Question: Represent the article for finding a single-sentence TLDR

Question: Effectively capturing graph node sequences in the form of vector embeddings is critical to many applications. We achieve this by (i) first learning vector embeddings of single graph nodes and (ii) then composing them to compactly represent node sequences. Specifically, we propose SENSE-S (Semantically Enhanced Node Sequence Embedding - for Single nodes), a skip-gram based novel embedding mechanism, for single graph nodes that co-learns graph structure as well as their textual descriptions. We demonstrate that SENSE-S vectors increase the accuracy of multi-label classification tasks by up to 50% and link-prediction tasks by up to 78% under a variety of scenarios using real datasets. Based on SENSE-S, we next propose generic SENSE to compute composite vectors that represent a sequence of nodes, where preserving the node order is important. We prove that this approach is efficient in embedding node sequences, and our experiments on real data confirm its high accuracy in node order decoding. Accurately learning vector embeddings for a sequence of nodes in a graph is critical to many scenarios, e.g., a set of Web pages regarding one specific topic that are linked together. Such a task is challenging as: (i) the embeddings may have to capture graph structure along with any available textual descriptions of the nodes, and moreover, (ii) nodes of interest may be associated with a specific order. For instance, (i) for a set of Wikipedia pages w.r.t. a topic, there exists a recommended reading sequence; (ii) an application may consist of a set of services/functions, which must be executed in a particular order (workflow composability); (iii) in source routing, the sender of a packet on the Internet specifies the path that the packet takes through the network or (iv) the general representation of any path in a graph or a network, e.g., shortest path. Node sequence embedding, thus, requires us to (i) learn embeddings for each individual node of the graph and (ii) compose them together to represent their sequences. To learn the right representation of individual nodes and also their sequences, we need to understand how these nodes are correlated with each other both functionally and structurally. A lot of work has only gone into learning single node embeddings (i.e., where node sequence length is 1), as they are essential in feature representations for applications like multi-label classification or link prediction. For instance, algorithms in BID22, BID4, BID28 and others try to extract features purely from the underlying graph structure; algorithms in BID12, BID19 and others learn vector representations of documents sharing a common vocabulary set. However, many applications would potentially benefit from representations that are able to capture both textual descriptions and the underlying graph structure simultaneously. For example, classification of nodes in a network not only depends on their inter-connections (i.e., graph structure), but also nodes' intrinsic properties (i.e., their textual descriptions); for product recommendations, if the product is new, it may not have many edges since not many users have interacted with it; however, using the textual descriptions along with the graph structure allows for efficient bootstrapping of the recommendation service. For general case of sequence lengths greater than 1, despite the importance in applications like workflow composability described above, there is generally a lack of efficient solutions. Intuitively, we can concatenate or add all involved node vectors; however, such a mechanism either takes too much space or loses the sequence information; thus unable to represent node sequences properly. We aim to learn node sequence embeddings by first first addressing the single node embedding issue, as a special case of node sequence embedding, by considering both the textual descriptions and the graph structure. We seek to answer two questions: How should we combine these two objectives? What framework should we use for feature learning? Works that jointly address these two questions either investigate them under different problem settings BID1 BID32, under restricted learning models BID13, ignore the word context within the document BID16, do not co-learn text and graph patterns or only consider linear combinations of text and graph BID3; this is elaborated further in Section 2. In contrast, we propose a generic neural-network-based model called SENSE-S (Semantically Enhanced Node Sequence Embeddings -for Single nodes) for computing vector representations of nodes with additional semantic information in a graph. SENSE-S is built on the foundation of skip-gram models. However, SENSE-S is significantly different from classic skipgram models in the following aspects: (i) For each word φ in the textual description of node v in the given graph, neighboring words of φ within v's textual description and neighboring nodes of v within the graph are sampled at the same time.(ii) The text and graph inputs are both reflected in the output layer in the form of probabilities of co-occurrence (in graph or text). (iii) Moreover, this joint optimization problem offers an opportunity to leverage the synergy between the graph and text inputs to ensure faster convergence. We evaluate the generated vectors on (i) to show that our SENSE-S model improves multi-label classification accuracy by up to 50% and (ii) Physics Citation dataset BID14 to show that SENSE-S improves link prediction accuracy by up to 78% over the state-of-the-art. Next, we propose SENSE for general feature representation of a sequence of nodes. This problem is more challenging in that (i) besides the original objectives in SENSE-S, we now face another representation goal, i.e., sequence representation while preserving the node order; (ii) it is important to represent the sequence in a compact manner; and (iii) more importantly, given a sequence vector, we need to be able to decipher which functional nodes are involved and in what order. To this end, we develop efficient schemes to combine individual vectors into complex sequence vectors that address all of the above challenges. The key technique we use here is vector cyclic shifting, and we prove that the different shifted vectors are orthogonal with high probability. This sequence embedding method is also evaluated on the Wikispeedia and Physics Citation datasets, and the accuracy of decoding a node sequence is shown to be close to 100% when the vector dimension is large. We overview the most related works by categorizing them as follows:Learning vector representation from text: Vector representation of words (Schütze, 1993) has been a long standing research topic. It has received significant attention in the recent times due to the advances in deep neural networks BID0 BID18. In particular, these neural-network-based schemes outperform n-gram-based techniques BID10 BID9 significantly as they are able to learn the similarities between words. Furthermore, paragraph2vec BID12 extends the well-established word2vec BID19 to learn representations of chunks of text. Learning vector representation from graphs: Lot of research has gone into learning graph representations by translating the network/graph into a set of words or documents BID22 BID23 BID24 BID29 BID28 BID4. Generic models incorporating both edge weight and direction information for graph embeddings are proposed in BID39, BID28 and BID4. Specifically, node2vec BID4 advances the state-of-the-art in this area by designing flexible node sampling methodologies to allow feature vectors to exhibit different properties. Subgraph2vec BID20 extends these schemes to learn vector representations of subgraphs. BID26 proposes techniques to represent graph sequences under the assumption that each node is represented by a random binary vector. Learning graph representation with auxiliary information: Broadly speaking, our work falls into the category of node embedding in graphs with auxiliary information. BID5, BID33, BID7 and others address the case where nodes are associated with labels. BID21 studies graph embedding when node/edge attributes are continuous. BID1 investigates phrase ambiguity resolution via leveraging hyperlinks. However, all these works operate under information or network constraints. On the other hand, BID32, BID36 and explore embedding strategies in the context of knowledge graphs, where the main goal is to maintain the entity relationships specified by semantic edges. In contrast, we consider a simpler network setting where only nodes are associated with semantic information. EP BID3 and GraphSAGE BID6 learn embeddings for structured graph data. However, the textual similarities are only captured by linear combinations. Planetoid computes node embeddings under semi-supervised settings; metapath2vec BID2 learns embeddings for heterogeneous networks (node can be author or paper); and Graph-Neural-Network-based embeddings are explored in BID11 and BID17. However, these papers do not explicitly learn graph structure and text patterns simultaneously, and thus are complementary to SENSE. SNE BID16, SNEA, TADW BID34, HSCA BID37, AANE BID8, ANRL BID38 and PLANE BID13 are more related to our work. However, unlike SENSE, these do not consider the relative context of the words w.r.t. the document. Furthermore, the objective of PLANE is to maximize the likelihood that neighboring nodes have similar embeddings, which is not always the case in practice because neighboring nodes may be semantically different; more critically, it relies on strong assumptions of statistical distributions of words and edges in the network. In this regard, we propose a generic embedding scheme that jointly considers network topology as well as the nodes' semantic information. To embed a general node sequence, we first consider a special case where each node sequence contains only one node. Such single node embedding is referred to as SENSE-S, which jointly learns node representations along with textual descriptions in graphs. Let G = (V, E) denote a given directed or undirected graph, where V is the set of nodes and E the set of edges. Each node in V is associated with a text description. We aim to embed each node v in V into a feature vector that captures both graphical (neighboring node inter-connections) and textual (semantic meaning of v) properties. Specifically, let φ denote a word in the text description of node v. Suppose we obtain a set of neighboring nodes of v in graph G via a specific node sampling strategy, e.g., biased random walk BID4, and a set of neighboring words of φ in the text description of v by a sliding window over consecutive words BID19. We then define N G (v) as a probabilistic event of observing the set of neighboring nodes of v in G (under the chosen model) and N T (φ|v) as an event of observing the set of neighboring words of φ in the text description of v. DISPLAYFORM0 be the embedding function that maps each node v in V into a d-dimensional vector. Our goal is to find function f that maximizes: DISPLAYFORM1 Since events N G (v) and N T (φ|v) are independent, can be rewritten as: DISPLAYFORM2 where w v is the number of words in the description of v. Given a word in a node, jointly captures the node neighborhood in the graph and the word neighborhood in text. We build SENSE-S on the foundation of the skip-gram model BID19. In particular, BID4 and BID22 leverage this skip-gram model to learn vector representation of nodes in a graph by performing biased random walks on the graph and treating each walk as equivalent to a sentence, aiming to predict a neighboring node given the current node in a graph. On the other hand, BID12 extends skip-gram models to learn embeddings of various chunks of text, e.g., sentences, paragraphs or entire documents, via sampling the neighboring words over a sliding window within the text. Motivated by the effectiveness of these models, we build two SENSE-S models, SENSE-S (add) and SENSE-S (concat), as detailed below. Let w denote the number of words (some uninformative words are skipped) in text descriptions across all nodes in the given graph G, i.e., w is the size of the entire vocabulary, and n the number of nodes in G. Then our model is formulated as a neural network, as shown in Figure 1. In this model, each input is a two-tuple (φ, v), i.e., word φ is picked from the description of node v. Then (φ, v) is mapped to two one-hot vectors, w-dimensional word vector φ and n-dimensional vertex vector v, where only the entries corresponding to φ and v are set to 1 and others are set to 0. Then as in typical fully connected neural network architectures, in the first layer, φ T and v T are multiplied (implemented as look-up for efficiency) by two matrices M w×d and M n×d, respectively. The ing vectors are then added together and multiplied by another matrix M d×(w+n) in the second layer, i.e., a (w DISPLAYFORM0 is obtained. Finally, unlike typical skip-gram models where all entries in the output vectors of the second layer are processed by a softmax function, we decompose vector h into two sub-vectors, h 1 consisting of the first w entries and h 2 consisting of the rest. Then h 1 and h 2 are fed to separate softmax functions, yielding h 1 and h 2, respectively (see Figure 1). The reason for this decomposition operation is that we use h 1 to represent the probability vector of neighboring words of φ in the description of node v, and h 2 to represent the probability vector of neighboring nodes of v in the graph. Using this neural network architecture, we aim to learn the values of all entries in the matrices M w×d, M n×d and M d×(w+n) such that the entries (i.e., probabilities) in h 1 and h 2 corresponding to neighboring words of φ (within the text description of node v) and neighboring nodes of v (within the given graph) are maximized; see the objective in. We use matrix M n×d as our final semantically augmented node embeddings, i.e., each row in M n×d corresponds to a node embedding vector. Note that a unique property of SENSE-S is that it is a conjoined model where the textual and graphical inputs both contribute to the learning of node embeddings. Moreover, there is an add operation for combining these features, and thus this model is called SENSE-S (add). This is in contrast to the concatenation operation that is used in a different implementation of SENSE-S; see below. FIG3. Clearly, SENSE-S (concat) is quite similar to SENSE-S (add), except that (i) the ing vectors generated by the first layer are concatenated, and (ii) the matrix dimension in the second layer becomes (2d) × (w + n). In SENSE-S, our focus has been on computing semantically enhanced embeddings for individual nodes. In this section, we propose the general SENSE to represent any set of nodes following a specified order using the node vectors generated by SENSE-S, called node sequence embedding. Given the original graph G = (V, E), let S = v 1 → v 2 → · · · → v q be a node sequence constructed with v i ∈ V (i = 1, 2, . . ., q). Note that S may contain repeated nodes, e.g., some functions need to be executed more than once in one application. Intuitively, node sequence S can be represented by a d × q matrix with column i being the vector representation of node v i. However, such a representation is costly in space. Alternatively, representing S by the vector sum q i=1 v i (v i corresponds to v i) in missing the node order information. Hence, in this section, we seek to find a low-dimensional vector representation such that (i) node properties in the original network G and (ii) the node order in S are both preserved. In this section, all node vectors are unit vectors obtained by normalizing the node vectors generated by SENSE-S; this property is critical in our node sequence embedding method (see Section 4.2).Node Sequence Vector Construction: Given a node sequence S = v 1 → v 2 → · · · → v q following in order from node v 1 to node v q, let v i be the unit node vector of node v i. We first perform positional encoding, via cyclic shift function. Specifically, given vector v of dimension-d and non-negative integer m, we define v. Note that for repeated nodes in S, they are cyclically shifted by different positions depending on the specific order in the node sequence. In S, by imposing the restriction q d, we ensure that wraparounds do not occur while shifting the vector, because it may lead to ambiguity of node positions within a node sequence. Simple as this embedding approach may seem, we show that it exhibits the following advantages. First, the dimension of node sequence vectors remains the same as that of the original node vectors. Second, given a node sequence vector S, we are able to infer which nodes are involved in S and their exact positions in S as explained below. Node Sequence Vector Decoding: The method for determining which nodes are included (and in which order) in a given node sequence vector is referred to as node sequence vector decoding. The basic idea in node sequence vector decoding is rooted in Theorem 2 (Section 4.2), which implies that using cyclic shifting, we essentially enable a preferable property that v. By this property, we make the following claim, assuming that all node vectors are unit vectors. Claim 1. Given a node sequence vector S, node v, whose node vector is v, is at the k-th position of this node sequence if the inner product S · v (k−1) ≈ 1.Claim 1 provides an efficient way to decode a given node sequence. In particular, to determine whether (and where) a node resides in a node sequence, it only takes quadratic complexity O(d 2).Example: Suppose we have a node sequence S = 1 → 2 → 3 consisting of three nodes. By SENSE-S and our encoding method, we construct its node sequence vector as S = v1 +v2 +v3. Then each node can compute the inner product of its cyclic shifted vector with S. If the is approximately 1, then its position in this node sequence is uniquely determined. For instance, DISPLAYFORM0 2 ≈ 0 + 1 + 0 = 1 (see Theorem 2). Thus, given the encoded node sequence S, we know node 2 is at the second position. We now present our theoretical , to support Claim 1. Proofs are presented in the appendix. = 0 happens with high probability when the vector dimension is sufficiently large, although v a and v b are originally related. This section evaluates SENSE under varying scenarios, with multiple datasets and applications. We evaluate SENSE on the following datasets that contain graph information along with textual descriptions:: it contains both Wikipedia plain text articles (text descriptions) and hyper links between articles (graph structure). It is a directed graph with 4, 604 nodes (each is a Wikipedia article) and 119, 882 hyper links. Citation Network BID14: it contains both textual descriptions (title, authors and abstract of the paper) and graphical structure (citations). It is a directed graph with 27, 770 nodes (papers) and 352, 807 edges (citations). To train our SENSE-S model, we first define the loss function based on. Suppose the vocabulary size associated with each node v ∈ V is similar, i.e., we assume ∀v ∈ V, w v ≈ c (c is a constant). Then let DISPLAYFORM0 We define the loss function as DISPLAYFORM1 We then use stochastic gradient descent to minimize our loss function. Let η be the learning rate. At each iteration, we update the model parameters by adding a fraction of −∇L, i.e.,−η∇L = cη∇F G + η∇F T. Since the per-node vocabulary size is much larger than 1, the node neighborhood sampling via random walk is much less frequent than the textual neighborhood sampling. Therefore we want to inject more input data consisting of only the nodes' graphical neighborhood information. To do so, we adjust the model parameter update rule −η∇L as DISPLAYFORM2 where β 1 and β 2 are the equivalent learning rates for graph inputs and text inputs, respectively. We first evaluate SENSE-S, which is compared against the following baseline solutions. To compare with schemes that use graph information alone, we use node2vec since it has been shown to be flexible to capture different graph properties. To compare against schemes that use textual information alone, we use semantic vectors from paragraph2vec since it outperforms other schemes such as Recursive Neural Tensor Networks BID27 ) for tasks like Sentiment Analysis. As in SENSE-S, we also study two implementations of paragraph2vec, i.e., addition and concatenation operations at the hidden layer, referred to as paragraph2vec (add) and paragraph2vec (concat), respectively. For joint text/graph learning, we compare with the following: 1) Initialize with semantic vectors: We learn embeddings using node2vec, but rather than using random initialization, we initialze the vectors using paragraph2vec.2) Initialize with graphical vectors: Here, we learn final embeddings using paragraph2vec, but initialize them with node2vec, i.e., just reverse of the scheme above.3) Iterative Vectorization: The above approaches only leverage semantic or graphical vectors for initializations. Here, we try to capture both iteratively. Specifically, in one iteration, we compute node embedding via node2vec with the embeddings from the previous iteration as initializations; the corresponding are then fed to paragraph2vec as initializations to further compute node embeddings, after which we go to the next iteration. We repeat this process multiple times (5 times in our experiment) to get the final embeddings. Here, we simply concatenate the vectors obtained from paragraph2vec and node2vec and use them as our node embedding vectors. Test splits of data. Document length = first 500 characters (n2v: node2vec, p2v (add)/(cat): paragraph2vec (add)/(concat), n2v w/ p2v: node2vec with paragraph2vec (add) initialization, p2v w/ n2v: paragraph2vec (add) with node2vec initialization, IteraVec: Iterative Vectorization, cat(p2v,n2v): concatenation of p2v and n2v vectors).Experimental Setup: We first learn the vector representations and then use these vector representations for two different tasks: (i) Multi-label classification: Wikipedia pages are classified into different categories, such as history, science, people, etc. This ground truth information is included in the Wikispeedia dataset (which is not used while learning the vectors). There are 15 different top level categories, and our multi-label classification task tries to classify a page into one or more of these categories based on the vectors obtained from different algorithms. We train the OneVsRestClassifier (SVM, linear kernel) from scikit-learn for this task.(ii) Link prediction: Since no category information is available for the Citation Network, we evaluate for link prediction. In particular, 1% of existing citations are removed, after which vectors are learned on this network. We use these removed links as positive samples for link prediction. For negative samples, we randomly sample the same number of pairs which are not linked via a citation in the original network. To obtain the similarity features w.r.t. a pair of nodes, after experimenting with several alternatives, we chose the element-wise absolute difference and train SVM classifier (linear kernel) for link prediction. Parameter settings: (i) We perform κ random walks starting from each node in the graph (κ is 10 for Wikispeedia and 3 for Citation Network, since Citation Network is larger); (ii) each walk is of length 80 as recommended by BID4; (iii) we use sliding window of size 5 for neighboring word sampling; (iv) the default node vector dimension is 128; (v) multi-label classification error is the misclassification rate over the test set; (vi) link prediction error is the percentage of incorrect link predictions over all pairs of papers in the test set; (vii) learning rates β 1 and β 2 in are selected based on the validation set, and the error is reported on the test set. The error of multi-label classification is reported in FIG4, where the first 500 characters of each Wikipedia page are selected as its textual description. Third, among the schemes that utilize both textual and graphical information, SENSE-S (add) and SENSE-S (concat) consistently perform the best. This is because we train the network to co-learn the textual and graphical information to ensure that both objectives converge. This is in contrast with other schemes where the two objectives, due to the loose coupling, are not guaranteed to converge. Finally, SENSE-S (add) (in FIG4 (c)) outperforms node2vec by over 40%, paragraph2vec (add) by over 55% and the closest baseline scheme that leverages both text and graph by 30%. This confirms the benefit of co-learning features from textual as well as graphical information under the SENSE-S architecture. We also see similar trends using the first 1, 000 characters and omit the for brevity. The error of link prediction in the Citation Network is reported in FIG5. We fix train:valid:test to 60%:20%:20% (based on number of links) and use the first 500 characters as text description. We make several interesting observations. First, schemes that use graph information alone (node2vec) substantially outperform schemes that use text descriptions alone (paragraph2vec) for link prediction task. Intuitively, this is because the neighborhood information, which is important for link prediction task, is captured effectively by the node embeddings obtained from node2vec-like techniques. Second, even in cases where the difference in accuracy using the two different sources of information is large, SENSE-S (add) and (cat) are robust, and can effectively extract useful information from text descriptions to further reduce the error of link prediction that uses graph structure alone. Finally, this is significant because it demonstrates the effectiveness of SENSE-S in a variety of scenarios, including cases where the two sources of information may not be equally valuable. We now evaluate the accuracy of encoding/decoding by SENSE for node sequences. We evaluate on three experiments via constructing node sequences in the following different ways: Experiment 1: the node at every position in a sequence is chosen uniformly at random from 4, 604 Wikispeedia nodes. Note that as mentioned earlier, this node sequence may contain repeated nodes (which is allowed), and may or may not be a subgraph of the original graph. Experiment 2: the node sequences are constructed by performing random walks on the Wikispeedia graph. Experiment 3: the node sequences are constructed by performing random walks on the Physics Citation Network. Note that in both Experiments 2 and 3, adjacent nodes in the node sequences will have related vector embeddings. Next, for such constructed node sequences, their vector representations are computed by SENSE-S. Given these sequence vectors, we then decode the node at each position. We evaluate the decoding accuracy under different sequence lengths and vector dimensions, as reported in FIG6.From FIG6 (a), we make the following observations. First, when the node sequence length is small, all vector dimensions lead to almost 100% decoding accuracy. Second, as node sequence length increases, the decoding accuracy declines sharply especially in cases where the node vector dimensions are relatively small, i.e., 128 and 256. This is because, by Theorem 2, correlations between the involved node vectors cause inevitable errors. Such error accumulates when the length of the node sequence is large. Nevertheless, with sufficiently large node vector dimension, i.e., 1024, even long sequences can be decoded perfectly, and with 512, we can decode a workflow of length 10 with over 90% accuracy. Interestingly, FIG6 and (c) also show similar trends. This is significant, as it shows that even if node sequences are constructed from correlated node vectors, i.e., picked from same graph neighborhood, the decoding still achieves high accuracy. This is because, as shown in Theorem 2, after cyclic shifting, the ing vectors are orthogonal with high probability when the vector dimension is large (even if these vectors are originally related). Finally, in Figures 5 (a) and (b) the decoding algorithm needs to find the best match from among 4, 592 nodes (Wikispeedia network). In contrast, in FIG6 (c), it is much more challenging to find the match among 27, 770 nodes. Yet, we are able to decode with high accuracy with theoretical guarantees. We presented SENSE that learns semantically enriched vector representations of graph node sequences. To achieve this, we first developed SENSE-S that learns single node embeddings via a multi-task learning formulation that jointly learns the co-occurrence probabilities of nodes within a graph and words within a node-associated document. We evaluated SENSE-S against state-ofthe-art approaches that leverage both graph and text inputs and showed that SENSE-S improves multi-label classification accuracy in Wikispeedia dataset by up to 50% and link prediction over Physics Citation network by up to 78%. We then developed SENSE that is able to employ provable schemes for vector composition to represent node sequences using the same dimension as the individual node vectors from SENSE-S. We demonstrated that the individual nodes within the sequence can be inferred with a high accuracy (close to 100%) from such composite SENSE vectors. A LEMMA FOR THE PROOF OF THEOREM 2 Proof. Since both x and y are unit vectors, we have x · y = ||x||2||y||2 cos θ = cos θ, where θ is the angle between x and y. Since x and y are not correlated and both x and y are uniformly distributed across the sphere surface, θ is also uniformly distributed, and thus E[x · y] = 0.As x · y is purely determined by the angle θ between x and y, without loss of generality, we select y =

represent the short summary for finding the corresponding scientific article (long)

Node sequence embedding mechanism that captures both graph and text properties.

represent this text to find a much much longer article about it with say hundreds-thousands of words.

This work aims to provide quantitative answers to the relative importance of concepts of interest via concept activation vectors (CAV). In particular, this framework enables non-machine learning experts to express concepts of interest and test hypotheses using examples (e.g., a set of pictures that illustrate  the concept). We show that CAV can be learned given a relatively small set of examples. Hypothesis testing with CAV can answer whether a particular concept (e.g., gender) is more important in predicting a given class (e.g., doctor) than other sets of concepts. Interpreting networks with CAV does not require any retraining or modification of the network.


Question: represent the text to find a very short summary (1-2 sentences max).

Question: Recent evidence shows that convolutional neural networks (CNNs) are biased towards textures so that CNNs are non-robust to adversarial perturbations over textures, while traditional robust visual features like SIFT (scale-invariant feature transforms) are designed to be robust across a substantial range of affine distortion, addition of noise, etc with the mimic of human perception nature. This paper aims to leverage good properties of SIFT to renovate CNN architectures towards better accuracy and robustness. We borrow the scale-space extreme value idea from SIFT, and propose EVPNet (extreme value preserving network) which contains three novel components to model the extreme values: parametric differences of Gaussian (DoG) to extract extrema, truncated ReLU to suppress non-stable extrema and projected normalization layer (PNL) to mimic PCA-SIFT like feature normalization. Experiments demonstrate that EVPNets can achieve similar or better accuracy than conventional CNNs, while achieving much better robustness on a set of adversarial attacks (FGSM,PGD,etc) even without adversarial training. Convolutional neural networks (CNNs) evolve very fast ever since AlexNet makes a great breakthrough on ImageNet image classification challenge in 2012. Various network architectures have been proposed to further boost classification performance since then, including VGGNet , GoogleNet, ResNet , DenseNet and SENet, etc. Recently, people even introduce network architecture search to automatically learn better network architectures . However, state-of-the-art CNNs are challenged by their robustness, especially vulnerability to adversarial attacks based on small, human-imperceptible modifications of the input . thoroughly study the robustness of 18 well-known ImageNet models using multiple metrics, and reveals that adversarial examples are widely existent. Many methods are proposed to improve network robustness, which can be roughly categorized into three perspectives: modifying input or intermediate features by transformation , denoising ), generative models ; modifying training by changing loss functions (; ;, network distillation , or adversarial training designing robust network architectures; ) and possible combinations of these basic categories. For more details of current status, please refer to a recent survey . Although it is known that adversarial examples are widely existent, some fundamental questions are still far from being well studied like what causes it, and how the factor impacts the performance, etc. One of the interesting findings in is that model architecture is a more critical factor to network robustness than model size (e.g. number of layers). Some recent works start to explore much deeper nature. For instance, both show that CNNs are trained to be strongly biased towards textures so that CNNs do not distinguish objects contours from other local or even noise edges, thus perform poorly on shape dominating object instances. On the contrary, there are no statistical difference for human behaviors on both texture rich objects and global shape dominating objects in psychophysical trials. further analyze and show that deep convolutional features can be categorized into robust and non-robust features, while non-robust features may even account for good generalization. However, non-robust features are not expected to have good model interpretability. It is thus an interesting topic to disentangle robust and non-robust features with certain kinds of human priors in the network designing or training process. In fact, human priors have been extensively used in handcraft designed robust visual features like SIFT . SIFT detects scale-space extrema from input images, and selects stable extrema to build robust descriptors with refined location and orientation, which achieves great success for many matching and recognition based vision tasks before CNN being reborn in 2012 . The scale-space extrema are efficiently implemented by using a difference-of-Gaussian (DoG) function to search over all scales and image locations, while the DoG operator is believed to biologically mimic the neural processing in the retina of the eye . Unfortunately, there is (at least explicitly) no such scale-space extrema operations in all existing CNNs. Our motivation is to study the possibility of leveraging good properties of SIFT to renovate CNN networks architectures towards better accuracy and robustness. In this paper, we borrow the scale-space extrema idea from SIFT, and propose extreme value preserving networks (EVPNet) to separate robust features from non-robust ones, with three novel architecture components to model the extreme values: parametric DoG (pDoG) to extract extreme values in scale-space for deep networks, truncated ReLU (tReLU) to suppress noise or non-stable extrema and projected normalization layer (PNL) to mimic PCA-SIFT like feature normalization. pDoG and tReLU are combined into one block named EVPConv, which could be used to replace all k × k (k > 1) conv-layers in existing CNNs. We conduct comprehensive experiments and ablation studies to verify the effectiveness of each component and the proposed EVPNet. Figure 1 illustrates a comparison of responses for standard convolution + ReLU and EVPConv in ResNet-50 trained on ImageNet, and shows that the proposed EVPConv produces less noises and more responses around object boundary than standard convolution + ReLU, which demonstrates the capability of EVPConv to separate robust features from non-robust ones. Our major contribution are: • To the best of our knowledge, we are the first to explicitly separate robust features from non-robust ones in deep neural networks from an architecture design perspective. • We propose three novel network architecture components to model extreme values in deep networks, including parametric DoG, truncated ReLU, and projected normalization layer, and verify their effectiveness through comprehensive ablation studies. • We propose extreme value preserving networks (EVPNets) to combine those three novel components, which are demonstrated to be not only more accurate, but also more robust to a set of adversarial attacks (FGSM, PGD, etc) even for clean model without adversarial training. Robust visual features. Most traditional robust visual feature algorithms like SIFT and SURF are based on the scale-space theory , while there is a close link between scale-space theory and biological vision , since many scalespace operations show a high degree of similarity with receptive field profiles recorded from the mammalian retina and the first stages in the visual cortex. For instance, DoG computes the difference of two Gaussian blurred images and is believed to mimic the neural processing in the retina . SIFT is one such kind of typical robust visual features, which consists of 4 major stages: scale-space extrema detection with DoG operations; Keypoints localization by their stability; Orientation and scale assignment based on primary local gradient direction; Histogram based keypoint description. We borrow the scale-space extrema idea from SIFT, and propose three novel and robust architecture components to mimic key stages of SIFT. Robust Network Architectures. Many research efforts have been devoted to network robustness especially on defending against adversarial attacks as summarized in. However, there are very limited works that tackle this problem from a network architecture design perspective. A major category of methods focus on designing new layers to perform denoising operations on the input image or the intermediate feature maps. Most of them are shown effective on black-box attacks, while are still vulnerable to white-box attacks. Non-local denoising layer proposed in is shown to improve robustness to white-box attack to an extent with adversarial training . Peer sample information is introduced in with a graph convolution layer to improve network robustness. Biologically inspired protection introduces highly non-linear saturated activation layer to replace ReLU layer, and demonstrates good robustness to adversarial attacks, while similar higher-order principal is also used in. However, these methods still lack a systematic architecture design guidance, and many are not robust to iterative attack methods like PGD under clean model setting. In this work, inspired by robust visual feature SIFT, we are able to design a series of innovative architecture components systematically for improving both model accuracy and robustness. We should stress that extreme value theory is a different concept to scale-space extremes, which tries to model the extreme in data distribution, and is used to design an attack-independent metric to measure robustness of DNNs by exploring input data distribution. Difference-of-Gaussian. Given an input image I and Gaussian kernel G(x, y, σ) as below where σ denotes the variance. Also, difference of Gaussian (DoG) is defined as where ⊗ is the convolution operation, and I 1 = G(x, y, σ) ⊗ I 0. Scale-space DoG repeatedly convolves input images with the same Gaussian kernels, and produces difference-of-Gaussian images by subtracting adjacent image scales. Scale-space extrema (maxima and minima) are detected in DoG images by comparing a pixel to its 26 neighbors in 3×3 grids at current and two adjacent scales . Adversarial Attacks. We use h(·) to denote the softmax output of classification networks, and h c (·) to denote the prediction probability of class c. Then given a classifier h(x) = y, the goal of adversarial attack is to find x adv such that the output of classifier deviates from the true label y: Attack Method. The most simple adversarial attack method is Fast Gradient Sign Method (FGSM) , a single-step method which takes the sign of the gradient on the input as the direction of the perturbation. L(·, ·) denotes the loss function defined by cross entropy. Specifically, the formation is as follows: where x is the clean input, y is the label. is the norm bound (||x − x adv || ≤, i.e. -ball) of the adversarial perturbation. Projected gradient descent (PGD) iteratively applies FGSM with a small step size α i (a;) with formulation as below: where i is the iteration number, α = /T with T being the number of iterations.' Proj' is the function to project the image back to -ball every step. Some advanced and complex attacks are further introduced in DeepFool , CW , MI-FGSM. Adversarial Training aims to inject adversarial examples into training procedure so that the trained networks can learn to classify adversarial examples correctly. Specifically, adversarial training solves the following empirical risk minimization problem: where A(x) denotes the area around x bounded by L ∞ /L 2 norm, and H is the hypothesis space. In this work, we employ both FGSM and PGD to generate adversarial examples for adversarial training. Inspired by traditional robust visual feature SIFT, this paper aims to improve model accuracy and robustness by introducing three novel network architecture components to mimic some key components in SIFT: parametric DoG (pDoG), truncated ReLU (tReLU), and projected normalization layer (PNL). Combining pDoG and tReLU constructs the so-called extreme value preserving convolution (EVPConv) block as shown in Figure 2 (a), which can be used to replace all k × k (k > 1) conv-layers in existing CNNs. PNL is a new and robust layer plugged in to replace global average pooling (GAP) layer as shown in Figure 2 (c). A network with all the three components is named as extreme value preserving network (EVPNet). In the following, we will describe these three components in details separately, and elaborate on how they are used to construct the EVPConv block and EVPNet. Parametric DoG (pDoG) is a network component we design to mimic DoG operation. Recall DoG in Equation 2, it repeatedly convolves input images with the same Gaussian kernel in which kernel size σ is designable, and then computes the differences for adjacent Gaussian blurred images. For CNNs, we mimic DoG with two considerations. First, we replace the Gaussian kernel with a learnable convolutional filter. Specifically, we treat each channel of feature map separately as one image, and convolve it with a learnable k × k kernel to mimic Gaussian convolution. Note that the learnable convolution kernel is not required to be symmetric since some recent evidence shows that non-symmetric DoG may perform even better (; Winnemöller, 2011). Applying the procedure to all the feature-map channels is equal to a depth-wise (DW) convolution. Second, we enforce successive depth-wise convolutions in the same block with shared weights since traditional DoG operation uses the same Gaussian kernel. As CNNs produce full scale information at different stages with a series of convolution and downsampling layers, each pDoG block just focuses on producing extrema for current scale, while not requiring to produce full octave extrema like SIFT. The shared DW convolution introduces minimum parameter overhead, and avoid "extrema drift" in the pDoG space so that it may help finding accurate local extrema. Formally, given input feature map f 0, a minimum of two successive depth-wise convolution is applied as where DW is depth-wise convolution with w as the shared weights. pDoG is thus computed as It is worth noting that the successive minus operations make the minus sign not able to be absorbed into w for replacing minus into addition operation. To the best of our knowledge, this is the first time, minus component has been introduced into deep neural networks, which brings totally new element for architecture design/search. Following SIFT, we compute local extrema (maxima and minimal) across the pDoG images using maxout operations : Note we do not compute local extrema in 3 × 3 spatial grids as in SIFT since we do not require keypoint localization in CNNs. Finally, to keep the module compatible to existing networks, we need ensure the output feature map to be of the same size (number of channels and resolution). Therefore, a maxout operation is taken over to merge two feature maps and obtain the final output of this block: Truncated ReLU (tReLU). The pDoG block keeps all the local extrema in the DoG space, while many local extrema are unstable because of small noise and even contrast changes. SIFT adopts a local structure fitting procedure to reject those unstable local extrema. To realize similar effect, we propose truncated ReLU (tReLU) to suppress non-robust local extrema. The basic idea is to truncate small extrema which correspond to noise and non-stable extrema in the pDoG space. This can be implemented by modifying the commonly used ReLU function as where θ is a learnable truncated parameter. Note that this function is discontinued at x = θ and x = −θ. We make a small modification to obtain a continuous version for easy training as below Figure 2(b) plots the tReLU function. Different from the original ReLU, tReLU introduces a threshold parameter θ and keeps elements with higher magnitude. θ can be either a block-level parameter (each block has one global threshold) or a channel-level parameter (each channel holds a separate threshold). By default, we take θ as a block-level parameter. tReLU is combined with pDoG not only to suppress non-robust extrema, but also to simplify the operations. When combining Equation 8 and Equation 9 together, there is nested maxout operation which satisfies commutative law, so that we could rewrite z 0 and z 1 as where | · | is element-wise absolute operation. With tReLU to suppress non-robust features, we have Hence, in practice, we use Equation 13 instead of Equation 8 to compute z 0 and z 1. Note that tReLU does improve robustness and accuracy for pDoG feature maps, while providing no benefits when replacing ReLU in standard CNNs according to our experiments (see Table 1). Projected Normalization Layer (PNL). SIFT computes gradient orientation histogram followed by L2 normalization to obtain final feature representation. This process does not take gradient pixel relationship into account. PCA-SIFT handles this issue by projecting each local gradient patch into a pre-computed eigen-space using PCA. We borrow the idea from PCA-SIFT to build projected normalization layer (PNL) to replace global average pooling (GAP) based feature generation in existing CNNs. Suppose the feature-map data matrix before GAP is X ∈ R d×c, where d = w × h corresponds to feature map resolution, and c is the number of channels, we obtain column vectors {x i ∈ R c} d i=1 from X to represent the i-th pixel values from all channels. The PNL contains three steps: We add a 1 × 1 conv-layer, which can be viewed as a PCA with learnable projection matrix W ∈ R c×p. The output is u i = W T x i, where u i ∈ R p further forms a data matrix U ∈ R d×p. We compute L2 norm for row vectors To eliminate contrast or scale impact, we normalize v to obtainṽ = v/ v p, while · p means the p norm. the normalized vectorṽ is fed into classification layer for prediction purpose. It is interesting to note that PNL actually computes a second order pooling similar as . Suppose w j ∈ R c is the j-th row of W, v j in step-2 can be rewritten as where is an auto-correlation matrix. Figure 2 (c) illustrates the PNL layer. Theoretically, GAP produces a hyper-cube, while PNL produces a hyper-ball. This is beneficial for robustness since hyper-ball is more smooth, and a smoothed surface is proven more robust . Our experiments also verify this point (see Table 1). With these three novel components, we can derive a novel convolution block named EVPConv, and the corresponding networks EVPNet. In details, EVPConv starts from the pDoG component, and replaces Equation 8 with tReLU as in Equation 13. In SIFT, the contribution of each pixel is weighted by the gradient magnitude. This idea can be extended to calibrate contributions from each feature-map channel. Fortunately, Squeeze-and-Excitation (SE) module proposed in provides the desired capability. We thus insert the SE block after tReLU, and compute the output of EVPConv as: where concat(·) means concatenating z 0 and z 1 together for a unified and unbiased calibration, SE(·) is the SE module, s 0 and s 1 are the calibration corresponding to z 0 and z 1, and max denotes an element-wise maximum operation. Figure 2 (a) illustrates the overall structure of EVPConv. EVPConv can be plugged to replace any k × k (k >1) conv-layers in existing CNNs, while the PNL layer can be plugged to replace the GAP layer for feature abstraction. The network consisting of both EVPConv block and the PNL layer is named as EVPNet. The EVPConv block introduces very few additional parameters: w for shared depth-wise convolution, θ for tReLU and parameters for SE module. Note that we allow each EVPConv block having its own w and θ. EVPConv brings relatively fewer additional parameters, which is about 7∼20% (see Appendix A) (smaller models more relative increasing). It also increases theoretic computing cost 3∼10% for a bunch of parameterfree operations like DoG and maxout. However, the added computing cost is non-negligible in practice (2× slower according to our training experiments) due to more memory cost for additional copy of feature-maps. Near memory computing architecture may provide efficient support for this new computing paradigm. Experimental Setup. We evaluate the proposed network components and EVPNet on CIFAR10 and SVHN datasets. CIFAR-10 is a widely used image classification dataset containing 60, 000 images of size 32×32 with 50, 000 for training and 10,000 for testing. SVHN is a digit recognition dataset containing 73,257 training images, 26,032 test images, all with size 32×32. We introduce our novel components into the well-known and widely used ResNet, and compare to the basic model on both clean accuracy and adversarial robustness. As the EVPConv block contains a SE module, to make a fair comparison, we set SE-ResNet as our comparison target. In details, we replace the input conv-layer and the first 3 × 3 conv-layer in the residual block with EVPConv, and replace the GAP layer with the proposed PNL layer. Following , for CIFAR-10, all the networks are trained with SGD using momentum 0.9, 160 epochs in total. The initial learning rate is 0.1, divided by 10 at 80 and 120 epochs. For SVHN, we use the same network architecture as CIFAR-10. The models are trained for 80 epochs, with initial learning rate 0.1, divided by 10 at 40 and 60 epochs. For tReLU, the channel-level parameter θ is initialized by uniformly sampling from. In this work, we consider adversarial perturbations constrained under l ∞ norm. The allowed perturbation norm is 8 pixels . We evaluate non-targeted attack adversarial robustness in three settings: normal training, FGSM adversarial training and PGD adversarial training . During adversarial training, we use the predicted label to generate adversarial examples to prevent label leaking effect (b). To avoid gradient masking , we use R-FGSM for FGSM adversarial training, which basically starts from a random point in the ball. , during training, PGD attacks generate adversarial examples by 7 PGD iterations with 2-pixel step size starting from random points in the allowed ball. We report accuracy on both whitebox and blackbox attack. We evaluate a set of well-known whitebox attacks, including FGSM, PGD, DeepFool, CW. We use'PGD-N' to denote attack with N PGD iterations of step size 2 pixels by default. Specifically, we compare for PGD-10 and PGD-40. For blackbox attack, we choose VGG-16 as the source model which is found by to exhibit high adversarial transferability, and choose FGSM as the method to generate adversarial examples from VGG-16 as it is shown to lead to better transferability. Ablation Study. This part conducts a thorough ablation study to show the effectiveness of each novel architecture component and how they interact to provide strong adversarial robustness. We conduct experiments on CIFAR-10 with SE-ResNet-20, which contains one input conv-layer, three residual stage each with two bottleneck residual blocks, and GAP layer followed by a classification layer. We evaluate the accuracy and robustness for all the possible combinations of the proposed three components under the normal training setting. For PGD attack, we use two step sizes: 1 pixel as in and 2 pixels as in. Table 1 lists full evaluation of each tested model. Several observations can be made from the table: Solely adding pDoG or PNL layer leads to significant robustness improvement. pDoG even yields clean accuracy improvement, while PNL yields slightly clean accuracy drops. tReLU does not bring benefit for standard convolution, while yields notable improvement on both clean accuracy and adversarial accuracy, when combining with pDoG. That verifies our previous claim that tReLU is suitable to work for the DoG space. Combining all the three components together obtains the best adversarial robustness, while still achieve 1.2% clean accuracy improvement over the model without these three components. Based on these observations, we incorporate all the three components into the CNNs to obtain the so-called EVPNet for the following experiments if not explicitly specified. As 2-pixels PGD attack is much stronger than 1-pixel PGD attack, we use it as default in the following studies. Benchmark Results. We conduct extensive experiments on CIFAR-10 and SVHN to compare the proposed EVPNet with the source networks. The two sources networks are For fair comparison, we use the SE extended ResNet as our baseline. Table 2 lists comprehensive comparison on CIFAR-10. We list 7 different kinds of accuracies: clean model accuracy, whilebox attack accuracies by FGSM/PGD-10/PGD-40/DeepFool/CW, and blackbox attack accuracy with adversarial examples generated by FGSM on the VGG-16 model. We can see that under normal training case, EVPNet outperforms baseline by a large margin in terms of robustness with FGSM, PGD, DeepFool, and CW attacks. Even under the strongest PGD-40 white box attack, our EVPNet still has non-zero accuracy without any adversarial training. For those cases with adversarial training, our EVPNet consistently beats baseline networks with noticeable margin. training performs worse on PGD-10/PGD-40 attacks than FGSM adversarial training, and even much worse than normal training on this dataset. This may be due to the fact that SVHN is a shape/edge dominating digit recognition dataset, which may generate a lot of difficult adversarial samples with broken edges. And it also coincides with the finding by. Our EVPNet shows better robustness on this dataset without adversarial training than CIFAR-10, which may suggest that EVPNet is more robust on shape/edge dominating object instances. All these evidences prove that the proposed EVPNet is a robust network architecture. Analysis. We make some further analysis to compare EVPNet to baseline networks. First, we plot the test error at different PGD iterations for different evaluated networks under normal training case on both CIFAR-10 and SVHN datasets as shown in Figure 3. It can be seen that EVPNet consistently performs significantly better than the corresponding baseline networks under all PGD iterations. Some may concern that the accuracy of EVPNet on the strongest PGD-40 attack is not satisfied (∼ 10%). We argue that from three aspects: The adversarial is remarkable as it is by the clean model without using any other tricks like adversarial training, adversarial loss, etc. The proposed components also brings consistent clean accuracy improvement even on large-scale dataset (see Appendix A). More importantly, the methodology we developed may shed some light on future studies in network robustness and network architecture design/search. Second, we further investigate the error amplification effect as. Specifically, we feed both benign examples and adversarial examples into the evaluated networks, and compute the normalized L 2 distance for each res-block outputs as γ = x − x 2 / x 2, where x is the response vector of benign example, and x is the response vector for the adversarial example. We randomly sample 64 images from the test set to generate adversarial examples using PGD-40. The models evaluated are trained without adversarial training. Figure 4 illustrates the . As we can see, EVPNet has much lower average normalized distance than the baseline models almost on all the blocks. It is interesting to see that the baseline models have a big jump for the normalized distance at the end of the networks on all the 4 sub-figures. This urges the adversarial learning researchers to make further investigation on the robustness especially for latter layers around GAP. Nevertheless, this analysis demonstrates that EVPNet significantly reduces the error amplification effect. Third, we compare the differences on feature responses between regular convolution + ReLU and EVPConv. This comparison is made on large-scale and relative high resolution (224 × 224) ImageNet dataset for better illustration. We train ResNet-50 and EVPNet-50 on ImageNet, and visualize their prediction responses for the first corresponding convolution block in Figure 1. It clearly shows that ResNet-50 has more noise responses, while EVPNet-50 gives more responses on object boundary. This demonstrates the capability of EVPConv to separate robust features from non-robust ones. Full benchmark on ImageNet are also very promising, see Appendix A for more details. This paper mimics good properties of robust visual feature SIFT to renovate CNN architectures with some novel architecture components, and proposes the extreme value preserving networks (EVPNet). Experiments demonstrate that EVPNets can achieve similar or better accuracy over conventional CNNs, while achieving much better robustness to a set of adversarial attacks (FGSM, PGD, etc) even for clean model without any other tricks like adversarial training. top-1 accuracy to near zero, while the EVP-ResNet variants keep 6∼10% top-1 accuracy. The gap in FGSM attacks is even larger. This improvement is remarkable considering that it is by clean model without adversarial training. For the MobileNet case, we also observe notable accuracy and robustness improvement. Please refer to Table 4 for more details. In summary, our solid and attempts may inspire future new ways for robust network architecture design or even automatic search. In the main paper, we demonstrate the great robustness of the proposed components on ResNet with bottleneck residual blocks. Here, we extend the proposed components to other state-of-the-art network architectures, and choose Wide-ResNet (Zagoruyko & komodakis, 2017) as an example since it is mostly studied on other adversarial training works (; ;). Wide-ResNet (WRN) has two successive wide-channel 3 × 3 conv-layers in residual block instead of the three conv-layer bottleneck structure based residual block. We use WRN-22-8 as the baseline network with depth 22 and widening factor 8. It is obvious that WRN-22-8 has much better clean accuracy than ResNet-20 and ResNet-56 used in the main paper. For EVPNet, We replace input conv-layer and the first 3 × 3 conv-layer in each wide residual block with EVPConv, and replace the GAP layer with our PNL layer. Table 5 shows the comparison on normal training case. We can see that EVPNet achieves similar clean accuracy, while performing significantly better on adversarial attacks with FGSM/PGD-10/PGD-40. Note that under PGD-10 and PGD-40 attacks, the baseline model drops accuracy to near 0, while the EVPNet remains a much higher accuracy, considering that no adversarial training is utilized in this study. This demonstrates the strong robustness of the proposed EVPNet.

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This paper aims to leverage good properties of robust visual features like SIFT to renovate CNN architectures towards better accuracy and robustness.

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We find that deep networks which generalize poorly are more reliant on single directions than those that generalize well, and evaluate the impact of dropout and batch normalization, as well as class selectivity on single direction reliance.


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Question: Compressed representations generalize better , which may be crucial when learning from limited or noisy labeled data. The Information Bottleneck (IB) method provides an insightful and principled approach for balancing compression and prediction in representation learning. The IB objective I(X; Z) − βI(Y ; Z) employs a Lagrange multiplier β to tune this trade-off. However, there is little theoretical guidance for how to select β. There is also a lack of theoretical understanding about the relationship between β, the dataset, model capacity, and learnability. In this work, we show that if β is improperly chosen, learning cannot happen: the trivial representation P(Z|X) = P(Z) becomes the global minimum of the IB objective. We show how this can be avoided, by identifying a sharp phase transition between the unlearnable and the learnable which arises as β varies. This phase transition defines the concept of IB-Learnability. We prove several sufficient conditions for IB-Learnability, providing theoretical guidance for selecting β. We further show that IB-learnability is determined by the largest confident, typical, and imbalanced subset of the training examples. We give a practical algorithm to estimate the minimum β for a given dataset. We test our theoretical on synthetic datasets, MNIST, and CIFAR10 with noisy labels, and make the surprising observation that accuracy may be non-monotonic in β. Compressed representations generalize better , which is likely to be particularly important when learning from limited or noisy labels, as otherwise we should expect our models to overfit to the noise. introduced the Information Bottleneck (IB) objective function which learns a representation Z of observed variables (X, Y) that retains as little information about X as possible, but simultaneously captures as much information about Y as possible:min IB β (X, Y ; Z) = min I(X; Z) − βI(Y ; Z)I(X; Y) = dx dy p(x, y)log p(x,y) p(x)p(y) is the mutual information. The hyperparameter β controls the trade-off between compression and prediction, in the same spirit as Rate-Distortion Theory , but with a learned representation function P (Z|X) that automatically captures some part of the "semantically meaningful" information, where the semantics are determined by the observed relationship between X and Y.The IB framework has been extended to and extensively studied in a variety of scenarios, including Gaussian variables BID6 ), meta-Gaussians , continuous variables via variational methods BID3; BID5 BID8 ), deterministic scenarios (Strouse & Schwab (2017a); BID12 ), geometric clustering (Strouse & Schwab (2017b) ), and is used for learning invariant and disentangled representations in deep neural nets BID0 b) ). However, a core issue remains: how should we select β? In the original work, the authors recommend sweeping β > 1, which can be prohibitively expensive in practice, but also leaves open interesting theoretical questions around the relationship between β, P (Z|X), and the observed data, P (X, Y). For example, under how much label noise will IB at a given β still be able to learn a useful representation?This work begins to answer some of those questions by characterizing the onset of learning. Specifically:• We show that improperly chosen β may in a failure to learn: the trivial solution P (Z|X) = P (Z) becomes the global minimum of the IB objective, even for β 1.• We introduce the concept of IB-Learnability, and show that when we vary β, the IB objective will undergo a phase transition from the inability to learn to the ability to learn.• Using the second-order variation, we derive sufficient conditions for IB-Learnability, which provide theoretical guidance for choosing a good β.• We show that IB-learnability is determined by the largest confident, typical, and imbalanced subset of the training examples, reveal its relationship with the slope of the Pareto frontier at the origin on the information plane I(Y ; Z) vs. I(X; Z), and discuss its relation with model capacity. We use our main to demonstrate on synthetic datasets, MNIST , and CIFAR10 BID13 ) under noisy labels that the theoretical prediction for IB-Learnability closely matches experiment. We present an algorithm for estimating the onset of IB-Learnability, and demonstrate that it does a good job of estimating the theoretical predictions and the empirical . Finally, we observe discontinuities in the Pareto frontier of the information plane as β increases, and those dicontinuities correspond to accuracy decreasing as β increases. We are given instances of (x, y) ∈ X × Y drawn from a distribution with probability (density) P (X, Y), where unless otherwise stated, both X and Y can be discrete or continuous variables. (X, Y) is our training data, and may be characterized by different types of noise. We can learn a representation Z of X with conditional probability 1 p(z|x), such that X, Y, Z obey the Markov chain Z ← X ↔ Y. Eq. above gives the IB objective with Lagrange multiplier β, IB β (X, Y ; Z), which is a functional of p(z|x): IB β (X, Y ; Z) = IB β [p(z|x)]. The IB learning task is to find a conditional probability p(z|x) that minimizes IB β (X, Y ; Z). The larger β, the more the objective favors making a good prediction for Y. Conversely, the smaller β, the more the objective favors learning a concise representation. How can we select β such that the IB objective learns a useful representation? In practice, the selection of β is done empirically. recommends "sweeping β". In this section, we provide theoretical guidance for choosing β by introducing the concept of IB-Learnability and providing a series of IB-learnable conditions. Definition 1 (IB β -Learnability). (X, Y) is IB β -learnable if there exists a Z given by some p 1 (z|x), such that DISPLAYFORM0, where p(z|x) = p(z) characterizes the trivial representation such that Z = Z trivial is independent of X.If (X; Y) is IB β -learnable, then when IB β (X, Y ; Z) is globally minimized, it will not learn a trivial representation. If (X; Y) is not IB β -learnable, then when IB β (X, Y ; Z) is globally minimized, it may learn a trivial representation. Necessary condition for IB-Learnability. From Definition 1, we can see that IB β -Learnability for any dataset (X; Y) requires β > 1. In fact, from the Markov chain Z ← X ↔ Y, we have I(Y ; Z) ≤ I(X; Z) via the dataprocessing inequality. If β ≤ 1, then since I(X; Z) ≥ 0 and I(Y ; Z) ≥ 0, we have that min(I(X; Z) − βI(Y ; Z)) = 0 = IB β (X, Y ; Z trivial). Hence (X, Y) is not IB β -learnable for β ≤ 1.Theorem 1 characterizes the IB β -Learnability range for β (see Appendix B for the proof): Theorem 1. If (X, Y) is IB β1 -learnable, then for any β 2 > β 1, it is IB β2 -learnable. Based on Theorem 1, the range of β such that (X, Y) is IB β -learnable has the form β ∈ (β 0, +∞). Thus, β 0 is the threshold of IB-Learnability. Furthermore, the trivial representation is a stationary solution for the IB objective: Lemma 1.1. p(z|x) = p(z) is a stationary solution for IB β (X, Y ; Z).The proof in Appendix E shows that the first-order variation δIB β [p(z|x)] = 0 vanishes at the trivial representation. Lemma 1.1 yields our strategy for finding sufficient conditions for learnability: find conditions such that p(z|x) = p(z) is not a local minimum for the functional IB β [p(z|x)]. By requiring that the second order variation δ 2 IB β [p(z|x)] < 0 at the trivial representation (Suff. Cond. 1, Appendix C), and constructing a special form of perturbation at the trivial representation (Suff. Cond. 2, Appendix F), we arrive at the key of this paper (see Appendix G for the proof) 2:Theorem 2 (Confident Subset Suff. Cond.). A sufficient condition for (X, Y) to be IB β -learnable is X and Y are not independent, and DISPLAYFORM1 where Ω x denotes the event that x ∈ Ω x, with probability p(Ω x). Moreover, (inf Ωx⊂X β 0 (Ω x)) −1 gives a lower bound on the slope of the Pareto frontier at the origin of the information plane I(Y ; Z) vs. I(X; Z).Characteristics of dataset leading to low β 0. From Eq., we see that three characteristics of the subset Ω x ⊂ X lead to low β 0: confidence: p(y|Ω x) is large; typicality and size: the number of elements in Ω x is large, or the elements in Ω x are typical, leading to a large probability of p(Ω x); imbalance: p(y) is small for the subset Ω x, but large for its complement. In summary, β 0 will be determined by the largest confident, typical and imbalanced subset of examples, or an equilibrium of those characteristics. Theorem 2 immediately leads to two important corollaries under special problem structures: classification with classconditional noisy labels BID4 ) and deterministic mappings. Corollary 2.1. Suppose that the true class labels are y *, and the input space belonging to each y * has no overlap. We only observe the corrupted labels y with class-conditional noise p(y|x, y *) = p(y|y *) = p(y). Then a sufficient condition for IB β -Learnability is: DISPLAYFORM2 Corollary 2.2. For classification problems, if Y is a deterministic function of X and not independent of X, then a necessary and sufficient condition for IB β -Learnability is β > β 0 = 1.Therefore, if we find that β 0 > 1 for a classification task, we may infer that Y is not a deterministic function of X, i.e. either some classes have overlap, or the labels are noisy. However, finite models may add effective class overlap if they have insufficient capacity for the learning task. This may translate into a higher observed β 0, even when learning deterministic functions. Proofs are provided in Appendix H. Based on Theorem 2, for general classification tasks we suggest Algorithm 1 in Appendix J to empirically estimate an upper-boundβ 0 ≥ β 0. Here, we give the intuition behind the algorithm. First, we train a single maximum likelihood model on the dataset. That model provides estimates for all p(y|x) in the training set. Since learnability is defined with respect to the training data, it is correct to directly use the empirical probability of p(x) and p(y) in the training data. Given p(x), p(y), and p(y|x), and the understanding that we are seaching for a confident subset Ω x, we can then perform an efficient targeted search of the exponential space of subsets of the training data. The algorithm returns the lowest estimate ofβ 0 found during that process. After estimatingβ 0, we can then use it for learning with IB, either directly, or as an anchor for a region where we can perform a much smaller sweep than we otherwise would have. This may be particularly important for very noisy datasets, where β 0 can be very large. To test our theoretical and Alg. 1, we perform experiments on synthetic datasets, MNIST, and CIFAR10. Additional experiment details are provided in Appendix K.Synthetic datasets. We generate a set of synthetic datasets with varying class-conditional noise rates. FIG0 shows the of sweeping β to find the empirical onset of learning, and compares that onset to the predicted onset using Eq.. Clearly the estimate provides a tight upper bound in this simple setting. Also note that β 0 grows exponentially as the label noise increases, underscoring that improperly-chosen β may in an inability to learn useful representations, and the importance of theoretically-guided β selection as opposed to sweeping β in general. MNIST. We perform binary classification with digits 0 and 1, but again add class-conditional noise to the labels with varying noise rates ρ. To explore how the model capacity influences the onset of learning, for each dataset we train two sets of Variational Information Bottleneck (VIB) models differing only by the number of neurons in their hidden layers of the encoder: one with n = 128 neurons, the other with n = 512 neurons. Insufficient capacity will in more uncertainty of Y given X from the point of view of the model, so we expect β 0,observed for the n = 128 model to be larger. FIG0 confirms this prediction. It also shows the β 0,estimated and β 0,predicted given by Algorithm 1 and Eq., respectively. We see that Algorithm 1 does a good job estimating the onset of learning for the large-capacity model, and that the estimated line up well with the theoretical predictions. CIFAR10 forgetting. For CIFAR10 BID13 ), we study how forgetting varies with β. In other words, given a VIB model trained at some high β 2, if we anneal it down to some much lower β 1, what accuracy does the model converge to? We estimated β 0 = 1.0483 on a version of CIFAR10 with 20% label noise using Alg. 1. The lowest β with performance above chance was β = 1.048. See Appendix K.1 for experiment details. As can be seen in FIG0, there are large discontinuities in the Pareto frontier, even though we vary β in very small increments. Those discontinuities start at points on the Pareto frontier where many values of β yield essentially the same I(X; Z) and I(Y ; Z), and end when β crosses apparent phase transitions that give large increases in both I(X; Z) and I(Y ; Z) (marked with red arrows). FIG0 shows that the lowest value of β in each such region tends to have the highest accuracy. A primary empirical of our work is the following: some datasets have non-monotonic performance in regions where multiple values of β cluster together. This surprising behavior is important to check for when training IB models. More thorough study is needed, but based on our initial , we may expect that reducing β to the minimal value that achieves a particular point on the information plane yields better representations. The phenomenon of discontinuities is also observed in prediction error vs. information in the model parameter BID0; BID2 ), I(c; X) vs. H(c) (c denotes clusters) in geometric clustering (Strouse & Schwab (2017b) ). Although these discontinuities (including ours) are observed via different axes, we conjecture that they may all have a shared root cause, which is an interesting topic for future research. In this paper, we have presented theoretical for predicting the onset of learning, and have shown that it is determined by the largest confident, typical and imbalanced subset of the examples. We gave a practical algorithm for predicting the transition, and showed that those predictions are accurate, even in cases of extreme label noise. We have also observed a surprising non-monotonic relationship between β and accuracy, and shown its relationship to discontinuities in the Pareto frontier of the information plane. We believe these will provide theoretical and practical guidance for choosing β in the IB framework for balancing prediction and compression. Our work also raises other questions, such as whether there are other phase transitions in learnability that might be identified. We hope to address some of those questions in future work. Mélanie Rey and Volker Roth. Meta-gaussian information bottleneck. In Advances in Neural Information Processing Systems, pp. 1916 Systems, pp. -1924 Systems, pp., 2012.Ohad Shamir, Sivan Sabato, and Naftali Tishby. Learning and generalization with the information bottleneck. The structure of the Appendix is as follows. In Appendix A, we provide preliminaries for the first-order and secondorder variations on functionals. Then we prove Theorem 1 in Appendix B. In Appendix C, we state and prove Sufficient Condition 1 for IB β -learnability. In Appendix D, we calculate the first and second variations of IB β [p(z|x)] at the trivial representation p(z|x) = p(z), which is used in proving the Sufficient Condition 2 IB β -learnability (Appendix F). After these preparations, we prove the key of this paper, Theorem 2, in Appendix G. Then two important corollaries 2.1, 2.2 are proved in Appendix H. We provide additional discussions and insights for Theorem 2 in Appendix I, and Algorithm 1 for estimation of an upper boundβ 0 ≥ β 0 in Appendix J. Finally in Appendix K, we provide details for the experiments. Let functional F [f (x)] be defined on some normed linear space R. Let us add a perturbative function h(x) to f (x), and now the functional F [f (x) + h(x)] can be expanded as DISPLAYFORM0 where ||h|| denotes the norm of h, DISPLAYFORM1 is a linear functional of h(x), and is called the first-order DISPLAYFORM2 is a quadratic functional of h(x), and is called the second- DISPLAYFORM3 B PROOF OF THEOREM 1Proof. At the trivial representation p(z|x) = p(z), we have I(X; Z) = 0, and I(Y ; Z) = 0 due to the Markov chain, so DISPLAYFORM4 C SUFFICIENT CONDITION 1 AND PROOFIn this section, we prove the Sufficient Condition 1 for IB β -learnability, which will lay the foundation for the Sufficient condition 2 (Appendix F) and the Confident Subset Sufficient condition (key of this paper, Theorem 2) that follow. Theorem 3 (Suff. Cond. 1). A sufficient condition for (X, Y) to be IB β -learnable is that there exists a perturbation function h(z|x) with 3 h(z|x)dz = 0, such that the second-order variation DISPLAYFORM5 Proof. To prove Theorem 3, we use the Theorem 1 of Chapter 5 of BID9 which gives a necessary condition for F [f (x)] to have a minimum at f 0 (x). Adapting to our notation, we have:Theorem 4 (Gelfand et al. FORMULA2). A necessary condition for the functional F [f (x)] to have a minimum at f (x) = f 0 (x) is that for f (x) = f 0 (x) and all admissible h(x), DISPLAYFORM6 DISPLAYFORM7, let us calculate the first and second-order variation of I(X; Z) and I(Y ; Z) w.r.t. p(z|x), respectively. Through this derivation, we use h(z|x) as a perturbative function, for ease of deciding different orders of variations. We will finally absorb into h(z|x). DISPLAYFORM8 We have DISPLAYFORM9 Expanding F 1 [p(z|x) + h(z|x)] to the second order of, we have DISPLAYFORM10 Collecting the first order terms of, we have DISPLAYFORM11 Collecting the second order terms of 2, we have DISPLAYFORM12 Now let us calculate the first and second-order variation of F 2 [p(z|x)] = I(Z; Y). We have DISPLAYFORM13 Using the Markov chain Z ← X ↔ Y, we have DISPLAYFORM14 Then expanding F 2 [p(z|x) + h(z|x)] to the second order of, we have DISPLAYFORM15 Collecting the first order terms of, we have DISPLAYFORM16 Collecting the second order terms of, we have DISPLAYFORM17 Finally, we have DISPLAYFORM18 DISPLAYFORM19 Absorb into h(z|x), we get rid of the factor and obtain the final expression in Lemma 4.1.E PROOF OF LEMMA 1.1Proof. Using Lemma 4.1, we have DISPLAYFORM20 Let p(z|x) = p(z) (the trivial representation), we have that log p(z|x) p(z) ≡ 0. Therefore, the two integrals are both 0. Hence, DISPLAYFORM21 F SUFFICIENT CONDITION 2 AND PROOF Theorem 5 (Suff. Cond. 2). A sufficient condition for (X, Y) to be IB β -learnable is X and Y are not independent, and β > inf DISPLAYFORM0 where the functional β 0 [h(x)] is given by DISPLAYFORM1 Moreover, we have that inf h(x) β[h(x)] −1 is a lower bound of the slope of the Pareto frontier in the information plane I(Y ; Z) vs. I(X; Z) at the origin. The proof is given in Appendix F, which also gives a construction for h(z|x) for Theorem 3 for any h(x) satisfying Theorem 5, and shows that the converse is also true: if there exists h(z|x) suth that the condition in Theorem 3 is true, then we can find h(x) satisfying the the condition in Theorem 5.The geometric meaning of (β 0 [h(x)]) −1 is as follows. It equals DISPLAYFORM2 under a perturbation function of the form h 1 (z|x) = h(x)h 2 (z) (satisfying h 2 (z)dz = 0 and δ 2 I(X;Z), which turns out to be equal to DISPLAYFORM3 under the class of perturbation functions h 1 (z|x) = h 2 (z)h(x), and provides a lower bound of sup h(z|x) DISPLAYFORM4, which is the slope of the Pareto frontier in the information plane I(Y ; Z) vs. I(X; Z) at the origin. Theorem 5 in essence states that as long as β −1 is lower than this lower bound of the slope of the Pareto frontier, (X; Y) is IB β -learnable. From Theorem 5, we see that it still has an infimum over an arbitrary function h(x), which is not easy to estimate. To get rid of h(x), we can use a specific functional form for h(x) in Eq. FORMULA26, and obtain a stronger sufficient condition for IB β -Learnability. But we want to choose h(x) as near to the infimum as possible. To do this, we note the following characteristics for the R.H.S of Eq.:• We can set h(x) to be nonzero if x ∈ Ω x for some region Ω x ⊂ X and 0 otherwise. Then we obtain the following sufficient condition: DISPLAYFORM5 • The numerator of the R.H.S. of Eq. attains its minimum when h(x) is a constant within Ω x. This can be proved using the Cauchy-Schwarz inequality: DISPLAYFORM6, and defining the inner product as u, v = u(x)v(x)dx. Therefore, the numerator of the R.H.S. of Eq. ≥ − 1, and attains equality when DISPLAYFORM7 Based on these observations, we can let h(x) be a nonzero constant inside some region Ω x ⊂ X and 0 otherwise, and the infimum over an arbitrary function h(x) is simplified to infimum over Ω x ⊂ X, and we obtain the confident subset sufficient condition (Theorem 2) for IB β -Learnability, which is a key of this paper. Proof. Firstly, from the necessary condition of β > 1 in Section 2, we have that any sufficient condition for IB β -learnability should be able to deduce β > 1. Now using Theorem 3, a sufficient condition for (X, Y) to be IB β -learnable is that there exists h(z|x) with h(z|x)dx = 0 such that DISPLAYFORM0 At the trivial representation, p(z|x) = p(z) and hence p(x, z) = p(x)p(z). Due to the Markov chain Z ← X ↔ Y, we have p(y, z) = p(y)p(z). Substituting them into the δ 2 IB β [p(z|x)] in Lemma 4.1, the condition becomes: there exists h(z|x) with h(z|x)dz = 0, such that DISPLAYFORM1 Rearranging terms and simplifying, we have DISPLAYFORM2 where DISPLAYFORM3 Now we prove that the condition that ∃h(z|x) s.t. p(z) dz > 0, and let h 1 (z|x) = h(x)h 2 (z). Now we have DISPLAYFORM4 DISPLAYFORM5 In other words, the condition Eq. FORMULA35 is equivalent to requiring that there exists an h(x) such that G[h(x)] < 0. Hence, a sufficient condition for IB β -learnability is that there exists an h(x) such that DISPLAYFORM6 When h(x) = C = const in the entire input space X, Eq. becomes: DISPLAYFORM7 which cannot be true. Therefore, h(x) = const cannot satisfy Eq..Rearranging terms and simplifying, and note that dxh(x)p(x) 2 > 0 due to h(x) ≡ 0 = const, we have DISPLAYFORM8 For the R.H.S. of Eq., let us show that it is greater than 0. Using Cauchy-Schwarz inequality: u, u v, v ≥ u, v 2, and setting u(x) = h(x) p(x), v(x) = p(x), and defining the inner product as DISPLAYFORM9 It attains equality when DISPLAYFORM10 v(x) = h(x) is constant. Since h(x) cannot be constant, we have that the R.H.S. of Eq. is greater than 0.For the L.H.S. of Eq., due to the necessary condition that β > 0, if FORMULA42 cannot hold. Then the h(x) such that Eq. holds is for those that satisfies DISPLAYFORM11 DISPLAYFORM12 We see this constraint contains the requirement that h(x) ≡ const. Written in the form of expectations, we have DISPLAYFORM13 Since the square function is convex, using Jensen's inequality on the outer expectation on the L.H.S. of Eq., we have DISPLAYFORM14 The equality holds iff E x∼p(x|y) [h(x)] is constant w.r.t. y, i.e. Y is independent of X. Therefore, in order for Eq. FORMULA0 to hold, we require that Y is not independent of X.Using Jensen's inequality on the innter expectation on the L.H.S. of Eq., we have DISPLAYFORM15 The equality holds when h(x) is a constant. Since we require that h(x) is not a constant, we have that the equality cannot be reached. Under the constraint that Y is not independent of X, we can divide both sides of Eq. 8, and obtain the condition: there exists an h(x) such that DISPLAYFORM16 Written in the form of expectations, we have DISPLAYFORM17 We can absorb the constraint Eq. into the above formula, and get DISPLAYFORM18 where DISPLAYFORM19 which proves the condition of Theorem 5.Furthermore, from Eq. FORMULA0 we have DISPLAYFORM20 for h(x) ≡ const, which satisfies the necessary condition of β > 1 in Section 2.Proof of lower bound of slope of the Pareto frontier at the origin: Inflection point for general Z: If we do not assume that Z is at the origin of the information plane, but at some general stationary solution Z * with p(z|x), we define DISPLAYFORM21 DISPLAYFORM22 DISPLAYFORM23 It becomes a non-stable solution (non-minimum), and we will have other Z that achieves a better IB β (X, Y ; Z) than the current Z *.Multiple phase transitions To discuss multiple phase transitions, let us first obtain the β for stationary solution for the IB objective. At a stationary solution for IB β [p(z|x)], for valid perturbation h(z|x) satisfying dzh(z|x) = 0 for any x, we have δ IB β [p(z|x)] − dzdxλ(x)p(z|x) = 0 as a constraint optimization with λ(x) as Lagrangian multipliers. Using Eq., we have DISPLAYFORM24 Therefore, we have DISPLAYFORM25 The last equality is due to that the first equality is always true for any function h(z|x). So we can take out the dxdzh(z|x) factor. λ(x) is used for normalization of p(z|x). Eq. FORMULA0 is equivalent to the of the self-consistent equation in.Eq. FORMULA0 and Eq. provide us with an ideal tool to study multiple phase transitions. For each β, at the minimization of the IB objective, Eq. FORMULA0 is satisfied by some Z * that is at the Pareto frontier on the I(Y ; Z) vs. I(X; Z) plane. As we increase β, the inf h(x) β [h(x)] may remain stable for a wide range of β, until β is greater than inf h(x) β [h(x)], at which point we will have a phase transition where suddenly there is a better Z = Z * * that achieves much lower IB β (X, Y ; Z) value. For example, we can rewrite Eq. FORMULA0 as DISPLAYFORM26 whereλ DISPLAYFORM27 p(x). By substituting into Eq. FORMULA0, we may proceed and get useful . Proof. According to Theorem 5, a sufficient condition for (X, Y) to be IB β -learnable is that X and Y are not independent, and DISPLAYFORM0 We can assume a specific form of h(x), and obtain a (potentially stronger) sufficient condition. Specifically, we let DISPLAYFORM1 for certain Ω x ⊂ X. Substituting into Eq., we have that a sufficient condition for (X, Y) to be IB β -learnable is DISPLAYFORM2 where DISPLAYFORM3 The denominator of Eq. FORMULA0 is DISPLAYFORM4 Using the inequality x − 1 ≥ logx, we have DISPLAYFORM5 Both equalities hold iff p(y|Ω x) ≡ p(y), at which the denominator of Eq. FORMULA0 is equal to 0 and the expression inside the infimum diverge, which will not contribute to the infimum. Except this scenario, the denominator is greater than 0. Substituting into Eq. FORMULA0, we have that a sufficient condition for (X, Y) to be IB β -learnable is DISPLAYFORM6 Since Ω x is a subset of X, by the definition of h(x) in Eq. FORMULA0, h(x) is not a constant in the entire X. Hence the numerator of Eq. FORMULA0 is positive. Since its denominator is also positive, we can then neglect the "> 0", and obtain the condition in Theorem 2.Since the h(x) used in this theorem is a subset of the h(x) used in Theorem 5, the infimum for Eq. FORMULA2 is greater than or equal to the infimum in Eq.. Therefore, according to the second statement of Theorem 5, we have that the (inf Ωx β 0 (Ω x)) −1 is also a lower bound of the slope for the Pareto frontier of I(Y ; Z) vs. I(X; Z) curve. Now we prove that the condition Eq. FORMULA2 is invariant to invertible mappings of X. In fact, if X = g(X) is a uniquely invertible map (if X is continuous, g is additionally required to be continuous), let X = {g(x)|x ∈ Ω x }, and denote DISPLAYFORM7 Additionally we have X = g(X). Then DISPLAYFORM8 For dataset (X, Y) = (g(X), Y ), applying Theorem 2 we have that a sufficient condition for it to be IB β -learnable is DISPLAYFORM9 where the equality is due to Eq.. Comparing with the condition for IB β -learnability for (X, Y) (Eq. FORMULA2), we see that they are the same. Therefore, the condition given by Theorem 2 is invariant to invertible mapping of X. Proof. We use Theorem 2.Let Ω x contain all elements x whose true class is y * for some certain y *, and 0 otherwise. Then we obtain a (potentially stronger) sufficient condition. Since the probability p(y|y *, x) = p(y|y *) is classconditional, we have DISPLAYFORM10, we obtain a sufficient condition for IB β learnability. Proof. We again use Theorem 2. Since Y is a deterministic function of X, let Y = f (X). Since it is classification problem, Y contains at least one value y such that its probability p(y) > 0, we let Ω x contain only x such that f (x) = y. Substituting into Eq., we have DISPLAYFORM0 Therefore, the sufficient condition becomes β > 1.Furthermore, since a necessary condition for IB β -learnability is β > 1 (Section 2), we have that β > β 0 = 1 is a necessary and sufficient condition. Similarity to information measures. The denominator of Eq. is closely related to mutual information. Using the inequality x − 1 ≥ log(x) for x > 0, it becomes: DISPLAYFORM0 whereĨ(Ω x ; Y) is the mutual information "density" at Ω x ⊂ X. Of course, this quantity is also D KL [p(y|Ω x)||p(y)], so we know that the denominator of Eq. FORMULA2 is non-negative. Incidentally, E y∼p (y|Ωx) p (y|Ωx) p(y) − 1 is the density of "rational mutual information" BID15 DISPLAYFORM1 Similarly, the numerator is related to the self-information of Ω x: DISPLAYFORM2 so we can estimate the phase transition as: DISPLAYFORM3 Since Eq. uses upper bounds on both the numerator and the denominator, it does not give us a bound on β 0.Multiple phase transitions. Based on this characterization of Ω x, we can hypothesize datasets with multiple learnability phase transitions. Specifically, consider a region Ω x0 that is small but "typical", consists of all elements confidently predicted as y 0 by p(y|x), and where y 0 is the least common class. By construction, this Ω x0 will dominate the infimum in Eq., ing in a small value of β 0. However, the remaining X − Ω x0 effectively form a new dataset, X 1. At exactly β 0, we may have that the current encoder, p 0 (z|x), has no mutual information with the remaining classes in X 1; i.e., I(Y 1 ; Z 0) = 0. In this case, Definition 1 applies to p 0 (z|x) with respect to I(X 1 ; Z 1). We might expect to see that, at β 0, learning will plateau until we get to some β 1 > β 0 that defines the phase transition for X 1. Clearly this process could repeat many times, with each new dataset X i being distinctly more difficult to learn than X i−1. The end of Appendix F gives a more detailed analysis on multiple phase transitions. Estimating model capacity. The observation that a model can't distinguish between cluster overlap in the data and its own lack of capacity gives an interesting way to use IB-Learnability to measure the capacity of a set of models relative to the task they are being used to solve. Learnability and the Information Plane. Many of our can be interpreted in terms of the geometry of the Pareto frontier illustrated in FIG1, which describes the trade-off between increasing I(Y ; Z) and decreasing I(X; Z). At any point on this frontier that minimizes IB min β ≡ min I(X; Z) − βI(Y ; Z), the frontier will have slope β −1 if it is differentiable. If the frontier is also concave (has negative second derivative), then this slope β −1 will take its maximum β −1 0 at the origin, which implies IB β -Learnability for β > β 0, so that the threshold for IB β -Learnability is simply the inverse slope of the frontier at the origin. More generally, as long as the Pareto frontier is differentiable, the threshold for IB β -learnability is the inverse of its maximum slope. Indeed, Theorem 2 gives lower bounds of the slope of the Pareto frontier at the origin. This means that we lack IB β -learnability for β < β 0, which makes the origin the optimal point. If the frontier is convex, then we achieve optimality at the upper right endpoint if β > β 1, otherwise on the frontier at the location between the two endpoints where the frontier slope is β −1. Learnability and contraction coefficient If we regard the true mapping from X to Y as a channel with transition kernel P Y |X, we can define contraction coefficient η KL (P Y |X) = sup Q;P:0<DKL(P ||Q)<∞ BID16 ) as a measure of how much it keeps the two distributions P and Q intact (as opposed to being drawn nearer measured by KL-divergence) after pushing forward through the channel. By BID16 DISPLAYFORM0 DISPLAYFORM1. Theorem 5 hence also provides a lower bound for the contraction coefficient DISPLAYFORM2 Similarly for Theorem 2. In Alg. 1 we present a detailed algorithm for estimating β 0.Algorithm 1 Estimating the upper bound for β 0 for IB β -Learnability Require: Dataset D = {(x i, y i)}, i = 1, 2,...N. The number of classes is C. Require ε: tolerance for estimating β 0 1: Learn a maximum likelihood model p θ (y|x) using the dataset D. DISPLAYFORM0. 5: j * = arg max j (P y|x) ij 6: Sort the rows of P y|x in decreasing values of (P y|x) ij *. 7: Search i upper untilβ 0 = Getβ(P y|x, p y, Ω) is minimal with tolerance ε, where Ω = {1, 2, ...i upper}. 8: returnβ 0 Subroutine Getβ(P y|x, p y, Ω) s1: (N, C, n) ← (number of rows of P y|x, number of columns of P y|x, number of elements of Ω). DISPLAYFORM1 We use the Variational Information Bottleneck (VIB) objective by BID3. For the synthetic experiment, the latent Z has dimension of 2. The encoder is a neural net with 2 hidden layers, each of which has 128 neurons with ReLU activation. The last layer has linear activation and 4 output neurons, with the first two parameterizes the mean of a Gaussian and the last two parameterizes the log variance of the Gaussian. The decoder is a neural net with 1 hidden layers with 128 neurons and ReLU activation. Its last layer has linear activation and outputs the logit for the class labels. It uses a mixture of Gaussian prior with 500 components (for the experiment with class overlap, 256 components), each of which is a 2D Gaussian with learnable mean and log variance, and the weights for the components are also learnable. For the MNIST experiment, the architecture is mostly the same, except the following: for Z, we let it have dimension of 256. For the prior, we use standard Gaussian with diagonal covariance matrix. For all experiments, we use Adam BID11 ) optimizer with default parameters. We do not add any regularization. We use learning rate of 10 −4 and have a learning rate decay of 1 1+0.01×epoch. We train in total 2000 epochs with batch size of 500. All experiments has train-test split of 5:1, and we report the accuracy on the test set, w.r.t. the true labels. For estimation of β 0,exp in FIG0, in the accuracy vs. β i curve, we take the mean and standard deviation of the accuracy for the lowest 5 β i values, denoting as µ β, σ β. When β i is greater than µ β + 3σ β, we regard it as learning a non-trivial representation, and take the average of β i and β i−1 as the experimentally estimated onset of learning. We also inspect manually and confirm that it is consistent with human intuition. For the estimating β 0,estimated using Alg. 1, at step 7 we use the following discrete search algorithm. We gradually narrow down the range We trained a deterministic 28x10 wide resnet BID10 ), using the open source implementation from BID7. However, we extended the final 10 dimensional logits of that model through another 3 layer MLP classifier, in order to keep the inference network architecture identical between this model and the VIB models we describe below. During training, we dynamically added label noise according to the class confusion matrix in Tab. K.1. The mean label noise averaged across the 10 classes is 20%. After that model had converged, we used it to estimate β 0 with Alg. 1. Even with 20% label noise, β 0 was estimated to be 1.0483.We then trained 73 different VIB models using the same 28x10 wide resnet architecture for the encoder, parameterizing the mean of a 10-dimensional unit variance Gaussian. Samples from the encoder distribution were fed to the same 3 layer MLP classifier architecture used in the deterministic model. The marginal distributions were mixtures of 500 fully covariate 10-dimensional Gaussians, all parameters of which are trained. The VIB models had β ranging from 1.02 to 2.0 by steps of 0.02, plus an extra set ranging from 1.04 to 1.06 by steps of 0.001 to ensure we captured the empirical β 0 with high precision. However, this particular VIB architecture does not start learning until β > 2.5, so none of these models would train as described. 4 Instead, we started them all at β = 100, and annealed β down to the corresponding target over 10,000 training gradient steps. The models continued to train for another 200,000 gradient steps after that. In all cases, the models converged to essentially their final accuracy within 20,000 additional gradient steps after annealing was completed. They were stable over the remaining ∼ 180, 000 gradient steps.

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Theory predicts the phase transition between unlearnable and learnable values of beta for the Information Bottleneck objective

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A backward model of previous (state, action) given the next state, i.e. P(s_t, a_t | s_{t+1}), can be used to simulate additional trajectories terminating at states of interest! Improves RL learning efficiency.