drop-transfer ?1 + 2 ? dropout ( ?2 - ?1 ) Find !2 , which makes !1:2 perform better Figure 1: Geometric illustration of incremental moment matching (IMM). Mean-IMM simply averages the parameters of two neural networks, whereas mode-IMM tries to find a maximum of the mixture of Gaussian posteriors. To make IMM be reasonable, the search space of the loss function between the posterior means ?1 and ?2 should be reasonably smooth and convex-like. To find a ?2 which satisfies this condition of a smooth and convex-like path from ?1 , we propose applying various transfer techniques for the IMM procedure. and each Gaussian posterior for the single task [11]. The other is mode-IMM, which merges the parameters of two networks using a Laplacian approximation [9] to approximate a mode of the mixture of two Gaussian posteriors, which represent the parameters of the two networks. In general, it is too na?ve to assume that the final posterior distribution for the whole task is Gaussian. To make our IMM work, the search space of the loss function between the posterior means needs to be smooth and convex-like. In other words, there should not be high cost barriers between the means of the two networks for an old and a new task. To make our assumption of Gaussian distribution for neural network reasonable, we applied three main transfer learning techniques on the IMM procedure: weight transfer, L2-norm of the old and the new parameters, and our newly proposed variant of dropout using the old parameters. The whole procedure of IMM is illustrated in Figure 1. 2 Previous Works on Catastrophic Forgetting One of the major approaches preventing catastrophic forgetting is to use an ensemble of neural networks. When a new task arrives, the algorithm makes a new network, and shares the representation between the tasks [12, 13]. However, this approach has a complexity issue, especially in inference, because the number of networks increases as the number of new tasks that need to be learned increases. Another approach studies the methods using implicit distributed storage of information, in typical stochastic gradient descent (SGD) learning. These methods use the idea of dropout, maxout, or neural module to distributively store the information for each task by making use of the large capacity of the neural network [4]. Unfortunately, most studies following this approach had limited success and failed to preserve performance on the old task when an extreme change to the environment occurred [3]. Alternatively, Fernando et al. [14] proposed PathNet, which extends the idea of the ensemble approach for parameter reuse [13] within a single network. In PathNet, a neural network has ten or twenty modules in each layer, and three or four modules are picked for one task in each layer by an evolutionary approach. This method alleviates the complexity issue of the ensemble approach to continual learning in a plausible way. The approach with a regularization term also has received attention. Learning without forgetting (LwF) is one example of this approach, which uses the pseudo-training data from the old task [7]. Before learning the new task, LwF puts the training data of the new task into the old network, and uses the output as pseudo-labels of the pseudo-training data. By optimizing both the pseudotraining data of the old task and the real data of the new task, LwF attempts to prevent catastrophic forgetting. This framework is promising where the properties of the pseudo training set is similar to the ideal training set. Elastic weight consolidation (EWC), another example of this approach, uses sequential Bayesian estimation to update neural networks for continual learning [8]. In EWC, the posterior distribution trained by the previous task is used to update the new prior distribution. This new prior is used for learning the new posterior distribution of the new task in a Bayesian manner. 2 EWC assumes that the covariance matrix of the posterior is diagonal and there are no correlations between the nodes. Though this assumption is fragile, EWC performs well in some domains. EWC is a monumental recent work that uses sequential Bayesian for continual learning of neural networks. However, updating the parameter of complex hierarchical models by sequential Bayesian estimation is not new [5]. Sequential Bayes was used to learn topic models from stream data by Broderick et al. [6]. Huang et al. applied sequential Bayesian to adapt a deep neural network to the specific user in the speech recognition domain [15, 16]. They assigned the layer for the user adaptation and applied MAP estimation to this single layer. Similar to our IMM method, Bayesian moment matching is used for sum-product networks, a kind of deep hierarchical probabilistic model [17]. Though sum-product networks are usually not scalable to large datasets, their online learning method is useful, and it achieves similar performance to the batch learner. Our method using moment matching focuses on continual learning and deals with significantly different statistics between tasks, unlike the previous method. 3 Incremental Moment Matching In incremental moment matching (IMM), the moments of posterior distributions are matched in an incremental way. In our work, we use a Gaussian distribution to approximate the posterior distribution of parameters. Given K sequential tasks, we want to find the optimal parameter ??1:K and ??1:K of the Gaussian approximation function q1:K from the posterior parameter for each kth task, (?k , ?k ). p1:K ? p(?|X1 , ? ? ? , XK , y1 , ? ? ? , yK ) ? q1:K ? q(?|?1:K , ?1:K ) pk ? p(?|Xk , yk ) ? qk ? q(?|?k , ?k ) (1) (2) q1:K denotes an approximation of the true posterior distribution p1:K for the whole task, and qk denotes an approximation of the true posterior distribution pk over the training dataset (Xk , yk ) for the kth task. ? denotes the vectorized parameter of the neural network. The dimension of ?k and ?1:k is D, and the dimension of ?k and ?1:k is D ? D, respectively, where D is the dimension of ?. For example, a multi-layer perceptrons (MLP) with [784-800-800-800-10] has the number of nodes, D = 1917610 including bias terms. Next, we explain two proposed moment matching algorithms for the continual learning of modern deep neural networks. The two algorithms generate two different moments of Gaussian with different objective functions for the same dataset. 3.1 Mean-based Incremental Moment Matching (mean-IMM) Mean-IMM averages the parameters of two networks in each layer, using mixing ratios ?k with PK k ?k = 1. The objective function of mean-IMM is to minimize the following local KL-distance or the weighted sum of KL-divergence between each qk and q1:K [11, 18]: ??1:K , ??1:K = argmin PK k ?1:K ,?1:K ??1:K = ??1:K = PK k PK k ?k KL(qk ||q1:K ) ?k ?k ?k (?k + (?k ? ??1:K )(?k (3) (4) ? ??1:K )T ) (5) ??1:K and ??1:K are the optimal solution of the local KL-distance. Notice that covariance information is not needed for mean-IMM, since calculating ??1:K does not require any ?k . A series of ?k is sufficient to perform the task. The idea of mean-IMM is commonly used in shallow networks [19, 20]. However, the contribution of this paper is to discover when and how mean-IMM can be applied in modern deep neural networks and to show it can performs better with other transfer techniques. Future works may include other measures to merge the networks, including the KL-divergence bePK tween q1:K and the mixture of each qk (i.e. KL(q1:K || k ?k qk )) [18]. 3 3.2 Mode-based Incremental Moment Matching (mode-IMM) Mode-IMM is a variant of mean-IMM which uses the covariance information of the posterior of Gaussian distribution. In general, a weighted average of two mean vectors of Gaussian distributions is not a mode of MoG. In discriminative learning, the maximum of the distribution is of primary interest. According to Ray and Lindsay [21], all the modes of MoG with K clusters lie on (K ? 1)PK PK P ?1 dimensional hypersurface {?|? = ( k ak ??1 ( k ak ??1 k ak = 1}. k ) k ?k ), 0 < ak < 1 and See Appendix A for more details. Motivated by the above description, a mode-IMM approximate MoG with Laplacian approximation, in which the logarithm of the function is expressed by the Taylor expansion [9]. Using Laplacian approximation, the MoG is approximated as follows: log q1:K ? PK k PK PK 1 ?1 0 ?k log qk + C = ? ?T ( k ?k ??1 k ?k ?k ?k )? + C k )? + ( 2 PK ??1:K = ??1:K ? ( k ?k ??1 k ?k ) PK ?1 ?1 ? ?1:K = ( k ?k ?k ) (6) (7) (8) For Equation 8, we add I to the term to be inverted in practice, with an identity matrix I and a small constant . Here, we assume diagonal covariance matrices, which means that there is no correlation among parameters. This diagonal assumption is useful, since it decreases the number of parameters for each covariance matrix from O(D2 ) to O(D) for the dimension of the parameters D. For covariance, we use the inverse of a Fisher information matrix, following [8, 22]. The main idea of this approximation is that the square of gradients for parameters is a good indicator of their precision, which is the inverse of the variance. The Fisher information matrix for the kth task, Fk is defined as:  ? ? T ln p(? y |x, ?k ) ? ln p(? y |x, ?k ) , Fk = E ??k ??k  (9) where the probability of the expectation follows x ? ?k and y? ? p(y|x, ?k ), where ?k denotes an empirical distribution of Xk . 4 Transfer Techniques for Incremental Moment Matching In general, the loss function of neural networks is not convex. Consider that shuffling nodes and their weights in a neural network preserves the original performance. If the parameters of two neural networks initialized independently are averaged, it might perform poorly because of the high cost barriers between the parameters of the two neural networks [23]. However, we will show that various transfer learning techniques can be used to ease this problem, and make the assumption of Gaussian distribution for neural networks reasonable. In this section, we introduce three practical techniques for IMM, including weight-transfer, L2-transfer, and drop-transfer. 4.1 Weight-Transfer Weight-transfer initialize the parameters for the new task ?k with the parameters of the previous task ?k?1 [24]. In our experiments, the use of weight-transfer was critical to the continual learning performance. For this reason, the experiments on IMM in this paper use the weight-transfer technique by default. The weight-transfer technique is motivated by the geometrical property of neural networks discovered in the previous work [23]. They found that there is a straight path from the initial point to the solution without any high cost barrier, in various types of neural networks and datasets. This discovery suggests that the weight-transfer from the previous task to the new task makes a smooth loss 4 Figure 2: Experimental results on visualizing the effect of weight-transfer. The geometric property of the parameter space of the neural network is analyzed. Brighter is better. ?1 , ?2 , and ?3 are the vectorized parameters of trained networks from randomly selected subsets of the CIFAR-10 dataset. This figure shows that there are better solutions between the three locally optimized parameters. surface between two solutions for the tasks, so that the optimal solution for both tasks lies on the interpolated point of the two solutions. To empirically validate the concept of weight-transfer, we use the linear path analysis proposed by Goodfellow et al. [23] (Figure 2). We randomly chose 18,000 instances from the training dataset of CIFAR-10, and divided them into three subsets of 6,000 instances each. These three subsets are used for sequential training by CNN models, parameterized by ?1 , ?2 , and ?3 , respectively. Here, ?2 is initialized from ?1 , and then ?3 is initialized from ?2 , in the same way as weight-transfer. In this analysis, each loss and accuracy is evaluated at a series of points ? = ?1 + ?(?2 ? ?1 ) + ?(?3 ? ?2 ), varying ? and ?. In Figure 2, the loss surface of the model on each online subset is nearly convex. The figure shows that the parameter at 31 (?1 + ?2 + ?3 ), which is the same as the solution of mean-IMM, performs better than any other reference points ?1 , ?2 , or ?3 . However, when ?2 is not initialized by ?1 , the convex-like shape disappears, since there is a high cost barrier between the loss function of ?1 and ?2 . 4.2 L2-transfer L2-transfer is a variant of L2-regularization. L2-transfer can be interpreted as a special case of EWC where the prior distribution is Gaussian with ?I as a covariance matrix. In L2-transfer, a regularization term of the distance between ?k?1 and ?k is added to the following objective function for finding ?k , where ? is a hyperparameter: log p(yk |Xk , ?k ) ? ? ? ||?k ? ?k?1 ||22 (10) The concept of L2-transfer is commonly used in transfer learning [25, 26] and continual learning [7, 8] with large ?. Unlike the previous usage of large ?, we use small ? for the IMM procedure. In other words, ?k is first trained by Equation 10 with small ?, and then merged to ?1:k in our IMM. Since we want to make the loss surface between ?k?1 and ?k smooth, and not to minimize the distance between ?k?1 and ?k . In convex optimization, the L2-regularizer makes the convex function strictly convex. Similarly, we hope L2-transfer with small ? help to find a ?k with a convexlike loss space between ?k?1 and ?k . 4.3 Drop-transfer Drop-transfer is a novel method devised in this paper. Drop-transfer is a variant of dropout where ?k?1 is the zero point of the dropout procedure. In the training phase, the following ? ?k,i is used for the weight vector corresponding to the ith node ?k,i : ? ?k,i ( ?k?1,i , = 1 1?p ? ?k,i ? p 1?p if ith node is turned off ? ?k?1,i , otherwise where p is the dropout ratio. Notice that the expectation of ? ?k,i is ?k,i . 5 (11) Table 1: The averaged accuracies on the disjoint MNIST for two sequential tasks (Top) and the shuffled MNIST for three sequential tasks (Bottom). The untuned setting refers to the most natural hyperparameter in the equation of each algorithm, whereas the tuned setting refers to using heuristic hand-tuned hyperparameters. Hyperparam denotes the main hyperparameter of each algorithm. For IMM with transfer, only ? is tuned. The numbers in the parentheses refer to standard deviation. Every IMM uses weight-transfer. Disjoint MNIST Experiment SGD [3] L2-transfer [25] Drop-transfer EWC [8] Mean-IMM Mode-IMM L2-transfer + Mean-IMM L2-transfer + Mode-IMM Drop-transfer + Mean-IMM Drop-transfer + Mode-IMM L2, Drop-transfer + Mean-IMM L2, Drop-transfer + Mode-IMM Explanation of Hyperparam epoch per dataset ? in (10) p in (11) ? in (20) ?2 in (4) ?2 in (7) ? / ?2 ? / ?2 p / ?2 p / ?2 ? / p / ?2 ? / p / ?2 Untuned Hyperparam Accuracy 10 47.72 (? 0.11) 0.5 51.72 (? 0.79) 1.0 47.84 (? 0.04) 0.50 90.45 (? 2.24) 0.50 91.49 (? 0.98) 0.001 / 0.50 78.34 (? 1.82) 0.001 / 0.50 92.52 (? 0.41) 0.5 / 0.50 80.75 (? 1.28) 0.5 / 0.50 93.35 (? 0.49) 0.001 / 0.5 / 0.50 66.10 (? 3.19) 0.001 / 0.5 / 0.50 93.97 (? 0.32) Tuned Hyperparam Accuracy 0.05 71.32 (? 1.54) 0.05 85.81 (? 0.52) 0.5 51.72 (? 0.79) 600M 52.72 (? 1.36) 0.55 91.92 (? 0.98) 0.45 92.02 (? 0.73) 0.001 / 0.60 92.62 (? 0.95) 0.001 / 0.45 92.73 (? 0.35) 0.5 / 0.60 92.64 (? 0.60) 0.5 / 0.50 93.35 (? 0.49) 0.001 / 0.5 / 0.75 93.97 (? 0.23) 0.001 / 0.5 / 0.45 94.12 (? 0.27) Shuffled MNIST Experiment SGD [3] L2-transfer [25] Drop-transfer EWC [8] Mean-IMM Mode-IMM L2-transfer + Mean-IMM L2-transfer + Mode-IMM Drop-transfer + Mean-IMM Drop-transfer + Mode-IMM L2, Drop-transfer + Mean-IMM L2, Drop-transfer + Mode-IMM epoch per dataset ? in (10) p in (11) ? in (20) ?3 in (4) ?3 in (7) ? / ?3 ? / ?3 p / ?3 p / ?3 ? / p / ?3 ? / p / ?3 Hyperparam 60 0.5 0.33 0.33 1e-4 / 0.33 1e-4 / 0.33 0.5 / 0.33 0.5 / 0.33 1e-4 / 0.5 / 0.33 1e-4 / 0.5 / 0.33 Hyperparam 1e-3 0.2 0.55 0.60 1e-4 / 0.65 1e-4 / 0.60 0.5 / 0.65 0.5 / 0.55 1e-4 / 0.5 / 0.90 1e-4 / 0.5 / 0.50 Accuracy 89.15 (? 2.34) 94.75 (? 0.62) 93.23 (? 1.37) 98.02 (? 0.05) 90.38 (? 1.74) 98.16 (? 0.08) 90.79 (? 1.30) 97.80 (? 0.07) 89.51 (? 2.85) 97.83 (? 0.10) Accuracy ?95.5 [8] 96.37 (? 0.62) 96.86 (? 0.21) ?98.2 [8] 95.02 (? 0.42) 98.08 (? 0.08) 95.93 (? 0.31) 98.30 (? 0.08) 96.49 (? 0.44) 97.95 (? 0.08) 97.36 (? 0.19) 97.92 (? 0.05) There are studies [27, 20] that have interpreted dropout as an exponential ensemble of weak learners. By this perspective, since the marginalization of output distribution over the whole weak learner is intractable, the parameters multiplied by the inverse of the dropout rate are used for the test phase in the procedure. In other words, the parameters of the weak learners are, in effect, simply averaged oversampled learners by dropout. At the process of drop-transfer in our continual learning setting, we hypothesize that the dropout process makes the averaged point of two arbitrary sampled points using Equation 11 a good estimator. We investigated the search space of the loss function of the MLP trained from the MNIST handwritten digit recognition dataset for with and without dropout regularization, to supplement the evidence of the described hypothesis. Dropout regularization makes the accuracy of a sampled point from dropout distribution and an average point of two sampled parameters, from 0.450 (? 0.084) to 0.950 (? 0.009) and 0.757 (? 0.065) to 0.974 (? 0.003), respectively. For the case of both with and without dropout, the space between two arbitrary samples is empirically convex, and fits to the second-order equation. Based on this experiment, we expect not only that the search space of the loss function between modern neural networks can be easily nearly convex [23], but also that regularizers, such as dropout, make the search space smooth and the point in the search space have a good accuracy in continual learning. 5 Experimental Results We evaluate our approach on four experiments, whose settings are intensively used in the previous works [4, 8, 7, 12]. For more details and experimental results, see Appendix D. Disjoint MNIST Experiment. The first experiment is the disjoint MNIST experiment [4]. In this experiment, the MNIST dataset is divided into two datasets: the first dataset consists of only digits {0, 1, 2, 3, 4} and the second dataset consists of the remaining digits {5, 6, 7, 8, 9}. Our task is 10class joint categorization, unlike the setting in the previous work, which considers two independent tasks of 5-class categorization. Because the inference should decide whether a new instance comes from the first or the second task, our task is more difficult than the task of the previous work. 6 The disjoint MNIST experiment The shuffled MNIST experiment The ImageNet2CUB experiment 1 First Task, Mean?IMM Second Task, Mean?IMM First Task, Mode?IMM Second Task, Mode?IMM 1.2 First Task, Mean?IMM Second Task, Mean?IMM First Task, Mode?IMM Second Task, Mode?IMM 0.995 0.99 First Task, Mean?IMM Second Task, Mean?IMM First Task, Mode?IMM Second Task, Mode?IMM 0.62 0.6 1 0.6 Test Accuracy Test Accuracy Test Accuracy 0.985 0.8 0.98 0.975 0.97 0.58 0.56 0.965 0.4 0.54 0.96 0.2 0.955 0 0 0.2 0.4 0.6 alpha, for weighing two networks 0.8 1 0.95 0.52 0 0.2 0.4 0.6 alpha, for weighing two networks 0.8 1 0 0.2 0.4 0.6 alpha, for weighing two networks 0.8 1 Figure 3: Test accuracies of two IMM models with weight-transfer on the disjoint MNIST (Left), the shuffled MNIST (Middle), and the ImageNet2CUB experiment (Right). ? is a hyperparameter that balances the information between the old and the new task. The disjoint MNIST experiment The disjoint MNIST experiment 0.95 0.95 0.9 0.9 0.85 0.85 0.8 Test Accuracy Test Accuracy 0.8 0.75 0.7 0.65 0.7 0.65 0.6 0 0.2 0.4 0.6 alpha, for weighing two networks 0.8 Mean?IMM Mode?IMM Drop?transfer + Mean?IMM Drop?transfer + Mode?IMM L2, Drop?transfer + Mean?IMM L2, Drop?transfer + Mode?IMM 0.6 Mean?IMM Mode?IMM L2?transfer + Mean?IMM L2?transfer + Mode?IMM 0.55 0.5 0.75 0.55 0.5 1 0 0.2 0.4 0.6 alpha, for weighing two networks 0.8 1 Figure 4: Test accuracies of IMM with various transfer techniques on the disjoint MNIST. Both L2transfer and drop-transfer boost the performance of IMM and make the optimal value of ? larger than 1/2. However, drop-transfer tends to make the accuracy curve more smooth than L2-transfer does. We evaluate the models both on the untuned setting and the tuned setting. The untuned setting refers to the most natural hyperparameter in the equation of each algorithm. The tuned setting refers to using heuristic hand-tuned hyperparameters. Consider that tuned hyperparameter setting is often used in previous works of continual learning as it is difficult to define a validation set in their setting. For example, when the model needs to learn from the new task after learning from the old task, a low learning rate or early stopping without a validation set, or arbitrary hyperparameter for balancing is used [3, 8]. We discover hyperparameters in the tuned setting not only to find the oracle performance of each algorithm, but also to show that there exist some paths consisting of the point that performs reasonably for both tasks. Hyperparam in Table 1 denotes hyperparameter mainly searched in the tuned setting. Table 1 (Top) and Figure 3 (Left) shows the experimental results from the disjoint MNIST experiment. In our experimental setting, the usual SGD-based optimizers always perform less than 50%, because the biases of the output layer for the old task are always pushed to large negative values, which implies that our task is difficult. Figure 4 also shows that mode-IMM is robust with ? and the optimal ? of mean-IMM is larger than 1/2 in the disjoint MNIST experiment. Shuffled MNIST Experiment. The second experiment is the shuffled MNIST experiment [3, 8] of three sequential tasks. In this experiment, the first dataset is the same as the original MNIST dataset. However, in the second dataset, the input pixels of all images are shuffled with a fixed, random permutation. In previous work, EWC reaches the performance level of the batch learner, and it is argued that EWC overcomes catastrophic forgetting in some domains. The experimental details are similar to the disjoint MNIST experiment, except all models are allowed to use dropout regularization. In the experiment, the first dataset is the same as the original MNIST dataset. However, in the second and the third dataset, the input pixels of all images are shuffled with a fixed, random permutation, respectively. Therefore, the difficulty of the three datasets is the same, though a different solution is required for each dataset. 7 Table 2: Experimental results on the Lifelog dataset among different classes (location, sub-location, and activity) and different subjects (A, B, C). Every IMM uses weight-transfer. Dual memory architecture [12] Mean-IMM Mode-IMM Location 78.11 77.60 77.14 Sub-location 72.36 73.78 75.76 Activity 52.92 52.74 54.07 A 67.02 67.03 67.97 B 58.80 57.73 60.12 C 77.57 79.35 78.89 Table 1 (Bottom) and Figure 3 (Middle) shows the experimental results from the shuffled MNIST experiment. Notice that accuracy of drop-transfer (p = 0.2) alone is 96.86 (? 0.21) and L2-transfer (? = 1e-4) + drop-transfer (p = 0.4) alone is 97.61 (? 0.15). These results are competitive to EWC without dropout, whose performance is around 97.0. ImageNet to CUB Dataset. The third experiment is the ImageNet2CUB experiment [7], the continual learning problem from the ImageNet dataset to the Caltech-UCSD Birds-200-2011 finegrained classification (CUB) dataset [28]. The numbers of classes of ImageNet and CUB dataset are around 1K and 200, and the numbers of training instances are 1M and 5K, respectively. In the ImageNet2CUB experiment, the last-layer is separated for the ImageNet and the CUB task. The structure of AlexNet is used for the trained model of ImageNet [29]. In our experiment, we match the moments of the last-layer fine-tuning model and the LwF model, with mean-IMM and modeIMM. Figure 3 (Right) shows that mean-IMM moderately balances the performance of two tasks between two networks. However, the balanced hyperparameter of mode-IMM is far from ? = 0.5. We think that it is because the scale of the Fisher matrix F is different between the ImageNet and the CUB task. Since the number of training data of the two tasks is different, the mean of the square of the gradient, which is the definition of F , tends to be different. This implies that the assumption of mode-IMM does not always hold for heterogeneous tasks. See Appendix D.3 for more information including the learning methods of IMM where a different class output layer or a different scale of the dataset is used. Our results of IMM with LwF exceed the previous state-of-the-art performance, whose model is also LwF. This is because, in the previous works, the LwF model is initialized by the last-layer finetuning model, not directly by the original AlexNet. In this case, the performance loss of the old task is not only decreased, but also the performance gain of the new task is decreased. The accuracies of our mean-IMM (? = 0.5) are 56.20 and 56.73 for the ImageNet task and the CUB task, respectively. The gains compared to the previous state-of-the-art are +1.13 and -1.14. In the case of mean-IMM (? = 0.8) and mode-IMM (? = 0.99), the accuracies are 55.08 and 59.08 (+0.01, +1.12), and 55.10 and 59.12 (+0.02, +1.35), respectively. Lifelog Dataset. Lastly, we evaluate the proposed methods on the Lifelog dataset [12]. The Lifelog dataset consists of 660,000 instances of egocentric video stream data, collected over 46 days from three participants using Google Glass [30]. Three class categories, location, sub-location, and activity, are labeled on each frame of video. In the Lifelog dataset, the class distribution changes continuously and new classes appear as the day passes. Table 2 shows that mean-IMM and mode-IMM are competitive to the dual-memory architecture, the previous state-of-the-art ensemble model, even though IMM uses single network. 6 Discussion A Shift of Optimal Hyperparameter of IMM. The tuned setting shows there often exists some ? which makes the performance of the mean-IMM close to the mode-IMM. However, in the untuned hyperparameter setting, mean-IMM performs worse when more transfer techniques are applied. Our Bayesian interpretation in IMM assumes that the SGD training of the k-th network ?k is mainly affected by the k-th task and is rarely affected by the information of the previous tasks. However, transfer techniques break this assumption; thus the optimal ? is shifted to larger than 1/k. Fortunately, mode-IMM works more robustly than mean-IMM where transfer techniques are applied. Figure 4 illustrates the change of the test accuracy curve corresponding to the applied transfer techniques and the following shift of the optimal ? in mean-IMM and mode-IMM. 8 Bayesian Approach on Continual Learning. Kirkpatrick et al. [8] interpreted that the Fisher matrix F as weight importance in explaining their EWC model. In the shuffled MNIST experiment, since a large number of pixels always have a value of zero, the corresponding elements of the Fisher matrix are also zero. Therefore, EWC does work by allowing weights to change, which are not used in the previous tasks. On the other hand, mode-IMM also works by selectively balancing between two weights using variance information. However, these assumptions on weight importance do not always hold, especially in the disjoint MNIST experiment. The most important weight in the disjoint MNIST experiment is the bias term in the output layer. Nevertheless, these bias parts of the Fisher matrix are not guaranteed to be the highest value nor can they be used to balance the class distribution between the first and second task. We believe that using only the diagonal of the covariance matrix in Bayesian neural networks is too na?ve in general and that this is why EWC failed in the disjoint MNIST experiment. We think it could be alleviated in future work by using a more complex prior, such as a matrix Gaussian distribution considering the correlations between nodes in the network [31]. Balancing the Information of an Old and a New Task. The IMM procedure produces a neural network without a performance loss for kth task ?k , which is better than the final solution ?1:k in terms of the performance of the kth task. Furthermore, IMM can easily weigh the importance of tasks in IMM models in real time. For example, ?t can be easily changed for the solution of meanPk IMM ?1:k = t ?t ?t . In actual service situations of IT companies, the importance of the old and the new task frequently changes in real time, and IMM can handle this problem. This property differentiates IMM from the other continual learning methods using the regularization approach, including LwF and EWC. 7 Conclusion Our contributions are four folds. First, we applied mean-IMM to the continual learning of modern deep neural networks. Mean-IMM makes competitive results to comparative models and balances the information between an old and a new network. We also interpreted the success of IMM by the Bayesian framework with Gaussian posterior. Second, we extended mean-IMM to mode-IMM with the interpretation of mode-finding in the mixture of Gaussian posterior. Mode-IMM outperforms mean-IMM and comparative models in various datasets. Third, we introduced drop-transfer, a novel method proposed in the paper. Experimental results showed that drop-transfer alone performs well and is similar to the EWC without dropout, in the domain where EWC rarely forgets. Fourth, We applied various transfer techniques in the IMM procedure to make our assumption of Gaussian distribution reasonable. We argued that not only the search space of the loss function among neural networks can easily be nearly convex, but also regularizers, such as dropout, make the search space smooth, and the point in the search space have good accuracy. Experimental results showed that applying transfer techniques often boost the performance of IMM. Overall, we made state-of-theart performance in various datasets of continual learning and explored geometrical properties and a Bayesian perspective of deep neural networks. Acknowledgments The authors would like to thank Jiseob Kim, Min-Oh Heo, Donghyun Kwak, Insu Jeon, Christina Baek, and Heidi Tessmer for helpful comments and editing. This work was supported by the Naver Corp. and partly by the Korean government (IITP-R0126-16-1072-SW.StarLab, KEIT-10044009HRI.MESSI, KEIT-10060086-RISF). Byoung-Tak Zhang is the corresponding author. References [1] Michael McCloskey and Neal J Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. Psychology of learning and motivation, 24:109?165, 1989. [2] Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3(4):128?135, 1999. 9 [3] Ian J Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, and Yoshua Bengio. An empirical investigation of catastrophic forgetting in gradient-based neural networks. arXiv preprint arXiv:1312.6211, 2013. [4] Rupesh K Srivastava, Jonathan Masci, Sohrob Kazerounian, Faustino Gomez, and J?rgen Schmidhuber. Compete to compute. In Advances in neural information processing systems, pages 2310?2318, 2013. [5] Zoubin Ghahramani. Online variational bayesian learning. 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Simplifying mixture models through function approximation. Neural Networks, IEEE Transactions on, 21(4):644?658, 2010. [19] Manas Pathak, Shantanu Rane, and Bhiksha Raj. Multiparty differential privacy via aggregation of locally trained classifiers. In Advances in Neural Information Processing Systems, pages 1876?1884, 2010. 10 [20] Pierre Baldi and Peter J Sadowski. Understanding dropout. In Advances in Neural Information Processing Systems, pages 2814?2822, 2013. [21] Surajit Ray and Bruce G Lindsay. The topography of multivariate normal mixtures. Annals of Statistics, pages 2042?2065, 2005. [22] Razvan Pascanu and Yoshua Bengio. Revisiting natural gradient for deep networks. arXiv preprint arXiv:1301.3584, 2013. [23] Ian J Goodfellow, Oriol Vinyals, and Andrew M Saxe. Qualitatively characterizing neural network optimization problems. arXiv preprint arXiv:1412.6544, 2014. [24] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pages 3320? 3328, 2014. [25] Theodoros Evgeniou and Massimiliano Pontil. Regularized multi?task learning. In Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 109?117. ACM, 2004. [26] Wolf Kienzle and Kumar Chellapilla. Personalized handwriting recognition via biased regularization. In Proceedings of the 23rd international conference on Machine learning, pages 457?464. ACM, 2006. [27] Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1):1929?1958, 2014. [28] Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. Tech. Rep. CNS-TR-2011-001, 2011. 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Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013. 11
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Balancing information exposure in social networks Kiran Garimella Aalto University & HIIT Helsinki, Finland [email protected] Aristides Gionis Aalto University & HIIT Helsinki, Finland [email protected] Nikos Parotsidis University of Rome Tor Vergata Rome, Italy [email protected] Nikolaj Tatti Aalto University & HIIT Helsinki, Finland [email protected] Abstract Social media has brought a revolution on how people are consuming news. Beyond the undoubtedly large number of advantages brought by social-media platforms, a point of criticism has been the creation of echo chambers and filter bubbles, caused by social homophily and algorithmic personalization. In this paper we address the problem of balancing the information exposure in a social network. We assume that two opposing campaigns (or viewpoints) are present in the network, and that network nodes have different preferences towards these campaigns. Our goal is to find two sets of nodes to employ in the respective campaigns, so that the overall information exposure for the two campaigns is balanced. We formally define the problem, characterize its hardness, develop approximation algorithms, and present experimental evaluation results. Our model is inspired by the literature on influence maximization, but there are significant differences from the standard model. First, balance of information exposure is modeled by a symmetric difference function, which is neither monotone nor submodular, and thus, not amenable to existing approaches. Second, while previous papers consider a setting with selfish agents and provide bounds on bestresponse strategies (i.e., move of the last player), we consider a setting with a centralized agent and provide bounds for a global objective function. 1 Introduction Social-media platforms have revolutionized many aspects of human culture, among others, the way people are exposed to information. A recent survey estimates that 62% of adults in the US get their news on social media [15]. Despite providing many desirable features, such as, searching, personalization, and recommendations, one point of criticism is that social media amplify echo chambers and filter bubbles: users get less exposure to conflicting viewpoints and are isolated in their own informational bubble. This phenomenon is contributed to social homophily and algorithmic personalization, and is more acute for controversial topics [9, 12, 14]. In this paper we address the problem of reducing the filter-bubble effect by balancing information exposure among users. We consider social-media discussions around a topic that are characterized by two or more conflicting viewpoints. Let us refer to these viewpoints as campaigns. Our approach follows the popular paradigm of influence propagation [18]: we want to select a small number of seed users for each campaign so as to maximize the number of users who are exposed to both campaigns. In contrast to existing work on competitive viral marketing, we do not consider the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. problem of finding an optimal selfish strategy for each campaign separately. Instead we consider a centralized agent responsible for balancing information exposure for the two campaigns Consider the following motivating examples. Example 1: Social-media companies have been called to act as arbiters so as to prevent ideological isolation and polarization in the society. The motivation for companies to assume this role could be for improving their public image or due to legislation.1 Consider a controversial topic being discussed in social-media platform X, which has led to polarization and filter bubbles. As part of a new filter-bubble bursting service, platform X would like to disseminate two high-quality and thought-provoking dueling op-eds, articles, one for each side, which present the arguments of the other side in a fair manner. Assume that X is interested in following a viral-marketing approach. Which users should X target, for each of the two articles, so that people in the network are informed in the most balanced way? Example 2: Government organization Y is initiating a program to help assimilate foreigners who have newly arrived in the country. Part of the initiative focuses on bringing the communities of foreigners and locals closer in social media. Organization Y is interested in identifying individuals who can help spreading news of one community into the other. From the technical standpoint, we consider the following problem setting: We assume that information is propagated in the network according to the independent-cascade model [18]. We assume that there are two opposing campaigns, and for each one there is a set of initial seed nodes, I1 and I2 , which are not necessarily distinct. Furthermore, we assume that the users in the network are exposed to information about campaign i via diffusion from the set of seed nodes Ii . The diffusion in the network occurs according to some information-propagation model. The objective is to recruit two additional sets of seed nodes, S1 and S2 , for the two campaigns, with |S1 | + |S2 | ? k, for a given budget k, so as to maximize the expected number of balanced users, i.e., the users who are exposed to information from both campaigns (or from none). We show that the problem of balancing the information exposure is NP-hard. We develop different approximation algorithms for the different settings we consider, as well as heuristic variants of the proposed algorithm. We experimentally evaluate our methods, on several real-world datasets. Although our approach is inspired by the large body of work on information propagation, and resembles previous problem formulations for competitive viral marketing, there are significant differences. In particular: ? This is the first paper to address the problem of balancing information exposure and breaking filter bubbles, using the information-propagation methodology. ? The objective function that best suits our problem setting is related to the size of the symmetric difference of users exposed to the two campaigns. This is in contrast to previous settings that consider functions related to the size of the coverage of the campaigns. ? As a technical consequence of the previous point, our objective function is neither monotone nor submodular making our problem more challenging. Yet we are able to analyze the problem structure and provide algorithms with approximation guarantees. ? While most previous papers consider selfish agents, and provide bounds on best-response strategies (i.e., move of the last player), we consider a centralized setting and provide bounds for a global objective function. Omitted proofs, figures, and tables are provided as supplementary material. Moreover, our datasets and implementations are publicly available.2 2 Related Work Detecting and breaking filter bubbles. Several studies have observed that users in online social networks prefer to associate with like-minded individuals and consume agreeable content. This phenomenon leads to filter bubbles, echo chambers [25], and to online polarization [1, 9, 12, 22]. 1 2 For instance, Germany is now fining Facebook for the spread of fake news. https://users.ics.aalto.fi/kiran/BalanceExposure/ 2 Once these filter bubbles are detected, the next step is to try to overcome them. One way to achieve this is by making recommendations to individuals of opposing viewpoints. This idea has been explored, in different ways, by a number of studies in the literature [13, 19]. However, previous studies address the problem of breaking filter bubbles by the means of content recommendation. To the best of our knowledge, this is the first paper that considers an information diffusion approach. Information diffusion. Following a large body of work, we model diffusion using the independentcascade model [18]. In the basic model a single item propagates in the network. An extension is when multiple items propagate simultaneously. All works that study optimization problems in the case of multiple items, consider that items compete for being adopted by users. In other words, every user adopts at most one of the existing items and participates in at most one cascade. Myers and Leskovec [23] argue that spreading processes may either cooperate or compete. Competing contagions decrease each other?s probability of diffusion, while cooperating ones help each other in being adopted. They propose a model that quantifies how different spreading cascades interact with each other. Carnes et al. [7] propose two models for competitive diffusion. Subsequently, several other models have been proposed [4, 10, 11, 17, 21, 27, 28]. Most of the work on competitive information diffusion consider the problem of selecting the best k seeds for one campaign, for a given objective, in the presence of competing campaigns [3, 6]. Bharathi et al. [3] show that, if all campaigns but one have fixed sets of seeds, the problem for selecting the seeds for the last player is submodular, and thus, obtain an approximation algorithm for the strategy of the last player. Game theoretic aspects of competitive cascades in social networks, including the investigation of conditions for the existence of Nash equilibrium, have also been studied [2, 16, 26]. The work that is most related to ours, in the sense of considering a centralized authority, is the one by Borodin et al. [5]. They study the problem where multiple campaigns wish to maximize their influence by selecting a set of seeds with bounded cardinality. They propose a centralized mechanism to allocate sets of seeds (possibly overlapping) to the campaigns so as to maximize the social welfare, defined as the sum of the individual?s selfish objective functions. One can choose any objective functions as long as it is submodular and non-decreasing. Under this assumption they provide strategyproof (truthful) algorithms that offer guarantees on the social welfare. Their framework applies for several competitive influence models. In our case, the number of balanced users is not submodular, and so we do not have any approximation guarantees. Nevertheless, we can use this framework as a heuristic baseline, which we do in the experimental section. 3 Problem Definition Preliminaries: We start with a directed graph G = (V, E, p1 , p2 ) representing a social network. We assume that there are two distinct campaigns that propagate through the network. Each edge e = (u, v) ? E is assigned two probabilities, p1 (e) and p2 (e), representing the probability that a post from vertex u will propagate (e.g., it will be reposted) to vertex v in the respective campaigns. Cascade model: We assume that information on the two campaigns propagates in the network following the independent-cascade model [18]. For instance, consider the first campaign (the process for the second campaign is analogous): we assume that there exists a set of seeds I1 from which the process begins. Propagation proceeds in rounds. At each round, there exists a set of active vertices A1 (initially, A1 = I1 ), where each vertex u ? A1 attempts to activate each vertex v ? / A1 , such that (u, v) ? E, with probability p1 (u, v). If the propagation attempt from a vertex u to a vertex v is successful, we say that v propagates the first campaign. At the end of each round, A1 is set to be the set of vertices that propagated the campaign during the current round. Given a seed set S, we write r1 (S) and r2 (S) for the vertices that are reached from S using the aforementioned cascade process, for the respective campaign. Note that since this process is random, both r1 (S) and r2 (S) are random variables. Computing the expected number of active vertices is a #P-hard problem [8], however, we can approximate it within an arbitrary small factor , with high probability, via Monte-Carlo simulations. Due to this obstacle, all approximation algorithms that evaluate an objective function over diffusion processes reduce their approximation by an additive . Throughout this work we avoid repeating this fact for the sake of simplicity of the notation. 3 Heterogeneous vs. correlated propagations: We also need to specify how the propagation on the two campaigns interact with each other. We consider two settings: In the first setting, we assume that the campaign messages propagate independently of each other. Given an edge e = (u, v), the vertex v is activated on the first campaign with probability p1 (e), given that vertex u is activated on the first campaign. Similarly, v is activated on the second campaign with probability p2 (e), given that u is activated on the second campaign. We refer to this setting as heterogeneous.3 In the second setting we assume that p1 (e) = p2 (e), for each edge e. We further assume that the coin flips for the propagation of the two campaigns are totally correlated. Namely, consider an edge e = (u, v), where u is reached by either or both campaigns. Then with probability p1 (e), any campaign that has reached u, will also reach v. We refer to this second setting as correlated. Note that in both settings, a vertex may be active by none, either, or both campaigns. This is in contrast to most existing work in competitive viral marketing, where it is assumed that a vertex can be activated by at most one campaign. The intuition is that in our setting activation means merely passing a message or posting an article, and it does not imply full commitment to the campaign. We also note that the heterogeneous setting is more realistic than the correlated, however, we also study the correlated model as it is mathematically simpler. Problem definition: We are now ready to state our problem for balancing information exposure (BALANCE). Given a directed graph, initial seed sets for both campaigns and a budget, we ask to find additional seeds that would balance the vertices. More formally: Problem 3.1 (BALANCE). Let G = (V, E, p1 , p2 ) be a directed graph, and two sets I1 and I2 of initial seeds of the two campaigns. Assume that we are given a budget k. Find two sets S1 and S2 , where |S1 | + |S2 | ? k maximizing ?(S1 , S2 ) = E[|V \ (r1 (I1 ? S1 ) 4 r2 (I2 ? S2 ))|] . The objective function ?(S1 , S2 ) is the expected number of vertices that are either reached by both campaigns or remain oblivious to both campaigns. Problem 3.1 is defined for both settings, heterogeneous and correlated. When we need to make explicit the underlying setting we refer to the respective problems by BALANCE -H and BALANCE -C. When referring to BALANCE -H, we denote the objective by ?H . Similarly, when referring to BALANCE -C, we write ?C . We drop the indices, when we are referring to both models simultaneously. Computational complexity: As expected, the optimization problem BALANCE turns out to be NP-hard for both settings, heterogeneous and correlated. A straightforward way to prove it is by setting I2 = V , so the problems reduce to standard influence maximization. However, we provide a stronger result. Note that instead of maximizing balanced vertices we can equivalently minimize the imbalanced vertices. However, this turns to be a more difficult problem. Proposition 1. Assume a graph G = (V, E, p1 , p2 ) with two sets I1 and I2 and a budget k. It is an NP-hard problem to decide whether there are sets S1 and S2 such that |S1 | + |S2 | ? k and E[|r1 (I1 ? S1 ) 4 r2 (I2 ? S2 )|] = 0. This result holds for both models, even when p1 = p2 = 1. This result implies that the minimization version of the problem is NP-hard, and there is no algorithm with multiplicative approximation guarantee. It also implies that BALANCE -H and BALANCE -C are also NP-hard. However, we will see later that we can obtain approximation guarantees for these maximization problems. 4 Greedy algorithms yielding approximation guarantees In this section we propose three greedy algorithms. The first algorithm yields an approximation guarantee of (1 ? 1/e)/2 for both models. The remaining two algorithms yield a guarantee for the correlated model only. Decomposing the objective: Recall that the objective function of the BALANCE problem is ?(S1 , S2 ). In order to show that this function admits an approximation guarantee, we decompose it into two components. To do that, assume that we are given initial seeds I1 and I2 , and let us write 3 Although independent is probably a better term than heterogeneous, we adopt the latter to avoid any confusion with the independent-cascade model. 4 X = r1 (I1 ) ? r2 (I2 ), Y = V \ X. Here X are vertices reached by any initial seed in the two campaigns and Y are the vertices that are not reached at all. Note that X and Y are random variables. Since X and Y partition V , we can decompose the score ?(S1 , S2 ) as ?(S1 , S2 ) = ?(S1 , S2 ) + ?(S1 , S2 ), where ?(S1 , S2 ) = E[|X \ (r1 (I1 ? S1 ) 4 r2 (I2 ? S2 ))|] , ?(S1 , S2 ) = E[|Y \ (r1 (I1 ? S1 ) 4 r2 (I2 ? S2 ))|] . We first show that ?(S1 , S2 ) is monotone and submodular. It is well-known that for maximizing a function that has these two properties under a size constraint, the greedy algorithm computes an (1 ? 1e ) approximate solution [24]. Lemma 2. ?(S1 , S2 ) is monotone and submodular. We are ready to discuss our algorithms. Algorithm 0: ignore ?. Our first algorithm is very simple: instead of maximizing ?, we maximize ?, i.e., we ignore any vertices that are made imbalanced during the process. Since ? is submodular and monotone we can use the greedy algorithm. If we then compare the obtained result with the empty solution, we get the promised approximation guarantee. We refer to this algorithm as Cover. Proposition 3. Let hS1? , S2? i be the optimal solution maximizing ?. Let hS1 , S2 i be the solution obtained via greedy algorithm maximizing ?. Then max{?(S1 , S2 ), ?(?, ?)} ? 1 ? 1/e ?(S1? , S2? ). 2 Algorithm 1: force common seeds. Ignoring the ? term may prove costly as it is possible to introduce a lot of new imbalanced vertices. The idea behind the second algorithm is to force ? = 0. We do this by either adding the same seeds to both campaigns, or adding a seed that is covered by an opposing campaign. This algorithm has guarantees only in the correlated setting with even budget k but in practice we can use the algorithm also for the heterogeneous setting. We refer to this algorithm as Common and the pseudo-code is given in Algorithm 1. Algorithm 1: Common, greedy algorithm that only adds common seeds 1 S1 ? S2 ? ?; 2 while |S1 | + |S2 | ? k do 3 c ? arg maxc ?(S1 ? {c} , S2 ? {c}); 4 s1 ? arg maxs?I1 ?(S1 , S2 ? {s}); 5 s2 ? arg maxs?I2 ?(S1 ? {s} , S2 ); 6 add the best option among hc, ci, h?, s1 i, hs2 , ?i to hS1 , S2 i while respecting the budget. We first show in the following lemma that adding common seeds may halve the score, in the worst case. Then, we use this lemma to prove the approximation guarantee Lemma 4. Let hS1 , S2 i be a solution to BALANCE -C, with an even budget k. There exists a solution hS10 , S20 i with S10 = S20 such that ?C (S10 , S20 ) ? ?C (S1 , S2 )/2. It is easy to see that the greedy algorithm satisfies the conditions of the following proposition. Proposition 5. Assume an iterative algorithm where at each iteration, we add one or two vertices to our solution until our constraints are met. Let S1i , S2i be the sets after the i-th iteration, S10 = S20 = ?. Let ?i = ?C (S1i , S2i ) be the cost after the i-th iteration. Assume that ?i ? ?i?1 . Assume further that for i = 1, . . . , k/2 it holds that ?i ? ?C (S1i?1 ? {c} , S2i?1 ? {c}). Then the algorithm yields (1 ? 1/e)/2 approximation. Algorithm 2: common seeds as baseline. Not allowing new imbalanced vertices may prove to be too restrictive. We can relax this condition by allowing new imbalanced vertices as long as the gain is at least as good as adding a common seed. We refer to this algorithm as Hedge and the pseudo-code is given in Algorithm 2. The approximation guarantee for this algorithm?in the correlated setting and with even budget?follows immediately from Proposition 5 as it also satisfies the conditions. 5 Algorithm 2: Hedge, greedy algorithm, where each step is as good as adding the best common seed 1 S1 ? S2 ? ?; 2 while |S1 | + |S2 | ? k do 3 c ? arg maxc ?(S1 ? {c} , S2 ? {c}); 4 s1 ? arg maxs ?(S1 , S2 ? {s}); 5 s2 ? arg maxs ?(S1 ? {s} , S2 ); 6 add the best option among hc, ci, h?, s1 i, hs2 , ?i, hs2 , s1 i, to hS1 , S2 i while respecting the budget. 5 Experimental evaluation In this section, we evaluate the effectiveness of our algorithms on real-world datasets. We focus on (i) analyzing the quality of the seeds picked by our algorithms in comparison to other heuristic approaches and baselines; (ii) analyzing the efficiency and the scalability of our algorithms; and (iii) providing anecdotal examples of the obtained results. Although we setup our experiments in order to mimic social behavior, we note that fully realistic experiments would entail the ability to intervene in the network, select seeds, and observe the resulting cascades. This, however, is well beyond our capacity and the scope of the paper. In all experiments we set k to range between 5 and 50 with a step of 5. We report averages over 1 000 random simulations of the cascade process. Datasets: To evaluate the effectiveness of our algorithms, we run experiments on real-world data collected from twitter. Let G = (V, E) be the twitter follower graph. A directed edge (u, v) ? E indicates that user v follows u; note that the edge direction indicates the ?information flow? from a user to their followers. We define a cascade GX = (X, EX ) as a graph over the set of users X ? V who have retweeted at least one hashtag related to a topic (e.g., US elections). An edge (u, v) ? EX ? E indicates that v retweeted u. We use datasets from six topics with opposing viewpoints, covering politics (US-elections, Brexit, ObamaCare), policy (Abortion, Fracking), and lifestyle (iPhone, focusing on iPhone vs. Samsung). All datasets are collected by filtering the twitter streaming API (1% random sample of all tweets) for a set of keywords used in previous work [20]. For each dataset, we identify two sides (indicating the two view-points) on the retweet graph, which has been shown to capture best the two opposing sides of a controversy [12]. Details on the statistics of the dataset can be found at the supplementary material. After building the graphs, we need to estimate the diffusion probabilities for the heterogeneous and correlated models. Note that the estimation of the diffusion probabilities is orthogonal to our contribution in this paper. For the sake of concreteness we have used the approach described below. One could use a different, more advanced, method; our methods are still applicable. Let q1 (v) and q2 (v) be an a priori probability of a user v retweeting sides 1 and 2, respectively. These are measured from the data by looking at how often a user retweets content from users and keywords that are discriminative of each side. For example, for US-elections, the discriminative users and keywords for side Hillary would be @hillaryclinton and #imwither, and for Trump, @realdonaldtrump and #makeamericagreatagain. The probability that user v retweets user u (cascade probability) is then defined as   R(u, v) + 1 pi (u, v) = ? qi (v) + (1 ? ?) , i = 1, 2, R(v) + 2 where R(u, v) is the number of times v has retweeted u, and R(v) is the total number of retweets of user v. The cascade probabilities pi capture the fact that users retweet content if they see it from their friends (term R(u,v)+1 R(v)+2 ) or based on their own biases (term qi (v)). The additive terms in the numerator and denominator provide an additive smoothing by Laplace?s rule of succession. We set the value of ? to 0.8 for the heterogeneous setting. For ? = 0 the edge probabilities become equal for the two campaigns, which is our assumption for the correlated setting. 6 ObamaCare US-elections 600 500 400 symm. diff. 2 500 symm. diff. symm. diff. iPhone 2 000 1 500 Cover Hedge Common Greedy 1 500 1 000 500 300 10 20 30 budget k 40 50 10 iPhone 40 50 10 40 20 40 50 2 000 400 symm. diff. symm. diff. 60 20 30 budget k US-elections ObamaCare 80 symm. diff. 20 30 budget k 200 0 1 500 1 000 500 0 10 20 30 budget k 40 50 10 20 30 budget k 40 50 10 20 30 budget k 40 50 Figure 1: Expected symmetric difference n ? ?C as a function of the budget k. Top row, heterogeneous model, bottom row: Correlated model. Low values are better. Baselines. We use 5 different baselines. The first baseline, BBLO, is an adaptation of the framework by Borodin et al. [5]. This framework requires an objective function as input, and here we use our objective function ?. The framework works as follows: The two campaigns are given a budget k/2 on the number of seeds that they can select. At each round, we select a vertex v for S1 , optimizing ?(S1 ? {v} , S2 ), and a vertex w for S2 , optimizing ?(S1 , S2 ? {w}). We should stress that the theoretical guarantees by [5] do not apply because our objective is not submodular. The next two heuristics add a set of common seeds to both campaigns. We run a greedy algorithm for campaign i = 1, 2 to select the set Si0 with the `  k vertices Pi that optimizes the function ri (Si0 ? Ii ). We consider two heuristics: Union selects S1 and S2 to be equal to the k/2 first distinct vertices in S10 ? S20 while Intersection selects S1 and S2 to be equal to k/2 first vertices in S10 ? S20 . Here the vertices are ordered based on their discovery time. Finally, HighDegree selects the vertices with the largest number of followers and assigns them alternately to the two cascades; and Random assigns k/2 random seeds to each campaign. In addition to the baselines, we also consider a simple greedy algorithm Greedy. The difference between Cover and Greedy is that, in each iteration, Cover adds the seed that maximizes ?, while Greedy adds the seed that maximizes ?. We can only show an approximation guarantee for Cover but Greedy is a more intuitive approach, and we use it as a heuristic. Comparison of the algorithms. We start by evaluating the quality of the sets of seeds computed by our algorithms, i.e., the number of equally-informed vertices. Heterogeneous setting. We consider first the case of heterogeneous networks. The results for the selected datasets are shown in Figure 1. Full results are shown in the supplementary material. Instead of plotting ?, we plot the number of the remaining unbalanced vertices, n??, as it makes the results easier to distinguish; i.e., an optimal solution achieves the value 0. The first observation is that the approximation algorithm Cover performs, in general, worse than the other two heuristics. This is due to the fact that Cover does not optimize directly the objective function. Hedge performs better than Greedy, in general, since it examines additional choices to select. The only deviation from this picture is for the US-elections dataset, where the Greedy outperforms Hedge by a small factor. This may due to the fact that while Hedge has more options, it allocates seeds in batches of two. Correlated setting. Next we consider correlated networks. We experiment with the three approximation algorithms Cover, Common, Hedge, and the heuristic Greedy. The results are shown in Figure 1. Cover performs again the worst since it is the only method that introduces new unbalanced vertices without caring about their cardinality. Its variant, Greedy, performs much better in practice even though it does not provide an approximation guarantee. The algorithms Common, Greedy, and Hedge perform very similar to each other without a clear winner. 7 Heterogeneous ?103 6 symm. diff. symm. diff. 4 4 2 0 Abortion Brexit Fracking iPhoneObamaCare 3 2 1 0 US Hedge BBLO Intersection Union HighDegree Random ?103 Correlated Abortion Brexit Fracking iPhoneObamaCare US Figure 2: Expected symm. diff. n ? ? of Hedge and the baselines. k = 20. Low values are better. Comparison with baselines. Our next step is to compare against the baselines. For simplicity, we focus on k = 20; the overall conclucions hold for other budgets. The results for Hedge versus the five baselines are shown in Figure 2. From the results we see that BBLO is the best competitor: its scores are the closest to Hedge, and it receives slightly better scores in 3 out of 12 cases. The competitiveness is not surprising because we specifically set the objective function in BBLO to be ?(S1 , S2 ). The Intersection and Union also perform well but are always worse than Hedge. Random is unpredictable but always worse than Hedge. In the case of heterogeneous networks, Hedge selects seeds that leave less unbalanced vertices, by a factor of two on average, compared to the seeds selected by the HighDegree method. For correlated networks, our method outperforms the two baselines by an order of magnitude. The actual values of this experiment can be found in the supplementary material. Running time. We proceed to evaluate the efficiency and the scalability of our algorithms. We observe that all algorithms have comparable running times and good scalability. More information can be found in the supplementary material. Use case with Fracking. We present a qualitative case-study analysis for the seeds selected by our algorithm. We highlight the Fracking dataset, even though we applied similar analysis to the other datasets as well (the results are given in the supplementary material of the paper). Recall that for each dataset we identify two sides with opposing views, and a set of initial seeds for each side (I1 and I2 ). We consider the users in the initial seeds I1 (side supporting fracking), and summarize the text of all their Twitter profile descriptions in a word cloud. The result, contains words that are used to emphasize the benefits of fracking (energy, oil, gas, etc.). We then draw a similar word cloud for the users identified by the Hedge algorithm as seed nodes in the sets S1 and S2 (k = 50). The result, contains a more balanced set of words, which includes many words used to underline the environmental dangers of fracking. We use word clouds as a qualitative case study to complement our quantitative results and to provide more intuition about our problem statement, rather than an alternative quantitative measure. 6 Conclusion We presented the first study of the problem of balancing information exposure in social networks using techniques from the area of information diffusion. Our approach has several novel aspects. In particular, we formulate our problem by seeking to optimize a symmetric difference function, which is neither monotone nor submodular, and thus, not amenable to existing approaches. Additionally, while previous studies consider a setting with selfish agents and provide bounds on best-response strategies (i.e., move of the last player), we consider a centralized setting and provide bounds for a global objective function. Our work provides several directions for future work. One interesting problem is to improve the approximation guarantee for the problem we define. Second, we would like to extend the problem definition for more than two campaigns and design approximation algorithms for that case. Finally, we believe that it is worth studying the BALANCE problem under complex diffusion models that capture more realistic social behavior in the presence of multiple campaigns. One such extension is to consider propagation probabilities on the edges that are dependent in the past behavior of the nodes with respect to the two campaigns, e.g., one could consider Hawkes processes [28]. Acknowledgments. This work has been supported by the Academy of Finland projects ?Nestor? (286211) and ?Agra? (313927), and the EC H2020 RIA project ?SoBigData? (654024). 8 References [1] L. A. Adamic and N. Glance. The political blogosphere and the 2004 us election: divided they blog. In LinkKDD, pages 36?43, 2005. [2] N. Alon, M. Feldman, A. D. Procaccia, and M. Tennenholtz. A note on competitive diffusion through social networks. IPL, 110(6):221?225, 2010. [3] S. 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SafetyNets: Verifiable Execution of Deep Neural Networks on an Untrusted Cloud Zahra Ghodsi, Tianyu Gu, Siddharth Garg New York University {zg451, tg1553, sg175}@nyu.edu Abstract Inference using deep neural networks is often outsourced to the cloud since it is a computationally demanding task. However, this raises a fundamental issue of trust. How can a client be sure that the cloud has performed inference correctly? A lazy cloud provider might use a simpler but less accurate model to reduce its own computational load, or worse, maliciously modify the inference results sent to the client. We propose SafetyNets, a framework that enables an untrusted server (the cloud) to provide a client with a short mathematical proof of the correctness of inference tasks that they perform on behalf of the client. Specifically, SafetyNets develops and implements a specialized interactive proof (IP) protocol for verifiable execution of a class of deep neural networks, i.e., those that can be represented as arithmetic circuits. Our empirical results on three- and four-layer deep neural networks demonstrate the run-time costs of SafetyNets for both the client and server are low. SafetyNets detects any incorrect computations of the neural network by the untrusted server with high probability, while achieving state-of-the-art accuracy on the MNIST digit recognition (99.4%) and TIMIT speech recognition tasks (75.22%). 1 Introduction Recent advances in deep learning have shown that multi-layer neural networks can achieve state-ofthe-art performance on a wide range of machine learning tasks. However, training and performing inference (using a trained neural network for predictions) can be computationally expensive. For this reason, several commercial vendors have begun offering ?machine learning as a service" (MLaaS) solutions that allow clients to outsource machine learning computations, both training and inference, to the cloud. While promising, the MLaaS model (and outsourced computing, in general) raises immediate security concerns, specifically relating to the integrity (or correctness) of computations performed by the cloud and the privacy of the client?s data [16]. This paper focuses on the former, i.e., the question of integrity. Specifically, how can a client perform inference using a deep neural network on an untrusted cloud, while obtaining strong assurance that the cloud has performed inference correctly? Indeed, there are compelling reasons for a client to be wary of a third-party cloud?s computations. For one, the cloud has a financial incentive to be ?lazy." A lazy cloud might use a simpler but less accurate model, for instance, a single-layer instead of a multi-layer neural network, to reduce its computational costs. Further the cloud could be compromised by malware that modifies the results sent back to the client with malicious intent. For instance, the cloud might always mis-classify a certain digit in a digit recognition task, or allow unauthorized access to certain users in a face recognition based authentication system. The security risks posed by cloud computing have spurred theoretical advances in the area of verifiable computing (VC) [21]. The idea is to enable a client to provably (and cheaply) verify that an untrusted 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. server has performed computations correctly. To do so, the server provides to the client (in addition to the result of computation) a mathematical proof of the correctness of the result. The client rejects, with high probability, any incorrectly computed results (or proofs) provided by the server, while always accepting correct results (and corresponding proofs) 1 . VC techniques aim for the following desirable properties: the size of the proof should be small, the client?s verification effort must be lower than performing the computation locally, and the server?s effort in generating proofs should not be too high. The advantage of proof-based VC is that it provides unconditional, mathematical guarantees on the integrity of computation performed by the server. Alternative solutions for verifiable execution require the client to make trust assumptions that are hard for the client to independently verify. Trusted platform modules [7], for instance, require the client to place trust on the hardware manufacturer, and assume that the hardware is tamper-proof. Audits based on the server?s execution time [15] require precise knowledge of the server?s hardware configuration and assume, for instance, that the server is not over-clocked. The work in this paper leverages powerful VC techniques referred to as indigit=4 5 teractive proof (IP) systems [5, 9, 18, Random challenge 1 19]. An IP system consists of two enChallenge Compute response 1 Response tities, a prover (P), i.e., the untrusted Verify server, and a verifier (V), i.e., the challenge n Random client. The framework is illustrated in Compute Challenge response n Figure 1. The verifier sends the prover Response Reject an input x, say a batch of test images, Reject and asks the prover to compute a funcFigure 1: High-level overview of the SafetyNets IP protocol. tion y = f (x). In our setting, f (.) is In this example, an untrusted server intentionally changes a trained multi-layer neural network the classification output from 4 to 5. that is known to both the verifier and prover, and y is the neural network?s classification output for each image in the batch. The prover performs the computation and sends the verifier a purported result y 0 (which is not equal to y if the prover cheats). The verifier and prover then engage in n rounds of interaction. In each round, the verifier sends the prover a randomly picked challenge, and the prover provides a response based on the IP protocol. The verifier accepts that y 0 is indeed equal to f (x) if it is satisfied with the prover?s response in each round, and rejects otherwise. Client (verifier) Input Image Execute Neural Network Untrusted Server (prover) ... A major criticism of IP systems (and, indeed, all existing VC techniques) when used for verifying general-purpose computations is that the prover?s overheads are large, often orders of magnitude more than just computing f (x) [21]. Recently, however, Thaler [18] showed that certain types of computations admit IP protocols with highly efficient verifiers and provers, which lays the foundations for the specialized IP protocols for deep neural networks that we develop in this paper. Paper Contributions. This paper introduces SafetyNets, a new (and to the best of our knowledge, the first) approach for verifiable execution of deep neural networks on untrusted clouds. Specifically, SafetyNets composes a new, specialized IP protocol for the neural network?s activation layers with Thaler?s IP protocol for matrix multiplication to achieve end-to-end verifiability, dramatically reducing the bandwidth costs versus a naive solution that verifies the execution of each layer of the neural network separately. SafetyNets applies to a certain class of neural networks that can be represented as arithmetic circuits that perform computations over finite fields (i.e., integers modulo a large prime p). Our implementation of SafetyNets addresses several practical challenges in this context, including the choice of the prime p, its relationship to accuracy of the neural network, and to the verifier and prover run-times. Empirical evaluations on the MNIST digit recognition and TIMIT speech recognition tasks illustrate that SafetyNets enables practical, low-cost verifiable outsourcing of deep neural network execution without compromising classification accuracy. Specifically, the client?s execution time is 8?-80? lower than executing the network locally, the server?s overhead in generating proofs is less than 5%, and the client/server exchange less than 8 KBytes of data during the IP protocol. SafetyNets? security 1 Note that the SafetyNets is not intended to and cannot catch any inherent mis-classifications due to the model itself, only those that result from incorrect computations of the model by the server. 2 guarantees ensure that a client can detect any incorrect computations performed by a malicious server with probability vanishingly close to 1. At the same time, SafetyNets achieves state-of-the-art classification accuracies of 99.4% and 75.22% on the MNIST and TIMIT datasets, respectively. 2 Background In this section, we begin by reviewing necessary background on IP systems, and then describe the restricted class of neural networks (those that can be represented as arithmetic circuits) that SafetyNets handles. 2.1 Interactive Proof Systems Existing IP systems proposed in literature [5, 9, 18, 19] use, at their heart, a protocol referred to as the sum-check protocol [13] that we describe here in some detail, and then discuss its applicability in verifying general-purpose computations expressed as arithmetic circuits. Sum-check Protocol Consider a d-degree n-variate polynomial g(x1 , x2 , . . . , xn ), where each variable xi ? Fp (Fp is the set of all natural numbers between zero and p ? 1, for a given prime p) and g : Fnp ? Fp . The prover P seeks to prove the following claim: X X X y= ... g(x1 , x2 , . . . , xn ) (1) x1 ?{0,1} x2 ?{0,1} xn ?{0,1} that is, the sum of g evaluated at 2n points is y. P and V now engage in a sum-check protocol to verify this claim. In the first round of the protocol, P sends the following unidimensional polynomial X X X h(x1 ) = ... g(x1 , x2 , . . . , xn ) (2) x2 ?{0,1} x3 ?{0,1} xn ?{0,1} to V in the form of its coefficients. V checks if h(0) + h(1) = y. If yes, it proceeds, otherwise it rejects P?s claim. Next, V picks a random value q1 ? Fp and evaluates h(q1 ) which, based on Equation 2, yields a new claim: X X X h(q1 ) = ... g(q1 , x2 , . . . , xn ). (3) x2 ?{0,1} x3 ?{0,1} xn ?{0,1} V now recursively calls the sum-check protocol to verify this new claim. By the final round of the sum-check protocol, P returns the value g(q1 , q2 , . . . , qn ) and the V checks if this value is correct by evaluating the polynomial by itself. If so, V accepts the original claim in Equation 1, otherwise it rejects the claim. Lemma 2.1. [2] V rejects an incorrect claim by P with probability greater than (1 ? ) where  = nd p is referred to as the soundness error. IPs for Verifying Arithmetic Circuits In their seminal work, Goldwasser et al. [9] demonstrated how sum-check can be used to verify the execution of arithmetic circuits using an IP protocol now referred to as GKR. An arithmetic circuit is a directed acyclic graph of computation over elements of a finite field Fp in which each node can perform either addition or multiplication operations (modulo p). While we refer the reader to [9] for further details of GKR, one important aspect of the protocol bears mention. GKR organizes nodes of an arithmetic circuit into layers; starting with the circuit inputs, the outputs of one layer feed the inputs of the next. The GKR proof protocol operates backwards from the circuit outputs to its inputs. Specifically, GKR uses sum-check to reduce the prover?s assertion about the circuit output into an assertion about the inputs of the output layer. This assertion is then reduced to an assertion about the inputs of the penultimate layer, and so on. The protocol continues iteratively till the verifier is left with an assertion about the circuit inputs, which it checks on its own. The layered nature of GKR?s prover aligns almost perfectly with the structure of a multi-layer neural network and motivates the use of an IP system based on GKR for SafetyNets. 3 2.2 Neural Networks as Arithmetic Circuits As mentioned before, SafetyNets applies to neural networks that can be expressed as arithmetic circuits. This requirement places the following restrictions on the neural network layers. Quadratic Activations The activation functions in SafetyNets must be polynomials with integer coefficients (or, more precisely, coefficients in the field Fp ). The simplest of these is the element-wise quadratic activation function whose output is simply the square of its input. Other commonly used activation functions such as ReLU, sigmoid or softmax activations are precluded, except in the final output layer. Prior work has shown that neural networks with quadratic activations have the same representation power as networks with threshold activations and can be efficiently trained [6, 12]. Sum Pooling Pooling layers are commonly used to reduce the network size, to prevent overfitting and provide translation invariance. SafetyNets uses sum pooling, wherein the output of the pooling layer is the sum of activations in each local region. However, techniques such as max pooling [10] and stochastic pooling [22] are not supported since max and divisions operations are not easily represented as arithmetic circuits. Finite Field Computations SafetyNets supports computations over elements of the field Fp , that p?1 is, integers in the range {? p?1 2 , . . . , 0, . . . , 2 }. The inputs, weights and all intermediate values computed in the network must lie in this range. Note that due to the use of quadratic activations and sum pooling, the values in the network can become quite large. In practice, we will pick large primes to support these large values. We note that this restriction applies to the inference phase only; the network can be trained with floating point inputs and weights. The inputs and weights are then re-scaled and quantized, as explained in Section 3.3, to finite field elements. We note that the restrictions above are shared by a recently proposed technique, CryptoNets [8], that seeks to perform neural network based inference on encrypted inputs so as to guarantee data privacy. However, Cryptonets does not guarantee integrity and compared to SafetyNets, incurs high costs for both the client and server (see Section 4.3 for a comparison). Conversely, SafetyNets is targeted towards applications where integrity is critical, but does not provide privacy. 2.3 Mathematical Model An L layer neural network with the constraints discussed above can be modeled, without loss of generality, as follows. The input to the network is x ? Fnp 0 ?b , where n0 is the dimension of each input and b is the batch size. Layer i ? [1, L] has ni output neurons2 , and is specified using a weight n ?n matrix wi?1 ? Fp i i?1 , and biases bi?1 ? Fnp i . The output of Layer i ? [1, L], yi ? Fnp i ?b is: yi = ?quad (wi?1 .yi?1 + bi?1 1T ) ?i ? [1, L ? 1]; yL = ?out (wL?1 .yL?1 + bL?1 1T ), (4) where ?quad (.) is the quadratic activation function, ?out (.) is the activation function of the output layer, and 1 ? Fbp is the vector of all ones. We will typically use softmax activations in the output n ?b layer. We will also find it convenient to introduce the variable zi ? Fp i+1 defined as zi = wi .yi + bi 1T ?i ? [0, L ? 1]. (5) The model captures both fully connected and convolutional layers; in the latter case the weight matrix is sparse. Further, without loss of generality, all successive linear transformations in a layer, for instance sum pooling followed by convolutions, are represented using a single weight matrix. With this model in place, the goal of SafetyNets is to enable the client to verify that yL was correctly computed by the server. We note that as in prior work [19], SafetyNets amortizes the prover and verifier costs over batches of inputs. If the server incorrectly computes the output corresponding to any input in a batch, the verifier rejects the entire batch of computations. 2 The 0th layer is defined to be input layer and thus y0 = x. 4 3 SafetyNets We now describe the design and implementation of our end-to-end IP protocol for verifying execution of deep networks. The SafetyNets protocol is a specialized form of the IP protocols developed by Thaler [18] for verifying ?regular" arithmetic circuits, that themselves specialize and refine prior work [5]. The starting point for the protocol is a polynomial representation of the network?s inputs and parameters, referred to as a multilinear extension. Multilinear Extensions Consider a matrix w ? Fn?n . Each row and column of w can be p referenced using m = log2 (n) bits, and consequently one can represent w as a function W : {0, 1}m ? {0, 1}m ? Fp . That is, given Boolean vectors t, u ? {0, 1}m , the function W (t, u) returns the element of w at the row and column specified by Boolean vectors t and u, respectively. m ? : Fm A multi-linear extension of W is a polynomial function W p ? Fp ? Fp that has the following m ? two properties: (1) given vectors t, u ? Fp such that W (t, u) = W (t, u) for all points on the unit ? has degree 1 in each of its variables. In the hyper-cube, that is, for all t, u ? {0, 1}m ; and (2) W ? ? ? i to refer to multi-linear extensions of remainder of this discussion, we will use X, Yi and Z?i and W x, yi , zi , and wi , respectively, for i ? [1, L]. We will also assume, for clarity of exposition, that the biases, bi are zero for all layers. The supplementary draft describes how biases are incorporated. Consistent with the IP literature, the description of our protocol refers to the client as the verifier and the server as the prover. Protocol Overview The verifier seeks to check the result yL provided by the prover corresponding to input x. Note that yL is the output of the final activation layer which, as discussed in Section 2.2, is the only layer that does not use quadratic activations, and is hence not amenable to an IP. Instead, in SafetyNets, the prover computes and sends zL?1 (the input of the final activation layer) as a result to the verifier. zL?1 has the same dimensions as yL and therefore this refinement has no impact on the server to client bandwidth. Furthermore, the verifier can easily compute yL = ?out (zL?1 ) locally. Now, the verifier needs to check whether the prover computed zL?1 correctly. As noted by Vu et al. [19], this check can be replaced by a check on whether the multilinear extension of zL?1 is correctly computed at a randomly picked point in the field, with minimal impact on the soundness log(n ) log(b) error. That is, the verifier picks two vectors, qL?1 ? Fp L and rL?1 ? Fp at random, ? evaluates ZL?1 (qL?1 , rL?1 ), and checks whether it was correctly computed using a specialized sum-check protocol for matrix multiplication due to Thaler [18] (described in Section 3.1). Since zL?1 depends on wL?1 and yL?1 , sum-check yields assertions on the values of L?1 ) ? L?1 (qL?1 , sL?1 ) and Y?L?1 (sL?1 , rL?1 ), where sL?1 ? Flog(n W is another random vector p picked by the verifier during sum-check. ? L?1 (qL?1 , sL?1 ) is an assertion about the weight of the final layer. This is checked by the verifier W locally since the weights are known to both the prover and verifier. Finally, the verifier uses our specialized sum-check protocol for activation layers (described in Section 3.2) to reduce the assertion on Y?L?1 (sL?1 , rL?1 ) to an assertion on Z?L?2 (qL?2 , sL?2 ). The protocol repeats till it reaches the ? 0 , r0 ), the multilinear extension of the input x. The input layer and produces an assertion on X(s verifier checks this locally. If at any point in the protocol, the verifier?s checks fail, it rejects the prover?s computation. Next, we describe the sum-check protocols for matrix multiplication and activation that SafetyNets uses. 3.1 Sum-check for Matrix Multiplication Since zi = wi .yi (recall we assumed zero biases for clarity), we can check an assertion about the multilinear extension of zi evaluated at randomly picked points qi and ri by expressing Z?i (qi , ri ) as [18]: X ? i (qi , j).Y?i (j, ri ) Z?i (qi , ri ) = W (6) j?{0,1}log(ni ) 5 Note that Equation 6 has the same form as the sum-check problem in Equation 1. Consequently the sum-check protocol described in Section 2.1 can be used to verify this assertion. At the end of the ? i which it checks locally, and Y?i which is sum-check rounds, the verifier will have assertions on W checked using the sum-check protocol for quadratic activations described in Section 3.2. The prover run-time for running the sum-check protocol in layer i is O(ni (ni?1 + b)), the verifier?s run-time is O(ni ni?1 ) and the prover/verifier exchange 4 log(ni ) field elements. 3.2 Sum-check for Quadratic Activation In this step, we check an assertion about the output of quadratic activation layer i, Y?i (si , ri ), by writing it in terms of the input of the activation layer as follows: X ? i , j)I(r ? i , k)Z? 2 (j, k), Y?i (si , ri ) = I(s (7) i?1 j?{0,1}log(ni ) ,k?{0,1}log(b) ? .) is the multilinear extension of the identity matrix. Equation 7 can also be verified using where I(., the sum-check protocol, and yields an assertion about Z?i?1 , i.e., the inputs to the activation layer. This assertion is in turn checked using the protocol described in Section 3.1. The prover run-time for running the sum-check protocol in layer i is O(bni ), the verifier?s runtime is O(log(bni )) and the prover/verifier exchange 5 log(bni ) field elements. This completes the theoertical description of the SafetyNets specialized IP protocol. Lemma 3.1. The SafetyNets verifier rejects incorrect computations with probability greater than P 3b L i=0 ni (1 ? ) where  = is the soundness error. p In practice, with p = 261 ? 1 the soundness error < sizes. 3.3 1 230 for practical network parameters and batch Implementation The fact that SafetyNets operates only on elements in a finite field Fp during inference imposes a practical challenge. That is, how do we convert floating point inputs and weights from training into field elements, and how do we select the size of the field p? Let wi0 ? Rni?1 ?ni and b0i ? Rni be the floating point parameters obtained from training for each layer i ? [1, L]. We convert the weights to integers by multiplying with a constant ? > 1 and rounding, i.e., wi = b?wi0 e. We do the same for inputs with a scaling factor ?, i.e., x = b?x0 e. Then, i?1 i?1 to ensure that all values in the network scale isotropically, we must set bi = b?2 ? (2 +1) b0i e. While larger ? and ? values imply lower quantization errors, they also result in large values in the network, especially in the layers closer to the output. Similar empirical observations were made by the CryptoNets work [8]. To ensure accuracy the values in the network must lie in the range p?1 [? p?1 2 , 2 ]; this influences the choice of the prime p. On the other hand, we note that large primes increase the verifier and prover run-time because of the higher cost of performing modular additions and multiplications. As in prior works [5, 18, 19], we restrict our choice of p to Mersenne primes since they afford efficient modular arithmetic implementations, and specifically to the primes p = 261 ? 1 and p = 2127 ? 1. For a given p, we explore and different values of ? and ? and use the validation dataset to the pick the p?1 ones that maximize accuracy while ensuring that the values in the network lie within [? p?1 2 , 2 ]. 4 Empirical Evaluation In this section, we present empirical evidence to support our claim that SafetyNets enables low-cost verifiable execution of deep neural networks on untrusted clouds without compromising classification accuracy. 6 10 CNN-2-ReLU Train CNN-2-ReLU Test CNN-2-Quad Train CNN-2-Quad Test 1.5 1 80 CNN-2-ReLU Train CNN-2-ReLU Test CNN-2-Quad Train CNN-2-Quad Test 8 Error (%) Error (%) 2 6 4 FcNN-3-ReLU Train FcNN-3-ReLU Test FcNN-3-Quad Train FcNN-3-Quad Test 70 60 Error (%) 2.5 50 40 30 0.5 0 2 200 400 600 800 Time (s) (a) MNIST 1000 1200 0 20 0 200 400 600 800 Time (s) 1000 1200 (b) MNIST-Back-Rand 10 10000 20000 Time (s) 30000 40000 (c) TIMIT Figure 2: Evolution of training and test error for the MNIST, MNIST-Back-Rand and TIMIT tasks. 4.1 Setup Datasets We evaluated SafetyNets on three classifications tasks. (1) Handwritten digit recognition on the MNIST dataset, using 50,000 training, 10,000 validation and 10,000 test images. (2) A more challenging version of digit recognition, MNIST-Back-Rand, an artificial dataset generated by inserting a random background into MNIST image [1]. The dataset has 10,000 training, 2,000 validation and 50,000 test images. ZCA whitening is applied to the raw dataset before training and testing [4]. (3) Speech recognition on the TIMIT dataset, split into a training set with 462 speakers, a validation set with 144 speakers and a testing set with 24 speakers. The raw audio samples are pre-processed as described by [3]. Each example includes its preceding and succeeding 7 frames, resulting in a 1845-dimensional input in total. During testing, all labels are mapped to 39 classes [11] for evaluation. Neural Networks For the two MNIST tasks, we used a convolutional neural network same as [23] with 2 convolutional layers with 5 ? 5 filters, a stride of 1 and a mapcount of 16 and 32 for the first and second layer respectively. Each convolutional layer is followed by quadratic activations and 2 ? 2 sum pooling with a stride of 2. The fully connected layer uses softmax activation. We refer to this network as CNN-2-Quad. For TIMIT, we use a four layer network described by [3] with 3 hidden, fully connected layers with 2000 neurons and quadratic activations. The output layer is fully connected with 183 output neurons and softmax activation. We refer to this network as FcNN-3-Quad. Since quadratic activations are not commonly used, we compare the performance of CNN-2-Quad and FcNN-3-Quad with baseline versions in which the quadratic activations are replaced by ReLUs. The baseline networks are CNN-2-ReLU and FcNN-3-ReLU. The hyper-parameters for training are selected based on the validation datasets. The Adam Optimizer is used for CNNs with learning rate 0.001, exponential decay and dropout probability 0.75. The AdaGrad optimizer is used for FcNNs with a learning rate of 0.01 and dropout probability 0.5. We found that norm gradient clipping was required for training the CNN-2-Quad and FcNN-3-Quad networks, since the gradient values for quadratic activations can become large. Our implementation of SafetyNets uses Thaler?s code for the IP protocol for matrix multiplication [18] and our own implementation of the IP for quadratic activations. We use an Intel Core i7-4600U CPU running at 2.10 GHz for benchmarking. 4.2 Classification Accuracy of SafetyNets SafetyNets places certain restrictions on the activation function (quadratic) and requires weights and inputs to be integers (in field Fp ). We begin by analyzing how (and if) these restrictions impact classification accuracy/error. Figure 2 compares training and test error of CNN-2-Quad/FcNN-3-Quad versus CNN-2-ReLU/FcNN-3-ReLU. For all three tasks, the networks with quadratic activations are competitive with networks that use ReLU activations. Further, we observe that the networks with quadratic activations appear to converge faster during training, possibly because their gradients are larger despite gradient clipping. Next, we used the scaling and rounding strategy proposed in Section 3.3 to convert weights and inputs to integers. Table 1 shows the impact of scaling factors ? and ? on the classification error and maximum values observed in the network during inference for MNIST-Back-Rand. The validation 7 Table 1: Validation error and maximum value observed in the network for MNIST-Rand-Back and different values of scaling parameters, ? and ?. Shown in bold red font are values of ? and ? that are infeasible because the maximum value exceeds that allowed by prime p = 261 ? 1. ? Err 0.188 0.194 0.188 0.186 0.185 4 8 16 32 64 ?=4 Max 4.0 ? 108 6.1 ? 109 9.4 ? 1010 1.5 ? 1012 2.5 ? 1013 Err 0.073 0.072 0.072 0.073 0.073 ?=8 Max 4.0 ? 1010 6.9 ? 1011 1.1 ? 1013 1.7 ? 1014 2.8 ? 1015 Err 0.042 0.039 0.036 0.038 0.038 ? = 16 Max 5.5 ? 1012 8.3 ? 1013 1.3 ? 1015 2.1 ? 1016 3.4 ? 1017 Err 0.039 0.038 0.037 0.037 0.037 ? = 32 Max 6.6 ? 1014 1.0 ? 1016 1.6 ? 1017 2.6 ? 1018 4.2 ? 1019 Err 0.04 0.037 0.035 0.036 0.036 ? = 64 Max 8.8 ? 1016 1.3 ? 1018 2.1 ? 1019 3.5 ? 1020 5.6 ? 1021 error drops as ? and ? are increased. On the other hand, for p = 261 ? 1, the largest value allowed is 1.35 ? 1018 ; this rules out ? and ? greater than 64, as shown in the table. For MNIST-Back-Rand, we pick ? = ? = 16 based on validation data, and obtain a test error of 4.67%. Following a similar methodology, we obtain a test error of 0.63% for MNIST (p = 261 ? 1) and 25.7% for TIMIT (p = 2127 ? 1). We note that SafetyNets does not support techniques such as Maxout [10] that have demonstrated lower error on MNIST (0.45%). Ba et al. [3] report an error of 18.5% for TIMIT using an ensemble of nine deep neural networks, which SafetyNets might be able to support by verifying each network individually and performing ensemble averaging at the client-side. 4.3 Verifier and Prover Run-times The relevant performance metrics for SafetyNets are (1) the client?s (or verifier?s) run-time, (2) the server?s runtime which includes baseline time to execute the neural network and overhead of generating proofs, and (3) the bandwidth required by the IP protocol. Ideally, these quantities should be small, and importantly, the client?s runtime should be smaller than the case in which it executes the network by itself. Figure 3 plots run-time data over input batch sizes ranging from 256 to 2048 for FcNNQuad-3. 1000 FcNN-Quad-3 Exe Time Additional Prover Time Verifier Time Running Time (s) 100 10 1 0.1 28 29 210 211 Input Batch Size 212 For FcNN-Quad-3, the client?s time for verifying proofs is 8? to 82? faster than the baseline in which it executes Figure 3: Run-time of verifier, prover FcNN-Quad-3 itself, and decreases with batch size. The and baseline execution time for the arithincrease in the server?s execution time due to the overmetic circuit representation of FcNNhead of generating proofs is only 5% over the baseline Quad-3 versus input batch size. unverified execution of FcNN-Quad-3. The prover and verifier exchange less than 8 KBytes of data during the IP protocol for a batch size of 2048, which is negligible (less than 2%) compared to the bandwidth required to communicate inputs and outputs back and forth. In all settings, the soundness error , i.e., the chance that the verifier fails to detect incorrect computations by the server is less than 2130 , a negligible value. We note SafetyNets has significantly lower bandwidth costs compared to an approach that separately verifies the execution of each layer using only the IP protocol for matrix multiplication. A closely related technique, CryptoNets [8], uses homomorphic encryption to provide privacy, but not integrity, for neural networks executing in the cloud. Since SafetyNets and CryptoNets target different security goals a direct comparison is not entirely meaningful. However, from the data presented in the CryptoNets paper, we note that the client?s run-time for MNIST using a CNN similar to ours and an input batch size b = 4096 is about 600 seconds, primarily because of the high cost of encryptions. For the same batch size, the client-side run-time of SafetyNets is less than 10 seconds. Recent work has also looked at how neural networks can be trained in the cloud without compromising the user?s training data [14], but the proposed techniques do not guarantee integrity. We expect that SafetyNets can be extended to address the verifiable neural network training problem as well. 5 Conclusion In this paper, we have presented SafetyNets, a new framework that allows a client to provably verify the correctness of deep neural network based inference running on an untrusted clouds. Building upon the rich literature on interactive proof systems for verifying general-purpose and specialized computations, we designed and implemented a specialized IP protocol tailored for a certain class 8 of deep neural networks, i.e., those that can be represented as arithmetic circuits. We showed that placing these restrictions did not impact the accuracy of the networks on real-world classification tasks like digit and speech recognition, while enabling a client to verifiably outsource inference to the cloud at low-cost. For our future work, we will apply SafetyNets to deeper networks and extend it to address both integrity and privacy. There are VC techniques [17] that guarantee both, but typically come at higher costs. Further, building on prior work on the use of IPs to build verifiable hardware [20], we intend to deploy the SafetyNets protocol in the design of a verifiable hardware accelerator for neural network inference. References [1] Variations on the MNIST digits. http://www.iro.umontreal.ca/~lisa/twiki/bin/ view.cgi/Public/MnistVariations. [2] S. Arora and B. Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. [3] J. Ba and R. Caruana. Do deep nets really need to be deep? In Advances in Neural Information Processing Systems, pages 2654?2662, 2014. [4] A. Coates, A. Ng, and H. Lee. An analysis of single-layer networks in unsupervised feature learning. In International Conference on Artificial Intelligence and Statistics, pages 215?223, 2011. [5] G. Cormode, J. Thaler, and K. Yi. Verifying computations with streaming interactive proofs. Proceedings of the Very Large Database Endowment, pages 25?36, 2011. [6] A. Gautier, Q. N. Nguyen, and M. Hein. Globally optimal training of generalized polynomial neural networks with nonlinear spectral methods. In Advances in Neural Information Processing Systems, pages 1687?1695, 2016. [7] R. Gennaro, C. Gentry, and B. Parno. Non-interactive verifiable computing: Outsourcing computation to untrusted workers. Annual Cryptology Conference, pages 465?482, 2010. [8] R. Gilad-Bachrach, N. Dowlin, K. Laine, K. Lauter, M. Naehrig, and J. Wernsing. Cryptonets: Applying neural networks to encrypted data with high throughput and accuracy. In International Conference on Machine Learning, pages 201?210, 2016. [9] S. Goldwasser, Y. T. Kalai, and G. N. Rothblum. Delegating computation: interactive proofs for muggles. Symposium on Theory of Computing, pages 113?122, 2008. [10] I. J. Goodfellow, D. Warde-Farley, M. Mirza, A. Courville, and Y. Bengio. Maxout networks. arXiv preprint arXiv:1302.4389, 2013. [11] K. Lee and H. Hon. Speaker-independent phone recognition using hidden markov models. IEEE Transactions on Acoustics, Speech, and Signal Processing, pages 1641?1648, 1989. [12] R. Livni, S. Shalev-Shwartz, and O. Shamir. On the computational efficiency of training neural networks. In Advances in Neural Information Processing Systems, pages 855?863, 2014. [13] C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, pages 859?868, 1992. [14] P. Mohassel and Y. Zhang. Secureml: A system for scalable privacy-preserving machine learning. IACR Cryptology ePrint Archive, 2017. [15] F. Monrose, P. Wyckoff, and A. D. Rubin. Distributed execution with remote audit. In Network and Distributed System Security Symposium, pages 3?5, 1999. [16] N. Papernot, P. McDaniel, A. Sinha, and M. Wellman. Towards the science of security and privacy in machine learning. arXiv preprint arXiv:1611.03814, 2016. [17] B. Parno, J. Howell, C. Gentry, and M. Raykova. Pinocchio: Nearly practical verifiable computation. In Symposium on Security and Privacy, pages 238?252, 2013. 9 [18] J. Thaler. Time-optimal interactive proofs for circuit evaluation. In International Cryptology Conference, pages 71?89, 2013. [19] V. Vu, S. Setty, A. J. Blumberg, and M. Walfish. A hybrid architecture for interactive verifiable computation. In Symposium on Security and Privacy, pages 223?237, 2013. [20] R. S. Wahby, M. Howald, S. Garg, A. Shelat, and M. Walfish. Verifiable asics. In Symposium on Security and Privacy, pages 759?778, 2016. [21] M. Walfish and A. J. Blumberg. Verifying computations without reexecuting them. Communications of the ACM, pages 74?84, 2015. [22] M. D. Zeiler and R. Fergus. Stochastic pooling for regularization of deep convolutional neural networks. arXiv preprint arXiv:1301.3557, 2013. [23] Y. Zhang, P. Liang, and M. J. Wainwright. Convexified convolutional neural networks. arXiv preprint arXiv:1609.01000, 2016. Proof of Lemma 3.1 Lemma 3.1 The SafetyNets verifier rejects incorrect computations with probability greater than P 3b L i=0 ni (1 ? ) where  = is the soundness error. p Proof. Verifying a multi-linear extension of the output sampled at a random point, instead of each value adds a soundness error of  = bnpL . Each instance of the sum-check protocol adds to the soundness error [19]. The IP protocol for matrix multiplication adds a soundness error of  = 2npi?1 i in Layer i [18]. Finally, the IP protocol for quadratic activations adds a soundness error of  = 3bn p in Layer i [18]. Summing together we get a total soundness error of final result is an upper bound on this value. 2 PL?1 i=0 ni +3 PL?1 i=1 p bni +bnL . The Handling Bias Variables We assumed that the bias variables were zero, allowing us to write bmzi = wi .yi while it should be bmzi = wi .yi + bi 1T . Let zi0 = wi .yi We seek to convert an assertion on Z?i (qi , ri ) to an assertion on Z?0 i . We can do so by noting that: X ? qi )(Z?0 i (j, ri ) + B ?i (j)) I(j, (8) Z?i (qi , ri ) = j?{0,1}log(ni ) ?i which the verifier checks which can be reduced to sum-check and thus yields an assertion on B 0 ? locally and Z i , which is passed to the IP protocol for matrix multiplication. 10
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Query Complexity of Clustering with Side Information Arya Mazumdar and Barna Saha College of Information and Computer Sciences University of Massachusetts Amherst Amherst, MA 01003 {arya,barna}@cs.umass.edu Abstract Suppose, we are given a set of n elements to be clustered into k (unknown) clusters, and an oracle/expert labeler that can interactively answer pair-wise queries of the form, ?do two elements u and v belong to the same cluster??. The goal is to recover the optimum clustering by asking the minimum number of queries. In this paper, we provide a rigorous theoretical study of this basic problem of query complexity of interactive clustering, and give strong information theoretic lower bounds, as well as nearly matching upper bounds. Most clustering problems come with a similarity matrix, which is used by an automated process to cluster similar points together. However, obtaining an ideal similarity function is extremely challenging due to ambiguity in data representation, poor data quality etc., and this is one of the primary reasons that makes clustering hard. To improve accuracy of clustering, a fruitful approach in recent years has been to ask a domain expert or crowd to obtain labeled data interactively. Many heuristics have been proposed, and all of these use a similarity function to come up with a querying strategy. Even so, there is a lack systematic theoretical study. Our main contribution in this paper is to show the dramatic power of side information aka similarity matrix on reducing the query complexity of clustering. A similarity matrix represents noisy pair-wise relationships such as one computed by some function on attributes of the elements. A natural noisy model is where similarity values are drawn independently from some arbitrary probability distribution f+ when the underlying pair of elements belong to the same cluster, and from some f? otherwise. We show that given such a similarity matrix, the query complexity reduces drastically from ?(nk) 2 n ) where H2 denotes the squared Hellinger (no similarity matrix) to O( H2k(flog + kf? ) divergence. Moreover, this is also information-theoretic optimal within an O(log n) factor. Our algorithms are all efficient, and parameter free, i.e., they work without any knowledge of k, f+ and f? , and only depend logarithmically with n. Our lower bounds could be of independent interest, and provide a general framework for proving lower bounds for classification problems in the interactive setting. Along the way, our work also reveals intriguing connection to popular community detection models such as the stochastic block model and opens up many avenues for interesting future research. 1 Introduction Clustering is one of the most fundamental and popular methods for data classification. In this paper we provide a rigorous theoretical study of clustering with the help of an oracle, a model that saw a recent surge of popular heuristic algorithms. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Suppose we are given a set of n points, that need to be clustered into k clusters where k is unknown to us. Suppose there is an oracle that either knows the true underlying clustering or can compute the best clustering under some optimization constraints. We are allowed to query the oracle whether any two points belong to the same cluster or not. How many such queries are needed to be asked at minimum to perform the clustering exactly? The motivation to this problem lies at the heart of modern machine learning applications where the goal is to facilitate more accurate learning from less data by interactively asking for labeled data, e.g., active learning and crowdsourcing. Specifically, automated clustering algorithms that rely just on a similarity matrix often return inaccurate results. Whereas, obtaining few labeled data adaptively can help in significantly improving its accuracy. Coupled with this observation, clustering with an oracle has generated tremendous interest in the last few years with increasing number of heuristics developed for this purpose [22, 40, 13, 42, 43, 18, 39, 12, 21, 29]. The number of queries is a natural measure of ?efficiency? here, as it directly relates to the amount of labeled data or the cost of using crowd workers ?however, theoretical guarantees on query complexity is lacking in the literature. On the theoretical side, query complexity or the decision tree complexity is a classical model of computation that has been extensively studied for different problems [16, 4, 8]. For the clustering problem, one can obtain an upper bound of O(nk) on the query complexity easily and it is achievable even when k is unknown [40, 13]: to cluster an element at any stage of the algorithm, ask one query per existing cluster with this element (this is sufficient due to transitivity), and start a new cluster if all queries are negative. It turns out that ?(nk) is also a lower bound, even for randomized algorithms (see, e.g., [13]). In contrast, the heuristics developed in practice often ask significantly less queries than nk. What could be a possible reason for this deviation between the theory and practice? Before delving into this question, let us look at a motivating application that drives this work. A Motivating Application: Entity Resolution. Entity resolution (ER, also known as record linkage) is a fundamental problem in data mining and has been studied since 1969 [17]. The goal of ER is to identify and link/group different manifestations of the same real world object, e.g., different ways of addressing (names, email address, Facebook accounts) the same person, Web pages with different descriptions of the same business, different photos of the same object etc. (see the excellent survey by Getoor and Machanavajjhala [20]). However, lack of an ideal similarity function to compare objects makes ER an extremely challenging task. For example, DBLP, the popular computer science bibliography dataset is filled with ER errors [30]. It is common for DBLP to merge publication records of different persons if they share similar attributes (e.g. same name), or split the publication record of a single person due to slight difference in representation (e.g. Marcus Weldon vs Marcus K. Weldon). In recent years, a popular trend to improve ER accuracy has been to incorporate human wisdom. The works of [42, 43, 40] (and many subsequent works) use a computer-generated similarity matrix to come up with a collection of pair-wise questions that are asked interactively to a crowd. The goal is to minimize the number of queries to the crowd while maximizing the accuracy. This is analogous to our interactive clustering framework. But intriguingly, as shown by extensive experiments on various real datasets, these heuristics use far less queries than nk [42, 43, 40]?barring the ?(nk) theoretical lower bound. On a close scrutiny, we find that all of these heuristics use some computer generated similarity matrix to guide in selecting the queries. Could these similarity matrices, aka side information, be the reason behind the deviation and significant reduction in query complexity? Let us call this clustering using side information, where the clustering algorithm has access to a similarity matrix. This can be generated directly from the raw data (e.g., by applying Jaccard similarity on the attributes), or using a crude classifier which is trained on a very small set of labelled samples. Let us assume the following generative model of side information: a noisy weighted upper-triangular similarity matrix W = {wi,j }, 1 ? i < j ? n, where wi,j is drawn from a probability distribution f+ if i, j, i 6= j, belong to the same cluster, and else from f? . However, the algorithm designer is given only the similarity matrix without any information on f+ and f? . In this work, one of our major contributions is to show the separation in query complexity of clustering with and without such side information. Indeed the recent works of [18, 33] analyze popular heuristic algorithms of [40, 43] where the probability distributions are obtained from real datasets which show that these heuristics are significantly suboptimal even for very simple distributions. To the best of our knowledge, before this work, there existed no algorithm that works for arbitrary unknown distributions f+ and f? with near-optimal performances. We develop a generic framework for proving information theoretic lower bounds for interactive clustering using side information, and design efficient algorithms for arbitrary 2 f+ and f? that nearly match the lower bound. Moreover, our algorithms are parameter free, that is they work without any knowledge of f+ , f? or k. Connection to popular community detection models. The model of side information considered in this paper is a direct and significant generalization of the planted partition model, also known as the stochastic block model (SBM) [28, 15, 14, 2, 1, 25, 24, 11, 36]. The stochastic block model is an extremely well-studied model of random graphs which is used for modeling communities in real world, and is a special case of a similarity matrix we consider. In SBM, two vertices within the same community share an edge with probability p, and two vertices in different communities share an edge with probability q, that is f+ is Bernoulli(p) and f? is Bernoulli(q). It is often assumed that k, the number of communities, is a constant (e.g. k = 2 is known as the planted bisection model and is studied extensively [1, 36, 15] or a slowly growing function of n (e.g. k = o(log n)). The points are assigned to clusters according to a probability distribution indicating the relative sizes of the clusters. In contrast, not only in our model f+ and f? can be arbitrary probability mass functions (pmfs), we do not have to make any assumption on k or the cluster size distribution, and can allow for any partitioning of the set of elements (i.e., adversarial setting). Moreover, f+ and f? are unknown. For SBM, parameter free algorithms are known relatively recently for constant number of linear sized clusters [3, 24]. There are extensive literature that characterize the threshold phenomenon in SBM in terms of p and q for exact and approximate recovery of clusters when relative cluster sizes are known and nearly balanced (e.g., see [2] and therein for many references). For k = 2 and equal sized clusters, sharp thresholds are derived in [1, 36] for a specific sparse region of p and q 1 . In a more general setting, the vertices in the ith and the jth communities are connected with probability qij and threshold results for the sparse region has been derived in [2] - our model can be allowed to have this as a special case when we have pmfs fi,j s denoting the distributions of the corresponding random variables. If an oracle gives us some of the pairwise binary relations between elements (whether they belong to the same cluster or not), the threshold of SBM must also change. But by what amount? This connection to SBM could be of independent interest to study query complexity of interactive clustering with side information, and our work opens up many possibilities for future direction. Developing lower bounds in the interactive setting appears to be significantly challenging, as algorithms may choose to get any deterministic information adaptively by querying, and standard lower bounding techniques based on Fano-type inequalities [9, 31] do not apply. One of our major contributions in this paper is to provide a general framework for proving information-theoretic lower bound for interactive clustering algorithms which holds even for randomized algorithms, and even with the full knowledge of f+ , f? and k. In contrast, our algorithms are computationally efficient and are parameter free (works without knowing f+ , f? and k). The technique that we introduce for our upper bounds could be useful for designing further parameter free algorithms which are extremely important in practice. Other Related works. The interactive framework of clustering model has been studied before where the oracle is given the entire clustering and the oracle can answer whether a cluster needs to be split or two clusters must be merged [7, 6]. Here we contain our attention to pair-wise queries, as in all practical applications that motivate this work [42, 43, 22, 40]. In most cases, an expert human or crowd serves as an oracle. Due to the scale of the data, it is often not possible for such an oracle to answer queries on large number of input data. Only recently, some heuristic algorithms with k-wise queries for small values of k but k > 2 have been proposed in [39], and a non-interactive algorithm that selects random triangle queries have been analyzed in [41]. Also recently, the stochastic block model with active label-queries have been studied in [19]. Perhaps conceptually closest to us is a recent work by [5] where they consider pair-wise queries for clustering. However, their setting is very different. They consider the specific NP-hard k-means objective with distance matrix which must be a metric and must satisfy a deterministic separation property. Their lower bounds are computational and not information theoretic; moreover their algorithm must know the parameters. There exists a significant gap between their lower and upper bounds:? log k vs k 2 , and it would be interesting if our techniques can be applied to improve this. Here we have assumed the oracle always returns the correct answer. To deal with the possibility that the crowdsourced oracle may give wrong answers, there are simple majority voting mechanisms or more complicated techniques [39, 12, 21, 29, 10, 41] to handle such errors. Our main objective is to 1 Most recent works consider the region of interest as p = 3 a log n n and q = b log n n for some a > b > 0. study the power of side information, and we do not consider the more complex scenarios of handling erroneous oracle answers. The related problem of clustering with noisy queries is studied by us in a companion work [34]. Most of the results of the two papers are available online in a more extensive version [32]. Contributions. Formally the problem we study in this paper can be described as follows. Problem 1 (Query-Cluster with an Oracle). Consider a set of elements V ? [n] with k latent clusters Vi , i = 1, . . . , k, where k is unknown. There is an oracle O : V ? V ? {?1}, that when queried with a pair of elements u, v ? V ? V , returns +1 iff u and v belong to the same cluster, and ?1 iff u and v belong to different clusters. The queries Q ? V ? V can be done adaptively. Consider the side information W = {wu,v : 1 ? u < v ? n}, where the (u, v)th entry of W , wu,v is a random variable drawn from a discrete probability distribution f+ if u, v belong to the same cluster, and is drawn from a discrete2 probability distribution f? 3 if u, v belong to different clusters. The parameters k, f+ and f? are unknown. Given V and W , find Q ? V ? V such that |Q| is minimum, and from the oracle answers and W it is possible to recover Vi , i = 1, 2, ..., k. Without side information, as noted earlier, it is easy to see an algorithm with query complexity O(nk) for Query-Cluster. When no side information is available, it is also not difficult to have a lower bound of ?(nk) on the query complexity. Our main contributions are to develop strong information theoretic lower bounds as well as nearly matching upper bounds when side information is available, and characterize the effect of side information on query complexity precisely. Upper Bound (Algorithms). We show that with side information W , a drastic reduction in query complexity of clustering is possible, even with unknown parameters f+ , f? , and k. We propose a 2 n Monte Carlo randomized algorithm that reduces the number of queries from O(nk) to O( H2k(flog ), + kf? ) where H(f kg) is the Hellinger divergence between the probability distributions f , and g, and recovers the clusters accurately with high probability (with success probability 1 ? n1 ) without knowing f+ , f? or k (see, Theorem 1). Depending on the value of k, this could be highly sublinear in n. Note that the squared Hellinger divergence between two pmfs f and g is defined to be, 2 p 1 X p H2 (f kg) = f (i) ? g(i) . 2 i We also develop a Las Vegas algorithm, that is one which recovers the clusters with probability 1 (and 2 n not just with high probability), with query complexity O(n log n + H2k(flog ). Since f+ and f? + kf? ) can be arbitrary, not knowing the distributions provides a major challenge, and we believe, our recipe could be fruitful for designing further parameter-free algorithms. We note that all our algorithms are computationally efficient - in fact, the time required is bounded by the size of the side information matrix, i.e., O(n2 ). Theorem 1. Let the number of clusters k be unknown and f+ and f? be unknown discrete distributions with fixed cardinality of support. There exists an efficient (polynomial-time) Monte Carlo 2 n algorithm for Query-Cluster that has query complexity O(min (nk, H2k(flog )) and recovers all + kf? ) 1 the clusters accurately with probability 1 ? o( n ). Moreover there exists an efficient Las Vegas 2 n algorithm that with probability 1 ? o( n1 ) has query complexity O(n log n + min (nk, H2k(flog )). + kf? ) Lower Bound. Our main lower bound result is information theoretic, and can be summarized in the following theorem. Note especially that, for lower bound we can assume the knowledge of k, f+ , f? in contrast to upper bounds, which makes the results stronger. In addition, f+ and f? can be discrete or continuous distributions. Note that when H2 (f+ kf? ) is close to 1, e.g., when the side information is perfect, no queries are required. However, that is not the case in practice, and we are interested in the region where f+ and f? are ?close?, that is H2 (f+ kf? ) is small. 1 Theorem 2. Assume H2 (f+ kf? ) ? 18 . Any (possibly randomized)  algorithm with the knowledge  2 of f+ , f? , and the number of clusters k, that does not perform ? min {nk, H2 (fk+ kf? ) } expected 2 Our lower bound holds for continuous distributions as well. For simplicity of expression, we treat the sample space to be of constant size. However, all our results extend to any finite sample space scaling linearly with its size. 3 4 number of queries, will be unable to return the correct clustering with probability at least16 ? O( ?1k ). And to recover the clusters with probability 1, the number of queries must be ? n +  2 min {nk, H2 (fk+ kf? ) } . The lower bound therefore matches the query complexity upper bound within a logarithmic factor. Note that when no querying is allowed, this turns out exactly to be the setting of stochastic block model though with much general distributions. We have analyzed this case in Appendix C. To see how the probability of error must scale, we have used a generalized version of Fano?s inequality (e.g., [23]). However, when the number of queries is greater than zero, plus when queries can be adaptive, any such standard technique fails. Hence, significant effort has to be put forth to construct a setting where information theoretic minimax bounds can be applied. This lower bound could be of independent interest, and provides a general framework for deriving lower bounds for fundamental problems of classification, hypothesis testing, distribution testing etc. in the interactive learning setting. They may also lead to new lower bound proving techniques in the related multi-round communication complexity model where information again gets revealed adaptively. Organization. The proof of the lower bound is provided in Section 2. The Monte Carlo algorithm is given in Section 3. The detailed proof of the Monte Carlo algorithm, and the Las Vegas algorithm and its proof are given in Appendix A and Appendix B respectively in the supplementary material for space constraint. 2 Lower Bound (Proof of Theorem 2) In this section, we develop our information theoretic lower bounds. We prove a more general result from which Theorem 2 follows. Lemma 1. Consider the case when we have k equally sized clusters of size a each (that is total number of elements is n = ka). Suppose we are allowed to make at most Q adaptive queries to the oracle. The probability of error for any algorithm for Query-Cluster is at least, r ? 2 4Q 2 4Q 1? ? ? 2 aH(f+ kf? ). 1+ k ak ak(k ? 1) The main high-level technique to prove Lemma 1 is the following. Suppose, a node is to be assigned to a cluster. This situation is obviously akin to a k-hypothesis testing problem, and we want to use a lower bound on the probability of error. The side information and the query answers constitute a random vector whose distributions (among the k possible) must be far apart for us to successfully identify the clustering. But the main challenge comes from the interactive nature of the algorithm since it reveals deterministic information and into characterizing the set of elements that are not queried much by the algorithm. Proof of Lemma 1. Since the total number of queries is Q, the average number of queries per element 4Q ak is at most 2Q ak . Therefore there exist at least 2 elements that are queried at most T < ak times. Let x be one such element. We just consider the problem of assignment of x to a cluster (all other elements have been correctly assigned already), and show that any algorithm will make wrong assignment with positive probability. Step 1: Setting up the hypotheses. Note that the side information matrix W = (wi,j ) is provided where the wi,j s are independent random variables. Now assume the scenario when we use an algorithm ALG to assign x to one of the k clusters, Vu , u = 1, . . . , k. Therefore, given x, ALG takes as input the random variables wi,x s where i ? tt Vt , makes some queries involving x and outputs a cluster index, which is an assignment for x. Based on the observations wi,x s, the task of ALG is thus a multi-hypothesis testing among k hypotheses. Let Hu , u = 1, . . . k denote the k different hypotheses Hu : x ? Vu . And let Pu , u = 1, . . . k denote the joint probability distributions of the random matrix W when x ? Vu . In short, for any event A, Pu (A) = Pr(A|Hu ). Going forward, the subscript of probabilities or expectations will denote the appropriate conditional distribution. Step 2: Finding ?weak? clusters. There must exist t ? {1, . . . , k} such that, k X Pt { a query made by ALG involving cluster Vv } ? Et {Number of queries made by ALG} ? T. v=1 5 We now find a subset of clusters, that are ?weak,? i.e., not queried enough if Ht were true. Consider 2T the set J 0 ? {v ? {1, . . . , k} : Pt { a query made by ALG involving cluster Vv } < k(1??) }, where ?? 1 ? . We must have, (k ? |J 0 |) ? 4Q 1+ 2T k(1??) ? T, which implies, |J 0 | ? (1+?)k . 2 ak Now, to output a cluster without using the side information, ALG has to either make a query to the actual cluster the element is from, or query at least k ? 1 times. In any other case, ALG must use the side information (in addition to using queries) to output a cluster. Let E u denote the event that 2 ALG outputs cluster Vu by using the side information. Let J 00 ? {u ? {1, . . . , k} : Pt (E u ) ? ?k }. Pk (2??)k ?k 2 00 u 00 Since u=1 Pt (E ) ? 1, we must have, (k ? |J |) ? ?k < 1, or |J | > k ? 2 = . 2 + (2??)k ? k = k2 . This means, {Vu : u ? J 0 ? J 00 } contains more We have, |J 0 ? J 00 | > (1+?)k 2 2 ak than ak 2 elements. Since there are 2 elements that are queried at most T times, these two sets must have nonzero intersection. Hence, we can assume that x ? V` for some ` ? J 0 ? J 00 , i.e., let H` be the true hypothesis. Now we characterize the error events of the algorithm ALG in assignment of x. Step 3: Characterizing error events for ?x?. We now consider the following two events. E1 = {a query made by ALG involving cluster V` }; E2 = {k ? 1 or more queries were made by ALG}. Note that if the algorithm ALG can correctly assign x to a cluster without using the side information then either of E1 or E2 must have to happen. Recall, E ` denotesSthe event S that ALG outputs cluster V` using the side information. Now consider the event E ? E ` E1 E2 . The probability of correct assignment is at most P` (E). We now bound this probability of correct recovery from above. Step 4: Bounding probability of correct recovery via Hellinger distance. We have, ? P` (E) ? Pt (E) + |P` (E) ? Pt (E)| ? Pt (E) + kP` ? Pt kT V ? Pt (E) + 2H(P` kPt ), where, kP ? QkT V ? supA |P (A) ? Q(A)| denotes the total variation distance between two probability distributions P and Q and in the last step we have used the relationship between total variation distance and the Hellinger divergence (see, for example, [38, Eq. (3)]). Now, recall that P` and Pt are the joint distributions of the independent random variables wi,x , i ? ?u Vu . Now, we use the fact that squared Hellinger divergence between product distribution of independent random variables are less than the sum of the squared Hellinger divergence between the individual distribution. We also note that the divergence between identical random variables are 0. We obtain p p ? 2H2 (P` kPt ) ? 2 ? 2aH2 (f+ kf? ) = 2 aH(f+ kf? ). This is true because the only times when wi,x differs ? under Pt and under P` is when x ? Vt or x ? V` . As a result we have, P` (E) ? Pt (E) + 2 aH(f+ kf? ). Now, using Markov inequality 4Q T Pt (E2 ) ? k?1 ? ak(k?1) . Therefore, 8Q 4Q 2 + + . ?k ak 2 (1 ? ?) ak(k ? 1) q 2  ? 4Q Therefore, putting the value of ? we get, P` (E) ? k2 1 + 4Q + ak(k?1) + 2 aH(f+ kf? ), ak which proves the lemma. Pt (E) ? Pt (E ` ) + Pt (E1 ) + Pt (E2 ) ? 2 Proof of Theorem 2. Consider two cases. In the first case, suppose, nk < 9H2 (fk+ kf? ) . Now consider the situation of Lemma 1, with a = nk . The probability of error of any algorithm must be at least, q 2  4Q 1 ? k2 1 + 4Q ? ak(k?1) ? 23 ? 16 ? O( ?1k ), if the number of queries Q ? nk ak 72 . 2 In the second case, suppose nk ? 9H2 (fk+ kf? ) . Assume, a = b 9H2 (f1+ kf? ) c. Then a ? 2, since 1 H2 (f+ kf? ) ? 18 . We have nk ? k 2 a. Consider the situation when we are already given a complete cluster Vk with n ? (k ? 1)a elements, remaining (k ? 1) clusters each has 1 element, and the rest (a ? 1)(k ? 1) elements are evenly distributed (but yet to be assigned) to the k ? 1 clusters. Now we 2 are exactly in the situation of Lemma 1 with k ? 1 playing the role of k. If we have Q < ak 72 , The 2 probability of error is at least 1 ? ok (1) ? 16 ? 23 = 16 ? O( ?1k ). Therefore Q must be ?( H2 (fk+ kf? ) ). Note that in this proof we have not in particular tried to optimize the constants. If we want to recover the clusters with probability 1, then ?(n) is a trivial lower bound. Hence, 2 coupled with the above we get a lower bound of ?(n + min {nk, H2 (fk+ kf? ) }) in that case. 6 3 Algorithms We propose two algorithms (Monte Carlo and Las Vegas) both of which are completely parameter free that is they work without any knowledge of k, f+ and f? , and meet the respective lower bounds within an O(log n) factor. Here we present the Monte Carlo algorithm which drastically reduces 2 n ) and recovers the clusters the number of queries from O(nk) (no side information) to O( H2k(flog + kf? ) exactly with probability at least 1 ? on (1). The detailed proof of it, as well as the Las Vegas algorithm are presented in Appendix A and Appendix B respectively in the supplementary material. Our algorithm uses a subroutine called Membership that takes as input an element v ? V and a subset of elements C ? V \ {v}. Assume that f+ , f? are discrete distributions over fixed set of q points a1 , a2 , . . . , aq ; that is wi,j takes value in the set {a1 , a2 , . . . , aq }. Define the empirical |{u?C:wu,v =ai }| ?inter? distribution pv,C for i = 1, . . . , q, pv,C (i) = Also compute the ?intra? dis|C| |{(u,v)?C?C:u6=v,w =a }| u,v i tribution pC for i = 1, . . . , q, pC (i) = . Then we use Membership(v, C) |C|(|C|?1) 2 = ?H (pv,C kpC ) as affinity of vertex v to C, where H(pv,C kpC ) denotes the Hellinger divergence between distributions. Note that since the membership is always negative, a higher membership implies that the ?inter? and ?intra? distributions are closer in terms of Hellinger distance. Designing a parameter free Monte Carlo algorithm seems to be highly challenging as here, the number of queries depends only logarithmically with n. Intuitively, if an element v has the highest membership in some cluster C, then v should be queried with C first. Also an estimation from side information is reliable when the cluster already has enough members. Unfortunately, we know neither whether the current cluster size is reliable, nor we are allowed to make even one query per element. To overcome this bottleneck, we propose an iterative-update algorithm which we believe will find more uses in developing parameter free algorithms. We start by querying a few points so that there is at least one cluster with ?(log n) points. Now based on these queried memberships, we learn two empirical distributions p1+ from intra-cluster similarity values, and p1? from inter-cluster similarity values. Given an element v which has not been clustered yet, and a cluster C with the highest number of current members, we would like to consider the submatrix of side information pertaining to v and all u ? C and determine whether that side information is generated from f+ or f? . We know if the statistical distance between f+ and f? is small, then we would need more members in C to successfully do this test. Since we do not know f+ and f? , we compute the squared Hellinger divergence between p1+ and p1? , and use that to compute a threshold ?1 on the size of C. If C crosses this size threshold, we just use the side information to determine if v should belong to C. Otherwise, we query further until there is one cluster with size ?1 , and re-estimate the empirical distributions p2+ and p2? . Again, we recompute a threshold ?2 , and stop if the cluster under consideration crosses this new threshold. If not we continue. Interestingly, we can show when the process converges, we have a very good estimate of H(f+ kf? ) and, moreover it converges fast. Algorithm. Phase 1. Initialization. We initialize the algorithm by selecting any element v and creating a singleton cluster {v}. We then keep selecting new elements randomly and uniformly that have not yet been clustered, and query the oracle with it by choosing exactly one element from each of the clusters formed so far. If the oracle returns +1 to any of these queries then we include the element in the corresponding cluster, else we create a new singleton cluster with it. We continue this process until one cluster has grown to a size of dC log ne, where C is a constant. Phase 2. Iterative Update. Let C1 , C2 , ...Clx be the set of clusters formed after the xth iteration for some lx ? k, where we consider Phase 1 as the 0-th iteration. We estimate p+,x = |{u, v ? Ci : u 6= v, wu,v = ai }| |{u ? Ci , v ? Cj , i < j, i, j ? [1, lx ] : wu,v = ai }| ; p?,x = Plx Plx P i=1 |Ci |(|Ci ? 1|) i=1 i<j |Ci ||Cj | C log n E Define MxE = H(p+,x kp?,x )2 . If there is no cluster of size at least Mx formed so far, we select a new element yet to be clustered and query it exactly once with the existing clusters (that is by selecting one arbitrary point from every cluster and querying the oracle with it and the new element), and include it in an existing cluster or create a new cluster with it based on the query answer. We then set x = x + 1 and move to the next iteration to get updated estimates of p+,x , p?,x , MxE and lx . Else if there is a cluster of size at least MxE , we stop and move to the next phase. 7 Phase 3. Processing the grown clusters. Once Phase 2 has converged, let p+ , p? , H(p+ kp? ), M E and l be the final estimates. For every cluster C of size |C| ? M E , call it grown and do the following. + kp? ) (3A.) For every unclustered element v, if Membership(v, C) ? ?( 4H(pC ? we include v in C without querying. 2H(p+ kp? )2 ? ), C log n + kp? ) ? (3B.) We create a new list Waiting(C), initially empty. If ?( 4H(pC + kp? ) ?( 4H(pC then 2H(p+ kp? )2 ? ) C log n > 2H(p+ kp? )2 ? ), C log n Membership(v, C) ? then we include v in Waiting(C). For every + element in Waiting(C), we query the oracle with it by choosing exactly one element from each of the clusters formed so far starting with C. If oracle returns answer ?yes? to any of these queries then we include the element in that cluster, else we create a new singleton cluster with it. We continue this until Waiting(C) is exhausted. We then call C completely grown, remove it from further consideration, and move to the next grown cluster. if there is no other grown cluster, then we move back to Phase 2. Analysis. The main steps of the analysis are as follows (for full analysis see Appendix A). 2 kp ) ?+ ? ] for a 1. First, Lemma 3 shows with high probability H(p+ kp? ) ? [H(f+ kf? ) ? 4H(p B log n suitable constant B that depends on C. Using it, we can show the process converges whenever log n a cluster has grown to a size of H4C 2 (f kf ) . The proof relies on adapting the Sanov?s Theorem + ? (see Lemma 2) of information theory. We are measuring the distance between distributions via Hellinger distance, as opposed to KL divergence (which would have been a natural choice because of its presence in the rate function in Sanov?s therem), because Hellinger distance is a metric which proves to be crucial in our analysis. 2. Lemma 5 and Corollary 1 show that every element that is included in C in Phase (3A) truly belongs to C, and elements that are not in Waiting(C) can not be in C with high probability. Once Phase 2 has converged, if the condition of (3A) is satisfied, the element must belong to C. There is a small gray region of confidence interval (3B) such that if an element belongs there, we cannot be sure either way, but if an element does not satisfy either (3A) or 3B, it cannot be part of C. 3. Lemma 6 shows that size of Waiting(C) is constant showing an anti-concentration property. This log n coupled with the fact that the process converges when a cluster reaches size H4C 2 (f kf ) gives the + ? desired query complexity bound in Lemma 7. 4 Experimental Results In this section, we report experimental results on a popular bibliographic dataset cora [35] consisting of 1879 nodes, 191 clusters and 1699612 edges out of which 62891 are intra-cluster edges. We remove any singleton node from the dataset ? the final number of vertices that we classify is 1812 with 124 clusters. We use the similarity function computation used by [18] to compute f+ and f? . The two distributions are shown in Figure 1 on the left. The Hellinger square divergence between the two distributions is 0.6. In order to observe the dependency of the algorithm performance on the learnt distributions, we perturb the exact distributions to obtain two approximate distributions as shown in Figure 1 (middle) with Hellinger square divergence being 0.4587. We consider three strategies. Suppose the cluster in which a node v must be included has already been initialized and exists in the current solution. Moreover, suppose the algorithm decides to use queries to find membership of v. Then in the best strategy, only one query is needed to identify the cluster in which v belongs. In the worst strategy, the algorithm finds the correct cluster after querying all the existing clusters whose current membership is not enough to take a decision using side information. In the greedy strategy, the algorithm queries the clusters in non-decreasing order of Hellinger square divergence between f+ (or approximate version of it) and the estimated distribution from side information between v and each existing clusters. Note that, in practice, we will follow the greedy strategy. Figure 2 shows the performance of each strategy. We plot the number of queries vs F1 Score which computes the harmonic mean of precision and recall. We observe that the performance of greedy strategy is very close to that of best. With just 1136 queries, greedy achieves 80% precision and close to 90% recall. The best strategy would need 962 queries to achieve that performance. The performance of our algorithm on the exact and approximate distributions are also very close which indicates it is enough to learn a distribution that is close to exact. For example, using the approximate distributions, 8 Figure 1: (left) Exact distributions of similarity values, (middle) approximate distributions of similarity values, (right) Number of Queries vs F1 Score for both distributions. to achieve similar precision and recall, the greedy strategy just uses 1148 queries, that is 12 queries more than when we use when the distributions are known. Figure 2: Number of Queries vs F1 Score using three strategies: best, greedy, worst. Discussion. This is the first rigorous theoretical study of interactive clustering with side information, and it unveils many interesting directions for future study of both theoretical and practical significance (see Appendix D for more details). Having arbitrary f+ , f? is a generalization of SBM. Also it raises an important question about how SBM recovery threshold changes with queries. For sparse region of 0 0 n n SBM, where f+ is Bernoulli( a log ) and f? is Bernoulli( b log ), a0 > b0 , Lemma 1 is not tight n n n yet. However, it shows the following trend. Let us set a = k in Lemma 1 with the above f+ , f? . ? We conjecture by ignoring the lower order terms and a log n factor that with Q queries,the sharp  ? ? ? ? ? ? Q recovery threshold of sparse SBM changes from ( a0 ? b0 ) ? k to ( a0 ? b0 ) ? k 1 ? nk . Proving this bound remains an exciting open question. We propose two computationally efficient algorithms that match the query complexity lower bound within log n factor and are completely parameter free. In particular, our iterative-update method to design Monte-Carlo algorithm provides a general recipe to develop any parameter-free algorithm, which are of extreme practical importance. The convergence result is established by extending Sanov?s theorem from the large deviation theory which gives bound only in terms of KL-divergence. Due to the generality of the distributions, the only tool we could use is Sanov?s theorem. However, Hellinger distance comes out to be the right measure both for lower and upper bounds. If f+ and f? are common distributions like Gaussian, Bernoulli etc., then other concentration results stronger than Sanov may be applied to improve the constants and a logarithm factor to show the trade-off between queries and thresholds as in sparse SBM. While some of our results apply to general fi,j s, a full picture with arbitrary fi,j s and closing the gap of log n between the lower and upper bound remain an important future direction. 9 Acknowledgement. This work is supported in part by NSF awards CCF 1642658, CCF 1642550, CCF 1464310, CCF 1652303, a Yahoo ACE Award and a Google Faculty Research Award. We are particularly thankful to an anonymous reviewer whose comments led to notable improvement of the presentation of the paper. References [1] E. Abbe, A. S. Bandeira, and G. Hall. 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QMDP-Net: Deep Learning for Planning under Partial Observability Peter Karkus1,2 1 David Hsu1,2 Wee Sun Lee2 NUS Graduate School for Integrative Sciences and Engineering 2 School of Computing National University of Singapore {karkus, dyhsu, leews}@comp.nus.edu.sg Abstract This paper introduces the QMDP-net, a neural network architecture for planning under partial observability. The QMDP-net combines the strengths of model-free learning and model-based planning. It is a recurrent policy network, but it represents a policy for a parameterized set of tasks by connecting a model with a planning algorithm that solves the model, thus embedding the solution structure of planning in a network learning architecture. The QMDP-net is fully differentiable and allows for end-to-end training. We train a QMDPnet on different tasks so that it can generalize to new ones in the parameterized task set and ?transfer? to other similar tasks beyond the set. In preliminary experiments, QMDP-net showed strong performance on several robotic tasks in simulation. Interestingly, while QMDP-net encodes the QMDP algorithm, it sometimes outperforms the QMDP algorithm in the experiments, as a result of end-to-end learning. 1 Introduction Decision-making under uncertainty is of fundamental importance, but it is computationally hard, especially under partial observability [24]. In a partially observable world, the agent cannot determine the state exactly based on the current observation; to plan optimal actions, it must integrate information over the past history of actions and observations. See Fig. 1 for an example. In the model-based approach, we may formulate the problem as a partially observable Markov decision process (POMDP). Solving POMDPs exactly is computationally intractable in the worst case [24]. Approximate POMDP algorithms have made dramatic progress on solving large-scale POMDPs [17, 25, 29, 32, 37]; however, manually constructing POMDP models or learning them from data remains difficult. In the model-free approach, we directly search for an optimal solution within a policy class. If we do not restrict the policy class, the difficulty is data and computational efficiency. We may choose a parameterized policy class. The effectiveness of policy search is then constrained by this a priori choice. Deep neural networks have brought unprecedented success in many domains [16, 21, 30] and provide a distinct new approach to decision-making under uncertainty. The deep Q-network (DQN), which consists of a convolutional neural network (CNN) together with a fully connected layer, has successfully tackled many Atari games with complex visual input [21]. Replacing the postconvolutional fully connected layer of DQN by a recurrent LSTM layer allows it to deal with partial observaiblity [10]. However, compared with planning, this approach fails to exploit the underlying sequential nature of decision-making. We introduce QMDP-net, a neural network architecture for planning under partial observability. QMDP-net combines the strengths of model-free learning and model-based planning. A QMDP-net is a recurrent policy network, but it represents a policy by connecting a POMDP model with an algorithm that solves the model, thus embedding the solution structure of planning in a network 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. (a) (b) (c) (d) Fig. 1: A robot learning to navigate in partially observable grid worlds. (a) The robot has a map. It has a belief over the initial state, but does not know the exact initial state. (b) Local observations are ambiguous and are insufficient to determine the exact state. (c, d) A policy trained on expert demonstrations in a set of randomly generated environments generalizes to a new environment. It also ?transfers? to a much larger real-life environment, represented as a LIDAR map [12]. learning architecture. Specifically, our network uses QMDP [18], a simple, but fast approximate POMDP algorithm, though other more sophisticated POMDP algorithms could be used as well. A QMDP-net consists of two main network modules (Fig. 2). One represents a Bayesian filter, which integrates the history of an agent?s actions and observations into a belief, i.e. a probabilistic estimate of the agent?s state. The other represents the QMDP algorithm, which chooses the action given the current belief. Both modules are differentiable, allowing the entire network to be trained end-to-end. We train a QMDP-net on expert demonstrations in a set of randomly generated environments. The trained policy generalizes to new environments and also ?transfers? to more complex environments (Fig. 1c?d). Preliminary experiments show that QMDP-net outperformed state-of-the-art network architectures on several robotic tasks in simulation. It successfully solved difficult POMDPs that require reasoning over many time steps, such as the well-known Hallway2 domain [18]. Interestingly, while QMDP-net encodes the QMDP algorithm, it sometimes outperformed the QMDP algorithm in our experiments, as a result of end-to-end learning. 2 2.1 Background Planning under Uncertainty A POMDP is formally defined as a tuple (S, A, O, T, Z, R), where S, A and O are the state, action, and observation space, respectively. The state-transition function T (s, a, s0 ) = P (s0 |s, a) defines the probability of the agent being in state s0 after taking action a in state s. The observation function Z(s, a, o) = p(o|s, a) defines the probability of receiving observation o after taking action a in state s. The reward function R(s, a) defines the immediate reward for taking action a in state s. In a partially observable world, the agent does not know its exact state. It maintains a belief, which is a probability distribution over S. The agent starts with an initial belief b0 and updates the belief bt at each time step t with a Bayesian filter: P bt (s0 ) = ? (bt 1 , at , ot ) = ?O(s0 , at , ot ) s2S T (s, at , s0 )bt 1 (s), (1) where ? is a normalizing constant. The belief bt recursively integrates information from the entire past history (a1 , o1 , a2 , o2 , . . . , at , ot ) for decision making. POMDP planning seeks a policy ? that maximizes the value, i.e., the expected total discounted reward: P1 t V? (b0 ) = E R(st , at+1 ) b0 , ? , (2) t=0 where st is the state at time t, at+1 = ?(bt ) is the action that the policy ? chooses at time t, and 2 (0, 1) is a discount factor. 2.2 Related Work To learn policies for decision making in partially observable domains, one approach is to learn models [6, 19, 26] and solve the models through planning. An alternative is to learn policies directly [2, 5]. Model learning is usually not end-to-end. While policy learning can be end-to-end, it does not exploit model information for effective generalization. Our proposed approach combines model-based and 2 model-free learning by embedding a model and a planning algorithm in a recurrent neural network (RNN) that represents a policy and then training the network end-to-end. RNNs have been used earlier for learning in partially observable domains [4, 10, 11]. In particular, Hausknecht and Stone extended DQN [21], a convolutional neural network (CNN), by replacing its post-convolutional fully connected layer with a recurrent LSTM layer [10]. Similarly, Mirowski et al. [20] considered learning to navigate in partially observable 3-D mazes. The learned policy generalizes over different goals, but in a fixed environment. Instead of using the generic LSTM, our approach embeds algorithmic structure specific to sequential decision making in the network architecture and aims to learn a policy that generalizes to new environments. The idea of embedding specific computation structures in the neural network architecture has been gaining attention recently. Tamar et al. implemented value iteration in a neural network, called Value Iteration Network (VIN), to solve Markov decision processes (MDPs) in fully observable domains, where an agent knows its exact state and does not require filtering [34]. Okada et al. addressed a related problem of path integral optimal control, which allows for continuous states and actions [23]. Neither addresses the issue of partial observability, which drastically increases the computational complexity of decision making [24]. Haarnoja et al. [9] and Jonschkowski and Brock [15] developed end-to-end trainable Bayesian filters for probabilistic state estimation. Silver et al. introduced Predictron for value estimation in Markov reward processes [31]. They do not deal with decision making or planning. Both Shankar et al. [28] and Gupta et al. [8] addressed planning under partial observability. The former focuses on learning a model rather than a policy. The learned model is trained on a fixed environment and does not generalize to new ones. The latter proposes a network learning approach to robot navigation in an unknown environment, with a focus on mapping. Its network architecture contains a hierarchical extension of VIN for planning and thus does not deal with partial observability during planning. The QMDP-net extends the prior work on network architectures for MDP planning and for Bayesian filtering. It imposes the POMDP model and computation structure priors on the entire network architecture for planning under partial observability. 3 Overview We want to learn a policy that enables an agent to act effectively in a diverse set of partially observable stochastic environments. Consider, for example, the robot navigation domain in Fig. 1. The environments may correspond to different buildings. The robot agent does not observe its own location directly, but estimates it based on noisy readings from a laser range finder. It has access to building maps, but does not have models of its own dynamics and sensors. While the buildings may differ significantly in their layouts, the underlying reasoning required for effective navigation is similar in all buildings. After training the robot in a few buildings, we want to place the robot in a new building and have it navigate effectively to a specified goal. Formally, the agent learns a policy for a parameterized set of tasks in partially observable stochastic environments: W? = {W (?) | ? 2 ?}, where ? is the set of all parameter values. The parameter value ? captures a wide variety of task characteristics that vary within the set, including environments, goals, and agents. In our robot navigation example, ? encodes a map of the environment, a goal, and a belief over the robot?s initial state. We assume that all tasks in W? share the same state space, action space, and observation space. The agent does not have prior models of its own dynamics, sensors, or task objectives. After training on tasks for some subset of values in ?, the agent learns a policy that solves W (?) for any given ? 2 ?. A key issue is a general representation of a policy for W? , without knowing the specifics of W? or its parametrization. We introduce the QMDP-net, a recurrent policy network. A QMDP-net represents a policy by connecting a parameterized POMDP model with an approximate POMDP algorithm and embedding both in a single, differentiable neural network. Embedding the model allows the policy to generalize over W? effectively. Embedding the algorithm allows us to train the entire network end-to-end and learn a model that compensates for the limitations of the approximate algorithm. Let M (?) = (S, A, O, fT (?|?), fZ (?|?), fR (?|?)) be the embedded POMDP model, where S, A and O are the shared state space, action space, observation space designed manually for all tasks in W? and fT (?|?), fZ (?|?), fR (?|?) are the state-transition, observation, and reward functions to be learned from data. It may appear that a perfect answer to our learning problem would have 3 (a) (b) (c) QMDP planner QMDP planner QMDP planner QMDP planner Policy Bayesian filter Bayesian filter Bayesian filter Bayesian filter Fig. 2: QMDP-net architecture. (a) A policy maps a history of actions and observations to a new action. (b) A QMDP-net is an RNN that imposes structure priors for sequential decision making under partial observability. It embeds a Bayesian filter and the QMDP algorithm in the network. The hidden state of the RNN encodes the belief for POMDP planning. (c) A QMDP-net unfolded in time. fT (?|?), fZ (?|?), and fR (?|?) represent the ?true? underlying models of dynamics, observation, and reward for the task W (?). This is true only if the embedded POMDP algorithm is exact, but not true in general. The agent may learn an alternative model to mitigate an approximate algorithm?s limitations and obtain an overall better policy. In this sense, while QMDP-net embeds a POMDP model in the network architecture, it aims to learn a good policy rather than a ?correct? model. A QMDP-net consists of two modules (Fig. 2). One encodes a Bayesian filter, which performs state estimation by integrating the past history of agent actions and observations into a belief. The other encodes QMDP, a simple, but fast approximate POMDP planner [18]. QMDP chooses the agent?s actions by solving the corresponding fully observable Markov decision process (MDP) and performing one-step look-ahead search on the MDP values weighted by the belief. We evaluate the proposed network architecture in an imitation learning setting. We train on a set of expert trajectories with randomly chosen task parameter values in ? and test with new parameter values. An expert trajectory consist of a sequence of demonstrated actions and observations (a1 , o1 , a2 , o2 , . . .) for some ? 2 ?. The agent does not access the ground-truth states or beliefs along the trajectory during the training. We define loss as the cross entropy between predicted and demonstrated action sequences and use RMSProp [35] for training. See Appendix C.7 for details. Our implementation in Tensorflow [1] is available online at http://github.com/AdaCompNUS/qmdp-net. 4 QMDP-Net We assume that all tasks in a parameterized set W? share the same underlying state space S, action space A, and observation space O. We want to learn a QMDP-net policy for W? , conditioned on the parameters ? 2 ?. A QMDP-net is a recurrent policy network. The inputs to a QMDP-net are the action at 2 A and the observation ot 2 O at time step t, as well as the task parameter ? 2 ?. The output is the action at+1 for time step t + 1. A QMDP-net encodes a parameterized POMDP model M (?) = (S, A, O, T = fT (?|?), Z = fZ (?|?), R = fR (?|?)) and the QMDP algorithm, which selects actions by solving the model approximately. We choose S, A, and O of M (?) manually, based on prior knowledge on W? , specifically, prior knowledge on S, A, and O. In general, S 6= S, A 6= A, and O 6= O. The model states, actions, and observations may be abstractions of their real-world counterparts in the task. In our robot navigation example (Fig. 1), while the robot moves in a continuous space, we choose S to be a grid of finite size. We can do the same for A and O, in order to reduce representational and computational complexity. The transition function T , observation function Z, and reward function R of M (?) are conditioned on ?, and are learned from data through end-to-end training. In this work, we assume that T is the same for all tasks in W? to simplify the network architecture. In other words, T does not depend on ?. End-to-end training is feasible, because a QMDP-net encodes both a model and the associated algorithm in a single, fully differentiable neural network. The main idea for embedding the algorithm in a neural network is to represent linear operations, such as matrix multiplication and summation, by convolutional layers and represent maximum operations by max-pooling layers. Below we provide some details on the QMDP-net?s architecture, which consists of two modules, a filter and a planner. 4 (a) Bayesian filter module (b) QMDP planner module Fig. 3: A QMDP-net consists of two modules. (a) The Bayesian filter module incorporates the current action at and observation ot into the belief. (b) The QMDP planner module selects the action according to the current belief bt . Filter module. The filter module (Fig. 3a) implements a Bayesian filter. It maps from a belief, action, and observation to a next belief, bt+1 = f (bt |at , ot ). The belief is updated in two steps. The first accounts for actions, the second for observations: P b0t (s) = s0 2S T (s, at , s0 )bt (s0 ), (3) bt+1 (s) = ?Z(s, ot )b0t (s), (4) where ot 2 O is the observation received after taking action at 2 A and ? is a normalization factor. We implement the Bayesian filter by transforming Eq. (3) and Eq. (4) to layers of a neural network. For ease of discussion consider our N ?N grid navigation task (Fig. 1a?c). The agent does not know its own state and only observes neighboring cells. It has access to the task parameter ? that encodes the obstacles, goal, and a belief over initial states. Given the task, we choose M (?) to have a N ?N state space. The belief, bt (s), is now an N ?N tensor. Eq. (3) is implemented as a convolutional layer with |A| convolutional filters. We denote the convolutional layer by fT . The kernel weights of fT encode the transition function T in M (?). The output of the convolutional layer, b0t (s, a), is a N ?N ?|A| tensor. b0t (s, a) encodes the updated belief after taking each of the actions, a 2 A. We need to select the belief corresponding to the last action taken by the agent, at . We can directly index b0t (s, a) by at if A = A. In general A 6= A, so we cannot use simple indexing. Instead, we will use ?soft indexing?. First we encode actions in A to actions in A through a learned function fA . fA maps from at to an indexing vector wta , a distribution over actions in A. We then weight b0t (s, a) by wta along the appropriate dimension, i.e. P b0t (s) = a2A b0t (s, a)wta . (5) Eq. (4) incorporates observations through an observation model Z(s, o). Now Z(s, o) is a N ?N ?|O| tensor that represents the probability of receiving observation o 2 O in state s 2 S. In our grid navigation task observations depend on the obstacle locations. We condition Z on the task parameter, Z(s, o) = fZ (s, o|?) for ? 2 ?. The function fZ is a neural network, mapping from ? to Z(s, o). In this paper fZ is a CNN. Z(s, o) encodes observation probabilities for each of the observations, o 2 O. We need the observation probabilities for the last observation ot . In general O 6= O and we cannot index Z(s, o) directly. Instead, we will use soft indexing again. We encode observations in O to observations in O through fO . fO is a function mapping from ot to an indexing vector, wto , a distribution over O. We then weight Z(s, o) by wto , i.e. P Z(s) = o2O Z(s, o)wto . (6) Finally, we obtain the updated belief, bt+1 (s), by multiplying b0t (s) and Z(s) element-wise, and normalizing over states. In our setting the initial belief for the task W (?) is encoded in ?. We initialize the belief in QMDP-net through an additional encoding function, b0 = fB (?). 5 Planner module. The QMDP planner (Fig. 3b) performs value iteration at its core. Q values are computed by iteratively applying Bellman updates, P 0 0 Qk+1 (s, a) = R(s, a) + (7) s0 2S T (s, a, s )Vk (s ), Vk (s) = maxa Qk (s, a). (8) Actions are then selected by weighting the Q values with the belief. We can implement value iteration using convolutional and max pooling layers [28, 34]. In our grid navigation task Q(s, a) is a N ?N ?|A| tensor. Eq. (8) is expressed by a max pooling layer, where Qk (s, a) is the input and Vk (s) is the output. Eq. (7) is a N ?N convolution with |A| convolutional filters, followed by an addition operation with R(s, a), the reward tensor. We denote the convolutional layer by fT0 . The kernel weights of fT0 encode the transition function T , similarly to fT in the filter. Rewards for a navigation task depend on the goal and obstacles. We condition rewards on the task parameter, R(s, a) = fR (s, a|?). fR maps from ? to R(s, a). In this paper fR is a CNN. We implement K iterations of Bellman updates by stacking the layers representing Eq. (7) and Eq. (8) K times with tied weights. After K iterations we get QK (s, a), the approximate Q values for each state-action pair. We weight the Q values by the belief to obtain action values, P q(a) = s2S QK (s, a)bt (s). (9) Finally, we choose the output action through a low-level policy function, f? , mapping from q(a) to the action output, at+1 . QMDP-net naturally extends to higher dimensional discrete state spaces (e.g. our maze navigation task) where n-dimensional convolutions can be used [14]. While M (?) is restricted to a discrete space, we can handle continuous tasks W? by simultaneously learning a discrete M (?) for planning, and fA , fO , fB , f? to map between states, actions and observations in W? and M (?). 5 Experiments The main objective of the experiments is to understand the benefits of structure priors on learning neural-network policies. We create several alternative network architectures by gradually relaxing the structure priors and evaluate the architectures on simulated robot navigation and manipulation tasks. While these tasks are simpler than, for example, Atari games, in terms of visual perception, they are in fact very challenging, because of the sophisticated long-term reasoning required to handle partial observability and distant future rewards. Since the exact state of the robot is unknown, a successful policy must reason over many steps to gather information and improve state estimation through partial and noisy observations. It also must reason about the trade-off between the cost of information gathering and the reward in the distance future. 5.1 Experimental Setup We compare the QMDP-net with a number of related alternative architectures. Two are QMDP-net variants. Untied QMDP-net relaxes the constraints on the planning module by untying the weights representing the state-transition function over the different CNN layers. LSTM QMDP-net replaces the filter module with a generic LSTM module. The other two architectures do not embed POMDP structure priors at all. CNN+LSTM is a state-of-the-art deep CNN connected to an LSTM. It is similar to the DRQN architecture proposed for reinforcement learning under partially observability [10]. RNN is a basic recurrent neural network with a single fully-connected hidden layer. RNN contains no structure specific to planning under partial observability. Each experimental domain contains a parameterized set of tasks W? . The parameters ? encode an environment, a goal, and a belief over the robot?s initial state. To train a policy for W? , we generate random environments, goals, and initial beliefs. We construct ground-truth POMDP models for the generated data and apply the QMDP algorithm. If the QMDP algorithm successfully reaches the goal, we then retain the resulting sequence of action and observations (a1 , o1 , a2 , o2 , . . .) as an expert trajectory, together with the corresponding environment, goal, and initial belief. It is important to note that the ground-truth POMDPs are used only for generating expert trajectories and not for learning the QMDP-net. 6 For fair comparison, we train all networks using the same set of expert trajectories in each domain. We perform basic search over training parameters, the number of layers, and the number of hidden units for each network architecture. Below we briefly describe the experimental domains. See Appendix C for implementation details. Grid-world navigation. A robot navigates in an unknown building given a floor map and a goal. The robot is uncertain of its own location. It is equipped with a LIDAR that detects obstacles in its direct neighborhood. The world is uncertain: the robot may fail to execute desired actions, possibly because of wheel slippage, and the LIDAR may produce false readings. We implemented a simplified version of this task in a discrete n?n grid world (Fig. 1c). The task parameter ? is represented as an n?n image with three channels. The first channel encodes the obstacles in the environment, the second channel encodes the goal, and the last channel encodes the belief over the robot?s initial state. The robot?s state represents its position in the grid. It has five actions: moving in each of the four canonical directions or staying put. The LIDAR observations are compressed into four binary values corresponding to obstacles in the four neighboring cells. We consider both a deterministic and a stochastic variant of the domain. The stochastic variant adds action and observation uncertainties. The robot fails to execute the specified move action and stays in place with probability 0.2. The observations are faulty with probability 0.1 independently in each direction. We trained a policy using expert trajectories from 10, 000 random environments, 5 trajectories from each environment. We then tested on a separate set of 500 random environments. Maze navigation. A differential-drive robot navigates in a maze with the help of a map, but it does not know its pose (Fig. 1d). This domain is similar to the grid-world navigation, but it is significant more challenging. The robot?s state contains both its position and orientation. The robot cannot move freely because of kinematic constraints. It has four actions: move forward, turn left, turn right and stay put. The observations are relative to the robot?s current orientation, and the increased ambiguity makes it more difficult to localize the robot, especially when the initial state is highly uncertain. Finally, successful trajectories in mazes are typically much longer than those in randomly-generated grid worlds. Again we trained on expert trajectories in 10, 000 randomly generated mazes and tested them in 500 new ones. Fig. 4: Highly ambiguous observations in a maze. The four observations (in red) are the same, despite that the robot states are all different. (a) (b) 2-D object grasping. A robot gripper picks up novel objects from a table using a two-finger hand with noisy touch sensors at the finger tips. The gripper uses the fingers to perform compliant motions while maintaining contact with the object or to grasp the object. It knows the shape of the object to be grasped, maybe from an object Fig. 5: Object grasping using touch sensdatabase. However, it does not know its own pose relative ing. (a) An example [3]. (b) Simplified to the object and relies on the touch sensors to localize 2-D object grasping. Objects from the itself. We implemented a simplified 2-D variant of this training set (top) and the test set (bottom). task, modeled as a POMDP [13]. The task parameter ? is an image with three channels encoding the object shape, the grasp point, and a belief over the gripper?s initial pose. The gripper has four actions, each moving in a canonical direction unless it touches the object or the environment boundary. Each finger has 3 binary touch sensors at the tip, resulting in 64 distinct observations. We trained on expert demonstration on 20 different objects with 500 randomly sampled poses for each object. We then tested on 10 previously unseen objects in random poses. 5.2 Choosing QMDP-Net Components for a Task Given a new task W? , we need to choose an appropriate neural network representation for M (?). More specifically, we need to choose S, A and O, and a representation for the functions fR , fT , fT0 , fZ , fO , fA , fB , f? . This provides an opportunity to incorporate domain knowledge in a principled way. For example, if W? has a local and spatially invariant connectivity structure, we can choose convolutions with small kernels to represent fT , fR and fZ . 7 In our experiments we use S = N ?N for N ?N grid navigation, and S = N ?N ?4 for N ?N maze navigation where the robot has 4 possible orientations. We use |A| = |A| and |O| = |O| for all tasks except for the object grasping task, where |O| = 64 and |O| = 16. We represent fT , fR and fZ by CNN components with 3?3 and 5?5 kernels depending on the task. We enforce that fT and fZ are proper probability distributions by using softmax and sigmoid activations on the convolutional kernels, respectively. Finally, fO is a small fully connected component, fA is a one-hot encoding function, f? is a single softmax layer, and fB is the identity function. We can adjust the amount of planning in a QMDP-net by setting K. A large K allows propagating information to more distant states without affecting the number of parameters to learn. However, it results in deeper networks that are computationally expensive to evaluate and more difficult to train. We used K = 20 . . . 116 depending on the problem size. We were able to transfer policies to larger environments by increasing K up to 450 when executing the policy. In our experiments the representation of the task parameter ? is isomorphic to the chosen state space S. While the architecture is not restricted to this setting, we rely on it to represent fT , fZ , fR by convolutions with small kernels. Experiments with a more general class of problems is an interesting direction for future work. 5.3 Results and Discussion The main results are reported in Table 1. Some additional results are reported in Appendix A. For each domain, we report the task success rate and the average number of time steps for task completion. Comparing the completion time is meaningful only when the success rates are similar. QMDP-net successfully learns policies that generalize to new environments. When evaluated on new environments, the QMDP-net has higher success rate and faster completion time than the alternatives in nearly all domains. To understand better the performance difference, we specifically compared the architectures in a fixed environment for navigation. Here only the initial state and the goal vary across the task instances, while the environment remains the same. See the results in the last row of Table 1. The QMDP-net and the alternatives have comparable performance. Even RNN performs very well. Why? In a fixed environment, a network may learn the features of an optimal policy directly, e.g., going straight towards the goal. In contrast, the QMDP-net learns a model for planning, i.e., generating a near-optimal policy for a given arbitrary environment. POMDP structure priors improve the performance of learning complex policies. Moving across Table 1 from left to right, we gradually relax the POMDP structure priors on the network architecture. As the structure priors weaken, so does the overall performance. However, strong priors sometimes over-constrain the network and result in degraded performance. For example, we found that tying the weights of fT in the filter and fT0 in the planner may lead to worse policies. While both fT and fT0 represent the same underlying transition dynamics, using different weights allows each to choose its own approximation and thus greater flexibility. We shed some light on this issue and visualize the learned POMDP model in Appendix B. QMDP-net learns ?incorrect?, but useful models. Planning under partial observability is intractable in general, and we must rely on approximation algorithms. A QMDP-net encodes both a POMDP model and QMDP, an approximate POMDP algorithm that solves the model. We then train the network end-to-end. This provides the opportunity to learn an ?incorrect?, but useful model that compensates the limitation of the approximation algorithm, in a way similar to reward shaping in reinforcement learning [22]. Indeed, our results show that the QMDP-net achieves higher success rate than QMDP in nearly all tasks. In particular, QMDP-net performs well on the well-known Hallway2 domain, which is designed to expose the weakness of QMDP resulting from its myopic planning horizon. The planning algorithm is the same for both the QMDP-net and QMDP, but the QMDP-net learns a more effective model from expert demonstrations. This is true even though QMDP generates the expert data for training. We note that the expert data contain only successful QMDP demonstrations. When both successful and unsuccessful QMDP demonstrations were used for training, the QMDP-net did not perform better than QMDP, as one would expect. QMDP-net policies learned in small environments transfer directly to larger environments. Learning a policy for large environments from scratch is often difficult. A more scalable approach 8 Table 1: Performance comparison of QMDP-net and alternative architectures for recurrent policy networks. SR is the success rate in percentage. Time is the average number of time steps for task completion. D-n and S-n denote deterministic and stochastic variants of a domain with environment size n?n. QMDP Domain QMDP-net Untied QMDP-net LSTM QMDP-net CNN +LSTM RNN SR Time SR Time SR Time SR Time SR Time SR Time Grid D-10 Grid D-18 Grid D-30 99.8 99.0 97.6 8.8 15.5 24.6 99.6 99.0 98.6 8.2 14.6 25.0 98.6 98.8 98.8 8.3 14.8 23.9 84.4 43.8 22.2 12.8 27.9 51.1 90.0 57.8 19.4 13.4 33.7 45.2 87.8 35.8 16.4 13.4 24.5 39.3 Grid S-18 98.1 23.9 98.8 23.9 95.9 24.0 23.8 55.6 41.4 65.9 34.0 64.1 Maze D-29 Maze S-19 63.2 63.1 54.1 50.5 98.0 93.9 56.5 60.4 95.4 98.7 62.5 57.1 9.8 18.9 57.2 79.0 9.2 19.2 41.4 80.8 9.8 19.6 47.0 82.1 Hallway2 37.3 28.2 82.9 64.4 69.6 104.4 82.8 89.7 77.8 99.5 68.0 108.8 Grasp 98.3 14.6 99.6 18.2 98.9 20.4 91.4 26.4 92.8 22.1 94.1 Intel Lab Freiburg 90.2 88.4 85.4 66.9 94.4 107.7 93.2 81.1 20.0 37.4 55.3 51.7 Fixed grid 98.8 17.4 98.6 99.8 17.0 17.6 97.0 19.7 98.4 25.7 - 19.9 98.0 19.8 would be to learn a policy in small environments and transfer it to large environments by repeating the reasoning process. To transfer a learned QMDP-net policy, we simply expand its planning module by adding more recurrent layers. Specifically, we trained a policy in randomly generated 30 ? 30 grid worlds with K = 90. We then set K = 450 and applied the learned policy to several real-life environments, including Intel Lab (100?101) and Freiburg (139?57), using their LIDAR maps (Fig. 1c) from the Robotics Data Set Repository [12]. See the results for these two environments in Table 1. Additional results with different K settings and other buildings are available in Appendix A. 6 Conclusion A QMDP-net is a deep recurrent policy network that embeds POMDP structure priors for planning under partial observability. While generic neural networks learn a direct mapping from inputs to outputs, QMDP-net learns how to model and solve a planning task. The network is fully differentiable and allows for end-to-end training. Experiments on several simulated robotic tasks show that learned QMDP-net policies successfully generalize to new environments and transfer to larger environments as well. The POMDP structure priors and end-to-end training substantially improve the performance of learned policies. Interestingly, while a QMDP-net encodes the QMDP algorithm for planning, learned QMDP-net policies sometimes outperform QMDP. There are many exciting directions for future exploration. First, a major limitation of our current approach is the state space representation. The value iteration algorithm used in QMDP iterates through the entire state space and is well known to suffer from the ?curse of dimensionality?. To alleviate this difficulty, the QMDP-net, through end-to-end training, may learn a much smaller abstract state space representation for planning. One may also incorporate hierarchical planning [8]. Second, QMDP makes strong approximations in order to reduce computational complexity. We want to explore the possibility of embedding more sophisticated POMDP algorithms in the network architecture. While these algorithms provide stronger planning performance, their algorithmic sophistication increases the difficulty of learning. Finally, we have so far restricted the work to imitation learning. It would be exciting to extend it to reinforcement learning. Based on earlier work [28, 34], this is indeed promising. Acknowledgments We thank Leslie Kaelbling and Tom?s Lozano-P?rez for insightful discussions that helped to improve our understanding of the problem. The work is supported in part by Singapore Ministry of Education AcRF grant MOE2016-T2-2-068 and National University of Singapore AcRF grant R-252-000-587112. 9 References [1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL http://tensorflow.org/. [2] J. A. Bagnell, S. Kakade, A. Y. Ng, and J. G. Schneider. Policy search by dynamic programming. In Advances in Neural Information Processing Systems, pages 831?838, 2003. [3] H. Bai, D. Hsu, W. S. Lee, and V. A. Ngo. Monte carlo value iteration for continuous-state POMDPs. In Algorithmic Foundations of Robotics IX, pages 175?191, 2010. [4] B. Bakker, V. Zhumatiy, G. Gruener, and J. Schmidhuber. A robot that reinforcement-learns to identify and memorize important previous observations. In International Conference on Intelligent Robots and Systems, pages 430?435, 2003. [5] J. Baxter and P. L. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial Intelligence Research, 15:319?350, 2001. [6] B. Boots, S. M. Siddiqi, and G. J. Gordon. Closing the learning-planning loop with predictive state representations. The International Journal of Robotics Research, 30(7):954?966, 2011. [7] K. Cho, B. Van Merri?nboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio. Learning phrase representations using RNN encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078, 2014. [8] S. Gupta, J. Davidson, S. Levine, R. Sukthankar, and J. Malik. Cognitive mapping and planning for visual navigation. arXiv preprint arXiv:1702.03920, 2017. [9] T. Haarnoja, A. Ajay, S. Levine, and P. Abbeel. Backprop kf: Learning discriminative deterministic state estimators. In Advances in Neural Information Processing Systems, pages 4376?4384, 2016. [10] M. J. Hausknecht and P. Stone. Deep recurrent Q-learning for partially observable MDPs. arXiv preprint, 2015. URL http://arxiv.org/abs/1507.06527. [11] S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735?1780, 1997. [12] A. Howard and N. Roy. The robotics data set repository (radish), 2003. URL http://radish. sourceforge.net/. [13] K. Hsiao, L. P. Kaelbling, and T. Lozano-P?rez. Grasping POMDPs. In International Conference on Robotics and Automation, pages 4685?4692, 2007. [14] S. Ji, W. Xu, M. Yang, and K. Yu. 3D convolutional neural networks for human action recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(1):221?231, 2013. [15] R. Jonschkowski and O. Brock. End-to-end learnable histogram filters. In Workshop on Deep Learning for Action and Interaction at NIPS, 2016. URL http://www.robotics.tu-berlin.de/fileadmin/ fg170/Publikationen_pdf/Jonschkowski-16-NIPS-WS.pdf. [16] A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pages 1097?1105, 2012. [17] H. Kurniawati, D. Hsu, and W. S. Lee. Sarsop: Efficient point-based POMDP planning by approximating optimally reachable belief spaces. In Robotics: Science and Systems, volume 2008, 2008. [18] M. L. Littman, A. R. Cassandra, and L. P. Kaelbling. Learning policies for partially observable environments: Scaling up. In International Conference on Machine Learning, pages 362?370, 1995. [19] M. L. Littman, R. S. Sutton, and S. Singh. Predictive representations of state. In Advances in Neural Information Processing Systems, pages 1555?1562, 2002. [20] P. Mirowski, R. Pascanu, F. Viola, H. Soyer, A. Ballard, A. Banino, M. Denil, R. Goroshin, L. Sifre, K. Kavukcuoglu, et al. Learning to navigate in complex environments. arXiv preprint arXiv:1611.03673, 2016. [21] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529?533, 2015. 10 [22] A. Y. Ng, D. Harada, and S. Russell. Policy invariance under reward transformations: Theory and application to reward shaping. In International Conference on Machine Learning, pages 278?287, 1999. [23] M. Okada, L. Rigazio, and T. Aoshima. Path integral networks: End-to-end differentiable optimal control. arXiv preprint arXiv:1706.09597, 2017. [24] C. H. Papadimitriou and J. N. Tsitsiklis. The complexity of Markov decision processes. Mathematics of Operations Research, 12(3):441?450, 1987. [25] J. Pineau, G. J. Gordon, and S. Thrun. Applying metric-trees to belief-point POMDPs. In Advances in Neural Information Processing Systems, page None, 2003. [26] G. Shani, R. I. Brafman, and S. E. Shimony. Model-based online learning of POMDPs. In European Conference on Machine Learning, pages 353?364, 2005. [27] G. Shani, J. Pineau, and R. Kaplow. A survey of point-based POMDP solvers. Autonomous Agents and Multi-agent Systems, 27(1):1?51, 2013. [28] T. Shankar, S. K. Dwivedy, and P. Guha. Reinforcement learning via recurrent convolutional neural networks. In International Conference on Pattern Recognition, pages 2592?2597, 2016. [29] D. Silver and J. Veness. Monte-carlo planning in large POMDPs. In Advances in Neural Information Processing Systems, pages 2164?2172, 2010. [30] D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, et al. Mastering the game of Go with deep neural networks and tree search. Nature, 529(7587):484?489, 2016. [31] D. Silver, H. van Hasselt, M. Hessel, T. Schaul, A. Guez, T. Harley, G. Dulac-Arnold, D. Reichert, N. Rabinowitz, A. Barreto, et al. The predictron: End-to-end learning and planning. arXiv preprint, 2016. URL https://arxiv.org/abs/1612.08810. [32] M. T. Spaan and N. Vlassis. Perseus: Randomized point-based value iteration for POMDPs. Journal of Artificial Intelligence Research, 24:195?220, 2005. [33] C. Stachniss. Robotics 2D-laser dataset. URL http://www.ipb.uni-bonn.de/datasets/. [34] A. Tamar, S. Levine, P. Abbeel, Y. Wu, and G. Thomas. Value iteration networks. In Advances in Neural Information Processing Systems, pages 2146?2154, 2016. [35] T. Tieleman and G. Hinton. Lecture 6.5 - rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural networks for machine learning, pages 26?31, 2012. [36] S. Xingjian, Z. Chen, H. Wang, D.-Y. Yeung, W.-k. Wong, and W.-c. Woo. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. In Advances in Neural Information Processing Systems, pages 802?810, 2015. [37] N. Ye, A. Somani, D. Hsu, and W. S. Lee. Despot: Online POMDP planning with regularization. Journal of Artificial Intelligence Research, 58:231?266, 2017. 11
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Robust Optimization for Non-Convex Objectives Robert Chen Computer Science Harvard University Brendan Lucier Microsoft Research New England Yaron Singer Computer Science Harvard University Vasilis Syrgkanis Microsoft Research New England Abstract We consider robust optimization problems, where the goal is to optimize in the worst case over a class of objective functions. We develop a reduction from robust improper optimization to stochastic optimization: given an oracle that returns ?-approximate solutions for distributions over objectives, we compute a distribution over solutions that is ?-approximate in the worst case. We show that derandomizing this solution is NP-hard in general, but can be done for a broad class of statistical learning tasks. We apply our results to robust neural network training and submodular optimization. We evaluate our approach experimentally on corrupted character classification, and robust influence maximization in networks. 1 Introduction In many learning tasks we face uncertainty about the loss we aim to optimize. Consider, for example, a classification task such as character recognition, required to perform well under various types of distortion. In some environments, such as recognizing characters in photos, the classifier must handle rotation and patterned backgrounds. In a different environment, such as low-resolution images, it is more likely to encounter noisy pixelation artifacts. Instead of training a separate classifier for each possible scenario, one seeks to optimize performance in the worst case over different forms of corruption (or combinations thereof) made available to the trainer as black-boxes. More generally, our goal is to find a minimax solution that optimizes in the worst case over a given family of functions. Even if each individual function can be optimized effectively, it is not clear such solutions would perform well in the worst case. In many cases of interest, individual objectives are non-convex and hence state-of-the-art methods are only approximate. In stochastic optimization, where one must optimize a distribution over loss functions, approximate stochastic optimization is often straightforward, since loss functions are commonly closed under convex combination. Can approximately optimal stochastic solutions yield an approximately optimal robust solution? In this paper we develop a reduction from robust optimization to stochastic optimization. Given an ?approximate oracle for stochastic optimization we show how to implement an ?-approximate solution for robust optimization under a necessary extension, and illustrate its effectiveness in applications. Main Results. Given an ?-approximate stochastic oracle for distributions over (potentially nonconvex) loss functions, we show how to solve ?-approximate robust optimization in a convexified solution space. This outcome is ?improper? in the sense that it may lie outside the original solution space, if the space is non-convex. This can be interpreted as computing a distribution over solutions. We show that the relaxation to improper learning is necessary in general: It is NP-hard to achieve robust optimization with respect to the original outcome space, even if stochastic optimization can be solved exactly, and even if there are only polynomially many loss functions. We complement this by showing that in any statistical learning scenario where loss is convex in the predicted dependent variable, we can find a single (deterministic) solution with matching performance guarantees. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Technical overview. Our approach employs an execution of no-regret dynamics on a zero-sum game, played between a learner equipped with an ?-approximate stochastic oracle, and an adversary who aims to find a distribution over loss functions that maximizes the learner?s loss. This game converges to an approximately robust solution, in which the learner and adversary settle upon an ?approximate minimax solution. This convergence is subject to an additive regret term that converges at a rate of T ?1/2 over T rounds of the learning dynamics. Applications. We illustrate the power of our reduction through two main examples. We first consider statistical learning via neural networks. Given an arbitrary training method, our reduction generates a net that optimizes robustly over a given class of loss functions. We evaluate our method experimentally on a character recognition task, where the loss functions correspond to different corruption models made available to the learner as black boxes. We verify experimentally that our approach significantly outperforms various baselines, including optimizing for average performance and optimizing for each loss separately. We also apply our reduction to influence maximization, where the goal is to maximize a concave function (the independent cascade model of influence [9]) over a non-convex space (subsets of vertices in a network). Previous work has studied robust influence maximization directly [7, 3, 12], focusing on particular, natural classes of functions (e.g., edge weights chosen within a given range) and establishing hardness and approximation results. In comparison, our method is agnostic to the particular class of functions, and achieves a strong approximation result by returning a distribution over solutions. We evaluate our method on real and synthetic datasets, with the goal of robustly optimizing a suite of random influence instantiations. We verify experimentally that our approach significantly outperforms natural baselines. Related work. There has recently been a great deal of interest in robust optimization in machine learning [16, 2, 13, 17]. For continuous optimization, the work that is closest to ours is perhaps that by Shalev-Shwartz and Wexler [16] and Namkoong and Duchi [13] that use robust optimization to train against convex loss functions. The main difference is that we assume a more general setting in which the loss functions are non-convex and one is only given access to the stochastic oracle. Hence, the proof techniques and general results from these papers do not apply to our setting. We note that our result generalizes these works, as they can be considered as the special case in which we have a distributional oracle whose approximation is optimal. In particular, [16, Theorem 1] applies to the realizable statistical learning setting where the oracle has small mistake bound C. Our applications require a more general framing that hold for any optimization setting with access to an approximate oracle, and approximation is in the multiplicative sense with respect to the optimal value. In submodular optimization there has been a great deal of interest in robust optimization as well [10, 8, 4]. The work closest to ours is that by He and Kempe [8] who consider a slightly different objective than ours. Kempe and He?s results apply to influence but do not extend to general submodular functions. Finally, we note that unlike recent work on non-convex optimization [5, 1, 6] our goal in this paper is not to optimize a non-convex function. Rather, we abstract the non-convex guarantees via the approximate stochastic oracle. 2 Robust Optimization with Approximate Stochastic Oracles We consider the following model of optimization that is robust to objective uncertainty. There is a space X over which to optimize, and a finite set of loss functions1 L = {L1 , . . . , Lm } where each Li ? L is a function from X to [0, 1]. Intuitively, our goal is to find some x ? X that achieves low loss in the worst-case over loss functions in L. For x ? X , write g(x) = maxi?[m] Li (x) for the worst-case loss of x. The minimax optimum ? is given by ? = min g(x) = min max Li (x). x?X x?X i?[m] (1) The goal of ?-approximate robust optimization is to find x such that g(x) ? ?? . Given a distribution P over solutions X , write g(P) = maxi?[m] Ex?P [Li (x)] for the worst-case expected loss of a solution drawn from P. A weaker version of robust approximation is improper robust optimization: find a distribution P over X such that g(P) ? ?? . 1 We describe an extension to infinite sets of loss functions in the full version of the paper. Our results also extend naturally to the goal of maximizing the minimum of a class of reward functions. 2 Algorithm 1 Oracle Efficient Improper Robust Optimization Input: Objectives L = {L1 , . . . , Lm }, Apx stochastic oracle M , parameters T, ? for each time step t ? [T ] do Set ( t?1 ) X wt [i] ? exp ? Li (x? ) (3) ? =1 Set xt = M (wt ) end for Output: the uniform distribution over {x1 , . . . , xT } Our results take the form of reductions to an approximate stochastic oracle, which finds a solution x ? X that approximately minimizes a given distribution over loss functions.2 Definition 1 (?-Approximate Stochastic Oracle). Given a distribution D over L, an ?-approximate stochastic oracle M (D) computes x? ? X such that EL?D [L(x? )] ? ? min EL?D [L(x)] . x?X 2.1 (2) Improper Robust Optimization with Oracles We first show that, given access to an ?-approximate stochastic oracle, it is possible to efficiently implement improper ?-approximate robust optimization, subject to a vanishing additive loss term. q Theorem 1. Given access to an ?-approximate stochastic oracle, Algorithm 1 with ? = log(m) 2T computes a distribution P over solutions, defined as a uniform distribution over a set {x1 , . . . , xT }, so that r 2 log(m) . (4) max Ex?P [Li (x)] ? ?? + T i?[m] Moreover, for any ? the distribution P computed by Algorithm 1 satisfies: max Ex?P [Li (x)] ? ?(1 + ?)? + i?[m] 2 log(m) . ?T (5) Proof. We give the proof of the first result and defer the second result to the full version of the paper. We can interpret Algorithm 1 in the following way. We define a zero-sum game between a learner and an adversary. The learner?s action set is equal to X and the adversary?s action set is equal to [m]. The loss of the learner when he picks x ? X and the adversary picks i ? [m] is defined as Li (x). The corresponding payoff of the adversary is Li (x). We will run no-regret dynamics on this zero-sum game, where at every iteration t = 1, . . . , T , the adversary will pick a distribution over functions and subsequently the learner picks a solution xt . For simpler notation we will denote with wt the probability density function on [m] associated with the distribution of the adversary. That is, wt [i] is the probability of picking function Li ? L. The adversary picks a distribution wt based on some arbitrary no-regret learning algorithm on the m actions in L. For concreteness consider the case where the adversary picks a distribution based on the multiplicative weight updates algorithm, i.e., (r ) t?1 log(m) X Li (x? ) . (6) wt [i] ? exp 2T ? =1 Subsequently the learner picks a solution xt that is the output of the ?-approximate stochastic oracle on the distribution selected by the adversary at time-step t. That is, xt = M (wt ) . (7) 2 All our results easily extend to the case where the oracle computes a solution that is approximately optimal up to an additive error, rather than a multiplicative one. For simplicity of exposition we present the multiplicative error case as it is more in line with the literature on approximation algorithms. 3 Write (T ) = that q 2 log(m) . T By the guarantees of the no-regret algorithm for the adversary, we have T T 1X 1X EI?wt [LI (xt )] ? max Li (xt ) ? (T ). T t=1 i?[m] T t=1 (8) Combining the above with the guarantee of the stochastic oracle we have ? = min max Li (x) ? min x?X i?[m] x?X ? T T 1X 1X EI?wt [LI (x)] ? min EI?wt [LI (x)] T t=1 T t=1 x?X T 1X1 ? EI?wt [LI (xt )] T t=1 ? 1 ? ? ? ! T 1X Li (xt ) ? (T ) . max i?[m] T t=1 (By oracle guarantee for each t) (By no-regret of adversary) Thus, if we define with P to be the uniform distribution over {x1 , . . . , xT }, then we have derived max Ex?P [Li (x)] ? ?? + (T ) i?[m] (9) as required. A corollary of Theorem 1 is that if the solution space X is convex and the objective functions Li ? L are all convex functions, then we can compute a single solution x? that is approximately minimax optimal. Of course, in this setting one can calculate and optimize the maximum loss directly in time proportional to |L|; this result therefore has the most bite when the set of functions is large. Corollary 2. If the space X is a convex space and each loss function Li ? L is a convex function, PT then the point x? = T1 t=1 xt ? X , where {x1 , . . . , xT } are the output of Algorithm 1, satisfies: r 2 log(m) ? (10) max Li (x ) ? ?? + T i?[m] Proof. By Theorem 1, we get that if P is the uniform distribution over {x1 , . . . , xT } then r 2 log(m) max Ex?P [Li (x)] ? ?? + . T i?[m] Since X is convex, the solution x? = Ex?P [x] is also part of X . Moreover, since each Li ? L is convex, we have that Ex?P [Li (x)] ? Li (Ex?P [x]) = Li (x? ). We therefore conclude r 2 log(m) ? max Li (x ) ? max Ex?P [Li (x)] ? ?? + T i?[m] i?[m] as required. 2.2 Robust Statistical Learning Next we apply our main theorem to statistical learning. Consider regression or classification settings where data points are pairs (z, y), z ? Z is a vector of features, and y ? Y is the dependent variable. The solution space X is then a space of hypotheses H, with each h ? H a function from Z to Y. We also assume that Y is a convex subset of a finite-dimensional vector space. We are given a set of loss functions L = {L1 , . . . , Lm }, where each Li ? L is a functional Li : H ? [0, 1]. Theorem 1 implies that, given an ?-approximate stochastic optimization oracle, we can compute a distribution over T hypotheses from H that achieves an ?-approximate minimax guarantee. If the loss functionals are convex over hypotheses, then we can compute a single ensemble hypothesis h? (possibly from a larger space of hypotheses, if H is non-convex) that achieves this guarantee. 4 Theorem 3. Suppose that L = {L1 , . . . , Lm } are convex functionals. Then the ensemble hypothPT esis h? = T1 t=1 h, where {h1 , . . . , hT } are the hypotheses output by Algorithm 1 given an ?-approximate stochastic oracle, satisfies r 2 log(m) ? . (11) max Li (h ) ? ? min max Li (h) + h?H i?[m] T i?[m] Proof. The proof is similar to the proof of Corollary 2. We emphasize that the convexity condition in Theorem 3 is over the class of hypotheses, rather than over features or any natural parameterization of H (such as weights in a neural network). This is a mild condition that applies to many examples in statistical learning theory. For instance, consider the case where each loss Li (h) is the expected value of some ex-post loss function `i (h(z), y) given a distribution Di over Z ? Y : Li (h) = E(z,y)?Di [`i (h(z), y)] . (12) In this case, it is enough for the function `i (?, ?) to be convex with respect to its first argument (i.e., the predicted dependent variable). This is satisfied by most loss functions used in machine P learning, such as multinomial logistic loss (cross-entropy loss) `(? y , y) = ? c?[k] yc log(? yc ) from 2 multi-class classification, the hinge or the square loss, or squared loss `(? y , y) = k? y ? yk as used in regression. For all these settings, Theorem 3 provides a tool for improper robust learning, where the final hypothesis h? is an ensemble of T base hypotheses from H. Again, the underlying optimization problem can be arbitrarily non-convex in the natural parameters of the hypothesis space; in Section 3.1 we will show how to apply this approach to robust training of neural networks, where the stochastic oracle is simply a standard network training method. For neural networks, the fact that we achieve improper learning (as opposed to standard learning) corresponds to training a neural network with a single extra layer relative to the networks generated by the oracle. 2.3 Robust Submodular Maximization In robust submodular maximization we are given a family of reward functions F = {f1 , . . . , fm }, where each fi ? F is a monotone submodular function from a ground set N of n elements to [0, 1]. Each function is assumed to be monotone and submodular, i.e., for any S ? T ? N , fi (S) ? fi (T ); and for any S, T ? N , f (S ? T ) + f (S ? T ) ? f (S) + f (T ). The goal is to select a set S ? N of size k whose worst-case value over i, i.e., g(S) = mini?[m] fi (S), is at least a 1/? factor of the minimax optimum ? = maxT :|T |?k mini?[m] fi (T ). This setting is a special case of our general robust optimization setting (phrased in terms of rewards rather than losses). The solution space X is equal to the set of subsets of size k among all elements in N and the set F is the set of possible objective functions. The stochastic oracle 1, instantiated in this of submodular functions F (S) = Pmsetting, asks for the following:?given a convex?combination 1 i=1 w[i] ? fi (S), compute a set S such that F (S ) ? ? maxS:|S|?k F (S). Computing the maximum value set of size k is NP-hard even for a single submodular function. The following very simple greedy algorithm computes a (1 ? 1/e)-approximate solution [15]: begin with Scur = ?, and at each iteration add to the current solution Scur the element j ? N ? Scur that has the largest marginal contribution: f ({j} ? Scur ) ? f (Scur ). Moreover, this approximation ratio is known to be the best possible in polynomial time [14]. Since a convex combination of monotone submodular functions is also a monotone submodular function, we immediately get that there exists a (1 ? 1/e)-approximate stochastic oracle that can be computed in polynomial time. The algorithm is formally given in Algorithm 2. Combining the above with Theorem 1 we get the following corollary. Corollary 4. Algorithm 1, with stochastic oracle Mgreedy , computes in time poly(T, n) a distribution P over sets of size k, defined as a uniform distribution over a set {S1 , . . . , ST }, such that   1 log(m) min ES?P [fi (S)] ? 1 ? (1 ? ?)? ? . (13) e ?T i?[m] We show in the full version of the paper that computing a single set S that achieves a (1 ? 1/e)approximation to ? is also N P -hard. This is true even if the functions fi are additive. However, by 5 Algorithm 2 Greedy stochastic Oracle for Submodular Maximization Mgreedy Input: Set of elements N , objectives F = {f1 , . . . , fm }, distribution over objectives w Set Scur = ? for j = 1 to k do Pm Let j ? = arg maxj?N ?Scur i=1 w[i] (fi ({j} ? Scur ) ? fi (Scur )) Set Scur = {j ? } ? Scur end for Figure 1: Sample MNIST image with each of the corruptions applied to it. Background Corruption Set & Shrink Corruption Set (top). Pixel Corruption Set & Mixed Corruption Set (bottom). allowing a randomized solution over sets we can achieve a constant factor approximation to ? in polynomial time. Since the functions are monotone, the above result implies a simple way of constructing a single set S ? that is of larger size than k, which deterministically achieves a constant factor approximation to ? . The latter holds by simply taking the union of the sets {S1 , . . . , ST } in the support of the distribution returned by Algorithm 1. We get the following bi-criterion approximation scheme. Corollary 5. Suppose that we run the reward version of Algorithm 1, with ? =  and for T = log(m) ? 2 , k log(m) ? returning {S1 , . . . , ST }. Then the set S = S1 ? . . . ? ST , which is of size at most ? 2 , satisfies   1 min fi (S ? ) ? 1 ? ? 2 ?. (14) e i?[m] 3 3.1 Experiments Robust Classification with Neural Networks A classic application of our robust optimization framework is classification with neural networks for corrupted or perturbed datasets. We have a data set Z of pairs (z, y) of an image z ? Z and label y ? Y that can be corrupted in m different ways which produces data sets Z1 , . . . , Zm . The hypothesis space H is the set of all neural nets of some fixed architecture and for each possible assignment of weights. We denote each such hypothesis with h(?; ?) : Z ? Y for ? ? Rd , with d being the number of parameters (weights) of the neural net. If we let Di be the uniform distribution over each corrupted data set Zi , then we are interested in minimizing the empirical cross-entropy (aka multinomial logistic) loss in the worst case over these different distributions Di . The latter is a special case of our robust statistical learning framework from Section 2.2. Training a neural network is a non-convex optimization problem and we have no guarantees on its performance. We instead assume that for any given distribution D over pairs (z, y) of images and labels and for any loss function `(h(z; ?), y), training a neural net with stochastic gradient descent run on images drawn from D can achieve an ? approximation to the optimal expected loss, i.e. min??Rd E(z,y)?D [`(h(z; ?), y)]. Notice that this implies an ?-approximate stochastic Oracle for the corrupted dataset robust training problem: for any distribution w over the different corruptions [m], the stochastic oracle asks to give an ?-approximation to the minimization problem: min ??Rd m X w[i] ? E(z,y)?Di [`(h(z; ?), y)] i=1 6 (15) The latter is simply another expected loss problem with distribution over images being the mixture distribution defined by first drawing a corruption index i from w and then drawing a corrupted image from distribution Di . Hence, our oracle assumption implies that SGD on this mixture is an ?-approximation. By linearity of expectation, an alternative way of viewing the stochastic oracle problem is that we are training a neural net on the original distribution of images, but with loss Pm function being the weighted combination of loss functions i=1 w[i] ? `(h(ci (z); ?), y), where ci (z) is the i-th corrupted version of image z. In our experiments we implemented both of these interpretations of the stochastic oracle, which we call the Hybrid Method and Composite Method, respectively, when designing our neural network training scheme (see the full version of the paper for further details). Finally, because we use the cross-entropy loss, which is convex in the prediction of the neural net, we can also apply Theorem 3 to get that the ensemble neural net, which takes the average of the predictions of the neural nets created at each iteration of the robust optimization, will also achieve good worst-case loss (we refer to this as Ensemble Bottleneck Loss). Experiment Setup. We use the MNIST handwritten digits data set containing 55000 training images, 5000 validation images, and 10000 test images, each image being a 28 ? 28 pixel grayscale image. The intensities of these 576 pixels (ranging from 0 to 1) are used as input to a neural network that has 1024 nodes in its one hidden layer. The output layer uses the softmax function to give a distribution over digits 0 to 9. The activation function is ReLU and the network is trained using Gradient Descent with learning parameter 0.5 through 500 iterations of mini-batches of size 100. In general, the corruptions can be any black-box corruption of the image. In our experiments, we consider four four types of corruption (m = 4). See the full version of the paper for further details about corruptions. Baselines. We consider three baselines: (i) Individual Corruption: for each corruption type i ? [m], we construct an oracle that trains a neural network using the training data perturbed by corruption i, and then returns the trained network weights as ?t , for every t = 1, . . . , T . This gives m baselines, one for each corruption type; (ii) Even Split: this baseline alternates between training with different corruption types between iterations. In particular, call the previous m baseline oracles O1 , ..., Om . Then this new baseline oracle will produce ?t with Oi+1 , where i ? t mod m, for every t = 1, ..., T ; (iii) Uniform Distribution: This more advanced baseline runs the robust optimization scheme with the Hybrid Method (see Appendix), but without the distribution updates. Instead, the distribution over 1 1 corruption types is fixed as the discrete uniform [ m , ..., m ] over all T iterations. This allows us to check if the multiplicative weight updates in the robust optimization algorithm are providing benefit. Results. The Hybrid and Composite Methods produce results far superior to all three baseline types, with differences both substantial in magnitude and statistically significant. The more sophisticated Composite Method outperforms the Hybrid Method. Increasing T improves performance, but with diminishing returns?largely because for sufficiently large T , the distribution over corruption types has moved from the initial uniform distribution to some more optimal stable distribution (see the full version for details). All these effects are consistent across the 4 different corruption sets tested. The Ensemble Bottleneck Loss is empirically much smaller than Individual Bottleneck Loss. For the best performing algorithm, the Composite Method, the mean Ensemble Bottleneck Loss (mean Individual Bottleneck Loss) with T = 50 was 0.34 (1.31) for Background Set, 0.28 (1.30) for Shrink Set, 0.19 (1.25) for Pixel Set, and 0.33 (1.25) for Mixed Set. Thus combining the T classifiers obtained from robust optimization is practical for making predictions on new data. 3.2 Robust Influence Maximization We apply the results of Section 2.3 to the robust influence maximization problem. Given a directed graph G = (V, E), the goal is to pick a seed set S of k nodes that maximize an influence function fG (S), where fG (S) is the expected number of individuals influenced by opinion of the members of S. We used fG (S) to be the number of nodes reachable from S (our results extend to other models). In robust influence maximization, the goal is to maximize influence in the worst-case (Bottleneck Influence) over m functions {f1 , . . . , fm }, corresponding to m graphs {G1 , . . . , Gm }, for some fixed seed set of size k. This is a special case of robust submodular maximization after rescaling to [0, 1]. 7 Background Set 2.0 Uniform Hybrid Composite 1.8 1.6 1.4 0 10 30 40 Indiv. Bottleneck Loss 1.32 1.30 1.28 1.26 1.24 0 10 20 30 Number of Iterations T 1.7 1.6 1.5 1.4 1.3 0 10 40 30 40 50 40 50 1.50 1.45 1.40 1.35 1.30 1.25 1.20 50 20 Number of Iterations T Mixed Set 1.55 1.34 1.22 1.8 1.2 50 Pixel Set 1.36 Indiv. Bottleneck Loss 20 Number of Iterations T Shrink Set 1.9 Indiv. Bottleneck Loss Indiv. Bottleneck Loss 2.2 0 10 20 30 Number of Iterations T Figure 2: Comparison of methods, showing mean of 10 independent runs and a 95% confidence band. The criterion is Individual Bottleneck Loss: min[m] E??P [`(h(z; ?), y)], where P is uniform over all solutions ?i for that method. Baselines (i) and (ii) are not shown as they produce significantly higher loss. Experiment Setup. Given a base directed graph G(V, E), we produce m graphs Gi = (V, Ei ) by randomly including each edge e ? E with some probability p. We consider two base graphs and two sets of parameters for each: (i) The Wikipedia Vote Graph [11]. In Experiment A, the parameters are |V | = 7115, |E| = 103689, m = 10, p = 0.01 and k = 10. In Experiment B, change p = 0.015 and k = 3. (ii) The Complete Directed Graph on |V | = 100 vertices. In Experiment A, the parameters are m = 50, p = 0.015 and k = 2. In Experiment B, change p = 0.01 and k = 4. Baselines. We compared our algorithm (Section 2.3) to three baselines: (i) Uniform over Individual Greedy Solutions: Apply greedy maximization (Algorithm 2) on each graph separately, to get g solutions {S1g , . . . , Sm }. Return the uniform distribution over these solutions; (ii) Greedy on Uniform Distribution over Graphs: Return the output of greedy submodular maximization (Algorithm 2) on the uniform distribution over influence functions. This can be viewed as maximizing expected influence; (iii) Uniform over Greedy Solutions on Multiple Perturbed Distributions: Generate T distributions {w?1 , . . . , w?T } over the m functions, by randomly perturbing the uniform distribution. Perturbation magnitudes were chosen s.t. w?t has the same expected `1 distance from uniform as the distribution returned by robust optimization at iteration t. Results. For both graph experiments, robust optimization outperforms all baselines on Bottleneck Influence; the difference is statistically significant as well as large in magnitude for all T > 50. Moreover, the individual seed sets generated at each iteration t of robust optimization themselves achieve empirically good influence as well; see the full version for further details. References [1] Zeyuan Allen Zhu and Elad Hazan. Variance reduction for faster non-convex optimization. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pages 699?707, 2016. [2] Sabyasachi Chatterjee, John C. Duchi, John D. Lafferty, and Yuancheng Zhu. Local minimax complexity of stochastic convex optimization. In Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 3423?3431, 2016. 8 Wikipedia Graph A 65 90 80 70 Robust Opt Perturbed Dist Uniform Dist Individual 60 50 0 50 100 150 Number of Iterations T 55 50 45 40 35 0 50 100 150 Number of Iterations T 200 Complete Graph B 20 18 Bottleneck Influence 35 Bottleneck Influence 60 30 200 Complete Graph A 40 30 25 20 15 10 5 0 Wikipedia Graph B 70 Bottleneck Influence Bottleneck Influence 100 0 50 100 150 Number of Iterations T 16 14 12 10 8 6 4 200 0 50 100 150 Number of Iterations T 200 Figure 3: Comparison for various T , showing mean Bottleneck Influence and 95% confidence on 10 runs. [3] Wei Chen, Tian Lin, Zihan Tan, Mingfei Zhao, and Xuren Zhou. Robust influence maximization. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pages 795?804, 2016. [4] Wei Chen, Tian Lin, Zihan Tan, Mingfei Zhao, and Xuren Zhou. Robust influence maximization. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pages 795?804, 2016. [5] Elad Hazan, Kfir Y. Levy, and Shai Shalev-Shwartz. Beyond convexity: Stochastic quasi-convex optimization. In Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pages 1594?1602, 2015. [6] Elad Hazan, Kfir Yehuda Levy, and Shai Shalev-Shwartz. On graduated optimization for stochastic non-convex problems. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pages 1833?1841, 2016. [7] Xinran He and David Kempe. Robust influence maximization. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pages 885?894, 2016. [8] Xinran He and David Kempe. Robust influence maximization. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pages 885?894, 2016. [9] David Kempe, Jon Kleinberg, and ?va Tardos. Maximizing the spread of influence through a social network. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ?03, pages 137?146, New York, NY, USA, 2003. ACM. [10] Andreas Krause, H. Brendan McMahan, Carlos Guestrin, and Anupam Gupta. Selecting observations against adversarial objectives. In Advances in Neural Information Processing Systems 20, Proceedings of the Twenty-First Annual Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 3-6, 2007, pages 777?784, 2007. [11] Jure Leskovec. Wikipedia vote network. Stanford Network Analysis Project. 9 [12] Meghna Lowalekar, Pradeep Varakantham, and Akshat Kumar. Robust influence maximization: (extended abstract). In Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems, Singapore, May 9-13, 2016, pages 1395?1396, 2016. [13] Hongseok Namkoong and John C. Duchi. Stochastic gradient methods for distributionally robust optimization with f-divergences. In Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 2208?2216, 2016. [14] G. L. Nemhauser and L. A. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of Operations Research, 3(3):177?188, 1978. [15] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions?i. Mathematical Programming, 14(1):265?294, 1978. [16] Shai Shalev-Shwartz and Yonatan Wexler. Minimizing the maximal loss: How and why. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pages 793?801, 2016. [17] Jacob Steinhardt and John C. Duchi. Minimax rates for memory-bounded sparse linear regression. In Proceedings of The 28th Conference on Learning Theory, COLT 2015, Paris, France, July 3-6, 2015, pages 1564?1587, 2015. 10
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Thy Friend is My Friend: Iterative Collaborative Filtering for Sparse Matrix Estimation Christian Borgs Jennifer Chayes Christina E. Lee [email protected] [email protected] [email protected] Microsoft Research New England One Memorial Drive, Cambridge MA, 02142 Devavrat Shah [email protected] Massachusetts Institute of Technology 77 Massachusetts Ave, Cambridge, MA 02139 Abstract The sparse matrix estimation problem consists of estimating the distribution of an n ? n matrix Y , from a sparsely observed single instance of this matrix where the entries of Y are independent random variables. This captures a wide array of problems; special instances include matrix completion in the context of recommendation systems, graphon estimation, and community detection in (mixed membership) stochastic block models. Inspired by classical collaborative filtering for recommendation systems, we propose a novel iterative, collaborative filteringstyle algorithm for matrix estimation in this generic setting. We show that the mean squared error (MSE) of our estimator goes to 0 as long as ?(d2 n) random entries from a total of n2 entries of Y are observed (uniformly sampled), E[Y ] has rank d, and the entries of Y have bounded support. The maximum squared error across all entries converges to 0 with high probability as long as we observe a little more, ?(d2 n ln2 (n)) entries. Our results are the best known sample complexity results in this generality. Our intuitive, easy to implement iterative nearest-neighbor style algorithm matches the conjectured sample complexity lower bound of d2 n for a computationally efficient algorithm for detection in the mixed membership stochastic block model. 1 Introduction In this work, we propose and analyze an iterative similarity-based collaborative filtering algorithm for the sparse matrix completion problem with noisily observed entries. As a prototype for such a problem, consider a noisy observation of a social network where observed interactions are signals of true underlying connections. We might want to predict the probability that two users would choose to connect if recommended by the platform, e.g. LinkedIn. As a second example, consider a recommendation system where we observe movie ratings provided by users, and we may want to predict the probability distribution over ratings for specific movie-user pairs. The classical collaborative filtering approach is to compute similarities between pairs of users by comparing their commonly rated movies. For a social network, similarities between users would be computed by comparing their sets of friends. We will be particularly interested in the very sparse case where most pairs of users have no common friends, or most pairs of users have no commonly rated movies; thus there is insufficient data to compute the traditional similarity metrics. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. To overcome this limitation, we propose a novel algorithm which computes similarities iteratively, incorporating information within a larger radius neighborhood. Whereas traditional collaborative filtering learns the preferences of a user through the ratings of her/his ?friends?, i.e. users who share similar ratings on commonly rated movies, our algorithm learns about a user through the ratings of the friends of her/his friends, i.e. users who may be connected through an indirect path in the data. For a social network, this intuition translates to computing similarities of two users by comparing the boundary of larger radius neighborhoods of their connections in the network. While an actual implementation of our algorithm will benefit from modifications to make it practical, we believe that our approach is very practical; indeed, we plan to implement it in a corporate setting. Like all such nearest-neighbor style algorithms, our algorithm can be accelerated and scaled to large datasets in practice by using a parallel implementation via an approximate nearest neighbor data structure. In this paper, however, our goal is to describe the basic setting and concept of the algorithm, and provide clear mathematical foundation and analysis. The theoretical results indicate that this method achieves consistency (i.e. guaranteed convergence to the correct solution) for very sparse datasets for a reasonably general Latent Variable Model with bounded entries. The problems discussed above can be mathematically formulated as a matrix estimation problem, where we observe a sparse subset of entries in an m ? n random matrix Y , and we wish to complete or de-noise the matrix by estimating the probability distribution of Yij for all (i, j). Suppose that Yij is categorical, taking values in [k] according to some unknown distribution. The task of estimating the distribution of Yij can be reduced to k ? 1 smaller tasks of estimating the expectation of a binary data matrix, e.g. Y t where Yijt = I(Yij = t) and E[Yijt ] = P(Yij = t). If the matrix that we would like to learn is asymmetric,  we can transform it to an equivalent symmetric model by defining a new data matrix Y 0 = Y0T Y0 . Therefore, for the remainder of the paper, we will assume a n ? n symmetric matrix which takes values in [0, 1] (real-valued or binary), but as argued above, our results apply more broadly to categorical-valued asymmetric matrices. We assume that the data is generated from a Latent Variable Model in which latent variables ?1 , . . . , ?n are sampled independently from U [0, 1], and the distribution of Yij is such that E[Yij |?i , ?j ] = f (?i , ?j ) ? Fij for some latent function f . Our goal is to estimate the matrix F . It is worth remarking that the Latent Variable Model is a canonical representation for exchangeable arrays as shown by Aldous and Hoover [5, 25, 7]. We present a novel algorithm for estimating F = [Fij ] from a sparsely sampled dataset {Yij }(i,j)?E where E ? [n] ? [n] is generated by assuming each entry is observed independently with probability p. We require that the latent function f when regarded as an integral operator has finite spectrum with rank d. We prove that the mean squared error (MSE) of our estimates converges to zero at a rate of O((pn)?1/5 ) as long as the sparsity p = ?(d2 n?1 ) (i.e. ?(d2 n) total observations). In addition, with high probability, the maximum squared error converges to zero at a rate of O((pn)?1/5 ) as long as the sparsity p = ?(d2 n?1 ln2 (n)). Our analysis applies to a generic noise setting as long as Yij has bounded support. Somewhat surprisingly, our simple nearest-neighbor style algorithm matches the conjectured sample complexity lower bound of total of d2 n samples for a computationally efficient algorithm, arising in the context of the mixed membership stochastic block model for detection (weaker than MSE going to 0). Our work takes inspiration from [1, 2, 3], which estimates clusters of the stochastic block model by computing distances from local neighborhoods around vertices. We improve upon their analysis to provide MSE bounds for the general latent variable model with finite spectrum, which includes a larger class of generative models such as mixed membership stochastic block models, while they consider the stochastic block model with non-overlapping communities. We show that our results hold even when the rank d increases with n, as long as d = o((pn)1/2 ). As compared to spectral methods such as [28, 39, 20, 19, 18], our analysis handles the general bounded noise model and holds for sparser regimes, only requiring p = ?(n?1 ). Related work. The matrix estimation problem introduced above includes as specific cases problems from different areas of literature: matrix completion popularized in the context of recommendation systems, graphon estimation arising from the asymptotic theory of graphs, and community detection using the stochastic block model or its generalization known as the mixed membership stochastic block model. The key representative results for each of these are mentioned in Table 1. We discuss the scaling of the sample complexity with respect to d (model complexity, usually rank) and n for polynomial time algorithms, including results for both mean squared error convergence, exact recovery in the noiseless setting, and convergence with high probability in the noisy setting. As can 2 Table 1: Sample Complexity of Related Literature grouped in sections according to the following areas ?matrix completion, 1-bit matrix completion, stochastic block model, mixed membership stochastic block model, graphon estimation, and our results Paper Sample Complexity Data/Noise Expected matrix Guarantee [27] [28] [37] [19] [18] [32] [17] [27] [39] ?(dn) ?(dn max(log n, d)), ?(dn) ?(dn log n) ?(n max(d, log2 n)) ?(dn log6 n) ?(n3/2 ) ?(dn log2 n max(d, log4 n)) ?(dn max(d, log n)) ?(dn log2 n) noiseless iid Gaussian iid Gaussian iid Gaussian indep bounded iid bounded noiseless noiseless noiseless rank d rank d rank d rank d rank d Lipschitz rank d rank d rank d MSE? 0 MSE? 0 MSE? 0 MSE? 0 MSE? 0 MSE? 0 exact recovery exact recovery exact recovery [19] [20] ?(n max(d log n, log2 n, d2 )) ?(n max(d, log n)), ?(dn) binary entries binary entries rank d rank d MSE? 0 MSE? 0 [1, 3] [1] ?(d2 n) ?(dn log n) binary entries binary entries d blocks d blocks (SBM) partial recovery exact recovery [6] [40] ?(d2 n polylog n) ?(d2 n) binary entries binary entries rank d rank d whp error ? 0 detection [4] [43] [10] ?(n2 ) ?(n2 ) ?(n) binary entries binary entries binary entries monotone row sum piecewise Lipschitz monotone row sum MSE? 0 MSE? 0 MSE? 0 this work ?(d2 n) ?(d2 n log2 n) indep bounded indep bounded rank d, Lipschitz rank d, Lipschitz MSE? 0 whp error ? 0 be seen from Table 1, our result provides the best sample complexity for the general matrix estimation problem with bounded entries noise model and rank d, as the other models either require extra log factors, or impose additional requirements on the noise model or the expected matrix. Similarly, ours is the best known sample complexity for high probability max-error convergence to 0 for the general rank d bounded entries setting, as other results either assume block constant or noiseless. It is worth comparing our results with the known lower bounds on the sample complexity. For the special case of matrix completion with an additive noise model, i.e. Yij = E[Yij ] + ?ij and ?ij are i.i.d. zero mean, [16, 20] showed that ?(dn) samples are needed for a consistent estimator, i.e. MSE convergence to 0, and [17] showed that dn log n samples are needed for exact recovery. There is a conjectured computational lower bound for the mixed membership stochastic block model of d2 n even for detection, which is weaker than MSE going to 0. Recently, [40] showed a partial result that this computational lower bound holds for algorithms that rely on fitting low-degree polynomials to the observed data. Given that these lower bounds apply to special cases of our setting, it seems that our result is nearly optimal if not optimal in terms of its dependence on both n and d for MSE convergence as well as high probability (near) exact recovery. Next we provide a brief overview of prior works reported in the Tables 1. In the context of matrix completion, there has been much progress under the low-rank assumption. Most theoretically founded methods are based on spectral decompositions or minimizing a loss function with respect to spectral constraints [27, 28, 15, 17, 39, 37, 20, 19, 18]. A work that is closely related to ours is by [32]. It proves that a similarity based collaborative filtering-style algorithm provides a consistent estimator for matrix completion under the generic model when the latent function is Lipschitz, not just low ? 3/2 ) samples. In a sense, ours can be viewed as an algorithmic rank; however, it requires O(n generalization of [32] that handles the sparse sampling regime and a generic noise model. Most of the results in matrix completion require additive noise models, which do not extend to setting when the observations are binary or quantized. The USVT estimator is able to handle general bounded noise, although it requires a few log factors more in its sample complexity [18]. Our work removes the extra log factors while still allowing for general bounded noise. 3 There is also a significant amount of literature which looks at the estimation problem when the data matrix is binary, also known as 1-bit matrix completion, stochastic block model (SBM) parameter estimation, or graphon estimation. The latter two terms are found within the context of community detection and network analysis, as the binary data matrix can alternatively be interpreted as the adjacency matrix of a graph ? which are symmetric, by definition. Under the SBM, each vertex is associated to one of d community types, and the probability of an edge is a function of the community types of both endpoints. Estimating the n ? n parameter matrix becomes an instance of matrix estimation. In SBM, the expected matrix is at most rank d due to its block structure. Precise thresholds for cluster detection (better than random) and estimation have been established by [1, 2, 3]. Our work, both algorithmically and technically, draws insight from this sequence of works, extending the analysis to a broader class of generative models through the design of an iterative algorithm, and improving the technical results with precise MSE bounds. The mixed membership stochastic block model (MMSBM) allows each vertex to be associated to a length d vector, which represents its weighted membership in each of the d communities. The probability of an edge is a function of the weighted community memberships vectors of both endpoints, resulting in an expected matrix with rank at most d. Recent work by [40] provides an algorithm for weak detection for MMSBM with sample complexity d2 n, when the community membership vectors are sparse and evenly weighted. They provide partial results to support a conjecture that d2 n is a computational lower bound, separated by a gap of d from the information theoretic lower bound of dn. This gap was first shown in the simpler context of the stochastic block model [21]. Our results also achieve this conjectured lower bound, with a sample complexity of ?(d2 n) in order to guarantee consistency, which is much stronger than weak detection. Graphon estimation extends SBM and MMSBM to the generic Latent Variable Model where the probability of an edge can be any measurable function f of real-valued types (or latent variables) associated to each endpoint. Graphons were first defined as the limiting object of a sequence of large dense graphs [14, 22, 34], with recent work extending the theory to sparse graphs [12, 13, 11, 41]. In the graphon estimation problem, we would like to estimate the function f given an instance of a graph generated from the graphon associated to f . [23, 29] provide minimax optimal rates for graphon estimation; however a majority of the proposed estimators are not computable in polynomial time, since they require optimizing over an exponentially large space (e.g. least squares or maximum likelihood) [42, 10, 9, 23, 29]. [10] provided a polynomial time method based on degree sorting in the special case when the expected degree function is monotonic. To our knowledge, existing positive results for sparse graphon estimation require either strong monotonicity assumptions [10], or rank constraints as assumed in the SBM, the 1-bit matrix completion, and in this work. We call special attention to the similarity based methods which are able to bypass the rank constraints, relying instead on smoothness properties of the latent function f (e.g. Lipschitz) [43, 32]. They hinge upon computing similarities between rows or columns by comparing commonly observed entries. Similarity based methods, also known in the literature as collaborative filtering, have been successfully employed across many large scale industry applications (Netflix, Amazon, Youtube) due to its simplicity and scalability [24, 33, 30, 38]; however the theoretical results have been relatively sparse. These recent results suggest that the practical success of these methods across a variety of applications may be due to its ability to capture local structure. A key limitation of this approach is that it requires a dense dataset with sufficient entries in order to compute similarity metrics, requiring that each pair of rows or columns has a growing number of overlapped observed entries, which does not hold when p = o(n?1/2 ). This work overcomes this limitation in an intuitive and simple way; rather than only considering directly overlapped entries, we consider longer ?paths? of data associated to each row, expanding the set of associated datapoints until there is sufficient overlap. Although we may initially be concerned that this would introduce bias and variance due to the sparse sampling, our analysis shows that in fact the estimate does converge to the true solution. The idea of comparing vertices by looking at larger radius neighborhoods was introduced in [1], and has connections to belief propagation [21, 3] and the non-backtracking operator [31, 26, 36, 35, 8]. The non-backtracking operator was introduced to overcome the issue of sparsity. For sparse graphs, vertices with high-degree dominate the spectrum, such that the informative components of the spectrum get hidden behind the high degree vertices. The non-backtracking operator avoids paths that immediately return to the previously visited vertex in a similar manner as belief propagation, and its spectrum has been shown to be more well-behaved, perhaps adjusting for the high degree vertices, which get visited very often by paths in the graph. In our algorithm, the neighborhood paths 4 are defined by first selecting a rooted tree at each vertex, thus enforcing that each vertex along a path in the tree is unique. This is important in our analysis, as it guarantees that the distribution of vertices at the boundary of each subsequent depth of the neighborhood is unbiased, since the sampled vertices are freshly visited. 2 Model We shall use graph and matrix notations in an interchangeable manner. For each pair of vertices (i.e. row or column indices) u, v ? [n], let Yuv ? [0, 1] denote its random realization. Let E denote the edges. If (u, v) ? E, Yuv is observed; otherwise it is unknown. ? Each vertex u ? [n] is associated to a latent variable ?u ? U [0, 1] sampled i.i.d. ? For each (u, v) ? [n] ? [n], Yuv = Yvu ? [0, 1] is a bounded random variable. Conditioned on {?i }i?[n] , the random variables {Yuv }1?u<v?n are independent.   ? Fuv := E Yuv | {?w }w?[n] = f (?u , ?v ) ? [0, 1] for a symmetric L-Lipschitz function f . ? The function f , when regarded as an integral operator, has finite spectrum with rank d. That is, Pd f (?u , ?v ) = k=1 ?k qk (?u )qk (?v ), where qk are orthonormal L2 -integrable basis functions. We assume that there exists some B such that |qk (y)| ? B for all k and y ? [0, 1]. ? For every (unordered) index pair (u, v), the entry is observed independently with probability p, i.e. (u, v) ? E and Muv = Mvu = Yuv . If (u, v) ? / E, then Muv = 0. The data (E, M ) can be viewed as a weighted undirected graph over n vertices with each (u, v) ? E having weights Muv . The goal is to estimate the matrix F = [Fuv ]u,v?[n] . Let ? denote the d ? d diagonal matrix with {?k }k?[d] as the diagonal entries. Let the eigenvalues be sorted in such a way that |?1 | ? |?2 | ? ? ? ? ? |?d | > 0. Let Q denote the d ? n matrix where Q(k, u) = qk (?u ). Since Q is a random matrix depending on the sampled ?, it is not guaranteed to be an orthonormal matrix (even though qk are orthonormal functions). By definition, it follows that F = QT ?Q. Let d0 be the ? denote be the d ? d0 matrix where ?(a, ? b) = ?a?1 . number of distinct valued eigenvalues. Let ? b Discussing Assumptions. The latent variable model imposes a natural and mild assumption, as Aldous and Hoover proved that if the network is exchangeable, i.e. the distribution over edges is invariant under permutations of vertex labels, then the network can be equivalently represented by a latent variable model [5, 25, 7]. Exchangeability is reasonable for anonymized datasets for which the identity of entities can be easily renamed. Our model additionally requires that the function is L-Lipschitz and has finite spectrum when regarded as an integral operator, i.e. F is low rank; this includes interesting scenarios such as the mixed membership stochastic block model and finite degree polynomials. We can also relax the condition to piecewise Lipschitz, as we only need to ensure that for every vertex u there are sufficiently many vertices v which are similar in function value to u. We assume observations are sampled independently with probability p; however, we discuss a possible solution for dealing with non-uniform sampling in Section 5. 3 Algorithm The algorithm that we propose uses the concept of local approximation, first determining which datapoints are similar in value, and then computing neighborhood averages for the final estimate. All similarity-based collaborative filtering methods have the following basic format: 1. Compute distances between pairs of vertices, e.g., R1 dist(u, a) ? 0 (f (?u , t) ? f (?a , t))2 dt. (1) 2. Form estimate by averaging over ?nearby? datapoints, P 1 F?uv = |Euv (a,b)?Euv Mab , | (2) where Euv := {(a, b) ? E s.t. dist(u, a) < ?n , dist(v, b) < ?n }. 5 The choice of ?n = (c1 pn)?1/5 will be small enough to drive the bias to zero, ensuring the included datapoints are close in value, yet large enough to reduce the variance, ensuring |Euv | diverges. Inutition. Various similarity-based algorithms differ in the distance computation (Step 1). For dense datasets, i.e. p = ?(n?1/2 ), previous works have proposed and analyzed algorithms which approximate the L2 distance of (1) by using variants of the finite sample approximation, P (3) dist(u, a) = |X1ua | y?Xua (Fuy ? Fay )2 , where y ? Xua iff (u, y) ? E and (a, y) ? E [4, 43, 32]. For sparse datasets, with high probability, Xua = ? for almost all pairs (u, a), such that this distance cannot be computed. In this paper we are interested in the sparse setting when p is significantly smaller than n?1/2 , down to the lowest threshold of p = ?(n?1 ). If we visualize the data via a graph with edge set E, then (3) corresponds to comparing common neighbors of vertices u and a. A natural extension when u and a have no common neighbors, is to instead compare the r-hop neighbors of u and a, i.e. vertices y which are at distance exactly r from both u and a. We compare the product of weights along edges in the path from u to y and a to y respectively, which in expectation approximates R Qr?2 P f (?u , t1 )( s=1 f (ts , ts+1 ))f (tr?1 , ?y )d~t = k ?rk qk (?u )qk (?y ) = eTu QT ?r Qey . (4) [0,1]r?1 We choose a large enough r such that there are sufficiently many ?common? vertices y which have paths to both u and a, guaranteeing that our distance can be computed from a sparse dataset. Algorithm Details. We present and discuss details of each step of the algorithm, which primarily involves computing pairwise distances (or similarities) between vertices. Step 1: Sample Splitting. We partition the datapoints into disjoint sets, which are used in different steps of the computation to minimize correlation across steps for the analysis. Each edge in E is independently placed into E1 , E2 , or E3 , with probabilities c1 , c2 , and 1 ? c1 ? c2 respectively. Matrices M1 , M2 , and M3 contain information from the subset of the data in M associated to E1 , E2 , and E3 respectively. M1 is used to define local neighborhoods of each vertex, M2 is used to compute similarities of these neighborhoods, and M3 is used to average over datapoints for the final estimate in (2). Step 2: Expanding the Neighborhood. We first expand local neighborhoods of radius r around each vertex. Let Su,s denote the set of vertices which are at distance s from vertex u in the graph defined by edge set E1 . Specifically, i ? Su,s if the shortest path in G1 = ([n], E1 ) from u to i has a length of s. Let Tu denote a breadth-first tree in G1 rooted at vertex u. The breadth-first property ensures that the length of the path from u to i within Tu is equal to the length of the shortest path from u to i in G1 . If there is more than one valid breadth-first tree rooted at u, choose one uniformly at random. Let Nu,r ? [0, 1]n denote the following vector with support on the boundary of the r-radius neighborhood of vertex u (we also call Nu,r the neighborhood boundary): (Q (a,b)?pathTu (u,i) M1 (a, b) if i ? Su,r , Nu,r (i) = 0 if i ? / Su,r , where pathTu (u, i) denotes the set of edges along the path from u to i in the tree Tu . The sparsity of Nu,r (i) is equal to Su,r , and the value of the coordinate Nu,r (i) is equal to the product of weights ?u,r denote the normalized neighborhood boundary such that along the path from u to i. Let N ln(1/p) ? . Nu,r = Nu,r /|Su,r |. We will choose radius r to be r = 86ln(c 1 pn) Step 3: Computing the distances. For each vertex, we present two variants for estimating the distance. 1. For each pair (u, v), compute dist1 (u, v) according to    1?c1 p ?u,r ? N ?v,r T M2 N ?u,r+1 ? N ?v,r+1 . N c2 p 2. For each pair (u, v), compute distance according to P dist2 (u, v) = i?[d0 ] zi ?uv (r, i), 6 where ?uv (r, i) is defined as  1?c1 p c2 p ?u,r ? N ?v,r N T  ?u,r+i ? N ?v,r+i , M2 N 0 ? T z = ?2 1. z always exists and is unique and z ? Rd is a vector that satisfies ?2r+2 ? T ?2r ? is a Vandermonde matrix, and ? 1 lies within the span of its columns. because ? Computing dist1 does not require knowledge of the spectrum of f . In our analysis we prove that the expected squared error of the estimate computed in (2) using dist1 converges to zero with n for p = ?(n?1+ ) for some  > 0, i.e. p must be polynomially larger than n?1 . Although computing dist2 requires knowledge of the spectrum of f to determine the vector z, the expected squared error of the estimate computed in (2) using dist2 conveges to zero for p = ?(n?1 ), which includes the sparser settings when p is only larger than n?1 by polylogarithmic factors. It seems plausible that the technique employed by [2] could be used to design a modified algorithm which does not need to have prior knowledge of the spectrium. They achieve this for the stochastic block model case by bootstrapping the algorithm with a method which estimates the spectrum first and then computes pairwise distances with the estimated eigenvalues. Step 4: Averaging datapoints to produce final estimate. The estimate F? (u, v) is computed by averaging over nearby points defined by the distance estimates dist1 (or dist2 ). Recall that B ? 1 was assumed in the model definition to upper bound supy?[0,1] |qk (y)|. Let Euv1 denote the set of undirected edges (a, b) such that (a, b) ? E3 and both dist1 (u, a) and dist1 (v, b) are less than ?1 (n) = (c1 pn)?1/5 . The final estimate F? (u, v) produced by using dist1 is computed by averaging over the undirected edge set Euv1 , X 1 M3 (a, b). (5) F? (u, v) = |Euv1 | (a,b)?Euv1 Let Euv2 denote the set of undirected edges (a, b) such that (a, b) ? E3 , and both dist2 (u, a) and dist2 (v, b) are less than ?2 (n) = (c1 pn)?1/5 . The final estimate F? (u, v) produced by using dist2 is computed by averaging over the undirected edge set Euv2 , X 1 M3 (a, b). (6) F? (u, v) = |Euv2 | (a,b)?Euv2 4 Main Results We prove bounds on the estimation error of our algorithm in terms of the mean squared error (MSE), h i P 1 2 ? MSE := E n(n?1) , u6=v (Fuv ? Fuv ) which averages the squared error over all edges. It follows from the model that R1 Pd (f (?u , y) ? f (?v , y))2 dy = k=1 ?2k (qk (?u ) ? qk (?v ))2 = k?Q(eu ? ev )k22 . 0 The key part of the analysis is to show that the computed distances are in fact good estimates of k?Q(eu ? ev )k22 . The analysis essentially relies on showing that the neighborhood growth around a vertex behaves according to its expectation, according to some properly defined notion. The radius r must be small enough to guarantee that the growth of the size of the neighborhood boundary is exponential, increasing at a factor of approximately c1 pn. However, if the radius is too small, then the boundaries of the respective neighborhoods of the two chosen vertices would have a small intersection, so that estimating the similarities based on the small intersection of datapoints would result in high variance. Therefore, the choice of r is critical to the algorithm and analysis. We are able to prove bounds on the squared error when r is chosen to satisfy the following conditions:     1 ln(1/c1 p) ln(1/c1 p) 6 ln(1/p) ln(1/p) r + d0 ? 8 7ln(9c = ? , r + ? (7) 2 c pn/8|? |) = ? ln(c pn) . pn/8) ln(c pn) 8 ln(7|? | 1 1 1 1 1 d 2 The parameter d0 denotes the number of distinct valued eigenvalues in the spectrum of f , (?1 . . . ?d ), and determines the number of different radius ?measurements? involved in computing dist2 (u, v). 7 Computing dist1 (u, v) only involves a single measurement, thus the left hand side of (7) can be reduced to r + 1 instead of r + d0 . When p is above a threshold, we choose c1 to decrease with n to ensure (7) can be satisfied, sparsifying the edge set E1 used for expanding the neighborhood around a vertex . When the sample probability is polynomially larger than n?1 , i.e. p = n?1+ for some ? ?1 ), we  > 0, these constraints imply that r is a constant with respect to n. However, if p = O(n will need r to grow with n according to a rate of 6 ln(1/p)/8 ln(c1 pn). Theorem 4.1. If p = n?1+ for some  > 0, with a choice of c1 such that c1 pn =  1 ? max(pn, (p6 n7 ) 19 ) , there exists a constant r (with respect to n) which satisfies (7). If d = o((c1 pn)1/2 ), then the estimate computed using dist1 with parameter r achieves     MSE = O |?d |?2r (c1 pn)?1/5 = O (c1 pn)?1/5 .   pn)1/2  With probability greater than 1 ? O d exp ? (c19B , the estimate satisfies 2d kF? ? F kmax := max |F?ij ? Fij | = O(|?d |?r (c1 pn)?1/10 ). i,j Theorem 4.1 proves that the mean squared error (MSE) of the estimate computed with dist1 is bounded by O(|?d |?2r (c1 pn)?1/5 ). Therefore, our algorithm with dist1 provides a consistent estimate when r is constant with respect to n, which occurs for p = n?1+ for some  > 0. In fact, the reason why the error blows up with a factor of |?d |?2r is because we compute the distance by summing product of weights over paths of length 2r. From (4), we see that in expectation, when we take the product of edge weights over a path of length r from u to y, instead of computing f (?u , ?y ) = eTu Q?Qey , the expression concentrates around eTu Q?r Qey , which contains extra factors of ?r?1 . Therefore, by computing over a radius r, the calculation in dist1 will approximate k?r+1 Q(eu ? ev )k22 rather than our intended k?Q(eu ? ev )k22 , thus leading to an error factor of |?d |?2r . It turns out that dist2 adjusts for this bias, as the multiple measurements ?uv (r, i) with different length paths allows us to separate out ek ?Q(eu ? ev ) for all k with distinct values of ?k .   1 Theorem 4.2. If p = O(n?2/3 ), with a choice of c1 such that c1 pn = ? max(pn, (p6 n7 ) (8d0 +11) ) , there exists a value for r which satisfies (7). If d = o((c1 pn)1/2 ) and d = o(r), then the estimate computed using dist2 with parameter r achieves   MSE = O (c1 pn)?1/5 .   pn)1/2  , the estimate satisfies If p = ?(n?1 d2 ln2 (n)), with probability 1 ? O d exp ? (c19B 2d kF? ? F kmax := max |F?ij ? Fij | = O((c1 pn)?1/10 ). i,j Theorem 4.2 proves that the mean squared error (MSE) of the estimate computed using dist2 is bounded by O((c1 pn)?1/5 ); and thus the estimate is consistent in the ultra sparse sampling regime of p = ?(d2 n?1 ). We also present high probability bounds on the squared error of each entry. Lemma 4.3. For any u, v ? [n], if d = o((c1 pn)1/2 ), with probability at least   2 pn)1/2  (c1 pn)?2/5  1 ? O d exp ? (c18B + exp ? c3 pn , 2d 48L2 |?1 |2r the squared error of the estimate computed with dist1 for parameter r satisfying (7) is bounded by (F?uv ? f (?u , ?v ))2 = O(|?d |?2r (c1 pn)?1/5 ). Lemma 4.4. For any u, v ? [n], assuming d = o((c1 pn)1/2 ) and d = o(r), with probability at least    pn)1/2 1 ? O d exp ? (c18B , 2d the squared error of the estimate computed with dist2 for parameter r satisfying (7) is bounded by (F?uv ? f (?u , ?v ))2 = O((c1 pn)?1/5 ). 8 5 Discussion In this work we presented a similarity based collaborative filtering algorithm which is provably consistent in sparse sampling regimes, as long as the sample probability p = ?(n?1 ). The algorithm computes similarity between two users by comparing their local neighborhoods. Our model assumes that the data matrix is generated according to a latent variable model, in which the weight on an observed edge (u, v) is equal in expectation to a function f over associated latent variables ?u and ?v . We presented two variants for computing similarities (or distances) between vertices. Computing dist1 does not require knowledge of the spectrum of f , but the estimate requires p to be polynomially larger than n in order to guarantee the expected squared error converges to zero. Computing dist2 uses the knowledge of the spectrum of f , but it provides an estimate that is provably consistent for a significantly sparse regime, only requiring that p = ?(n?1 ). The mean squared error of both algorithms is bounded by O((pn)?1/5 ). Since the computation is based on of comparing local neighborhoods within the graph, the algorithm can be easily implemented for large scale datasets where the data may be stored in a distributed fashion optimized for local graph computations. Practical implementation. In practice, we do not know the model parameters, and we would use cross validation to tune the radius r and threshold ?n . If r is either too small or too large, then the vector Nu,r will be too sparse. The threshold ?n trades off between bias and variance of the final estimate. Since we do not know the spectrum, dist1 may be easier to compute, and still enjoys good properties as long as r is not too large. When the sampled observations are not uniform across entries, the algorithm may require more modifications to properly normalize for high degree hub vertices, as the optimal choice of r may differ depending on the local sparsity. The key computational step of our algorithm involves comparing the expanded local neighborhoods of pairs of vertices to find the ?nearest neighbors?. The local neighborhoods can be computed in parallel, as they are independent computations. Furthermore, the local neighborhood computations are suitable for systems in which the data is distributed across different machines in a way that optimizes local neighborhood queries. The most expensive part of our algorithm involves computing similarities for all pairs of vertices in order to determine the set of nearest neighbors. However, it would be possible to use approximate nearest neighbor techniques to greatly reduce the computation such that approximate nearest neighbor sets could be computed with significantly fewer than n2 pairwise comparisons. Non-uniform sampling. In reality, the probability that entries are observed is not be uniform across all pairs (i, j). However, we believe that an extension of our result can also handle variations in the sample probability as long as the sample probability is a function of the latent variables and scales in the same way with respect to n across all entries. Suppose that the probability of observing (i, j) is given by pg(?i , ?j ), where p is the scaling factor (contains the dependence upon n), and g allows for constant factor variations in the sample probability across entries as a function of the latent variables. If we let matrix X indicate the presence of an observation or not, then we can apply our algorithm twice, first on matrix X to estimate function g, and then on data matrix M to estimate f times g. We can simply divide by the estimate for g to obtain the estimate for f . The limitation is that if g(?i , ?j ) is very small, then the error in estimating the corresponding f (?i , ?j ) will have higher variance. However, it is expected that error increases for edge types with fewer samples. Acknowledgments This work is supported in parts by NSF under grants CMMI-1462158 and CMMI-1634259, by DARPA under grant W911NF-16-1-0551, and additionally by a NSF Graduate Fellowship and Claude E. Shannon Research Assistantship. References [1] Emmanuel Abbe and Colin Sandon. Community detection in general stochastic block models: Fundamental limits and efficient algorithms for recovery. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 670?688. IEEE, 2015. 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Community detection thresholds and the weak ramanujan property. In Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, STOC ?14, pages 694?703, New York, NY, USA, 2014. ACM. [36] Elchanan Mossel, Joe Neeman, and Allan Sly. A proof of the block model threshold conjecture. Combinatorica, Aug 2017. [37] Sahand Negahban and Martin J Wainwright. Estimation of (near) low-rank matrices with noise and high-dimensional scaling. The Annals of Statistics, pages 1069?1097, 2011. [38] Xia Ning, Christian Desrosiers, and George Karypis. Recommender Systems Handbook, chapter A Comprehensive Survey of Neighborhood-Based Recommendation Methods, pages 37?76. Springer US, 2015. [39] Benjamin Recht. A simpler approach to matrix completion. Journal of Machine Learning Research, 12(Dec):3413?3430, 2011. [40] David Steurer and Sam Hopkins. Bayesian estimation from few samples: community detection and related problems. https://arxiv.org/abs/1710.00264, 2017. [41] Victor Veitch and Daniel M Roy. The class of random graphs arising from exchangeable random measures. arXiv preprint arXiv:1512.03099, 2015. 11 [42] Patrick J Wolfe and Sofia C Olhede. Nonparametric graphon estimation. arXiv preprint arXiv:1309.5936, 2013. [43] Yuan Zhang, Elizaveta Levina, and Ji Zhu. Estimating network edge probabilities by neighborhood smoothing. arXiv preprint arXiv:1509.08588, 2015. 12
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Adaptive Classification for Prediction Under a Budget Venkatesh Saligrama Electrical Engineering Boston University Boston, MA 02215 [email protected] Feng Nan Systems Engineering Boston University Boston, MA 02215 [email protected] Abstract We propose a novel adaptive approximation approach for test-time resourceconstrained prediction motivated by Mobile, IoT, health, security and other applications, where constraints in the form of computation, communication, latency and feature acquisition costs arise. We learn an adaptive low-cost system by training a gating and prediction model that limits utilization of a high-cost model to hard input instances and gates easy-to-handle input instances to a low-cost model. Our method is based on adaptively approximating the high-cost model in regions where low-cost models suffice for making highly accurate predictions. We pose an empirical loss minimization problem with cost constraints to jointly train gating and prediction models. On a number of benchmark datasets our method outperforms state-of-the-art achieving higher accuracy for the same cost. 1 Introduction Resource costs arise during test-time prediction in a number of machine learning applications. Feature costs in Internet, Healthcare, and Surveillance applications arise due to to feature extraction time [23], and feature/sensor acquisition [19]. In addition to feature acquisition costs, communication and latency costs pose a key challenge in the design of mobile computing, or the Internet-ofThings(IoT) applications, where a large number of sensors/camera/watches/phones (known as edge devices) are connected to a cloud. Adaptive System: Rather than having the edge devices constantly transmit measurements/images to the cloud where a centralized model makes prediction, a more efficient approach is to allow the edge devices make predictions locally [12], whenever possible, saving the high communication cost and reducing latency. Due to the memory, computing and battery constraints, the prediction models on the edge devices are limited to low complexity. Consequently, to maintain high-accuracy, adaptive systems are desirable. Such systems identify easy-to-handle input instances where local edge models suffice, thus limiting the utilization cloud services for only hard instances. We propose to learn an adaptive system by training on fully annotated training data. Our objective is to maintain high accuracy while meeting average resource constraints during prediction-time. There have been a number of promising approaches that focus on methods for reducing costs while improving overall accuracy [9, 24, 19, 20, 13, 15]. These methods are adaptive in that, at testtime, resources (features, computation etc) are allocated adaptively depending on the difficulty of the input. Many of these methods train models in a top-down manner, namely, attempt to build out the model by selectively adding the most cost-effective features to improve accuracy. In contrast we propose a novel bottom-up approach. We train adaptive models on annotated training data by selectively identifying parts of the input space for which high accuracy can be maintained at a lower cost. The principle advantage of our method is twofold. First, our approach can be readily applied to cases where it is desirable to reduce costs of an existing high-cost legacy system. Second, training top-down models in case of feature costs leads to fundamental combinatorial issues in multi- stage search over all feature subsets (see Sec. 2). In contrast, we bypass many of these issues by posing a natural adaptive approximation objective to partition the input space into easy and hard cases. In particular, when no legacy system is available, our method consists of first learning a high-accuracy model that minimizes the empirical loss regardless of costs. The resulting high prediction-cost model (HPC) can be readily trained using any of the existing methods. For example, this could be a large neural network in the cloud that achieves the state-of-the-art accuracy. Next, we jointly learn a low-cost gating function as well as a low prediction-cost (LPC) model so as to adaptively approximate the high-accuracy model by identifying regions of input space where a low-cost gating and LPC model are adequate to achieve high-accuracy. In IoT applications, such low-complexity models can be deployed on the edge devices to perform gating and prediction. At test-time, for each input instance, the gating function decides whether or not the LPC model is adequate for accurate classification. Intuitively, ?easy? examples can be correctly classified using only an LPC model while ?hard? examples require HPC model. By identifying which of the input instances can be classified accurately with LPCs we bypass the utilization of HPC model, thus reducing average prediction cost. The upper part of Figure 1 is a schematic of our approach, where x is feature vector and y is the predicted label; we aim to learn g and an LPC model to adaptively approximate the HPC. The key observation as depicted in the lower figure is that the probability of correct classification given x for a HPC model is in general a highly complex function with higher values than that of a LPC model. Yet there exists regions of the input space where the LPC has competitive accuracy (as shown to the right of the gating threshold). Sending examples in such regions (according to the gating function) to the LPC results in no loss of prediction accuracy while reducing prediction costs. Figure 1: Upper: single stage schematic of our approach. We learn low-cost gating g and a LPC model to adaptively approximate a HPC model. Lower: Key insight for adaptive approximation. x-axis represents feature space; y-axis represents conditional probability of correct prediction; LPC can match HPC?s prediction in the input region corresponding to the right of the gating threshold but performs poorly otherwise. Our goal is to learn a low-cost gating function that attempts to send examples on the right to LPC and the left to HPC. The problem would be simpler if our task were to primarily partition the input space into regions where LPC models would suffice. The difficulty is that we must also learn a low-cost gating function capable of identifying input instances for which LPC suffices. Since both prediction and gating account for cost, we favor design strategies that lead to shared features and decision architectures between the gating function and the LPC model. We pose the problem as a discriminative empirical risk minimization problem that jointly optimizes for gating and prediction models in terms of a joint margin-based objective function. The resulting objective is separately convex in gating and prediction functions. We propose an alternating minimization scheme that is guaranteed to converge since with appropriate choice of loss-functions (for instance, logistic loss), each optimization step amounts to a probabilistic approximation/projection (I-projection/M-projection) onto a probability space. While our method can be recursively applied in multiple stages to successively approximate the adaptive system obtained in the previous stage, thereby refining accuracy-cost trade-off, we observe that on benchmark datasets even a single stage of our method outperforms state-of-art in accuracy-cost performance. 2 Related Work Learning decision rules to minimize error subject to a budget constraint during prediction-time is an area of active interest[9, 17, 24, 19, 22, 20, 21, 13, 16]. Pre-trained Models: In one instantiation 2 of these methods it is assumed that there exists a collection of prediction models with amortized costs [22, 19, 1] so that a natural ordering of prediction models can be imposed. In other instances, the feature dimension is assumed to be sufficiently low so as to admit an exhaustive enumeration of all the combinatorial possibilities [20, 21]. These methods then learn a policy to choose amongst the ordered prediction models. In contrast we do not impose any of these restrictions. Top-Down Methods: For high-dimensional spaces, many existing approaches focus on learning complex adaptive decision functions top-down [9, 24, 13, 21]. Conceptually, during training, top-down methods acquire new features based on their utility value. This requires exploration of partitions of the input space together with different combinatorial low-cost feature subsets that would result in higher accuracy. These methods are based on multi-stage exploration leading to combinatorially hard problems. Different novel relaxations and greedy heuristics have been developed in this context. Bottom-up Methods: Our work is somewhat related to [16], who propose to prune a fully trained random forests (RF) to reduce costs. Nevertheless, in contrast to our adaptive system, their perspective is to compress the original model and utilize the pruned forest as a stand-alone model for test-time prediction. Furthermore, their method is specifically tailored to random forests. Another set of related work includes classifier cascade [5] and decision DAG [3], both of which aim to re-weight/re-order a set of pre-trained base learners to reduce prediction budget. Our method, on the other hand, only requires to pre-train a high-accuracy model and jointly learns the low-cost models to approximate it; therefore ours can be viewed as complementary to the existing work. The teacher-student framework [14] is also related to our bottom-up approach; a low-cost student model learns to approximate the teacher model so as to meet test-time budget. However, the goal there is to learn a better stand-alone student model. In contrast, we make use of both the lowcost (student) and high-accuracy (teacher) model during prediction via a gating function, which learns the limitation of the low-cost (student) model and consult the high-accuracy (teacher) model if necessary, thereby avoiding accuracy loss. Our composite system is also related to HME [10], which learns the composite system based on max-likelihood estimation of models. A major difference is that HME does not address budget constraints. A fundamental aspect of budget constraints is the resulting asymmetry, whereby, we start with an HPC model and sequentially approximate with LPCs. This asymmetry leads us to propose a bottom-up strategy where the high-accuracy predictor can be separately estimated and is critical to posing a direct empirical loss minimization problem. 3 Problem Setup We consider the standard learning scenario of resource constrained prediction with feature costs. A training sample S = {(x(i) , y (i) ) : i = 1, . . . , N } is generated i.i.d. from an unknown distribution, where x(i) ? <K is the feature vector with an acquisition cost c? ? 0 assigned to each of the features ? = 1, . . . , K and y (i) is the label for the ith example. In the case of multi-class classification y ? {1, . . . , M }, where M is the number of classes. Let us consider a single stage of our training method in order to formalize our setup. The model, f0 , is a high prediction-cost (HPC) model, which is either a priori known, or which we train to high-accuracy regardless of cost considerations. We would like to learn an alternative low prediction-cost (LPC) model f1 . Given an example x, at test-time, we have the option of selecting which model, f0 or f1 , to utilize to make a prediction. The accuracy of a prediction model fz is modeled by a loss function `(fz (x), y), z ? {0, 1}. We exclusively employ the logistic loss function in binary classification: `(fz (x), y) = log(1 + exp(?yfz (x)), although our framework allows other loss models. For a given x, we assume that once it pays the cost to acquire a feature, its value can be efficiently cached; its subsequent use does not incur additional cost. Thus, the cost of utilizing a particular prediction model, denoted by c(fz , x), is computed as the sum of the acquisition cost of unique features required by fz . Oracle Gating: Consider a general gating likelihood function q(z|x) with z ? {0, 1}, that outputs the likelihood of sending the input x to a prediction model, fz . The overall empirical loss is:   ESn Eq(z|x) [`(fz (x), y)] = ESn [`(f0 (x), y)] + ESn q(1|x) (`(f1 (x), y) ? `(f0 (x), y)) | {z } Excess Loss 3 The first term only depends on f0 , and from our perspective a constant. Similar to average loss we can write the average cost as (assuming gating cost is negligible for now): ESn Eq(z|x) [c(fz , x)] = ESn [c(f0 , x)] ? ESn [q(1|x) (c(f0 , x) ? c(f1 , x))], | {z } Cost Reduction where the first term is again constant. We can characterize the optimal gating function (see [19]) that minimizes the overall average loss subject to average cost constraint: Cost reduction Excess loss z }| { q(1|x)=0 z }| { `(f1 , x) ? `(f0 , x) > < ? (c(f0 , x) ? c(f1 , x)) q(1|x)=1 for a suitable choice ? ? R. This characterization encodes the important principle that if the marginal cost reduction is smaller than the excess loss, we opt for the HPC model. Nevertheless, this characterization is generally infeasible. Note that the LHS depends on knowing how well HPC performs on the input instance. Since this information is unavailable, this target can be unreachable with low-cost gating. Gating Approximation: Rather than directly enforcing a low-cost structure on q, we decouple the constraint and introduce a parameterized family of gating functions g ? G that attempts to mimic (or approximate) q. To ensure such approximation, we can minimize some distance measure D(q(?|x), g(x)). A natural choice for an approximation metric is the Kullback-Leibler (KL) divergence although other P choices are possible. The KL divergence between q and g is given by DKL (q(?|x)kg(x)) = z q(z|x) log(q(z|x)/?(sgn(0.5 ? z)g(x))), where ?(s) = 1/(1 + e?s ) is the sigmoid function. Besides KL divergence, we have also proposed another symmetrized metric fitting g directly to the log odds ratio of q. See Suppl. Material for details. Budget Constraint: With the gating function g, the cost of predicting x depends on whether the example is sent to f0 or f1 . Let c(f0 , g, x) denote the feature cost of passing x to f0 through g. As discussed, this is equal to the sum of the acquisition cost of unique features required by f0 and g for x. Similarly c(f1 , g, x) denotes the cost if x is sent to f1 through g. In many cases the cost c(fz , g, x) is independent of the example x and depends primarily on the model being used. This is true for linear models where each x must be processed through the same collection of features. For these cases c(fz , g, x) , c(fz , g). The total budget simplifies to: ESn [q(0|x)]c(f0 , g) + (1 ? ESn [q(0|x)])c(f1 , g) = c(f1 , g) + ESn [q(0|x)](c(f0 , g) ? c(f1 , g)). The budget thus depends on 3 quantities: ESn [q(0|x)], c(f1 , g) and c(f0 , g). Often f0 is a high-cost model that requires most, if not all, of features so c(f0 , g) can be considered a large constant. Thus, to meet the budget constraint, we would like to have (a) low-cost g and f1 (small c(f1 , g)); and (b) small fraction of examples being sent to the high-accuracy model (small ESn [q(0|x)]). We can therefore split theP budget constraint into two separate objectives: (a) ensure low-cost through penalty ?(f1 , g) = ? ? c? kV? + W? k0 , where ? is a tradeoff parameter and the indicator variables V? , W? ? {0, 1} denote whether or not the feature ? is required by f1 and g, respectively. Depending on the model parameterization, we can approximate ?(f1 , g) using a group-sparse norm or in a stage-wise manner as we will see in Algorithms 1 and 2. (b) Ensure only Pfull fraction of examples are sent to f0 via the constraint ESn [q(0|x)] ? Pfull . Putting Together: We are now ready to pose our general optimization problem: Losses min f1 ?F ,g?G,q ESn z X }| { z Gating}|Approx { z Costs }| { [q(z|x)`(fz (x), y)] + D(q(?|x), g(x)) + ?(f1 , g) (OPT) z subject to: ESn [q(0|x)] ? Pfull . (F raction to f0 ) The objective function penalizes excess loss and ensures through the second term that this excess loss can be enforced through admissible gating functions. The third term penalizes the feature cost usage of f1 and g. The budget constraint limits the fraction of examples sent to the costly model f0 . Remark 1: Directly parameterizing q leads to non-convexity. Average loss is q-weighted sum of losses from HPC and LPC; while the space of probability distributions is convex, a finitedimensional parameterization is generally non-convex (e.g. sigmoid). What we have done is to keep q in non-parametric form to avoid non-convexity and only parameterize g, connecting both via 4 a KL term. Thus, (OPT) is now convex with respect to the f1 and g for a fixed q. It is again convex in q for a fixed f1 and g. Otherwise it would introduce non-convexity as in prior work. For instance, in [5] a non-convex problem is solved in each inner loop iteration (line 7 of their Algorithm 1). Remark 2: We presented the case for a single stage approximation system. However, it is straightforward to recursively continue this process. We can then view the composite system f0 , (g, f1 , f0 ) as a black-box predictor and train a new pair of gating and prediction models to approximate the composite system. Remark 3: To limit the scope of our paper, we focus on reducing feature acquisition cost during prediction as it is a more challenging (combinatorial) problem. However, other prediction-time costs such as computation cost can be encoded in the choice of functional classes F and G in (OPT). Surrogate Upper Bound of Composite System: We can get better insight for the first two terms of the objective in P (OPT) if we view z ? {0, 1} as a latent variable and consider the composite system Pr(y|x) = z Pr(z|x; g) Pr(y|x, fz ). A standard application of Jensen?s inequality reveals that, ? log(Pr(y|x)) ? Eq(z|x) `(fz (x), y) + DKL (q(z|x)k Pr(z|x; g)). Therefore, the conditionalentropy of the composite system is bounded by the expected value of our loss function (we overload notation and represent random-variables in lower-case format): H(y | x) , E[? log(Pr(y|x))] ? Ex?y [Eq(z|x) `(fz (x), y) + DKL (q(z|x)k Pr(z|x; g))]. This implies that the first two terms of our objective attempt to bound the loss of the composite system; the third term in the objective together with the constraint serve to enforce budget limits on the composite system. Group Sparsity: Since the cost for feature re-use is zero we encourage feature re-use among gating and prediction models. So the fundamental question here is: How to choose a common, sparse (low-cost) subset of features on which both g and f1 operate, such that g can effective gate examples between f1 and f0 for accurate prediction? This is a hard combinatorial problem. The main contribution of our paper is to address it using the general optimization framework of (OPT). 4 Algorithms To be concrete, we instantiate our general framework (OPT) into two algorithms via different parameterizations of g, f1 : A DAPT- LIN for the linear class and A DAPT-G BRT for the non-parametric class. Both of them use the KL-divergence as distance measure. We also provide a third algorithm Algorithm 1 A DAPT-L IN A DAPT-L STSQ that uses the symmetrized disInput: (x(i) , y (i) ), Pfull , ? tance in the Suppl. Material. All of the alTrain f0 . Initialize g, f1 . gorithms perform alternating minimization of repeat (OPT) over q, g, f1 . Note that convergence Solve (OPT1) for q given g, f1 . of alternating minimization follows as in [8]. Solve (OPT2) for g, f1 given q. Common to all of our algorithms, we use two until convergence parameters to control cost: Pfull and ?. In practice they are swept to generate various cost- Algorithm 2 A DAPT-G BRT accuracy tradeoffs and we choose the best one Input: (x(i) , y (i) ), Pfull , ? satisfying the budget B using validation data. Train f0 . Initialize g, f1 . A DAPT- LIN: Let g(x) = g T x and f1 (x) = repeat T f1 x be linear classifiers. A feature is used if the Solve (OPT1) for q given g, f1 . corresponding component is non-zero: V? = 1 for t = 1 to T do if f1,? 6= 0, and W? = 1 if g? 6= 0. The miniFind f1t using CART to minimize (1). mization for q solves the following problem: f1 = f1 + f1t . P N For each feature ? used, set u? = 0. min N1 i=1 [(1 ? qi )Ai + qi Bi ? H(qi )] q Find g t using CART to minimize (2). P N s.t. N1 i=1 qi ? Pfull , g = g + gt . For each feature ? used, set u? = 0. (OPT1) end for where we have used shorthand notations qi = until convergence q(z = 0|x(i) ), H(qi ) = ?qi log(qi ) ? (1 ? (i) T (i) qi ) log(1 ? qi ), Ai = log(1 + e?y f1 x ) + 5 T (i) T (i) log(1 + eg x ) and Bi = ? log p(y (i) |z (i) = 0; f0 ) + log(1 + e?g x ). This optimization has a closed form solution: qi = 1/(1 + eBi ?Ai +? ) for some non-negative constant ? such that the constraint is satisfied. This optimization is also known as I-Projection in information geometry because of the entropy term [8]. Having optimized q, we hold it constant and minimizePwith respect to g, f1 by solving the problem (OPT2), where we have relaxed the non-convex cost ? c? kV? + W? k0 into a L2,1 norm for group sparsity and a tradeoff parameter ? to make sure the feature budget is satisfied. Once we solve for g, f1 , we can hold them constant and minimize with respect to q again. A DAPT-L IN is summarized in Algorithm 1. min g,f1 N   i Xq (i) T (i) T (i) T (i) 1 Xh 2 . (1 ? qi ) log(1 + e?y f1 x ) + log(1 + eg x ) + qi log(1 + e?g x ) + ? g?2 + f1,? N i=1 ? (OPT2) A DAPT-G BRT: We can also consider the non-parametric family of classifiers such as gradient PT PT t t t t boosted trees [7]: g(x) = t=1 g (x) and f1 (x) = t=1 f1 (x), where g and f1 are limiteddepth regression trees. Since the trees are limited to low depth, we assume that the feature utility of each tree is example-independent: V?,t (x) u V?,t , W?,t (x) u W?,t , ?x. V?,t = 1 if feature ? appears in f1t , otherwise V?,t = 0, similarly for W?,t . The optimization over q still solves (i) (i) (i) (OPT1). We modify Ai = log(1 + e?y f1 (x ) ) + log(1 + eg(x ) ) and Bi = ? log p(y (i) |z (i) = (i) 0; f0 ) + log(1 + e?g(x ) ). Next, to minimize over g, f1 , denote loss: " # N   1 X ?y (i) f1 (x(i) ) g(x(i) ) ?g(x(i) ) (1 ? qi ) ? log(1 + e ) + log(1 + e ) + qi log(1 + e ) , `(f1 , g) = N i=1 which is essentially the same as the first part of the objective in (OPT2). Thus, we need to minimize `(f1 , g) + ?(f1 , g) with respect to f1 and g. Since both f1 and g are gradient boosted trees, we naturally adopt a stage-wise approximation for the objective. In particular, we define an impurity function which on the one hand approximates the negative gradient of `(f1 , g) with the squared loss, and on the other hand penalizes the initial acquisition of features by their cost c? . To capture the initial acquisition penalty, we let u? ? {0, 1} indicates if feature ? has already been used in previous trees (u? = 0), or not (u? = 1). u? is updated after adding each tree. Thus we arrive at the following impurity for f1 and g, respectively: N X 1 X ?`(f1 , g) t (i) 2 (? ? f (x )) + ? u? c? V?,t , 1 2 i=1 ?f1 (x(i) ) ? (1) N X 1 X ?`(f1 , g) (? ? g t (x(i) ))2 + ? u? c? W?,t . (i) 2 i=1 ?g(x ) ? (2) Minimizing such impurity functions balances the need to minimize loss and re-using the already acquired features. Classification and Regression Tree (CART) [2] can be used to construct decision trees with such an impurity function. A DAPT-GBRT is summarized in Algorithm 2. Note that a similar impurity is used in G REEDY M ISER [24]. Interestingly, if Pfull is set to 0, all the examples are forced to f1 , then A DAPT-G BRT exactly recovers the G REEDY M ISER. In this sense, G REEDY M ISER is a special case of our algorithm. As we will see in the next section, thanks to the bottom-up approach, A DAPT-G BRT benefits from high-accuracy initialization and is able to perform accuracy-cost tradeoff in accuracy levels beyond what is possible for G REEDY M ISER. 5 Experiments BASELINE A LGORITHMS: We consider the following simple L1 baseline approach for learning f1 and g: first perform a L1-regularized logistic regression on all data to identify a relevant, sparse subset of features; then learn f1 using training data restricted to the identified feature(s); finally, learn g based on the correctness of f1 predictions as pseudo labels (i.e. assign pseudo label 1 to example x if f1 (x) agrees with the true label y and 0 otherwise). We also compare with two stateof-the-art feature-budgeted algorithms: G REEDY M ISER[24] - a top-down method that builds out an 6 ensemble of gradient boosted trees with feature cost budget; and B UDGET P RUNE[16] - a bottom-up method that prunes a random forest with feature cost budget. A number of other methods such as ASTC [13] and CSTC [23] are omitted as they have been shown to under-perform G REEDY M ISER on the same set of datasets [15]. Detailed experiment setups can be found in the Suppl. Material. We first visualize/verify the adaptive approximation ability of A DAPT-L IN and A DAPT-G BRT on the Synthetic-1 dataset without feature costs. Next, we illustrate the key difference between A DAPT-L IN and the L1 baseline approach on the Synthetic-2 as well as the Letters datasets. Finally, we compare A DAPT-G BRT with state-of-the-art methods on several resource constraint benchmark datasets. (a) Input Data (b) Lin Initialization (c) Lin after 10 iterations (d) RBF Contour (e) Gbrt Initialization (f) Gbrt after 10 iterations Figure 2: Synthetic-1 experiment without feature cost. (a): input data. (d): decision contour of RBF-SVM as f0 . (b) and (c): decision boundaries of linear g and f1 at initialization and after 10 iterations of A DAPT-L IN. (e) and (f): decision boundaries of boosted tree g and f1 at initialization and after 10 iterations of A DAPT-G BRT. Examples in the beige areas are sent to f0 by the g. P OWER OF A DAPTATION : We construct a 2D binary classification dataset (Synthetic-1) as shown in (a) of Figure 2. We learn an RBF-SVM as the high-accuracy classifier f0 as in (d). To better visualize the adaptive approximation process in 2D, we turn off the feature costs (i.e. set ?(f1 , g) to 0 in (OPT)) and run A DAPT-L IN and A DAPT-G BRT. The initializations of g and f1 in (b) results in wrong predictions for many red points in the blue region. After 10 iterations of A DAPT-L IN, f1 adapts much better to the local region assigned by g while g sends about 60% (Pfull ) of examples to f0 . Similarly, the initialization in (e) results in wrong predictions in the blue region. A DAPTG BRT is able to identify the ambiguous region in the center and send those examples to f0 via g. Both of our algorithms maintain the same level of prediction accuracy as f0 yet are able to classify large fractions of examples via much simpler models. Figure 3: A 2-D synthetic example for P OWER OF J OINT O PTIMIZATION: We return to the adaptive feature acquisition. On the left: problem of prediction under feature budget constrains. data distributed in four clusters. The We illustrate why a simple L1 baseline approach for two features correspond to x and y colearning f1 and g would not work using a 2D dataset ordinates, respectively. On the right: (Synthetic-2) as shown in Figure 3 (left). The data points accuracy-cost tradeoff curves. Our alare distributed in four clusters, with black triangles and gorithm can recover the optimal adapred circles representing two class labels. Let both feature tive system whereas a L1-based ap1 and 2 carry unit acquisition cost. A complex classifier proach cannot. f0 that acquires both features can achieve full accuracy at the cost of 2. In our synthetic example, clusters 1 and 2 are given more data points so that the L1-regularized logistic regression would produce the vertical red dashed line, separating cluster 1 from the others. So feature 1 is acquired for both g and f1 . The best such an adaptive system can 7 0.138 0.930 0.90 0.136 0.925 Adapt_Gbrt GreedyMiser(Xu et al. 2012) BudgetPrune (Nan et al. 2016) 0.920 15 20 25 30 35 40 Average Feature Cost (a) MiniBooNE 45 0.88 0.86 0.84 50 10 15 20 25 Average Feature Cost 30 (b) Forest Covertype 0.80 Test Accuracy 0.92 Average Precision@5 0.935 Test Accuracy Test Accuracy do is to send cluster 1 to f1 and the other three clusters to the complex classifier f0 , incurring an average cost of 1.75, which is sub-optimal. A DAPT-L IN, on the other hand, optimizing between q, g, f1 in an alternating manner, is able to recover the horizontal lines in Figure 3 (left) for g and f1 . g sends the first two clusters to the full classifier and the last two clusters to f1 . f1 correctly classifies clusters 3 and 4. So all of the examples are correctly classified by the adaptive system; yet only feature 2 needs to be acquired for cluster 3 and 4 so the overall average feature cost is 1.5, as shown by the solid curve in the accuracy-cost tradeoff plot on the right of Figure 3. This example shows that the L1 baseline approach is sub-optimal as it doesnot optimize the selection of feature subsets jointly for g and f1 . 0.134 0.132 0.130 0.128 0.75 0.70 0.65 40 60 80 100 120 140 160 180 Average Feature Cost (c) Yahoo! Rank 0 50 100 150 200 250 300 350 400 Average Feature Cost (d) CIFAR10 Figure 4: Comparison of A DAPT-G BRT against G REEDY M ISER and B UDGET P RUNE on four benchmark datasets. RF is used as f0 for A DAPT-G BRT in (a-c) while an RBF-SVM is used as f0 in (d). A DAPT-G BRT achieves better accuracy-cost tradeoff than other methods. The gap is significant in (b) (c) and (d). Note the accuracy of G REEDY M ISER in (b) never exceeds 0.86 and its precision in (c) slowly rises to 0.138 at cost of 658. We limit the cost range for a clearer comparison. R EAL DATASETS: We test various aspects Table 1: Dataset Statistics of our algorithms and compare with stateDataset #Train #Validation #Test #Features Feature Costs of-the-art feature-budgeted algorithms on five Letters 12000 4000 4000 16 Uniform 45523 19510 65031 50 Uniform real world benchmark datasets: Letters, Mini- MiniBooNE Forest 36603 15688 58101 54 Uniform BooNE Particle Identification, Forest Cover- CIFAR10 19761 8468 10000 400 Uniform Yahoo! 141397 146769 184968 519 CPU units type datasets from the UCI repository [6], CIFAR-10 [11] and Yahoo! Learning to Rank[4]. Yahoo! is a ranking dataset where each example is associated with features of a querydocument pair together with the relevance rank of the document to the query. There are 519 such features in total; each is associated with an acquisition cost in the set {1,5,20,50,100,150,200}, which represents the units of CPU time required to extract the feature and is provided by a Yahoo! employee. The labels are binarized into relevant or not relevant. The task is to learn a model that takes a new query and its associated documents and produce a relevance ranking so that the relevant documents come on top, and to do this using as little feature cost as possible. The performance metric is Average Precision @ 5 following [16]. The other datasets have unknown feature costs so we assign costs to be 1 for all features; the aim is to show A DAPT-G BRT successfully selects sparse subset of ?usefull? features for f1 and g. We summarize the statistics of these datasets in Table 1. Next, we highlight the key insights from the real dataset experiments. Generality of Approximation: Our framework allows approximation of powerful classifiers such as RBF-SVM and Random Forests as shown in Figure 5 as red and black curves, respectively. In particular, A DAPT-G BRT can well maintain high accuracy while reducing cost. This is a key advantage for our algorithms because we can choose to approximate the f0 that achieves the best accuracy. A DAPT-L IN Vs L1: Figure 5 shows that A DAPT-L IN outperforms L1 baseline method on real dataset as well. Again, this confirms the intuition we have in the Synthetic-2 example as A DAPT-L IN is able to iteratively select the common subset of features jointly for g and f1 . A DAPTG BRT Vs A DAPT-L IN: A DAPT-G BRT leads to significantly better performance than A DAPT-L IN in approximating both RBF-SVM and RF as shown in Figure 5. This is expected as the non-parametric non-linear classifiers are much more powerful than linear ones. A DAPT-G BRT Vs B UDGET P RUNE: Both are bottom-up approaches that benefit from good initializations. In (a), (b) and (c) of Figure 4 we let f0 in A DAPT-G BRT be the same RF that B UDGETP RUNE starts with. A DAPT-G BRT is able to maintain high accuracy longer as the budget decreases. 8 Thus, A DAPT-G BRT improves state-of-the-art bottom-up method. Notice in (c) of Figure 4 around the cost of 100, B UDGET P RUNE has a spike in precision. We believe this is because the initial pruning improved the generalization performance of RF. But in the cost region of 40-80, A DAPT-G BRT maintains much better accuracy than B UDGETP RUNE. Furthermore, A DAPT-G BRT has the freedom to approximate the best f0 given the problem. So in (d) of Figure 4 we see that with f0 being RBF-SVM, A DAPT-G BRT can achieve much higher accuracy than B UDGET P RUNE. Significant Cost Reduction: Without sacrificing top accuracies (within 1%), A DAPT-G BRT reduces average feature costs during test-time by around 63%, 32%, 58%, 12% and 31% on MiniBooNE, Forest, Yahoo, Cifar10 and Letters datasets, respectively. 6 Conclusions 0.98 0.96 Test Accuracy A DAPT-G BRT Vs G REEDY M ISER: A DAPT-G BRT outperforms G REEDY M ISER on all the datasets. The gaps in Figure 5, (b) (c) and (d) of Figure 4 are especially significant. 0.94 0.92 0.90 0.88 11 Adapt_Gbrt RF Adapt_Lin RF L1 RF Adapt_Gbrt RBF 12 Adapt_Lin RBF L1 RBF GreedyMiser(Xu et al 2012) 13 14 Average Feature Cost 15 16 Figure 5: Compare the L1 baseline approach, A DAPT-L IN and A DAPT-G BRT based on RBF-SVM and RF as f0 ?s on the Letters dataset. We presented an adaptive approximation approach to account for prediction costs that arise in various applications. At test-time our method uses a gating function to identify a prediction model among a collection of models that is adapted to the input. The overall goal is to reduce costs without sacrificing accuracy. We learn gating and prediction models by means of a bottom-up strategy that trains low prediction-cost models to approximate high prediction-cost models in regions where low-cost models suffice. On a number of benchmark datasets our method leads to an average of 40% cost reduction without sacrificing test accuracy (within 1%). It outperforms state-of-the-art top-down and bottom-up budgeted learning algorithms, with a significant margin in several cases. Acknowledgments Feng Nan would like to thank Dr Ofer Dekel for ideas and discussions on resource constrained machine learning during an internship in Microsoft Research in summer 2016. Familiarity and intuition gained during the internship contributed to the motivation and formulation in this paper. We also thank Dr Joseph Wang and Tolga Bolukbasi for discussions and helps in experiments. This material is based upon work supported in part by NSF Grants CCF: 1320566, CNS: 1330008, CCF: 1527618, DHS 2013-ST-061-ED0001, NGA Grant HM1582-09-1-0037 and ONR Grant N0001413-C-0288. References [1] Tolga Bolukbasi, Joseph Wang, Ofer Dekel, and Venkatesh Saligrama. Adaptive neural networks for efficient inference. In Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 527?536, International Convention Centre, Sydney, Australia, 06?11 Aug 2017. PMLR. [2] Leo Breiman, Jerome Friedman, Charles J Stone, and Richard A Olshen. Classification and regression trees. CRC press, 1984. [3] R?bert Busa-Fekete, Djalel Benbouzid, and Bal?zs K?gl. Fast classification using sparse decision dags. In Proceedings of the 29th International Conference on Machine Learning, ICML 2012, Edinburgh, Scotland, UK, June 26 - July 1, 2012, 2012. [4] O Chapelle, Y Chang, and T Liu, editors. Proceedings of the Yahoo! Learning to Rank Challenge, held at ICML 2010, Haifa, Israel, June 25, 2010, 2011. 9 [5] Minmin Chen, Zhixiang Eddie Xu, Kilian Q. Weinberger, Olivier Chapelle, and Dor Kedem. Classifier cascade for minimizing feature evaluation cost. In Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, AISTATS 2012, La Palma, Canary Islands, April 21-23, 2012, pages 218?226, 2012. [6] A. Frank and A. Asuncion. UCI machine learning repository, 2010. [7] Jerome H. Friedman. Greedy function approximation: A gradient boosting machine. Annals of Statistics, 29:1189?1232, 2000. [8] Kuzman Ganchev, Ben Taskar, and Jo ao Gama. Expectation maximization and posterior constraints. In J. C. Platt, D. Koller, Y. Singer, and S. T. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 569?576. Curran Associates, Inc., 2008. [9] T. Gao and D. Koller. Active classification based on value of classifier. In Advances in Neural Information Processing Systems (NIPS 2011), 2011. [10] Michael I. Jordan and Robert A. Jacobs. Hierarchical mixtures of experts and the em algorithm. Neural Comput., 6(2):181?214, March 1994. [11] Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images. Master?s thesis, 2009. [12] Ashish Kumar, Saurabh Goyal, and Manik Varma. Resource-efficient machine learning in 2 KB RAM for the internet of things. In Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 1935? 1944, International Convention Centre, Sydney, Australia, 06?11 Aug 2017. PMLR. [13] M Kusner, W Chen, Q Zhou, E Zhixiang, K Weinberger, and Y Chen. Feature-cost sensitive learning with submodular trees of classifiers. In AAAI Conference on Artificial Intelligence, 2014. [14] D. Lopez-Paz, B. Sch?lkopf, L. Bottou, and V. Vapnik. Unifying distillation and privileged information. In International Conference on Learning Representations, 2016. [15] Feng Nan, Joseph Wang, and Venkatesh Saligrama. Feature-budgeted random forest. In David Blei and Francis Bach, editors, Proceedings of the 32nd International Conference on Machine Learning (ICML-15), pages 1983?1991. JMLR Workshop and Conference Proceedings, 2015. [16] Feng Nan, Joseph Wang, and Venkatesh Saligrama. Pruning random forests for prediction on a budget. In Advances in Neural Information Processing Systems 29, pages 2334?2342. Curran Associates, Inc., 2016. [17] Feng Nan, Joseph Wang, Kirill Trapeznikov, and Venkatesh Saligrama. Fast margin-based cost-sensitive classification. In IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2014, Florence, Italy, May 4-9, 2014, 2014. [18] Daniel P. Robinson and Suchi Saria. Trading-off cost of deployment versus accuracy in learning predictive models. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI?16, pages 1974?1982. AAAI Press, 2016. [19] K Trapeznikov and V Saligrama. Supervised sequential classification under budget constraints. In International Conference on Artificial Intelligence and Statistics, pages 581?589, 2013. [20] Joseph Wang, Tolga Bolukbasi, Kirill Trapeznikov, and Venkatesh Saligrama. Model Selection by Linear Programming, pages 647?662. Springer International Publishing, Cham, 2014. [21] Joseph Wang, Kirill Trapeznikov, and Venkatesh Saligrama. Efficient learning by directed acyclic graph for resource constrained prediction. In Advances in Neural Information Processing Systems 28, pages 2143?2151. Curran Associates, Inc., 2015. [22] D. Weiss, B. Sapp, and B. Taskar. Dynamic structured model selection. In 2013 IEEE International Conference on Computer Vision, pages 2656?2663, Dec 2013. 10 [23] Z Xu, M Kusner, M Chen, and K. Q Weinberger. Cost-sensitive tree of classifiers. In Proceedings of the 30th International Conference on Machine Learning, 2013. [24] Zhixiang Eddie Xu, Kilian Q. Weinberger, and Olivier Chapelle. The greedy miser: Learning under test-time budgets. In Proceedings of the 29th International Conference on Machine Learning, ICML, 2012. 11
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Convergence rates of a partition based Bayesian multivariate density estimation method Linxi Liu ? Department of Statistics Columbia University [email protected] Dangna Li ICME Stanford University [email protected] Wing Hung Wong Department of Statistics Stanford University [email protected] Abstract We study a class of non-parametric density estimators under Bayesian settings. The estimators are obtained by adaptively partitioning the sample space. Under a suitable prior, we analyze the concentration rate of the posterior distribution, and demonstrate that the rate does not directly depend on the dimension of the problem in several special cases. Another advantage of this class of Bayesian density estimators is that it can adapt to the unknown smoothness of the true density function, thus achieving the optimal convergence rate without artificial conditions on the density. We also validate the theoretical results on a variety of simulated data sets. 1 Introduction In this paper, we study the asymptotic behavior of posterior distributions of a class of Bayesian density estimators based on adaptive partitioning. Density estimation is a building block for many other statistical methods, such as classification, nonparametric testing, clustering, and data compression. With univariate (or bivariate) data, the most basic non-parametric method for density estimation is the histogram method. In this method, the sample space is partitioned into regular intervals (or rectangles), and the density is estimated by the relative frequency of data points falling into each interval (rectangle). However, this method is of limited utility in higher dimensional spaces because the number of cells in a regular partition of a p-dimensional space will grow exponentially with p, which makes the relative frequency highly variable unless the sample size is extremely large. In this situation the histogram may be improved by adapting the partition to the data so that larger rectangles are used in the parts of the sample space where data is sparse. Motivated by this consideration, researchers have recently developed several multivariate density estimation methods based on adaptive partitioning [13, 12]. For example, by generalizing the classical P?lya Tree construction [7, 22] developed the Optional P?lya Tree (OPT) prior on the space of simple functions. Computational issues related to OPT density estimates were discussed in [13], where efficient algorithms were developed to compute the OPT estimate. The method performs quite well when the dimension is moderately large (from 10 to 50). The purpose of the current paper is to address the following questions on such Bayesian density estimates based on partition-learning. Question 1: what is the class of density functions that can be ?well estimated? by the partition-learning based methods. Question 2: what is the rate at which the posterior distribution is concentrated around the true density as the sample size increases. Our main contributions lie in the following aspects: ? Work was done while the author was at Stanford University. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. ? We impose a suitable prior on the space of density functions defined on binary partitions, and calculate the posterior concentration rate under the Hellinger distance with mild assumptions. The rate is adaptive to the unknown smoothness of the true density. ? For two dimensional density functions of bounded variation, the posterior contraction rate of our method is n?1/4 (log n)3 . ? For H?lder continuous (one-dimensional case) or mixture H?lder continuous (multidimensional case) density functions with regularity parameter ? in (0, 1], the posterior p ? concentration rate is n? 2?+p (log n)2+ 2? , whereas the minimax rate for one-dimensional H?lder continuous functions is (n/ log n)??/(2?+1) . ? When the true density function is sparse in the sense that the Haar wavelet coefficients satisfy q?1/2 1 a weak-lq (q > 1/2) constraint, the posterior concentration rate is n? 2q (log n)2+ 2q?1 . ? We can use a computationally efficient algorithm to sample from the posterior distribution. We demonstrate the theoretical results on several simulated data sets. 1.1 Related work An important feature of our method is that it can adapt to the unknown smoothness of the true density function. The adaptivity of Bayesian approaches has drawn great attention in recent years. In terms of density estimation, there are mainly two categories of adaptive Bayesian nonparametric approaches. The first category of work relies on basis expansion of the density function and typically imposes a random series prior [15, 17]. When the prior on the coefficients of the expansion is set to be normal [4], it is also a Gaussian process prior. In the multivariate case, most existing work [4, 17] uses tensor-product basis. Our improvement over these methods mainly lies in the adaptive structure. In fact, as the dimension increases the number of tensor-product basis functions can be prohibitively large, which imposes a great challenge on computation. By introducing adaptive partition, we are able to handle the multivariate case even when the dimension is 30 (Example 2 in Section 4). Another line of work considers mixture priors [16, 11, 18]. Although the mixture distributions have good approximation properties and naturally lead to adaptivity to very high smoothness levels, they may fail to detect or characterize the local features. On the other hand, by learning a partition of the sample space, the partition based approaches can provide an informative summary of the structure, and allow us to examine the density at different resolutions [14, 21]. The paper is organized as follows. In Section 2 we provide more details of the density functions on binary partitions and define the prior distribution. Section 3 summarizes the theoretical results on posterior concentration rates. The results are further validated in Section 4 by several experiments. 2 Bayesian multivariate density estimation We focus on density estimation problems in p-dimensional Euclidean space. Let (?, B) be a measurable space and f0 be a compactly supported density function with respect to the Lebesgue measure ?. Y1 , Y2 , ? ? ? , Yn is a sequence of independent variables distributed according to f0 . After translation and scaling, we can always assume that the support of f0 is contained in the unit 1 2 p l cube in Rp . Translating this into notations, we assume R that ? = {(y , y , ? ? ? , y ) : y ? [0, 1]}. F = {f is a nonnegative measurable function on ? : ? f d? = 1} denotes the collection of all the density functions on (?, B, ?). Then F constitutes the parameter space in this problem. Note that F is an infinite dimensional parameter space. 2.1 Densities on binary partitions To address the infinite dimensionality of F, we construct a sequence of finite dimensional approximating spaces ?1 , ?2 , ? ? ? , ?I , ? ? ? based on binary partitions. With growing complexity, these spaces provide more and more accurate approximations to the initial parameter space F. Here, we use a recursive procedure to define a binary partition with I subregions of the unit cube in Rp . Let ? = {(y 1 , y 2 , ? ? ? , y p ) : y l ? [0, 1]} be the unit cube in Rp . In the first step, we choose one of the coordinates y l and cut ? into two subregions along the midpoint of the range of y l . That is, ? = ?l0 ? ?l1 , where ?l0 = {y ? ? : y l ? 1/2} and ?l1 = ?\?l0 . In this way, we get a partition 2 with two subregions. Note that the total number of possible partitions after the first step is equal to the dimension p. Suppose after I ? 1 steps of the recursion, we have obtained a partition {?i }Ii=1 with I subregions. In the I-th step, further partitioning of the region is defined as follows: 1. Choose a region from ?1 , ? ? ? , ?I . Denote it as ?i0 . 2. Choose one coordinate y l and divide ?i0 into two subregions along the midpoint of the range of y l . Such a partition obtained by I ? 1 recursive steps is called a binary partition of size I. Figure 1 displays all possible two dimensional binary partitions when I is 1, 2 and 3. Figure 1: Binary partitions Now, let ?I = {f : f = I I X X ?i 1?i , ?i = 1, {?i }Ii=1 is a binary partition of ?.} |?i | i=1 i=1 where |?i | is the volume of ?i . Then, ?I is the collection of the density functions supported by the binary partitions of size I. They constitute a sequence of approximating spaces (i.e. a sieve, see [10, 20] for background on sieve theory). Let ? = ?? I=1 ?I be the space containing all the density functions supported by the binary partitions. Then ? is an approximation of the initial parameter space F to certain approximation error which will be characterized later. We take the metric on F, ? and ?I to be Hellinger distance, which is defined as Z p p ?(f, g) = ( ( f (y) ? g(y))2 dy)1/2 , f, g ? F. (1) ? 2.2 Prior distribution An ideal prior ? on ? = ?? I=1 ?I is supposed to be capable of balancing the approximation error and the complexity of ?. The prior in this paper penalizes the size of the partition in the sense that the probability mass on each ?I is proportional to exp(??I log I). Given a sample of size n, n/ log n we restrict our attention to ?n = ?I=1 ?I , because in practice we need enough samples within each subregion to get a meaningful estimate of the density. This is to say, when I ? n/ log n, ?(?I ) ? exp(??I log I), otherwise ?(?I ) = 0. If we use TI to denote the total number of possible partitions of size I, then it is not hard to see that log TI ? c? I log I, where c? is a constant. Within each ?I , the prior is uniform across all binary partitions. In other words, let {?i }Ii=1 be a binary partition of ? of size I, and F({?i }Ii=1 ) is the collection of piecewise constant density functions on this partition (i.e. F({?i }Ii=1 ) = {f = PI PI ?i i=1 |?i | 1?i : i=1 ?i = 1 and ?i ? 0, i = 1, . . . , I}), then  ? F {?i }Ii=1 ? exp(??I log I)/TI . (2) 3 Given a partition {?i }Ii=1 , the weights ?i on the subregions follow a Dirichlet distribution with PI parameters all equal to ? (? < 1). This is to say, for x1 , ? ? ? , xI ? 0 and i=1 xi = 1, ! I I X Y  ?i 1 I ? f= 1?i : ?1 ? dx1 , ? ? ? , ?I ? dxI F {?i }i=1 = xi??1 , (3) |? | D(?, ? ? ? , ?) i i=1 i=1 QI PI where D(?1 , ? ? ? , ?I ) = i=1 ?(?i )/?( i=1 ?i ). Let ?n (?|Y1 , ? ? ? , Yn ) to denote the posterior distribution. After  integrating out the weights ?i , we can compute the marginal posterior probability of F {?i }Ii=1 : ! Z Y I   n ?n F({?i }Ii=1 ) Y1 , ? ? ? , Yn ? ? F({?i }Ii=1 ) (?i /|?i |) i i=1 ? I Y 1 ???1 D(?, ? ? ? , ?) i=1 i ! d?1 ? ? ? d?I I ? exp(??I log I) D(? + n1 , ? ? ? , ? + nI ) Y 1 ? , (4) TI D(?, ? ? ? , ?) |?i |ni i=1 where ni is the number of observations in ?i . Under the prior introduced in [13], the marginal posterior distribution is: I  ??n F({?i }Ii=1 ) Y1 , ? ? ? , Yn ? exp(??I) D(? + n1 , ? ? ? , ? + nI ) Y 1 , D(?, ? ? ? , ?) |?i |ni i=1 while the maximum log-likelihood achieved by histograms on the partition {?i }ni=1 is:   I X ni I ni log ln (F({?i }i=1 )) := max ln (f ) = . n|?i | f ?F ({?i }Ii=1 ) i=1 (5) (6) From a model selection perspective, we may treat the histograms on each binary partition as a model of the data. When I  n, asymptotically,  1 (7) log ??n F({?i }Ii=1 ) Y1 , ? ? ? , Yn  ln (F({?i }Ii=1 )) ? (I ? 1) log n. 2 This is to say, in [13], selecting the partition which maximizes the marginal posterior distribution is equivalent to applying the Bayesian information criterion (BIC) to perform model selection. However, if we allow I to increase with n, (7) will not hold any more. But if we use the prior introduced in this section, in the case when nI ? ? ? (0, 1) as n ? ?, we still have  log ?n F({?i }Ii=1 ) Y1 , ? ? ? , Yn  ln (F({?i }Ii=1 )) ? ?I log I. (8) From a model selection perspective, this is closer to the risk inflation criterion (RIC, [8]). 3 Posterior concentration rates We are interested in how fast the posterior probability measure concentrates around the true the density f0 . Under the prior specified above, the posterior probability is the random measure given by R Qn j=1 f (Yj )d?(f ) ?n (B|Y1 , ? ? ? , Yn ) = RB Qn . j=1 f (Yj )d?(f ) ? A Bayesian estimator is said to be consistent if the posterior distribution concentrates on arbitrarily small neighborhoods of f0 , with probability tending to 1 under P0n (P0 is the probability measure corresponding to the density function f0 ). The posterior concentration rate refers to the rate at which these neighborhoods shrink to zero while still possessing most of the posterior mass. More explicitly, we want to find a sequence n ? 0, such that for sufficiently large M , ?n ({f : ?(f, f0 ) ? M n }|Y1 , ? ? ? , Yn ) ? 0 in P0n ? probability. 4 In [6] and [2], the authors demonstrated that it is impossible to find an estimator which works uniformly well for every f in F. This is the case because for any estimator f?, there always exists f ? F for which f? is inconsistent. Given the minimaxity of the Bayes estimator, we have to restrict our attention to a subset of the original parameter space F. Here, we focus on the class of density functions that can be well approximated by ?I ?s. To be more rigorous, a density function f ? F is said to be well approximated by elements in ?, if there exits a sequence of fI ? ?I , satisfying that ?(fI , f ) = O(I ?r )(r > 0). Let F0 be the collection of these density functions. We will first derive posterior concentration rate for the elements in F0 as a function of r. For different function classes, this approximation rate r can be calculated explicitly. In addition to this, we also assume that f0 has finite second moment. The following theorem gives the posterior concentration rate under the prior introduced in Section 2.2. Theorem 3.1. Y1 , ? ? ? , Yn is a sequence of independent random variables distributed according to f0 . P0 is the probability measure corresponding to f0 . ? is the collection of p-dimensional density functions supported by the binary partitions as defined in Section 2.1. With the modified prior r 1 distribution, if f0 ? F0 , then the posterior concentration rate is n = n? 2r+1 (log n)2+ 2r . The strategy to show this theorem is to write the posterior probability of the shrinking ball as P? R Qn f (Yj ) I=1 {f :?(f,f0 )?M n }??I j=1 f0 (Yj ) d?(f ) . ?({f : ?(f, f0 ) ? M n }|Y1 , ? ? ? , Yn ) = P? R Qn f (Yj ) I=1 ?I j=1 f0 (Yj ) d?(f ) (9) The proof employs the mechanism developed in the landmark works [9] and [19]. We first obtain the upper bounds for the items in the numerator by dividing them into three blocks, each of which accounts for bias, variance, and rapidly decaying prior respectively, and calculate the upper bound for each block separately. Then we provide the prior thickness result, i.e., we bound the prior mass of a ball around the true density from below. Due to space constraints, the details of the proof will be provided in the appendix. This theorem suggests the following two take-away messages: 1. The rate is adaptive to the unknown r 1 smoothness of the true density. 2. The posterior contraction rate is n? 2r+1 (log n)2+ 2r , which does not directly depend on the dimension p. For some density functions, r may depend on p. But in several special cases, like the density function is spatially sparse or the density function lies in a low dimensional subspace, we will show that the rate will not be affected by the full dimension of the problem. In the following three subsections, we will calculate the explicit rates for three density classes. Again, all proofs are given in the appendix. 3.1 Spatial adaptation First, we assume that the density concentrates spatially. Mathematically, this implies the density function satisfies a type of sparsity. In the past two decades, sparsity has become one of the most discussed types of structure under which we are able to overcome the curse of dimensionality. A remarkable example is that it allows us to solve high-dimensional linear models, especially when the system is underdetermined. Let f be a ? p dimensional density function and ? the p-dimensional Haar basis. We will work with P g = f first. Note that g ? L2 ([0, 1]p ). Thus we can expand g with respect to ? as g = ??? hg, ?i?, ? ? ?. We rearrange this summation by the size of wavelet coefficients. In other words, we order the coefficients as the following |hg, ?(1) i| ? |hg, ?(2) i| ? ? ? ? ? |hg, ?(k) i| ? ? ? ? , then the sparsity condition imposed on the density functions is that the decay of the wavelet coefficients follows a power law, |hg, ?(k) i| ? Ck ?q for all k ? N and q > 1/2, where C is a constant. 5 (10) We call such a constraint a weak-lq constraint. The condition has been widely used to characterize the sparsity of signals and images [1, 3]. In particular, in [5], it was shown that for two-dimensional cases, when q > 1/2, this condition reasonably captures the sparsity of real world images. Corollary 3.2. (Application to spatial adaptation) Suppose f0 is a p-dimensional density function and satisfies the condition (10). If we apply our approaches to this type of density functions, the q?1/2 1 posterior concentration rate is n? 2q (log n)2+ 2q?1 . 3.2 Density functions of bounded variation Let ? = [0, 1)2 be a domain in R2 . We first characterize the space BV (?) of functions of bounded variation on ?. For a vector ? ? R2 , the difference operator ?? along the direction ? is defined by ?? (f, y) := f (y + ?) ? f (y). For functions f defined on ?, ?? (f, y) is defined whenever y ? ?(?), where ?(?) := {y : [y, y + ?] ? ?} and [y, y + ?] is the line segment connecting y and y + ?. Denote by el , l = 1, 2 the two coordinate vectors in R2 . We say that a function f ? L1 (?) is in BV (?) if and only if V? (f ) := sup h?1 h>0 2 X k?hel (f, ?)kL1 (?(hel )) = lim h?1 h?0 l=1 2 X k?hel (f, ?)kL1 (?(hel )) l=1 is finite. The quantity V? (f ) is the variation of f over ?. Corollary 3.3. Assume that f0 ? BV (?). If we apply the Bayesian multivariate density estimator based on adaptive partitioning here to estimate f0 , the posterior concentration rate is n?1/4 (log n)3 . 3.3 H?lder space In one-dimensional case, the class of H?lder functions H(L, ?) with regularity parameter ? is defined as the following: let ? be the largest integer smaller than ?, and denote by f (?) its ?th derivative. H(L, ?) = {f : [0, 1] ? R : |f (?) (x) ? f (?) (y)| ? L|x ? y|??? }. In multi-dimensional cases, we introduce the Mixed-H?lder continuity. In order to simplify the notation, we give the definition when the dimension is two. It can be easily generalized to highdimensional cases. A real-valued function f on R2 is called Mixed-H?lder continuous for some nonnegative constant C and ? ? (0, 1], if for any (x1 , y1 ), (x1 , y2 ) ? R2 , |f (x2 , y2 ) ? f (x2 , y1 ) ? f (x1 , y2 ) + f (x1 , y1 )| ? C|x1 ? x2 |? |y1 ? y2 |? . ? Corollary 3.4. Let f0 be the p-dimensional density function. If f0 is H?lder continuous (when p = 1) or mixed-H?lder continuous (when p ? 2) with regularity parameter ? ? (0, 1], then the ? p posterior concentration rate of the Bayes estimator is n? 2?+p (log n)2+ 2? . This result also implies that if f0 only depends on p? variable where p? < p, but we do not know in advance which p? variables, then the rate of this method is determined by the effective dimension p? of the problem, since the smoothness parameter r is only a function of p?. In next section, we will use a simulated data set to illustrate this point. 4 4.1 Simulation Sequential importance sampling Each partition AI = {?i }Ii=1 is obtained by recursively partitioning the sample space. We can use a sequence of partitions A1 , A2 , ? ? ? , AI to keep track of the path leading to AI . Let ?n (?) denote the posterior distribution ?n (?|Y1 , ? ? ? , Yn ) for simplicity, and ?In be the posterior distribution conditioning on ?I . Then ?In (AI ) can be decomposed as ?In (AI ) = ?In (A1 )?In (A2 |A1 ) ? ? ? ?In (AI |AI?1 ). 6 Figure 2: Heatmap of the density and plots of the 2-dimensional Haar coefficients. For the plot on the right, the left panel is the plot of the Haar coefficients from low resolution to high resolution up to level 6. The middle one is the plot of the sorted coefficients according to their absolute values. And the right one is the same as the middle plot but with the abscissa in log scale. The conditional distribution ?In (Ai+1 |Ai ) can be calculated by ?In (Ai+1 )/?In (Ai ). However, the computation of the marginal distribution ?In (Ai ) is sometimes infeasible, especially when both I and I ? i are large, because we need to sum the marginal posterior probability over all binary partitions of size I for which the first i steps in the partition generating path are the same as those of Ai . Therefore, we adopt the sequential importance algorithm proposed in [13]. In order to build a sequence of binary partitions, at each step, the conditional distribution is approximated by ?i+1 n (Ai+1 |Ai ). The obtained partition is assigned a weight to compensate the approximation, where the weight is wI (AI ) = ?In (AI ) . ?1n (A1 )?2n (A2 |A1 ) ? ? ? ?In (AI |AI?1 ) In order to make the data points as uniform as possible, we apply a copula transformation to each variable in advance whenever the dimension exceeds 3. More specifically, we estimate the marginal distribution of each variable Xj by our approach, denoted as f?j (we use F?j to denote the cdf of Xj ), and transform each point (y 1 , ? ? ? , y p ) to (F1 (y 1 ), ? ? ? , Fp (y p )). Another advantage of this transformation is that after the transformation the sample space naturally becomes [0, 1]p . Example 1 Assume that the two-dimensional density function is         2 3 0.25 0.75 Y1 2 2 ? N , 0.05 I2?2 + N , 0.05 I2?2 . 0.25 0.75 Y2 5 5 This density function both satisfies the spatial sparsity condition and belongs to the space of functions of bounded variation. Figure 2 shows the heatmap of the density function and its Haar coefficients. The last panel in the second plot displays the sorted coefficients with the abscissa in log-scale. From this we can clearly see that the power-law decay defined in Section 3.1 is satisfied. We apply the adaptive partitioning approach to estimate the density, and allow the sample size increase from 102 to 105 . In Figure 3, the left plot is the density estimation result based on a sample with 10000 data points. The right one is the plot of Kullback-Leibler (KL) divergence from the estimated density to f0 vs. sample size in log-scale. The sample sizes are set to be 100, 500, 1000, 5000, 104 , and 105 . The linear trend in the plot validates the posterior concentrate rates calculated in Section 3. The reason why we use KL divergence instead of the Hellinger distance is that for any f0 ? F0 and f? ? ?, we can show that the KL divergence and the Hellinger distance are of the same order. But KL divergence is relatively easier to compute in our setting, since we can show that it is linear in the logarithm of the posterior marginal probability of a partition. The proof will be provided in the appendix. For each fixed sample size, we run the experiment 10 times and estimate the standard error, which is shown by the lighter blue part in the plot. Example 2 In the second example we work with a density function of moderately high dimension. Assume that the first five random variables Y1 , ? ? ? Y5 are generated from the following location 7 KL divergence 1.00 0.80 0.60 0.40 0.20 0.01 1e2 5e2 1e3 5e3 1e4 1e5 sample size Figure 3: Plot of the estimated density and KL divergence against sample size. We use the posterior mean as the estimate. The right plot is on log-log scale, while the labels of x and y axes still represent the sample size and the KL divergence before we take the logarithm. Figure 4: KL divergence vs. sample size. The blue, purple and red curves correspond to the cases when p = 5, p = 10 and p = 30 respectively. The slopes of the three lines are almost the same, implying that the concentration rate only depends on the effective dimension of the problem (which is 5 in this example). mixture of the Gaussian distribution: ? ! ! ? 0.052 Y1 0.25 1 ? Y2 0.25 , ?0.032 ? N 2 Y 0.25 0 3 Y4 , Y5 ? 0.032 0.052 0 ?? 0 1 0 ?? + N 2 2 0.05 ! ! 0.75 2 0.75 , 0.05 I3?3 , 0.75 N (0.5, 0.1), the other components Y6 , ? ? ? , Yp are independently uniformly distributed. We run experiments for p = 5, 10, and 30. For a fixed p, we generate n ? {500, 1000, 5000, 104 , 105 } data points. For each pair of p and n, we repeat the experiment 10 times and calculate the standard error. Figure 4 displays the plot of the KL divergence vs. the sample size on log-log scale. The density function is continuous differentiable. Therefore, it satisfies the mixed-H?lder continuity condition. The effective dimension of this example is p? = 5, and this is reflected in the plot: the slopes of the three lines, which correspond to the concentration rates under different dimensions, almost remain the same as we increase the full dimension of the problem. 5 Conclusion In this paper, we study the posterior concentration rate of a class of Bayesian density estimators based on adaptive partitioning. We obtain explicit rates when the density function is spatially sparse, belongs to the space of bounded variation, or is H?lder continuous. For the last case, the rate is minimax up to a logarithmic term. When the density function is sparse or lies in a low-dimensional subspace, the rate will not be affected by the dimension of the problem. Another advantage of this method is that it can adapt to the unknown smoothness of the underlying density function. 8 Bibliography [1] Felix Abramovich, Yoav Benjamini, David L. Donoho, and Iain M. Johnstone. Adapting to unknown sparsity by controlling the false discovery rate. The Annals of Statistics, 34(2):584?653, 04 2006. [2] Lucien Birg? and Pascal Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli, 4(3):329?375, 09 1998. [3] E.J. Cand?s and T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies? Information Theory, IEEE Transactions on, 52(12):5406?5425, Dec 2006. [4] R. de Jonge and J.H. van Zanten. Adaptive estimation of multivariate functions using conditionally gaussian tensor-product spline priors. Electron. J. Statist., 6:1984?2001, 2012. [5] R.A. DeVore, B. Jawerth, and B.J. Lucier. Image compression through wavelet transform coding. Information Theory, IEEE Transactions on, 38(2):719?746, March 1992. [6] R. H. Farrell. On the lack of a uniformly consistent sequence of estimators of a density function in certain cases. The Annals of Mathematical Statistics, 38(2):471?474, 04 1967. [7] Thomas S. Ferguson. Prior distributions on spaces of probability measures. Ann. Statist., 2:615?629, 1974. [8] Dean P. Foster and Edward I. George. The risk inflation criterion for multiple regression. Ann. Statist., 22(4):1947?1975, 12 1994. [9] Subhashis Ghosal, Jayanta K. Ghosh, and Aad W. van der Vaart. Convergence rates of posterior distributions. The Annals of Statistics, 28(2):500?531, 04 2000. [10] U. Grenander. Abstract Inference. Probability and Statistics Series. John Wiley & Sons, 1981. [11] Willem Kruijer, Judith Rousseau, and Aad van der Vaart. Adaptive bayesian density estimation with location-scale mixtures. Electron. J. Statist., 4:1225?1257, 2010. [12] Dangna Li, Kun Yang, and Wing Hung Wong. Density estimation via discrepancy based adaptive sequential partition. 30th Conference on Neural Information Processing Systems (NIPS 2016), 2016. [13] Luo Lu, Hui Jiang, and Wing H. Wong. Multivariate density estimation by bayesian sequential partitioning. Journal of the American Statistical Association, 108(504):1402?1410, 2013. [14] Li Ma and Wing Hung Wong. Coupling optional p?lya trees and the two sample problem. Journal of the American Statistical Association, 106(496):1553?1565, 2011. [15] Vincent Rivoirard and Judith Rousseau. Posterior concentration rates for infinite dimensional exponential families. Bayesian Anal., 7(2):311?334, 06 2012. [16] Judith Rousseau. Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density. The Annals of Statistics, 38(1):146?180, 02 2010. [17] Weining Shen and Subhashis Ghosal. Adaptive bayesian procedures using random series priors. Scandinavian Journal of Statistics, 42(4):1194?1213, 2015. 10.1111/sjos.12159. [18] Weining Shen, Surya T. Tokdar, and Subhashis Ghosal. Adaptive bayesian multivariate density estimation with dirichlet mixtures. Biometrika, 100(3):623?640, 2013. [19] Xiaotong Shen and Larry Wasserman. Rates of convergence of posterior distributions. The Annals of Statistics, 29(3):687?714, 06 2001. [20] Xiaotong Shen and Wing Hung Wong. Convergence rate of sieve estimates. The Annals of Statistics, 22(2):pp. 580?615, 1994. [21] Jacopo Soriano and Li Ma. Probabilistic multi-resolution scanning for two-sample differences. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(2):547?572, 2017. [22] Wing H. Wong and Li Ma. Optional p?lya tree and bayesian inference. The Annals of Statistics, 38(3):1433? 1459, 06 2010. 9
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Synchronization and Grammatical Inference in an Oscillating Elman Net Bill Baird Dept Mathematics, U .C.Berkeley, Berkeley, Ca. 94720, [email protected] Todd Troyer Dept Mathematics, U .C.Berkeley, Berkeley, Ca. 94720 Frank Eeckman Lawrence Livermore National Laboratory, P.O. Box 808 (L-426), Livermore, Ca. 94551 Abstract We have designed an architecture to span the gap between biophysics and cognitive science to address and explore issues of how a discrete symbol processing system can arise from the continuum, and how complex dynamics like oscillation and synchronization can then be employed in its operation and affect its learning. We show how a discrete-time recurrent "Elman" network architecture can be constructed from recurrently connected oscillatory associative memory modules described by continuous nonlinear ordinary differential equations. The modules can learn connection weights between themselves which will cause the system to evolve under a clocked "machine cycle" by a sequence of transitions of attractors within the modules, much as a digital computer evolves by transitions of its binary flip-flop attractors. The architecture thus employs the principle of "computing with attractors" used by macroscopic systems for reliable computation in the presence of noise. We have specifically constructed a system which functions as a finite state automaton that recognizes or generates the infinite set of six symbol strings that are defined by a Reber grammar. It is a symbol processing system, but with analog input and oscillatory subsymbolic representations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. This holds input and "context" modules clamped at their attractors while 'hidden and output modules change state, then clamps hidden and output states while context modules are released to load those states as the new context for the next cycle of input. Superior noise immunity has been demonstrated for systems with dynamic attractors over systems with static attractors, and synchronization ("binding") between coupled oscillatory attractors in different modules has been shown to be important for effecting reliable transitions. 236 Synchronization and Grammatical Inference in an Oscillating Elman Net 1 Introduction Patterns of 40 to 80 Hz oscillation have been observed in the large scale activity (local field potentials) of olfactory cortex [Freeman and Baird, 1987] and visual neocortex [Gray and Singer, 1987], and shown to predict the olfactory [Freeman and Baird, 1987] and visual pattern recognition responses of a trained animal. Similar observations of 40 Hz oscillation in auditory and motor cortex (in primates), and in the retina and EMG have been reported. It thus appears that cortical computation in general may occur by dynamical interaction of resonant modes, as has been thought to be the case in the olfactory system. The oscillation can serve a macroscopic clocking function and entrain or "bind" the relevant microscopic activity of disparate cortical regions into a well defined phase coherent collective state or "gestalt". This can overide irrelevant microscopic activity and produce coordinated motor output. There is further evidence that although the oscillatory activity appears to be roughly periodic, it is actually chaotic when examined in detail. If this view is correct, then oscillatoryI chaotic network modules form the actual cortical substrate of the diverse sensory, motor, and cognitive operations now studied in static networks. It must then be shown how those functions can be accomplished with oscillatory and chaotic dynamics, and what advantages are gained thereby. It is our expectation that nature makes ~ood use of this dynamical complexity, and our intent is to search here for novel deSign principles that may underly the superior computational performance of biological systems over man made devices in many task domains. These principles may then be applied in artificial systems to engineering problems to advance the art of computation. We have therefore constructed a parallel distributed processing architecture that is inspired by the structure and dynamics of cerebral cortex, and applied it to the problem of grammatical inference. The construction assumes that cortex is a set of coupled oscillatory associative memories, and is also guided by the principle that at tractors must be used by macroscopic systems for reliable computation in the presence of noise. Present day digital computers are built of flip-flops which, at the level of their transistors, are continuous dissipative dynamical systems with different attractors underlying the symbols we call "0" and "1". 2 Oscillatory Network Modules The network modules of this architecture were developed previously as models of olfactory cortex, or caricatures of "patches"of neocortex [Baird, 1990a]. A particular subnetwork is formed by a set of neural populations whose interconnections also contain higher order synapses. These synapses determine at tractors for that subnetwork independent of other subnetworks. Each subnetwork module assumes only minimal coupling justified by known olfactory anatomy. An N node module can be shown to function as an associative memory for up to N 12 oscillatory and NI3 chaotic memory attractors [Baird, 1990b, Baird and Eeckman, 1992b). Single modules with static, oscillatory, and three types of chaotic attractors - Lorenz, Roessler, Ruelle-Takens - have been sucessfully used for recognition of handwritten characters [Baird and Eeckman, 1992b]. We have shown in these modules a superior stability of oscillatory attractors over static attractors in the presence of additive Gaussian noise perturbations with the 1If spectral character of the noise found experimentally by Freeman in the brain[Baird and Eeckman, 1992a]. This may be one reason why the brain uses dynamic attractors. An oscillatory attractor acts like a a bandpass filter and is 237 238 Baird, Troyer, and Eeckman effectively immune to the many slower macroscopic bias perturbations in the thetaalpha-beta range (3 - 25 Hz) below its 40 -80 Hz passband, and the more microscopic perturbations of single neuron spikes in the 100 - 1000 Hz range. The mathematical foundation for the construction of network modules is contained in the normal form projection algorithm[Baird, 1990b]. This is a learning algorithm for recurrent analog neural networks which allows associative memory storage of analog patterns, continuous periodic sequences, and chaotic attractors in the same network. A key feature of a net constructed by this algorithm is that the underlying dynamics is explicitly isomorphic to any of a class of standard, well understood nonlinear dynamical systems - a "normal form" [Guckenheimer and Holmes, 1983]. This system is chosen in advance, independent of both the patterns to be stored and the learning algorithm to be used. This control over the dynamics permits the design of important aspects of the network dynamics independent of the particular patterns to be stored. Stability, basin geometry, and rates of convergence to attractors can be programmed in the standard dynamical system. By analyzing the network in the polar form of these "normal form coordinates", the amplitude and phase dynamics have a particularly simple interaction. When the input to a module is synchronized with its intrinsic oscillation, the amplitudes of the periodic activity may be considered separately from the phase rotation, and the network of the module may be viewed as a static network with these amplitudes as its activity. We can further show analytically that the network modules we have constructed have a strong tendency to synchronize as required. 3 Oscillatory Elman Architecture Because we work with this class of mathematically well-understood associative memory networks, we can take a constructive approach to building a cortical computer architecture, using these networks as modules in the same way that digital computers are designed from well behaved continuous analog flip-flop circuits. The architecture is such that the larger system is itself a special case of the type of network of the submodules, and can be analysed with the same tools used to design the subnetwork modules. Each module is described in normal form or "mode" coordinates as a k-winnertake-all network where the winning set of units may have static, periodic or chaotic dynamics. By choosing modules to have only two attractors, networks can be built which are similar to networks using binary units. There can be fully recurrent connections between modules. The entire super-network of connected modules, however, is itself a polynomial network that can be projected into standard network coordinates. The attractors within the modules may then be distributed patterns like those described for the biological model [Baird, 1990a], and observed experimentally in the olfactory system [Freeman and Baird, 1987]. The system is still equivalent to the architecture of modules in normal form, however, and may easily be designed, simulated, and theoretically evaluated in these coordinates. In this paper all networks are discussed in normal form coordinates. As a benchmark for the capabilities of the system, and to create a point of contact to standard network architectures, we have constructed a discrete-time recurrent "Elman" network [Elman, 1991] from oscillatory modules defined by ordinary differential equations. We have at present a system which functions as a finite state automaton that perfectly recognizes or generates the infinite set of strings defined by the Reber grammar described in Cleeremans et. al. [Cleeremans et al., 1989]. The connections for this network were found by psuedo-inverting to find the connection matrices between a set of pre-chosen automata states for the hidden layer modules Synchronization and Grammatical Inference in an Oscillating Elman Net and the proper possible output symbols of the Reber grammar, and between the proper next hidden state and each legal combination of a new input symbol and the present state contained in the context modules. We use two types of modules in implementing the Elman network architecture. The input and output layer each consist of a single associative memory module with six oscillatory at tractors (six competing oscillatory modes), one for each of the six possible symbols in the grammar. An attractor in these winner-take-all normal form cordinates is one oscillator at its maximum amplitude, with the others near zero amplitude. The hidden and context layers consist of binary "units" composed of a two competing oscillator module. We think of one mode within the unit as representing "I" and the other as representing"O" (see fig. 1). A "weight" for this unit is simply defined to be the weight of a driving unit to the input of the 1 attractor. The weights for the 0 side of the unit are then given as the compliment of these, w O = A - WI. This forces the input to the 0 side of the unit be the complement of the input to the 1 side, If A - If, where A is a bias constant chosen to divide input equally between the oscillators at the midpoint of activation. = .--------------------------- Figure 1. OUTPUT I : I : \(!)@00<V@)1 ??? ? ? ? ?. ?????????? ;? ??. . ? ? ? ) HIDDEN fr : I 1""'"tJ~~I~~t~_~J~~!~~U ,I --------~-------------~------- ~\:.: . - - - - .:.' .. - - I : 1@)01@)01@)01@)01@)01 1~(!)@Cge~1 : L I ________________________________________ CONTEXT INPUT ~ I Information flow in the network is controlled by a "machine cycle" implemented by the sinusoidal clocking of a bifurcation parameter which controls the level of inhibitory inter-mode coupling or "competition" between the individual oscillatory modes within each winner-take-all module. For illustration, we use a binary module represnting either a single hidden or context unit; the behavior of the larger input and output modules is similar. Such a unit is defined in polar normal form coordinates by the following equations: rli Uirli - crli(rii + (d - bsin(wc/od: t ?r5i) + L: wijIj cos(Oj - Oli) j rOi UirOi - croi(r5i + (d - bsin(wc/ockt?rii) + L:(A - Wij )Ij cos(Oj - OOi) j Oli Wi +L Wij(Ij !rli) sin(Oj - Oli) j 00i Wi + I)A j Wij )(Ij !rOi) sin(Oj - Ood 239 240 Baird, Troyer, and Eeckman The clocked parameter bsin(wclockt) has lower (1/10) frequency than the intrinsic frequency of the unit Wi. Asuming that all inputs to the unit are phase-locked, examination of the phase equations shows that the unit will synchronize with this input. When the oscillators are phase-locked to the input, (J; - (Jli 0, and the phase terms cos((J; - (Jli) cos(O) 1 dissappear. This leaves the amplitude equations rli and rOi with static inputs E; wi;I; and E;(A - wi;)I;. The phase equations show a strong tendency to phase-lock, since there is an attractor at zero phase difference </> = (Jo - (JI = (Jo - wIt = 0, and a repellor at 180 degrees in the phase difference equations ;p for either side of a unit driven by an input of the same frequency, WI - Wo o. = = = = ? = -sin-1[(ro/rI)(WI - ;p = Wo -WI + (rI/ro)sin(-</?, so, wo)] Thus we have a network module which approximates a static network unit in its amplitude activity when fully phase-locked. Amplitude information is transmitted between modules, with an oscillatory carrier. If the frequencies of attractors in the architecture are randomly dispersed by a significant amount, phase-lags appear, then synchronization is lost and improper transitions begin to occur. For the remainder of the paper we assume the entire system is operating in the synchronized regime and examine the flow of information characterized by the pattern of amplitudes of the oscillatory modes within the network. 4 Machine Cycle by Clocked Bifurcation Given this assumption of a phase-locked system, the amplitude dynamics behave as a gradient dynamical system for an energy function given by = = E; wii Ii and B E; Ii. Figures 2a and 2b show the where the total input I energy landscape with no external input for minimal and maximal levels of competition respectively. External input simply adds a linear "tilt" to the landscape, with large I giving a larger tilt toward the rli axis and small I a larger tilt toward the rOi axis. Note that for low levels of competition, there is a broad circular valley. When tilted by external input, there is a unique equilibrium that is determined by the bias in tilt alon~ one axis over the other. Thinking of rli as the "acitivity" of the unit, this acitlvity becomes an increasing function of I. The module behaves as analog connectionist unit whose transfer function can be approximated by a sigmoid. With high levels of competition, the unit will behave as a binary (bistable) "digital" flip-flop element. There are two deep valleys, one on each axis. Hence the final steady state of the unit is determined by which basin contains the initial state of the system reached during the analog mode of operation before competition is increased by the clock. This state changes little under the influence of external input: a tilt will move the location of the valleys only slightly. Hence the unit performs a winner-take-all choice on the coordinates of its initial state and maintains that choice independent of external input. Synchronization and Grammatical Inference in an Oscillating Elman Net Figure 2a. Figure 2b. Low Competition High Competition We use this bifurcation in the behavior of the modules to control information flow within the network. We think of the input and context modules as "sensory", and the hidden and output modules as "motor" modules. The action of the clock is applied reciprocally to these two sets (grouped by dotted lines in fig.1) so that they alternatively open to receive input from each other and make transitions of attractors. This enables a network completely defined as a set of ordinary differential equations to implement the discrete-time recurrent Elman network. At the beginning of a machine cycle, the input and context layers are at high competition and hence their activity is "clamped" at the bottom of deep attractors. The hidden and output modules are at low competition and therefore behave as traditional feedforward network free to take on analog values. Then the situation reverses. As the competition comes up in the output module, it makes a winnertake-all choice as to the next symbol. Meanwhile high competition has quantized and clamped the activity in the hidden layer to a fixed binary vector. Then competition is lowered in the input and context layers, freeing these modules from their attractors. Identity mappings from hidden to context and from output to input (gray arrows in fig.1) "load" the binarized activity of the hidden layer to the context layer for the next cycle, and "place" the generated output symbol into the input layer. For a Reber grammar there are always two equally possible next symbols being generated in the output layer, and we apply noise to break this symmetry and let the winnertake-all dynamics of the output module chose one. For the recognition mode of operation, these symbols are thought of as "predicted" by the output, and one of them must always match the next actual input of a string to be recognized or the string is instantly rejected. Note that even though the clocking is sinusiodal and these transitions are not sharp, the system is robust and reliable. It is only necessary to set the rates of convergence within modules to be faster than the rate of change of the clocked bifurcation parameter, so that the modules are operating "adiabatically" - i.e. always internally relaxed to an equilibrium that is moved slowly by the clocked parameter. It is the bifurcation in the phase portrait of a module from one to two attractors that contributes the essential "digitization" of the system in time and state. A bifurcation is a discontinuous (topologically sharp) change in the phase portrait of possibilities for the continuous dynamical behavior of a system that occurs as a bifurcation parameter reaches a "critical" value. We can think of the analog mode for a module as allowing input to prepare its initial state for the binary "decision" between attractor basins that occurs as competition rises and the double potential well appears. The feedback between sensory and motor modules is effectively cut when one set is clamped at high competition. The system can thus be viewed as operating in discrete time by alternating transitions between a finite set of attracting states. This kind of clocking and "buffering" (clamping) of some states while other states 241 242 Baird, Troyer, and Eeckman relax is essential to the reliable operation of digital architectures. The clock input on a flip-flop clamps it's state until its signal inputs have settled and the choice of transition can be made with the proper information available. In our simulations, if we clock all modules to transition at once, the programmed sequences lose stability, and we get transitions to unprogrammed fixed points and simple limit cycles for the whole system. 5 Training When the input and context modules are clamped at their attractors, and the hidden and output modules are in the analog operating mode and synchronized to their inputs, the network approximates the behavior of a standard feedforward network in terms of its amplitude activities. Thus a real valued error can be defined for the hidden and output units and standard learning algorithms like back propagation can be used to train the connections. We can use techniques of Giles et. aI. [Giles et aI., 1992] who have trained simple recurrent networks to become finite state automata that can recognize the regular Tomita languages and others. If the context units are clamped with high competition, they are essentially "quantized" to take on only their 0 or 1 attractor values, and the feedback connections from the hidden units cannot affect them. While Giles, et. aI. often do not quantize their units until the end of training to extract a finite state automaton, they find that quantizing of the context units during training like this increases learning speed in many cases[Giles et aI., 1992]. In preparation for learning in the dynamic architecture, we have sucessfully trained the back propogation network of Cleermans et. aI. with digititized context units and a shifted sigmoid activation function that approximates the one calculated for our oscillatory units. In the dynamic architecture, we have also the option of leaving the competition within the context units at intermediate levels to allow them to take on analog values in a variable sized neighborhood of the 0 or 1 attractors. Since our system is recurrently connected by an identity map from hidden to context units, it will relax to some equilibrium determined by the impact of the context units and the clamped input on the hidden unit states, and the effect of the feedback from those hidden states on the context states. We can thus further explore the impact on learning of this range of operation between discrete time and space automaton and continuous analog recurrent network. 6 Discusion The ability to operate as an finite automaton with oscillatory/chaotic "states" is an important benchmark for this architecture, but only a subset of its capabilities. At low to zero competition, the supra-system reverts to one large continuous dynamical system. We expect that this kind of variation of the operational regime, especially with chaotic attractors inside the modules, though unreliable for habitual behaviors, may nontheless be very useful in other areas such as the search process of reinforcement learning. An important element of intra-cortical communication in the brain, and between modules in this architecture, is the ability of a module to detect and respond to the proper input signal from a particular module, when inputs from other modules which is irrelevant to the present computation are contributing cross-talk and noise. This is smilar to the problem of coding messages in a computer architecture like the Synchronization and Grammatical Inference in an Oscillating Elman Net connection machine so that they can be picked up from the common communication buss line by the proper receiving module. We believe that sychronization is one important aspect of how the brain solves this coding problem. Attractors in modules of the architecture may be frequency coded during learning so that they will sychronize only with the appropriate active attractors in other modules that have a similar resonant frequency. The same hardware (or "wetware") and connection matrix can thus subserve many different networks of interaction between modules at the same time without cross-talk problems. This type of computing architecture and its learning algorithms for computation with oscillatory spatial modes may be ideal for implementation in optical systems, where electromagnetic oscillations, very high dimensional modes, and high processing speeds are available. The mathematical expressions for optical mode competition are nearly identical to our normal forms. Acknowledgements Supported by AFOSR-91-0325, and a grant from LLNL . It is a pleasure to acknowledge the invaluable assistance of Morris Hirsch and Walter Freeman. References [Baird, 1990a] Baird, B. (1990a) . Bifurcation and learning in network models of oscillating cortex. In Forest, S., editor, Emergent Computation, pages 365-384. North Holland. also in Physica D, 42. [Baird, 1990b] Baird, B. (1990b) . A learning rule for cam storage of continuous periodic sequences. In Proc. Int. Joint Conf. on Neural Networks, San Diego, pages 3: 493-498. [Baird and Eeckman, 1992a] Baird, B. and Eeckman, F. H. (1992a). A hierarchical sensory-motor architecture of oscillating cortical area subnetworks. In Eeckman, F. H., editor, Analysis and Modeling of Neural Systems II, pages 96-204, Norwell, Ma. Kluwer. [Baird and Eeckman, 1992b] Baird, B. and Eeckman, F . H. (1992b). A normal form projection algorithm for associative memory. In Hassoun, M. 11., editor, Associative Neural Memories: Theory and Implementation, New York, NY. Oxford University Press. in press. [Cleeremans et al., 1989] Cleeremans, A., Servan-Schreiber, D., and McClelland, J. (1989). Finite state automata and simple recurrent networks. Neural Computation, 1(3):372-381. [Elman, 1991] Elman, J. (1991). Distributed representations, simple recurrent networks and grammatical structure. Machine Learning, 7(2/3):91. [Freeman and Baird, 19871 Freeman, W. and Baird, B. (1987) . Relation of olfactory eeg to behavior: Spatiaf analysis. Behavioral Neuroscience, 101:393-408 . [Giles et al., 1992] Giles, C., Miller, C.B.and Chen, D., Chen, H., Sun, G. , and Lee, Y. (1992). Learning and extracting finite state automata with second order recurrent neural networks. Neural Computation, pages 393-405. [Gray and Singer, 1987] Gray, C. M. and Singer, W . (1987) . Stimulus dependent neuronal oscillations in the cat visual cortex area 17. Neuroscience [Supplj, 22:1301P. [Guckenheimer and Holmes, 1983] Guckenheimer, J. and Holmes, D. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York. 243
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