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3,900 | 4,530 | Latent Graphical Model Selection: Efficient Methods
for Locally Tree-like Graphs
Ragupathyraj Valluvan
UC Irvine
[email protected]
Animashree Anandkumar
UC Irvine
[email protected]
Abstract
Graphical model selection refers to the problem of estimating the unknown graph
structure given observations at the nodes in the model. We consider a challenging
instance of this problem when some of the nodes are latent or hidden. We characterize conditions for tractable graph estimation and develop efficient methods
with provable guarantees. We consider the class of Ising models Markov on locally tree-like graphs, which are in the regime of correlation decay. We propose
an efficient method for graph estimation, and establish its structural consistency
???(?+1)?2
when the number of samples n scales as n = ?(?min
log p), where ?min
is the minimum edge potential, ? is the depth (i.e., distance from a hidden node to
the nearest observed nodes), and ? is a parameter which depends on the minimum
and maximum node and edge potentials in the Ising model. The proposed method
is practical to implement and provides flexibility to control the number of latent
variables and the cycle lengths in the output graph. We also present necessary
conditions for graph estimation by any method and show that our method nearly
matches the lower bound on sample requirements.
Keywords: Graphical model selection, latent variables, quartet methods, locally tree-like graphs.
1
Introduction
It is widely recognized that the process of fitting observed data to a statistical model needs to incorporate latent or hidden factors, which are not directly observed. Learning latent variable models
involves mainly two tasks: discovering structural relationships among the observed and hidden variables, and estimating the strength of such relationships. One of the simplest models is the latent
class model (LCM), which incorporates a single hidden variable and the observed variables are
conditionally independent given the hidden variable. Latent tree models extend this model class
to incorporate many hidden variables in a hierarchical fashion. Latent trees have been effective in
modeling data in a variety of domains, such as phylogenetics [1]. Their computational tractability:
upon learning the latent tree model, enables the inference to be carried out efficiently through belief
propagation. There has been extensive work on learning latent trees, including some of the recent
works, e.g. [2?4], demonstrate efficient learning in high dimensions. However, despite the advantages, the assumption of an underlying tree structure may be too restrictive. For instance, consider
the example of topic-word models, where topics (which are hidden) are discovered using information about word co-occurrences. In this case, a latent tree model does not accurately represent the
hierarchy of topics and words, since there are many common words across different topics. Here, we
relax the latent tree assumption to incorporate cycles in the latent graphical model while retaining
many advantages of latent tree models, including tractable learning and inference. Relaxing the tree
constraint leads to many challenges: in general, learning these models is NP-hard, even when there
are no latent variables, and developing tractable methods for such models is itself an area of active
research, e.g. [5?7]. We consider structure estimation in latent graphical models Markov on locally
1
tree-like graphs. These extensions of latent tree models are relevant in many settings: for instance,
when there is a small overlap among different hierarchies of variables, the resulting graph has mostly
long cycles. There are many questions to be addressed: are there parameter regimes where these
models can be learnt consistently and efficiently? If so, are there practical learning algorithms?
Are learning guarantees for loopy models comparable to those for latent trees? How does learning
depend on various graph attributes such as node degrees, girth of the graph, and so on?
Our Approach: We consider learning Ising models with latent variables Markov on locally treelike graphs. We assume that the model parameters are in the regime of correlation decay. In this
regime, there are no long-range correlations, and the local statistics converge to a tree limit. Hence,
we can employ the available latent tree methods to learn ?local? subgraphs consistently, as long as
they do not contain any cycles. However, merging these estimated local subgraphs (i.e., latent trees)
remains a non-trivial challenge. It is not clear whether an efficient approach is possible for matching
latent nodes during this process. We employ a different philosophy for building locally tree-like
graphs with latent variables. We decouple the process of introducing cycles and latent variables in
the output model. We initialize a loopy graph consisting of only the observed variables, and then
iteratively add latent variables to local neighborhoods of the graph. We establish correctness of our
method under a set of natural conditions. We establish that our method is structurally consistent
???(?+1)?2
when the number of samples n scales as n = ?(?min
log p), where p is the number of
observed variables, ?min is the minimum edge potential, ? is the depth (i.e., graph distance from a
hidden node to the nearest observed nodes), and ? is a parameter which depends on the minimum
and maximum node and edge potentials of the Ising model (? = 1 for homogeneous models).
The sample requirement for our method is comparable to the requirement for many popular latent
tree methods, e.g. [2?4]. Moreover, note that when there are no hidden variables (? = 1), the
?2
sample complexity of our method is strengthened to n = ?(?min
log p), which matches with the
sample complexity of existing algorithms for learning fully-observed Ising models [5?7]. Thus, we
present an efficient method which bridges structure estimation in latent trees with estimation in fully
observed loopy graphical models. Finally, we present necessary conditions for graph estimation by
any method and show that our method nearly matches the lower bound. Our method has a number
of attractive features: it is amenable to parallelization making it efficient on large datasets, provides
flexibility to control the length of cycles and the number of latent variables in the output model, and
it can incorporate penalty scores such as the Bayesian information criterion (BIC) [8] to tradeoff
model complexity and fidelity. Preliminary experiments on the newsgroup dataset suggests that
the method can discover intuitive relationships efficiently, and also compares well with the popular
latent Dirichlet allocation (LDA) [9] in terms of topic coherence and perplexity.
Related Work: Learning latent trees has been studied extensively, mainly in the context of phylogenetics. Efficient algorithms with provable guarantees are available (e.g. [2?4]). Our proposed
method for learning loopy models is inspired by the efficient latent tree learning algorithm of [4].
Works on high-dimensional graphical model selection are more recent. They can be mainly classified into two groups: non-convex local approaches [5, 6, 10] and those based on convex optimization [7, 11, 12]. There is a general agreement that the success of these methods is related to the
presence of correlation decay in the model [13]. This work makes the connection explicit: it relates the extent of correlation decay with the learning efficiency for latent models on large girth
graphs. An analogous study of the effect of correlation decay for learning fully observed models
is presented in [5]. This paper is the first work to provide provable guarantees for learning discrete
graphical models on loopy graphs with latent variables (which can also be easily extended to Gaussian models, see Remark following Theorem 1). The work in [12] considers learning latent Gaussian
graphical models using a convex relaxation method, by exploiting a sparse-low rank decomposition
of the Gaussian precision matrix. However, the method cannot be easily extended to discrete models. Moreover, the ?incoherence? conditions required for the success of convex methods are hard to
interpret and verify in general. In contrast, our conditions for success are transparent and based on
the presence of correlation decay in the model.
2
System Model
Ising Models: A graphical model is a family of multivariate distributions Markov in accordance
to a fixed undirected graph [14]. Each node in the graph i ? W is associated to a random variable Xi
2
taking value in a set X . The set of edges E captures the set of conditional independence relations
among the random variables. We say that a set of random variables XW := {Xi , i ? W } with
probability mass function (pmf) P is Markov on the graph G if
P (xi |xN (i) ) = P (xi |xW \i )
(1)
holds for all nodes i ? W , where N (i) are the neighbors of node i in graph G. The HammersleyClifford theorem [14] states that under the positivity condition, given by P (xW ) > 0, for all
xW ? X |W | , a distribution P satisfies the Markov property according to a graph G iff. it factorizes
according to the cliques of G. A special case of graphical models is the class of Ising models, where
each node consists of a binary variable over {?1, +1} and there are only pairwise interactions in
the model. In this case, the joint distribution factorizes as
!
X
X
P (xW ) = exp
?i,j xi xj +
?i xi ? A(?) ,
(2)
e?E
i?V
where ? := {?i,j } and ? := {?i } are known as edge and the node potentials, and A(?) is known
as the log-partition function, which serves to normalize the probability distribution. We consider
latent graphical models in which a subset of nodes is latent or hidden. Let H ? W denote the
hidden nodes and V ? W denote the observed nodes. Our goal is to discover the presence of hidden
variables XH and learn the unknown graph structure G(W ), given n i.i.d. samples from observed
variables XV . Let p := |V | denote the number of observed nodes and m := |W | denote the total
number of nodes.
Tractable Models for Learning: In general, structure estimation of graphical models is NP-hard.
We now characterize a tractable class of models for which we can provide guarantees on graph
estimation.
Girth-Constrained Graph Families: We consider the family of graphs with a bound on the girth,
which is the length of the shortest cycle in the graph. Let GGirth (m; g) denote the ensemble of
graphs with girth at most g. There are many graph constructions which lead to a bound on girth.
For example, the bipartite Ramanujan graph [15] and the random Cayley graphs [16] have bounds
on the girth. Theoretical guarantees for our learning algorithm will depend on the girth of the graph.
However, our experiments reveal that our method is able to construct models with short cycles as
well.
Regime of Correlation Decay: This work establishes tractable learning when the graphical model
converges locally to a tree limit. A sufficient condition for the existence of such limits is the regime
of correlation decay, which refers to the property that there are no long-range correlations in the
model [5]. In this regime, the marginal distribution at a node is asymptotically independent of the
configuration of a growing boundary. For the class of Ising models in (2), the regime of correlation
decay can be explicitly characterized, in terms of the maximum edge potential ?max of the model
and the maximum node degree ?max . Define ? := ?max tanh ?max . When ? < 1, the model is in
the regime of correlation decay, and we provide learning guarantees in this regime.
3
Method, Guarantees and Necessary Conditions
Background on Learning Latent Trees: Most latent tree learning methods are distance based,
meaning they are based on the presence of an additive tree metric between any two nodes in the
tree model. For Ising model (and more generally, any discrete model), the ?information? distance
between any two nodes i and j in a tree T is defined as
d(i, j; T ) := ? log | det(Pi,j )|,
(3)
where Pi,j denotes the joint probability distribution between nodes i and j. On a tree model T , it can
be established that {d(i, j)} is additive along any path in T . Learning latent trees can thus be reforb j) : i, j ? V }
mulated as learning tree structure T given end-to-end (estimated) distances d := {d(i,
between the observed nodes V . Various methods with performance guarantees have been proposed,
e.g. [2?4]. They are usually based on local tests such as quartet tests, involving groups of four nodes.
3
In [4], the so-called CLGrouping method is proposed, which organically grows the tree structure by
adding latent nodes to local neighborhoods. In the initial step, the method constructs the minimum
spanning tree MST(V ; d) over the observed nodes V using distances d. The method then iteratively
visits local neighborhoods of MST(V ; d) and adds latent nodes by conducting local distance tests.
Since a tree structure is maintained in every iteration of the algorithm, we can parsimoniously add
hidden variables by selecting neighborhoods which locally maximize scores such as the Bayesian
information criterion (BIC) [8]. This method also allows for fast implementation by parallelization
of latent tree reconstruction in different neighborhoods, see [17] for details.
Proposed Algorithm: We now propose a method for learning loopy latent graphical models. As
in the case of latent tree methods, our method is also based on estimated information distances
n
dbn (i, j; G) := ? log | det(Pbi,j
)|, ?i, j ? V,
(4)
n
where Pbi,j
denotes the empirical probability distribution at nodes i and j computed using n i.i.d.
samples. The presence of correlation decay in the Ising model implies that dbn (i, j; G) is approximately a tree metric when nodes i and j are ?close? on graph G (compared to the girth g of the
graph). Thus, intuitively, local neighborhoods of G can be constructed through latent tree methb
ods. However, the challenge is in merging these local estimates together to get a global estimate G:
the presence of latent nodes in the local estimates makes merging challenging. Moreover, such a
merging-based method cannot easily incorporate global penalties for the number of latent variables
added in the output model, which is relevant to obtain parsimonious representations on real datasets.
We overcome the above challenges as follows: our proposed method decouples the process of adding
b 0 and then iteratively adds
cycles and latent nodes to the output model. It initializes a loopy graph G
latent variables to local neighborhoods. Given a parameter r > 0, for every node i ? V , consider
b n ) := {j : dbn (i, j) < r}. The initial graph estimate G
b 0 is obtained by taking
the set of nodes Br (i; d
the union of local minimum spanning trees:
b n ); d
b n ).
b 0 ? ?i?V MST(Br (i; d
G
(5)
b 0 and running a
The method then adds latent variables by considering only local neighborhoods in G
b is obtained.
latent tree reconstruction routine. By visiting all the neighborhoods, a graph estimate G
Implementation details about the algorithm are available in [17]. We subsequently establish that
correctness of the proposed method under a set of natural conditions. We require that the parameter
r, which determines the set Br (i; d) for each node i, needs to be chosen as a function of the depth ?
(i.e., distance from a hidden node to its closest observed nodes) and girth g of the graph. In practice,
the parameter r provides flexibility in tuning the length of cycles added to the graph estimate. When
r is large enough, we obtain a latent tree, while for small r, the graph estimate can contain many
short cycles (and potentially many components). In experiments, we evaluate the performance of
our method for different values of r. For more details, see Section 4.
3.1
Conditions for Recovery
We present a set of natural conditions on the graph structure and model parameters under which our
proposed method succeeds in structure estimation.
(A1) Minimum Degree of Latent Nodes: We require that all latent nodes have degree at least
three, which is a natural assumption for identifiability of hidden variables. Otherwise, the
latent nodes can be marginalized to obtain an equivalent representation of the observed
statistics.
(A2) Bounded Potentials: The edge potentials ? := {?i,j } of the Ising model are bounded,
and let
?min ? |?i,j | ? ?max , ? (i, j) ? G.
(6)
Similarly assume bounded node potentials.
(A3) Correlation Decay: As described in Section 2, we assume correlation decay in the Ising
model. We require
? := ?max tanh ?max < 1,
4
?g/2
?(?+1)+2
?min
= o(1),
(7)
where ?max is the maximum node degree, g is the girth and ?min , ?max are the minimum
and maximum (absolute) edge potentials in the model.
(A4) Distance Bounds: We now define certain quantities which depend on the edge potential
bounds. Given an Ising model P with edge potentials ? = {?i,j } and node potentials
? := {|?i,j |} and node
? = {?i }, consider its attractive counterpart P? with edge potentials ?
0
?
?
? is the expectation
potentials ? := {|?i |}. Let ?max := maxi?V atanh(E(Xi )), where E
?
with respect to the distribution P . Let P (X1,2 ; {?, ?1 , ?2 }) denote an Ising model on two
nodes {1, 2} with edge potential ? and node potentials {?1 , ?2 }. Our learning guarantees
depend on dmin and dmax defined below.
dmin := ? log|det P (X1,2 ; {?max , ?0max , ?0max })|,
dmax := ? log|det P (X1,2 ; {?min , 0, 0})|,
? :=
dmax
.
dmin
(A5) Girth vs. Depth: The depth ? characterizes how close the latent nodes are to observed
nodes on graph G: for each hidden node h ? H, find a set of four observed nodes which
form the shortest quartet with h as one of the middle nodes, and consider the largest graph
distance in that quartet. The depth ? is the worst-case distance over all hidden nodes. We
require the following tradeoff between the girth g and the depth ?:
g
? ?? (? + 1) = ?(1),
(8)
4
Further, the parameter r in our algorithm is chosen as
r > ? (? + 1) dmax + ,
for some > 0,
g
dmin ? r = ?(1).
4
(9)
(A1) is a natural assumption on the minimum degree of the hidden nodes for identifiability. (A2)
assumes bounds on the edge potentials. It is natural that the sample requirement of any graph estimation algorithm depends on the ?weakest? edge characterized by the minimum edge potential
?min . Further, the maximum edge potential ?max characterizes the presence/absence of long range
correlations in the model, and is made exact in (A3). Intuitively, there is a tradeoff between the
maximum degree ?max and the maximum edge potential ?max of the model. Moreover, (A3) prescribes that the extent of correlation decay be strong enough (i.e., a small ? and a large enough girth
g) compared to the weakest edge in the model. Similar conditions have been imposed before for
graphical model selection in the regime of correlation decay when there are no hidden variables [5].
(A4) defines certain distance bounds. Intuitively, dmin and dmax are bounds on information distances given by the local tree approximation of the loopy model. Note that e?dmax = ?(?min ) and
e?dmin = O(?max ). (A5) provides the tradeoff between the girth g and the depth ?. Intuitively, the
depth needs to be smaller than the girth to avoid encountering cycles during the process of graph reconstruction. Recall that the parameter r in our algorithm determines the neighborhood over which
local MSTs are built in the first step. It is chosen such that it is roughly larger than the depth ? in
order for all the hidden nodes to be discovered. The upper bound on r ensures that the distortion
from an additive metric is not too large. The parameters for latent tree learning routines (such as
confidence intervals for quartet tests) are chosen appropriately depending on dmin and dmax , see [17]
for details.
3.2
Guarantees
We now provide the main result of this paper that the proposed method correctly estimates the graph
structure of a loopy latent graphical model in high dimensions. Recall that ? is the depth (distance
from a hidden node to its closest observed nodes), ?min is the minimum (absolute) edge potential
is the ratio of distance bounds.
and ? = ddmax
min
Theorem 1 (Structural Consistency and Sample Requirements) Under (A1)?(A5), the probability that the proposed method is structurally consistent tends to one, when the number of samples
scales as
???(?+1)?2
n = ? ?min
log p .
(10)
5
Thus, for learning Ising models on locally tree-like graphs, the sample complexity is dependent both
on the minimum edge potential ?min and on the depth ?. Our method is efficient in high dimensions
since the sample requirement is only logarithmic in the number of nodes p.
Dependence on Maximum Degree: For the correlation decay to hold (A3), we require ?min ?
??(?+1)+2
?max = ?(1/?max ). This implies that the sample complexity is at least n = ?(?max
log p).
Comparison with Fully Observed Models: In the special case when all the nodes are observed1
(? = 1), we strengthen the results for our method and establish that the sample complexity is
?2
n = ?(?min
log p). This matches the best known sample complexity for learning fully observed
Ising models [5, 6].
Comparison with Learning Latent Trees: Our method is an extension of latent tree methods
for learning locally tree-like graphs. The sample complexity of our method matches the sample
requirements for learning general latent tree models [2?4]. Thus, we establish that learning locally
tree-like graphs is akin to learning latent trees in the regime of correlation decay.
Extensions: We strengthen the above results to provide non-asymptotic sample complexity
bounds and also consider general discrete models, see [17] for details. The above results can also
be easily extended to Gaussian models using the notion of walk-summability in place of correlation
decay (see [18]) and the negative logarithm of the correlation coefficient as the additive tree metric
(see [4]).
Dependence on Fraction of Observed Nodes: In the special case when a fraction ? of the nodes
are uniformly selected as observed nodes, we can provide probabilistic bounds on the depth ? in the
resulting latent model, see [17] for details. For ? = 1 (homogeneous models) and regular graphs
?min = ?max = ?, the sample complexity simplifies to n = ? ?2 ??2 (log p)3 . Thus, we can
characterize an explicit dependence on the fraction of observed nodes ?.
3.3
Necessary Conditions for Graph Estimation
We have so far provided sufficient conditions for recovering locally tree-like graphs in latent Ising
models. We now provide necessary conditions on the number of samples required by any algorithm
b n : (X |V | )n ? Gm denote any deterministic graph estimator using
to reconstruct the graph. Let G
n i.i.d. samples from the observed node set V and Gm is the set of all possible graphs on m nodes.
We first define the notion of the graph edit distance.
b be two graphs2 with adjacency matrices AG , A b , and let
Definition 1 (Edit Distance) Let G, G
G
V be the set of labeled vertices in both the graphs (with identical labels). Then the edit distance
b is defined as
between G, G
b G; V ) := min ||A b ? ?(AG )||1 ,
dist(G,
G
?
where ? is any permutation on the unlabeled nodes while keeping the labeled nodes fixed.
In other words, the edit distance is the minimum number of entries that are different in AGb and in
any permutation of AG over the unlabeled nodes. In our context, the labeled nodes correspond to
the observed nodes V while the unlabeled nodes correspond to latent nodes H. We now provide
necessary conditions for graph reconstruction up to certain edit distance.
bm :
Theorem 2 (Necessary Condition for Graph Estimation) For any deterministic estimator G
?mn
2
7? Gm based on n i.i.d. samples, where ? ? [0, 1] is the fraction of observed nodes and m is
1
In the trivial case, when all the nodes are observed and the graph is locally tree-like, our method reduces
to thresholding of information distances at each node, and building local MSTs. The threshold can be chosen
as r = dmax + , for some > 0.
2
We consider inexact graph matching where the unlabeled nodes can be unmatched. This is done by adding
required number of isolated unlabeled nodes in the other graph, and considering the modified adjacency matrices [19].
6
the total number of nodes of an Ising model Markov on graph Gm ? GGirth (m; g, ?min , ?max ) on
m nodes with girth g, minimum degree ?min and maximum degree ?max , for all > 0, we have
2n?m m(2+1)m 3m
,
m0.5?min m (m ? g?gmax )0.5?min m
under any sampling process used to choose the observed nodes.
b m , Gm ; V ) > m] ? 1 ?
P[dist(G
Proof:
The proof is based on counting arguments. See [17] for details.
Lower Bound on Sample Requirements: The above result states that roughly
n = ? ?min ??1 log p
(11)
2
(12)
samples are required for structural consistency under any estimation method. Thus, when ? =
?(1) (constant fraction of observed nodes), polylogarithmic number of samples are necessary (n =
?(poly log p)), while when ? = ?(m?? ) for some ? > 0 (i.e., a vanishing fraction of observed
nodes), polynomial number of samples are necessary for reconstruction (n = ?(poly(p)).
Comparison with Sample Complexity of Proposed Method: For Ising models, under uniform
sampling of observed nodes, we established that the sample complexity of the proposed method
scales as n = ?(?2 ??2 (log p)3 ) for regular graphs with degree ?. Thus, we nearly match the
lower bound on sample complexity in (12).
4
Experiments
We employ latent graphical models for topic modeling. Each hidden variable in the model can
be thought of as representing a topic, and topics and words in a document are drawn jointly from
the graphical model. We conduct some preliminary experiments on 20 newsgroup dataset with
16,242 binary samples of 100 selected keywords. Each binary sample indicates the appearance
of the given words in each posting, these samples are divided in to two equal groups for learning
and testing purposes. We compare the performance with popular latent Dirichlet allocation (LDA)
model [9]. We evaluate performance in terms of perplexity and topic coherence. In addition, we also
study tradeoff between model complexity and data fitting through the Bayesian information criterion
(BIC) [8].
Methods: We consider a regularized variant of the proposed method for latent graphical model
selection. Here, in every iteration, the decision to add hidden variables to a local neighborhood is
based on the improvement of the overall BIC score. This allows us to tradeoff model complexity
and data fitting. Note that our proposed method only deals with structure estimation and we use
expectation maximization (EM) for parameter estimation. We compare the proposed method with
the LDA model3 . This method is implemented in MATLAB. We used the modules for LBP, made
available with UGM4 package. The LDA models are learnt using the lda package5 .
Performance Evaluation: We evaluate performance based on the test perplexity [20] given by
"
#
n
1 X
Perp-LL := exp ?
log P (xtest (k)) ,
(13)
np
k=1
where n is the number of test samples and p is the number of observed variables (i.e., words). Thus
the perplexity is monotonically decreasing in the test likelihood and a lower perplexity indicates a
better generalization performance. On lines of (13), we also define
n
X
1
Perp-BIC := exp ?
BIC(xtest ) , BIC(xtest ) :=
log P (xtest (k)) ? 0.5(df) log n,
np
k=1
(14)
3
Typically, LDA models the counts of different words in documents. Here, since we have binary data, we
consider a binary LDA model where the observed variables are binary.
4
These codes are available at http://www.di.ens.fr/?mschmidt/Software/UGM.html
5
http://chasen.org/?daiti-m/dist/lda/
7
Method
Proposed
Proposed
Proposed
Proposed
LDA
LDA
LDA
LDA
r
7
9
11
13
NA
NA
NA
NA
Hidden
32
24
26
24
10
20
30
40
Edges
183
129
125
123
NA
NA
NA
NA
PMI
0.4313
0.6037
0.4585
0.4289
0.2921
0.1919
0.1653
0.1470
Perp-LL
1.1498
1.1543
1.1555
1.1560
1.1480
1.1348
1.1421
1.1494
Perp-BIC
1.1518
1.1560
1.1571
1.1576
1.1544
1.1474
1.1612
1.1752
Table 1: Comparison of proposed method under different thresholds (r) with LDA under different number of topics (i.e., number of hidden variables) on 20 newsgroup data. For definition of
perplexity based on test likelihood and BIC scores, and PMI, see (13), (14), and (15).
where df is the degrees of freedom in the model. For a graphical model, we set df GM := m +
|E|, where m is the total number of variables (both observed and hidden) and |E| is the number
of edges in the model. For the LDA model, we set df LDA := (p(m ? p) ? 1), where p is the
number of observed variables (i.e., words) and m ? p is the number of hidden variables (i.e., topics).
This is because a LDA model is parameterized by a p ? (m ? p) topic probability matrix and a
(m ? p)-length Dirichlet prior. Thus, the BIC perplexity in (14) is monotonically decreasing in
the BIC score, and a lower BIC perplexity indicates better tradeoff between model complexity and
data fitting. However, the likelihood and BIC score in (13) and (14) are not tractable for exact
evaluation in general graphical models since they involve the partition function. We employ loopy
belief propagation (LBP) to evaluate them. Note that it is exact on a tree model and approximate
for loopy models. In addition, we also evaluate topic coherence, frequently considered in topic
modeling. It is based on the average pointwise mutual information (PMI) score
1 X X
P (Xi = 1, Xj = 1)
PMI :=
PMI(Xi ; Xj ), PMI(Xi ; Xj ) := log
, (15)
45|H|
P (Xi = 1)P (Xj = 1)
h?H i,j?A(h)
i<j
where the set A(h) represents the
?top-10? words associated with topic h ? H. The number of such
word pairs for each topic is 10
2 = 45, and is used for normalization. In [21], it is found that the
PMI scores are a good measure of human evaluated topic coherence when it is computed using an
external corpus. We compute PMI scores based on NYT articles bag-of-words dataset [22].
Experimental Results: We learn the graph structures under different thresholds r ?
{7, 9, 11, 13}, which controls the length of cycles. At r = 13, we obtain a latent tree and for all
other values, we obtain loopy models. The the first long cycle appears at r = 9. At r = 7, we find a
combination of short and long cycles. We find that models with cycles are more effective in discovering intuitive relationships. For instance, in the latent tree (r = 13), the link between ?computer?
and ?software? is missing due to the tree constraint, but is discovered when r ? 9. Moreover, we
see that common words across different topics tend to connect the local subgraphs, and thus loopy
models are better at discovering such relationships. The graph structures from the experiments are
available in [17]. In Table 1, we present results under our method and under LDA modeling. For the
LDA model, we vary the number of hidden variables (i.e., topics) as {10, 20, 30, 40}. In contrast,
our method is designed to optimize for the number of hidden variables, and does not need this input.
We note that our method is competitive in terms of both perplexity and topic coherence. We find
that topic coherence (i.e., PMI) for our method is optimal at r = 9, where the graph has a single
long cycle and a few short cycles.
The above experiments confirm the effectiveness of our approach for discovering hidden topics, and
are in line with the theoretical guarantees established earlier in the paper. Our analysis reveals that
a large class of loopy graphical models with latent variables can be learnt efficiently.
Acknowledgement
This work is supported by NSF Award CCF-1219234, AFOSR Award FA9550-10-1-0310, ARO
Award W911NF-12-1-0404, the setup funds at UCI, and ONR award N00014-08-1-1015.
8
References
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9
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3,901 | 4,531 | Sparse Approximate Manifolds for
Differential Geometric MCMC
Ben Calderhead?
CoMPLEX
University College London
London, WC1E 6BT, UK
[email protected]
M?ty?s A. Sustik
Department of Computer Sciences
University of Texas at Austin
Austin, TX 78712, USA
[email protected]
Abstract
One of the enduring challenges in Markov chain Monte Carlo methodology is
the development of proposal mechanisms to make moves distant from the current
point, that are accepted with high probability and at low computational cost. The
recent introduction of locally adaptive MCMC methods based on the natural underlying Riemannian geometry of such models goes some way to alleviating these
problems for certain classes of models for which the metric tensor is analytically
tractable, however computational efficiency is not assured due to the necessity of
potentially high-dimensional matrix operations at each iteration.
In this paper we firstly investigate a sampling-based approach for approximating
the metric tensor and suggest a valid MCMC algorithm that extends the applicability of Riemannian Manifold MCMC methods to statistical models that do
not admit an analytically computable metric tensor. Secondly, we show how the
approximation scheme we consider naturally motivates the use of `1 regularisation to improve estimates and obtain a sparse approximate inverse of the metric,
which enables stable and sparse approximations of the local geometry to be made.
We demonstrate the application of this algorithm for inferring the parameters of a
realistic system of ordinary differential equations using a biologically motivated
robust Student-t error model, for which the Expected Fisher Information is analytically intractable.
1
Introduction
The use of Markov chain Monte Carlo methods can be extremely challenging in many modern day
applications. This difficulty arises from the more frequent use of complex and nonlinear statistical
models that induce strong correlation structures in their often high-dimensional parameter spaces.
The exact structure of the target distribution is generally not known in advance and local correlation
structure between different parameters may vary across the space, particularly as the chain moves
from the transient phase, exploring areas of negligible probability mass, to the stationary phase
exploring higher density regions [1].
Constructing a Markov chain that adapts to the target distribution while still drawing samples from
the correct stationary distribution is challenging, although much research over the last 15 years has
resulted in a variety of approaches and theoretical results. Adaptive MCMC for example, allows
for global adaptation based on the partial or full history of a chain; this breaks its Markov property,
although it has been shown that subject to some technical conditions [2,3] the resulting chain will
still converge to the desired stationary distribution. Most recently, advances in Riemannian Manifold MCMC allow locally changing, position specific proposals to be made based on the underlying
?
http://www.2020science.net/people/ben-calderhead
1
geometry of the target distribution [1]. This directly takes into account the changing sensitivities
of the model for different parameter values and enables very efficient inference over a number of
popular statistical models. It is useful for inference over large numbers of strongly covarying parameters, however this methodology is still not suitable for all statistical models; in its current form
it is only applicable to models that admit an analytic expression for the metric tensor. In practice,
there are many commonly used models for which the Expected Fisher Information is not analytically
tractable, such as when a robust Student-t error model is employed to construct the likelihood.
In this paper we propose the use of a locally adaptive MCMC algorithm that approximates the local
Riemannian geometry at each point in the target space. This extends the applicability of Riemannian
Manifold MCMC to a much wider class of statistical models than at present. In particular, we do so
by estimating the covariance structure of the tangent vectors at a point on the Riemannian manifold
induced by the statistical model. Considering this geometric problem as one of inverse covariance
estimation naturally leads us to the use of an `1 regularised maximum likelihood estimator. This
approximate inverse approach allows the required geometry to be estimated with few samples, enabling good proposals for the Markov chain while inducing a natural sparsity in the inverse metric
tensor that reduces the associated computational cost.
We first give a brief characterisation of current adaptive approaches to MCMC, making a distinction between locally and globally adaptive methods, since these two approaches have very different
requirements in terms of proving convergence to the stationary distribution. We then discuss the
use of geometry in MCMC and the interpretation of such methods as being locally adaptive, before
giving the necessary background on Riemannian geometry and MCMC algorithms defined on induced Riemannian manifolds. We focus on the manifold MALA sampler, which is derived from a
Langevin diffusion process that takes into account local non-Euclidean geometry, and we discuss
simplifications that may be made for computational efficiency. Finally we present a valid MCMC
algorithm that estimates the Riemannian geometry at each iteration based on covariance estimates
of random vectors tangent to the manifold at the chain?s current point. We demonstrate the use of
`1 regularisation to calculate sparse approximate inverses of the metric tensor and investigate the
sampling properties of the algorithm on an extremely challenging statistical model for which the
Expected Fisher Information is analytically intractable.
2
Background
We wish to sample from some arbitrary target density ?(x) defined on a continuous state space XD ,
which may be high-dimensional. We may define a Markov chain that converges to the correct stationary distribution in the usual manner by proposing a new position x? from the current position
xn via some fixed proposal distribution q(x? |xn ); we accept the new move setting xn+1 = x? with
?(x? ) q(xn |x? )
probability ?(x? |xn ) = min( ?(x
, 1) and set xn+1 = xn otherwise. In a Bayesian con?
n ) q(x |xn )
text, we will often have a posterior distribution as our target ?(x) = p(?|y), where y is the data and
? are the parameters of a statistical model. The choice of proposal distribution is the critical factor in
determining how efficiently the Markov chain can explore the space and whether new moves will be
accepted with high probability and be sufficiently far from the current point to keep autocorrelation
of the samples to a minimum. There is a lot of flexibility in the choice of proposal distribution, in
that it may depend on the current point in a deterministic manner.
We note that Adaptive MCMC approaches attempt to change their proposal mechanism throughout
the running of the algorithm, and for the purpose of proving convergence to the stationary distribution it is useful to categorise them as follows; locally adaptive MCMC methods make proposals
based only on the current position of the chain, whereas globally adaptive MCMC methods use previously collected samples in the chain?s history to generate a new proposal mechanism. This is an
important distinction since globally adaptive methods lose their Markov property and convergence
to the stationary distribution must be proven in an alternative manner. It has been shown that such
chains may still be usefully employed as long as they satisfy some technical conditions, namely
diminishing adaptation and bounded convergence [2]. In practice these algorithms represent a step
towards MCMC as a ?black box? method and may be very useful for sampling from target distributions for which there is no derivative or higher order geometric information available, however
there are simple examples of standard Adaptive MCMC methods requiring hundreds of thousands
of iterations in higher dimensions before adapting to a suitable proposal distribution [3]. In addi2
tion, if there is more information about the target density available, then there seems little point in
trying to guess the geometric structure when it may be calculated directly. In this paper we focus
on locally adaptive methods that employ proposals constructed deterministically from information
at the current position of the Markov chain.
2.1
Locally Adaptive MCMC
Many geometric-based MCMC methods may be categorised as being locally adaptive. When
the derivative of the target density is available, MCMC methods such as the Metropolis-adjusted
Langevin Algorithm (MALA) [4] allow local adaptation based on the geometry at the current point,
but unlike globally adaptive MCMC, they retain their Markovian property and therefore converge to
the correct stationary distribution using a standard Metropolis-Hastings step and without the need to
satisfy further technical conditions.
In general, we can define position-specific proposal densities based on deterministic functions that
depend only on the current point. This idea has been previously employed to develop approaches for
sampling multimodal distributions whereby large initial jumps followed by deterministic optimisation functions were used to create mode-jumping proposal mechanisms [5]. In some instances, the
use of first order geometric information may drastically speed up the convergence to a stationary distribution, however in other cases such algorithms exhibit very slow convergence, due to the gradients
not being isotropic in magnitude [6]; in practice gradients may vary greatly in different directions
and the rate of exploration of the target density may in addition be dependent on the problem-specific
choice of parameterisation [1]. Methods using the standard gradient implicitly assume that the slope
in each direction is approximately constant over a small distance, when in fact these gradients may
rapidly change over short distances. Incorporating higher order geometry often helps although at an
increased computational cost.
A number of Hessian-based MCMC methods have been proposed as a solution [7]. While such
approaches have been shown to work very well for selected problems there are a number of problems
with this use of geometry; ad hoc methods are often necessary to deal with the fact that the Hessian
might not be everywhere positive-definite, and second derivatives can be challenging and costly to
compute. We can also exploit higher order information through the use of Riemannian geometry.
Using a metric tensor instead of a Hessian matrix lends us nice properties such as invariance to
reparameterisation of our statistical model, and positive-definiteness is also assured. Riemannian
geometry has been useful in a variety of other machine learning and statistical contexts [8] however
the limiting factor is usually analytic or computational tractability.
3
Differential Geometric MCMC
During the 1940s, Jeffreys and Rao demonstrated that the Expected Fisher Information has the
same properties as a metric tensor and indeed induces a natural Riemannian structure for a statistical
model [11, 10], providing a fascinating link between statistics and differential geometry. Much work
has been done since then elucidating the relationship between statistics and Riemannian geometry,
in particular examining geometric concepts such as distance, curvature and geodesics on statistical
manifolds, within a field that has become known as Information Geometry [6]. We first provide an
overview of Riemannian geometry and MCMC algorithms defined on Riemannian manifolds. We
then describe a sampling scheme that allows the local geometry to be estimated at each iteration for
statistical models that do not admit an analytically tractable metric tensor.
3.1
Riemannian Geometry
Informally, a manifold is an n-dimensional space that is locally Euclidean; it is locally equivalent
to Rn via some smooth transformation. At each point ? ? Rn on a Riemannian manifold M there
exists a tangent space, which we denote as T? M . We can think of this as a linear approximation
to the Riemannian manifold at the point ? and this is simply a standard vector space, whose origin
is the current point on the manifold and whose
h vectors are itangent to this point. The vector space
T? M is spanned by the differential operators ??? 1 , . . . , ???n , which act on functions defining paths
on the underlying manifold [9]. In the context of MCMC we can consider the target density as the
3
log-likelihood of a statistical model given some data, such that at a particular point ?, the derivatives
of
are tangent to the manifold and these are just the score vectors at ?, ?? L =
h the log-likelihood
i
?
?
??1 , . . . , ??n . The tangent space at each point ? arises when we equip a differentiable manifold
with an inner product at each point, which we can use to measure distance and angles between
vectors. This inner product is defined in terms of a metric tensor, G? , which defines a basis on each
tangent space T? M . The tangent space is therefore a linear approximation of the manifold at a given
point and it has the same dimensionality. A natural inner product for this vector space is given by
the covariance of the basis score vectors, since the covariance function satisfies the same properties
as a metric tensor, namely symmetry, bilinearity and positive-definiteness [9]. This inner product
then turns out to be equivalent to the Expected Fisher Information, following from the fact that the
expectation of the score is zero, with the [i, j]th component of the tensor given by
Gi,j
= Cov
?L ?L
,
??i ??j
?L T ?L
??i ??j
= Ep(x|?)
!
= ?Ep(x|?)
?2L
??i ??j
(1)
Each tangent vector, t1 ? T? M , at a point on the manifold, ? ? M , has a length ||t1 || ? R+ ,
whose square is given by the inner product, such that ||t1 ||2G? = ht1 , t1 i? = tT1 G? t1 . This squared
distance is known as the first fundamental form in Riemannian geometry [9], is invariant to reparameterisations of the coordinates, and importantly for MCMC provides a local measure of distance
that takes into account the local 2nd order sensitivity of the statistical model. We note that when the
metric tensor is constant for all values of ? then the Riemannian manifold is equivalent to a vector
space with constant inner product; further, if the metric tensor is an identity matrix then the manifold
simply becomes a Euclidean space.
3.2
Manifold MCMC
We consider the manifold version of the MALA sampling algorithm, which proposes moves based
on a stochastic differential equation defining a Langevin diffusion [4]. It turns out we can also define
such a diffusion on a Riemannian manifold [12], and so in a similar manner we can derive a sampling
algorithm that takes the underlying geometric structure into account when making proposals. It is
based on the Laplace-Beltrami operator, which simply measures the divergence of a vector field on
a manifold. The stochastic differential equation defining the Langevin diffusion on a Riemannian
? where the natural gradient [6] is the gradient of a
? ? L(?(t))dt + db(t),
manifold is d?(t) = 12 ?
function transformed into the tangent space at the current point by a linear transformation using the
? ? L(?(t)) = G?1 (?(t))?? L(?(t)), and the Brownian
basis defined by the metric tensor, such that ?
motion on the Riemannian manifold is defined as
? i (t) = |G(?(t))|? 21
db
D
p
X
1
?
(G?1 (?(t))ij |G(?(t))| 2 )dt +
G?1 (?(t))db(t)
?? j
i
j=1
(2)
The first part of the right hand side of Equation 2 represents the 1st order terms of the LaplaceBeltrami operator and these relate to the local curvature of the manifold, reducing to zero if the
metric is everywhere constant. The second term on the right hand side provides a position specific
linear transformation of the Brownian motion b(t) based on the local metric. Employing a first order
Euler integrator, the discrete form of the Langevin diffusion on a Riemannian manifold follows as
? n+1
i
=
? ni
D
n
X
2 ?1 n
n
2
?1 n ?G(? ) ?1 n
+ (G (? )?? L(? ))i ?
G (? )
G (? )
2
?? j
ij
j=1
D
p
2 X ?1 n
?G(? n )
G (? ) ij T r G?1 (? n )
+ G?1 (? n )zn
2 j=1
?? j
i
p
= ?(? n , )i + G?1 (? n )zn
+
i
4
(3)
which defines a proposal mechanism with density q(? ? |? n ) = N (? ? |?(? n , ), 2 G?1 (? n )) and acceptance probability min{1, p(? ? )q(? n |? ? )/p(? n )q(? ? |? n )} to ensure convergence to the invariant
density p(?). We note that this deterministically defines a position-specific proposal distribution at
each point on the manifold; we may categorise this as another locally adaptive MCMC method and
convergence to the invariant density follows from using the standard Metropolis-Hastings ratio.
It may be computationally expensive to calculate the 3rd order derivatives needed for working out the
rate of change of the metric tensor, and so an obvious approximation is to assume these derivatives
are zero for each step. In other words, for each step we can assume that the metric is locally constant.
Of course even if the curvature of the manifold is not constant, this simplified proposal mechanism
still defines a correct MCMC method which converges to the target measure, as we accept or reject
moves using a Metropolis-Hastings ratio. This is equivalent to a position-specific pre-conditioned
MALA proposal, where the pre-conditioning is dependent on the current parameter values
? n+1 = ? n +
p
2 ?1 n
G (? )?? L(? n ) + G?1 (? n )zn
2
(4)
For a manifold whose metric tensor is globally constant, this reduces further to a pre-conditioned
MALA proposal, where the pre-conditioning is effectively independent of the current parameter
values. In this context, such pre-conditioning no longer needs to be chosen arbitrarily, but rather it
may be informed by the geometry of the distribution we are exploring.
We point out that any approximations of the metric tensor would be best employed in the simplified
mMALA scheme, defining the covariance of the proposal distribution, or as a flat approximation to
a manifold. In the case of full mMALA, or even Hamiltonian Monte Carlo defined on a Riemannian
manifold [1], Christoffel symbols are also used, incorporating the derivatives of the metric tensor
as it changes across the surface of the manifold - in many cases the extra expense of computing or
estimating such higher order information is not sufficiently supported by the increase in sampling
efficiency [1] and for this reason we do not consider such methods further.
In the next section we consider the representation of the metric tensor as the covariance of the tangent
vectors at each point. We consider a method of estimating this such that convergence is guaranteed
by extending the state-space and introducing auxiliary variables that are conditioned on the current
point and we demonstrate its potential within a Riemannian geometric context.
4
Approximate Geometry for MCMC Proposals
We first derive an acceptance ratio on an extended state-space that enables convergence to the stationary distribution before describing the implications for developing new differential geometric
MCMC methods. Following [13, 14] we can employ the oft-used trick of defining an extended state
space X ? D. We may of course choose D to be of any size, however in our particular case we
shall choose D to be Rm?s , where m is the dimension of the data and s is the number of samples;
the reasons for this shall become clear. We therefore sample from this extended state space, whose
joint distribution follows as ? ? = ?(x)?
? (d|x). Given the current states [xn , dn ], we may propose a
new state q(x? |xn , dn ) and the MCMC algorithm will satisfy detailed balance and hence converge
to the stationary distribution if we accept joint proposals with Metropolis-Hastings probability ratio,
? (dn |xn )
? ? (x? , d? ) q(xn |x? , d? ) ?
?(x , d |xn , dn ) = min 1, ?
? (xn , dn ) q(x? |xn , dn ) ?
? (d? |x? )
?
? ?
?
?(x ) ?
? (d |x ) q(xn |x , d? ) ?
? (dn |xn )
= min 1,
?(xn ) ?
? (dn |xn ) q(x? |xn , dn ) ?
? (d? |x? )
?
?
?
?(x ) q(xn |x , d )
= min 1,
?(xn ) q(x? |xn , dn )
?
?
(5)
This is a reversible transition on ?(x, d), from which we can sample to obtain ?(x) as the marginal
distribution. The key point here is that we may define our proposal distribution q(x? |xn , dn ) in
almost any deterministic manner we wish. In particular, choosing ?
? (d|x) to be the same distribution
5
as the log-likelihood for our statistical model, the s samples from the extended state space D may
be thought of as pseudo-data, from which we can deterministically calculate an estimate of the
Expected Fisher Information to use as the covariance of a proposal distribution. Specifically, each
sampled pseudo-data can be used deterministically to give a sample of ?L
d? given the current ?, all of
which may then be used deterministically to obtain an approximation of the covariance of tangent
vectors at the current point. This approximation, unlike the Hessian, will always be positive definite,
and gives us an approximation of the metric tensor defining the local geometry. Further, we may use
additional deterministic procedures, given xn and dn , to construct better proposals; we consider a
sparsity inducing approach in the next section.
5
Stability and Sparsity via `1 Regularisation
We have two motivations for using an `1 regularisation approach for computing the inverse of the
metric tensor; firstly, since the metric is equivalent to the covariance of tangent vectors, we may
obtain more stable estimates of the inverse metric tensor using smaller numbers of samples, and
secondly, it induces a natural sparsity in the inverse metric, which may be exploited to decrease the
computational cost associated with repeated Cholesky factorisations and matrix-vector multiplications. We adopted the graphical lasso [15, 16], in which the maximum likelihood solution results in
the matrix optimisation problem,
arg min{? log det(A) + tr(AG) + ?
A 0
X
|Aij |}
(6)
i6=j
where G is an empirical covariance matrix and ? is a regularisation parameter. This convex optimisation problem aims to find A, the regularised maximum likelihood estimate for the inverse of
the covariance matrix. Importantly, the optimisation algorithm we employ is deterministic given our
tangent vectors, and therefore does not affect the validity of our MCMC algorithm; indeed we note
that we may use any deterministic sparse matrix inverse estimation approaches within this MCMC
algorithm. The use of the `1 regularisation promotes sparsity [23]; larger values for the regularisation
parameter matrix ? results in a solution that is more sparse, on the other hand when ? approaches
zero, the solution converges to the inverse of G (assuming it exists). It is also worth noting that the
`1 regularisation helps to recover a sparse structure in a high dimensional setting where the number
of samples is less than the number of parameters [17].
In order to achieve sufficiently fast computation we carefully implemented the graphical lasso algorithm tailored to this problem. We used no penalisation for the diagonal and uniform regularisation
parameter value for the off-diagonal elements. The motivation for not penalising the diagonal is
that it has been shown in the covariance estimation setting that the true inverse is approached as the
number of samples is increased [18], and the structure is learned more accurately [19]. The simple
regularisation structure allowed code simplification and reduction in memory use. We refactored
the graphical lasso algorithm of [15] and implemented it directly in F ORTRAN which we then called
from M ATLAB, making sure to minimise matrix copying due to M ATLAB processing. This code is
available as a software package, G LASSOFAST [20].
In the current context, the use of this approach allows us to obtain sparse approximations to the
inverse metric tensor, which may then be used in an MCMC proposal. Indeed, even if we have
access to an analytic metric tensor we need not use the full inverse for our proposals; we could still
obtain an approximate sparse representation, which may be beneficial computationally. The metric
tensor varies smoothly across a Riemannian manifold and, theoretically, if we are calculating the
inverse of 2 metric tensors that are close to each other, they may be numerically similar enough to
be able to use the solution of one to speed up convergence of solution for the other, although in the
simulations in this paper we found no benefit in doing so, i.e. the metric tensor varied too much as
the MCMC sampler took large steps across the manifold.
6
Simulation Study
We consider a challenging class of statistical models that severely tests the sampling capability
of MCMC methods; in particular, two examples based on nonlinear differential equations using a
6
(a) Exact full inverse
(b) Approximate sparse inverse
Figure 1: In this comparison we plotted the exact and the sparse approximate inverses of a typical
metric tensor G; we note that only subsets of parameters are typically strongly correlated in the statistical models we consider here and that the sparse approximation still captures the main correlation
structure present. Here the dimension is p = 25, and the regularisation parameter ? is 0.05 ? ||G||? .
Table 1: Summary of results for the Fitzhugh-Nagumo model with 10 runs of each parameter sampling scheme and 5000 posterior samples.
Sampling
Time (s)
Mean ESS
Total Time/
Relative
Method
(a, b, c)
(Min mean ESS)
Speed
Metropolis
14.5
139, 18.2, 23.4
0.80
?1.1
MALA
24.9
119.3, 28.7, 52.3
0.87
?1.0
mMALA Simp.
35.9
283.4, 136.6, 173.7
0.26
?3.4
biologically motivated robust Student-t likelihood, which renders the metric tensor analytically intractable. We examine the efficiency of our MCMC method with approximate metric on a well studied toy example, the Fitzhugh-Nagumo model, before examining a realistic, nonlinear and highly
challenging example describing enzymatic circadian control in the plant Arabidopsis thaliana [22].
6.1
Nonlinear Ordinary Differential Equations
Statistical modelling using systems of nonlinear ordinary differential equations plays a vital role in
unravelling the structure and behaviour of biological processes at a molecular level. The well-used
Gaussian error model however is often inappropriate, particularly in molecular biology where limited measurements may not be repeated under exactly the same conditions and are susceptible to
bias and systematic errors. The use of a Student-t distribution as a likelihood may help the robustness of the model with respect to possible outliers in the data. This presents a problem for standard
manifold MCMC algorithms as it makes the metric tensor analytically intractable. We consider
first the Fitzhugh-Nagumo model [1]. This synthetic dataset consisted of 200 time points simulated
from the model between t = [0, 20] with parameters [a, b, c] = [0.2, 0.2, 3], to which Gaussian distributed noise was added with variance ? 2 = 0.25. We employed a Student-t likelihood with scaling
parameter v = 3, and compared M-H and MALA (both employing scaled isotropic covariances),
and simplified mMALA with approximate metric. The stepsize for each was automatically adjusted
during the burn-in phase to obtain the theoretically optimal acceptance rate.
Table 1 shows the results including time-normalised effective sample size (ESS) as a measure of
sampling efficiency excluding burn-in [1]. The approximate manifold sampler offers a modest improvement on the other two samplers; despite taking longer to run because of the computational
cost of estimating the metric, the samples it draws exhibit lower autocorrelation, and as such the
approximate manifold sampler offers the highest time-normalised ESS.
The toy Fitzhugh-Nagumo model is however rather simple, and despite being a popular example is
rather unlike many realistic models used nowadays in the molecular modelling community. As such
we consider another larger model that describes the enzymatic control of the circadian networks in
Arabidopsis thaliana [21]. This is an extremely challenging, highly nonlinear model. We consider
7
Table 2: Comparison of pseudodata sample size on the quality of metric tensor estimation, and
hence on sampling efficiency, using the circadian network example model, with 10 runs and 10,000
posterior samples.
Number of Time (s) Min Mean ESS
Total Time/
Relative
Samples
(Min mean ESS)
Speed
10
155.6
85.1
1.90
?1.0
20
163.2
171.9
0.95
?2.0
30
168.9
209.1
0.81
?2.35
40
175.2
208.3
0.84
?2.26
Table 3: Summary of results for the circadian network model with 10 runs of each parameter sampling scheme and 10,000 posterior samples.
Sampling
Time (s) Min Mean ESS
Total Time/
Relative
Method
(Min mean ESS)
Speed
Metropolis
37.1
6.0
6.2
?4.4
MALA
101.3
3.7
27.4
?1.0
Adaptive MCMC
110.4
46.7
2.34
?11.7
mMALA Simp.
168.9
209.1
0.81
?33.8
inferring the 6 rate parameters that control production and decay of proteins in the nucleus and
cytoplasm (see [22] for the equations and full details of the model), again employing a Student-t
likelihood for which the Expected Fisher Information is analytically intractable. We used parameter
values from [22] to simulate observations for each of the six species at 48 time points representing
48 hours in the model. Student-t distributed noise was then added to obtain the data for inference.
We first investigated the effect that the tangent vector sample size for covariance estimation has on
the sampling efficiency of simplified mMALA. The results in Table 2 show that there is a threshold
above which a more accurate estimate of the metric tensor does not result in additional sampling
advantage. The threshold for this particular example model is around 30 pseudodata samples. Table
3 shows the time normalised statistical efficiency for each of the sampling methods; this time we
also compare an Adaptive MCMC algorithm [2] with M-H, MALA, and simplified mMALA with
approximate geometry. Both the M-H and MALA algorithms fail to explore the target distribution
and have severe difficulties with the extreme scalings and nonlinear correlation structure present in
the manifold. The Adaptive MCMC method works reasonably well after taking 2000 samples to
learn the covariance structure, although its performance is still poorer than the simplified mMALA
scheme, which converges almost immediately with no adaptation time required; the approximation
mMALA makes of the local geometry allows it to adequately deal with the different scalings and
correlations that occur in different parts of the space.
7
Conclusions
The use of Riemannian geometry can be very useful for enabling efficient sampling from arbitrary
probability densities. The metric tensor may be used for creating position-specific proposal mechanisms that allow MCMC methods to automatically adapt to the local correlation structure induced
by the sensitivities of the parameters of a statistical model. The metric tensor may conveniently be
defined as the Expected Fisher Information, however this quantity is often either difficult or impossible to compute analytically. We have presented a sampling scheme that approximates the Expected
Fisher Information by estimating the covariance structure of the tangent vectors at each point on the
manifold. By considering this problem as one of inverse covariance estimation, this naturally led
us to consider the use of `1 regularisation to improve the estimation procedure. This had the added
benefit of inducing sparsity into the metric tensor, which may offer computational advantages when
proposing MCMC moves across the manifold. For future work it will be exciting to investigate the
potential impact of approximate, sparse metric tensors for high dimensional problems.
8
Ben Calderhead gratefully acknowledges his Research Fellowship through the 2020 Science programme, funded by EPSRC grant number EP/I017909/1 and supported by Microsoft Research.
References
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3,902 | 4,532 | Learning to Discover Social Circles in Ego Networks
Jure Leskovec
Stanford, USA
[email protected]
Julian McAuley
Stanford, USA
[email protected]
Abstract
Our personal social networks are big and cluttered, and currently there is no good
way to organize them. Social networking sites allow users to manually categorize
their friends into social circles (e.g. ?circles? on Google+, and ?lists? on Facebook
and Twitter), however they are laborious to construct and must be updated whenever a user?s network grows. We define a novel machine learning task of identifying users? social circles. We pose the problem as a node clustering problem on
a user?s ego-network, a network of connections between her friends. We develop
a model for detecting circles that combines network structure as well as user profile information. For each circle we learn its members and the circle-specific user
profile similarity metric. Modeling node membership to multiple circles allows us
to detect overlapping as well as hierarchically nested circles. Experiments show
that our model accurately identifies circles on a diverse set of data from Facebook,
Google+, and Twitter for all of which we obtain hand-labeled ground-truth.
1
Introduction
Online social networks allow users to follow streams of posts generated by hundreds of their friends
and acquaintances. Users? friends generate overwhelming volumes of information and to cope with
the ?information overload? they need to organize their personal social networks. One of the main
mechanisms for users of social networking sites to organize their networks and the content generated by them is to categorize their friends into what we refer to as social circles. Practically all
major social networks provide such functionality, for example, ?circles? on Google+, and ?lists? on
Facebook and Twitter. Once a user creates her circles, they can be used for content filtering (e.g. to
filter status updates posted by distant acquaintances), for privacy (e.g. to hide personal information
from coworkers), and for sharing groups of users that others may wish to follow.
Currently, users in Facebook, Google+ and Twitter identify their circles either manually, or in a
na??ve fashion by identifying friends sharing a common attribute. Neither approach is particularly
satisfactory: the former is time consuming and does not update automatically as a user adds more
friends, while the latter fails to capture individual aspects of users? communities, and may function
poorly when profile information is missing or withheld.
In this paper we study the problem of automatically discovering users? social circles. In particular,
given a single user with her personal social network, our goal is to identify her circles, each of which
is a subset of her friends. Circles are user-specific as each user organizes her personal network of
friends independently of all other users to whom she is not connected. This means that we can
formulate the problem of circle detection as a clustering problem on her ego-network, the network
of friendships between her friends. In Figure 1 we are given a single user u and we form a network
between her friends vi . We refer to the user u as the ego and to the nodes vi as alters. The task then
is to identify the circles to which each alter vi belongs, as in Figure 1. In other words, the goal is to
find nested as well as overlapping communities/clusters in u?s ego-network.
Generally, there are two useful sources of data that help with this task. The first is the set of edges
of the ego-network. We expect that circles are formed by densely-connected sets of alters [20].
1
Figure 1: An ego-network with labeled circles. This network shows typical behavior that we observe in our data: Approximately 25% of our ground-truth circles (from Facebook) are contained
completely within another circle, 50% overlap with another circle, and 25% of the circles have no
members in common with any other circle. The goal is to discover these circles given only the
network between the ego?s friends. We aim to discover circle memberships and to find common
properties around which circles form.
However, different circles overlap heavily, i.e., alters belong to multiple circles simultaneously [1,
21, 28, 29], and many circles are hierarchically nested in larger ones (Figure 1). Thus it is important
to model an alter?s memberships to multiple circles. Secondly, we expect that each circle is not only
densely connected but its members also share common properties or traits [18, 28]. Thus we need
to explicitly model different dimensions of user profiles along which each circle emerges.
We model circle affiliations as latent variables, and similarity between alters as a function of common profile information. We propose an unsupervised method to learn which dimensions of profile
similarity lead to densely linked circles. Our model has two innovations: First, in contrast to mixedmembership models [2] we predict hard assignment of a node to multiple circles, which proves
critical for good performance. Second, by proposing a parameterized definition of profile similarity, we learn the dimensions of similarity along which links emerge. This extends the notion of
homophily [12] by allowing different circles to form along different social dimensions, an idea related to the concept of Blau spaces [16]. We achieve this by allowing each circle to have a different
definition of profile similarity, so that one circle might form around friends from the same school,
and another around friends from the same location. We learn the model by simultaneously choosing
node circle memberships and profile similarity functions so as to best explain the observed data.
We introduce a dataset of 1,143 ego-networks from Facebook, Google+, and Twitter, for which we
obtain hand-labeled ground-truth from 5,636 different circles.1 Experimental results show that by
simultaneously considering social network structure as well as user profile information our method
performs significantly better than natural alternatives and the current state-of-the-art. Besides being
more accurate our method also allows us to generate automatic explanations of why certain nodes
belong to common communities. Our method is completely unsupervised, and is able to automatically determine both the number of circles as well as the circles themselves.
Further Related Work. Topic-modeling techniques have been used to uncover ?mixedmemberships? of nodes to multiple groups [2], and extensions allow entities to be attributed with
text information [3, 5, 11, 13, 26]. Classical algorithms tend to identify communities based on node
features [9] or graph structure [1, 21], but rarely use both in concert. Our work is related to [30] in
the sense that it performs clustering on social-network data, and [23], which models memberships
to multiple communities. Finally, there are works that model network data similar to ours [6, 17],
though the underlying models do not form communities. As we shall see, our problem has unique
characteristics that require a new model. An extended version of our article appears in [15].
2
A Generative Model for Friendships in Social Circles
We desire a model of circle formation with the following properties: (1) Nodes within circles should
have common properties, or ?aspects?. (2) Different circles should be formed by different aspects,
e.g. one circle might be formed by family members, and another by students who attended the same
university. (3) Circles should be allowed to overlap, and ?stronger? circles should be allowed to form
within ?weaker? ones, e.g. a circle of friends from the same degree program may form within a circle
1
http://snap.stanford.edu/data/
2
from the same university, as in Figure 1. (4) We would like to leverage both profile information and
network structure in order to identify the circles. Ideally we would like to be able to pinpoint which
aspects of a profile caused a circle to form, so that the model is interpretable by the user.
The input to our model is an ego-network G = (V, E), along with ?profiles? for each user v ? V .
The ?center? node u of the ego-network (the ?ego?) is not included in G, but rather G consists only of
u?s friends (the ?alters?). We define the ego-network in this way precisely because creators of circles
do not themselves appear in their own circles. For each ego-network, our goal is to predict a set of
circles C = {C1 . . . CK }, Ck ? V , and associated parameter vectors ?k that encode how each circle
emerged. We encode ?user profiles? into pairwise features ?(x, y) that in some way capture what
properties the users x and y have in common. We first describe our model, which can be applied
using arbitrary feature vectors ?(x, y), and in Section 5 we describe several ways to construct feature
vectors ?(x, y) that are suited to our particular application.
We describe a model of social circles that treats circle memberships as latent variables. Nodes within
a common circle are given an opportunity to form an edge, which naturally leads to hierarchical and
overlapping circles. We will then devise an unsupervised algorithm to jointly optimize the latent
variables and the profile similarity parameters so as to best explain the observed network data.
Our model of social circles is defined as follows. Given an ego-network G and a set of K circles
C = {C1 . . . CK }, we model the probability that a pair of nodes (x, y) ? V ? V form an edge as
(
)
X
X
p((x, y) ? E) ? exp
h?(x, y), ?k i ?
?k h?(x, y), ?k i .
(1)
Ck ?{x,y}
|
Ck +{x,y}
{z
}
|
circles containing both nodes
{z
all other circles
}
For each circle Ck , ?k is the profile similarity parameter that we will learn. The idea is that
h?(x, y), ?k i is high if both nodes belong to Ck , and low if either of them do not (?k trades-off
these two effects). Since the feature vector ?(x, y) encodes the similarity between the profiles of
two users x and y, the parameter vector ?k encodes what dimensions of profile similarity caused the
circle to form, so that nodes within a circle Ck should ?look similar? according to ?k .
Considering that edges e = (x, y) are generated independently, we can write the probability of G as
Y
Y
P? (G; C) =
p(e ? E) ?
p(e ?
/ E),
(2)
e6?E
e?E
where ? = {(?k , ?k )}
k=1...K
is our set of model parameters. Defining the shorthand notation
X
dk (e) = ?(e ? Ck ) ? ?k ?(e ?
/ Ck ), ?(e) =
dk (e) h?(e), ?k i
Ck ?C
allows us to write the log-likelihood of G:
X
X
l? (G; C) =
?(e) ?
log 1 + e?(e) .
(3)
e?V ?V
e?E
Next, we describe how to optimize node circle memberships C as well as the parameters of the user
profile similarity functions ? = {(?k , ?k )} (k = 1 . . . K) given a graph G and user profiles.
3
Unsupervised Learning of Model Parameters
??
? = {?,
Treating circles C as latent variables, we aim to find ?
? } so as to maximize the regularized
log-likelihood of (eq. 3), i.e.,
? C? = argmax l? (G; C) ? ??(?).
?,
(4)
?,C
We solve this problem using coordinate ascent on ? and C [14]:
Ct
?t+1
=
argmax l?t (G; C)
(5)
=
argmax l? (G; C t ) ? ??(?).
(6)
C
?
3
Noting that (eq. 3) is concave in ?, we optimize (eq. 6) through gradient ascent, where partial derivatives are given by
?l
??k
=
?l
??k
=
X
?de (k)?k
e?V ?V
X
X
e?(e)
??
+
dk (e)?k ?
??k
1 + e?(e) e?E
?(e ?
/ Ck ) h?(e), ?k i
e?V ?V
X
e?(e)
?
?(e ?
/ Ck ) h?(e), ?k i .
?(e)
1+e
e?E
For fixed C \ Ci we note that solving argmaxCi l? (G; C \ Ci ) can be expressed as pseudo-boolean
optimization in a pairwise graphical model [4], i.e., it can be written as
X
Ck = argmax
E(x,y) (?(x ? C), ?(y ? C)).
(7)
C
(x,y)?V ?V
In words, we want edges with high weight
P(under ?k ) to appear in Ck , and edges with low weight to
appear outside of Ck . Defining ok (e) = Ck ?C\Ci dk (e) h?(e), ?k i the energy Ee of (eq. 7) is
Ee (0, 0) = Ee (0, 1) = Ee (1, 0)
Ee (1, 1)
=
?
ok (e) ? ?k h?(e), ?k i ? log(1 + eok (e)??k h?(e),?k i ),
? log(1 + eok (e)??k h?(e),?k i ),
=
?
ok (e) + h?(e), ?k i ? log(1 + eok (e)+h?(e),?k i ),
? log(1 + eok (e)+h?(e),?k i ),
e?E
e?
/E
e?E
.
e?
/E
By expressing the problem in this form we can draw upon existing work on pseudo-boolean optimization. We use the publicly-available ?QPBO? software described in [22], which is able to
accurately approximate problems of the form shown in (eq. 7). We solve (eq. 7) for each Ck in a
random order.
The two optimization steps of (eq. 5) and (eq. 6) are repeated until convergence, i.e., until C t+1 = C t .
PK P|?k |
We regularize (eq. 4) using the `1 norm, i.e., ?(?) = k=1 i=1
|?ki |, which leads to sparse (and
readily interpretable) parameters. Since ego-networks are naturally relatively small, our algorithm
can readily handle problems at the scale required. In the case of Facebook, the average ego-network
has around 190 nodes [24], while the largest network we encountered has 4,964 nodes. Note that
since the method is unsupervised, inference is performed independently for each ego-network. This
means that our method could be run on the full Facebook graph (for example), as circles are independently detected for each user, and the ego-networks typically contain only hundreds of nodes.
Hyperparameter estimation. To choose the optimal number of circles, we choose K so as to
minimize an approximation to the Bayesian Information Criterion (BIC) [2, 8, 25],
? = argmin BIC (K; ?K )
K
(8)
K
where ?K is the set of parameters predicted for a particular number of communities K, and
BIC (K; ?K ) ' ?2l?K (G; C) + |?K | log |E|.
(9)
The regularization parameter ? ? {0, 1, 10, 100} was determined using leave-one-out cross validation, though in our experience did not significantly impact performance.
4
Dataset Description
Our goal is to evaluate our unsupervised method on ground-truth data. We expended significant time,
effort, and resources to obtain high quality hand-labeled data.2 We were able to obtain ego-networks
and ground-truth from three major social networking sites: Facebook, Google+, and Twitter.
From Facebook we obtained profile and network data from 10 ego-networks, consisting of 193 circles and 4,039 users. To do so we developed our own Facebook application and conducted a survey
of ten users, who were asked to manually identify all the circles to which their friends belonged. On
average, users identified 19 circles in their ego-networks, with an average circle size of 22 friends.
Examples of such circles include students of common universities, sports teams, relatives, etc.
2
http://snap.stanford.edu/data/
4
?rst name
Alan
last name
Turing
position
company
name
work
type
name
education
type
?rst name
Dilly
last name
Knox
position
company
position
work
education
company
name
type
Cryptanalyst
GC&CS
Cambridge
College
1 ? ?x,y
Princeton
Graduate School
Cryptanalyst
GC&CS
Cryptanalyst
Royal Navy
0
1 ? ?x,y
Cambridge
College
203first name : Dilly
607last name : Knox
607first name : Alan
6 7
607last name : Turing
6 7
617work : position : Cryptanalyst
6 7
7
=6
617work : location : GC &CS
607work : location : Royal Navy
6 7
617education : name : Cambridge
6 7
617education : type : College
4 5
0 education : name : Princeton
0 education : type : Graduate School
2 3
0 first name
607last name
6 7
617work : position
=6 7
617work : location
415education : name
1 education : type
Figure 2: Feature construction. Profiles are tree-structured, and we construct features by comparing paths in those trees. Examples of trees for two users x (blue) and y (pink) are shown at
left. Two schemes for constructing feature vectors from these profiles are shown at right: (1) (top
right) we construct binary indicators measuring the difference between leaves in the two trees, e.g.
?work?position?Cryptanalyst? appears in both trees. (2) (bottom right) we sum over the leaf nodes
in the first scheme, maintaining the fact that the two users worked at the same institution, but discarding the identity of that institution.
For the other two datasets we obtained publicly accessible data. From Google+ we obtained data
from 133 ego-networks, consisting of 479 circles and 106,674 users. The 133 ego-networks represent all 133 Google+ users who had shared at least two circles, and whose network information
was publicly accessible at the time of our crawl. The Google+ circles are quite different to those
from Facebook, in the sense that their creators have chosen to release them publicly, and because
Google+ is a directed network (note that our model can very naturally be applied to both to directed
and undirected networks). For example, one circle contains candidates from the 2012 republican
primary, who presumably do not follow their followers, nor each other. Finally, from Twitter we
obtained data from 1,000 ego-networks, consisting of 4,869 circles (or ?lists? [10, 19, 27, 31]) and
81,362 users. The ego-networks we obtained range in size from 10 to 4,964 nodes.
Taken together our data contains 1,143 different ego-networks, 5,541 circles, and 192,075 users.
The size differences between these datasets simply reflects the availability of data from each of the
three sources. Our Facebook data is fully labeled, in the sense that we obtain every circle that a
user considers to be a cohesive community, whereas our Google+ and Twitter data is only partially
labeled, in the sense that we only have access to public circles. We design our evaluation procedure
in Section 6 so that partial labels cause no issues.
5
Constructing Features from User Profiles
Profile information in all of our datasets can be represented as a tree where each level encodes
increasingly specific information (Figure 2, left). From Google+ we collect data from six categories
(gender, last name, job titles, institutions, universities, and places lived). From Facebook we collect
data from 26 categories, including hometowns, birthdays, colleagues, political affiliations, etc. For
Twitter, many choices exist as proxies for user profiles; we simply collect data from two categories,
namely the set of hashtags and mentions used by each user during two-weeks? worth of tweets.
?Categories? correspond to parents of leaf nodes in a profile tree, as shown in Figure 2.
We first describe a difference vector to encode the relationship between two profiles. A non-technical
description is given in Figure 2. Suppose that users v ? V each have an associated profile tree Tv ,
and that l ? Tv is a leaf in that tree. We define the difference vector ?x,y between two users x and y
as a binary indicator encoding the profile aspects where users x and y differ (Figure 2, top right):
?x,y [l] = ?((l ? Tx ) 6= (l ? Ty )).
(10)
Note that feature descriptors are defined per ego-network: while many thousands of high schools
(for example) exist among all Facebook users, only a small number appear among any particular
user?s friends.
Although the above difference vector has the advantage that it encodes profile information at a fine
granularity, it has the disadvantage that it is high-dimensional (up to 4,122 dimensions in the data
5
we considered). One way to address this is to form difference vectors based on the parents of leaf
nodes: this way, we encode what profile categories two users have in common, but disregard specific
values (Figure 2, bottom right). For example, we encode how many hashtags two users tweeted in
common, but discard which hashtags they tweeted:
P
0
?x,y
[p] = l?children(p) ?x,y [l].
(11)
This scheme has the advantage that it requires a constant number of dimensions, regardless of the
size of the ego-network (26 for Facebook, 6 for Google+, 2 for Twitter, as described above).
0
Based on the difference vectors ?x,y (and ?x,y
) we now describe how to construct edge features
?(x, y). The first property we wish to model is that members of circles should have common relationships with each other:
?1 (x, y) = (1; ??x,y ).
(12)
The second property we wish to model is that members of circles should have common relationships
to the ego of the ego-network. In this case, we consider the profile tree Tu from the ego user u. We
then define our features in terms of that user:
?2 (x, y) = (1; ? ?x,u ? ?y,u )
(13)
(|?x,u ? ?y,u | is taken elementwise). These two parameterizations allow us to assess which mechanism better captures users? subjective definition of a circle. In both cases, we include a constant feature (?1?), which controls the probability that edges form within circles, or equivalently it measures
the extent to which circles are made up of friends. Importantly, this allows us to predict memberships
even for users who have no profile information, simply due to their patterns of connectivity.
0
, we define
Similarly, for the ?compressed? difference vector ?x,y
0
0
0
? ?y,u
).
? 1 (x, y) = (1; ??x,y
), ? 2 (x, y) = (1; ? ?x,u
(14)
To summarize, we have identified four ways of representing the compatibility between different
aspects of profiles for two users. We considered two ways of constructing a difference vector (?x,y
0
) and two ways of capturing the compatibility of a pair of profiles (?(x, y) vs. ?(x, y)).
vs. ?x,y
6
Experiments
Although our method is unsupervised, we can evaluate it on ground-truth data by examining the
maximum-likelihood assignments of the latent circles C = {C1 . . . CK } after convergence. Our
goal is that for a properly regularized model, the latent variables will align closely with the human
labeled ground-truth circles C? = {C?1 . . . C?K? }.
Evaluation metrics. To measure the alignment between a predicted circle C and a ground-truth
? we compute the Balanced Error Rate (BER) between the two circles [7], BER(C, C)
? =
circle
C,
?
?
|C\
C|
|
C\C|
1
. This measure assigns equal importance to false positives and false negatives,
?
2
|C| + |C|
so that trivial or random predictions incur an error of 0.5 on average. Such a measure is preferable to
the 0/1 loss (for example), which assigns extremely low error to trivial predictions. We also report
the F1 score, which we find produces qualitatively similar results.
Aligning predicted and ground-truth circles. Since we do not know the correspondence between
? we compute the optimal match via linear assignment by maximizing:
circles in C and C,
1 X
max
(1 ? BER(C, f (C))),
(15)
?
f :C?C |f |
C?dom(f )
? That is, if the number of predicted circles |C|
where f is a (partial) correspondence between C and C.
? then every circle C ? C must have a match C? ? C,
?
is less than the number of ground-truth circles |C|,
?
but if |C| > |C|, we do not incur a penalty for additional predictions that could have been circles
but were not included in the ground-truth. We use established techniques to estimate the number of
? = |C|, nor can any
circles, so that none of the baselines suffers a disadvantage by mispredicting K
method predict the ?trivial? solution of returning the powerset of all users. We note that removing the
bijectivity requirement (i.e., forcing all circles to be aligned by allowing multiple predicted circles
to match a single groundtruth circle or vice versa) lead to qualitatively similar results.
6
Accuracy (1 - BER)
Accuracy (F1 score)
Accuracy on detected communities (1 - Balanced Error Rate, higher is better)
1.0
multi-assignment clustering (Streich, Frank, et al.)
low-rank embedding (Yoshida)
block-LDA (Balasubramanyan and Cohen)
.84
.77
.72
.72
.70
.70
our model (friend-to-friend features ?1 , eq. 12)
our model (friend-to-user features ?2 , eq. 13)
our model (compressed features ? 1 , eq. 14)
0.5
our model (compressed features ? 2 , eq. 14)
Facebook
Twitter
Google+
Accuracy on detected communities (F1 score, higher is better)
1.0
multi-assignment clustering (Streich, Frank, et al.)
low-rank embedding (Yoshida)
block-LDA (Balasubramanyan and Cohen)
.59
.40
.38
.38
.34
.34
our model (friend-to-friend features ?1 , eq. 12)
our model (friend-to-user features ?2 , eq. 13)
our model (compressed features ? 1 , eq. 14)
0.0
our model (compressed features ? 2 , eq. 14)
Facebook
Google+
Twitter
Figure 3: Performance on Facebook, Google+, and Twitter, in terms of the Balanced Error Rate
(top), and the F1 score (bottom). Higher is better. Error bars show standard error. The improvement
of our best features ?1 compared to the nearest competitor are significant at the 1% level or better.
Baselines. We considered a wide number of baseline methods, including those that consider only
network structure, those that consider only profile information, and those that consider both. First
we experimented with Mixed Membership Stochastic Block Models [2], which consider only network information, and variants that also consider text attributes [5, 6, 13]. For each node, mixedmembership models predict a stochastic vector encoding partial circle memberships, which we
threshold to generate ?hard? assignments. We also considered Block-LDA [3], where we generate
?documents? by treating aspects of user profiles as words in a bag-of-words model.
Secondly, we experimented with classical clustering algorithms, such as K-means and Hierarchical
Clustering [9], that form clusters based only on node profiles, but ignore the network. Conversely we
considered Link Clustering [1] and Clique Percolation [21], which use network information, but ignore profiles. We also considered the Low-Rank Embedding approach of [30], where node attributes
and edge information are projected into a feature space where classical clustering techniques can
be applied. Finally we considered Multi-Assignment Clustering [23], which is promising in that it
predicts hard assignments to multiple clusters, though it does so without using the network.
Of the eight baselines highlighted above we report the three whose overall performance was the best,
namely Block-LDA [3] (which slightly outperformed mixed membership stochastic block models
[2]), Low-Rank Embedding [30], and Multi-Assignment Clustering [23].
Performance on Facebook, Google+, and Twitter Data. Figure 3 shows results on our Facebook,
Google+, and Twitter data. Circles were aligned as described in (eq. 15), with the number of circles
? determined as described in Section 3. For non-probabilistic baselines, we chose K
? so as to
K
maximize the modularity, as described in [20]. In terms of absolute performance our best model
?1 achieves BER scores of 0.84 on Facebook, 0.72 on Google+ and 0.70 on Twitter (F1 scores are
0.59, 0.38, and 0.34, respectively). The lower F1 scores on Google+ and Twitter are explained by the
fact that many circles have not been maintained since they were initially created: we achieve high
recall (we recover the friends in each circle), but at low precision (we recover additional friends who
appeared after the circle was created).
Comparing our method to baselines we notice that we outperform all baselines on all datasets by a
statistically significant margin. Compared to the nearest competitors, our best performing features
?1 improve on the BER by 43% on Facebook, 26% on Google+, and 16% on Twitter (improvements
in terms of the F1 score are similar). Regarding the performance of the baseline methods, we
note that good performance seems to depend critically on predicting hard memberships to multiple
circles, using a combination of node and edge information; none of the baselines exhibit precisely
this combination, a shortcoming our model addresses.
Both of the features we propose (friend-to-friend features ?1 and friend-to-user features ?2 ) perform
similarly, revealing that both schemes ultimately encode similar information, which is not surprising,
7
studied the same degree
speak the same languages
feature index for ?i1
Americans
weight ?4,i
1
feature index for ?1i
weight ?2,i
weight ?1,i
feature index for ?1i
1
Germans
who went to school in 1997
1
1
studied the same degree
feature index for ?i1
1
same level of education
feature index for ?i1
college educated people
working at a particular institute
feature index for ?1i
feature index for ?1i
weight ?4,i
living in S.F. or Stanford
1
weight ?3,i
people with PhDs
weight ?3,i
1
weight ?2,i
weight ?1,i
Figure 4: Three detected circles on a small ego-network from Facebook, compared to three groundtruth circles (BER ' 0.81). Blue nodes: true positives. Grey: true negatives. Red: false positives.
Yellow: false negatives. Our method correctly identifies the largest circle (left), a sub-circle contained within it (center), and a third circle that significantly overlaps with it (right).
1
worked for the same employer
at the same time
feature index for ?i1
Figure 5: Parameter vectors of four communities for a particular Facebook user. The top four plots
show ?complete? features ?1 , while the bottom four plots show ?compressed? features ? 1 (in both
cases, BER ' 0.78). For example the former features encode the fact that members of a particular
community tend to speak German, while the latter features encode the fact that they speak the same
language. (Personally identifiable annotations have been suppressed.)
since users and their friends have similar profiles. Using the ?compressed? features ? 1 and ? 2 does
not significantly impact performance, which is promising since they have far lower dimension than
the full features; what this reveals is that it is sufficient to model categories of attributes that users
have in common (e.g. same school, same town), rather than the attribute values themselves.
We found that all algorithms perform significantly better on Facebook than on Google+ or Twitter.
There are a few explanations: Firstly, our Facebook data is complete, in the sense that survey participants manually labeled every circle in their ego-networks, whereas in other datasets we only observe
publicly-visible circles, which may not be up-to-date. Secondly, the 26 profile categories available
from Facebook are more informative than the 6 categories from Google+, or the tweet-based profiles
we build from Twitter. A more basic difference lies in the nature of the networks themselves: edges
in Facebook encode mutual ties, whereas edges in Google+ and Twitter encode follower relationships, which changes the role that circles serve [27]. The latter two points explain why algorithms
that use either edge or profile information in isolation are unlikely to perform well on this data.
Qualitative analysis. Finally we examine the output of our model in greater detail. Figure 4 shows
results of our method on an example ego-network from Facebook. Different colors indicate true-,
false- positives and negatives. Our method is correctly able to identify overlapping circles as well
as sub-circles (circles within circles). Figure 5 shows parameter vectors learned for four circles for
a particular Facebook user. Positive weights indicate properties that users in a particular circle have
in common. Notice how the model naturally learns the social dimensions that lead to a social circle.
Moreover, the first parameter that corresponds to a constant feature ?1? has the highest weight; this
reveals that membership to the same community provides the strongest signal that edges will form,
while profile data provides a weaker (but still relevant) signal.
Acknowledgements. This research has been supported in part by NSF IIS-1016909, CNS-1010921,
IIS-1159679, DARPA XDATA, DARPA GRAPHS, Albert Yu & Mary Bechmann Foundation, Boeing, Allyes, Samsung, Intel, Alfred P. Sloan Fellowship and the Microsoft Faculty Fellowship.
8
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9
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3,903 | 4,533 | Perceptron Learning of SAT
Matthew B. Blaschko
Center for Visual Computing
Ecole Centrale Paris
[email protected]
Alex Flint
Department of Engineering Science
University of Oxford
[email protected]
Abstract
Boolean satisfiability (SAT) as a canonical NP-complete decision problem is one
of the most important problems in computer science. In practice, real-world SAT
sentences are drawn from a distribution that may result in efficient algorithms for
their solution. Such SAT instances are likely to have shared characteristics and
substructures. This work approaches the exploration of a family of SAT solvers as
a learning problem. In particular, we relate polynomial time solvability of a SAT
subset to a notion of margin between sentences mapped by a feature function into
a Hilbert space. Provided this mapping is based on polynomial time computable
statistics of a sentence, we show that the existance of a margin between these data
points implies the existance of a polynomial time solver for that SAT subset based
on the Davis-Putnam-Logemann-Loveland algorithm. Furthermore, we show that
a simple perceptron-style learning rule will find an optimal SAT solver with a
bounded number of training updates. We derive a linear time computable set of
features and show analytically that margins exist for important polynomial special
cases of SAT. Empirical results show an order of magnitude improvement over a
state-of-the-art SAT solver on a hardware verification task.
1
Introduction
SAT was originally shown to be a canonical NP-complete problem in Cook?s seminal work [5]. SAT
is of practical interest for solving a number of critical problems in applications such as theorem proving [8], model checking [2], planning [19], and bioinformatics [22]. That it is NP-complete indicates
that an efficient learning procedure is unlikely to exist to solve arbitrary instances of SAT. Nevertheless, SAT instances resulting from real world applications are likely to have shared characteristics
and substructures. We may view them as being drawn from a distribution over SAT instances, and
for key problems this distribution may be benign in that a learning algorithm can enable quick determination of SAT. In this work, we explore the application of a perceptron inspired learning algorithm
applied to branching heuristics in the Davis-Putnam-Logemann-Loveland algorithm [8, 7].
The Davis-Putnam-Logemann-Loveland (DPLL) algorithm formulates SAT as a search problem,
resulting in a valuation of variables that satisfies the sentence, or a tree resolution refutation proof
indicating that the sentence is not satisfiable. The branching rule in this depth-first search procedure
is a key determinant of the efficiency of the algorithm, and numerous heuristics have been proposed
in the SAT literature [15, 16, 26, 18, 13]. Inspired by the recent framing of learning as search optimization [6], we explore here the application of a perceptron inspired learning rule to application
specific samples of the SAT problem. Efficient learning of SAT has profound implications for algorithm development across computer science as a vast number of important problems are polynomial
time reducable to SAT.
A number of authors have considered learning branching rules for SAT solvers. Ruml applied reinforcement learning to find valuations of satisfiable sentences [25]. An approach that has performed
well in SAT competitions in recent years is based on selecting a heuristic from a fixed set and apply1
ing it on a per-sentence basis [27, 17]. The relationship between learnability and NP-completeness
has long been considered in the literature, e.g. [20]. Closely related to our approach is the learning
as search optimization framework [6]. That approach makes perceptron-style updates to a heuristic
function in A? search, but to our knowledge has not been applied to SAT, and requires a level of
supervision that is not available in a typical SAT setting. A similar approach to learning heuristics
for search was explored in [12].
2
Theorem Proving as a Search Problem
The SAT problem [5] is to determine whether a sentence ? in propositional logic is satisfiable. First
we introduce some notation. A binary variable q takes on one of two possible values, {0, 1}. A
literal p is a proposition of the form q (a ?positive literal?) or ?q (a ?negative literal?). A clause ?k
is a disjunction of nk literals, p1 ? p2 ? ? ? ? ? pnk . A unit clause contains exactly one literal. A
sentence ? in conjunctive normal form (CNF) [15] is a conjunction of m clauses, ?1 ??2 ?? ? ???m .
A valuation B for ? assigns to each variable in ? a value bi ? {0, 1}. A variable is free under B if
B does not assign it a value. A sentence ? is satisfiable iff there exists a valuation under which ? is
true. CNF is considered a canonical representation for automated reasoning systems. All sentences
in propositional logic can be transformed to CNF [15].
2.1
The Davis?Putnam?Logemann?Loveland algorithm
Davis et al. [7] proposed a simple procedure for recognising satisfiabile CNF sentences on N
variables. Their algorithm is essentially a depth first search over all possible 2N valuations over the
input sentence, with specialized criteria to prune the search and transformation rules to simplify the
sentence. We summarise the DPLL procedure below.
if ? contains only unit clauses and no contradictions then
return YES
end if
if ? contains an empty clause then
return NO
end if
for all unit clauses ? ? ? do
? := UnitPropagate(?, ?)
end for
for all literals p such that ?p ?
/ ? do
remove all clauses containing p from ?
end for
p :=PickBranch(?)
return DPLL(? ? p) ? DPLL(? ? ?p)
UnitPropagate simplifies ? under the assumption p. PickBranch applies a heuristic to choose a
literal in ?. Many modern SAT algorithms contain the DPLL procedure at their core [15, 16, 26, 18,
13], including top performers at recent SAT competitions [21]. Much recent work has focussed on
choosing heuristics for the selection of branching literals since good heuristics have been empirically
shown to reduce processing time by several orders of magnitude [28, 16, 13].
In this paper we learn heuristics by optimizing over a family of the form, argmaxp f (x, p) where
x is a node in the search tree, p is a candidate literal, and f is a priority function mapping possible
branches to real numbers. The state x will contain at least a CNF sentence and possibly pointers to
ancestor nodes or statistics of the local search region. Given this relaxed notion of the search state,
we are unaware of any branching heuristics in the literature that cannot be expressed in this form.
We explicitly describe several in section 4.
3
Perceptron Learning of SAT
We propose to learn f from a sequence of sentences drawn from some distribution determined by
a given application. We identify f with an element of a Hilbert space, H, the properties of which
2
are determined by a set of statistics polynomial time computable from a SAT instance, ?. We apply
stochastic updates to our estimate of f in order to reduce our expected search time. We use xj
to denote a node that is visited in the application of the DPLL algorithm, and ?i (xj ) to denote the
feature map associated with instantiating literal pi . Using reproducing kernel Hilbert space notation,
our decision function at xj takes the form
argmaxhf, ?i (xj )iH .
(1)
i
We would like to learn f such that the expected search time is reduced. We define yij to be +1
if the instantiation of pi at xj leads to the shortest possible proof, and ?1 otherwise. Our learning
procedure therefore will ideally learn a setting of f that only instantiates literals for which yij is +1.
We define a margin in a standard way:
max ? s.t. hf, ?i (xj )iH ? hf, ?k (xl )iH ? ?
3.1
?{(i, j)|yij = +1}, {(k, l)|ykl = ?1}
(2)
Restriction to Satisfiable Sentences
If we had access to all yij , the application of any binary learning algorithm to the problem of learning
SAT would be straightforward. Unfortunately, the identity of yij is only known in the worst case
after an exhaustive enumeration of all 2N variable assignments. We do note, however, that the DPLL
algorithm is a depth?first search over literal valuations. Furthermore, for satisfiable sentences the
length of the shortest proof is bounded by the number of variables. Consequently, in this case, all
nodes visited on a branch of the search tree that resolved to unsatisfiable have yij = ?1 and the
nodes on the branch leading to satisfiable have yij = +1. We may run the DPLL algorithm with a
current setting of f and if the sentence is satisfiable, update f using the inferred yij .
This learning framework is capable of computing in polynomial time valuations of satisfiable sentences in the following sense.
Theorem 1 ? a polynomial time computable ? with ? > 0 ?? ? belongs to a subset of satisfiable
sentences for which there exists a polynomial time algorithm to find a valid valuation.
Proof Necessity is shown by noting that the argmax in each step of the DPLL algorithm is computable in time polynomial in the sentence length by computing ? for all literals, and that there exists
a setting of f such that there will be at most a number of steps equal to the number of variables.
Sufficiency is shown by noting that we may run the polynomial algorithm to find a valid valuation
and use that valuation to construct a feature space with ? ? 0 in polynomial time. Concretely,
choose a canonical ordering of literals indexed by i and let ?i (xj ) be a scalar. Set ?i (xj ) = +i
if literal pi is instantiated in the solution found by the polynomial algorithm, ?1 otherwise. When
f = 1, ? = 2.
2
Corollary 1 ? polynomial time computable feature space with ? > 0 for SAT ?? P = N P
Proof If P = N P there is a polynomial time solution to SAT, meaning that there is a polynomial
time solution to finding valuations satisfiable sentences. For satisfiable sentences, this indicates that
there is a non-negative margin. For unsatisfiable sentences, either a proof exists with length less
than the number of variables, or we may terminate the DPLL procedure after N + 1 steps and return
unstatisfiable.
2
While Theorem 1 is positive for finding variable settings that satisfy sentences, unsatisfiable sentences remain problematic when we are unsure that there exists ? > 0 or if we have an incorrect
setting of f . We are unaware of an efficient method to determine all yij for visited nodes in proofs
of unsatisfiable sentences. However, we expect that similar substructures will exist in satisfiable
and unsatisfiable sentences resulting from the same application. Early iterations of our learning
algorithm will mistakenly explore branches of the search tree for satisfiable sentences and these
branches will share important characteristics with inefficient branches of proofs of unsatisfiability.
Consequently, proofs of unsatisfiability may additionally benefit from a learning procedure applied
only to satisfiable sentences. In the case that we analyitically know that ? > 0 and we have a correct
setting of f , we may use the termination procedure in Corollary 1.
3
Figure 2: Geometry of the feature space. Positive
and negative nodes are separated by a margin of
?. Given the current estimate of f , a threshold,
T , is selected as described in section 3.2. The
positive nodes with a score less than T are averaged, as are negative nodes with a score greater
than T . The resulting means lie within the respective convex hulls of the positive and negative sets, ensuring that the geometric conditions
of the proof of Theorem 2 are fulfilled.
Figure 1: Generation of training samples from
the search tree. Nodes labeled in red result in
backtracking and therefore have negative label,
while those coloured blue lie on the path to a
proof of satisfiability.
3.2
Davis-Putnam-Logemann-Loveland Stochastic Gradient
We use a modified perceptron style update based on the learning as search optimization framework
proposed in [6]. In contrast to that work, we do not have a notion of ?good? and ?bad? nodes at each
search step. Instead, we must run the DPLL algorithm to completion with a fixed model, ft . We
know that nodes on a path to a valuation that satisfies the sentence have positive labels, and those
nodes that require backtracking have negative labels (Figure 1). If the sentence is satisfiable, we may
compute a DPLL stochastic gradient, ?DPLL , and update f . We define two sets of nodes, S+ and
S? , such that all nodes in S+ have positive label and lower score than all nodes in S? (Figure 2). In
this work, we have used the sufficient condition of defining these sets by setting a score threshold,
T , such that fk (?i (xj )) < T ?(i, j) ? S+ , fk (?i (xj )) > T ?(i, j) ? S? , and |S+ | ? |S? | is
maximized. The DPLL stochastic gradient update is defined as follows:
X ?i (xj )
X ?k (xl )
?DPLL =
?
,
ft+1 = ft ? ??DPLL
(3)
|S? |
|S+ |
(i,j)?S?
(k,l)?S+
where ? is a learning rate. While poor settings of f0 may result in a very long proof before learning
can occur, we show in Section 4 that we can initialize f0 to emulate the behavior of current state-ofthe-art SAT solvers. Subsequent updates improve performance over the baseline.
We define R to be a positive real value such that ?i, j, k, l k?i (xj ) ? ?k (xl )k ? R
Theorem 2 For any training sequence that is separable by a margin of size ? with kf k = 1, using
the update rule in Equation (3) with ? = 1, the number of errors (updates) made during training on
satisfiable sentences is bounded above by R2 /? 2 .
Proof Let f1 (?(x)) = 0 ??(x). Considering the kth update,
kfk+1 k2 = kfk ? ?DPLL k2 = kfk k2 ? 2hfk , ?DPLL i + k?DPLL k2 ? kfk k2 + 0 + R2 .
(4)
We note that it is the case that hfk , ?DPLL i ? 0 for any selection of training examples such that the
average of the negative examples score higher than the average of the positive examples generated
by running a DPLL search. It is possible that some negative examples with lower scores than the
some positive nodes will be visited during the depth first search of the DPLL algorithm, but we are
guaranteed that at least one of them will have higher score. Similarly, some positive examples may
have higher scores than the highest scoring negative example. In both cases, we may simply discard
4
Feature
Dimensions
Description
is-positive
lit-unit-clauses
var-unit-clauses
lit-counts
var-counts
bohm-max
bohm-min
lit-total
neg-lit-total
var-total
lit-smallest
neg-lit-smallest
jw
jw-neg
activity
time-since-active
has-been-active
1
1
1
3
3
3
3
1
1
1
1
1
1
1
1
1
1
1 if p is positive, 0 otherwise
C1 (p), occurences of literal in unit clauses
C1 (q), occurences of variable in unit clauses
Ci (p) for i = 2, 3, 4, occurences in small clauses
Ci (q) for i = 2, 3, 4, as above, by variable
max(Ci (p), Ci (?p)), i = 2, 3, 4
max(Ci (p), Ci (?p)), i = 2, 3, 4
C(p), total occurences by literal
C(?p), total occurences of negated literal
C(q), total occurences by variable
Cm (p), where m is the size of the smallest unsatisfied clause
Cm (?p), as above, for negated literal
J(p) Jeroslow?Wang cue, see main text
J(?p) Jeroslow?Wang cue, see main text
minisat activity measure
t ? T (p) time since last activity (see main text)
1 if this p has ever appeared in a conflict clause; 0 otherwise
Figure 3: Summary of our feature space. Features are computed as a function of a sentence ? and a
literal p. q implicitly refers to the variable within p.
such instances from the training algorithm (as described in Section 3.2) guaranteeing the desired
inequality. By induction, kfk+1 k2 ? kR2 .
Let u be an element of H that obtains a margin of ? on the training set. We next obtain a lower bound
on hu, fk+1 i = hu, fk i ? hu, ?DPLL i ? hu, fk i + ?. That ?hu, ?DPLL i ? ? follows from the fact
that the means of the positive and negative training examples lie in the convex hull of the positive
and negative sets, respectively, and that u achieves a margin of ?. By induction, hu, fk+1 i ? k?.
?
Putting the two results together gives kR ? kfk+1 k ? hu, fk+1 i ? k? which, after some algebra,
yields k ? (R/?)2 .
2
The proof of this theorem closely mirrors those of the mistake bounds in [24, 6]. We note also that
an extension to approximate large-margin updates is straightforward to implement, resulting in an
alternate mistake bound (c.f. [6, Theorem 4]). For simplicity we consider only the perceptron style
updates of Equation (3) in the sequel.
4
Feature Space
In this section we describe our feature space. Recall that each node xj consists of a CNF sentence
? together with a valuation for zero or more variables. Our feature function ?(x, p) maps a node x
and a candidate branching literal p to a real vector ?. Many heuristics involve counting occurences
of literals and variables. For notational convenience let C(p) be the number of occurences of p in ?
and let Ck (p) be the number of occurences of p among clauses of size k. Table 4 summarizes our
feature space.
4.1
Relationship to previous branching heuristics
Many branching heuristics have been proposed in the SAT literature [28, 13, 18, 26]. Our features
were selected from the most successful of these and our system is hence able to emulate many other
systems for particular priority functions f .
Literal counting. Silva [26] suggested two simple heuristics based directly on literal counts. The
first was to always branch on the literal that maximizes C(p) and the second was to maximize
C(p) + C(?p). Our features ?lit-total? and ?neg-lit-total? capture these cues.
MOM. Freeman [13] proposed a heuristic that identified the size of the smallest unsatisfied clause,
m = min |?|, ? ? ?, and then identified the literal appearing most frequently amongst clauses of
size m. This is the motivation for our features ?lit-smallest? and ?neg-lit-smallest?.
BOHM. Bohm [3] proposed a heuristic that selects the literal maximizing
? max Ck (p, xj ), Ck (?p, xj ) + ? min Ck (p, xj ), Ck (?p, xj ) ,
(5)
5
with k = 2, or in the case of a tie, with k = 3 (and so on until all ties are broken). In practice we
found that ties are almost always broken by considering just k ? 4; hence we include ?bohm-max?
and ?bohm-min? in our feature space.
Jeroslow?Wang. Jerosolow and Wang [18] proposed a voting scheme in which clauses vote for
their components with weight 2?k , where k is the length of the clause. The total votes for a literal p
is
X
J(p) =
2?|?|
(6)
where the sum is over clauses ? that contain p. The Jeroslow?Wang rule chooses branches that
maximize J(p). Three variants were studied by Hooker [16]. Our features ?jw? and ?jw-neg? are
sufficient to span the original rule as well as the variants.
Dynamic activity measures. Many modern SAT solvers use boolean constraint propagation (BCP)
to speed up the search process [23]. One component of BCP generates new clauses as a result of
conflicts encountered during the search. Several modern SAT solvers use the time since a variable
was last added to a conflict clause to measure the ?activity? of that variable . Empirically, resolving
variables that have most recently appeared in conflict clauses results in an efficient search[14]. We
include several activity?related cues in our feature vector, which we compute as follows. Each
decision is is given a sequential time index t. After each decision we update the most?recent?
activity table T (p) := t for each p added to a conflict clause during that iteration. We include
the difference between the current iteration and the last iteration at which a variable was active in
the feature ?time-since-active?. We also include the boolean feature ?has-been-active? to indicate
whether a variable has ever been active. The feature ?activity? is a related cue used by minisat [10].
5
Polynomial special cases
In this section we discuss special cases of SAT for which polynomial?time algorithms are known.
For each we show that a margin exists in our feature space.
5.1
Horn
A Horn clause [4] is a disjunction containing at most one positive literal, ?q1 ? ? ? ? ? ?qk?1 ? qk .
A sentence ? is a Horn formula iff it is a conjunction of Horn clauses. There are polynomial time
algorithms for deciding satisfiability of Horn formulae [4, 9]. One simple algorithm based on unit
propagation [4] operates as follows. If there are no unit clauses in ? then ? is trivially satisfiable
by setting all variables to false. Otherwise, let {p} be a unit clause in ?. Delete any clause from ?
that contains p and remove ?p wherever it appears. Repeat until either a trivial contradiction q ? ?q
is produced (in which case ? is unsatisfiable) or until no further simplification is possible (in which
case ? is satisfiable) [4].
Theorem 3 There is a margin for Horn clauses in our feature space.
Proof We will show that there is a margin for Horn clauses in our feature space by showing that for
a particular priority function f0 , our algorithm will emulate the unit propagation algorithm above.
Let f0 be zero everywhere except for the following elements:1 ?is-positive? = ?, ?lit-unit-clauses?
= 1. Let H be the decision heuristic corresponding to f0 . Consider a node x and let ? be the
input sentence ?0 simplified according to the (perhaps partial) valuation at x. If ? contains no unit
clauses then clearly h?(x, p), f0 i will be maximized for a negative literal p = ?q. If ? does contain
unit clauses then for literals p which appear in unit clauses we have h?(x, p), f0 i ? 1, while for all
other literals we have h?(x, p), f0 i < 1. Therefore H will select a unit literal if ? contains one.
For satisfiable ?, this exactly emulates the unit propagation algorithm, and since that algorithm never
back?tracks [4], our algorithm makes no mistakes. For unsatisfiable ? our algorithm will behave as
follows. First note that every sentence encountered contains at least one unit clause, since, if not,
that sentence would be trivially satisfiable by setting all variables to false and this would contradict
the assumption that ? is unsatisfiable. So at each node x, the algorithm will first branch on some
unit clause p, then later will back?track to x and branch on ?p. But since p appears in a unit clause
at x this will immediately generate a contradiction and no further nodes will be expanded along that
path. Therefore the algorithm expands no more than 2N nodes, where N is the length of ?.
2
1
For concreteness let =
1
K+1
where K is the length of the input sentence ?
6
(a) Performance for planar graph colouring
(b) Performance for hardware verification
Figure 4: Results for our algorithm applied to (a) planar graph colouring; (b) hardware verification.
Both figures show the mistake rate as a function of the training iteration. In figure (a) we report
the mistake rate on the current training example since no training example is ever repeated, while
in figure (b) it is computed on a seperate validation set (see figure 5). The red line shows the
performance of minisat on the validation set (which does not change over time).
5.2
2?CNF
A 2?CNF sentence is a CNF sentence in which every clause contains exactly two literals. In this section we show that a function exists in our feature space for recognising satisfiable 2?CNF sentences
in polynomial time.
A simple polynomial?time solution to 2?CNF proposed by Even et al. [11] operates as follows. If
there are no unit clauses in ? then pick any literal and add it to ?. Otherwise, let {p} be a unit
clause in ? and apply unit propagation to p as described in the previous section. If a contradiction
is generated then back?track to the last branch and negate the literal added there. If there is no such
branch, then ? is unsatisfiable. Even et al. showed that this algorithm never back?tracks over more
than one branch, and therefore completes in polynomial time.
Theorem 4 Under our feature space, H contains a priority function that recognizes 2?SAT sentences in polynomial time.
Proof By construction. Let f0 be a weight vector with all elements set to zero except for the element
corrersponding to the ?appears-in-unit-clause? feature, which is set to 1. When using this weight
vector, our algorithm will branch on a unit literal whenever one is present. This exactly emulates the
behaviour of the algorithm due to Even et al. described above, and hence completes in polynomial
time for all 2?SAT sentences.
2
6
Empirical Results
Planar Graph Colouring: We applied our algorithm on the problem of planar graph colouring, for
which polynomial time algorithms are known [1]. Working in this domain allowed us to generate an
unlimited number of problems with a consistent but non?trivial structure on which to validate our
algorithm. By allowing up to four colours we also ensured that all instances were satisfiable [1].
We generated instances as follows. Starting with an empty L ? L grid we sampled K cells at
random and labelled them 1 . . . K. We then repeatedly picked a labelled cell with at least one
unlabelled neighbour and copied its label to its neighbour until all cells were labelled. Next we
formed a K ? K adjacency matrix A with Aij = 1 iff there is a pair of adjacent cells with labels
i and j. Finally we generated a SAT sentence over 4K variables (each variable corresponds to a
particular colouring of a particular vertex), with clauses expressing the constraints that each vertex
must be assigned one and only one colours and that no pair of adjacent vertices may be assigned the
same colour.
In our experiments we used K = 8, L = 5 and a learning rate of 0.1. We ran 40 training iterations of
our algorithm. No training instance was repeated. The number of mistakes (branching decision that
7
Training
Validation
Problem
Clauses
Problem
Clauses
ferry11
ferry11u
ferry9
ferry9u
ferry12u
26106
25500
16210
15748
31516
ferry10
ferry10u
ferry8
ferry8u
ferry12
20792
20260
12312
11916
32200
Figure 5: Instances in training and validation sets.
were later reversed by back?tracking) made at each iteration is shown in figure 4(a). Our algorithm
converged after 18 iterations and never made a mistake after that point.
Hardware Verification: We applied our algorithm to a selection of problems from a well?known
SAT competition [21]. We selected training and validation examples from the same suite of problems; this is in line with our goal of learning the statistical structure of particular subsets of SAT
problems. The problems selected for training and validation are from the 2003 SAT competition and
are listed in figure 5.
Due to the large size of these problems we extended an existing high?performance SAT solver,
minisat [10], replacing its decision heuristic with our perceptron strategy. We executed our algorithm
on each training problem sequentially for a total of 8 passes through the training set (40 iterations
in total). We performed a perceptron update (3) after solving each problem. After each update we
evaluated the current priority function on the entire validation set. The average mistake rate on the
validation set are shown for each training iteration in figure 4(b).
7
Discussion
Section 6 empirically shows that several important theoretical results of our learning algorithm hold
in practice. The experiments reported in Figure 4(a) show in practice that for a polynomial time solvable subset of SAT, the algorithm indeed has a bounded number of mistakes during training. Planar
graph colouring is a known polynomial time computable problem, but it is difficult to characterize
theoretically and an automated theorem prover was employed in the proof of polynomial solvability.
The hardware verification problem explored in Figure 4(b) shows that the algorithm learns a setting
of f that gives performance an order of magnitude faster than the state of the art Minisat solver. It
does so after relatively few training iterations and then maintains good performance.
Several approaches present themselves as good opportunites of extensions to learning SAT. In this
work, we argued that learning on positive examples is sufficient if the subset of SAT sentences
generated by our application has a positive margin. However, it is of interest to consider learning
in the absense of a positive margin, and learning may be accelerated by making updates based on
unsatisfiable sentences. One potential approach would be to consider a stochastic finite difference
approximation to the risk gradient by running the DPLL algorithm a second time with a perturbed f .
Additionally, we may consider updates to f during a run of the DPLL algorithm when the algorithm
backtracks from a branch of the search tree for which we can prove that all yij = ?1. This, however,
requires care in ensuring that the implicit empirical risk minimization is not biased.
In this work, we have shown that a perceptron-style algorithm is capable of learning all polynomial
solvable SAT subsets in bounded time. This has important implications for learning real-world
SAT applications such as theorem proving, model checking, planning, hardware verification, and
bioinformatics. We have shown empirically that our theoretical results hold, and that state-of-theart computation time can be achieved with our learning rule on a real-world hardware verification
problem. As SAT is a canonical NP-complete problem, we expect that the efficient solution of
important subsets of SAT may have much broader implications for the solution of many real-world
problems.
Acknowledgements: This work is partially funded by the European Research Council under the
European Community?s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement
number 259112, and by the Royal Academy of Engineering under the Newton Fellowship Alumni
Scheme.
8
References
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9
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3,904 | 4,534 | Truncation-free Stochastic Variational Inference for
Bayesian Nonparametric Models
Chong Wang?
Machine Learning Department
Carnegie Mellon University
[email protected]
David M. Blei
Computer Science Department
Princeton Univeristy
[email protected]
Abstract
We present a truncation-free stochastic variational inference algorithm for Bayesian
nonparametric models. While traditional variational inference algorithms require
truncations for the model or the variational distribution, our method adapts model
complexity on the fly. We studied our method with Dirichlet process mixture
models and hierarchical Dirichlet process topic models on two large data sets. Our
method performs better than previous stochastic variational inference algorithms.
1
Introduction
Bayesian nonparametric (BNP) models [1] have emerged as an important tool for building probability
models with flexible latent structure and complexity. BNP models use posterior inference to adapt
the model complexity to the data. For example, as more data are observed, Dirichlet process (DP)
mixture models [2] can create new mixture components and hierarchical Dirichlet process (HDP)
topic models [3] can create new topics.
In general, posterior inference in BNP models is intractable and we must approximate the posterior.
The most widely-used approaches are Markov chain Monte Carlo (MCMC) [4] and variational
inference [5]. For BNP models, the advantage of MCMC is that it directly operates in the unbounded
latent space; whether to increase model complexity (such as adding a new mixture component)
naturally folds in to the sampling steps [6, 3]. However MCMC does not easily scale?it requires
storing many configurations of hidden variables, each one on the order of the number of data
points. For scalable MCMC one typically needs parallel hardware, and even then the computational
complexity scales linearly with the data, which is not fast enough for massive data [7, 8, 9].
The alternative is variational inference, which finds the member of a simplified family of distributions
to approximate the true posterior [5, 10]. This is generally faster than MCMC, and recent innovations
let us use stochastic optimization to approximate posteriors with massive and streaming data [11,
12, 13]. Unlike MCMC, however, variational inference algorithms for BNP models do not operate
in an unbounded latent space. Rather, they truncate the model or the variational distribution to a
maximum model complexity [13, 14, 15, 16, 17, 18].1 This is particularly limiting in the stochastic
approach, where we might hope for a Bayesian nonparametric posterior seamlessly adapting its model
complexity to an endless stream of data.
In this paper, we develop a truncation-free stochastic variational inference algorithm for BNP models.
This lets us more easily apply Bayesian nonparametric data analysis to massive and streaming data.
?
Work was done when the author was with Princeton University.
In [17, 18], split-merge techniques were used to grow/shrink truncations. However, split-merge operations
are model-specific and difficult to design. It is also unknown how to apply these to the stochastic variational
inference setting we consider.
1
1
In particular, we present a new general inference algorithm, locally collapsed variational inference.
When applied to BNP models, it does not require truncations and gives a principled mechanism
for increasing model complexity on the fly. We demonstrate our algorithm on DP mixture models
and HDP topic models with two large data sets, showing improved performance over truncated
algorithms.
2 Truncation-free stochastic variational inference for BNP models
Although our goal is to develop an efficient stochastic variational inference algorithm for BNP
models, it is more succinct to describe our algorithm for a wider class of hierarchical Bayesian
models [19]. We will show how we apply our algorithm for BNP models in ?2.3.
We consider the general class of hierarchical Bayesian models shown in Figure 1. Let the global
hidden variables be ? with prior p(? | ?) (? is the hyperparameter) and local variables for each data
sample be zi (hidden) and xi (observed) for i = 1, . . . , n. The joint distribution of all variables
(hidden and observed) factorizes as,
Qn
Qn
p(?, z1:n , x1:n | ?) = p(? | ?) i=1 p(xi , zi | ?) = p(? | ?) i=1 p(xi | zi , ?)p(zi | ?).
(1)
The idea behind the nomenclature is that the local variables are conditionally independent of each
other given the global variables. For convenience, we assume global variables ? are continuous and
local variables zi are discrete. (This assumption is not necessary.) A large range of models can be
represented using this form, e.g., mixture models [20, 21], mixed-membership models [3, 22], latent
factor models [23, 24] and tree-based hierarchical models [25].
As an example, consider a DP mixture model for document clustering. Each document is modeled
as a bag of words drawn from a distribution over the vocabulary. The mixture components are the
distributions over the vocabulary ? and the mixture proportions ? are represented with a stick-breaking
process [26]. The global variables ? , (?, ?) contain the proportions and components, and the local
variables zi are the mixture assignments for each document xi . The generative process is:
1. Draw mixture component ?k and sticks ?k for k = 1, 2, ? ? ? ,
Qk?1
?k ? Dirichlet(?), ?k = ?
?k `=1 (1 ? ?
?` ), ?
?k ? Beta(1, a).
2. For each document xi ,
(a) Draw mixture assignment zi ? Mult(?).
(b) For each word xij , draw the word xij ? Mult(?zi ).
We now return to the general model in Eq. 1. In inference, we are interested in the posterior of the
hidden variables ? and z1:n given the observed data x1:n , i.e., p(?, z1:n | x1:n , ?). For many models,
this posterior is intractable. We will approximate it using mean-field variational inference.
2.1
Variational inference
In variational inference we try to find a distribution in a simple family that is close to the true posterior.
We describe the mean-field approach, the simplest variational inference algorithm [5]. It assumes the
fully factorized family of distributions over the hidden variables,
Qn
q(?, z1:n ) = q(?) i=1 q(zi ).
(2)
We call q(?) the global variational distribution and q(zi ) the local variational distribution. We want
to minimize the KL-divergence between this variational distribution and the true posterior. Under the
standard variational theory [5], this is equivalent to maximizing a lower bound of the log marginal
likelihood of the observed data x1:n . We obtain this bound with Jensen?s inequality,
RP
log p(x1:n | ?) = log
z1:n p(x1:n , z1:n , ? | ?)d?
Pn
? Eq [log p(?) ? log q(?) + i=1 log p(xi , zi |?) ? log q(zi )] , L(q). (3)
2
A: q(theta1)=Dirichlet(0.1,1,...,1)
30
-4
20
10
-6
-8
log odds
our method
0
-1
Figure 1:
Graphical model
for hierarchical Bayesian models
with global hidden variables ?,
local hidden and observed variables zi and xi , i = 1, . . . , n.
Hyperparameter ? is fixed, not a
random variable.
5
4
3
2
1
5
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4
xi
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-1
zi
B: q(theta1)=Dirichlet(1,1,...,1)
-2
b
0
h
w1: frequency of word 1 in the new doc
Figure 2: Results on assigning document d = {w1 , 0, . . . , 0} to q(?1 ) (case
A and B, shown in the figure above) or q(?2 ) = Dirichlet(0.1, 0.1, . . . , 0.1).
The y axis is the log-odds of q(z = 1) to q(z = 2)?if it is larger than 0,
it is more likely to be assigned to component 1. The mean-field approach
underestimates the uncertainty around ?2 , assigning d incorrectly for case B.
The locally collapsed approach does it correctly in both cases.
Algorithm 1 Mean-field variational inference. Algorithm 2 Locally collapsed variational inference.
1: Initialize q(?).
2: for iter = 1 to M do
3:
for i = 1 to n do
4:
Set local
variational distribution
q(zi ) ?
exp Eq(?) [log p(xi , zi | ?)] .
5:
end for
6:
Set global
variational distribution q(?)
?
exp Eq(z1:n ) [log p(x1:n , z1:n , ?)] .
7: end for
8: return q(?).
1: Initialize q(?).
2: for iter = 1 to M do
3:
for i = 1 to n do
4:
Set local distribution q(zi ) ? Eq(?) [p(xi , zi | ?)].
5:
Sample from q(zi ) to obtain its empirical q?(zi ).
6:
end for
7:
Set global variational distribution
q(?)
?
exp Eq?(z1:n ) [log p(x1:n , z1:n , ?)] .
8: end for
9: return q(?).
Maximizing L(q) w.r.t. q(?, z1:n ) defined in Eq. 2 (with the optimal conditions given in [27]) gives
q(?) ? exp Eq(z1:n ) [log p(x1:n , z1:n , ? | ?)]
(4)
q(zi ) ? exp Eq(?) [log p(xi , zi | ?)] .
(5)
Typically these equations are used in a coordinate ascent algorithm, iteratively optimizing each factor
while holding the others fixed (see Algorithm 1). The factorization into global and local variables
ensures that the local updates only depend on the global factors, which facilitates speed-ups like
parallel [28] and stochastic variational inference [11, 12, 13, 29].
In BNP models, however, the value of zi is potentially unbounded (e.g., the mixture assignment in a
DP mixture). Thus we need to truncate the variational distribution [13, 14]. Truncation is necessary
in variational inference because of the mathematical structure of BNP models. Moreover, it is difficult
to grow the truncation in mean-field variational inference even in an ad-hoc way because it tends to
underestimate posterior variance [30, 31]. In contrast, its mathematical structure and that it gets the
variance right in the conditional distribution allow Gibbs sampling for BNP models to effectively
explore the unbounded latent space [6].
2.2
Locally collapsed variational inference
We now describe locally collapsed variational inference, which mitigates the problem of underestimating posterior variance in mean-field variational inference. Further, when applied to BNP models,
it is truncation-free?it gives a good mechanism to increase truncation on the fly.
Algorithm 2 outlines the approach. The difference between traditional mean-field variational inference
and our algorithm lies in the update of the local distribution q(zi ). In our algorithm, it is
q(zi ) ? Eq(?) [p(xi , zi | ?)] ,
(6)
as opposed to the mean-field update in Eq. 5. Because we collapse out the global variational
distribution q(?) locally, we call this method locally collapsed variational inference. Note the two
algorithms are similar when q(?) has low variance. However, when the uncertainty modeled in q(?)
is high, these two approaches lead to different approximations of the posterior.
3
In our implementation, we use a collapsed Gibbs sampler to sample from Equation 6. This is a local
Gibbs sampling step and thus is very fast. Further, this is where our algorithm does not require
truncation because Gibbs samplers for BNP models can operate in an unbounded space [6, 3].
Now we update q(?). Suppose we have a set of samples from q(zi ) to construct its empirical
distribution q?(zi ). Plugging this into Eq. 3 gives the solution to q(?),
q(?) ? exp Eq?(z1:n ) [log p(x1:n , z1:n , ? | ?)] ,
(7)
which has the same form as in Eq. 4 for the mean-field approach. This finishes Algorithm 2.
To give an intuitive comparison of locally collapsed (Algorithm 2) and mean-field (Algorithm 1)
variational inference, we consider a toy document clustering problem with vocabulary size V = 10.
We use a two-component Bayesian mixture model with fixed and equal prior proportions ?1 = ?2 =
1/2. Suppose at some stage, component 1 has some documents assignments while component 2
has not yet and we have obtained the (approximate) posterior for the two component parameters ?1
and ?2 as q(?1 ) and q(?2 ). For ?1 , we consider two cases, A) q(?1 ) = Dirichlet(0.1, 1, . . . , 1); B)
q(?1 ) = Dirichlet(1, 1, . . . , 1). For ?2 , we only consider q(?2 ) = Dirichlet(0.1, 0.1, . . . , 0.1). In
both cases, q(?1 ) has relatively low variance while q(?2 ) has high variance. The difference is that the
q(?1 ) in case A has a lower probability on word 1 than that in case B. Now we have a new document
d = {w1 , 0, . . . , 0}, where word 1 is the only word and its frequency is w1 . In both cases, document
d is more likely be assigned to component 2 when w1 becomes larger. Figure 2 shows the difference
between mean-field and locally collapsed variational inference. In case A, the mean-field approach
does it correctly, since word 1 already has a very low probability in ?1 . But in case B, it ignores the
uncertainty around ?2 , resulting in incorrect clustering. Our approach does it correctly in both cases.
What justifies this approach? Alas, as for some other adaptations of variational inference, we do not
yet have an airtight justification [32, 33, 34]. We are not optimizing q(zi ) and so the corresponding
lower bound must be looser than the optimized lower bound from the mean-field approach if the
issue of local modes is excluded. However, our experiments show that we find a better predictive
distribution than mean-field inference. One possible explanation is outlined in S.1 (section 1 of the
supplement), where we show that our algorithm can be understood as an approximate Expectation
Propagation (EP) algorithm [35].
Related algorithms. Our algorithm is closely related to collapsed variational inference (CVI) [15,
16, 36, 32, 33]. CVI applies variational inference to the marginalized model, integrating out the
global hidden variable ?. This gives better estimates of posterior variance. In CVI, however, the
optimization for each local variable zi depends on all other local variables, and this makes it difficult
to apply it at large scale. Our algorithm is akin to applying CVI for the intermediate model that treats
q(?) as a prior and considers a single data point xi with its hidden structure zi . This lets us develop
stochastic algorithms that can be fit to massive data sets (as we show below).
Our algorithm is also related to the recently proposed a hybrid approach of using Gibbs sampling
inside stochastic variational inference to take advantage of the sparsity in text documents in topic
modeling [37]. Their approach still uses the mean-field update as in Eq. 5, where all local hidden
topic variables (for a document) are grouped together and the optimal q(zi ) is approximated by a
Gibbs sampler. With some adaptations, their fast sparse update idea can be used inside our algorithm.
Stochastic locally collapsed variational inference. We now extend our algorithm to stochastic
variational inference, allowing us to fit approximate posteriors to massive data sets. To do this, we
assume the model in Figure 1 is in the exponential family and satisfies the conditional conjugacy [11,
13, 29]?the global distribution p(? | ?) is the conjugate prior for the local distribution p(xi , zi | ?),
p(? | ?) = h(?) exp ? > t(?) ? a(?) ,
(8)
>
p(xi , zi | ?) = h(xi , zi ) exp ? t(xi , zi ) ? a(?) ,
(9)
where we overload the notation for base measures h(?), sufficient statistics t(?), and log normalizers
a(?). (These will often be different for the two families.) Due to the conjugacy, the term t(?) has
the form t(?) = [?; ?a(?)]. Also assume the global variational distribution q(? | ?) is in the same
family as the prior q(? | ?). Given these conditions, the batch update for q(? | ?) in Eq. 7 is
Pn
? = ? + i=1 Eq?(zi ) [t?(xi , zi )] .
(10)
4
The term t?(xi , zi ) is defined as t?(xi , zi ) , [t(xi , zi ); 1].
Analysis in [12, 13, 29] shows that given the conditional conjugacy assumption, the batch update
of parameter ? in Eq. 10 can be easily turned into a stochastic update using natural gradient [38].
Suppose our parameter is ?t at step t. Given a random observation xt , we sample from q(zt | xt , ?t )
to obtain the empirical distribution q?(zt ). With an appropriate learning rate ?t , we have
?t+1 ? ?t + ?t ??t + ? + nEq?(zi ) [t?(xt , zt )] .
(11)
This corresponds to an stochastic update using the noisy natural gradient to optimize the lower bound
in Eq. 3 [39]. (We note that the natural gradient is an approximation since our q(zi ) in Eq. 6 is
suboptimal for the lower bound Eq. 3.)
Mini-batch. A common strategy used in stochastic variational inference [12, 13] is to use a small
batch of samples at each update. Suppose we have a batch size S, and the set of samples xt , t ? S.
Using our formulation, the q(zt , t ? S) becomes
Q
q(zt,t?S ) ? Eq(? | ?t )
t?S p(xt , zt |?) .
We choose not to factorize zt,t?S , since factorization will potentially lead to the label-switching
problem when new components are instantiated for BNP models [7].
2.3
Truncation-free stochastic variational inference for BNP models
We have described locally collapsed variational inference in a general setting. Our main interests in
this paper are BNP models, and we now show how this approach leads to truncation-free variational
algorithms. We describe the approach for a DP mixture model [21], whose full description was
presented in the beginning of ?2.1. See S.2 for the details on the HDP topic models [3].
The global variational distribution. The variational distribution for the global hidden variables,
mixture components ? and stick proportions ?
? is
Q
q(?, ?
? | ?, u, v) = k q(? | ?k )q(?
?k | uk , vk ),
where ?k is the Dirichlet parameter and (uk , vk ) is the Beta parameter. The sufficient statistic term
t(xi , zi ) defined in Eq. 9 can be summarized as
P
P
t(xi , zi )?kw = 1[zi =k] j 1[xij =w] ; t(xi , zi )uk = 1[zi =k] , t(xi , zi )vk = j=k+1 1[zi =j] ,
where 1[?] is the indicator function. Suppose at time t, we have obtained the empirical distribution
q?(zi ) for observation xi , we use Eq. 11 to update Dirichlet parameter ? and Beta parameter (u, v),
P
?kw ? ?kw + ?t (??kw + ? + n?
q (zi = k) j 1[xij =w] )
uk ? uk + ?t (?uk + 1 + n?
q (zi = k))
P
vk ? vk + ?t (?vk + a + n `=k+1 q?(zi = `)).
Although we have a unbounded number of mixture components, we do not need to represent them
explicitly. Suppose we have T components that are associated with some data. These updates indicate
q(?k | ?k ) = Dirichlet(?) and q(?
?k ) = Beta(1, a), i.e., their prior distributions, when k > T .
Similar to a Gibbs sampler [6], the model is ?truncated? automatically. (We re-ordered the sticks
according to their sizes [15].)
The local empirical distribution q?(zi ). Since the mixture assignment zi is the only hidden variable,
we obtain its analytical form using Eq. 6,
R
q(zi = k) ? p(xi | ?k )p(zi = k | ?)q(?k | ?k )q(?
? )d?k d?
?
Q
P
P
Qk?1 v`
?( w ?kw ) w ?(?kw + j 1[xij =w]) uk
P
= Q ?(?kw )
`=1 u` +v` ,
?(
?kw +|xi |)
uk +vk
w
w
where |xi | is the document length and ?(?) is the Gamma function. (For mini batches, we do not have
an analytical form, but we can sample from it.) The probability of creating a new component is
Q
P
) w ?(?+ j 1[xij =w]) QT
vk
q(zi > T ) ? ?(?V
k=1 uk +vk .
?(?V +|xi |)
?V (?)
We sample from q(zi ) to obtain its empirical distribution q?(zi ). If zi > T , we create a new component.
5
Discussion. Why is ?locally collapsed? enough? This is analogous to the collapsed Gibbs sampling
algorithm in DP mixture models [6]? whether or not exploring a new mixture component is initialized
by one single sample. The locally collapsed variational inference is powerful enough to trigger this.
In the toy example above, the role of distribution q(?2 ) = Dirichlet(0.1, . . . , 0.1) is similar to that
of the potential new component we want to maintain in Gibbs sampling for DP mixture models.
Note the difference between this approach and those found in [17, 18], which use mean-field methods
that can grow or shrink the truncation using split-merge moves. These approaches are model-specific
and difficult to design. Further, they do not transfer to the stochastic setting. In contrast, the approach
presented here grows the truncation as a natural consequence of the inference algorithm and is easily
adapted to stochastic inference.
3
Experiments
We evaluate our methods on DP mixtures and HDP topic models, comparing them to truncation-based
stochastic mean-field variational inference. We focus on stochastic methods and large data sets.
Datasets. We analyzed two large document collections. The Nature data contains about 350K
documents from the journal Nature from years 1869 to 2008, with about 58M tokens and a vocabulary
size of 4,253. The New York Times dataset contains about 1.8M documents from the years 1987 to
2007, with about 461M tokens and a vocabulary size of 8,000. Standard stop words and those words
that appear less than 10 times or in more than 20 percent of the documents are removed, and the
final vocabulary is chosen by TF-IDF. We set aside a test set of 10K documents from each corpus on
which to evaluate its predictive power; these test sets were not given for training.
Evaluation Metric. We evaluate the different algorithms using held-out per-word likelihood,
P
likelihoodpw , log p(Dtest | Dtrain )/ xi ?Dtest |xi |,
Higher likelihood is better. Since exact computing the held-out likelihood is intractable, we use
approximations. See S.3 for details of approximating the likelihood. There is some question as to
the meaningfulness of held-out likelihood as a metric for comparing different models [40]. Held-out
likelihood metrics are nonetheless suited to measuring how well an inference algorithm accomplishes
the specific optimization task defined by a model.
Experimental Settings. For DP mixtures, we set component Dirichlet parameter ? = 0.5 and the
concentration parameter of DP a = 1. For HDP topic models, we set topic Dirichlet parameter
? = 0.01, and the first-level and second-level concentration parameters of DP a = b = 1 as in [13].
(See S.2 for the full description of HDP topic models.) For stochastic mean-field variational inference,
we set the truncation level at 300 for both DP and HDP. We run all algorithms for 10 hours and took
the model at the final stage as the output, without assessing the convergence. We vary the mini-batch
size S = {1, 2, 5, 10, 50, 100, 500}. (We do not intend to compare DP and HDP; we want to show
our algorithm works on both methods.)
For stochastic mean-field approach, we set the learning rate according to [13] with ?t = (?0 + t)??
with ? = 0.6 and ?0 = 1. We start our new method with 0 components without seeing any data. We
cannot use the learning rate schedule as in [13], since it gives very large weights to the first several
components, effectively leaving no room for creating new components on the fly. We set the learning
rate ?t = S/nt , for nt < n, where nt is the size of corpus that the algorithm has seen at time t. After
we see all the documents, nt = n. For both stochastic mean-field and our algorithm, we set the lower
bound of learning rate as S/n. We found this works well in practice. This mimics the usual trick
of running Gibbs sampler?one uses sequential prediction for initialization and after all data points
have been initialized, one runs the full Gibbs sampler [41]. We remove components with fewer than
1 document for DP and topics with fewer than 1 word for HDP topic models each time when we
process 20K documents.
6
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method
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Batchsize=100
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ourtopic
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likelihood
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(a)both
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corpora (b) on Nature
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corpora
(b) on New York Times
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0
(a) on both corpora
(a) on both corpora
Nature
Nature
Figure 3: Results on DP mixtures. (a) Held-out likelihood comparison on both corpora. Our approach is more
robust to batch sizes and gives better predictive performance. (b) The inferred number of mixtures on New York
Times. (Nature is similar.) The left of figure (b) shows the number of mixture components inferred after 10
hours; our method tends to give more mixtures. Small batch sizes for the stochastic mean-field approach do not
really work, resulting in very small number of mixtures. The right of figure (b) shows how different methods
infer the number of mixtures. The stochastic mean field approach shrinks it while our approach grows it.
(a) on
both corpora (b) on Nature
(a) on both
corpora
method
HDPourtopic
methodmodels
mean-field
method
50 100 150 200 250 300
50 100 150 200 250 300
method
Results. Figure 3 shows the results for DP mixture models. Figure 3(a) shows the held-out
likelihood comparison on both datasets. Our approach is more robust to batch sizes and usually gives
better predictive performance. Small batch sizes of the stochastic mean-field approach do not work
well. Figure 3(b) shows the inferred number of mixtures on New York Times. (Nature is similar.)
Our method tends to give more mixtures than the stochastic mean-field approach. The stochastic
mean-field approach shrinks the preset truncation; our approach does not need a truncation and grows
the number of mixtures when data requires.
Figure 4 shows the results for HDP topic models. Figure 4(a) shows the held-out likelihood comparison on both datasets. Similar to DP mixtures, our approach is more robust to batch sizes and
gives better predictive performance most of time. And small batch sizes of the stochastic mean-field
approach do not work well. Figure 4(b) shows the inferred number of topics on Nature. (New York
Times is similar.) This is also similar to DP. Our method tends to give more topics than the stochastic
mean-field approach. The stochastic mean-field approach shrinks the preset truncation while our
approach grows the number of topics when data requires.
One possible explanation that our method gives better results than the truncation-based stochastic
mean-field approach is as follows. For truncation-based approach, the algorithm relies more on
the random initialization placed on the parameters within the preset truncations. If the random
initialization is not used well, performance degrades. This also explains that smaller batch sizes
in stochastic mean-fields tend to work much worse?the first fewer samples might dominate the
effect from the random initialization, leaving no room for later samples. Our approach mitigates this
problem by allowing new components/topics to be created as data requires.
If we compare DP and HDP, the best result of DP is better than that of HDP. But this comparison is
not meaningful. Besides the different settings of hyperparameters, computing the held-out likelihood
for DP is tractable, but intractable for HDP. We used importance sampling to approximate. (See S.3
160
time (hours)
time (hours)
method
mean-field mean-field
our methodour method
more robust to batch sizes and gives better predictive performance most of time. (b) The inferred number of
topics on Nature. (New York Times is similar.) The left of figure (b) show the number of topics inferred after 10
hours; our method tends to give more topics. Small batch sizes for the stochastic mean-field approach do not
really work, resulting in very small number of topics. The right of figure (b) shows how different methods infer
the number of topics. Similar to DP, the stochastic mean field approach shrinks it while our approach grows it.
4
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Figure 4: Results on HDP topic models. (a) Held-out likelihood comparison on both corpora. Our approach is
7
Batchsize=10
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(a) on both corpora
Nature
for details.) [42] shows that importance sampling usually gives the correct ranking of different topic
models but significantly underestimates the probability.
4
Conclusion and future work
In this paper, we have developed truncation-free stochastic variational inference algorithms for
Bayesian nonparametric models (BNP models) and applied them to two large datasets. Extensions
to other BNP models, such as Pitman-Yor process [43], Indian buffet process (IBP) [23, 24] and
the nested Chinese restaurant process [18, 25] are straightforward by using their stick-breaking
constructions. Exploring how this algorithm behaves in the true streaming setting where the program
never stops?a ?never-ending learning machine? [44]?is an interesting future direction.
Acknowledgements. Chong Wang was supported by Google PhD and Siebel Scholar Fellowships.
David M. Blei is supported by ONR N00014-11-1-0651, NSF CAREER 0745520, AFOSR FA955009-1-0668, the Alfred P. Sloan foundation, and a grant from Google.
References
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[2] Antoniak, C. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. The
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9
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3,905 | 4,535 | Fast Bayesian Inference for Non-Conjugate
Gaussian Process Regression
Mohammad Emtiyaz Khan, Shakir Mohamed, and Kevin P. Murphy
Department of Computer Science, University of British Columbia
Abstract
We present a new variational inference algorithm for Gaussian process regression with non-conjugate likelihood functions, with application to a wide array of
problems including binary and multi-class classification, and ordinal regression.
Our method constructs a concave lower bound that is optimized using an efficient
fixed-point updating algorithm. We show that the new algorithm has highly competitive computational complexity, matching that of alternative approximate inference methods. We also prove that the use of concave variational bounds provides
stable and guaranteed convergence ? a property not available to other approaches.
We show empirically for both binary and multi-class classification that our new
algorithm converges much faster than existing variational methods, and without
any degradation in performance.
1
Introduction
Gaussian processes (GP) are a popular non-parametric prior for function estimation. For real-valued
outputs, we can combine the GP prior with a Gaussian likelihood and perform exact posterior inference in closed form. However, in other cases, such as classification, the likelihood is no longer
conjugate to the GP prior, and exact inference is no longer tractable.
Various approaches are available to deal with this intractability. One approach is Markov Chain
Monte Carlo (MCMC) techniques [1, 11, 22, 9]. Although this can be accurate, it is often quite
slow, and assessing convergence is challenging. There is therefore great interest in deterministic approximate inference methods. One recent approach is the Integrated Nested Laplace Approximation
(INLA) [21], which uses numerical integration to approximate the marginal likelihood. Unfortunately, this method is limited to six or fewer hyperparameters, and is thus not suitable for models
with a large number of hyperparameters. Expectation propagation (EP) [17] is a popular alternative, and is a method that approximates the posterior distribution by maintaining expectations and
iterating until these expectations are consistent for all variables. Although this is fast and accurate
for the case of binary classification [15, 18], there are difficulties extending EP to many other cases,
such as multi-class classification and parameter learning [24, 13]. In addition, EP is known to have
convergence issues and can be numerically unstable.
In this paper, we use a variational approach, where we compute a lower bound to the log marginal
likelihood using Jensen?s inequality. Unlike EP, this approach does not suffer from numerical issues
and convergence problems, and can easily handle multi-class and other likelihoods. This is an active
area of research and many solutions have been proposed, see for example, [23, 6, 5, 19, 14]. Unfortunately, most of these methods are slow, since they attempt to solve for the posterior covariance
matrix, which has size O(N 2 ), where N is the number of data points. In [19], a reparameterization was proposed that only requires computing O(N ) variational parameters. Unfortunately, this
method relies on a non-concave lower bound. In this paper, we propose a new lower bound that is
concave, and derive an efficient iterative algorithm for its maximization. Since the original objective
is unimodal, we reach the same global optimum as the other methods, but we do so much faster.
1
p(z|X, ?) = N (z|?, ?)
p(y|z) =
N
Y
(1)
p(yn |zn )
(2)
n=1
Type
Binary
Categorical
Ordinal
Distribution
Bernoulli logit
Multinomial logit
Cumulative logit
p(y|z)
p(y = 1|z) = ?(z)
p(y = k|z) = ezk ?lse(z)
p(y ? k|z) = ?(?k ? z)
Count
Poisson
p(y = k|z) =
?
X
?
?
z1
z2
zN
y1
y2
yN
z
e?e ekz
k!
Table 1: Gaussian process regression (top left) and its graphical model (right), along with the example likelihoods for outputs (bottom left). Here, ?(z) = 1/(1 + e?z ), lse(?) is the log-sum-exp function, k indexes over discrete output values, and ?k are real numbers such that ?1 < ?2 < . . . < ?K
for K ordered categories.
2
Gaussian Process Regression
Gaussian process (GP) regression is a powerful method for non-parametric regression that has gained
a great deal of attention as a flexible and accurate modeling approach. Consider N data points with
the n?th observation denoted by yn , with corresponding features xn . A Gaussian process model uses
a non-linear latent function z(x) to obtain the distribution of the observation y using an appropriate
likelihood [15, 18]. For example, when y is binary, a Bernoulli logit/probit likelihood is appropriate.
Similarly, for count observations, a Poisson distribution can be used.
A Gaussian process [20] specifies a distribution over z(x), and is a stochastic process that is characterized by a mean function ?(x) and a covariance function ?(x, x0 ), which are specified using a
kernel function that depends on the observed features x. Assuming a GP prior over z(x) implies that
a random vector is associated with every input x, such that given all inputs X = [x1 , x2 , . . . , xN ],
the joint distribution over z = [z(x1 ), z(x2 ), . . . , z(xN )] is Gaussian.
The GP prior is shown in Eq. 1. Here, ? is a vector with ?(xi ) as its i?th element, ? is a matrix with
?(xi , xj ) as the (i, j)?th entry, and ? are the hyperparameters of the mean and covariance functions.
We assume throughout a zero mean-function and a squared-exponential covariance function (also
known as radial-basis function or Gaussian) defined as: ?(xi , xj ) = ? 2 exp[?(xi ? xj )T (xi ?
xj )/(2s)]. The set of hyperparameters is ? = (s, ?). We also define ? = ??1 .
Given the GP prior, the observations are modeled using the likelihood shown in Eq. 2. The exact
form of the distribution p(yn |zn ) depends on the type of observations and different choices instantiates many existing models for GP regression [15, 18, 10, 14]. We consider frequently encountered
data such as binary, ordinal, categorical and count observations, and describe their likelihoods in Table 1. For the case of categorical observations, the latent function z is a vector whose k?th element
is the latent function for k?th category. A graphical model for Gaussian process regression is also
shown.
Given these models, there are three tasks that are to be performed: posterior inference, prediction
at test inputs, and model selection. In all cases, the likelihoods we consider are not conjugate to
the Gaussian prior distribution and as a result, the posterior distribution is intractable. Similarly,
the integrations required in computing the predictive distribution and the marginal likelihood are
intractable. To deal with this intractability we make use of variational methods.
3
Variational Lower Bound to the Log Marginal Likelihood
Inference and model selection are always problematic in any Gaussian process regression using nonconjugate likelihoods due to the fact that the marginal likelihood contains an intractable integral. In
this section, we derive a tractable variational lower bound to the marginal likelihood. We show
2
that the lower bound takes a well known form and can be maximized using concave optimization.
Throughout the section, we assume scalar zn , with extension to the vector case being straightforward.
We begin with the intractable log marginal likelihood L(?) in Eq. 3 and introduce a variational
posterior distribution q(z|?). We use a Gaussian posterior with mean m and covariance V. The
full set of variational parameters is thus ? = {m, V}. As log is a concave function, we obtain a
lower bound LJ (?, ?) using Jensen?s inequality, given in Eq. 4. The first integral is simply the
Kullback?Leibler (KL) divergence from the variational Gaussian posterior q(z|m, V) to the GP
prior p(z|?, ?) as shown in Eq. 5, and has a closed-form expression that we substitute to get the
first term in Eq. 6 (inside square brackets), with ? = ??1 .
The second integral can be expressed in terms of the expectation with respect to the marginal
q(zn |mn , Vnn ) as shown in the second term of Eq. 5. Here mn is the n?th element of m and
Vnn is the n?th diagonal element of V, the two variables collectively denoted by ?n . The lower
bound LJ is still intractable since the expectation of log p(yn |zn ) is not available in closed form for
the distributions listed in Table 1. To derive a tractable lower bound, we make use of local variational
bounds (LVB) fb , defined such that E[log p(yn |zn )] ? fb (yn , mn , Vnn ), giving us Eq. 6.
Z
Z
p(z|?)p(y|z)
L(?) = log p(z|?)p(y|z)dz = log q(z|?)
dz
(3)
q(z|?)
Zz
Zz
q(z|?)
? LJ (?, ?) := ? q(z|?) log
dz +
q(z|?) log p(y|z)dz
(4)
p(z|?)
z
z
N
X
Eq(zn |?n ) [log p(yn |zn )]
(5)
= ?DKL [q(z|?)||p(z|?)]+
n=1
N
X
? LJ (?, ?) := 12 log |V?|?tr(V?) ?(m??)T ?(m??)+N +
fb (yn , mn ,Vnn ).
(6)
n=1
We discuss the choice of LVBs in the next section, but first discuss the well-known form that the
lower bound of Eq. 6 takes. Given V, the optimization function with respect to m is a nonlinear
least-squares function. Similarly, the function with respect to V is similar to the graphical lasso
[8] or covariance selection problem [7], but is different in that the argument is a covariance matrix
instead of a precision matrix [8]. These two objective functions are coupled through the non-linear
term fb (?). Usually this term arises due to the prior distribution and may be non-smooth, for example, in graphical lasso. In our case, this term arises from the likelihood, and is smooth and concave
as we discuss in next section.
It is straightforward to show that the variational lower bound is strictly concave with respect to
? if fb is jointly concave with respect to mn and Vnn . Strict concavity of terms other than fb is
well-known since both the least squares and covariance selection problems are concave. Similar
concavity results have been discussed by Braun and McAuliffe [5] for the discrete choice model,
and more recently by Challis and Barber [6] for the Bayesian linear model, who consider concavity
with respect to the Cholesky factor of V. We consider concavity with respect to V instead of its
Cholesky factor, which allows us to exploit the special structure of V, as explained in Section 5.
4
Concave Local Variational Bounds
In this section, we describe concave LVBs for various likelihoods. For simplicity, we suppress
the dependence on n and consider the log-likelihood of a scalar observation y given a predictor z
distributed according to q(z|?) = N (z|m, v) with ? = {m, v}. We describe the LVBs for the
likelihoods given in Table 1 with z being a scalar for count, binary, and ordinal data, but a vector of
length K for categorical data, K being the number of classes. When V is a matrix, we denote its
diagonal by v.
For the Poison distribution, the expectation is available in closed form and we do not need any
bounding: E[log p(y|?)] = ym ? exp(m + v/2) ? log y!. This function is jointly concave with
respect to m and v since the exponential is a convex function.
3
For binary data, we use the piecewise linear/quadratic bounds proposed by [16], which is a bound
on the logistic-log-partition (LLP) function log(1 + exp(x)) and can be used to obtain a bound over
the sigmoid function ?(x). The final bound can be expressed as sum of R pieces: E(log p(y|?)) =
PR
fb (y, m, v) = ym ? r=1 fbr (m, v) where fbr is the expectation of r?th quadratic piece. The
function fbr is jointly concave with respect to m, v and their gradients are available in closed-form.
An important property of the piecewise bound is that its maximum error is bounded and can be
driven to zero by increasing the number of pieces. This means that the lower bound in Eq. 6 can
be made arbitrarily tight by increasing the number of pieces. For this reason, this bound always
performs better than other existing bounds, such as Jaakola?s bound [12], given that the number
of pieces is chosen appropriately. Finally, the cumulative logit likeilhood for ordinal observations
depends on ?(x) and its expectation can be bounded using piecewise bounds in a similar way.
For the multinomial logit distribution, we can use the bounds proposed by [3] and [4], both leading
to concave LVBs. The first bound takes the form fb (y, m, V) = yT m ? lse(m + v/2) with y
represented using a 1-of-K encoding. This function is jointly concave with respect to m and v,
which can be shown by noting the fact that the log-sum-exp function is convex. The second bound
is the product of sigmoids bound proposed by [4] which bounds the likelihood with product of
sigmoids (see Eq. 3 in [4]), with each sigmoid bounded using Jaakkola?s bound [12]. We can also
use piecewise linear/quadratic bound to bound each sigmoid. Alternatively, we can use the recently
proposed stick-breaking likelihood of [14] which uses piecewise bounds as well.
Finally, note that the original log-likelihood may not be concave itself, but if it is such that LJ has
a unique solution, then designing a concave variational lower bound will allow us to use concave
optimization to efficiently maximize the lower bound.
5
Existing Algorithms for Variational Inference
In this section, we assume that for each output yn there is a corresponding scalar latent function zn .
All our results can be easily extended to the case of multi-class outputs where the latent function is a
vector. In variational inference, we find the approximate Gaussian posterior distribution with mean
m and covariance V that maximizes Eq. 6. The simplest approach is to use gradient-based methods
for optimization, but this can be problematic since the number of variational parameters is quadratic
in N due to the covariance matrix V. The authors of [19] speculate that this may perhaps be the
reason behind limited use of Gaussian variational approximations.
We now show that the problem is simpler than it appears to be, and in fact the number of parameters
can be reduced to O(N ) from O(N 2 ). First, we write the gradients with respect to m and v in Eq.
7 and 8 and equate to zero, using gnm := ?fb (yn , mn , vn )/?mn and gnv := ?fb (yn , mn , vn )/?vn .
Also, gm and gv are the vectors of these gradients, and diag(gv ) is the matrix with gv as its diagonal.
1
2
??(m ? ?) + gm = 0
V?1 ? ? + diag(gv ) = 0
(7)
(8)
v
At the solution, we see that V is completely specified if g is known. This property can be exploited
to reduce the number of variational parameters.
Opper and Archambeau [19] (and [18]) propose a reparameterization to reduce the number of parameters to O(N ). From the fixed-point equation, we note that at the solution m and V will have
the following form,
V = (??1 + diag(?))?1
m = ? + ??,
(9)
(10)
where ? and ? are real vectors with ?d > 0, ?d. At the maximum (but not everywhere), ? and ?
will be equal to gm and gv respectively. Therefore, instead of solving the fixed-point equations to
obtain m and V, we can reparameterize the lower bound with respect to ? and ?. Substituting Eq.
9 and 10 in Eq. 6 and after simplification using the matrix inversion and determinant lemmas, we
get the following new objective function (for a detailed derivation, see [18]),
1
2
N
X
T
? log(|B? ||diag(?)|) + Tr(B?1
?)
?
?
??
+
fb (yn , mn , Vnn ),
?
n=1
4
(11)
with B? = diag(?)?1 + ?. Since the mapping between {?, ?} and {m, V} is one-to-one, we can
recover the latter given the former. The one-to-one relationship also implies that the new objective
function has a unique maximum. The new lower bound involves vectors of size N , reducing the
number of variational parameters to O(N ).
The problem with this reparameterization is that the new lower bound is no longer concave, even
though it has a unique maximum. To see this, consider the 1-D case. We collect all the terms
involving V from Eq. 6, except the LVB term, to define the function f (V ) = [log(V ??1 ) ?
V ??1 ]/2. We substitute the reparameterization V = (??1 + ?)?1 to get a new function f (?) =
[? log(1 + ??) ? (1 + ??)?1 ]/2. The second derivative of this function is f 00 (?) = 21 [?/(1 +
??)]2 (?? ? 1). Clearly, this derivative is negative for ? < 1/? and non-negative otherwise, making
the function neither concave nor convex.
The objective function is still unimodal and the maximum of (11) is equal to the maximum of
(6). With the reparameterization, we loose concavity and therefore the algorithm may have slow
convergence. Our experimental results (Section 7) confirm the slow convergence.
6
Fast Convergent Variational Inference using Coordinate Ascent
We now derive an algorithm that reduces the number of variational parameters to 2N while maintaining concavity. Our algorithm uses simple scalar fixed-point updates to obtain the diagonal elements
of V. The complete algorithm is shown in Algorithm 1.
To derive the algorithm, we first note that the fixed-point equation Eq. 8 has an attractive property:
at the solution, the off-diagonal elements of V?1 are the same as the off-diagonal elements of ?,
i.e. if we denote K := V?1 , then Kij = ?ij . We need only find the diagonal elements of K to get
the full V. This is difficult, however, since the gradient gv depends on v.
We take the approach of optimizing each diagonal element Kii fixing all others (and fixing m as
well). We partition V as shown on the left side of Eq. 12, indexing the last row by 2 and rest of the
rows by 1. We consider a similar partitioning of K and ?. Our goal is to compute v22 and k22 given
all other elements of K. Matrices K and V are related through the blockwise inversion, as shown
below.
?
?
T
?1
K?1
K?1
?1
11 k12
11 k12 k12 K11
?
K
+
T
T
?1
?1
V11 v12
? 11
k22 ?k12 K11 k12
k22 ?k12 K11 k12 ?
=?
(12)
?
kT12 K?1
vT12 v22
1
11
?
T
T
?1
?1
k22 ?k12 K11 k12
k22 ?k12 K11 k12
From the right bottom corner, we have the first relation below, which we simplify further.
v22 = 1/(k22 ? kT12 K?1
11 k12 )
?
k22 = e
k22 + 1/v22
(13)
where we define e
k22 := kT12 K?1
11 k12 . We also know from the fixed point Eq. 8 that the optimal v22
v
and k22 satisfy Eq. 14 at the solution, where g22
is the gradient of fb with respect to v22 . Substitute
the value of k22 from Eq. 13 in Eq. 14 to get Eq. 15. It is easy to check (by taking derivative) that
the value v22 that satisfies this fixed-point can be found by maximizing the function defined in Eq.
16.
v
0 = k22 ? ?22 + 2g22
0=e
k22 + 1/v22 ? ?22 + 2g v
22
f (v) = log(v) ? (?22 ? e
k22 )v + 2fb (y2 , m22 , v)
(14)
(15)
(16)
The function f (v) is a strictly concave function and can be optimized by iterating the following
v
update: v22 ? 1/(?22 ? e
k22 ? 2g22
). We will refer to this as a ?fixed-point iteration?.
Since all elements of K, except k22 , are fixed, e
k22 can be computed beforehand and need not be
evaluated at every fixed-point iteration. In fact, we do not need to compute it explicitly, since we
can obtain its value using Eq. 13: e
k22 = k22 ? 1/v22 , and we do this before starting a fixed-point
v
iteration. The complexity of these iterations depends on the number of gradient evaluations g22
,
which is usually constant and very low.
5
After convergence of the fixed-point iterations, we update V using Eq. 12. It turns out that this is a
rank-one update, the complexity of which is O(N 2 ). To show these updates, let us denote the new
new
new
values obtained after the fixed-point iterations by k22
and v22
respectively. and denote the old
old
old
values by k22 and v22 . We use the right top corner of Eq. 12 to get first equality in Eq. 17. Using
Eq. 13, we get the second equality. Similarly, we use the top left corner of Eq. 12 to get the first
equality in Eq. 18, and use Eq. 13 and 17 to get the second equality.
old
old
old
old
e
K?1
11 k12 = ?(k22 ? k22 )v12 = ?v12 /v22
old
K?1
11 = V11 ?
T
?1
K?1
11 k12 k12 K11
old ? e
k22
k22
old
old T
old
= Vold
11 ? v12 (v12 ) /v22
(17)
(18)
Note that both K?1
11 and k12 do not change after the fixed point iteration. We use this fact to obtain
Vnew . We use Eq. 12 to write updates for Vnew and use 17, 18, and 13 to simplify.
vnew
12 =
K?1
v new old
11 k12
v
= ? 22
old 12
v22
k new ? e
k22
(19)
22
?1
Vnew
11 = K11 +
T
?1
new
old
K?1
v22
? v22
11 k12 k12 K11
old T
+
= Vold
vold
11
12 (v12 )
old )2
new ? e
(v22
k22
k22
(20)
After updating V, we update m by optimizing the following non-linear least squares problem,
max ? 21 (m ? ?)T ?(m ? ?) +
m
N
X
fb (yn , mn , Vnn )
(21)
n=1
We use Newton?s method, the cost of which is O(N 3 ).
6.1
Computational complexity
The final procedure is shown in Algorithm 1. The main advantage of our algorithm is its fast
convergence
P as we show this in the results section. The overall computational complexity is
O(N 3 + n Inf p ). First term is due to O(N 2 ) update of V for all n and also due to the optimization of m. Second term is for Inf p fixed-point iterations, the total cost of which is linear in N
due to the summation. In all our experiments, Inf p is usually 3 to 5, adding very little cost.
6.2
Proof of convergence
Proposition 2.7.1 in [2] states that the coordinate ascent algorithm converges if the maximization
with respect to each coordinate is uniquely attained. This is indeed the case for us since each fixed
point iteration solves a concave problem of the form given by Eq. 16. Similarly, optimization with
respect to m is also strictly concave. Hence, convergence of our algorithm is assured.
Proof that V will always be positive definite
6.3
Let us assume that we start with a positive definite K, for example, we can initialize it with ?. Now
new
consider the update of v22 and k22 . Note that v22
will be positive since it is the maximum of Eq.
new
16 which involves the log term. Using this and Eq. 13, we get k22
> kT12 K?1
11 k12 . Hence, the
T
?1
new
Schur complement k22 ? k12 K11 k12 > 0. Using this and the fact that K11 is positive definite, it
follows that Knew will also be positive definite, and hence Vnew will be positive definite.
7
Results
We now show that the proposed algorithm leads to a significant gain in the speed of Gaussian process
regression. The software to reproduce the results of this section are available online1 . We evaluate
the performance of our fast variational inference algorithm against existing inference methods for
1
http://www.cs.ubc.ca/emtiyaz/software/codeNIPS2012.html
6
Algorithm 1 Fast convergent coordinate-ascent algorithm
1. Initialize K ? ?, V ? ??1 , m ? ?, where ? := ??1 .
2. Alternate between updating the diagonal of V and then m until convergence, as follows:
(a) Update the i?th diagonal of V for all i = 1, . . . , N :
i. Rearrange V and ? so that the i?th column is the last one.
ii. e
k22 ? k22 ? 1/v22 .
old
iii. Store old value v22
? v22 .
v
iv. Run fixed-point iterations for a few steps: v22 ? 1/(?22 ? e
k22 ? 2g22
).
v. Update V.
old
old 2
A. V11 ? V11 + (v22 ? v22
)v12 vT12 /(v22
) .
old
B. v12 ? ?v22 v12 /v22 .
vi. Update k22 ? e
k22 + 1/v22 .
(b) Update m by maximizing the least-squares problem of Eq. 21.
binary and multi-class classification. For binary classification, we use the UCI ionosphere data (with
351 data examples containing 34 features). For multi-class classification, we use the UCI forensic
glass data set with 214 data examples each with 6 category output and features of length 8. In both
cases, we use 80% of the dataset for training and the rest for testing.
We consider GP classification using the Bernoulli logit likelihood, for which we use the piecewise
bound of [16] with 20 pieces. We compare our algorithm with the approach of Opper and Archambeau [19] (Eq. 11). For the latter, we use L-BFGS method for optimization. We also compared to
the naive method of optimizing with respect to full m and V, e.g. method of [5], but do not present
these results since these algorithms have very slow convergence.
We examine the computational cost for each method in terms of the number of floating point operations (flops) for four hyperparameter settings ? = {log(s), log(?)}. This comparison is shown in
Figure 1(a). The y-axis shows (negative of) the value of the lower bound, and the x-axis shows the
number of flops. We draw markers at iteration 1,2,4,50 and in steps of 50 from then on. In all cases,
due to non-concavity, the optimization of the Opper and Archambeau reparameterization (black
curve with squares) convergence slowly, passing through flat regions of the objective and requiring
a large number of computations to reach convergence. The proposed algorithm (blue curve with
circles) has consistently faster convergence than the existing method. For this dataset, our algorithm
always converged in 5 iterations.
We also compare the total cost to convergence, where we count the total number of flops until
successive increase in the objective function is below 10?3 . Each entry is a different setting of
{log(s), log(?)}. Rows correspond to values of log(s) while columns correspond to log(?), with
units M,G,T denoting Mega-, Giga-, and Terra-flops. We can see that the proposed algorithm takes
a much smaller number of operations compared to the existing algorithm.
Opper and Archambeau
-1
1
3
-1
20G 212G 6T
1
101G
24T
24T
3
38G
1T
24T
Proposed Algorithm
-1
1
3
-1
6M
7M
7M
1
26M 20M 22M
3
47M 81M 75M
We also applied our method to two more datasets of [18], namely ?sonar? and ?usps-3vs5? dataset
and observed similar behavior.
Next, we apply our algorithm to the problem of multi-class classification, following [14], using the
stick-breaking likelihood, and compare to inference using the approach of Opper and Archambeau
[19] (Eq. 11). We show results comparing the lower bound vs the number of flops taken in Figure
1(b), for four hyperparameter settings {log(s), log(?)}. We show markers at iterations 1, 2, 10,
100 and every 100th iteration thereafter. The results follow those discussed for binary classification,
7
(?1.0,?1.0)
(?1.0, ?1.0)
(?1.0,2.5)
(?1.0, 2.5)
320
2000
600
300
300
290
280
1500
1000
500
270
260
134
0
300
600
900
0
1000
2000
0
3000
1000
2000
3000
4000
0
30K
40K
Mega?flops
Mega?flops
(1.0,1.0)
(2.5, 2.5)
(1.0, 1.0)
200
350
Neg?LogLik
110
300
250
100
15K
20K
0
2000
Mega?Flops
4000
6000
0
8000
400
300
200
200
80
10K
proposed
Opper?Arch
500
Neg?LogLik
neg?LogLik
170
140
50K
600
400
Opper?Arch
proposed
5K
20K
Mega?Flops
(3.5,3.5)
300
0
10K
Mega?Flops
200
neg?LogLik
Neg?LogLik
138
Neg?LogLik
neg?LogLik
neg?LogLik
310
900
142
20K
40K
60K
80K
100K
0
10K
Mega?flops
Mega?Flops
(a) Ionosphere data
20K
30K
40K
50K
Mega?flops
(b) Forensic glass data
Figure 1: Convergence results for (a) the binary classification on the ionosphere data set and (b) the
multi-class classification on the glass dataset. We plot the negative of the lower bound vs the number
of flops. Each plot shows the progress of algorithms for a hyperparameter setting {log(s), log(?)}
shown at the top of the plot. The proposed algorithm always converges faster than the other method,
in fact, in less than 5 iterations.
where both methods reach the same lower bound value, but the existing approach converging much
slower, with our algorithm always converged within 20 iterations.
8
Discussion
In this paper we have presented a new variational inference algorithm for non-conjugate GP regression. We derived a concave variational lower bound to the log marginal likelihood, and used
concavity to develop an efficient optimization algorithm. We demonstrated the efficacy of our new
algorithm on both binary and multiclass GP classification, demonstrating significant improvement
in convergence.
Our proposed algorithm is related to many existing methods for GP regression. For example, the
objective function that we consider is exactly the KL minimization method discussed in [18], for
which a gradient based optimization was used. Our algorithm uses an efficient approach where we
update the marginals of the posterior and then do a rank one update of the covariance matrix. Our
results show that this leads to fast convergence.
Our algorithm also takes a similar form to the popular EP algorithm [17], e.g. see Algorithm 3.5 in
[20]. Both EP and our algorithm update posterior marginals, followed by a rank-one update of
the covariance. Therefore, the computational complexity of our approach is similar to that of EP.
The advantage of our approach is that, unlike EP, it does not suffer from any numerical issues (for
example, no negative variances) and is guaranteed to converge.
The derivation of our algorithm is based on the observation that the posterior covariance has a special
structure, and does not directly use the concavity of the lower bound. An alternate derivation based
on the Fenchel duality exists and shows that the fixed-point iterations compute dual variables which
are related to the gradients of fb . We skip this derivation since it is tedious, and present the more
intuitive derivation instead. The alternative derivation will be made available in an online appendix.
Acknowledgements
We thank the reviewers for their valuable suggestions. SM is supported by the Canadian Institute
for Advanced Research (CIFAR).
8
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9
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3,906 | 4,536 | A Nonparametric Conjugate Prior Distribution for
the Maximizing Argument of a Noisy Function
Pedro A. Ortega
Max Planck Institute for Intelligent Systems
Max Planck Institute for Biolog. Cybernetics
[email protected]
Jordi Grau-Moya
Max Planck Institute for Intelligent Systems
Max Planck Institute for Biolog. Cybernetics
[email protected]
Tim Genewein
Max Planck Institute for Intelligent Systems
Max Planck Institute for Biolog. Cybernetics
[email protected]
David Balduzzi
Max Planck Institute for Intelligent Systems
[email protected]
Daniel A. Braun
Max Planck Institute for Intelligent Systems
Max Planck Institute for Biolog. Cybernetics
[email protected]
Abstract
We propose a novel Bayesian approach to solve stochastic optimization problems
that involve finding extrema of noisy, nonlinear functions. Previous work has
focused on representing possible functions explicitly, which leads to a two-step
procedure of first, doing inference over the function space and second, finding
the extrema of these functions. Here we skip the representation step and directly
model the distribution over extrema. To this end, we devise a non-parametric
conjugate prior based on a kernel regressor. The resulting posterior distribution
directly captures the uncertainty over the maximum of the unknown function.
Given t observations of the function, the posterior can be evaluated efficiently
in time O(t2 ) up to a multiplicative constant. Finally, we show how to apply our
model to optimize a noisy, non-convex, high-dimensional objective function.
1 Introduction
Historically, the fields of statistical inference and stochastic optimization have often developed their
own specific methods and approaches. Recently, however, there has been a growing interest in
applying inference-based methods to optimization problems and vice versa [1?4]. Here we consider
stochastic optimization problems where we observe noise-contaminated values from an unknown
nonlinear function and we want to find the input that maximizes the expected value of this function.
The problem statement is as follows. Let X be a metric space. Consider a stochastic function
f :X
R mapping a test point x ? X to real values y ? R characterized by the conditional pdf
P (y|x). Consider the mean function
Z
?
f (x) := E[y|x] = yP (y|x) dy.
(1)
The goal consists in modeling the optimal test point
x? := arg max{f?(x)}.
x
1
(2)
a)
2
5
b)
1.5
4
1
0.5
3
0
2
?0.5
?1
1
?1.5
?2
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
Figure 1: a) Given an estimate h of the mean function f? (left), a simple probability density function
over the location of the maximum x? is obtained using the transformation P (x? ) ? exp{?h(x? )},
where ? > 0 plays the role of the precision (right). b) Illustration of the Gramian matrix for different
test locations. Locations that are close to each other produce large off-diagonal entries.
Classic approaches to solve this problem are often based on stochastic approximation methods [5].
Within the context of statistical inference, Bayesian optimization methods have been developed
where a prior distribution over the space of functions is assumed and uncertainty is tracked during
the entire optimization process [6, 7]. In particular, non-parametric Bayesian approaches such as
Gaussian Processes have been applied for derivative-free optimization [8, 9], also within the context
of the continuum-armed bandit problem [10]. Typically, these Bayesian approaches aim to explicitly
represent the unknown objective function of (1) by entertaining a posterior distribution over the
space of objective functions. In contrast, we aim to model directly the distribution of the maximum
of (2) conditioned on observations.
2 Brief Description
Our model is intuitively straightforward and easy to implement1 . Let h(x) : X ? R be an estimate
of the mean f?(x) constructed from data Dt := {(xi , yi )}ti=1 (Figure 1a, left). This estimate can
easily be converted into a posterior pdf over the location of the maximum by first multiplying it with
a precision parameter ? > 0 and then taking the normalized exponential (Figure 1a, right)
P (x? |Dt ) ? exp{? ? h(x? )}.
In this transformation, the precision parameter ? controls the certainty we have over our estimate of
the maximizing argument: ? ? 0 expresses almost no certainty, while ? ? ? expresses certainty.
The rationale for the precision is: the more distinct inputs we test, the higher the precision?testing
the same (or similar) inputs only provides local information and therefore should not increase our
knowledge about the global maximum. A simple and effective way of implementing this idea is
given by
P
P
K(xi , xi )
K(xi , x? )yi + K0 (x? )y0 (x? )
?
i
i
P
P (x |Dt ) ? exp ? ? ? + t ? P P
, (3)
?
?
?
i
j K(xi , xj )
i K(xi , x ) + K0 (x )
{z
}
|
{z
}
|
estimate of f?(x? )
effective # of locations
where ?, ?, K, K0 and y0 are parameters of the estimator: ? > 0 is the precision we gain for each
new distinct observation; ? > 0 is the number of prior points; K : X ? X ? R+ is a symmetric
kernel function; K0 : X ? R+ is a prior precision function; and y0 : X ? R is a prior estimate of
f?.
In (3), the mean function f? is estimated with a kernel regressor [11] that combines the function
observations with a prior estimate of the function, and the total effective number of locations is
calculated as the sum of the prior locations ? and the number of distinct locations in the data Dt .
The latter is estimated by multiplying the number of data points t with the coefficient
P
i K(xi , xi )
P P
? (0, 1],
(4)
i
j K(xi , xj )
1
Implementations can be downloaded from http://www.adaptiveagents.org/argmaxprior
2
Noisy Function
10 Data Points
100 Data Points
1000 Data Points
20
20
20
20
15
15
15
15
10
10
10
10
5
5
5
5
0
0
0
0
?5
?5
?5
?5
?10
?10
?10
?10
?15
?15
?15
?15
?20
?20
?20
0
0.5
1
1.5
2
2.5
0
3
0.5
1
1.5
2
2.5
3
?20
0
0.5
1
1.5
2
2.5
3
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Figure 2: Illustration of the posterior distribution over the maximizing argument for 10, 100 and
1000 observations drawn from a function with varying noise. The top-left panel illustrates the function and the variance bounds (one standard deviation). The observations in the center region close
to x = 1.5 are very noisy. It can be seen that the prior gets progressively washed out with more
observations.
i.e. the ratio between the trace of the Gramian matrix (K(xi , xj ))i,j and the sum of its entries.
Inputs that are very close to each other will have overlapping kernels, resulting in large off-diagonal
entries of the Gramian matrix?hence decreasing the number of distinct locations (Figure 1b). For
example, if we have t observations from n ? t locations, and each location has t/n observations,
then the coefficient (4) is equal to n/t and hence the number of distinct locations is exactly n, as
expected.
Figure 2 illustrates the behavior of the posterior distribution. The expression for the posterior can
be calculated up to a constant factor in O(t) time. The computation of the normalizing constant
is in general intractable. Therefore, our proposed posterior can be easily combined with Markov
chain Monte Carlo methods (MCMC) to implement stochastic optimizers as will be illustrated in
Section 4.
3 Derivation
3.1 Function-Based, Indirect Model
Our first task is to derive an indirect Bayesian model for the optimal test point that builds its estimate
via the underlying function space. Let G be the set of hypotheses, and assume that each hypothesis
g ? G corresponds to a stochasticQmapping g : X
R. Let P (g) be the prior2 over G and let the
likelihood be P ({yt }|g, {xt }) = t P (yt |g, xt ). Then, the posterior of g is given by
Q
P (g) t P (yt |g, xt )
P (g)P ({yt }|g, {xt })
P (g|{yt }, {xt }) =
=
.
(5)
P ({yt }|{xt })
P ({yt }|{xt })
For each x? ? X , let G(x? ) ? G be the subset of functions such that for all g ? G(x? ), x? =
arg maxx {?
g(x)}3 . Then, the posterior over the optimal test point x? is given by
Z
P (x? |{yt }, {xt }) =
P (g|{yt }, {xt }) dg,
(6)
G(x? )
This model has two important drawbacks: (a) it relies on modeling the entire function space G,
which is potentially much more complex than necessary; (b) it requires calculating the integral (6),
which is intractable for virtually all real-world problems.
2
3
For the sake of simplicity, we neglect issues of measurability of G.
Note that we assume that the mean function g? is bounded and that it has a unique maximizing test point.
3
3.2 Domain-Based, Direct Model
We want to arrive at a Bayesian model that bypasses the integration step suggested by (6) and directly
models the location of optimal test point x? . The following theorem explains how this direct model
relates to the previous model.
Theorem 1. The Bayesian model for the optimal test point x? is given by
Z
P (g) dg
P (x? ) =
(prior)
G(x? )
?
P (yt |x , xt , Dt?1 ) =
R
Qt?1
P (yt |g, xt )P (g) k=1 P (yk |g, xk ) dg
,
R
Qt?1
k=1 P (yk |g, xk ) dg
G(x? ) P (g)
G(x? )
(likelihood)
where Dt := {(xk , yk )}tk=1 is the set of past tests.
Proof. Using Bayes? rule, the posterior distribution P (x? |{yt }, {xt }) can be rewritten as
Q
P (x? ) t P (yt |x? , xt , Dt?1 )
.
P ({yt }|{xt })
(7)
Since this posterior is equal to (6), one concludes (using (5)) that
Z
Y
Y
?
?
P (x )
P (g)
P (yt |x , xt , Dt?1 ) =
P (yt |g, xt ) dg.
G(x? )
t
t
?
Note that this expression corresponds to the joint P (x , {yt }|{xt }). The prior P (x? ) is obtained by
setting t = 0. The likelihood is obtained as the fraction
P (yt |x? , xt , Dt?1 ) =
P (x? , {yk }tk=1 |{xk }tk=1 )
t?1 ,
P (x? , {yk }t?1
k=1 |{xk }k=1 )
t?1
where it shall be noted that the denominator P (x? , {yk }t?1
k=1 |{xk }k=1 ) doesn?t change if we add the
condition xt .
From Theorem 1 it is seen that although the likelihood model P (yt |g, xt ) for the indirect model
is i.i.d. at each test point, the likelihood model P (yt |x? , xt , Dt?1 ) for the direct model depends
on the past tests Dt?1 , that is, it is adaptive. More critically though, the likelihood function?s
internal structure of the direct model corresponds to an integration over function space as well?
thus inheriting all the difficulties of the indirect model.
3.3 Abstract Properties of the Likelihood Function
There is a way to bypass modeling the function space explicitly if we make a few additional assumptions. We assume that for any g ? G(x? ), the mean function g? is continuous and has a unique
maximum. Then, the crucial insight consists in realizing that the value of the mean function g? inside
a sufficiently small neighborhood of x? is larger than the value outside of it (see Figure 3a).
We assume that, for any ? > 0 and any z ? X , let B? (z) denote the open ?-ball centered on z. The
functions in G fulfill the following properties:
a. Continuous: Every function g ? G is such that its mean g? is continuous and bounded.
b. Maximum: For any x? ? X , the functions g ? G(x? ) are such that for all ? > 0 and all
z?
/ B? (x? ), g?(x? ) > g?(z).
Furthermore, we impose a symmetry condition on the likelihood function. Let x?1 and x?2 be in X ,
and consider their associated equivalence classes G(x?1 ) and G(x?2 ). There is no reason for them to
be very different: in fact, they should virtually be indistinguishable outside of the neighborhoods
of x?1 and x?2 . It is only inside of the neighborhood of x?1 when G(x?1 ) becomes distinguishable from
the other equivalence classes because the functions in G(x?1 ) systematically predict higher values
4
a)
c)
b)
0
Figure 3: Illustration of assumptions. a) Three functions from G(x? ). They all have their maximum
located at x? ? X . b) Schematic representation of the likelihood function of x? ? X conditioned
on a few observations. The curve corresponds to the mean and the shaded area to the confidence
bounds. The density inside of the neighborhood is unique to the hypothesis x? , while the density
outside is shared amongst all the hypotheses. c) The log-likelihood ratio of the hypotheses x?1 and
x?2 as a function of the test point x. The kernel used in the plot is Gaussian.
than the rest. This assumption is illustrated in Figure 3b. In fact, taking the log-likelihood ratio of
two competing hypotheses
P (yt |x?1 , xt , Dt?1 )
log
P (yt |x?2 , xt , Dt?1 )
for a given test location xt should give a value equal to zero unless xt is inside of the vicinity of x?1
or x?2 (see Figure 3c). In other words, the amount of evidence a hypothesis gets when the test point
is outside of its neighborhood is essentially zero (i.e. it is the same as the amount of evidence that
most of the other hypotheses get).
3.4 Likelihood and Conjugate Prior
Following our previous discussion, we propose the following likelihood model. Given the previous
data Dt?1 and a test point xt ? X , the likelihood of the observation yt is
1
P (yt |x? , xt , Dt?1 ) =
?(yt |xt , Dt?1 ) exp ?t ? ht (x? ) ? ?t?1 ? ht?1 (x? ) , (8)
Z(xt , Dt?1 )
where: Z(xt , Dt?1 ) is a normalizing constant; ?(yt |xt , Dt?1 ) is a posterior probability over yt
given xt and the data Dt?1 ; ?t is a precision measuring the knowledge we have about the whole
function; and and ht is an estimate of the mean function f?. We have chosen the precision ?t as
P
i K(xi , xi )
?t := ? ? ? + P P
i
j K(xi , xj )
where ? > 0 is a scaling parameter; ? > 0 is a parameter representing the number prior locations
tested; and K : X ? X ? R+ is a symmetric kernel function4. For the estimate ht , we have chosen
a Naradaya-Watson kernel regressor [11]
Pt
K(xi , x? )yi + K0 (x? )y0 (x? )
?
ht (x ) := i=1
.
Pt
?
?
i=1 K(xi , x ) + K0 (x )
In the last expression, y0 corresponds to a prior estimate of f? with prior precision K0 . Inspecting (8),
we see that the likelihood model favours positive changes to the estimated mean function from new,
unseen test locations. The pdf ?(yt |xt , Dt?1 ) does not need to be explicitly defined, as it will later
drop out when computing the posterior. The only formal requirement is that it should be independent
of the hypothesis x? .
We propose the conjugate prior
P (x? ) =
4
1
1
exp{?0 ? g0 (x? )} =
exp{? ? y0 (x? )}.
Z0
Z0
We refer the reader to the kernel regression literature for an analysis of the choice of kernel functions.
5
(9)
The conjugate prior just encodes a prior estimate of the mean function. In a practical optimization
application, it serves the purpose of guiding the exploration of the domain, as locations x? with high
prior value y0 (x? ) are more likely to contain the maximizing argument.
Given a set of data points Dt , the prior (9) and the likelihood (8) lead to a posterior given by
Qt
P (x? ) k=1 P (yk |x? , xk , Dk?1 )
P (x? |Dt ) = R
Qt
?
?
?
k=1 P (yk |x , xk , Dk?1 ) dx
X P (x )
Pt
?1 Qt
?
?
Z(xk , Dk?1 )?1
exp
k=1 ?k ? hk (x ) ? ?k?1 ? hk?1 (x ) Z0
= R
Pt
?1 Qtk=1
?1 dx?
?k ? hk (x? ) ? ?k?1 ? hk?1 (x? ) Z0
k=1 Z(xk , Dk?1 )
X exp
k=1 ?
exp ?t ? ht (x )
= R
.
(10)
exp
?t ? ht (x? ) dx?
X
Thus, the particular choice of the likelihood function guarantees an analytically compact posterior
expression. In general, the normalizing constant in (10) is intractable, which is why the expression is
only practical for relative comparisons of test locations. Substituting the precision ?t and the mean
function estimate ht yields
P
P
K(xi , x? )yi + K0 (x? )y0 (x? )
i K(xi , xi )
P (x? |Dt ) ? exp ? ? ? + t ? P P
? i P
.
?
?
i
j K(xi , xj )
i K(xi , x ) + K0 (x )
4 Experimental Results
4.1 Parameters.
We have investigated the influence of the parameters on the resulting posterior probability distribution. We have used the Gaussian kernel
o
n 1
(11)
K(x, x? ) = exp ? 2 (x ? x? )2 .
2?
In this figure, 7 data points are shown, which were drawn as y ? N (f (x), 0.3), where the mean
function is
f (x) = cos(2x + 23 ?) + sin(6x + 32 ?).
(12)
The prior precision K0 and the prior estimate of the mean function y0 were chosen as
K0 (x) = 1
and
y0 (x) = ?
1
(x ? ?0 )2 ,
2?02
(13)
where the latter corresponds to the logarithm of a Gaussian with mean ?0 = 1.5 and variance ?02 = 5. This prior favours the region close to ?.
Figure 4 shows how the choice of the precision scale ? and the kernel width ? affect the shape of
the posterior probability density. Here, it is seen that a larger kernel width ? increases the region of
influence of a particular data point, and hence produce smoother posterior densities. The precision
scale parameter ? controls the precision per distinct data point: higher values for ? lead to sharper
updates of the posterior distribution.
4.2 Application to Optimization.
The main motivation behind our proposed model is its application to the optimization of noisy
functions. Because of the noise, choosing new test locations requires carefully balancing explorative
and exploitative tests?a problem well known in the multiarmed bandits literature. To overcome
this, one can apply the Bayesian control rule/Thompson sampling [12, 13]: the next test location
is chosen by sampling it from the posterior. We have carried out two experiments, described in the
following.
6
a)
b)
c)
Figure 4: Effect of the change of parameters on the posterior density over the location of the maximizing test point. Panel (a) shows the 7 data points drawn from the noisy function (solid curve).
Panel (b) shows the effect of increasing the width of the kernel (here, Gaussian). The solid and
dotted curves correspond to ? = 0.01 and ? = 0.1 respectively. Panel (c) shows the effect of diminishing the precision on the posterior, where solid and shaded curves correspond to ? = 0.2 and
? = 0.1 respectively.
Average Value
Average Value
1.5
1.5
1
1
0.5
0.5
y
y
obs
0
obs
0
?0.5
?0.5
?1
?1
?1.5
?1.5
?2
0
50
100
150
# of samples
200
?2
0
50
100
150
# of samples
200
Figure 5: Observation values obtained by sampling from the posterior over the maximizing argument
(left panel) and according to GP-UCB (right panel). The solid blue curve corresponds to the timeaveraged function value, averaged over ten runs. The gray area corresponds to the error bounds
(1 standard deviation), and the dashed curve in red shows the time-average of a single run.
Comparison to Gaussian Process UCB. We have used the model to optimize the same function (12) as in our preliminary tests but with higher additive noise equal to one. This is done by sampling the next test point xt directly from the posterior density over the optimum location P (x? |Dt ),
and then using the resulting pair (xt , yt ) to recursively update the model. Essentially, this procedure
corresponds to Bayesian control rule/Thompson sampling.
We compared our method against a Gaussian Process optimization method using an upper confidence bound (UCB) criterion [10]. The parameters for the GP-UCB were set to the following
values: observation noise ?n = 0.3 and length scale ? = 0.3. For the constant that trades off exploration and exploitation we followed Theorem 1 in [10] which states ?t = 2 log(|D|t2 ? 2 /6?)
with ? = 0.5. We have implemented our proposed method with a Gaussian kernel as in (11) with
width ? 2 = 0.05. The prior sufficient statistics are exactly as in (13). The precision parameter was
set to ? = 0.3.
Simulation results over ten independent runs are summarized in Figure 5. We show the timeaveraged observation values y of the noisy function evaluated at test locations sampled from the
posterior. Qualitatively, both methods show very similar convergence (on average), however our
method converges faster and with a slightly higher variance.
High-Dimensional Problem. To test our proposed method on a challenging problem, we have
designed a non-convex, high-dimensional noisy function with multiple local optima. This Noisy
Ripples function is defined as
1
f (x) = ? 1000
kx ? ?k2 + cos( 32 ?kx ? ?k)
where ? ? X is the location of the global maximum, and where observations have additive Gaussian
noise with zero mean and variance 0.1. The advantage of this function is that it generalizes well to
any number of dimensions of the domain. Figure 6a illustrates the function for the 2-dimensional
7
a)
b)
Average Value
0
-100
5
0
-200
?5
0
?10
15
50
Samples
100
150
100
150
Regret
10
8000
5
6000
15
10
0
4000
5
?5
0
2000
?5
?10
?10
?15
0
?15
50
Samples
Figure 6: a) The Noisy Ripples objective function in 2 dimensions. b) The time-averaged value and
the regret obtained by the optimization algorithm on a 50-dimensional version of the Noisy Ripples
function.
input domain. This function is difficult to optimize because it requires averaging the noisy observations and smoothing the ridged landscape in order to detect the underlying quadratic form.
We optimized the 50-dimensional version of this function using a Metropolis-Hastings scheme to
sample the next test locations from the posterior over the maximizing argument. The Markov chain
was started at [20, 20, ? ? ? , 20]T , executing 120 isotropic Gaussian steps of variance 0.07 before
the point was used as an actual test location. For the arg-max prior, we used a Gaussian kernel
with lengthscale l = 2, precision factor ? = 1.5, prior precision K0 (x? ) = 1 and prior mean
2
estimate y0 (x? ) = ? 1000
kx + 5k2 . The goal ? was located at the origin.
The result of one run is presented in Figure 6b. It can be seen that the optimizer manages to quickly
(? 100 samples) reach near-optimal performance, overcoming the difficulties associated with the
high-dimensionality of the input space and the numerous local optima. Crucial for this success was
the choice of a kernel that is wide enough to accurately estimate the mean function. The authors are
not aware of any method capable of solving a problem of similar characteristics.
5 Conclusions
Our goal was to design a probabilistic model over the maximizing argument that is algorithmically
efficient and statistically robust even for large, high-dimensional noisy functions. To this end, we
have derived a Bayesian model that directly captures the uncertainty over the maximizing argument,
thereby bypassing having to model the underlying function space?a much harder problem.
Our proposed model is computationally very efficient when compared to Gaussian process-based
(which have cubic time complexity) or models based on upper confidence bounds (which require
finding the input maximizing the bound?a generally intractable operation). In our model, evaluating the posterior up to a constant factor scales quadratically with the size of the data.
In practice, we have found that one of the main difficulties associated with our proposed method is
the choice of the parameters. As in any kernel-based estimation method, choosing the appropriate
kernel bandwidth can significantly change the estimate and affect the performance of optimizers that
rely on the model. There is no clear rule on how to choose a good bandwidth.
In a future research, it will be interesting to investigate the theoretical properties of the proposed
nonparametric model, such as the convergence speed of the estimator and its relation to the extensive
literature on active learning and bandits.
8
References
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[3] A. Shapiro. Probabilistic Constrained Optimization: Methodology and Applications, chapter
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[4] H.J. Kappen, V. G?omez, and M. Opper. Optimal control as a graphical model inference problem. Machine Learning, 87(2):159?182, 2012.
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[8] D.R. Jones, M. Schonlau, and W.J. Welch. Efficient global optimization of expensive blackbox functions. Journal of Global Optimization, 13(4):455?492, 1998.
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3,907 | 4,537 | Sparse Prediction with the k-Support Norm
Andreas Argyriou
?
Ecole
Centrale Paris
[email protected]
Rina Foygel
Department of Statistics, Stanford University
[email protected]
Nathan Srebro
Toyota Technological Institute at Chicago
[email protected]
Abstract
We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an `2 penalty. We show that this new k-support norm provides
a tighter relaxation than the elastic net and can thus be advantageous in in sparse
prediction problems. We also bound the looseness of the elastic net, thus shedding
new light on it and providing justification for its use.
1
Introduction
Regularizing with the `1 norm, when we expect a sparse solution to a regression problem, is often
justified by kwk1 being the ?convex envelope? of kwk0 (the number of non-zero coordinates of a
vector w ? Rd ). That is, kwk1 is the tightest convex lower bound on kwk0 . But we must be careful
with this statement?for sparse vectors with large entries, kwk0 can be small while kwk1 is large.
In order to discuss convex lower bounds on kwk0 , we must impose some scale constraint. A more
accurate statement is that kwk1 ? kwk? kwk0 , and so, when the magnitudes of entries in w are
bounded by 1, then kwk1 ? kwk0 , and indeed it is the largest such convex lower bound. Viewed as
a convex outer relaxation,
(?)
Sk := w kwk0 ? k, kwk? ? 1 ? w kwk1 ? k .
Intersecting the right-hand-side with the `? unit ball, we get the tightest convex outer bound (convex
(?)
hull) of Sk :
(?)
w kwk1 ? k, kwk? ? 1 = conv(Sk ) .
However, in our view, this relationship between kwk1 and kwk0 yields disappointing learning guarantees, and does not appropriately capture the success of the `1 norm as a surrogate for sparsity. In
particular, the sample complexity1 of learning a linear predictor with k non-zero entries by empirical
risk minimization inside this class (an NP-hard optimization problem) scales as O(k log d), but relaxing to the constraint kwk1 ? k yields a sample complexity which scales as O(k 2 log d), because
the sample complexity of `1 -regularized learning scales quadratically with the `1 norm [11, 20].
Perhaps a better reason for the `1 norm being a good surrogate for sparsity is that, not only do we
expect the magnitude of each entry of w to be bounded, but we further expect kwk2 to be small. In
a regression setting, with a vector of features x, this can be justified when E[(x> w)2 ] is bounded
(a reasonable assumption) and the features are not too correlated?see, e.g. [15]. More broadly,
1
We define this as the number of observations needed in order to ensure expected prediction error no more
than worse than that of the best k-sparse predictor, for an arbitrary constant (that is, we suppress the
dependence on and focus on the dependence on the sparsity k and dimensionality d).
1
especially in the presence of correlations, we might require this as a modeling
assumption to aid
p
in robustness and generalization. In any case, we have kwk1 ? kwk2 kwk0 , and so if we are
interested in predictors with bounded `2 norm, we can motivate the `1 norm through the following
relaxation of sparsity, where the scale is now set by the `2 norm:
?
w kwk0 ? k, kwk2 ? B ? w kwk1 ? B k .
The sample complexity when using the relaxation now scales as2 O(k log d).
Sparse + `2 constraint. Our starting point is then that of combining sparsity and `2 regularization,
and learning a sparse predictor with small `2 norm. We are thus interested in classes of the form
(2)
Sk := w kwk0 ? k, kwk2 ? 1 .
?
As discussed above, the class {kwk1 ? k} (corresponding to the standard Lasso) provides a
(2)
convex relaxation of Sk . But clearly we can get a tighter relaxation by keeping the `2 constraint:
n
o n
? o
?
(2)
conv(Sk ) ? w kwk1 ? k, kwk2 ? 1 ( w kwk1 ? k .
(1)
Constraining (or equivalently, penalizing) both the `1 and `2 norms, as in (1), is known as the ?elastic
net? [5, 21] and has indeed been advocated as a better alternative to the Lasso. In this paper, we ask
(2)
whether the elastic net is the tightest convex relaxation to sparsity plus `2 (that is, to Sk ) or whether
a tighter, and better, convex relaxation is possible.
A new norm.
(2)
We consider the convex hull (tightest convex outer bound) of Sk ,
(2)
Ck := conv(Sk ) = conv w kwk0 ? k, kwk2 ? 1 .
(2)
We study the gauge function associated with this convex set, that is, the norm whose unit ball is
given by (2), which we call the k-support norm. We show that, for k > 1, this is indeed a tighter
convex relaxation than the elastic net (that is, both inequalities in (1) are in fact strict inequalities),
and is therefore a better convex constraint than the elastic net when seeking a sparse, low `2 -norm
linear predictor. We thus advocate using it as a replacement for the elastic net.
However,
we also show that the gap between the elastic net and the k-support norm is at most a factor
?
of 2, corresponding to a factor of two difference in the sample complexity. Thus, our work can
also be interpreted as justifying the use of the elastic net, viewing it as a fairly good approximation
to the tightest possible convex relaxation of sparsity intersected with an `2 constraint. Still, even a
factor of two should not necessarily be ignored and, as we show in our experiments, using the tighter
k-support norm can indeed be beneficial.
To better understand the k-support norm, we show in Section2 that it can also be described as
the group lasso with overlaps norm [10] corresponding to all kd subsets of k features. Despite the
exponential number of groups in this description, we show that the k-support norm can be calculated
efficiently in time O(d log d) and that its dual is given simply by the `2 norm of the k largest entries.
We also provide efficient first-order optimization algorithms for learning with the k-support norm.
Related Work In many learning problems of interest, Lasso has been observed to shrink too many
of the variables of w to zero. In particular, in many applications, when a group of variables is highly
correlated, the Lasso may prefer a sparse solution, but we might gain more predictive accuracy by
including all the correlated variables in our model. These drawbacks have recently motivated the use
of various other regularization methods, such as the elastic net [21], which penalizes the regression
coefficients w with a combination of `1 and `2 norms:
1
kXw ? yk2 + ?1 kwk1 + ?2 kwk22 : w ? Rd ,
(3)
min
2
2
More precisely, the sample complexity is O(B 2 k log d), where the dependence on B 2 is to be expected.
Note that if feature vectors are `? -bounded (i.e. individual features are bounded), the sample complexity when
using only kwk2 ? B (without a sparsity or `1 constraint) scales as O(B 2 d). That is, even after identifying
the correct support, we still need a sample complexity that scales with B 2 .
2
where for a sample of size n, y ? Rn is the vector of response values, and X ? Rn?d is a matrix
with column j containing the values of feature j.
The elastic net can be viewed as a trade-off between `1 regularization (the Lasso) and `2 regularization (Ridge regression [9]), depending on the relative values of ?1 and ?2 . In particular, when
?2 = 0, (3) is equivalent to the Lasso. This method, and the other methods discussed below, have
been observed to significantly outperform Lasso in many real applications.
The pairwise elastic net (PEN) [13] is a penalty function that accounts for similarity among features:
EN
kwkP
= kwk22 + kwk21 ? |w|> R|w| ,
R
where R ? [0, 1]p?p is a matrix with Rjk measuring similarity between features Xj and Xk . The
trace Lasso [6] is a second method proposed to handle correlations within X, defined by
kwktrace
= kXdiag(w)k? ,
X
where k ? k? denotes the matrix trace-norm (the sum of the singular values) and promotes a low-rank
solution. If the features are orthogonal, then both the PEN and the Trace Lasso are equivalent to
the Lasso. If the features are all identical, then both penalties are equivalent to Ridge regression
(penalizing kwk2 ). Another existing penalty is OSCAR [3], given by
X
kwkOSCAR
= kwk1 + c
max{|wj |, |wk |} .
c
j<k
Like the elastic net, each one of these three methods also ?prefers? averaging similar features over
selecting a single feature.
The k-Support Norm
2
One argument for the elastic net has been the flexibility of tuning the cardinality k of the regression vector w. Thus, when groups of correlated variables are present, a larger k may be learned,
which corresponds to a higher ?2 in (3). A more natural way to obtain such an effect of tuning the
cardinality is to consider the convex hull of cardinality k vectors,
(2)
Ck = conv(Sk ) = conv{w ? Rd kwk0 ? k, kwk2 ? 1}.
Clearly the sets Ck are nested, and C1 and Cd are the unit balls for the `1 and `2 norms, respectively.
Consequently we define the k-support norm as the norm whose unit ball equals Ck (the gauge
function associated with the Ck ball).3 An equivalent definition is the following variational formula:
d
Definition 2.1. Let k ? {1, . . . , d}. The k-support norm k ? ksp
k is defined, for every w ? R , as
(
)
X
X
kwksp
kvI k2 : supp(vI ) ? I,
vI = w ,
k := min
I?Gk
I?Gk
where Gk denotes the set of all subsets of {1, . . . , d} of cardinality at most k.
The equivalence
P is immediate by rewriting vI = ?I zI in the above definition, where ?I ? 0, zI ?
Ck , ?I ? Gk , I?Gk ?I = 1. In addition, this immediately implies that k ? ksp
k is indeed a norm. In
fact, the k-support norm is equivalent to the norm used by the group lasso with overlaps [10], when
the set of overlapping groups is chosen to be Gk (however, the group lasso has traditionally been
used for applications with some specific known group structure, unlike the case considered here).
Although the variational definition 2.1 is not amenable to computation because of the exponential growth of the set of groups Gk , the k-support norm is computationally very tractable, with an
O(d log d) algorithm described in Section 2.2.
sp
As already mentioned, k ? ksp
1 = k ? k1 and k ? kd = k ? k2 . The unit ball of this new norm in
R3 for k = 2 is depicted in Figure 1. We immediately notice several differences between this unit
ball and the elastic net unit ball. For example, at points with cardinality k and `2 norm equal to 1,
the k-support norm is not differentiable, but unlike the `1 or elastic-net norm, it is differentiable at
points with cardinality less than k. Thus, the k-support norm is less ?biased? towards sparse vectors
than the elastic net and the `1 norm.
3
The gauge function ?Ck : Rd ? R ? {+?} is defined as ?Ck (x) = inf{? ? R+ : x ? ?Ck }.
3
Figure 1: Unit ball of the 2-support norm (left) and of the elastic net (right) on R3 .
2.1
The Dual Norm
It is interesting and useful to compute the dual of the k-support norm. For w ? Rd , denote |w| for
the vector of absolute values, and wi? for the i-th largest element of w [2]. We have
?
?
! 12
! 21
k
? X
?
X
?
(2)
kukksp = max {hw, ui : kwksp
u2i
=: kuk(k) .
: I ? Gk =
(|u|?i )2
k ? 1} = max ?
?
i?I
i=1
This is the `2 -norm of the largest k entries in u, and is known as the 2-k symmetric gauge norm [2].
Not surprisingly, this dual norm interpolates between the `2 norm (when k = d and all entries
are taken) and the `? norm (when k = 1 and only the largest entry is taken). This parallels the
interpolation of the k-support norm between the `1 and `2 norms.
2.2
Computation of the Norm
In this section, we derive an alternative formula for the k-support norm, which leads to computation
of the value of the norm in O(d log d) steps.
?
!2 ? 12
k?r?1
d
P
P
1
?
Proposition 2.1. For every w ? Rd , kwksp
(|w|?i )2 + r+1
|w|?i ? ,
k =
i=1
where, letting
|w|?0
i=k?r
denote +?, r is the unique integer in {0, . . . , k ? 1} satisfying
|w|?k?r?1 >
d
X
1
|w|?i ? |w|?k?r .
r+1
(4)
i=k?r
This result shows that k ? ksp
k trades off between the `1 and `2 norms in a way that favors sparse
vectors but allows for cardinality larger than k. It combines the uniform shrinkage of an `2 penalty
for the largest components, with the sparse shrinkage of an `1 penalty for the smallest components.
Proof of Proposition 2.1. We will use the inequality hw, ui ? hw? , u? i [7]. We have
( d
k
X
1
1X 2
1
(2) 2
sp 2
d
(kwkk ) = max hu, wi ? (kuk(k) ) : u ? R = max
?i |w|?i ?
? :
2
2
2 i=1 i
i=1
)
(k?1
)
d
k
X
X
1X 2
?
?
?1 ? ? ? ? ? ?d ? 0 = max
?i |w|i + ?k
|w|i ?
? : ?1 ? ? ? ? ? ?k ? 0 .
2 i=1 i
i=1
i=k
Let Ar :=
d
P
i=k?r
|w|?i for r ? {0, . . . , k ? 1}. If A0 < |w|?k?1 then the solution ? is given by
?i = |w|?i for i = 1, . . . , (k ? 1), ?i = A0 for i = k, . . . , d. If A0 ? |w|?k?1 then the optimal ?k ,
?k?1 lie between |w|?k?1 and A0 , and have to be equal. So, the maximization becomes
(k?2
)
k?2
X
1X 2
?
2
max
?i |w|i ?
? + A1 ?k?1 ? ?k?1 : ?1 ? ? ? ? ? ?k?1 ? 0 .
2 i=1 i
i=1
4
If A0 ? |w|?k?1 and |w|?k?2 > A21 then the solution is ?i = |w|?i for i = 1, . . . , (k ? 2), ?i = A21
for i = (k ? 1), . . . , d. Otherwise we proceed as before and continue this process. At stage r the
Ar
? |w|?k?r , r+1
< |w|?k?r?1 and all but the last two
process terminates if A0 ? |w|?k?1 , . . . , Ar?1
r
inequalities are redundant. Hence the condition can be rewritten as (4). One optimal solution is
Ar
?i = |w|?i for i = 1, . . . , k ? r ? 1, ?i = r+1
for i = k ? r, . . . , d. This proves the claim.
2.3
Learning with the k-support norm
We thus propose using learning rules with k-support norm regularization. These are appropriate
when we would like to learn a sparse predictor that also has low `2 norm, and are especially relevant
when features might be correlated (that is, in almost all learning tasks) but the correlation structure
is not known in advance. E.g., for squared error regression problems we have:
1
?
sp 2
2
d
min
kXw ? yk + (kwkk ) : w ? R
(5)
2
2
with ? > 0 a regularization parameter and k ? {1, . . . , d} also a parameter to be tuned. As typical
in regularization-based methods, both ? and k can be selected by cross validation [8]. Despite the
(2)
relationship to Sk , the parameter k does not necessarily correspond to the sparsity of the actual
minimizer of (5), and should be chosen via cross-validation rather than set to the desired sparsity.
3
Relation to the Elastic Net
Recall that the elastic net with penalty parameters ?1 and ?2 selects a vector of coefficients given by
1
kXw ? yk2 + ?1 kwk1 + ?2 kwk22 .
arg min
(6)
2
For ease of comparison with the k-support norm, we first show that the set of optimal solutions for
the elastic net, when the parameters are varied, is the same as for the norm
n
? o
kwkel
:=
max
kwk
,
kwk
/
k ,
2
1
k
when k ? [1, d], corresponding to the unit ball in (1) (note that k is not necessarily an integer). To
see this, let w
? be a solution to (6), and let k := (kwk
? 1 /kwk
? 2 )2 ? [1, d] . Now for any w 6= w,
?
el
el
if kwkk ? kwk
? k , then kwkp ? kwk
? p for p = 1, 2. Since w
? is a solution to (6), therefore,
kXw ? yk22 ? kX w
? ? yk22 . This proves that, for some constraint parameter B,
1
w
? = arg min
kXw ? yk22 : kwkel
?
B
.
k
n
Like the k-support norm, the elastic net interpolates between the `1 and `2 norms. In fact, when k
is an integer, any k-sparse unit vector w ? Rd must lie in the unit ball of k ? kel
k . Since the k-support
norm gives the convex hull of all k-sparse unit vectors, this immediately implies that
sp
kwkel
? w ? Rd .
k ? kwkk
The two norms are not equal, however. The difference between the two is illustrated in Figure 1,
where we see that the k-support norm is more ?rounded?.
To see an example where the two norms are not equal, we set d = 1 + k 2 for some large k, and let
w = (k 1.5 , 1, 1, . . . , 1)> ? Rd . Then
p
k 1.5 + k 2
1
1.5
3 + k2 ,
?
?
kwkel
=
max
k
=
k
1
+
.
k
k
k
(2)
Taking u = ( ?12 , ?12k , ?12k , . . . , ?12k )> , we have kuk(k) < 1, and recalling this norm is dual to the
k-support norm:
?
k 1.5
1
? + k 2 ? ? = 2 ? k 1.5 .
kwksp
>
hw,
ui
=
k
2
2k
?
In this example, we see that the two norms can differ by as much as a factor of 2. We now show
that this is actually the most by which they can differ.
5
sp
Proposition 3.1. k ? kel
k ? k ? kk <
?
2 k ? kel
k.
Proof. We show that these bounds hold in the duals of the two norms. First, since k ? kel
k is a
maximum over the `1 and `2 norms, its dual is given by
n
o
?
(el)?
kukk
:= inf kak2 + k ? ku ? ak?
a?Rd
(el)?
(2)
Now take any u ? Rd . First we show kuk(k) ? kukk
u1 ? ? ? ? ? ud ? 0. For any a ? Rd ,
. Without loss of generality, we take
(2)
kuk(k) = ku1:k k2 ? ka1:k k2 + ku1:k ? a1:k k2 ? kak2 +
(el)?
(el)?
kku ? ak? .
(2)
2 kuk(k) . Let a = (u1 ? uk+1 , . . . , uk ? uk+1 , 0, . . . , 0)> . Then
v
u k
uX
?
?
? kak2 + k ? ku ? ak? = t (ui ? uk+1 )2 + k ? |uk+1 |
Finally, we show that kukk
kukk
?
?
<
i=1
v
u k
uX
? t (u2 ? u2
i
k+1 )
+
q
k u2k+1 ?
?
v
u k
uX
?
(2)
2 ? t (u2i ? u2k+1 ) + k u2k+1 = 2 kuk(k) .
i=1
i=1
Furthermore, this yields a strict inequality, because if u1 > uk+1 , the next-to-last inequality is strict,
while if u1 = ? ? ? = uk+1 , then the last inequality is strict.
4
Optimization
Solving the optimization problem (5) efficiently can be done with a first-order proximal algorithm.
Proximal methods ? see [1, 4, 14, 18, 19] and references therein ? are used to solve composite
problems of the form min{f (x) + ?(x) : x ? Rd }, where the loss function f (x) and the regularizer
?(x) are convex functions, and f is smooth with an L-Lipschitz gradient. These methods require
fast computation of the gradient ?f and the proximity operator
1
prox? (x) := argmin
ku ? xk2 + ?(u) : u ? Rd .
2
To obtain a proximal method for k-support regularization, it suffices to compute the proximity map
L
1
2
(k ? ksp
of g = 2?
k ) , for any ? > 0 (in particular, for problem (5) ? corresponds to ? ). This
computation can be done in O(d(k + log d)) steps with Algorithm 1.
Algorithm 1 Computation of the proximity operator.
Input v ? Rd
1
Output q = prox 2?
2 (v)
(k?ksp
k )
Find r ? {0, . . . , k ? 1}, ` ? {k, . . . , d} such that
1
?+1 zk?r?1
z` >
>
Tr,`
`?k+(?+1)r+?+1
Tr,`
`?k+(?+1)r+?+1
?
? z`+1
`
P
where z := |v|? , z0 := +?, zd+1 := ??, Tr,` :=
zi
i=k?r
? ?
?
if i = 1, . . . , k ? r ? 1
? ?+1 zi
Tr,`
qi ? zi ? `?k+(?+1)r+?+1
if i = k ? r, . . . , `
?
?
0
if i = ` + 1, . . . , d
Reorder and change signs of q to conform with v
6
1
?+1 zk?r
(7)
(8)
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
35
40
40
45
45
50
50
5
10
15
20
25
30
35
40
40
45
50
5
10
15
20
25
30
35
40
5
10
15
20
25
30
35
40
Figure 2: Solutions learned for the synthetic data. Left to right: k-support, Lasso and elastic net.
Proof of Correctness of Algorithm 1. Since the support-norm is sign and permutation invariant,
proxg (v) has the same ordering and signs as v. Hence, without loss of generality, we may assume
that v1 ? ? ? ? ? vd ? 0 and require that q1 ? ? ? ? ? qd ? 0, which follows from inequality (7) and
the fact that z is ordered.
2
Now, q = proxg (v) is equivalent to ?z ? ?q = ?v ? ?q ? ? 12 (k ? ksp
k ) (q). It suffices to show
that, for w = q, ?z ? ?q is an optimal ? in the proof of Proposition 2.1. Indeed, Ar corresponds to
d
`
P
P
Tr,`
(`?k+r+1)Tr,`
? Tr,`
qi =
zi ? `?k+(?+1)r+?+1
= Tr,` ? `?k+(?+1)r+?+1
= (r + 1) `?k+(?+1)r+?+1
i=k?r
i=k?r
and (4) is equivalent to condition (7). For i ? k ? r ? 1, we have ?zi ? ?qi = qi . For k ? r ? i ? `,
1
1
we have ?zi ? ?qi = r+1
Ar . For i ? ` + 1, since qi = 0, we only need ?zi ? ?qi ? r+1
Ar , which
is true by (8).
We can now apply a standard accelerated proximal method, such as FISTA [1], to (5), at each
iteration using the gradient of the loss and performing a prox step using Algorithm 1. The FISTA
guarantee ensures us that, with appropriate step sizes, after T such iterations, we have:
!
1
?
?
1
2Lkw? ? w1 k2
sp 2
2
?
2
? sp 2
kXwT ? yk + (kwT kk ) ?
kXw ? yk + (kw kk ) +
.
2
2
2
2
(T + 1)2
5
Empirical Comparisons
Our?theoretical analysis indicates that the k-support norm and the elastic net differ by at most a factor
of 2, corresponding to at most a factor of two difference in their sample complexities and generalization guarantees. We thus do not expect huge differences between their actual performances, but
would still like to see whether the tighter relaxation of the k-support norm does yield some gains.
Synthetic Data For the first simulation we follow [21, Sec. 5, example 4]. In this experimental
protocol, the target (oracle) vector equals w? = (3, . . . , 3, 0 . . . , 0), with y = (w? )> x + N (0, 1).
| {z } | {z }
15
25
The input data X were generated from a normal distribution such that components 1, . . . , 5 have the
same random mean Z1 ? N (0, 1), components 6, . . . , 10 have mean Z2 ? N (0, 1) and components
11, . . . , 15 have mean Z3 ? N (0, 1). A total of 50 data sets were created in this way, each containing
50 training points, 50 validation points and 350 test points. The goal is to achieve good prediction
performance on the test data.
We compared the k-support norm with Lasso and the elastic net. We considered the ranges k =
{1, . . . , d} for k-support norm regularization, ? = 10i , i = {?15, . . . , 5}, for the regularization
parameter of Lasso and k-support regularization and the same range for the ?1 , ?2 of the elastic net.
For each method, the optimal set of parameters was selected based on mean squared error on the
validation set. The error reported in Table 5 is the mean squared error with respect to the oracle w? ,
namely M SE = (w
? ? w? )> V (w
? ? w? ), where V is the population covariance matrix of Xtest .
To further illustrate the effect of the k-support norm, in Figure 5 we show the coefficients learned
by each method, in absolute value. For each image, one row corresponds to the w learned for one
of the 50 data sets. Whereas all three methods distinguish the 15 relevant variables, the elastic net
result varies less within these variables.
South African Heart Data This is a classification task which has been used in [8]. There are
9 variables and 462 examples, and the response is presence/absence of coronary heart disease. We
7
Table 1: Mean squared errors and classification accuracy for the synthetic data (median over 50 repetition),
SA heart data (median over 50 replications) and for the ?20 newsgroups? data set. (SE = standard error)
Method
Lasso
Elastic net
k-support
Synthetic
MSE (SE)
0.2685 (0.02)
0.2274 (0.02)
0.2143 (0.02)
Heart
MSE (SE)
Accuracy (SE)
0.18 (0.005)
66.41 (0.53)
0.18 (0.005)
66.41 (0.53)
0.18 (0.005)
66.41 (0.53)
Newsgroups
MSE Accuracy
0.70
73.02
0.70
73.02
0.69
73.40
normalized the data so that each predictor variable has zero mean and unit variance. We then split the
data 50 times randomly into training, validation, and test sets of sizes 400, 30, and 32 respectively.
For each method, parameters were selected using the validation data. In Tables 5, we report the
MSE and accuracy of each method on the test data. We observe that all three methods have identical
performance.
20 Newsgroups This is a binary classification version of 20 newsgroups created in [12] which can
be found in the LIBSVM data repository.4 The positive class consists of the 10 groups with names of
form sci.*, comp.*, or misc.forsale and the negative class consists of the other 10 groups. To reduce
the number of features, we removed the words which appear in less than 3 documents. We randomly
split the data into a training, a validation and a test set of sizes 14000,1000 and 4996, respectively.
We report MSE and accuracy on the test data in Table 5. We found that k-support regularization
gave improved prediction accuracy over both other methods.5
6
Summary
We introduced the k-support norm as the tightest convex relaxation of sparsity
plus `2 regularization,
?
and showed that it is tighter than the elastic net by exactly a factor of 2. In our view, this sheds
light on the elastic net as a close approximation to this tightest possible convex relaxation, and
motivates using the k-support norm when a tighter relaxation is sought. This is also demonstrated
in our empirical results.
We note that the k-support norm has better prediction properties, but not necessarily better sparsityinducing properties, as evident from its more rounded unit ball. It is well understood that there
is often a tradeoff between sparsity and good prediction, and that even if the population optimal
predictor is sparse, a denser predictor often yields better predictive performance [3, 10, 21]. For
example, in the presence of correlated features, it is often beneficial to include several highly correlated features rather than a single representative feature. This is exactly the behavior encouraged by
`2 norm regularization, and the elastic net is already known to yield less sparse (but more predictive)
solutions. The k-support norm goes a step further in this direction, often yielding solutions that are
even less sparse (but more predictive) compared to the elastic net.
Nevertheless, it is interesting to consider whether compressed sensing results, where `1 regularization is of course central, can be refined by using the k-support norm, which might be able to handle
more correlation structure within the set of features.
Acknowledgements The construction
showing that the gap between the elastic net and the k?
overlap norm can be as large as 2 is due to joint work with Ohad Shamir. Rina Foygel was
supported by NSF grant DMS-1203762.
References
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4
http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets/
Regarding other sparse prediction methods, we did not manage to compare with OSCAR, due to memory
limitations, or to PEN or trace Lasso, which do not have code available online.
5
8
[3] H.D. Bondell and B.J. Reich. Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR. Biometrics, 64(1):115?123, 2008.
[4] P.L. Combettes and V.R. Wajs. Signal recovery by proximal forward-backward splitting. Multiscale Modeling and Simulation, 4(4):1168?1200, 2006.
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[7] G. H. Hardy, J. E. Littlewood, and G. P?olya. Inequalities. Cambridge University Press, 1934.
[8] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining,
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[12] S. S. Keerthi and D. DeCoste. A modified finite Newton method for fast solution of large scale
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in sparse regression via the pairwise elastic net. In Proceedings of the 13th International
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[14] Y. Nesterov. Gradient methods for minimizing composite objective function. CORE, 2007.
[15] N. Srebro, K. Sridharan, and A. Tewari. Smoothness, low-noise and fast rates. In Advances in
Neural Information Processing Systems 23, 2010.
[16] T. Suzuki and R. Tomioka. SpicyMKL: a fast algorithm for multiple kernel learning with
thousands of kernels. Machine learning, pages 1?32, 2011.
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Preprint, 2008.
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optimization. Mathematical Programming, 125(2):263?295, 2010.
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[21] H. Zou and T. Hastie. Regularization and variable selection via the elastic net. Journal of the
Royal Statistical Society: Series B (Statistical Methodology), 67(2):301?320, 2005.
9
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3,908 | 4,538 | A Convex Formulation for Learning Scale-Free
Networks via Submodular Relaxation
Tiberio S. Caetano
NICTA/ANU/University of Sydney
Canberra and Sydney, Australia
[email protected]
Aaron J. Defazio
NICTA/Australian National University
Canberra, ACT, Australia
[email protected]
Abstract
A key problem in statistics and machine learning is the determination of network
structure from data. We consider the case where the structure of the graph to be
reconstructed is known to be scale-free. We show that in such cases it is natural
to formulate structured sparsity inducing priors using submodular functions, and
we use their Lov?asz extension to obtain a convex relaxation. For tractable classes
such as Gaussian graphical models, this leads to a convex optimization problem
that can be efficiently solved. We show that our method results in an improvement
in the accuracy of reconstructed networks for synthetic data. We also show how
our prior encourages scale-free reconstructions on a bioinfomatics dataset.
Introduction
Structure learning for graphical models is a problem that arises in many contexts. In applied statistics, undirected graphical models can be used as a tool for understanding the underlying conditional
independence relations between variables in a dataset. For example, in bioinfomatics Gaussian
graphical models are fitted to data resulting from micro-array experiments, where the fitted graph
can be interpreted as a gene expression network [9].
In the context of Gaussian models, the structure learning problem is known as covariance selection [8]. The most common approach is the application of sparsity inducing regularization to the
maximum likelihood objective. There is a significant body of literature, more than 30 papers by
our count, on various methods of optimizing the L1 regularized covariance selection objective alone
(see the recent review by Scheinberg and Ma [17]).
Recent research has seen the development of structured sparsity, where more complex prior knowledge about a sparsity pattern can be encoded. Examples include group sparsity [22], where parameters are linked so that they are regularized in groups. More complex sparsity patterns, such as region
shape constraints in the case of pixels in an image [13], or hierarchical constraints [12] have also
been explored.
In this paper, we study the problem of recovering the structure of a Gaussian graphical model under
the assumption that the graph recovered should be scale-free. Many real-world networks are known
a priori to be scale-free and therefore enforcing that knowledge through a prior seems a natural
idea. Recent work has offered an approach to deal with this problem which results in a non-convex
formulation [14]. Here we present a convex formulation. We show that scale-free networks can be
induced by enforcing submodular priors on the network?s degree distribution, and then using their
convex envelope (the Lov?asz extension) as a convex relaxation [2]. The resulting relaxed prior has an
interesting non-differentiable structure, which poses challenges to optimization. We outline a few
options for solving the optimisation problem via proximal operators [3], in particular an efficient
dual decomposition method. Experiments on both synthetic data produced by scale-free network
models and a real bioinformatics dataset suggest that the convex relaxation is not weak: we can
infer scale-free networks with similar or superior accuracy than in [14].
1
1
Combinatorial Objective
Consider an undirected graph with edge set E and node set V , where n is the number of nodes. We
denote the degree of node v as dE (v), and the complete graph with n nodes as Kn . We are concerned
with placing priors on the degree distributions of graphs such as (V, E). By degree distribution, we
mean the bag of degrees {dE (v)|v ? V }.
A natural prior on degree distributions can be formed from the family of exponential random graphs
[21]. Exponential random graph (ERG) models assign a probability to each n node graph using an
exponential family model. The probability of each graph depends on a small set of sufficient statistics, in our case we only consider the degree statistics. A ERG distribution with degree parametrization takes the form:
"
#
X
1
p(G = (V, E); h) ?
exp ?
h(dE (v)) ,
(1)
Z(h)
v?V
The degree weighting function h : Z+ ? R encodes the preference for each particular degree. The
function Z is chosen so that the distribution is correctly normalized over n node graphs.
A number of choices for h are reasonable; A geometric series h(i) ? 1 ? ?i with ? ? (0, 1) has
been proposed by Snijders et al. [20] and has been widely adopted. However for encouraging scale
free graphs we require a more rapidly increasing sequence. It is instructive to observe that, under
the strong assumption that each node?s degree is independent of the rest, h grows logarithmically.
To see this, take a scale free model with scale ?; the joint distribution takes the form:
p(G = (V, E); , ?) ?
Y
1
(dE (v) + )?? ,
Z(, ?)
v?V
where > 0 is added to prevent infinite weights. Putting this into ERG form gives the weight
sequence h(i) = ? log(i + ). We will consider this and other functions h in Section 4. We intend
to perform maximum a posteriori (MAP) estimation of a graph structure using such a distribution as
a prior, so the object of our attention is the negative log-posterior, which we denote F :
X
F (E) =
h(dE (v)) + const.
(2)
v?V
So far we have defined a function on edge sets only, however in practice we want to optimize over
a weighted graph, which is intractable when using discontinuous functions such as F . We now
consider the properties of h that lead to a convex relaxation of F .
2
Submodularity
A set function F : 2E ? R on E is a non-decreasing submodular function if for all A ? B ? E
and x ? E\B the following conditions hold:
F (A ? {x}) ? F (A) ? F (B ? {x}) ? F (B)
and F (A) ? F (B).
(submodularity)
(non-decreasing)
The first condition can be interpreted as a diminishing returns condition; adding x to a set A increases
F (A) by more than adding it to a larger set B, if B contains A.
We now consider a set of conditions that can be placed on h so that F is submodular.
Proposition 1. Denote h as tractable if h is non-decreasing, concave and h(0) = 0. For tractable
h, F is a non-decreasing submodular function.
Proof. First note that the degree function is a set cardinality function, and hence modular. A concave
transformation of a modular function is submodular [1], and the sum of submodular functions is
submodular.
2
The concavity restriction we impose on h is the key ingredient that allows us to use submodularity
to enforce a prior for scale-free networks; any prior favouring long tailed degree distributions must
place a lower weight on new edges joining highly connected nodes than on those joining other nodes.
As far as we are aware, this is a novel way of mathematically modelling the ?preferential attachment?
rule [4] that gives rise to scale-free networks: through non-decreasing submodular functions on the
degree distribution.
Let X denote a symmetric matrix of edge weights. A natural convex relaxation of F would be
the convex envelope of F (Supp(X)) under some restricted domain. For tractable h, we have by
construction that F satisfies the conditions of Proposition 1 in [2], so that the convex envelope
of F (Supp(X)) on the L? ball is precisely the Lov?asz extension evaluated on |X|. The Lov?asz
extension for our function is easy to determine as it is a sum of ?functions of cardinality? which are
considered in [2]. Below is the result from [2] adapted to our problem.
Proposition 2. Let Xi,(j) be the weight of the jth edge connected to i, under a decreasing ordering
by absolute value (i.e |Xi,(0) | ? |Xi,(1) | ? ... ? |Xi,(n?1) |). The notation (i) maps from sorted
order to the natural ordering, with the diagonal not included. Then the convex envelope of F for
tractable h over the L? norm unit ball is:
n n?1
X
X
(h(k + 1) ? h(k)) |Xi,(k) |.
?(X) =
i=0 k=0
This function is piece-wise linear and convex.
The form of ? is quite intuitive. It behaves like a L1 norm with an additional weight on each edge
that depends on how the edge ranks with respect to the other edges of its neighbouring nodes.
3
Optimization
We are interested in using ? as a prior, for optimizations of the form
minimizeX f (X) = g(X) + ??(X),
for convex functions g and prior strength parameters ? ? R+ , over symmetric X. We will focus
on the simplest structure learning problem that occurs in graphical model training, that of Gaussian
models. In which case we have
g(X) = hX, Ci ? log det X,
where C is the observed covariance matrix of our data. The support of X will then be the set
of edges in the undirected graphical model together with the node precisions. This function is a
rescaling of the maximum likelihood objective. In order for the resulting X to define a normalizable
distribution, X must be restricted to the cone of positive definite matrices. This is not a problem
in practice as g(X) is infinite on the boundary of the PSD cone, and hence the constraint can be
handled by restricting optimization steps to the interior of the cone. In fact X can be shown to be
in a strictly smaller cone, X ? aI, for a derivable from C [15]. This restricted domain is useful
as g(X) has Lipschitz continuous gradients over X aI but not over all positive definite matrices
[18].
There are a number of possible algorithms that can be applied for optimizing a convex nondifferentiable objective such as f . Bach [2] suggests two approaches to optimizing functions involving submodular relaxation priors; a subgradient approach and a proximal approach.
Subgradient methods are the simplest class of methods for optimizing non-smooth convex functions.
They provide a good baseline for comparison with other methods. For our objective, a subgradient
is simple to evaluate at any point, due to the piecewise continuous nature of ?(X). Unfortunately
(primal) subgradient methods for our problem will not return sparse solutions except in the limit of
convergence. They will instead give intermediate values that oscillate around their limiting values.
An alternative is the use of proximal methods [2]. Proximal methods exhibit superior convergence
in comparison to subgradient methods, and produce sparse solutions. Proximal methods rely on
solving a simpler optimization problem, known as the proximal operator at each iteration:
1
2
arg min ??(X) + kX ? Zk2 ,
2
X
3
where Z is a variable that varies at each iteration. For many problems of interest, the proximal
operator can be evaluated using a closed form solution. For non-decreasing submodular relaxations,
the proximal operator can be evaluated by solving a submodular minimization on a related (not
necessarily non-decreasing) submodular function [2].
Bach [2] considers several example problems where the proximal operator can be evaluated using
fast graph cut methods. For the class of functions we consider, graph-cut methods are not applicable.
Generic submodular minimization algorithms could be as slow as O(n12 ) for a n-vertex graph,
which is clearly impractical [11]. We will instead propose a dual decomposition method for solving
this proximal operator problem in Section 3.2.
For solving our optimisation problem, instead of using the standard proximal method (sometimes
known as ISTA), which involves a gradient step followed by the proximal operator, we propose to
use the alternating direction method of multipliers (ADMM), which has shown good results when
applied to the standard L1 regularized covariance selection problem [18]. Next we show how to
apply ADMM to our problem.
3.1
Alternating direction method of multipliers
The alternating direction method of multipliers (ADMM, Boyd et al. [6]) is one approach to optimizing our objective that has a number of advantages over the basic proximal method. Let U be the
matrix of dual variables for the decoupled problem:
minimizeX g(X) + ??(Y ),
s.t. X = Y.
Following the presentation of the algorithm in Boyd et al. [6], given the values Y (l) and U (l) from
iteration l, with U (0) = 0n and Y (0) = In the ADMM updates for iteration l + 1 are:
h
i
?
X (l+1) = arg min hX, Ci ? log det X + ||X ? Y (l) + U (l) ||22
2
X
i
h
?
(l+1)
(l+1)
? Y + U (l) ||22
Y
= arg min ??(Y ) + ||X
2
Y
U (l+1) = U (l) + X (l+1) ? Y (l+1) ,
where ? > 0 is a fixed step-size parameter (we used ? = 0.5). The advantage of this form is that
both the X and Y updates are a proximal operation. It turns out that the proximal operator for g (i.e.
the X (l+1) update) actually has a simple solution [18] that can be computed by taking an eigenvalue
decomposition QT ?Q = ?(Y ? U ) ? C, where ? = diag(?1 , . . . , ?n ) and updating the eigenvalues
using the formula
p
?i + ?2i + 4?
?0i :=
2?
to give X = QT ?0 Q. The stopping criterion we used was ||X (l+1) ? Y (l+1) || < and
||Y (l+1) ? Y (l) || < . In practice the ADMM method is one of the fastest methods for L1 regularized covariance selection. Scheinbert et al. [18] show that convergence is guaranteed if additional
cone restrictions are placed on the minimization with respect to X, and small enough step sizes are
used. For our degree prior regularizer, the difficultly is in computing the proximal operator for ?, as
the rest of the algorithm is identical to that presented in Boyd et al. [6]. We now show how we solve
the problem of computing the proximal operator for ?.
3.2
Proximal operator using dual decomposition
Here we describe the optimisation algorithm that we effectively use for computing the proximal
operator. The regularizer ? has a quite complicated structure due to the interplay between the
terms involving the two end points for each edge. We can decouple these terms using the dual
decomposition technique, by writing the proximal operation for a given Z = Y ? U as:
n n?1
1
?XX
minimizeX =
(h(k + 1) ? h(k)) Xi,(k) + ||X ? Z||22
? i
2
k
s.t.
X = XT .
4
The only difference so far is that we have made the symmetry constraint explicit. Taking the dual
gives a formulation where the upper and lower triangle are treated as separate variables. The dual
variable matrix V corresponds to the Lagrange multipliers of the symmetry constraint, which for
notational convenience we store in an anti-symmetric matrix. The dual decomposition method is
given in Algorithm 1.
Algorithm 1 Dual decomposition main
input: matrix Z, constants ?, ?
input: step-size 0 < ? < 1
initialize: X = Z
initialize: V = 0n
repeat
for l = 0 until n ? 1 do
Xl? = solveSubproblem(Zl? , Vl? ) # Algorithm 2
end for
V = V + ?(X ? X T )
until ||X ? X T || < 10?6
X = 21 (X + X T ) # symmetrize
round: any |Xij | < 10?15 to 0
return X
We use the notation Xi? to denote the ith row of X. Since this is a dual method, the primal variables
X are not feasible (i.e. symmetric) until convergence. Essentially we have decomposed the original
problem, so that now we only need to solve the proximal operation for each node in isolation, namely
the subproblems:
n?1
?X
(l)
(l+1)
(h(k + 1) ? h(k)) x(k) + ||x ? Zi? + Vi? ||22 .
(3)
?i. Xi?
= arg min
x
?
k
Note that the dual variable has been integrated into the quadratic term by completing the square.
As the diagonal elements of X are not included in the sort ordering, they will be minimized by
Xii = Zii , for all i. Each subproblem is strongly convex as they consist of convex terms plus a
positive quadratic term. This implies that the dual problem is differentiable (as the subdifferential
contains only one subgradient), hence the V update is actually gradient ascent. Since a fixed step
size is used, and the dual is Lipschitz continuous, for sufficiently small step-size convergence is
guaranteed. In practice we used ? = 0.9 for all our tests.
This dual decomposition subproblem can also be interpreted as just a step within the ADMM framework. If applied in a standard way, only one dual variable update would be performed before another
expensive eigenvalue decomposition step. Since each iteration of the dual decomposition is much
faster than the eigenvalue decomposition, it makes more sense to treat it as a separate problem as
we propose here. It also ensures that the eigenvalue decomposition is only performed on symmetric
matrices.
Each subproblem in our decomposition is still a non-trivial problem. They do have a closed form
solution, involving a sort and several passes over the node?s edges, as described in Algorithm 2.
Proposition 3. Algorithm 2 solves the subproblem in equation 3.
Proof: See Appendix 1 in the supplementary material. The main subtlety is the grouping together
of elements induced at the non-differentiable points. If multiple edges connected to the same node
have the same absolute value, their subdifferential becomes the same, and they behave as a single
point whose weight is the average. To handle this grouping, we use a disjoint-set data-structure,
where each xj is either in a singleton set, or grouped in a set with other elements, whose absolute
value is the same.
4
Alternative degree priors
Under the restrictions on h detailed in Proposition 1, several other choices seem reasonable. The
scale free prior can be smoothed somewhat, by the addition of a linear term, giving
h,? (i) = log(i + ) + ?i,
5
Algorithm 2 Dual decomposition subproblem (solveSubproblem)
input: vectors z, v
initialize: Disjoint-set datastructure with set membership function ?
w = z ? v # w gives the sort order
u = 0n
build: sorted-to-original position function ? under descending absolute value order of w, excluding the
diagonal
for k = 0 until n ? 1 do
j = ?(k)
uj = |wj | ? ?? (h(k + 1) ? h(k))
?(j).value = uj
r=k
while r > 1 and ?(?(r)).value ? ?(?(r ? 1)).value do
join: the sets containing ?(r)
P and ?(r ? 1)
1
?(?(r)).value = |?(?(r))|
i??(?(r)) ui
set: r to the first element of ?(?(r)) by the sort ordering
end while
end for
for i = 1 to N do
xi = ?(i).value
if xi < 0 then
xj = 0 # negative values imply shrinkage to 0
end if
if wi < 0 then
xj = ?xj # Correct orthant
end if
end for
return x
where ? controls the strength of the smoothing. A slower diminishing choice would be a square-root
function such as
1
h? (i) = (i + 1) 2 ? 1 + ?i.
This requires the linear term in order to correspond to a normalizable prior.
Ideally we would choose h so that the expected degree distribution under the ERG model matches
the particular form we wish to encourage. Finding such a h for a particular graph size and degree
distribution amounts to maximum likelihood parameter learning, which for ERG models is a hard
learning problem. The most common approach is to use sampling based inference. Approaches
based on Markov chain Monte Carlo techniques have been applied widely to ERG models [19] and
are therefore applicable to our model.
5
Related Work
The covariance selection problem has recently been addressed by Liu and Ihler [14] using
reweighted L1 regularization. They minimize the following objective:
X
X
f (X) = hX, Ci ? log det X + ?
log (kX?v k + ) + ?
|Xvv | .
v?V
v
The regularizer is split into an off diagonal term which is designed to encourage sparsity in the edge
parameters, and a more traditional diagonal term. Essentially they use kX?v k as the continuous
counterpart of node v?s degree. The biggest difficulty with this objective is the log term, which
makes f highly non-convex. This can be contrasted to our approach, where we start with essentially
the same combinatorial prior, but we use an alternative, convex relaxation.
The reweighted L1 [7] aspect refers to the method of optimization applied. A double loop method is
used, in the same class as EM methods and difference of convex programming, where each L1 inner
problem gives a monotonically improving lower bound on the true solution.
6
1.00
0.95
0.95
0.90
0.90
True Positives
True Positives
1.00
0.85
0.80
L1
Reweighted L1
Submodular log
Submodular root
0.75
0.70
0.00
0.05
0.10
0.15
False Positives
0.20
0.85
0.80
L1
Reweighted L1
Submodular log
Submodular root
0.75
0.70
0.00
0.25
0.05
0.10
0.15
False Positives
0.20
0.25
Figure 1: ROC curves for BA model (left) and fixed degree distribution model (right)
Figure 2: Reconstruction of a gene association network using L1 (left), submodular relaxation (middle), and
reweighted L1 (right) methods
6
Experiments
Reconstruction of synthetic networks. We performed a comparison against the reweighted L1
method of Liu and Ihler [14], and a standard L1 regularized method, both implemented using
ADMM for optimization. Although Liu and Ihler [14] use the glasso [10] method for the inner
loop, ADMM will give identical results, and is usually faster [18]. Graphs with 60 nodes were generated using both the Barabasi-Albert model [4] and a predefined degree distribution model sampled
using the method from Bayati et al. [5] implemented in the NetworkX software package. Both methods generate scale-free graphs; the BA model exhibits a scale parameter of 3.0, whereas we fixed
the scale parameter at 2.0 for the other model. To define a valid GaussianP
model, edge weights of
Xij = ?0.2 were assigned, and the node weights were set at Xii = 0.5 ? i6=j Xij so as to make
the resulting precision matrix diagonally dominant. The resulting Gaussian graphical model was
sampled 500 times. The covariance matrix of these samples was formed, then normalized to have
diagonal uniformly 1.0. We tested with the two h sequences described in section 4. The parameters for the degree weight sequences were chosen by grid search on random instances separate from
those we tested on. The resulting ROC curves for the Hamming reconstruction loss are shown in
Figure 1. Results were averaged over 30 randomly generated graphs for each each figure.
We can see from the plots that our method with the square-root weighting presents results superior
to those from Liu and Ihler [14] for these datasets. This is encouraging particularly since our formulation is convex while the one from Liu and Ihler [14] isn?t. Interestingly, the log based weights
give very similar but not identical results to the reweighting scheme which also uses a log term. The
only case where it gives inferior reconstructions is when it is forced to give a sparser reconstruction
than the original graph.
Reconstruction of a gene activation network. A common application of sparse covariance selection is the estimation of gene association networks from experimental data. A covariance matrix of
gene co-activations from a number of independent micro-array experiments is typically formed, on
which a number of methods, including sparse covariance selection, can be applied. Sparse estimation is key for a consistent reconstruction due to the small number of experiments performed. Many
biological networks are conjectured to be scale-free, and additionally ERG modelling techniques are
known to produce good results on biological networks [16]. So we consider micro-array datasets a
natural test-bed for our method. We ran our method and the L1 reconstruction method on the first
7
500 genes from the GDS1429 dataset (http://www.ncbi.nlm.nih.gov/gds/1429), which contains 69
samples for 8565 genes. The parameters for both methods were tuned to produce a network with
near to 50 edges for visualization purposes. The major connected component for each is shown in
Figure 2.
While these networks are too small for valid statistical analysis of the degree distribution, the submodular relaxation method produces a network with structure that is commonly seen in scale free
networks. The star subgraph centered around gene 60 is more clearly defined in the submodular
relaxation reconstruction, and the tight cluster of genes in the right is less clustered in the L1 reconstruction. The reweighted L1 method produced a quite different reconstruction, with greater
clustering.
Runtime comparison: different proximal operator methods. We performed a comparison against
10
two other methods for computing the proximal operator: subgradient descent and the minimum norm point
10
(MNP) algorithm. The MNP algorithm is a submodu10
lar minimization method that can be adapted for com10
puting the proximal operator [2]. We took the input pa10
rameters from the last invocation of the proximal oper10
ator in the BA test, at a prior strength of 0.7. We then
plotted the convergence rate of each of the methods,
10
Dual decomp
Subgradient
shown in Figure 3. As the tests are on randomly gen10
MNP
erated graphs, we present only a representative exam10
0
20
40
60
80
100
ple. It is clear from this and similar tests that we perIteration
formed that the subgradient descent method converges
too slowly to be of practical applicability for this prob- Figure 3: Comparison of proximal operators
lem. Subgradient methods can be a good choice when
only a low accuracy solution is required; for convergence of ADMM the error in the proximal operator needs to be smaller than what can be obtained by the subgradient method. The MNP method also
converges slowly for this problem, however it achieves a low but usable accuracy quickly enough
that it could be used in practice. The dual decomposition method achieves a much better rate of
convergence, converging quickly enough to be of use even for strong accuracy requirements.
0
Distance from solution
-1
-2
-3
-4
-5
-6
-7
-8
The time for individual iterations of each of the methods was 0.65ms for subgradient descent, 0.82ms
for dual decomposition and 15ms for the MNP method. The speed difference is small between a
subgradient iteration and a dual decomposition iteration as both are dominated by the cost of a sort
operation. The cost of a MNP iteration is dominated by two least squares solves, whose running
time in the worst case is proportional to the square of the current iteration number. Overall, it is
clear that our dual decomposition method is significantly more efficient.
Runtime comparison: submodular relaxation against other approaches. The running time of
the three methods we tested is highly dependent on implementation details, so the following speed
comparison should be taken as a rough guide. For a sparse reconstruction of a BA model graph with
100 vertices and 200 edges, the average running time per 10?4 error reconstruction over 10 random
graphs was 16 seconds for the reweighted L1 method and 5.0 seconds for the submodular relaxation
method. This accuracy level was chosen so that the active edge set for both methods had stabilized
between iterations. For comparison, the standard L1 method was significantly faster, taking only
0.72 seconds on average.
Conclusion
We have presented a new prior for graph reconstruction, which enforces the recovery of scale-free
networks. This prior falls within the growing class of structured sparsity methods. Unlike previous
approaches to regularizing the degree distribution, our proposed prior is convex, making training
tractable and convergence predictable. Our method can be directly applied in contexts where sparse
covariance selection is currently used, where it may improve the reconstruction quality.
Acknowledgements
NICTA is funded by the Australian Government as represented by the Department of Broadband,
Communications and the Digital Economy and the Australian Research Council through the ICT
Centre of Excellence program.
8
References
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INRIA, 2010.
[2] Francis Bach. Structured sparsity-inducing norms through submodular functions. NIPS, 2010.
[3] Francis Bach, Rodolphe Jenatton, Julien Mairal, and Guillaume Obozinski. Optimization with sparsityinducing penalties. Foundations and Trends in Machine Learning, 2012.
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3,909 | 4,539 | A Geometric take on Metric Learning
S?ren Hauberg
MPI for Intelligent Systems
T?ubingen, Germany
Oren Freifeld
Brown University
Providence, US
Michael J. Black
MPI for Intelligent Systems
T?ubingen, Germany
[email protected]
[email protected]
[email protected]
Abstract
Multi-metric learning techniques learn local metric tensors in different parts of a
feature space. With such an approach, even simple classifiers can be competitive
with the state-of-the-art because the distance measure locally adapts to the structure of the data. The learned distance measure is, however, non-metric, which has
prevented multi-metric learning from generalizing to tasks such as dimensionality reduction and regression in a principled way. We prove that, with appropriate
changes, multi-metric learning corresponds to learning the structure of a Riemannian manifold. We then show that this structure gives us a principled way to
perform dimensionality reduction and regression according to the learned metrics.
Algorithmically, we provide the first practical algorithm for computing geodesics
according to the learned metrics, as well as algorithms for computing exponential
and logarithmic maps on the Riemannian manifold. Together, these tools let many
Euclidean algorithms take advantage of multi-metric learning. We illustrate the
approach on regression and dimensionality reduction tasks that involve predicting
measurements of the human body from shape data.
1
Learning and Computing Distances
Statistics relies on measuring distances. When the Euclidean metric is insufficient, as is the case in
many real problems, standard methods break down. This is a key motivation behind metric learning,
which strives to learn good distance measures from data. In the most simple scenarios a single
metric tensor is learned, but in recent years, several methods have proposed learning multiple metric
tensors, such that different distance measures are applied in different parts of the feature space. This
has proven to be a very powerful approach for classification tasks [1, 2], but the approach has not
generalized to other tasks. Here we consider the generalization of Principal Component Analysis
(PCA) and linear regression; see Fig. 1 for an illustration of our approach. The main problem with
generalizing multi-metric learning is that it is based on assumptions that make the feature space both
non-smooth and non-metric. Specifically, it is often assumed that straight lines form geodesic curves
and that the metric tensor stays constant along these lines. These assumptions are made because it
is believed that computing the actual geodesics is intractable, requiring a discretization of the entire
feature space [3]. We solve these problems by smoothing the transitions between different metric
tensors, which ensures a metric space where geodesics can be computed.
In this paper, we consider the scenario where the metric tensor at a given point in feature space is
defined as the weighted average of a set of learned metric tensors. In this model, we prove that the
feature space becomes a chart for a Riemannian manifold. This ensures a metric feature space, i.e.
dist(x, y) = 0 ? x = y ,
dist(x, y) = dist(y, x)
(symmetry),
(1)
dist(x, z) ? dist(x, y) + dist(y, z)
(triangle inequality).
To compute statistics according to the learned metric, we need to be able to compute distances,
which implies that we need to compute geodesics. Based on the observation that geodesics are
1
(a) Local Metrics & Geodesics
(b) Tangent Space Representation
(c) First Principal Geodesic
Figure 1: Illustration of Principal Geodesic Analysis. (a) Geodesics are computed between the
mean and each data point. (b) Data is mapped to the Euclidean tangent space and the first principal
component is computed. (c) The principal component is mapped back to the feature space.
smooth curves in Riemannian spaces, we derive an algorithm for computing geodesics that only
requires a discretization of the geodesic rather than the entire feature space. Furthermore, we show
how to compute the exponential and logarithmic maps of the manifold. With this we can map any
point back and forth between a Euclidean tangent space and the manifold. This gives us a general
strategy for incorporating the learned metric tensors in many Euclidean algorithms: map the data to
the tangent of the manifold, perform the Euclidean analysis and map the results back to the manifold.
Before deriving the algorithms (Sec. 3) we set the scene by an analysis of the shortcomings of current
state-of-the-art methods (Sec. 2), which motivate our final model. The model is general and can be
used for many problems. Here we illustrate it with several challenging problems in 3D body shape
modeling and analysis (Sec. 4). All proofs can be found in the supplementary material along with
algorithmic details and further experimental results.
2
Background and Related Work
Single-metric learning learns a metric tensor, M, such that distances are measured as
dist2 (xi , xj ) = kxi ? xj k2M ? (xi ? xj )T M(xi ? xj ) ,
(2)
where M is a symmetric and positive definite D ? D matrix. Classic approaches for finding such a
metric tensor include PCA, where the metric is given by the inverse covariance matrix of the training
?1
data; and linear discriminant analysis (LDA), where the metric tensor is M = S?1
W SB SW , with Sw
and SB being the within class scatter and the between class scatter respectively [9].
A more recent approach tries to learn a metric tensor from triplets of data points (xi , xj , xk ), where
the metric should obey the constraint that dist(xi , xj ) < dist(xi , xk ). Here the constraints are often
chosen such that xi and xj belong to the same class, while xi and xk do not. Various relaxed versions of this idea have been suggested such that the metric can be learned by solving a semi-definite
or a quadratic program [1, 2, 4?8]. Among the most popular approaches is the Large Margin Nearest Neighbor (LMNN) classifier [5], which finds a linear transformation that satisfies local distance
constraints, making the approach suitable for multi-modal classes.
For many problems, a single global metric tensor is not enough, which motivates learning several
local metric tensors. The classic work by Hastie and Tibshirani [9] advocates locally learning metric
tensors according to LDA and using these as part of a kNN classifier. In a somewhat similar fashion,
Weinberger and Saul [5] cluster the training data and learn a separate metric tensor for each cluster
using LMNN. A more extreme point of view was taken by Frome et al. [1, 2], who learn a diagonal
metric tensor for every point in the training set, such that distance rankings are preserved. Similarly,
Malisiewicz and Efros [6] find a diagonal metric tensor for each training point such that the distance
to a subset of the training data from the same class is kept small.
Once a set of metric tensors {M1 , . . . , MR } has been learned, the distance dist(a, b) is measured
according to (2) where ?the nearest? metric tensor is used, i.e.
R
X
w
?r (x)
1 kx ? xr k2Mr ? kx ? xj k2Mj , ?j
P
M(x) =
Mr , where w
?r (x) =
, (3)
0 otherwise
?j (x)
jw
r=1
where x is either a or b depending on the algorithm. Note that this gives a non-metric distance
function as it is not symmetric. To derive this equation, it is necessary to assume that 1) geodesics
2
?8
?8
Assumed Geodesics
Location of Metric Tensors
Test Points
?6
?8
Actual Geodesics
Location of Metric Tensors
Test Points
?6
?4
?4
?4
?2
?2
?2
0
0
0
2
2
2
4
4
4
6
?8
6
?8
?6
?4
?2
0
(a)
2
4
6
?6
?4
?2
0
2
4
Riemannian Geodesics
Location of Metric Tensors
Test Points
?6
6
6
?8
?6
(b)
?4
?2
(c)
0
2
4
6
(d)
Figure 2: (a)?(b) An illustrative example where straight lines do not form geodesics and where
the metric tensor does not stay constant along lines; see text for details. The background color
is proportional to the trace of the metric tensor, such that light grey corresponds to regions where
paths are short (M1 ), and dark grey corresponds to regions they are long (M2 ). (c) The suggested
geometric model along with the geodesics. Again, background colour is proportional to the trace
of the metric tensor; the colour scale is the same is used in (a) and (b). (d) An illustration of the
exponential and logarithmic maps.
form straight lines, and 2) the metric tensor stays constant along these lines [3]. Both assumptions
are problematic, which we illustrate with a simple example in Fig. 2a?c.
Assume we are given two metric tensors M1 = 2I and M2 = I positioned at x1 = (2, 2)T and
x2 = (4, 4)T respectively. This gives rise to two regions in feature space in which x1 is nearest
in the first and x2 is nearest in the second, according to (3). This is illustrated in Fig. 2a. In the
same figure, we also show the assumed straight-line geodesics between selected points in space. As
can be seen, two of the lines goes through both regions, such that the assumption of constant metric
tensors along the line is violated. Hence, it would seem natural to measure the length of the line,
by adding the length of the line segments which pass through the different regions of feature space.
This was suggested by Ramanan and Baker [3] who also proposed a polynomial time algorithm for
measuring these line lengths. This gives a symmetric distance function.
Properly computing line lengths according to the local metrics is, however, not enough to ensure that
the distance function is metric. As can be seen in Fig. 2a the straight line does not form a geodesic
as a shorter path can be found by circumventing the region with the ?expensive? metric tensor M1
as illustrated in Fig. 2b. This issue makes it trivial to construct cases where the triangle inequality is
violated, which again makes the line length measure non-metric.
In summary, if we want a metric feature space, we can neither assume that geodesics are straight
lines nor that the metric tensor stays constant along such lines. In practice, good results have been
reported using (3) [1,3,5], so it seems obvious to ask: is metricity required? For kNN classifiers this
does not appear to be the case, with many successes based on dissimilarities rather than distances
[10]. We, however, want to generalize PCA and linear regression, which both seek to minimize the
reconstruction error of points projected onto a subspace. As the notion of projection is hard to define
sensibly in non-metric spaces, we consider metricity essential.
In order to build a model with a metric feature space, we change the weights in (3) to be smooth
functions. This impose a well-behaved geometric structure on the feature space, which we take
advantage of in order to perform statistical analysis according to the learned metrics. However, first
we review the basics of Riemannian geometry as this provides the theoretical foundation of our
work.
2.1
Geodesics and Riemannian Geometry
We start by defining Riemannian manifolds, which intuitively are smoothly curved spaces equipped
with an inner product. Formally, they are smooth manifolds endowed with a Riemannian metric [11]:
Definition A Riemannian metric M on a manifold M is a smoothly varying inner product
< a, b >x = aT M(x)b in the tangent space Tx M of each point x ? M .
3
Often Riemannian manifolds are represented by a chart; i.e. a parameter space for the curved surface. An example chart is the spherical coordinate system often used to represent spheres. While
such charts are often flat spaces, the curvature of the manifold arises from the smooth changes in the
metric.
On a Riemannian manifold M, the length of a smooth curve c : [0, 1] ? M is defined as the
integral of the norm of the tangent vector (interpreted as speed) along the curve:
Z 1
Z 1q
0
Length(c) =
kc (?)kM(c(?)) d? =
c0 (?)T M(c(?))c0 (?)d? ,
(4)
0
0
where c0 denotes the derivative of c and M(c(?)) is the metric tensor at c(?). A geodesic curve is
then a length-minimizing curve connecting two given points x and y, i.e.
(5)
cgeo = arg min Length(c) with c(0) = x and c(1) = y .
c
The distance between x and y is defined as the length of the geodesic.
Given a tangent vector v ? Tx M, there exists a unique geodesic cv (t) with initial velocity v at x.
The Riemannian exponential map, Expx , maps v to a point on the manifold along the geodesic cv
at t = 1. This mapping preserves distances such that dist(cv (0), cv (1)) = kvk. The inverse of
the exponential map is the Riemannian logarithmic map denoted Logx . Informally, the exponential
and logarithmic maps move points back and forth between the manifold and the tangent space while
preserving distances (see Fig. 2d for an illustration). This provides a general strategy for generalizing
many Euclidean techniques to Riemannian domains: data points are mapped to the tangent space,
where ordinary Euclidean techniques are applied and the results are mapped back to the manifold.
3
A Metric Feature Space
With the preliminaries settled we define the new model. Let C = RD denote the feature space. We
endow C with a metric tensor in every point x, which we define akin to (3),
R
X
w
?r (x)
M(x) =
wr (x)Mr , where wr (x) = PR
,
(6)
?j (x)
r=1
j=1 w
with w
?r > 0. The only difference from (3) is that we shall not restrict ourselves to binary weight
functions w
?r . We assume the metric tensors Mr have already been learned; Sec. 4 contain examples
where they have been learned using LMNN [5] and LDA [9].
From the definition of a Riemannian metric, we trivially have the following result:
Lemma 1 The space C = RD endowed with the metric tensor from (6) is a chart of a Riemannian
manifold, iff the weights wr (x) change smoothly with x.
Hence, by only considering smooth weight functions w
?r we get a well-studied geometric structure
on the feature space, which ensures us that it is metric. To illustrate the implications we return to the
example in Fig. 2. We change the weight functions from binary to squared exponentials, which gives
the feature space shown in Fig. 2c. As can be seen, the metric tensor now changes smoothly, which
also makes the geodesics smooth curves (a property we will use when computing the geodesics).
It is worth noting that Ramanan and Baker [3] also consider the idea of smoothly averaging the
metric tensor. They, however, only evaluate the metric tensor at the test point of their classifier
and then assume straight line geodesics with a constant metric tensor. Such assumptions violate the
premise of a smoothly changing metric tensor and, again, the distance measure becomes non-metric.
Lemma 1 shows that metric learning can be viewed as manifold learning. The main difference between our approach and techniques such as Isomap [12] is that, while Isomap learns an embedding
of the data points, we learn the actual manifold structure. This gives us the benefit that we can
compute geodesics as well as the exponential and logarithmic maps. These provide us with mappings back and forth between the manifold and Euclidean representation of the data, which preserve
distances as well as possible. The availability of such mappings is in stark contrast to e.g. Isomap.
In the next section we will derive a system of ordinary differential equations (ODE?s) that geodesics
in C have to satisfy, which provides us with algorithms for computing geodesics as well as exponential and logarithmic maps. With these we can generalize many Euclidean techniques.
4
3.1
Computing Geodesics, Maps and Statistics
At minima of (4) we know that the Euler-Lagrange equation must hold [11], i.e.
?L
d ?L
, where L(?, c, c0 ) = c0 (?)T M(c(?))c0 (?) .
=
?c
d? ?c0
As we have an explicit expression for the metric tensor we can compute (7) in closed form:
Theorem 2 Geodesic curves in C satisfy the following system of 2nd order ODE?s
T
1 ?vec [M(c(?))]
M(c(?))c00 (?) = ?
(c0 (?) ? c0 (?)) ,
2
?c(?)
(7)
(8)
where ? denotes the Kronecker product and vec [?] stacks the columns of a matrix into a vector [13].
Proof See supplementary material.
This result holds for any smooth weight functions w
?r . We, however, still need to compute ?vec[M]
,
?c
which depends on the specific choice of w
?r . Any smooth weighting scheme is applicable, but we
restrict ourselves to the obvious smooth generalization of (3) and use squared exponentials. From
this assumption, we get the following result
Theorem 3 For w
?r (x) = exp ? ?2 kx ? xr k2Mr the derivative of the metric tensor from (6) is
?vec [M(c)]
?
= P
?c
R
j=1
R
X
w
?j
2
w
?r vec [Mr ]
r=1
R
X
T
T
w
?j (c ? xj ) Mj ? (c ? xr ) Mr . (9)
j=1
Proof See supplementary material.
Computing Geodesics. Any geodesic curve must be a solution to (8). Hence, to compute a
geodesic between x and y, we can solve (8) subject to the constraints
c(0) = x and c(1) = y .
(10)
This is a boundary value problem, which has a smooth solution. This allows us to solve the problem numerically using a standard three-stage Lobatto IIIa formula, which provides a fourth-order
accurate C 1 ?continuous solution [14].
Ramanan and Baker [3] discuss the possibility of computing geodesics, but arrive at the conclusion
that this is intractable based on the assumption that it requires discretizing the entire feature space.
Our solution avoids discretizing the feature space by discretizing the geodesic curve instead. As this
is always one-dimensional the approach remains tractable in high-dimensional feature spaces.
Computing Logarithmic Maps. Once a geodesic c is found, it follows from the definition of the
logarithmic map, Logx (y), that it can be computed as
v = Logx (y) =
c0 (0)
Length(c) .
kc0 (0)k
(11)
In practice, we solve (8) by rewriting it as a system of first order ODE?s, such that we compute both
c and c0 simultaneously (see supplementary material for details).
Computing Exponential Maps. Given a starting point x on the manifold and a vector v in the
tangent space, the exponential map, Expx (v), finds the unique geodesic starting at x with initial
velocity v. As the geodesic must fulfill (8), we can compute the exponential map by solving this
system of ODE?s with the initial conditions
c(0) = x and c0 (0) = v .
(12)
This initial value problem has a unique solution, which we find numerically using a standard RungeKutta scheme [15].
5
3.1.1
Generalizing PCA and Regression
At this stage, we know that the feature space is Riemannian and we know how to compute geodesics
and exponential and logarithmic maps. We now seek to generalize PCA and linear regression,
which becomes straightforward since solutions are available in Riemannian spaces [16, 17]. These
generalizations can be summarized as mapping the data to the tangent space at the mean, performing
standard Euclidean analysis in the tangent and mapping the results back.
The first step is to compute the mean value on the manifold, which is defined as the point that
minimizes the sum-of-squares distances to the data points. Pennec [18] provides an efficient gradient
descent approach for computing this point, which we also summarize in the supplementary material.
The empirical covariance of a set of points is defined as the ordinary Euclidean covariance in the
tangent space at the mean value [18]. With this in mind, it is not surprising that the principal
components of a dataset have been generalized as the geodesics starting at the mean with initial
velocity corresponding to the eigenvectors of the covariance [16],
?vd (t) = Exp? (tvd ) ,
(13)
th
where vd denotes the d eigenvector of the covariance. This approach is called Principal Geodesic
Analysis (PGA), and the geodesic curve ?vd is called the principal geodesic. An illustration of the
approach can be seen in Fig. 1 and more algorithmic details are in the supplementary material.
Linear regression has been generalized in a similar way [17] by performing regression in the tangent
of the mean and mapping the resulting line back to the manifold using the exponential map.
The idea of working in the tangent space is both efficient and convenient, but comes with an element
of approximation as the logarithmic map is only guarantied to preserve distances to the origin of the
tangent and not between all pairs of data points. Practical experience, however, indicates that this is
a good tradeoff; see [19] for a more in-depth discussion of when the approximation is suitable.
4
Experiments
To illustrate the framework1 we consider an example in human body analysis, and then we analyze
the scalability of the approach. But first, to build intuition, Fig. 3a show synthetically generated
data samples from two classes. We sample random points xr and learn a local LDA metric [9] by
considering all data points within a radius; this locally pushes the two classes apart. We combine the
local metrics using (6) and Fig. 3b show the data in the tangent space of the resulting manifold. As
can be seen the two classes are now globally further apart, which shows the effect of local metrics.
4.1
Human Body Shape
We consider a regression example concerning human body shape analysis. We study 986 female
body laser scans from the CAESAR [20] data set; each shape is represented using the leading 35
principal components of the data learned using a SCAPE-like model [21, 22]. Each shape is associated with anthropometric measurements such as body height, shoe size, etc. We show results for
shoulder to wrist distance and shoulder breadth, but results for more measurements are in the supplementary material. To predict the measurements from shape coefficients, we learn local metrics
and perform linear regression according to these. As a further experiment, we use PGA to reduce
the dimensionality of the shape coefficients according to the local metrics, and measure the quality
of the reduction by performing linear regression to predict the measurements. As a baseline we use
the corresponding Euclidean techniques.
To learn the local metric we do the following. First we whiten the data such that the variance
captured by PGA will only be due to the change of metric; this allows easy visualization of the
impact of the learned metrics. We then cluster the body shapes into equal-sized clusters according
to the measurement and learn a LMNN metric for each cluster [5], which we associate with the
mean of each class. These push the clusters apart, which introduces variance along the directions
where the measurement changes. From this we construct a Riemannian manifold according to (6),
1
Our software implementation for computing geodesics and performing manifold statistics is available at
http://ps.is.tue.mpg.de/project/Smooth Metric Learning
6
30
Euclidean Model
Riemannian Model
24
20
18
16
20
15
10
5
14
12
0
(a)
25
22
Running Time (sec.)
Average Prediction Error
26
10
(b)
20
Dimensionality
0
0
30
50
(c)
100
Dimensionality
150
(d)
4
3
3
2
2
1
1
0
0
?1
?1
?2
?2
?3
?3
?4
?4
?3
?2
?1
0
1
2
3
Shoulder breadth
4
3
25
Euclidean Model
Riemannian Model
Prediction Error
4
20
15
10
0
?4
?5
0
5
10
5
4
17
3
16
15
20
Dimensionality
25
30
35
Euclidean Model
Riemannian Model
2
15
2
1
1
Prediction Error
Shoulder to wrist distance
Figure 3: Left panels: Synthetic data. (a) Samples from two classes along with illustratively
sampled metric tensors from (6). (b) The data represented in the tangent of a manifold constructed
from local LDA metrics learned at random positions. Right panels: Real data. (c) Average error
of linearly predicted body measurements (mm). (d) Running time (sec) of the geodesic computation
as a function of dimensionality.
0
0
?1
?2
?1
?3
14
13
12
11
?2
?4
?3
?4
?4
10
?5
?3
?2
?1
0
1
Euclidean PCA
2
3
?6
?4
9
0
?2
0
2
4
Tangent Space PCA (PGA)
6
5
10
15
20
Dimensionality
25
30
35
Regression Error
Figure 4: Left: body shape data in the first two principal components according to the Euclidean
metric. Point color indicates cluster membership. Center: As on the left, but according to the
Riemannian model. Right: regression error as a function of the dimensionality of the shape space;
again the Euclidean metric and the Riemannian metric are compared.
compute the mean value on the manifold, map the data to the tangent space at the mean and perform
linear regression in the tangent space.
As a first visualization we plot the data expressed in the leading two dimensions of PGA in Fig. 4; as
can be seen the learned metrics provide principal geodesics, which are more strongly related with the
measurements than the Euclidean model. In order to predict the measurements from the body shape,
we perform linear regression, both directly in the shape space according to the Euclidean metric
and in the tangent space of the manifold corresponding to the learned metrics (using the logarithmic
map from (11)). We measure the prediction error using leave-one-out cross-validation. To further
illustrate the power of the PGA model, we repeat this experiment for different dimensionalities of
the data. The results are plotted in Fig. 4, showing that regression according to the learned metrics
outperforms the Euclidean model.
To verify that the learned metrics improve accuracy, we average the prediction errors over all millimeter measurements. The result in Fig. 3c shows that much can be gained in lower dimensions by
using the local metrics.
To provide visual insights into the behavior of the learned metrics, we uniformly sample body shape
along the first principal geodesic (in the range ?7 times the standard deviation) according to the
different metrics. The results are available as a movie in the supplementary material, but are also
shown in Fig. 5. As can be seen, the learned metrics pick up intuitive relationships between body
shape and the measurements, e.g. shoulder to wrist distance is related to overall body size, while
shoulder breadth is related to body weight.
7
Shoulder to wrist distance
Shoulder breadth
Figure 5: Shapes corresponding to the mean (center) and ?7 times the standard deviations along the
principal geodesics (left and right). Movies are available in the supplementary material.
4.2
Scalability
The human body data set is small enough (986 samples in 35 dimensions) that computing a geodesic
only takes a few seconds. To show that the current unoptimized Matlab implementation can handle
somewhat larger datasets, we briefly consider a dimensionality reduction task on the classic MNIST
handwritten digit data set. We use the preprocessed data available with [3] where the original 28?28
gray scale images were deskewed and projected onto their leading 164 Euclidean principal components (which captures 95% of the variance in the original data).
We learn one diagonal LMNN metric per class, which we associate with the mean of the class. From
this we construct a Riemannian manifold from (6), compute the mean value on the manifold and
compute geodesics between the mean and each data point; this is the computationally expensive part
of performing PGA. Fig. 3d plots the average running time (sec) for the computation of geodesics as
a function of the dimensionality of the training data. A geodesic can be computed in 100 dimensions
in approximately 5 sec., whereas in 150 dimensions it takes about 30 sec.
In this experiment, we train a PGA model on 60,000 data points, and test a nearest neighbor classifier
in the tangent space as we decrease the dimensionality of the model. Compared to a Euclidean
model, this gives a modest improvement in classification accuracy of 2.3 percent, when averaged
across different dimensionalities. Plots of the results can be found in the supplementary material.
5
Discussion
This work shows that multi-metric learning techniques are indeed applicable outside the realm of
kNN classifiers. The idea of defining the metric tensor at any given point as the weighted average of
a finite set of learned metrics is quite natural from a modeling point of view, which is also validated
by the Riemannian structure of the resulting space. This opens both a theoretical and a practical
toolbox for analyzing and developing algorithms that use local metric tensors. Specifically, we
show how to use local metric tensors for both regression and dimensionality reduction tasks.
Others have attempted to solve non-classification problems using local metrics, but we feel that our
approach is the first to have a solid theoretical backing. For example, Hastie and Tibshirani [9] use
local LDA metrics for dimensionality reduction by averaging the local metrics and using the resulting metric as part of a Euclidean PCA, which essentially is a linear approach. Another approach
was suggested by Hong et al. [23] who simply compute the principal components according to each
metric separately, such that one low dimensional model is learned per metric.
The suggested approach is, however, not difficulty-free in its current implementation. Currently, we
are using off-the-shelf numerical solvers for computing geodesics, which can be computationally
demanding. While we managed to analyze medium-sized datasets, we believe that the run-time can
be drastically improved by developing specialized numerical solvers.
In the experiments, we learned local metrics using techniques specialized for classification tasks as
this is all the current literature provides. We expect improvements by learning the metrics specifically for regression and dimensionality reduction, but doing so is currently an open problem.
Acknowledgments: S?ren Hauberg is supported in part by the Villum Foundation, and Oren Freifeld is
supported in part by NIH-NINDS EUREKA (R01-NS066311).
8
References
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[3] Deva Ramanan and Simon Baker. Local distance functions: A taxonomy, new algorithms, and an evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(4):794?806, 2011.
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Econometrics. John Wiley & Sons, 2007.
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PSE. ACM Transactions on Mathematical Software, 27(3):299?316, 2001.
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[16] P. Thomas Fletcher, Conglin Lu, Stephen M. Pizer, and Sarang Joshi. Principal Geodesic Analysis for the
study of Nonlinear Statistics of Shape. IEEE Transactions on Medical Imaging, 23(8):995?1005, 2004.
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1987.
[18] Xavier Pennec. Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements. In Proceedings of Nonlinear Signal and Image Processing, pages 194?198, 1999.
[19] Stefan Sommer, Franc?ois Lauze, S?ren Hauberg, and Mads Nielsen. Manifold valued statistics, exact
principal geodesic analysis and the effect of linear approximations. In European Conference on Computer
Vision (ECCV), pages 43?56, 2010.
[20] Kathleen M. Robinette, Hein Daanen, and Eric Paquet. The CAESAR project: a 3-D surface anthropometry survey. In 3-D Digital Imaging and Modeling, pages 380?386, 1999.
[21] Dragomir Anguelov, Praveen Srinivasan, Daphne Koller, Sebastian Thrun, Jim Rodgers, and James Davis.
Scape: shape completion and animation of people. ACM Transactions on Graphics, 24(3):408?416, 2005.
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9
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3,910 | 454 | Gradient Descent: Second-Order Momentum
and Saturating Error
Barak Pearlmutter
Department of Psychology
P.O. Box llA Yale Station
New Haven, CT 06520-7447
[email protected]
Abstract
=
Batch gradient descent, ~w(t)
-7JdE/dw(t) , conver~es to a minimum
of quadratic form with a time constant no better than '4Amax/ Amin where
Amin and Amax are the minimum and maximum eigenvalues of the Hessian
matrix of E with respect to w. It was recently shown that adding a
momentum term ~w(t) = -7JdE/dw(t) + Q'~w(t - 1) improves this to
~ VAmax/ Amin, although only in the batch case. Here we show that secondorder momentum, ~w(t) -7JdE/dw(t) + Q'~w(t -1) + (3~w(t - 2), can
lower this no further. We then regard gradient descent with momentum
as a dynamic system and explore a non quadratic error surface, showing
that saturation of the error accounts for a variety of effects observed in
simulations and justifies some popular heuristics.
=
1
INTRODUCTION
Gradient descent is the bread-and-butter optimization technique in neural networks.
Some people build special purpose hardware to accelerate gradient descent optimization of backpropagation networks. Understanding the dynamics of gradient descent
on such surfaces is therefore of great practical value.
Here we briefly review the known results in the convergence of batch gradient descent; show that second-order momentum does not give any speedup; simulate a
real network and observe some effect not predicted by theory; and account for these
effects by analyzing gradient descent with momentum on a saturating error surface.
887
888
Pearl mutter
1.1
SIMPLE GRADIENT DESCENT
First, let us review the bounds on the convergence rate of simple gradient descent
without momentum to a minimum of quadratic form [11,1]. Let w* be the minimum
of E, the error, H = d2 E/dw 2 (w*), and Ai, vi be the eigenvalues and eigenvectors
of H. The weight change equation
.6.w =
(where .6.f(t)
= f(t + 1) -
dE
dw
(1)
-T}-
f(t? is limited by
o < T} < 2/ Amax
(2)
We can substitute T} = 2/ Amax into the weight change equation to obtain convergence that tightly bounds any achievable in practice, getting a time constant of
convergence of -1/log(1 - 2s) = (2s)-1 + 0(1), or
E - E*
~
exp(-4st)
(3)
where we use s = Amin/ Amax for the inverse eigenvalues spread of H and
"asymptotically converges to zero more slowly than."
1.2
~
is read
FIRST-ORDER MOMENTUM
Sometimes a momentum term is used, the weight update (1) being modified to
incorporate a momentum term a < 1 [5, equation 16],
.6.w(t)
dE
= -T} dw (t) + a.6.w(t -
1).
(4)
The Momentum LMS algorithm, MLMS, has been analyzed by Shynk and Roy [6],
who have shown that the momentum term can not speed convergence in the online,
or stochastic gradient, case. In the batch case, which we consider here, Tugay and
Tanik [9] have shown that momentum is stable when
a < 1 and 0 < TJ < 2(a+ 1)/Amax
(5)
which speeds convergence to
E - E* ~ exp(-(4VS + O(s)) t)
(6)
by
*
a =
2
2 - 4Js(1 - s)
(1- 2s)2
-1 = 1- 4VS+ 0 (s),
T}*=2(a*+1)/Amax.
(7)
SECOND-ORDER MOMENTUM
The time constant of asymptotic convergence can be changed from O(Amax/ Amin) to
O( JA max / Amin) by going from a first-order system, (1), to a second-order system,
(4). Making a physical analogy, the first-order system corresponds to a circuit with
Gradient Descent: Second Order Momentum and Saturating Error
....
Figure 1: Second-order momentum converges if 7]Amax is less than the value plotted
as "eta," as a function of a and (3. The region of convergence is bounded by four
smooth surfaces: three planes and one hyperbola. One of the planes is parallel
to the 7] axis, even though the sampling of the plotting program makes it appear
slightly sloped. Another is at 7] = 0 and thus hidden. The peak is at 4.
a resistor, and the second-order system adds a capacitor to make an RC oscillator.
One might ask whether further gains can be had by going to a third-order system,
dE
~w(t)
-7] dw + a~w(t - 1) + (3~w(t - 2) .
(8)
=
For convergence, all the eigenvalues of the matrix
Mi =
(~6
- {3 -a + {3
~)
1 - 7]Ai + a
in (c,(t - 1) Ci(t) Ci(t + l))T ~ M,(Ci(t - 2) c,(t - 1) ci(t)f must have absolute
value less than or equal to 1, which occurs precisely when
-1
~
(3
~
~
7]Ad2 - (1 - (3)
<
7]
a
~
~
o
1
4({3 + 1) / Ai
{37]Ai/2 + (1 - (3).
For {3 ~ 0 this is most restrictive for Amax, but for {3 > 0 Amin also comes into play.
Taking the limit as Amin -.0, this gives convergence conditions for gradient descent
with second-order momentum of
-1<
{3-1~
when a
~
~1-{3
3{3 + 1 :
0<
when a
{3
a
(9)
7]
~
2
-(1+a-{3)
Amax
7]
~
f
2: 3{3+ 1:
0<
+ ~(a + (3 - 1)
max
889
890
Pearlmutter
a region shown in figure 1.
Fastest convergence for Amin within this region lies along the ridge a = 3{3 + 1,
T}
2(1 + a - {3)/ Amax. Unfortunately, although convergence is slightly faster than
with first-order momentum, the relative advantage tends to zero as 8 --+- 0, giving
negligible speedup when Amax ~ Amin. For small 8, the optimal settings of the
parameters are
=
1-
9
4: vIS + 0(8)
3
--vIS
+ 0(8)
4
4(1 - vIS) + 0(8)
(10)
where a? is as in (7).
3
SIMULATIONS
We constructed a standard three layer backpropagation network with 10 input units,
3 sigmoidal hidden units, and 10 sigmoidal output units. 15 associations between
random 10 bit binary input and output vectors were constructed, and the weights
were initialized to uniformly chosen random values between -1 and +1. Training
was performed with a square error measure, batch weight updates, targets of 0 and
1, and a weight decay coefficient of 0.01.
=
To get past the initial transients, the network was run at T}
0.45, a = 0 for
150 epochs, and at T} 0.3, a
0.9 for another 200 epochs. The weights were then
saved, and the network run for 200 epochs for T} ranging from 0 to 0.5 and a ranging
from 0 to 1 from that starting point.
=
=
Figure 3 shows that the region of convergence has the shape predicted by theory.
Calculation of the eigenvalues of d 2 E / dw 2 confirms that the location of the boundary is correctly predicted. Figure 2 shows that momentum speeded convergence by
the amount predicted by theory. Figure 3 shows that the parameter setting that
give the most rapid convergence in practice are the settings predicted by theory.
However, within the region that does not converge to the minimum, there appear
to be two regimes: one that is characterized by apparently chaotic fluctuations
of the error, and one which slopes up gradually from the global minimum. Since
this phenomenon is so atypical of a quadratic minimum in a linear system, which
either converges or diverges, and this phenomenon seems important in practice, we
decided to investigate a simple system to see if this behavior could be replicated
and understood, which is the subject of the next section.
4
GRADIENT DESCENT WITH SATURATING ERROR
The analysis of the sections above may be objected to on the grounds that it assumes
the minimum to have quadratic form and then performs an analysis in the neighborhood of that minimum, which is equivalent to analyzing a linear unit. Surely
our nonlinear backpropagation networks are richer than that.
Gradient Descent: Second Order Momentum and Saturating Error
0.1195
! 0.690~~11~---------__---:l
0.611:1
O . 680L......~~-----.J'---'-~~------.l~~~-..J.~~~..........
350
400
450
epoch
500
Figure 2: Error plotted as a function of time for two settings of the learning parameters, both determined empirically: the one that minimized the error the most, and
the one with a = 0 that minimized the error the most. There exists a less aggressive
setting of the parameters that converges nearly as fast as the quickly converging
curve but does not oscillate.
A clue that this might be the case was shown in figure 3. The region where the
system converges to the minimum is of the expected shape, but rather than simply
diverging outside of this region, as would a linear system, more complex phenomena
are observed, in particular a sloping region .
Acting on the hypothesis that this region is caused by Amax being maximal at
the minimum, and gradually decreasing away from it (it must decrease to zero in
the limit, since the hidden units saturate and the squared error is thus bounded)
we decided to perform a dynamic systems analysis of the convergence of gradient
descent on a one dimensional nonquadratic error surface. We chose
1
E=l-l
+w 2
(11)
which is shown in figure 4, as this results in a bounded E.
Letting
f( ) =
W
W
_ E'( ) _ w(l - 2T} + 2w 2 + w 4 )
T}
W (1 + w 2 )2
(12)
be our transfer function, a local analysis at the minimum gives Amax = E"(O) = 2
which limits convergence to T} < 1. Since the gradient towards the minimum is
always less than predicted by a second-order series at the minimum, such T} are in
fact globally convergent. As T} passes 1 the fixedpoint bifurcates into the limit cycle
w
= ?j.,;ry- 1,
=
(13)
which remains stable until T} --+- 16/9 1.77777 ... , at which point the single symmetric binary limit cycle splits into two asymmetric limit cycles, each still of period
two. These in turn remain stable until T} --+- 2.0732261475-, at which point repeated
period doubling to chaos occurs. This progression is shown in figure 7.
891
892
Pearlmutter
"i 0.40
!!
~
0.30
'c
5 0.20
.!l
-; 0.10
O.OOC=:==""""",_--=~
0.0
0.2
Q
0.4
0.6 0.8
(mom.... lum)
' .0
Figure 3: (Left) the error at epoch 550 as a function of the learning regime. Shading
is based on the height, but most of the vertical scale is devoted to nonconvergent networks in order to show the mysterious non convergent sloping region. The minimum,
corresponding to the most darkly shaded point, is on the plateau of convergence
at the location predicted by the theory. (Center) the region in which the network
is convergent, as measured by a strictly monotonically decreasing error. Learning
parameter settings for which the error was strictly decreasing have a low value while
those for which it was not have a high one. The lip at 7] 0 has a value of 0, given
where the error did not change. The rim at a = 1 corresponds to damped oscillation
caused by 7] > 4aA/(1 - a)2. (Right) contour plot of the convergent plateau shows
that the regions of equal error have linear boundaries in the nonoscillatory region
in the center, as predicted by theory.
=
As usual in a bifurcation, w rises sharply as 7] passes 1. But recall that figure 3,
with the smooth sloping region, plotted the error E rather than the weights. The
analogous graph here is shown in figure 6 where we see the same qualitative feature
of a smooth gradual rise, which first begins to jitter as the limit cycle becomes
asymmetric, and then becomes more and more jagged as the period doubles its way
to chaos. From figure 7 it is clear that for higher 7] the peak error of the attractor
will continue to rise gently until it saturates.
Next, we add momentum to the system. This simple one dimensional system duplicates the phenomena we found earlier, as can be seen by comparing figure 3 with
figure 5. We see that momentum delays the bifurcation of the fixed point attractor
at the minimum by the amount predicted by (5), namely until 7] approaches 1 + a.
At this point the fixed point bifurcates into a symmetric limit cycle of period 2 at
(14)
a formula of which (13) is a special case. This limit cycle is stable for
16
7]<
g(1+a),
(15)
but as 7] reaches this limit, which happens at the same time that w reaches ?1/V3
(the inflection point of E where E
1/4) the limit cycle becomes unstable. However, for a near 1 the cycle breaks down more quickly in practice, as it becomes
haloed by more complex attractors which make it progressively less likely that a
sequence of iterations will actually converge to the limit cycle in question. Both
boundaries of this strip, 7] = 1 + a and 7] = 196 (1 + a), are visible in figure 5,
=
Gradient Descent: Second Order Momentum and Saturating Error
1
E
Figure 6: E as a funco.
tion of 7J with a
When convergent, the final value is shown; otherwise E after 100 iterations from a starting
point of w
1.0. This a
more detailed graph of a
slice of figure 5 at a = O.
=
o
o
3
w
Figure 4: A one dimensional tulip-shaped nonlinear error surface E
1- (1 + w2)-1.
-3
=
Figure 5: E after 50 iterations from a starting
point of 0.05, as a function of 7J and a.
0.'
=
1.'
-1
-I
Figure 7: The attractor in was a function of 7J is shown, with the progression from a
single attract or at the minimum of E to a limit cycle of period two, which bifurcates
and then doubles to chaos . a = 0 (left) and a = 0.8 (right). For the numerical
simulations portions of the graphs, iterations 100 through 150 from a starting point
of w 1 or w 0.05 are shown.
=
=
particularly since in the region between them E obeys
E= J1 ~a
1-
(16)
The bifurcation and subsequent transition to chaos with momentum is shown for
a
0.8 in figure 7. This a is high enough that the limit cycle fails to be reached
by the iteration procedure long before it actually becomes unstable. Note that this
diagram was made with w started near the minimum. If it had been started far
from it, the system would usually not reach the attractor at w = 0 but instead
enter a halo attractor. This accounts for the policy of backpropagation experts,
who gradually raise momentum as the optimization proceeds.
=
893
894
Pearlmutter
5
CONCLUSIONS
The convergence bounds derived assume that the learning parameters are set optimally. Finding these optimal values in practice is beyond the scope of this paper, but some techniques for achieving nearly optimal learning rates are available
[4, 10, 8, 7, 3]. Adjusting the momentum feels easier to practitioners than adjusting the learning rate, as too high a value leads to small oscillations rather than
divergence, and techniques from control theory can be applied to the problem [2].
However, because error surfaces in practice saturate, techniques for adjusting the
learning parameters automatically as learning proceeds can not be derived under
the quadratic minimum assumption, but must take into account the bifurcation and
limit cycle and the sloping region of the error, or they may mistake this regime of
stable error for convergence, leading to premature termination.
References
[1] S. Thomas Alexander. Adaptive Signal Processing. Springer-Verlag, 1986.
[2] H. S. Dabis and T. J. Moir. Least mean squares as a control system. International Journal of Control, 54(2):321-335, 1991.
[3] Yan Fang and Terrence J. Sejnowski. Faster learning for dynamic recurrent
backpropagation. Neural Computation, 2(3):270-273, 1990.
[4] Robert A. Jacobs. Increased rates of convergence through learning rate adaptation. Neural Networks, 1(4):295-307,1988.
[5] David E. Rumelhart, Geoffrey E. Hinton, and R. J. Williams. Learning internal
representations by error propagation. In D. E. Rumelhart, J. 1. McClelland,
and the PDP research group., editors, Parallel distributed processing: Explorations in the microstructure of cognition, Volume 1: Foundations. MIT Press,
1986.
[6] J. J. Shynk and S. Roy. The LMS algorithm with momentum updating. In
Proceedings of the IEEE International Symposium on Circuits and Systems,
pages 2651-2654, June 6-9 1988.
[7] F. M. Silva and L. B. Almeida. Acceleration techniques for the backpropagation algorithm. In L. B. Almeida and C. J. Wellekens, editors, Proceedings of
the 1990 EURASIP Workshop on Neural Networks. Springer-Verlag, February
1990. (Lecture Notes in Computer Science series).
[8] Tom Tollenaere. SuperSAB: Fast adaptive back propagation with good scaling
properties. Neural Networks, 3(5):561-573, 1990.
[9] Mehmet Ali Tugay and Yal~in Tanik. Properties of the momentum LMS algorithm. Signal Processing, 18(2):117-127, October 1989.
[10] T. P. Vogl, J. K. Mangis, A. K. Zigler, W. T. Zink, and D. L. Alkon. Accelerating the convergence of the back-propagation method. Biological Cybernetics,
59:257-263, September 1988.
[11] B. Widrow, J. M. McCool, M. G. Larimore, and C. R. Johnson Jr. Stational and
nonstationary learning characteristics of the LMS adaptive filter. Proceedings
of the IEEE, 64:1151-1162, 1979.
| 454 |@word briefly:1 achievable:1 seems:1 termination:1 d2:1 confirms:1 simulation:3 gradual:1 jacob:1 shading:1 initial:1 series:2 past:1 comparing:1 must:3 visible:1 numerical:1 j1:1 subsequent:1 shape:2 plot:1 update:2 progressively:1 v:2 plane:2 location:2 sigmoidal:2 height:1 rc:1 along:1 constructed:2 symposium:1 qualitative:1 expected:1 rapid:1 behavior:1 ry:1 globally:1 decreasing:3 automatically:1 becomes:5 begin:1 bounded:3 circuit:2 sloping:4 finding:1 control:3 unit:5 appear:2 before:1 negligible:1 understood:1 local:1 tends:1 limit:15 mistake:1 analyzing:2 fluctuation:1 might:2 chose:1 shaded:1 fastest:1 limited:1 speeded:1 tulip:1 obeys:1 decided:2 practical:1 practice:6 backpropagation:6 chaotic:1 ooc:1 procedure:1 lla:1 yan:1 get:1 equivalent:1 center:2 williams:1 starting:4 amax:15 fang:1 dw:8 analogous:1 feel:1 target:1 play:1 hypothesis:1 secondorder:1 roy:2 rumelhart:2 particularly:1 updating:1 asymmetric:2 observed:2 region:16 cycle:12 decrease:1 dynamic:4 raise:1 ali:1 conver:1 accelerate:1 tugay:2 fast:2 sejnowski:1 neighborhood:1 outside:1 heuristic:1 richer:1 otherwise:1 final:1 online:1 advantage:1 eigenvalue:5 sequence:1 bifurcates:3 maximal:1 adaptation:1 amin:10 getting:1 convergence:22 double:2 diverges:1 converges:5 recurrent:1 widrow:1 objected:1 measured:1 predicted:9 come:1 saved:1 filter:1 stochastic:1 exploration:1 transient:1 ja:1 microstructure:1 biological:1 strictly:2 ad2:1 ground:1 exp:2 great:1 scope:1 cognition:1 lm:4 purpose:1 mit:1 always:1 modified:1 rather:3 derived:2 june:1 inflection:1 attract:1 hidden:3 going:2 special:2 bifurcation:4 equal:2 shaped:1 sampling:1 nearly:2 minimized:2 haven:1 duplicate:1 alkon:1 tightly:1 divergence:1 attractor:6 investigate:1 analyzed:1 tj:1 devoted:1 damped:1 initialized:1 plotted:3 increased:1 earlier:1 eta:1 delay:1 johnson:1 too:1 optimally:1 st:1 peak:2 international:2 terrence:1 quickly:2 squared:1 slowly:1 tanik:2 expert:1 leading:1 account:4 aggressive:1 de:3 coefficient:1 jagged:1 caused:2 vi:4 performed:1 break:1 tion:1 apparently:1 portion:1 reached:1 parallel:2 slope:1 square:2 who:2 characteristic:1 hyperbola:1 cybernetics:1 plateau:2 reach:3 strip:1 mysterious:1 mi:1 gain:1 adjusting:3 popular:1 ask:1 recall:1 improves:1 rim:1 actually:2 back:2 higher:1 tom:1 mutter:1 box:1 though:1 until:4 nonlinear:2 propagation:3 effect:3 read:1 symmetric:2 ridge:1 pearlmutter:5 performs:1 silva:1 ranging:2 chaos:4 recently:1 physical:1 empirically:1 volume:1 gently:1 association:1 ai:4 enter:1 had:2 stable:5 surface:7 add:2 j:1 verlag:2 binary:2 continue:1 seen:1 minimum:19 mccool:1 surely:1 converge:2 v3:1 period:5 monotonically:1 signal:2 bread:1 smooth:3 faster:2 characterized:1 calculation:1 long:1 converging:1 iteration:5 sometimes:1 diagram:1 w2:1 jde:3 pass:2 subject:1 capacitor:1 practitioner:1 nonstationary:1 near:2 split:1 enough:1 variety:1 psychology:1 whether:1 accelerating:1 nonquadratic:1 hessian:1 oscillate:1 clear:1 eigenvectors:1 detailed:1 amount:2 hardware:1 mcclelland:1 correctly:1 group:1 four:1 achieving:1 asymptotically:1 graph:3 run:2 inverse:1 jitter:1 oscillation:2 scaling:1 bit:1 layer:1 bound:3 ct:1 convergent:5 yale:2 quadratic:6 precisely:1 sharply:1 larimore:1 simulate:1 speed:2 speedup:2 department:1 jr:1 remain:1 slightly:2 making:1 happens:1 gradually:3 equation:3 wellekens:1 remains:1 turn:1 letting:1 available:1 observe:1 progression:2 away:1 batch:5 supersab:1 substitute:1 thomas:1 assumes:1 giving:1 restrictive:1 build:1 february:1 question:1 occurs:2 usual:1 september:1 gradient:18 unstable:2 sloped:1 unfortunately:1 october:1 robert:1 rise:3 policy:1 perform:1 vertical:1 descent:16 saturates:1 halo:1 hinton:1 pdp:1 station:1 david:1 namely:1 darkly:1 pearl:1 beyond:1 proceeds:2 usually:1 regime:3 zink:1 saturation:1 program:1 max:2 axis:1 started:2 shynk:2 lum:1 mehmet:1 review:2 understanding:1 epoch:5 mom:1 asymptotic:1 relative:1 lecture:1 analogy:1 geoffrey:1 foundation:1 plotting:1 editor:2 changed:1 barak:2 taking:1 absolute:1 distributed:1 regard:1 boundary:3 curve:1 slice:1 transition:1 contour:1 made:1 clue:1 replicated:1 fixedpoint:1 premature:1 adaptive:3 far:1 global:1 lip:1 transfer:1 funco:1 complex:2 did:1 spread:1 repeated:1 fails:1 momentum:28 resistor:1 lie:1 atypical:1 third:1 saturate:2 formula:1 down:1 nonconvergent:1 showing:1 decay:1 exists:1 workshop:1 adding:1 ci:4 justifies:1 easier:1 simply:1 explore:1 likely:1 saturating:6 doubling:1 springer:2 aa:1 corresponds:2 vogl:1 acceleration:1 towards:1 oscillator:1 butter:1 change:3 eurasip:1 determined:1 uniformly:1 acting:1 e:1 diverging:1 tollenaere:1 internal:1 people:1 almeida:2 alexander:1 incorporate:1 phenomenon:4 |
3,911 | 4,540 | Sketch-Based Linear Value Function Approximation
Marc G. Bellemare
University of Alberta
Joel Veness
University of Alberta
Michael Bowling
University of Alberta
[email protected]
[email protected]
[email protected]
Abstract
Hashing is a common method to reduce large, potentially infinite feature vectors
to a fixed-size table. In reinforcement learning, hashing is often used in conjunction with tile coding to represent states in continuous spaces. Hashing is also
a promising approach to value function approximation in large discrete domains
such as Go and Hearts, where feature vectors can be constructed by exhaustively
combining a set of atomic features. Unfortunately, the typical use of hashing in
value function approximation results in biased value estimates due to the possibility of collisions. Recent work in data stream summaries has led to the development
of the tug-of-war sketch, an unbiased estimator for approximating inner products.
Our work investigates the application of this new data structure to linear value
function approximation. Although in the reinforcement learning setting the use of
the tug-of-war sketch leads to biased value estimates, we show that this bias can
be orders of magnitude less than that of standard hashing. We provide empirical
results on two RL benchmark domains and fifty-five Atari 2600 games to highlight
the superior learning performance obtained when using tug-of-war hashing.
1
Introduction
Recent value-based reinforcement learning applications have shown the benefit of exhaustively generating features, both in discrete and continuous state domains. In discrete domains, exhaustive
feature generation combines atomic features into logical predicates. In the game of Go, Silver et al.
[19] showed that good features could be generated by enumerating all stone patterns up to a certain
size. Sturtevant and White [21] similarly obtained promising reinforcement learning results using a
feature generation method that enumerated all 2, 3 and 4-wise combinations of a set of 60 atomic
features. In continuous-state RL domains, tile coding [23] is a canonical example of exhaustive
feature generation; tile coding has been successfully applied to benchmark domains [22], to learn to
play keepaway soccer [20], in multiagent robot learning [4], to train bipedal robots to walk [18, 24]
and to learn mixed strategies in the game of Goofspiel [3].
Exhaustive feature generation, however, can result in feature vectors that are too large to be represented in memory, especially when applied to continuous spaces. Although such feature vectors are
too large to be represented explicitly, in many domains of interest they are also sparse. For example,
most stone patterns are absent from any particular Go position. Given a fixed memory budget, the
standard approach is to hash features into a fixed-size table, with collisions implicitly handled by
the learning algorithm; all but one of the applications discussed above use some form of hashing.
With respect to its typical use for linear value function approximation, hashing lacks theoretical
guarantees. In order to improve on the basic hashing idea, we turn to sketches: state-of-the-art
methods for approximately storing large vectors [6]. Our goal is to show that one such sketch,
the tug-of-war sketch [7], is particularly well-suited for linear value function approximation. Our
work is related to recent developments on the use of random projections in reinforcement learning
[11] and least-squares regression [16, 10]. Hashing, however, possesses a computational advantage
over traditional random projections: each feature is hashed exactly once. In comparison, even sparse
1
random projection methods [1, 14] carry a per-feature cost that increases with the size of the reduced
space. Tug-of-war hashing seeks to reconcile the computational efficiency that makes hashing a
practical method for linear value function approximation on large feature spaces, while preserving
the theoretical appeal of random projection methods.
A natural concern when using hashing in RL is that hash collisions irremediably degrade learning. In
this paper we argue that tug-of-war hashing addresses this concern by providing us with a low-error
approximation of large feature vectors at a fraction of the memory cost. To quote Sutton and Barto
[23], ?Hashing frees us from the curse of dimensionality in the sense that memory requirements
need not be exponential in the number of dimensions, but need merely match the real demands of
the task.?
2
Background
We consider the reinforcement learning framework of Sutton and Barto [23]. An MDP M is a
tuple hS, A, P, R, ?i, where S is the set of states, A is the set of actions, P : S ? A ? S ? [0, 1]
is the transition probability function, R : S ? A ? R is the reward function and ? ? [0, 1] is
the discount factor. At time step t the agent observes state st ? S, selects an action at ? A
and receives a reward rt := R(st , at ). The agent then observes the new state st+1 distributed
according to P (?|st , at ).P
From state st , the agent?s
goal is to maximize the expected discounted
?
i
sum of future rewards E
?
R(s
,
a
)
.
A
typical
approach is to learn state-action values
t+i t+i
i=0
Q? (s, a), where the stationary policy ? : S ? A ? [0, 1] represents the agent?s behaviour. Q? (s, a)
is recursively defined as:
"
#
X
?
0 0
? 0 0
Q (s, a) := R(s, a) + ?Es0 ?P (?|s,a)
?(a |s )Q (s , a )
(1)
a0 ?A
A special case of this equation is the optimal value function Q? (s, a) := R(s, a) +
?Es0 [maxa0 Q? (s0 , a0 )]. The optimal value function corresponds to the value under an optimal
policy ? ? . For a fixed ?, The SARSA(?) algorithm [23] learns Q? from sample transitions
(st , at , rt , st+1 , at+1 ). In domains where S is large (or infinite), learning Q? exactly is impractical and one must rely on value function approximation. A common value function approximation scheme in reinforcement learning is linear approximation. Given ? : S ? A ? Rn
mapping state-action pairs to feature vectors, we represent Q? with the linear approximation
Qt (s, a) := ?t ? ?(s, a), where ?t ? Rn is a weight vector. The gradient descent SARSA(?)
update is defined as:
?t
et
?t+1
? rt + ??t ? ?(st+1 , at+1 ) ? ?t ? ?(st , at )
? ??et?1 + ?(st , at )
? ?t + ??t et ,
(2)
where ? ? [0, 1] is a step-size parameter and ? ? [0, 1] controls the degree to which changes in
the value function are propagated back in time. Throughout the rest of this paper Q? (s, a) refers to
the exact value function computed from Equation 1 and we use Qt (s, a) to refer to the linear approximation ?t ? ?(s, a); ?gradient descent SARSA(?) with linear approximation? is always implied
when referring to SARSA(?). We call ?(s, a) the full feature vector and Qt (s, a) the full-vector
value function.
Asymptotically, SARSA(?) is guaranteed to find the best solution within the span of ?(s, a), up to
a multiplicative constant that depends on ? [25]. If we let ? ? R|S||A|?n denote the matrix of full
feature vectors ?(s, a), and let ? : S ? A ? [0, 1] denote the steady state distribution over stateaction pairs induced by ? and P then, under mild assumptions, we can guarantee the existence and
uniqueness of ?. We denote by h?, ?i? the inner product induced by ?, i.e. hx, yi? := xT Dy, where
x, y ? R|S||A| and D ? R|S||A|?|S||A| is a diagonal matrix with entries ?(s, a). The norm k?k? is
p
defined as h?, ?i? . We assume the following: 1) S and A are finite, 2) the Markov chain induced
by ? and P is irreducible and aperiodic, and 3) ? has full rank. The following theorem bounds the
error of SARSA(?):
Theorem 1 (Restated from Tsitsiklis and Van Roy [25]). Let M = hS, A, P, R, ?i be an MDP and
? : S ? A ? [0, 1] be a policy. Denote by ? ? R|S||A|?n the matrix of full feature vectors and
2
by ? the stationary distribution on (S, A) induced by ? and P . Under assumptions 1-3), SARSA(?)
converges to a unique ?? ? Rn with probability one and
k??? ? Q? k? ?
1 ? ??
k?Q? ? Q? k? ,
1??
where Q? ? R|S||A| is a vector representing the exact solution to Equation 1 and
? := ?(?T D?)?1 ?T D is the projection operator.
Because ? is the projector operator for ?, for any ? we have k?? ? Q? k? ? k?Q? ? Q? k? ;
Theorem 1 thus implies that SARSA(1) converges to ?? = arg min? k?? ? Q? k? .
2.1
Hashing in Reinforcement Learning
As discussed previously, it is often impractical to store the full weight vector ?t in memory. A
typical example of this is tile coding on continuous-state domains [22], which generates a number
of features exponential in the dimensionality of the state space. In such cases, hashing can effectively be used to approximate Q? (s, a) using a fixed memory budget. Let h be a hash function
h : {1, . . . , n} ? {1, . . . , m}, mapping full feature vector indices into hash table indices, where
? a) whose
m n is the hash table size. We define standard hashing features as the feature map ?(s,
ith component is defined as:
n
X
??i (s, a) :=
I[h(j)=i] ?j (s, a) ,
(3)
j=1
th
where ?j (s, a) denotes the j component of ?(s, a) and I[x] denotes the indicator function. We
assume that our hash function h is drawn from a universal family: for any i, j ? {1, . . . , n}, i 6= j,
1 1
? a),
? t (s, a) := ??t ? ?(s,
Pr(h(i) = h(j)) ? m
. We define the standard hashing value function Q
m
?
?
where ?t ? R is a weight vector, and ?(s, a) is the hashed vector. Because of hashing collisions,
? t (s, a)] 6=
the standard hashing value function is a biased estimator of Qt (s, a), i.e., in general Eh [Q
Qt (s, a). For example, consider the extreme case where m = 1: all features share the same weight.
We return to the issue of the bias introduced by standard hashing in Section 4.1.
2.2
Tug-of-War Hashing
The tug-of-war sketch, also known as the Fast-AGMS, was recently introduced as a powerful method
for approximating inner products of large vectors [7]. The name ?sketch? refers to the data structure?s function as a summary of a stream of data. In the canonical sketch setting, we summarize a
count vector ? ? Rn using a sketch vector ?? ? Rm . At each time step a vector ?t ? Rn is received.
Pt?1
The purpose of the sketch vector is to approximate the count vector ?t := i=0 ?i . Given two hash
functions, h and ? : {1, . . . , n} ? {?1, 1}, ?t is mapped to a vector ??t whose ith component is
??t,i :=
n
X
I[h(j)=i] ?t,j ?(j)
(4)
j=1
The tug-of-war sketch vector is then updated as ??t+1 ? ??t + ??t . In addition to h being drawn
from a universal family of hash functions, ? is drawn from a four-wise independent family of hash
functions: for all sets of four unique indices {i1 , i2 , i3 , i4 }, Pr? (?(i1 ) = k1 , ?(i2 ) = k2 , ?(i3 ) =
1
with k1 . . . k4 ? {?1, 1}. For an arbitrary ? ? Rn and its corresponding
k3 , ?(i4 ) = k4 ) = 16
? = ?t ? ?: the tug-of-war sketch produces unbiased estimates
tug-of-war vector ?? ? Rm , Eh,? [??t ? ?]
Pt?1
of inner products [7]. This unbiasedness property can be derived as follows. First let ??t = i=0 ??i .
1
While it may seem odd to randomly select your hash function, this can equivalently be thought as sampling
an indexing assignment for the MDP?s features. While a particular hash function may be well- (or poorly-)
suited for a particular MDP, it is hard to imagine how this could be known a priori. By considering a randomly
selected hash function (or random permutation of the features), we are simulating the uncertainty of using a
particular hash function on a never before encountered MDP.
3
Pt?1
Then ??t ? ??t0 = i=0 ??i ? ??t0 and
?
Eh,? [??i ? ??t0 ]
= Eh,? ?
n X
n
X
?
I[h(j1 )=h(j2 )] ?i,j1 ?t0 ,j2 ?(j1 )?(j2 )?
j1 =1 j2 =1
E? [?(j1 )?(j2 )]
=
1
0
if j1 = j2
otherwise
(by four-wise independence)
The result follows by noting that I[h(j1 )=h(j2 )] is independent from ?(j1 )?(j2 ) given j1 , j2 .
3
Tug-of-War with Linear Value Function Approximation
We now extend the tug-of-war sketch to the reinforcement
setting by defining the tug-of-war
Plearning
n
hashing features as ?? : S ? A ? Rm with ??i (s, a) := j=1 I[h(j)=i] ?j (s, a)?(j). The SARSA(?)
update becomes:
??t
e?t
?
?t+1
? t+1 , at+1 ) ? ??t ? ?(s
? t , at )
? rt + ? ??t ? ?(s
? t , at )
? ???
et?1 + ?(s
? ??t + ???t e?t .
(5)
? a) with ??t ? Rm and refer to
? t (s, a) := ??t ? ?(s,
We also define the tug-of-war value function Q
? a) as the tug-of-war vector.
?(s,
3.1
Value Function Approximation with Tug-of-War Hashing
Intuitively, one might hope that the unbiasedness of the tug-of-war sketch for approximating inner
products carries over to the case of linear value function approximation. Unfortunately, this is not
the case. However, it is still possible to bound the error of the tug-of-war value function learned with
SARSA(1) in terms of the full-vector value function. Our bound relies on interpreting tug-of-war
hashing as a special kind of Johnson-Lindenstrauss transform [8].
We define a ?-universal family of hash functions H such that for any set of indices i1 , i2 , . . . , il
Pr(h(i1 ) = k1 , . . . , h(il ) = kl ) ? |C1|l , where C ? N and h ? H : {1, . . . , n} ? C .
Lemma 1 (Dasgupta et al. [8], Theorem 2). Let h : {1, . . . , n} ? {1, . . . , m} and ? : {1, . . . , n} ?
{?1, 1} be two independent hash functions chosen uniformly at random from ?-universal families
1
,
and let H ? {0, ?1}m?n be a matrix with entries Hij = I[h(j)=i] ?(j). Let < 1, ? < 10
2 m
12
1
16
1
1
n
?
m = 2 log ? and c = log ? log ? . For any given vector x ? R such that kxk? ? c ,
with probability 1 ? 3?, H satisfies the following property:
2
2
2
(1 ? ) kxk2 ? kHxk2 ? (1 + ) kxk2 .
Lemma 1 states that, under certain conditions on the input vector x, tug-of-war hashing approximately preserves the norm of x. When ? and are constant, the requirement on kxk? can be waived
by applying Theorem 1 to the normalized vector u = kxkx ?c . A clear discussion on hashing as
2
a Johnson-Lindenstrauss transform can be found in the work of Kane and Nelson [13], who also
improve Lemma 1 and extend it to the case where the family of hash functions is k-universal rather
than ?-universal.
Lemma 2 (Based on Maillard and Munos [16], Proposition 1). Let x1 . . . xK and y be vectors in
Rn . Let H ? {0, ?1}m?n , , ? and m be defined as in Lemma 1. With probability at least 1 ? 6K?,
for all k ? {1, . . . , K},
xk ? y ? kxk k2 kyk2 ? Hxk ? Hy ? xk ? y + kxk k2 kyk2 .
Proof (Sketch). The proof follows the steps of Maillard and Munos [16]. Given two unit vectors
2
2
u, v ? Rn , we can relate (Hu) ? (Hv) to kHu + Hvk2 and kHu ? Hvk2 using the parallelogram
law. We then apply Lemma 1 to bound both sides of each squared norm and substitute xk for u and
y for w to bound Hxk ? Hy. Applying the union bound yields the desired statement.
4
We are now in a position to bound the asymptotic error of SARSA(1) with tug-of-war hashing.
Given hash functions h and ? defined as per Lemma 1, we denote by H ? Rm?n the matrix whose
? a) = H?(s, a). We also denote by ?
? := ?H T the
entries are Hij := I[h(j)=i] ?(j), such that ?(s,
matrix of tug-of-war vectors. We again assume that 1) S and A are finite, that 2) ? and P induce an
irreducible, aperiodic Markov chain and that 3) ? has full rank. For simplicity of argument, we also
? := ?H T has full rank; when ?
? is rank-deficient, SARSA(1) converges to a set of
assume that 4) ?
?
? satisfying the bound of Theorem 2, rather than to a unique ??? .
solutions ?
Theorem 2. Let M = hS, A, P, R, ?i be an MDP and ? : S ? A ? [0, 1] be a policy. Let
? ? R|S||A|?m be the matrix of tug-of? ? R|S||A|?n be the matrix of full feature vectors and ?
war vectors. Denote by ? the stationary distribution on (S, A) induced by ? and P . Let < 1,
?
1
? < 1, ? 0 = 6|S||A|
and m ? 12
2 log ? 0 . Under assumptions 1-4), gradient-descent SARSA(1) with
tug-of-war hashing converges to a unique ??? ? Rm and with probability at least 1 ? ?
?
? ?? ? Q?
? k??? ? Q? k + k?? k
?
sup k?(s, a)k2 ,
?
2
?
s?S,a?A
where Q? is the exact solution to Equation 1 and ?? = arg min? k?? ? Q? k? .
Proof. First note that Theorem 1 implies the convergence of SARSA(1) with tug-of-war hashing to
a unique solution, which we denote ??? . We first apply Lemma 2 to the set {?(s, a) : (s, a) ? S ?A}
and ?? ; note that we can safely assume |S||A| > 1, and therefore ? 0 < 1/10. By our choice of m,
for all (s, a) ? S ? A, |H?(s, a) ? H?? ? ?(s, a) ? ?? | ? k?(s, a)k2 k?? k2 with probability at least
? ? Q? k? ;
1 ? 6|S||A|? 0 = 1 ? ?. As previously noted, SARSA(1) converges to ??? = arg min? k??
?
?
?
?
?
?
compared to ?? , the solution ?H := ?H? is thus an equal or worse approximation to Q? . It
follows that
?
?
?
? ?? ? Q?
? H ? ???
+
??? ? Q?
? H ? Q?
?
??
?
?
??
?
?
?
?
s X
2
=
?(s, a) H?(s, a) ? H?? ? ?(s, a) ? ?? + k??? ? Q? k?
s?S,a?A
?
s X
2
?(s, a) k?(s, a)k2 k?? k2 + k??? ? Q? k?
(Lemma 2)
s?S,a?A
? k?? k2
sup
s?S,a?A
k?(s, a)k2 + k??? ? Q? k? ,
as desired.
Our proof of Theorem 2 critically requires the use of ? = 1. A natural next step would be to attempt
to drop this restriction on ?. It also seems likely that the finite-sample analysis of LSTD with random
projections [11] can be extended to cover the case of tug-of-war hashing. Theorem 2 suggests that,
under the right conditions, the tug-of-war value function is a good approximation to the full-vector
value function. A natural question now arises: does tug-of-war hashing lead to improved linear
value function approximation compared with standard hashing? More importantly, does tug-of-war
hashing result in better learned policies? These are the questions we investigate empirically in the
next section.
4
Experimental Study
In the sketch setting, the appeal of tug-of-war hashing over standard hashing lies in its unbiasedness.
We therefore begin with an empirical study of the magnitude of the bias when applying different
hashing methods in a value function approximation setting.
4.1
Value Function Bias
We used standard hashing, tug-of-war hashing, and no hashing to learn a value function over a
short trajectory in the Mountain Car domain [22]. Our evaluation uses a standard implementation
available online [15].
5
10
100
1
Mean Squared Error
Standard
Bias
0.1
0.01
Tug-of-War
0.001
0.0001
1e-05
1e-06
10
Standard
1
0.1
Tug-of-War
0.01
0.001
0.0001
0
100 200 300 400 500 600 700 800 900 1000
0
Time Steps
100 200 300 400 500 600 700 800 900 1000
Time Steps
Figure 1: Bias and Mean Squared Error of value estimates using standard and tug-of-war hashing in
1,000 learning steps of Mountain Car. Note the log scale of the y axis.
We generated a 1,000-step trajectory using an -greedy policy [23]. For this fixed trajectory we
updated a full feature weight vector ?t using SARSA(0) with ? = 1.0 and ? = 0.01. We focus on
SARSA(0) as it is commonly used in practice for its ease of implementation and its faster update
speed in sparse settings. Parallel to the full-vector update we also updated both a tug-of-war weight
vector ??t and a standard hashing weight vector ??t , with the same values of ? and ?. Both methods
use a hash table size of m = 100 and the same randomly selected hash function. This hash function
is defined as (ax + b) mod p mod m, where p is a large prime and a, b < p are random integers
? t (st , at )
[5]. At every step we compute the difference in value between the hashed value functions Q
?
and Qt (st , at ), and the full-vector value function Qt (st , at ). We repeated this experiment using 1
million hash functions selected uniformly at random. Figure 1 shows for each time step, estimates
? t (st , at )] ? Qt (st , at ) and E[Q
? t (st , at )] ? Qt (st , at ) as well as
of the magnitude of the biases E[Q
2
? t (st , at ) ? Qt (st , at )) ] and E[(Q
? t (st , at ) ? Qt (st , at ))2 ]
estimates of the mean squared errors E[(Q
using the different hash functions. To provide a sense of scale, the estimate of the value of the final
state when using no hashing is approximately ?4; note that the y-axis uses a logarithmic scale.
The tug-of-war value function has a small, almost negligible bias. In comparison, the bias of standard hashing is orders of magnitude larger ? almost as large as the value it is trying to estimate.
The mean square error estimates show a similar trend. Furthermore, the same experiment on the
Acrobot domain [22] yielded qualitatively similar results. Our results confirm the insights provided
in Section 2: the tug-of-war value function can be significantly less biased than the standard hashing
value function.
4.2
Reinforcement Learning Performance
Having smaller bias and mean square error in the Q-value estimates does not necessarily imply
improved agent performance. In reinforcement learning, actions are selected based on relative Qvalues, so a consistent bias may be harmless. In this section we evaluate the performance (cumulative
reward per episode) of learning agents using both tug-of-war and standard hashing.
4.2.1
Tile Coding
We first studied the performance of agents using each of the two hashing methods in conjunction
with tile coding. Our study is based on Mountain Car and Acrobot, two standard RL benchmark
domains. For both domains we used the standard environment dynamics [22]; we used the fixed
starting-state version of Mountain Car to reduce the variance in our results. We compared the two
hashing methods using -greedy policies and the SARSA(?) algorithm.
For each domain and each hashing method we performed a parameter sweep over the learning rate
? and selected the best value which did not cause the value estimates to divergence. The Acrobot
state was represented using 48 6 ? 6 ? 6 ? 6 tilings and the Mountain Car state, 10 9 ? 9 tilings.
Other parameters were set to ? = 1.0, ? = 0.9, = 0.0; the learning rate was further divided by the
number of tilings.
6
Acrobot
2500
Mountain Car
5000
Random Agent
Standard Hashing
4000
Steps to Goal
Steps to Goal
2000
Random Agent
1500
Tug-of-War Hashing
1000
500
0
3000
2000
Standard Hashing
Tug-of-War Hashing
1000
100
500
1000
1500
0
2000
Hash Table Size
200
400
600
800
1000
Hash Table Size
Figure 2: Performance of standard hashing and tug-of-war hashing in two benchmark domains. The
performance of the random agent is provided as reference.
We experimented with hash table sizes m ? [20, 1000] for Mountain Car and m ? [100, 2000] for
Acrobot. Each experiment consisted of 100 trials, sampling a new hash function for each trial. Each
trial consisted of 10,000 episodes, and episodes were restricted to 5,000 steps. At the end of each
trial, we disabled learning by setting ? = 0 and evaluated the agent on an additional 500 episodes.
Figure 2 shows the performance of standard hashing and tug-of-war hashing as a function of the
hash table size. The conclusion is clear: when the hashed vector is small relative to the full vector,
tug-of-war hashing performs better than standard hashing. This is especially true in Acrobot, where
the number of features (over 62,000) necessarily results in harmful collisions.
4.2.2
Atari
We next evaluated tug-of-war hashing and standard hashing on a suite of Atari 2600 games. The
Atari domain was proposed as a game-independent platform for AI research by Naddaf [17]. Atari
games pose a variety of challenges for learning agents. The learning agent?s observation space is the
game screen: 160x210 pixels, each taking on one of 128 colors. In the game-independent setting,
agents are tuned using a small number of training games and subsequently evaluated over a large
number of games for which no game-specific tuning takes place. The game-independent setting
forces us to use features that are common to all games, for example, by encoding the presence
of color patterns in game screens; such an encoding is a form of exhaustive feature generation.
Different learning methods have been evaluated on the Atari 2600 platform [9, 26, 12]. We based
our evaluation on prior work on a suite of Atari 2600 games [2], to which we refer the reader for full
details on handling Atari 2600 games as RL domains. We performed parameter sweeps over five
training games, and tested our algorithms on fifty testing games.
We used models of contingency awareness to locate the player avatar [2]. From a given game,
we generate feature sets by exhaustively enumerating all single-color patterns of size 1x1 (single
pixels), 2x2, and 3x3. The presence of each different pattern within a 4x5 tile is encoded as a binary
feature. We also encode the relative presence of patterns with respect to the player avatar location.
This procedures gives rise to 569,856,000 different features, of which 5,000 to 15,000 are active at
a given time step.
We trained -greedy SARSA(0) agents using both standard hashing and tug-of-war hashing with
hash tables of size m=1,000, 5,000 and 20,000. We chose the step-size ? using a parameter sweep
over the training games: we selected the best-performing ? which never resulted in divergence in
the value function. For standard hashing, ? = 0.01, 0.05, 0.2 for m = 1,000, 5,000 and 20,000,
respectively. For tug-of-war hashing, ? = 0.5 across table sizes. We set ? = 0.999 and = 0.05.
Each experiment was repeated over ten trials lasting 10,000 episodes each; we limited episodes to
18,000 frames to avoid issues with non-terminating policies.
7
Hash Table Size: 1000
1.0
1.0
Hash Table Size: 5000
1.0
0.5
Tug-of-War
Fraction of games
Fraction of games
Fraction of games
Tug-of-War
0.5
Standard
Hash Table Size: 20,000
Tug-of-War
Standard
0.5
Standard
0.0
1.0
0.8
0.6
0.4
Inter-algorithm score
0.2
0
0.0
1.0
0.8
0.6
0.4
Inter-algorithm score
0.2
0
0.0
1.0
0.8
0.6
0.4
0.2
0
Inter-algorithm score
Figure 3: Inter-algorithm score distributions over fifty-five Atari games. Higher curves reflect higher
normalized scores.
Accurately comparing methods across fifty-five games poses a challenge, as each game exhibits a
different reward function and game dynamics. We compared methods using inter-algorithm score
distributions [2]. For each game, we extracted the average score achieved by our agents over the
last 500 episodes of training, yielding six different scores (three per hashing method) per game.
Denoting these scores by sg,i , i = 1 . . . 6, we defined the inter-algorithm normalized score zg,i :=
(sg,i ? rg,min )/(rg,max ? rg,min ) with rg,min := mini {sg,i } and rg,max := maxi {sg,i }. Thus
zg,i = 1.0 indicates that the ith score was the highest for game g, and zg,i = 0.0 similarly indicates
the lowest score. For each combination of hashing method and memory size, its inter-algorithm
score distribution shows the fraction of games for which the corresponding agent achieves a certain
normalized score or better.
Figure 3 compares the score distributions of agents using either standard hashing or tug-of-war
hashing for m = 1,000, 5,000 and 20,000. Tug-of-war hashing consistently outperforms standard
hashing across hash table sizes. For each m and each game, we also performed a two-tailed Welch?s
t-test with 99% confidence intervals to determine the statistical significance of the average score
difference between the two methods. For m = 1,000, tug-of-war hashing performed statistically
better in 38 games and worse in 5; for m = 5,000, it performed better in 41 games and worse in 7;
and for m = 20,000 it performed better in 35 games and worse in 5. Our results on Atari games
confirm what we observed on Mountain Car and Acrobot: in practice, tug-of-war hashing performs
much better than standard hashing. Furthermore, computing the ? function took less than 0.3% of
the total experiment time, a negligible cost in comparison to the benefits of using tug-of-war hashing.
5
Conclusion
In this paper, we cast the tug-of-war sketch into the reinforcement learning framework. We showed
that, although the tug-of-war sketch is unbiased in the setting for which it was developed [7], the
self-referential component of reinforcement learning induces a small bias. We showed that this bias
can be much smaller than the bias that results from standard hashing and provided empirical results
confirming the superiority of tug-of-war hashing for value function approximation.
As increasingly more complex reinforcement learning problems arise and strain against the boundaries of practicality, so the need for fast and reliable approximation methods grows. If standard
hashing frees us from the curse of dimensionality, then tug-of-war hashing goes a step further by
ensuring, when the demands of the task exceed available resources, a robust and principled shift
from the exact solution to its approximation.
Acknowledgements
?
We would like to thank Bernardo Avila
Pires, Martha White, Yasin Abbasi-Yadkori and Csaba
Szepesv?ari for the help they provided with the theoretical aspects of this paper, as well as Adam
White and Rich Sutton for insightful discussions on hashing and tile coding. This research was
supported by the Alberta Innovates Technology Futures and the Alberta Innovates Centre for Machine Learning at the University of Alberta. Invaluable computational resources were provided by
Compute/Calcul Canada.
8
References
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9
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3,912 | 4,541 | Causal discovery with scale-mixture model for
spatiotemporal variance dependencies
Zhitang Chen* , Kun Zhang? , and Laiwan Chan*
*
Department of Computer Science and Engineering, Chinese University of Hong Kong, Hong Kong
{ztchen,lwchan}@cse.cuhk.edu.hk
?
Max Planck Institute for Intelligent Systems, T?ubingen, Germany
[email protected]
Abstract
In conventional causal discovery, structural equation models (SEM) are directly
applied to the observed variables, meaning that the causal effect can be represented
as a function of the direct causes themselves. However, in many real world problems, there are significant dependencies in the variances or energies, which indicates that causality may possibly take place at the level of variances or energies. In
this paper, we propose a probabilistic causal scale-mixture model with spatiotemporal variance dependencies to represent a specific type of generating mechanism
of the observations. In particular, the causal mechanism including contemporaneous and temporal causal relations in variances or energies is represented by a
Structural Vector AutoRegressive model (SVAR). We prove the identifiability of
this model under the non-Gaussian assumption on the innovation processes. We
also propose algorithms to estimate the involved parameters and discover the contemporaneous causal structure. Experiments on synthetic and real world data are
conducted to show the applicability of the proposed model and algorithms.
1 Introduction
Causal discovery aims to discover the underlying generating mechanism of the observed data, and
consequently, the causal relations allow us to predict the effects of interventions on the system
[15, 19]. For example, if we know the causal structure of a stock market, we are able to predict the
reactions of other stocks against the sudden collapse of one share price in the market. A traditional
way to infer the causal structure is by controlled experiments. However, controlled experiments
are in general expensive, time consuming, technically infeasible and/or ethically prohibited. Thus,
causal discovery from non-experimental data is of great importance and has drawn considerable
attention in the past decades [15, 19, 16, 17, 12, 22, 2]. Probabilistic models such as Bayesian
Networks (BNs) and Linear Non-Gaussian Acyclic Models (LiNGAM) have been proposed and
applied to many real world problems [18, 13, 14, 21].
Conventional models such as LiNGAM assume that the causal relations are of a linear form, i.e., if
the observed variable x is the cause of another observed variable y, we model the causal relation as
y = ?x + e, where ? is a constant coefficient and e is the additive noise independent of x. However,
in many types of natural signals or time series such as MEG/EEG data [23] and financial data [20],
a common form of nonlinear dependency, as seen from the correlation in variances or energies, is
found [5]. This observation indicates that causality may take place at the level of variances or energies instead of the observed variables themselves. Generally speaking, traditional methods cannot
detect this type of causal relations. Another restriction of conventional causal models is that these
models assume constant variances of the observations; this assumption is unrealistic for those data
with strong heteroscedasticity [1].
1
In this paper, we propose a new probabilistic model called Causal Scale-Mixture model with SpatioTemporal Variance Dependencies (CSM-STVD) incorporating the spatial and temporal variance
or energy dependencies among the observed data. The main feature of the new model is that we
model the spatiotemporal variance dependencies based on the Structural Vector AutoRegressive
(SVAR) model, in particular the Non-Gaussian SVAR [11]. The contributions of this study are
two-fold. First, we provide an alternative way to model the causal relations among the observations, i.e., causality in variances or energies. In this model, causality takes place at the level of
variances or energies, i.e., the variance or energy of one observed series at time instant t0 is influenced by the variances or energies of other variables at time instants t ? t0 and its past values at
time instants t < t0 . Thus, both contemporaneous and temporal causal relations in variances are
considered. Secondly, we prove the identifiability of this model and more specifically, we show
that Non-Gaussianity makes the model fully identifiable. Furthermore, we propose a method which
directly estimates such causal structures without explicitly estimating the variances.
2
Related work
To model the variance or energy dependencies of the observations, a classic method is to use a scalemixture model [5, 23, 9, 8]. Mathematically, we can represent a signal as si = ui ?i , where ui is a
signal with zero mean and constant variance, and ?i is a positive factor which is independent of ui
and modulates the variance or energy of si [5]. For multivariate case, we have
s = u ? ?,
(1)
where ? means element-wise multiplication. In basic scale-mixture model, u and ? are statistically
independent and the components ui are spatiatemporally independent, i.e. ui,t?1 ?
? uj,t?2 , ?t?1 , t?2 .
However, ?i , the standard deviations of the observations, are dependent across i. The observation
x, in many situations, is assumed to be a linear mixture of the source s, i.e., x = As, where A is a
mixing matrix.
In [5], Hirayama and Hyv?arinen proposed a two-stage model. The first stage is a classic ICA model
[3, 10], where the observation x is a linear mixture of the hidden source s, i.e., x = As. On the
second stage, the variance dependencies are modeled by applying a linear Non-Gaussian (LiN) SEM
to the log-energies of the sources.
?
yi =
hij yj + hi0 + ri , i = 1, 2, ? ? ? , d,
j
where yi = log ?(?i ) are the log-energies of sources si and the nonlinear function ? is any appropriate measure of energy; ri are non-Gaussian distributed and independent of yj . To make the problem
tractable, they assumed that ui are binary, i.e., ui ? {?1, 1} and uniformly distributed. The parameters of this two-stage model including A and hij are estimated by maximum likelihood without
approximation due to the uniform binary distribution assumption of u. However, this assumption is
restrictive and thus may not fit real world observations well. Furthermore, they assumed that ?i are
spatially dependent but temporally white. However, many time series show strong heterosecadasticity and temporal variance dependencies such as financial time series and brain signals. Taking into
account of temporal variance dependencies would improve the quality of the estimated underlying
structure of the observed data.
Another two-stage model for magnetoencephalography (MEG) or electroencephalography (EEG)
data was propsoed earlier in [23]. The first stage also performs linear separation; they proposed
a blind source separation algorithm by exploiting the autocorrelations and time-varying variances
of the sources. In the second stage, si (t) are modeled by autoregressive processes with L lags
(AR(L)) driven by innovations ei (t). The innovation processes ei (t) are mutually uncorrelated and
temporally white. However, ei (t) are not necessarily independent. They modeled ei (t) as follows:
ei (t) = ?it zi (t), where zi (t) ? N (0, 1).
(2)
Two different methods are used to model the dependencies among the variances of the innovations.
2
The first method is causal-in-variance GARCH (CausalVar-GARCH). Specifically ?it
are modeled
by a multivariate GARCH model. The advantage of this model is that we are able to estimate
the temporal causal structure in variances. However, this model provides no information about the
2
contemporaneous causal relations among the sources if there indeed exist such causal relations. The
second method to model the variance dependencies is applying a factor model to the log-energies
2
(log ?it
) of the sources. The disadvantage of this method is that we cannot model the causal relations
among the sources which are more interesting to us.
In many real world observations, there are causal influences in variances among the observed variables. For instance, there are significant mutual influences among the volatilities of the observed
stock prices. We are more interested in investigating the underlying causal structure among the
variances of the observed data. Consequently, in this paper, we consider the situation where the
correlation in the variances of the observed data is interesting. That is, the first stage of [5, 23] is
not needed, and we focus on the second stage, i.e., modeling the spatiotemporal variance dependencies and causal mechanism among the observations. In the following sections, we propose our
probabilistic model based on SVAR to describe the spatiotemporal variance dependencies among
the observations. Our model is, as shown in later sections, closely related to the models introduced
in [5, 23], but has significant advantages: (1) both contemporaneous and temporal causal relations
can be modeled; (2) this model is fully identifiable under certain assumptions.
3
Causal scale-mixture model with spatiotemporal variances dependencies
We propose the causal scale-mixture model with spatiotemporal variance dependencies as follows.
Let z(t) be the m ? 1 observed vector with components zi (t), which are assumed to be generated
according to the scale-mixture model:
zi (t) = ui (t)?i (t).
(3)
Here we assume that ui (t) are temporally independent processes, i.e., ui (t?1 ) ?
? uj (t?2 ), ?t?1 ?= t?2
but unlike basic scale-mixture model, here ui (t) may be contemporarily dependent, i.e., ui (t) ??
?
uj (t), ?i ?= j. ?(t) is spatially and temporally independent of u(t). Using vector notation,
zt = ut ? ?t .
(4)
Here ?it > 0 are related to the variances or energies of the observations zt and are assumed to be
spatiotemporally dependent. As in [5, 23], let yt = log ?t . In this paper, we model the spatiotemporal variance dependencies by a Structural Vector AutoRegressive model (SVAR), i.e.,
yt = A0 yt +
L
?
B? yt?? + ?t ,
(5)
? =1
where A0 contains the contemporaneous causal strengths among the variances of the observations,
i.e., if [A0 ]ij ?= 0, we say that yit is contemporaneously affected by yjt ; B? contains the temporal
(time-lag) causal relations, i.e., if [B? ]ij ?= 0, we say that yi,t is affected by yj,t?? . Here, ?t are
i.i.d. mutually independent innovations. Let xt = log |zt | (In this model, we assume that ui (t) do
not take value zero) and ?t = log |ut |.Take log of the absolute values of both sides of equation (4),
then we have the following model:
xt = yt + ?t ,
yt = A0 yt +
L
?
B? yt?? + ?t .
(6)
? =1
We make the following assumptions on the model:
A1 Both ?t and ?t are temporally white with zero means. The components of ?t are not necessarily independent, and we assume that the covariance matrix of ?t is ?? . The components
of ?t are independent and ?? = I1 .
A2 The contemporaneous causal structure is acyclic, i.e., by simultaneous row and column
permutations, A0 can be permuted to a strictly lower triangular matrix. BL is of full rank.
1
Note that ?? = I is assumed just for convenience. A0 and B? can also be correctly estimated if ?? is a
general diagonal covariance matrix. The explanation why the scaling indeterminacy can be eliminated is the
same as LiNGAM given in [16].
3
A3 The innovations ?t are non-Gaussian, and ?t are either Gaussian or non-Gaussian.
Inspired by the identifiability results of the Non-Gaussian state-space model in [24], we show that
our model is identifiable. Note that our new model and the state-space model proposed in [24] are
two different models, while interestingly by simple re-parameterization we can prove the following
Lemma 3.1 and Theorem 3.1 following [24].
Lemma 3.1 Given the log-transformed observation xt = log |zt | generated by Equations (6), if the
assumptions A1 ? A2 hold, by solving simple linear equations involving the autocovariances of xt ,
the covariance ?? and AB? can be uniquely determined, where A = (I ? A0 )?1 ; furthermore, A
and B? can be identified up to some rotation transformations. That is, suppose that two models with
? ? L
? ? ) generate the same observation xt , then we
parameters (A, {B? }L
? =1 , ?? ) and (A, {B? }? =1 , ??
T
?
?
?
have ?? = ??? , A = AU, B? = U B? , where U is an orthogonal matrix.
Non-Gaussianity of the innovations ?t makes the model fully identifiable, as seen in the following
theorem.
Theorem 3.1 Given the log-transformed observation xt = log |zt | generated by Equations (6) and
given L, if assumptions A1 ? A3 hold, then the model is identifiable. In other words, suppose
? ? L
? ? ) generate the same
that two models with parameters (A, {B? }L
? =1 , ?? ) and (A, {B? }? =1 , ??
?
? = A, B
? ? = B? ,
observation xt ; then these two models are identical, i.e., we have ??? = ?? , A
? t = yt .
and y
4
Parameter learning and causal discovery
In this section, we propose an effective algorithm to estimate the contemporaneous causal structure
matrix A0 and temporal causal structure matrices B? , ? = 1, ? ? ? , L (see (6)).
4.1
Estimation of AB?
We have shown that AB? can be uniquely determined, where A = (I ? A0 )?1 . The proof of
Lemma 3.1 also suggests a way to estimate AB? , as given below. Readers can refer to the appendix
for the detailed mathematical derivation. Although we are aware that this method might not be statistically efficient, we adopt this estimation method due to its great computational efficiency. Given
the log-transformed observations xt = log |zt |, denoted by Rx (k) the autocovariance function of
xt at lag k, we have Rx (k) = E(xt xTt+k ). Based on the model assumptions A1 and A2 , we have
the following linear equations of the autocovarainces of xt .
?? T
C1
?
? ?
??
?
R
(L
+
2)
R
(L
+
1)
R
(L)
?
?
?
R
(2)
? x
? ? x
? ? CT2
x
x
?
? ?
??
=
?
?
?
?? .
..
..
..
..
..
?
? ?
?? .
.
?
? ?
?? .
.
.
.
.
Rx (2L)
Rx (2L ? 1) Rx (2L ? 2) ? ? ? Rx (L)
CTL
|
{z
}
?
Rx (L + 1)
?
?
Rx (L)
Rx (L ? 1) ? ? ? Rx (1)
?
?
?
?
?,
?
?
?
(7)
,H
where C? = AB? (? = 1, ? ? ? , L). As shown in the proof of Lemma 3.1, H is invertible. We can
easily estimate AB? by solving the linear Equations (7).
4.2
Estimation of A0
The estimations of AB? (? = 1, ? ? ? , L) still contain the mixing information of the causal structures
A0 and B? . In order to further obtain the contemporaneous and temporal causal relations, we need
to estimate both A0 and B? (? = 1, ? ? ? , L). Here, we show that the estimation of A0 can be reduced
to solving a Linear Non-Gaussian Acyclic Models with latent confounders.
Substituting yt = xt ? ?t into Equations (6), we have
xt ? ?t =
L
?
AB? (xt?? ? ?t?? ) + A?t .
? =1
4
(8)
Since AB? can be uniquely determined according to Lemma 3.1 or more specifically Equations (7),
?L
we can easily obtain ?t = xt ? ? =1 AB? xt?? , then we have:
?t = A?t + ?t ?
L
?
AB? ?t?? .
(9)
? =1
This is exactly a Linear Non-Gaussian Acyclic Model with latent confounders and the estimation of
A is a very challenging problem [6, 2]. To make to problem tractable, we further have the following
two assumptions on the model:
? A4 If the components of ?t are not independent, we assume that ?t follows a factor model:
?t = Dft , where the components of ft are spatially and temporally independent Gaussian
factors and D is the factor loading matrix (not necessarily square).
? A5 The components of ?t are simultaneously super-Gaussian or sub-Gaussian.
By replacing ?t with Dft , we have:
?t = A?t + Dft ?
|
L
?
AB? Dft?? .
(10)
? =1
{z
}
confounding effects
To identify the matrix A which contains the contemporaneous causal information of the observed
variables, we treat ft and ft?? as latent confounders and the interpretation of assumption A4 is that
we can treat the independent factors ft as some external factors outside the system. The Gaussian
assumption of ft can be interpreted hierarchically as the result of central limit theorem because these
factors themselves represent the ensemble effects of numerous factors from the whole environment.
On the contrary, the disturbances ?it are local factors that describe the intrinsic behaviors of the
observed variables [4]. Since they are local and thus not regarded as the ensembles of large amount
of factors. In this case, the disturbances ?it are assumed to be non-Gaussian.
The LiNGAM-GC model [2] takes into the consideration of latent confounders. In that model, the
confounders are assumed to follow Gaussian distribution, which was interpreted as the result of
central limit theorem. It mainly focuses on the following cause-effect pair:
x = e1 + ?c,
y = ?x + e2 + ?c,
(11)
where e1 and e2 are non-Gaussian and mutually independent, and c is the latent Gaussian confounder
independent of the disturbances e1 and e2 . To tackle the causal discovery problem of LiNGAMGC, it was firstly shown that if x and y are standardized to unit absolute kurtosis then |?| < 1
based on the assumption that e1 and e2 are simultaneously super-Gaussian or sub-Gaussian. Note
that assumption A5 is a natural extension of this assumption. It holds in many practical problems,
? xy
especially for financial data. After the standardization, the following cumulant-based measure R
was proposed [2]:
? xy = (Cxy + Cyx )(Cxy ? Cyx ), where
R
? 3 y} ? 3E{xy}
?
? 2 },
Cxy = E{x
E{x
3
?
?
? 2 },
Cyx = E{xy
} ? 3E{xy}
E{y
(12)
? means sample average. It was shown that the causal direction can be identified simply by
and E
? xy , i.e., if R
? xy > 0, x ? y is concluded; otherwise if R
? xy < 0, y ?
examining the sign of R
x is concluded. Once the causal direction has been identified, the estimation of causal strength
is straightforward. The work can be extended to multivariate causal network discovery following
DirectLiNGAM framework [17].
Here we adopt LiNGAM-GC-UK, the algorithm proposed in [2], to find the contemporaneous casual
? ? = (I ? A
? 0 )C
??,
structure matrix A0 . Once A0 has been estimated, B? can be easily obtained by B
? 0 and C
? ? are the estimations of A0 and AB? , respectively. The algorithm for learning the
where A
model is summarized in the following algorithm.
5
Algorithm 1 Causal discovery with scale-mixture model for spatiotemporal variance dependencies
1: Given the observations zt , compute xt = log |zt |.
? t from xt , i.e., xt = xt ? x
?t
2: Subtract the mean x
3: Choose an appropriate lag L for the SVAR and then estimate AB? where A = (I ? A0 )?1 and
? = 1, ? ? ? , L, using Equations(7).
?L
4: Obtain the residues by ?t = xt ? ? =1 AB? xt?? .
5: Apply LiNGAM-GC algorithms to ?t and obtain the estimation of A0 and B? (? = 1, ? ? ? , L)
and the corresponding comtemporaneous and temporal causal orderings.
5
Experiment
We conduct experiments using synthetic data and real world data to investigate the effectiveness of
our proposed model and algorithms.
5.1
Synthetic data
We generate the observations according to the following model:
zt = r ? ut ? ?t ,
r is a m?1 scale vector of which the elements are randomly selected from interval [1.0, 6.0]; ut > 0
and ?t = log ut follows a factor model:
?t = Dft ,
where D is m ? m and the elements of D are randomly selected from [0.2, 0.4] . fit are i.i.d. and
fit ? N (0, 0.5). Denoted by yt = log ?t , we model the spatiotemporal variance dependencies of
the observations xt by an SVAR(1):
yt = A0 yt + B1 yt?1 + ?t ,
where A0 is a m ? m strictly lower triangular matrix of which the elements are randomly selected
from [0.1, 0.2] or [?0.2, ?0.1]; B1 is a m ? m matrix of which the diagonal elements [B1 ]ii are
randomly selected from [0.7, 0.8], 80% of the off-diagonal elements [B1 ]i?=j are zero and the remaining 20% are randomly selected from [?0.1, 0.1]; ?it are i.i.d. super-Gaussian generated by
?it = sign(nit )|nit |2 (nit ? N (0, 1)) and normalized to unit variance. The generated observations
are permuted to a random order. The task of this experiment is to investigate the performance of our
algorithms in estimating the coefficient matrix (I ? A0 )?1 B1 and also the contemporaneous causal
ordering induced by A0 . We estimate the matrix (I ? A0 )?1 B1 using Lemma 3.1 or specifically
Equations (7). We use different algorithms: LiNGAM-GC-UK proposed in [2], C-M proposed in
[7] and LiNGAM [16] to estimate the contemporaneous causal structure. We investigate the performances of different algorithms in the scenarios of m = 4 with sample size from 500 to 4000 and
m = 8 with sample size from 1000 to 10000. For each scenario, we randomly conduct 100 independent trials and discard those trials where the SVAR processes are not stable. We calculate the
accuracies of LiNGAM-GC-UK, C-M and LiNGAM in finding (1) whole causal ordering (2) exogenous variable (root) of the causal network. We also calculate the sum square error Err of estimated
causal strength matrix
? of different algorithms with respect to the true one. The average SNR defined
V ar(? )
?
as SN R = 10 log i V ar(fii ) is about 13.85 dB. The experimental results are shown in Figure 1 and
i
Table 1. Figure 1 shows the plots of the estimated entries of (I?A0 )?1 B1 versus the true ones when
the dimension of the observations m = 8. From Figure 1, we can see that the matrix (I ? A0 )?1 B1
is estimated well enough when the sample size is only 1000. This confirms the correctness of our
theoretical analysis of the proposed model. From Table 1, we can see that when the dimension of
the observations is small (m = 4), all algorithms have acceptable performances. The performance
of LiNGAM is the best when the sample size is small. This is because C-M and LiNGAM-GC-UK
are cumulant-based methods which need sufficiently large sample size. When the dimension of the
observations m increases to 8, we can see that the performances of C-M and LiNGAM degrade
dramatically. While LiNGAM-GC-UK still successfully finds the exogenous variable (root) or even
the whole contemporaneous causal ordering among the variances of the observations if the sample
size is sufficiently large enough. This is mainly due to the fact that when the dimension increases,
6
sample size: 2000
sample size: 4000
1
1
0.5
0.5
FTSE
?1
?1
0
?1
?1
1
sample size: 8000
0
sample size: 10000
1
1
1
0.5
0.5
0.5
0
0
0
0.8
GDAXI
estimated parameters
4
?1
?1
?0.5
0
1
DJI
?0.5
?1
?1
?1
?1
0
1
true parameters
7
0.9833
?0.5
42
1
-0.624
sample size: 6000
FCHI
0.4798
?1
?1
0
1
true parameters
40
0
?0.5
0.7
0
?0.5
05
0
?0.5
1.0
estimated parameters
sample size: 1000
1
0.5
0
NDX
1
Figure 1: Estimated entries causal strength matrix (I ?
A0 )?1 B1 vs the true ones (m = 8)
Figure 2: Contemporaneous causal network of the selected stock indices
Table 1: Accuracy of finding the causal ordering
sample size
m=4
500
1000
2000
3000
4000
m=8
1000
2000
4000
6000
8000
10000
whole causal ordering
first variable found
Err
C-M
LiNGAM
LiNGAM-GC-UK
C-M
LiNGAM
LiNGAM-GC-UK
C-M
LiNGAM
LiNGAM-GC-UK
37%
47%
74%
67%
63%
70%
75%
86%
78%
83%
28%
25%
81%
90%
90%
61%
25%
82%
79%
81%
85%
92%
90%
88%
92%
60%
72%
92%
96%
94%
0.1101
0.0865
0.0679
0.0716
0.0669
0.0326
0.024
0.02
0.0201
0.0193
0.0938
0.0444
0.0199
0.0126
0.0109
0%
1.14%
0%
0%
2.20%
0%
23.08%
26.14%
31.87%
25.29%
30.77%
23.53%
8.79%
25%
58.24%
83.91%
80.22%
91.76%
20.88%
25%
19.78%
25.29%
17.58%
12.94%
75.82%
70.45%
82.41%
75.86%
79.12%
68.24%
65.93%
75%
86.81%
96.55%
91.21%
97.64%
0.8516
0.7866
0.7537
0.7638
0.7735
0.7794
0.2318
0.2082
0.1916
0.1843
0.1824
0.194
0.3017
0.1396
0.0634
0.0341
0.029
0.0199
the confounding effects of Dft ? (I ? A)?1 B1 Dft?1 become more problematic such that the performances of C-M and LiNGAM are strongly affected by confounding effect. Table 1 also shows
the estimation accuracies of the compared methods. Among them, LiNGAM-GC-UK significantly
outperforms other methods given sufficiently large sample size.
In order to investigate the robustness of our methods against the Gaussian assumption on the external factors ft , we conduct the following experiment. The experimental setting is the same as
that in the above experiment but here the external factors ft are non-Gaussian, and more specifically fit = sign(nit )|nit |p , where nit ? N (0, 0.5). When p > 1, the factor is super-Gaussian
and when p < 1 the factor is sub-Gaussian. We investigate the performances of LiNGAMGC-UK, LiNGAM and C-M in finding the whole causal ordering in difference scenarios where
p = {0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6} with sample size of 6000. The results in Figure 3 show that
LiNGAM-GC-UK achieved satisfying results compared to LiNGAM and C-M. This suggests that although LiNGAM-GC is developed based on the assumption that the latent confounders are Gaussian
distributed, it is still robust in the scenarios where the latent confounders are mildly non-Gaussian
with mild causal strengths.
whole causal ordering
100
accuracy(%)
80
60
40
20
0
0.2
0.4
0.6
0.8
kurtosis of f
1
p
1.2
1.4
1.6
1.8
t
0.4
LiNGAM?GC?UK
C?M
LiNGAM
kurtosis
0.2
0
?0.2
?0.4
0.5
1
p
1.5
Figure 3: Robustness against Gaussianity of ft
7
5.2
Real world data
In this section, we use our new model to discover the causal relations among five major world stocks
indices: (1) Dow Jones Industrial Average (DJI) (2) FTSE 100 (FTSE) (3) Nasdaq-100 (NDX) (4)
CAC 40 (FCHI) (5) DAX (GDAXI), where DJI and NDX are stock indices in US, and FTSE, FCHI
and GDAXI are indices in Europe. Note that because of the time difference, we believe that the
causal relations among these stock indices are mainly acyclic, as we assumed in this paper. We
collect the adjusted close prices of these selected indices from May 2nd, 2006 to April 12th, 2012,
and use linear interpolation to estimate the prices on those dates when the data are not available.
We apply our proposed model with SVAR(1) to model the spatiotemporal variance dependencies
of the data. For the contemporaneous causal structure discovery, we use LiNGAM-GC-UK, C-M,
LiNGAM2 and Direct-LiNGAM3 to estimate the causal ordering. The discovered causal orderings
of different algorithms are shown in Table 2. From Table 2, we see that in the causal ordering
Table 2: Contemporaneous causal ordering of the selected stock indices
algorithm
LiNGAM-GC-UK
C-M
LiNGAM
Direct-LiNGAM
causal ordering
{2} ? {4} ? {5} ? {1} ? {3}
{1} ? {2} ? {4} ? {5}
{1} ? {3}
{2} ? {5} ? {3} ? {1}
{2} ? {4}
{3} ? {1} ? {5} ? {4} ? {2}
discovered by LiNGAM-GC-UK and LiNGAM, the stock indices in US, i.e., DJI and NDX are contemporaneously affected by the indices in Europe. Note that each stock index is given in local time.
Because of the time difference between Europe and America and the efficient market hypothesis
(the market is quick to absorb new information and adjust stock prices relative to that), the contemporaneous causal relations should be from Europe to America, if they exist. This is consistent with
the results our method and LiNGAM produced. Another interesting finding is that in the graphs
obtained by LiNGAM-GC-UK and LiNGAM, we can see that FTSE is the root, which is consistent
with the fact that London is the financial centre of Europe and FTSE is regarded as Europe?s most
important index. However, in results by C-M and DirectLiNGAM, we have the opposite direction,
i.e., the stock indices in US is contemporaneously the cause of the indices in Europe, which is difficult to interpret. The contemporaneous causal network of the stock indices are shown in Figure 2.
Further interpretation on the discovered causal strengths needs expertise knowledge.
6
Conclusion
In this paper, we investigate the causal discovery problem where causality takes place at the level
of variances or energies instead of the observed variables themselves. We propose a causal scalemixture model with spatiotemporal variance dependencies to describe this type of causal mechanism. We show that the model is fully identifiable under the non-Gaussian assumption of the
innovations. In addition, we propose algorithms to estimate the parameters, especially the contemporaneous causal structure of this model. Experimental results on synthetic data verify the practical
usefulness of our model and the effectiveness of our algorithms. Results using real world data further suggest that our new model can possibly explain the underlying interaction mechanism of major
world stock markets.
Acknowledgments
The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, China.
2
LiNGAM converges to several local optima. We only show one of the discovered causal ordering here.
The code is available at:http://www.cs.helsinki.fi/group/neuroinf/lingam/
3
http://www.ar.sanken.osaka-u.ac.jp/?inazumi/dlingam.html
8
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9
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3,913 | 4,542 | Tight Bounds on Profile Redundancy and Distinguishability
Jayadev Acharya
ECE, UCSD
[email protected]
Hirakendu Das
Yahoo!
[email protected]
Alon Orlitsky
ECE & CSE, UCSD
[email protected]
Abstract
The minimax KL-divergence of any distribution from all distributions in a collection P has several
practical implications. In compression, it is called redundancy and represents the least additional
number of bits over the entropy needed to encode the output of any distribution in P. In online estimation and learning, it is the lowest expected log-loss regret when guessing a sequence of random
values generated by a distribution in P. In hypothesis testing, it upper bounds the largest number of
distinguishable distributions in P. Motivated by problems ranging from population estimation to text
classification and speech recognition, several machine-learning and information-theory researchers
have recently considered label-invariant observations and properties induced by i.i.d. distributions.
A sufficient statistic for all these properties is the data?s profile, the multiset of the number of times
each data element appears. Improving on a sequence of previous works, we show that the redundancy of the collection of distributions induced over profiles by length-n i.i.d. sequences is between
0.3 ? n1/3 and n1/3 log2 n, in particular, establishing its exact growth power.
1
Introduction
Information theory, machine learning, and statistics, are closely related disciplines. One of their main intersection
areas is the confluence of universal compression, online learning, and hypothesis testing. We consider two concepts in
this overlap. The minimax KL divergence?a fundamental measure for, among other things, how difficult distributions
are to compress, predict, and classify, and profiles?a relatively new approach for compression, classification, and
property testing over large alphabets. Improving on several previous results, we determine the exact growth power of
the KL-divergence minimax of profiles of i.i.d. distributions over any alphabet.
1.1
Minimax KL divergence
As is well known in information theory, the expected number of bits required to compress data X generated according
to a known distribution P is the distribution?s entropy, H(P ) = EP log 1/P (X), and is achieved by encoding X using
roughly log 1/P (X) bits. However, in many applications P is unknown, except that it belongs to a known collection
P of distributions, for example the collection of all i.i.d., or all Markov distributions. This uncertainty typically raises
the number of bits above the entropy and is studied in Universal compression [9, 13]. Any encoding corresponds to
some distribution Q over the encoded symbols. Hence the increase in the expected number of bits used to encode the
output of P is EP log 1/Q(X) ? H(P ) = D(P ||Q), the KL divergence between P and Q. Typically one is interested
in the highest increase for any distribution P ? P, and finds the encoding that minimizes it. The resulting quantity,
called the (expected) redundancy of P, e.g., [8, Chap. 13], is therefore the KL minimax
def
R(P) = min max D(P ||Q).
Q
P ?P
The same quantity arises in online-learning, e.g., [5, Ch. 9], where the probabilities of random elements X1 , . . . , Xn
are sequentially
Pn estimated. One of the most popular measures for the performance of an estimator Q is the per-symbol
log loss n1 i=1 log Q(Xi |X i?1 ). As in compression, for underlying distribution P ? P, the expected log loss is
EP log 1/Q(X), and the log-loss regret is EP log 1/Q(X) ? H(P ) = D(P ||Q). The maximal expected regret for
any distribution in P, minimized over all estimators Q is again the KL minimax, namely, redundancy.
1
In statistics, redundancy arises in multiple hypothesis testing. Consider the largest number of distributions that can
be distinguished from their observations. For example, the largest number of topics distinguishable based on text of a
given length. Let P be a collection of distributions over a support set X . As in [18], a sub-collection S ? P of the
distributions is -distinguishable if there is a mapping f : X ? S such that if X is generated by a distribution S ? S,
then P (f (X) 6= S) ? . Let M (P, ) be the largest number of -distinguishable distributions in P, and let h() be
the binary entropy function. In Section 4 we show that for all P,
(1 ? ) log M (P, ) ? R(P) + h(),
(1)
and in many cases, like the one considered here, the inequality is close to equality.
Redundancy has many other connections to data compression [27, 28], the minimum-description-length principle [3,
16, 17], sequential prediction [21], and gambling [20]. Because of the fundamental nature of R(P), and since tight
bounds on it often reveal the structure of P, the value of R(P) has been studied extensively in all three communities,
e.g., the above references as well as [29, 37] and a related minimax in [6].
1.2
Redundancy of i.i.d. distributions
The most extensively studied collections are independently, identically distributed (i.i.d.). For example, for the collection Ikn of length-n i.i.d. distributions over alphabets of size k, a string of works [7, 10, 11, 28, 33, 35, 36] determined
the redundancy up to a diminishing additive term,
R(Ikn ) =
k?1
log n + Ck + o(1),
2
(2)
where the constant Ck was determined exactly in terms of k. For compression this shows that the extra number of
bits per symbol required to encode an i.i.d. sequence when the underlying distribution is unknown diminishes to zero
as (k ? 1) log n/(2n). For online learning this shows that these distributions can be learned (or approximated) and
that this approximation can be done at the above rate. In hypothesis testing this shows that there are roughly n(k?1)/2
distinguishable i.i.d. distributions of alphabet size k and length n.
Unfortunately, while R(Ikn ) increases logarithmically in the sequence length n, it grows linearly in the alphabet size k.
For sufficiently large k, this value even exceeds n itself, showing that general distributions over large alphabets cannot
be compressed or learned at a uniform rate over all alphabet sizes, and as the alphabet size increases, progressively
larger lengths are needed to achieve a given redundancy, learning rate, or test error.
1.3
Patterns
Partly motivated by redundancy?s fast increase with the alphabet size, a new approach was recently proposed to address
compression, estimation, classification, and property testing over large alphabets.
The pattern [25] of a sequence represents the relative order in which its symbols appear. For example, the pattern of
abracadabra is 12314151231. A natural method to compress a sequence over a large alphabet is to compress its pattern
as well as the dictionary that maps the order to the original symbols. For example, for abracadabra, 1 ? a, 2 ? b,
3 ? r, 4 ? c, 5 ? d.
It can be shown [15, 26] that for all i.i.d. distributions, over any alphabet, even infinitely large, as the sequence
length increases, essentially all the entropy lies in the pattern, and practically none is in the dictionary. Hence [25]
focused on the redundancy of compressing patterns. They showed, e.g., Subsection 1.5, that the although, as in (2),
i.i.d. sequences over large alphabets have arbitrarily high per-symbol redundancy, and although as above patterns
contain essentially all the information of long sequences, the per-symbol redundancy of patterns diminishes to zero at
a uniform rate independent of the alphabet size.
In online learning, patterns correspond to estimating the probabilities of each observed symbol, and of all unseen ones
combined. For example, after observing the sequence dad, with pattern 121, we estimate the probabilities of 1, 2, and
3. The probability we assign to 1 is that of d, the probability we assign to 2 is that of a, and the probability we assign
to 3 is the probability of all remaining letters combined. The aforementioned results imply that while distributions
over large alphabets cannot be learned with uniformly diminishing per-symbol log loss, if we would like to estimate
the probability of each seen element, but combine together the probabilities of all unseen ones, then the per symbol
log loss diminishes to zero uniformly regardless of the alphabet size.
2
1.4
Profiles
Improving on existing pattern-redundancy bounds seems easier to accomplish via profiles. Since we consider i.i.d.
distributions, the order of the elements in a pattern does not affect its probability. For example, for every distribution
P , P (112) = P (121). It is easy to see that the probability of a pattern is determined by the fingerprint [4] or
profile [25] of the pattern, the multiset of the number of appearances of the symbols in the pattern. For example, the
profile of the pattern 121 is {1, 2} and all patterns with this profile, 112, 121, 122 will have the same probability under
any distribution P . Similarly, the profile of 1213 is {1, 1, 2} and all patterns with this profile, 1123, 1213, 1231, 1223,
1232, and 1233, will have the same probability under any distribution.
It is easy to see that since all patterns of a given profile have the same probability, the ratio between the actual and
estimated probability of a profile is the same as this ratio for each of its patterns. Hence pattern redundancy is the
same as profile redundancy [25]. Therefore from now on we consider only profile redundancy, and begin by defining
it more formally.
The multiplicity ?(a) of a symbol a in a sequence is the number of times it appears. The profile ?(x) of a sequence
x is the multiset of multiplicities of all symbols appearing in it [24, 25]. The profile of the sequence is the multiset of
multiplicities. For example, the sequence ababcde has multiplicities ?(a) = ?(b) = 2, ?(c) = ?(d) = ?(e) = 1, and
profile {1, 1, 1, 2, 2}. The prevalence ?? of a multiplicity ? is the number of elements with multiplicity ?.
Let ?n denote the collection of all profiles of length-n sequences. For example, for sequences of length one there is a
single element appearing once, hence ?1 = {{1}}, for length two, either one element appears twice, or each of two
elements appear once, hence ?2 = {{2}, {1, 1}}, similarly ?3 = {{3}, {2, 1}, {1, 1, 1}}, etc.
We consider the distributions induced on ?n by all discrete i.i.d. distributions over any alphabet. The probability
def Qn
that an i.i.d. distribution P generates an n-element sequence x is P (x) = i=1 P (xi ). The probability of a profile
def P
? ? ?n is the sum of the probabilities of all sequences of this profile, P (?) =
x:?(x)=? P (x). For example, if P is
B(2/3) over h and t, then for n = 3, P ({3}) = P (hhh) + P (ttt) = 1/3, P ({2, 1}) = P (hht) + P (hth) + P (thh) +
P (tth) + P (tht) + P (htt) = 2/3, and P ({1, 1, 1} = 0 as this P is binary hence at most two symbols can appear.
On the other hand, if P is a roll of a fair die, then P ({3}) = 1/36, P ({2, 1}) = 5/12, and P ({1, 1, 1} = 5/9. We let
I?n = {P (?) : P is a discrete i.i.d. distribution} be the collection of all distributions on ?n induced by any discrete
i.i.d. distribution over any alphabet, possibly even infinite.
It is easy to see that any relabeling of the elements in an i.i.d. distribution will leave the profile distribution unchanged,
for example, if instead of h and t above, we have a distribution over 0?s and 1?s. Furthermore, profiles are sufficient
statistics for every label-invariant property. While many theoretical properties of profiles are known, even calculating
the profile probabilities for a given distribution and a profile seems hard [23, 38] in general.
Profile redundancy arises in at least two other machine-learning applications, closeness-testing and classification.In
closeness testing [4], we try to determine if two sequences are generated by same or different distributions. In classification, we try to assign a test sequence to one of two training sequences. Joint profiles and quantities related to
profile redundancy are used to construct competitive closeness tests and classifiers that perform almost as well as the
best possible [1, 2].
Profiles also arise in statistics, in estimating symmetric or label-invariant properties of i.i.d. distributions ([34] and
references therein). For example the support size, entropy, moments, or number of heavy hitters. All these properties
depend only on the multiset of probability values in the distribution. For example, the entropy of the distribution
p(heads) = .6, p(tails) = .4, depends only on the probability multiset {.6, .4}. For all these properties, profiles are
a sufficient statistic.
1.5
Previous Results
As patterns and profiles have the same redundancy, we describe the results for profiles.
Instead of the expected redundancy R(I?n ) that reflects the increase in the expected number of bits, [25] bounded the
? n ), reflecting the increase in the worst-case number of
more stringent but closely-related worst-case redundancy, R(I
?
bits, namely over all sequences. Using bounds [19] on the partition function, they showed that
r !
1/3
n
? ? ) ? ? 2 n1/2 .
?(n ) ? R(I
3
3
These bounds do not involve the alphabet size, hence show that unlike the sequences themselves, patterns (whose redundancy equals that of profiles), though containing essentially all the information of the sequence, can be compressed
and learned with redundancy and log-loss diminishing as n?1/2 , uniformly over all alphabet sizes.
Note however that by contrast to i.i.d. distributions, where the redundancy (2) was determined up to a diminishing
additive constant, here not even the power was known. Consequently several papers considered improvements of these
bounds, mostly for expected redundancy, the minimax KL divergence.
Since expected redundancy is at most the worst-case redundancy, the upper bound applies also for expected redundancy. Subsequently [31] described a partial proof-outline that could potentially show the following tighter upper
bound on expected redundancy, and [14] proved the following lower bound, strengthening one in [32],
1/3
n
1.84
? R(I?n ) ? n0.4 .
(3)
log n
1.6
New results
In Theorem 15 we use error-correcting codes to exhibit a larger class of distinguishable distributions in I?n than was
known before, thereby removing the log n factor from the lower bound in (3). In Theorem 11 we demonstrate a small
number of distributions such that every distribution in I?n is within a small KL divergence from one of them, thereby
reducing the upper bound to have the same power as the lower bound. Combining these results we obtain,
0.3 ? n1/3 ? (1 ? ) log M (I?n , ) ? R(I?n ) ? n1/3 log2 n.
(4)
These results close the power gap between the upper and lower bounds that existed in the literature. They show that
when a pattern is compressed or a sequence is estimated (with all unseen elements combined into new), the per-symbol
redundancy and log-loss decrease to 0 uniformly over all distributions faster than log2 n/n2/3 , a rate that is optimal
up to a log2 n factor. They also show that for length-n profiles, the redundancy R(I?n ) is essentially the logarithm
log M (I?n , ) of the number of distinguishable distributions.
1.7
Outline
In the next section we describe properties of Poisson sampling and redundancy that will be used later in the paper.
In Section 3 we establish the upper bound and in Section 4, the lower bound. Most of the proofs are provided in the
Appendix.
2
Preliminaries
We describe some techniques and results used in the proofs.
2.1
Poisson sampling
When a distribution is sampled i.i.d. exactly n times, the multiplicities are dependent, complicating the analysis of
many properties. A standard approach [22] to overcome the dependence is to sample the distribution a random poi(n)
times, the Poisson distribution with parameter n, resulting in sequences of random length near close to n. We let
def
poi(?, ?) = e?? ?? /?! denote the probability that a poi(?) random variable attains the value ?.
The following basic properties of Poisson sampling help simplify the analysis and relate it to fixed-length sampling.
Lemma 1. If a discrete i.i.d. distribution is sampled poi(n) times then: (1) the number of appearances of different
symbols are independent; (2) a symbol with probability p appears poi(np) times; (3) for any fixed n0 , conditioned on
the length poi(n) ? n0 , the first n0 elements are distributed identically to sampling P exactly n0 times.
We now express profile probabilities and redundancy under Poisson sampling. As we saw, the probability of a profile is
determined by just the multiset of probability value and the symbol labels are irrelevant. For convenience, we assume
that the distribution is over the positive integers, and we replace the distribution parameters {pi } by the Poisson
def
parameters {npi }. For a distribution P = {p1 , p2 , . . .}, let ?i = npi , and ? = {?1 , ?2 , . . .}. The profile generated
4
by this distribution is a multiset ? = {?1 , ?2 , . . .}, where each ?i generated independently according to poi(?i ). The
probability that ? generates ? is [1, 25],
XY
1
?(?) = Q?
poi(??(i) , ?i ).
(5)
?=0 ?? ! ?
i
where the summation is over all permutations of the support set.
For example, for ? = {?1 , ?2 , ?3 }, the profile ? = {2, 2, 3} can be generated by specifying which element appears
three times. This is reflected by the ?2 ! in the denominator, and each of the repeated terms in the numerator are
counted only once.
?
poi(n)
Similar to I?n , we use I?
to denote the class of distributions induced on ?? = ?0 ? ?1 ? ?2 ? . . . when sequences
poi(n)
of length poi(n) are generated i.i.d.. It is easy to see that a distribution in I?
is a collection of ?i ?s summing to n.
poi(n)
poi(n)
The redundancy R(I?
), and -distinguishability M (I?
, ) are defined as before. The following lemma shows
poi(n)
poi(n)
) is sufficient to bound R(I?n ).
that bounding M (I?
, ) and R(I?
Lemma 2. For any fixed > 0,
?
n? n log n
(1 ? o(1))R(I?
poi(n)
) ? R(I?
)
and
poi(n)
M (I?
?
n+ n log n
, ) ? M (I?
, 2).
in n. Combining this with the fact that
Proof Sketch. It is easy to show that R(I?n )?and M (I?n , ) are non-decreasing
?
the probability that poi(n) is less than n ? n log n or greater than n + n log n goes to 0 yields the bounds.
Finally, the next lemma, proved in the Appendix, provides a simple formula for cross expectations of Poisson distributions.
Lemma 3. For any ?0 , ?1 , ?2 > 0,
E??poi(?1 )
2.2
poi(?2 , ?)
(?1 ? ?0 )(?2 ? ?0 )
= exp
.
poi(?0 , ?)
?0
Redundancy
We state some basic properties of redundancy.
For a distribution P over A and a function f : A ? B, let f (P ) be the distribution over B that assigns to b ? B
the probability P (f ?1 (b)). Similarly, for a collection P of distributions over A, let f (P) = {f (P ) : P ? P}. The
convexity of KL-divergence shows that D(f (P )||f (Q)) ? D(P ||Q), and can be used to show
Lemma 4 (Function Redundancy). R(f (P)) ? R(P).
For a collection P of distributions over A ? B, let PA and PB be the collection of marginal distributions over A and B,
respectively. In general, R(P) can be larger or smaller than R(PA ) + R(PB ). However, when P consists of product
distributions, namely P (a, b) = PA (a) ? PB (b), the redundancy of the product is at most the sum of the marginal
redundancies. The proof is given in the Appendix.
Lemma 5 (Redundancy of products). If P be a collection of product distributions over A ? B, then
R(P) ? R(PA ) + R(PB ).
For a prefix-free code C : A ? {0, 1}? , let EP [|C|] be the expected length of C under distribution P . Redundancy is
the extra number of bits above the entropy needed to encode the output of any distribution in P. Hence,
Lemma 6. For every prefix-free code C, R(P) ? maxP ?P EP [|C|].
Lemma 7 (Redundancy of unions). If P1 , . . . , PT are distribution collections, then
[
R(
Pi ) ? max R(Pi ) + log T.
1?i?k
1?i?T
5
3
Upper bound
poi(n)
A distribution in ? ? I?
is a multiset of ??s adding to n. For any such distribution, let
def
def
def
?low = {? ? ? : ? ? n1/3 }, ?med = {? ? ? : n1/3 < ? ? n2/3 }, ?high = {? ? ? : ? > n2/3 },
and let ?low , ?med , ?high denote the corresponding profile each subset generates. Then ? = ?low ? ?med ? ?high . Let
poi(n)
I?low = {?low : ? ? I?
} be the collection of all ?low . Note that n is implicit here and in the rest of the paper.
A distribution in I?low is a multiset of ??s such that each is ? n1/3 and they sum to either n or to ? n ? n1/3 . I?med
and I?high are defined similarly.
? is determined by the triple (?low , ?med , ?high ), and by Poisson sampling, ?low , ?med and ?high are independent.
Hence by Lemmas 4 and 5,
R(I?n ) ? R(I(?low ,?med ,?high ) ) ? R(I?low ) + R(I?med ) + R(I?high ).
In Subsection 3.1 we show that I?low < 4n1/3 log n and I?high < 4n1/3 log n. In Subsection 3.2 we show that
I?med < 12 n1/3 log2 n.
In the next two subsections we elaborate on the overview and sketch some proof details.
3.1
Bounds on R(I?low ) and R(I?high )
Elias Codes [12] are prefix-free codes that encode a positive integer n using at most log n + log(log n + 1) + 1 bits.
We use Elias codes and design explicit coding schemes for distributions in I?low and I?high , and prove the following
result.
Lemma 8. R(I?low ) < 4n1/3 log n, and R(I?high ) < 2n1/3 log n.
Proof. Any distribution ?high ? I?high consists of ??s that are > n2/3 and add to ? n. Hence |?high | is < n1/3 , and
so is the number of multiplicities in ?high . Each multiplicity is a poi(?) random variable, and is encoded separately
using Elias code. For example, the profile {100, 100, 200, 250, 500} is encoded by coding the sequence 100, 100, 200,
250, 500 all using Elias scheme. For ? > 10, the number of bits needed to encode a poi(?) random variable using
Elias codes can be shown to be at most 2 log ?. The expected code-length is at most n1/3 ? 2 log n. Applying Lemma 6
gives R(I?high ) < 2n1/3 log n.
A distribution ?low ? I?low consists of ??s less that < n1/3 and sum at most n. We encode distinct multiplicities along
with their prevalences, using two integers for each distinct multiplicity. For example, ? = {1, 1, 1, 1, 1, 2, 2, 2, 5} is
coded as 1, 5, 2, 3, 5, 1. Using Poisson tail bounds, we bound the largest multiplicity in ?low , and use arguments similar
to I?high to obtain R(I?low ) < 4n1/3 log n.
3.2
Bound on R I?med
We partition the interval (n1/3 , n2/3 ] into B = n1/3 bins. For each distribution in I?med , we divide the ??s in it
according to these bins. We show that within each interval, there is a uniform distribution such that the KL divergence
between the underlying distribution and the induced uniform distribution is small. We then show that the number of
uniform distributions needed is at most exp(n1/3 log n). We expand on these ideas and bound R(I?med ).
We partition I?med into T ? exp(n1/3 log n) classes, upper bound the redundancy of each class, and then invoke
Lemma 7 to obtain an upper bound on R I?med . A distribution ? = {?1 , ?2 , . . . , ?r } ? I?med is such that ?i ?
Pr
[n1/3 , n2/3 ] and i=1 ?i ? n.
def
Consider any partition of (n1/3 , n2/3 ] into B = n1/3 consecutive intervals I1 , I2 , . . . , IB of lengths
def
def
?1 , ?2 , . . . , ?B .For each distribution ? ? I?med , let ?j = {?j,l : l = 1, 2, . . . , mj } = {? : ? ? ? ? Ij } be
def
def
the set of elements of ? in Ij where mj = mj (?) = |?j | is the number of elements of ? in Ij . Let
def
? (?) = (m1 , m2 , . . . , mB )
6
be the B?tuple of the counts of ??s in each interval.
For example, if n = 1000, then n1/3 = 10 and n2/3 = 100. For simplicity, we choose B = 3 instead of
n1/3 and ?1 = 10, ?2 = 30, ?3 = 50, so the intervals are I1 = (10, 20], I2 = (20, 50], I3 = (50, 100].
Suppose, ? = {12, 15, 25, 35, 32, 43, 46, 73}, then ?1 = {12, 15}, ?2 = {25, 35, 32, 43, 46}, ?3 = {73} and
? (?) = (m1 , m2 , m3 ) = (2, 5, 1).
We partition I?med , such that two distributions ? and ?0 are in the same class if and only if ? (?) = ? (?0 ). Thus each
class of distributions is characterized by a B-tuple of integers ? = (m1 , m2 , . . . , mB ) and let I? denote this class. Let
def
T = T (?) be the set of all possible different ? (such that I? is non-empty), and T = |T | be the numberP
of classes.
We first bound T below. Observe that for any ? ? I?med , and any j, we have mj < n2/3 , otherwise ??? ? >
1/3
mj ? n1/3 = n. So, each mj in ? can take at most n2/3 < n values. So, T < (n2/3 )B < nn = exp(n1/3 log n).
Pj?1
def 1/3
For any choice of ?, let ??
+ i=1 ?i be the left end point of the interval Ij for j = 1, 2, . . . , B. We upper
j = n
bound R(I? ) of any particular class ? = (m1 , m2 , . . . , mB ) in the following result.
Lemma 9. For all choices of ? = (?1 , . . . , ?B ), and all classes I? such that ? = (m1 , . . . , mB ) ? T (?),
R(I? ) ?
B
X
mj
j=1
?2j
??
j
.
Proof Sketch. For any choice of ?, ? = (m1 , . . . , mB ) ? T (?), we show a distribution ?? ? I? such that for
PB
?2j
all ? ? I? , D(?||?? ) ?
. Recall that for ? ? I? , ?j is the set of elements of ? in Ij . Let ?j
j=1 mj ??
j
be the profile generated by ?j . Then, ?med = ?1 ? . . . ? ?B . The distribution ?? is chosen to be of the form
{??1 ?m1 , ??2 ?m2 , . . . , ??B ?mB }, i.e., each ??j is uniform. The result follows from Lemma 3, and the details are in
the Appendix .
We now prove that R(I?med ) < 12 n1/3 log2 n.
By Lemma 7 it suffices to bound R(I? ). From Theorem 9 it follows that the choice of ? determines the bound on
R(I? ). A solution to the following optimization problem yields a bound :
B
B
X
X
?2j
mj ? , subject to
min max
mj ??
j ? n.
?
?
?
j
j=1
j=1
Instead of minimizing over all partitions, we choose the endpoints of the intervals as a geometric series as a bound for
?
?
1/3
the expression. The left-end point of Ij is ??
. We let ??
j , so ?1 = n
j+1 = ?j (1 + c). The constant c is chosen to
B
1/3
ensure that ??
(1+c)n
1 (1+c) = n
?
?
Now, ?j = ??
j+1 ? ?j = c?j , so
1/3
?2j
??
j
= n2/3 , the right end-point of IB . This yields, c < 2 log(1+c) =
2 log(n1/3 )
.
n1/3
= c2 ??
j . This translates the objective function to the constraint, and is in fact
the optimal intervals for the optimization problem (details omitted). Using this, for any ? = (m1 , . . . , mB ) ? T (?),
B
X
j=1
mj
?2j
??
j
= c2
B
X
2
mj ??
j ? c n<
j=1
2 log(n1/3 )
n1/3
2
n=
This, along with Lemma 7 gives the following Corollary for sufficiently large n.
Corollary 10. For large n, R(I?med ) <
1
2
? n1/3 log2 n.
Combining Lemma 8 with this result yields,
Theorem 11. For sufficiently large n,
R(I?n ) ? n1/3 log2 n.
7
4 1/3
n log2 n.
9
4
Lower bound
We use error-correcting codes to construct a collection of 20.3n
rithmic factor the bound in [14, 31].
1/3
distinguishable distributions, improving by a loga-
The convexity of KL-divergence can be used to show
Lemma 12. Let P and Q be distributions
on A. Suppose A1 ? A be such that P (A1 ) ? 1 ? > 1/2, Q(A1 ) ? ? <
1/2. Then, D(P ||Q) ? (1 ? ) log 1? ? h().
def
We use this result to show that (1 ? ) log M (P, ) ? R(P). Recall that for P over A, M = M (P, ) is the largest
number of ?distinguishable distributions in P. Let P1 , P2 , . . . , PM in P and A1 , A2 , . . . , AM be a partition of A
PM
such that Pj (Aj ) ? 1?. Let Q0 be the distribution such that, R(P) = supP ?P D(P ||Q0 ). Since j=1 Q0 (Aj ) = 1,
1
for some m ? {1, . . . , M }. Also, Pm (Am ) ? 1 ? . Plugging in P = Pm , Q = Q0 , A1 = Am , and
Q0 (Am ) < M
? = 1/M in the Lemma 12,
R(P) ? D(Pm ||Q0 ) ? (1 ? ) log (M (P, )) ? h().
def
def
We now describe the class of distinguishable distributions. Fix C > 0. Let ??i = Ci2 , K = b(3n/C)1/3 c, and
def
S = {??i : 1 ? i ? K}. K is chosen so that sum of elements in S is at most n. Let x = x1 x2 . . . xK be a binary
string and
n
o
X
def
?x = {??i : xi = 1} ? n ?
??i xi .
The distribution contains ??i whenever xi = 1, and the last element ensures that the elements add up to n. A binary
code of length k and minimum distance dmin is a collection of k?length binary strings with Hamming distance
between any two strings is at least dmin . The size of the code is the number of elements (codewords) in it. The
following shows the existence of codes with a specified minimum distance and size.
Lemma 13 ([30]). Let
1
2
> ? > 0. There exists a code with dmin ? ?k and size ? 2k(1?h(?)?o(1)) .
Let C be a code satisfying Lemma 13 for k = K and let L = {?c : c ? C} be the set of distributions generated by
using the strings in C. The following result shows that distributions in L are distinguishable and is proved in Appendix
.
?C/4
Lemma 14. The set L is 2e ? ?distinguishable.
Plugging ? = 5 ? 10?5 and C = 60, then Lemma 13 and Equation (1) yields,
Theorem 15. For sufficiently large n,
0.3 ? n1/3 ? R(I?n ).
Acknowledgments
The authors thank Ashkan Jafarpour and Ananda Theertha Suresh for many helpful discussions.
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9
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3,914 | 4,543 | Optimal Regularized Dual Averaging Methods for
Stochastic Optimization
Xi Chen
Machine Learning Department
Carnegie Mellon University
[email protected]
?
Qihang Lin
Javier Pena
Tepper School of Business
Carnegie Mellon University
{qihangl,jfp}@andrew.cmu.edu
Abstract
This paper considers a wide spectrum of regularized stochastic optimization problems where both the loss function and regularizer can be non-smooth. We develop
a novel algorithm based on the regularized dual averaging (RDA) method, that
can simultaneously achieve the optimal convergence rates for both convex and
strongly convex loss. In particular, for strongly convex loss, it achieves the optimal rate of O( N1 + N12 ) for N iterations, which improves the rate O( logNN ) for previous regularized dual averaging algorithms. In addition, our method constructs
the final solution directly from the proximal mapping instead of averaging of all
previous iterates. For widely used sparsity-inducing regularizers (e.g., `1 -norm),
it has the advantage of encouraging sparser solutions. We further develop a multistage extension using the proposed algorithm as a subroutine, which achieves the
uniformly-optimal rate O( N1 + exp{?N }) for strongly convex loss.
1
Introduction
Many risk minimization problems in machine learning can be formulated into a regularized stochastic optimization problem of the following form:
minx?X {?(x) := f (x) + h(x)}.
(1)
Here, the set of feasible solutions X is a convex set in Rn , which is endowed with a norm k ? k and
the dual norm k ? k? . The regularizer h(x) is assumed to be convex, but could be non-differentiable.
Popular examples of h(x) include `1 -norm and related
R sparsity-inducing regularizers. The loss
function f (x) takes the form: f (x) := E? (F (x, ?)) = F (x, ?)dP (?), where ? is a random vector
with the distribution P . In typical regression or classification tasks, ? is the input and response (or
class label) pair. We assume that for every random vector ?, F (x, ?) is a convex and continuous
function in x ? X . Therefore, f (x) is also convex. Furthermore, we assume that there exist
constants L ? 0, M ? 0 and ?
e ? 0 such that
L
?
e
kx ? yk2 ? f (y) ? f (x) ? hy ? x, f 0 (x)i ? kx ? yk2 + M kx ? yk,
2
2
?x, y ? X ,
(2)
where f 0 (x) ? ?f (x), the subdifferential of f . We note that this assumption allows us to adopt a
wide class of loss functions. For example, if f (x) is smooth and its gradient f 0 (x) = ?f (x) is
Lipschitz continuous, we have L > 0 and M = 0 (e.g., squared or logistic loss). If f (x) is nonsmooth but Lipschitz continuous, we have L = 0 and M > 0 (e.g., hinge loss). If ?
e > 0, f (x) is
strongly convex and ?
e is the so-called strong convexity parameter.
In general, the optimization problem in Eq.(1) is challenging since the integration in f (x) is computationally intractable for high-dimensional P . In many learning problems, we do not even know the
underlying distribution P but can only generate i.i.d. samples ? from P . A traditional approach is to
1
consider empirical loss minimization problem where the expectation in fP
(x) is replaced by its emm
1
pirical average on a set of training samples {?1 , . . . , ?m }: femp (x) := m
i=1 F (x, ?i ). However,
for modern data-intensive applications, minimization of empirical loss with an off-line optimization
solver could suffer from very poor scalability.
In the past few years, many stochastic (sub)gradient methods [6, 5, 8, 12, 14, 10, 9, 11, 7, 18] have
been developed to directly solve the stochastic optimization problem in Eq.(1), which enjoy low periteration complexity and the capability of scaling up to very large data sets. In particular, at the t-th
iteration with the current iterate xt , these methods randomly draw a sample ?t from P ; then compute the so-called ?stochastic subgradient? G(xt , ?t ) ? ?x F (xt , ?t ) where ?x F (xt , ?t ) denotes the
subdifferential of F (x, ?t ) with respect to x at xt ; and update xt using G(xt , ?t ). These algorithms
fall into the class of stochastic approximation methods. Recently, Xiao [21] proposed the regularized dual averaging (RDA) method and its accelerated version (AC-RDA) based on Nesterov?s
primal-dual method [17]. Instead of only utilizing a single stochastic subgradient G(xt , ?t ) of the
current iteration, it updates the parameter vector using the average of all past stochastic subgradients
{G(xi , ?i )}ti=1 and hence leads to improved empirical performances.
In this paper, we propose a novel regularized dual averaging method, called optimal RDA or ORDA,
which achieves the optimal expected convergence rate of E[?(b
x) ? ?(x? )], where x
b is the solution
?
from ORDA and x is the optimal solution of Eq.(1). As compared to previous dual averaging
methods, it has three main advantages:
1. For strongly convex f (x), ORDA improves the convergence rate of stochastic
dual aver 2
? 2 log N
log N
? +M 2
L
aging methods O( ?eN ) ? O( ?eN ) [17, 21] to an optimal rate O
+
?
2
?
eN
N
O ?e1N , where ? 2 is the variance of the stochastic subgradient, N is the number of iterations, and the parameters ?
e, M and L of f (x) are defined in Eq.(2).
2. ORDA is a self-adaptive and optimal algorithm for solving both convex and strongly convex f (x) with the strong convexity parameter ?
e as an input. When ?
e = 0, ORDA reduces
to a variant of AC-RDA in [21] with the optimal rate for solving convex f (x). Furthermore, our analysis allows f (x) to be non-smooth while AC-RDA requires the smoothness
of f (x).
e > 0, our algorithm achieves the optimal rate of
For strongly convex f (x) with ?
1
?
eN while AC-RDA does not utilize the advantage of strong convexity.
3. Existing RDA methods [21] and many other stochastic gradient methods (e.g., [14, 10])
PN
PN
can only show the convergence rate for the averaged iterates: x
?N = t=1 %t xt / t=1 %t ,
where the {%t } are nonnegative weights. However, in general, the average iterates x
?N
cannot keep the structure that the regularizer tends to enforce (e.g., sparsity, low-rank,
etc). For example, when h(x) is a sparsity-inducing regularizer (`1 -norm), although xt
computed from proximal mapping will be sparse as t goes large, the averaged solution
could be non-sparse. In contrast, our method directly generates the final solution from the
proximal mapping, which leads to sparser solutions.
In addition to the rate of convergence, we also provide high probability bounds on the error of
objective values. Utilizing a technical lemma from [3], we could show the same high probability
bound as in RDA [21] but under a weaker assumption.
Furthermore, using ORDA
we develop
the multi-stage ORDA which obtains the
2 as 2a subroutine,
p
+
exp{?
?
e
/LN
}
for
strongly convex f (x). Recall that ORDA
convergence rate of O ? ?e+M
N
2
2
has the rate O ? ?e+M
+ NL2 for strongly convex f (x). The rate of muli-stage ORDA improves
N
p
the second term in the rate of ORDA from O NL2 to O exp{? ?
e/LN } and achieves the socalled ?uniformly-optimal ? rate [15]. Although the improvement is on the non-dominating term,
multi-stage ORDA is an optimal algorithm for both stochastic and deterministic optimization. In
particular, for deterministic strongly convex and smooth f (x) (M = 0), one can use the same algorithm but only replaces the stochastic subgradient G(x, ?) by the deterministic gradient ?f (x).
2
2
Then, the variance of the stochastic subgradient ? = 0. Now the term ? ?e+M
in the rate equals
N
to 0 and multi-stage ORDA becomes an optimal deterministic solver with the exponential rate
2
Algorithm 1 Optimal Regularized Dual Averaging Method: ORDA(x0 , N, ?, c)
Input Parameters: Starting point x0 ? X , the number of iterations N , constants ? ? L and c ? 0.
Parameters for f (x): Constants L, M and ?
e for f (x) in Eq. (2) and set ? = ?
e/? .
2
2
Initialization: Set ?t = t+2
; ?t = t+1
; ?t = c(t + 1)3/2 + ? ?; z0 = x0 .
Iterate for t = 0, 1, 2, . . . , N :
1. yt =
(1??t )(?+?t2 ?t )
x
?t2 ?t +(1??t2 )? t
+
(1??t )?t ?+?t3 ?t
z
?t2 ?t +(1??t2 )? t
2. Sample ?t from the distribution P (?) and compute the stochastic subgradient G(yt , ?t ).
P
t
G(yi ,?i )
3. gt = ?t ?t
i=0
?i
n
P
o
t
?V (x,yi )
4. zt+1 = arg minx?X hx, gt i + h(x) + ?t ?t
+ ?t ?t ?t+1 V (x, x0 )
i=0
?i
n
o
5. xt+1 = arg minx?X hx, G(yt , ?t )i + h(x) + ???2 + ??t V (x, yt )
t
Output: xN +1
p
O exp{? ?
e/LN } . This is the reason why such a rate is ?uniformly-optimal?, i.e., optimal
with respect to both stochastic and deterministic optimization.
2
Preliminary and Notations
In the framework of first-order stochastic optimization, the only available information of f (x) is the
stochastic subgradient. Formally speaking, stochastic subgradient of f (x) at x, G(x, ?), is a vectorvalued function such that E? G(x, ?) = f 0 (x) ? ?f (x). Following the existing literature, a standard
assumption on G(x, ?) is made throughout the paper : there exists a constant ? such that ?x ? X ,
E? (kG(x, ?) ? f 0 (x)k2? ) ? ? 2 .
(3)
A key updating step in dual averaging methods, the proximal mapping, utilizes the Bregman divergence. Let ?(x) : X ? R be a strongly convex and differentiable function, the Bregman divergence
associated with ?(x) is defined as:
V (x, y) := ?(x) ? ?(y) ? h??(y), x ? yi.
(4)
1
2
2 kxk2
together with V (x, y) = 12 kx ? yk22 . One may
scale ?(x) so that V (x, y) ? 21 kx ? yk2 for all
One typical and simple example is ?(x) =
refer to [21] for more examples. We can always
x, y ? X . Following the assumption in [10]: we assume that V (x, y) grows quadratically with the
parameter ? > 1, i.e., V (x, y) ? ?2 kx ? yk2 with ? > 1 for all x, y ? X . In fact, we could simply
choose ?(x) with a ? -Lipschitz continuous gradient so that the quadratic growth assumption will be
automatically satisfied.
3
Optimal Regularized Dual Averaging Method
In dual averaging methods [17, 21], the key proximal mapping step utilizes the average of all past
stochastic subgradients
to update the parameter
vector. In particular, it takes the form: zt+1 =
n
o
Pt
1
arg minx?X hgt , xi + h(x) + ?tt V (x, x0 ) , where ?t is the step-size and gt = t+1
i=0 G(zi , ?i ).
2
N
For strongly convex f (x), the current dual averaging methods achieve a rate of O( ? ?elog
N ), which
is suboptimal. In this section, we propose a new dual averaging algorithm which adapts to both
strongly and non-strongly convex f (x) via the strong convexity parameter ?
e and achieves optimal
rates in both cases. In addition, for previous dual averaging methods, to guarantee the convergence,
PN
the final solution takes the form: x
b = N1+1 t=0 zt and hence is not sparse in nature for sparsityinducing regularizers. Instead of taking the average, we introduce another proximal mapping and
generate the final solution directly from the second proximal mapping. This strategy will provide us
sparser solutions in practice. It is worthy to note that in RDA, zN has been proved to achieve the desirable sparsity pattern (i.e., manifold identification property) [13]. However, according to [13], the
3
convergence of ?(zN ) to the optimal ?(x? ) is established only under a more restrictive assumption
that x? is a strong local minimizer of ? relative to the optimal manifold and the convergence rate is
quite slow. Without this assumption, the convergence of ?(zN ) is still unknown.
The proposed optimal RDA (ORDA) method is presented in Algorithm 1. To simplify our notations,
we define the parameter ? = ?
e/? , which scales the strong convexity parameter ?
e by ?1 , where ? is
the quadratic growth constant. In general, the constant ? which defines the step-size parameter ?t
is set to L. However, we allow ? to be an arbitrary constant greater than or equal to L to facilitate
the introduction of the multi-stage ORDA in the later section. The parameter c is set to achieve the
optimal rates for both convex and strongly convex loss.? When ? > 0 (or equivalently, ?
e > 0), c is
)
?
.
Since
x
is
unknown
in practice,
set to 0 so that ?t ? ? ? ? ? L; while for ? = 0, c = ?? (?+M
?
2
V (x ,x0 )
one might replace V (x? , x0 ) in c by a tuning parameter.
Here, we make a few more explanations of Algorithm 1. In Step 1, the intermediate point yt is
a convex combination of xt and zt and when ? = 0, yt = (1 ? ?t )xt + ?t zt . The choice of the
combination
weights is inspired by [10]. Second, with our choice of ?t and ?t , it is easy to prove that
Pt 1
1
t
=
i=0 ?i
?t ?t . Therefore, gt in Step 3 is a convex combination of {G(yi , ?i )}i=0 . As compared
to RDA which uses the average of past subgradients, gt in ORDA is a weighted average of all
past stochastic subgradients and the subgradient from the larger iteration has a larger weight (i.e.,
2(i+1)
G(yi , ?i ) has the weight (t+1)(t+2)
). In practice, instead of storing all past stochastic subgradients,
gt?1
G(yt ,?t )
+
. We also note that
gt could be simply updated based on gt?1 : gt = ?t ?t ?t?1
?t?1
?t
since the error in the stochastic subgradient G(yt , ?t ) will affect the sparsity of xt+1 via the second
proximal mapping, to obtain stable sparsity recovery performances, it would be better to construct
the stochastic subgradient with a small batch of samples [21, 1]. This could help to reduce the noise
of the stochastic subgradient.
3.1
Convergence Rate
We present the convergence rate for ORDA. We start by presenting a general theorem without plugging the values of the parameters. To simplify our notations, we define ?t := G(yt , ?t ) ? f 0 (yt ).
Theorem 1 For ORDA, if we require c > 0 when ?
e = 0, then for any t ? 0:
?(xt+1 ) ? ?(x? ) ? ?t ?t ?t+1 V (x? , x0 ) +
t
?t ? t X
2 i=0
(k?i k? + M )2
?
? ?i
+
?i ?i
?
+ ?t ?t
? ?i L ?i
t
X
hx? ? zbi , ?i i
, (5)
?i
i=0
(1?? )?+? ? 2
?t ?
t
t t
where zbt = ?+?
zt , is a convex combination of yt and zt ; and zbt = zt when
2 yt +
?+?t ?t2
t ?t
? = 0. Taking the expectation on both sides of Eq.(5):
E?(xt+1 ) ? ?(x? ) ? ?t ?t ?t+1 V (x? , x0 ) + (? 2 + M 2 )?t ?t
t
X
1
i=0
?
? ?i
+
?i ?i
?
.
? ?i L ? i
(6)
The proof of Theorem 1 is given in Appendix. In the next two corollaries, we establish the rates of
convergence in expectation for ORDA by choosing different values for c based on ?
e.
?
)
Corollary 1 For convex f (x) with ?
e = 0 , by setting c = ?? (?+M
and ? = L, we obtain:
?
2
E?(xN +1 ) ? ?(x? ) ?
V (x ,x0 )
p
4? LV (x? , x0 ) 8(? + M ) ? V (x? , x0 )
?
+
.
N2
N
(7)
Based on Eq.(6), the proof of Corollary 1 is straightforward with the details in Appendix. Since x?
is unknown in practice, one could set c by replacing V (x? , x0 ) in c with any value D? ? V (x? , x0 ).
By doing so, Eq.(7) remains valid after replacing all V (x? , x0 ) by D? . For convex f (x) with ?
e = 0,
the rate in Eq.(7) has achieved the uniformly-optimal rate according to [15]. In fact, if f (x) is a
deterministic and smooth function with ? = M = 0 (e.g., smooth empirical loss), one only needs
4
to change the stochastic subgradient G(yt , ?t ) to ?f (yt ). The resulting algorithm, which reduces to
?
,x0 )
Algorithm 3 in [20], is an optimal deterministic first-order method with the rate O( LV (x
).
N2
We note that the quadratic growth assumption of V (x, y) is not necessary for convex f (x).
If one doesn not assume this assumption
andreplaces the
o last step in ORDA by xt+1 =
?
?t
2
arg minx?X hx, G(yt , ?t )i + h(x) + 2?2 + 2 kx ? yt k , we can achieve the same rate as in
t
Eq.(7) but just removing all ? from the right hand side. But the quadratic growth assumption is
indeed required for showing the convergence for strongly convex f (x) as in the next corollary.
Corollary 2 For strongly convex f (x) with ?
e > 0, we set c = 0 and ? = L and obtain that:
4? LV (x? , x0 ) 4? (? 2 + M 2 )
+
.
(8)
N2
?N
N
1
, is optimal and better than the O log
rate for previous
The dominating term in Eq.(8), O ?N
?N
dual averaging methods. However, ORDA has not achieved the uniformly-optimal rate, which takes
p?
2
+M 2
the form of O( ? ?N
+exp(? L
N )). In particular, for deterministic smooth and strongly convex
f (x) (i.e., empirical loss with ? = M = 0), ORDA only achieves the rate of O( NL2 ) while the
p?
optimal deterministic rate should be O exp(? L
N ) [16]. Inspired by the multi-restart technique
in [7, 11], we present a multi-stage extension of ORDA in Section 4 which achieves the uniformlyoptimal convergence rate.
E?(xN +1 ) ? ?(x? ) ?
3.2
High Probability Bounds
For stochastic optimization problems, another important evaluation criterion is the confidence
level of the objective value. In particular, it is of great interest to find (N, ?) as a monotonically decreasing function in both N and ? ? (0, 1) such that the solution xN +1 satisfies
Pr (?(xN +1 ) ? ?(x? ) ? (N, ?)) ? ?. In other words, we want to show that with probability
at least 1 ? ?, ?(xN +1 ) ? ?(x? ) < (N, ?). According to Markov inequality, for any > 0,
?
?
Pr(?(xN +1 ) ? ?(x? ) ? ) ? E(?(xN +1)??(x )) . Therefore, we have (N, ?) = E?(xN +1?)??(x ) .
Under the basic assumption in Eq.(3), namely
?) ? f 0 (x)k2? ) ? ? 2 , and according to
? ? E? (kG(x,
2
(?+M ) V (x ,x0 )
? +M 2
?
Corollary 1 and 2, (N, ?) = O
for
convex
f
(x),
and
(N,
?)
=
O
?N ?
N?
for strongly convex f (x).
However, the above bounds are quite loose. To obtain tighter bounds, we strengthen the basic
assumption of the stochastic
to the ?light-tail? assumption [14]. In particular,
subgradient in Eq. (3)
we assume that E exp kG(x, ?) ? f 0 (x)k2? /? 2
? exp{1}, ?x ? X . By further making the
boundedness assumption (kx? ? zbt k ? D) and utilizing?a technical lemma from [3], we obtain a
ln(1/?)D?
?
much tighter high probability bound with (N, ?) = O
for both convex and strongly
N
convex f (x). The details are presented in Appendix.
4
Multi-stage ORDA for Stochastic Strongly Convex Optimization
As we show in Section 3.1, for convex f (x), ORDA achieves the uniformly-optimal rate. However, for strongly convex f (x), although the dominating term of the convergence rate in Eq.(8) is
optimal, the overall rate is not uniformly-optimal. Inspired by the multi-stage stochastic approximation methods [7, 9, 11], we propose the multi-stage extension of ORDA in Algorithm 2 for
stochastic strongly convex optimization. For each stage 1 ? k ? K, we run ORDA in Algorithm
1 as a sub-routine for Nk iterations with the parameter ?t = c(t + 1)3/2 + ? ? with c = 0 and
? = ?k + L. Roughly speaking, we set Nk = 2Nk?1 and ?k = 4?k?1 . In other words, we double
the number of iterations for the next stage but reduce the step-size. The multi-stage ORDA has
achieved uniformly-optimal convergence rate as shown in Theorem 2 with the proof in Appendix.
The proof technique follows the one in [11]. Due this specialized proof technique, instead of showing E(?(xN )) ? ?(x? ) ? (N ) as in ORDA, we show the number of iterations N () to achieve the
-accurate solution: E(?(xN () )) ? ?(x? ) ? . But the two convergence rates are equivalent.
5
Algorithm 2 Multi-stage ORDA for Stochastic Strongly Convex Optimization
Initialization: x0 ? X , a constant V0 ? ?(x0 ) ? ?(x? ) and the number of stages K.
Iterate for k = 1, 2, . . . , K:
n q
o
k+9
(? 2 +M 2 )
1. Set Nk = max 4 ??L , 2 ??V
0
q
3/2
2k?1 ?(? 2 +M 2 )
2. Set ?k = Nk
? V0
3. Generate x
ek by calling the sub-routine ORDA(e
xk?1 , Nk , ? = ?k + L, c = 0)
Output: x
eK
Theorem 2 If we run multi-stage ORDA for K stages with K = log2 V0 for any given , we have
E(?(e
xK )) ? ?(x? ) ? and the total number of iterations is upper bounded by:
s
K
X
?L
V0
1024? (? 2 + M 2 )
log2
+
.
(9)
N=
Nk ? 4
?
?
k=1
5
Related Works
In the last few years, a number of stochastic gradient methods [6, 5, 8, 12, 14, 21, 10, 11, 7, 4, 3] have
been developed to solve Eq.(1), especially for a sparsity-inducing h(x). In Table 1, we compare the
proposed ORDA and its multi-stage extension with some widely used stochastic gradient methods
using the following metrics. For the ease of comparison, we assume f (x) is smooth with M = 0.
1. The convergence rate for solving (non-strongly) convex f (x) and whether this rate has
achieved the uniformly-optimal (Uni-opt) rate.
2. The convergence rate for solving
convex f (x) and whether (1) the dominating
strongly
term of rate is optimal, i.e., O
?2
?
eN
and (2) the overall rate is uniformly-optimal.
3. Whether the final solution x
b, on which the results of convergence are built, is generated
from the weighted average of previous iterates (Avg) or from the proximal mapping (Prox).
For sparsity-inducing regularizers, the solution directly from the proximal mapping is often
sparser than the averaged solution.
4. Whether an algorithm allows to use a general Bregman divergence in proximal mapping or
it only allows the Euclidean distance V (x, y) = 21 kx ? yk22 .
In Table 1, the algorithms in the first 7 rows are stochastic approximation algorithms where only
the current stochastic gradient is used at each iteration. The last 4 rows are dual averaging methods
where all past subgradients are used. Some algorithms in Table 1 make a more restrictive assumption
on the stochastic gradient: ?G > 0, EkG(x, ?)k2? ? G2 , ?x ? X . It is easy to verify that this
assumption implies our basic assumption in Eq.(3) by Jensen?s inequality.
As we can see from Table 1, the proposed ORDA possesses all good properties except that the
convergence rate for strongly convex f (x) is not uniformly-optimal. Multi-stage ORDA further
improves this rate to be uniformly-optimal. In particular, SAGE [8] achieves a nearly optimal rate
since the parameter D in the convergence rate is chosen such that E kxt ? x? k22 ? D for all t ? 0
and it could be much larger than V ? V (x? , x0 ). In addition, SAGE requires the boundedness of the
domain X , the smoothness of f (x), and only allows the Euclidean distance in proximal mapping.
As compared to AC-SA [10] and multi-stage AC-SA [11], our methods do not require the final
averaging step; and as shown in our experiments, ORDA has better empirical performances due
to the usage of all past stochastic subgradients. Furthermore, we improve the rates of RDA and
extend AC-RDA to an optimal algorithm for both convex and strongly convex f (x). Another highly
relevant work is [9]. Juditsky et al. [9] proposed multi-stage algorithms to achieve the optimal
strongly convex rate based on non-accelerated dual averaging methods. However, the algorithms in
[9] assume that ?(x) is a Lipschitz continuous function, i.e., the subgradient of ?(x) is bounded.
Therefore, when the domain X is unbounded, the algorithms in [9] cannot be directly applied.
6
FOBOS [6]
COMID [5]
SAGE [8]
AC-SA [10]
Convex f (x)
Rate
Uni-opt
?
V
G
?
O
NO
?N
V
G
O ?N
NO
?
?? D
LD
O
+ N 2 NEARLY
?N
?? V
O
+ LV
YES
N2
N
M-AC-SA [11] NA
NA
Epoch-GD [7] NA
?
RDA [21]
O G?NV
?
AC-RDA [21] O ??NV +
?
ORDA
O ??NV +
NA
M-ORDA
NA
NO
LV
N2
YES
LV
N2
YES
NA
Strongly Convex f (x)
Rate
Opt
2
G log N
O
NO
2?eN
G log N
O
NO
2?eN
?
LD
O ?eN + N 2
YES
2
?
LV
O ?eN + N 2
YES
q
2
?
e
N } YES
O ?e?N + exp{? L
2
O ?eGN
YES
2
G log N
O
NO
?
eN
Uni-opt
Final x
b Bregman
NO
Prox
NO
NO
Prox
YES
NO
Prox
NO
NO
Avg
YES
YES
Avg
YES
NO
Avg
NO
NO
Avg
YES
Avg
YES
Prox
YES
Prox
YES
NA
NA NA
2
?
LV
O ?eN + N 2
YES NO
q
2
?
e
O ?e?N + exp{? L
N } YES YES
Table 1: Summary for different stochastic gradient algorithms. V is short for V (x? , x0 ); AC for ?accelerated?;
M for ?multi-stage? and NA stands for either ?not applicable? or ?no analysis of the rate?.
2
Recently, the paper [18] develops another stochastic gradient method which achieves the rate O( ?eGN )
for strongly convex f (x). However, for non-smooth f (x), it requires the averaging of the last a few
iterates and this rate is not uniformly-optimal.
6
Simulated Experiments
In this section, we conduct simulated experiments to demonstrate the performance of ORDA and
its multi-stage extension (M ORDA). We compare our ORDA and M ORDA (only for strongly
convex loss) with several state-of-the-art stochastic gradient methods, including RDA and AC-RDA
[21], AC-SA [10], FOBOS [6] and SAGE [8]. For a fair comparison, we compare all different
methods using solutions which have expected convergence guarantees. For all algorithms, we tune
the parameter related to step-size (e.g., c in ORDA for convex loss) within an appropriate range and
choose the one that leads to the minimum objective value.
In this experiment, we solve a sparse linear regression problem: minx?Rn f (x)+h(x) where f (x) =
?
1
T
2
2
2 Ea,b ((a x ? b) ) + 2 kxk2 and h(x) = ?kxk1 . The input vector a is generated from N (0, In?n )
T ?
and the response b = a x + , where x?i = 1 for 1 ? i ? n/2 and 0 otherwise and the noise
? N (0, 1). When ? = 0, th problem is the well known Lasso [19] and when ? > 0, it is known
as Elastic-net [22]. The regularization parameter ? is tuned so that a deterministic solver on all the
samples can correctly recover the underlying sparsity pattern. We set n = 100 and create a large
pool of samples for generating stochastic gradients and evaluating objective values. The number
of iterations N is set to 500. Since we focus on stochastic optimization instead of online learning,
we could randomly draw samples from an underlying distribution. So we construct the stochastic
gradient using the mini-batch strategy [2, 1] with the batch size 50. We run each algorithm for 100
times and report the mean of the objective value and the F1-score for sparsity recovery performance.
Pp
Pp
precision?recall
where precision =
F1-score is defined as 2 precision+recall
xi =1,x?
=1} /
xi =1} and
i=1 1{b
i=1 1{b
i
Pp
Pp
?
?
recall = i=1 1{bxi =1,xi =1} / i=1 1{xi =1} . The higher the F1-score is, the better the recovery
ability of the sparsity pattern. The standard deviations for both objective value and the F1-score in
100 runs are very small and thus omitted here due to space limitations.
We first set ? = 0 to test algorithms for (non-strongly) convex f (x). The result is presented in
Table 2 (the first two columns). We also plot the decrease of the objective values for the first 200
iterations in Figure 1. From Table 2, ORDA performs the best in both objective value and recovery
ability of sparsity pattern. For those optimal algorithms (e.g., AC-RDA, AC-SA, SAGE, ORDA),
they achieve lower final objective values and the rates of the decrease are also faster. We note that
for dual averaging methods, the solution generated from the (first) proximal mapping (e.g., zt in
7
Table 2: Comparisons in objective
value and F1-score.
28
31
RDA
AC?RDA
AC?SA
FOBOS
SAGE
ORDA
27
26
25
24
23
29
28
27
26
22
25
21
24
20
50
100
150
200
RDA
AC?RDA
AC?SA
FOBOS
SAGE
ORDA
30
Objective
?=1
Obj F1
21.57 0.67
21.12 0.67
21.01 0.67
21.19 0.84
21.09 0.73
20.97 0.87
20.98 0.88
Objective
?=0
Obj F1
RDA
20.87 0.67
AC-RDA 20.67 0.67
AC-SA
20.66 0.67
FOBOS 20.98 0.83
SAGE
20.65 0.82
ORDA
20.56 0.92
M ORDA N.A. N.A.
23
50
100
150
200
Iteration
Iteration
Figure 1: Obj for Lasso.
Figure 2: Obj for Elastic-Net.
ORDA) has almost perfect sparsity recovery performance. However, since here is no convergence
guarantee for that solution, we do not report results here.
Objective
Then we set ? = 1 to test algorithms for solving
strongly convex f (x). The results are presented
30
in Table 2 (the last two columns) and Figure
ORDA
M_ORDA
2 and 3. As we can see from Table 2, ORDA
28
and M ORDA perform the best. Although
M ORDA achieves the theoretical uniformly26
optimal convergence rate, the empirical per24
formance of M ORDA is almost identical to
that of ORDA. This observation is consistent
22
with our theoretical analysis since the improvement of the convergence rate only appears on
20
the non-dominating term. In addition, ORDA,
100
200
300
400
500
Iteration
M ORDA, AC-SA and SAGE with the convergence rate O( ?e1N ) achieve lower objective valFigure 3: ORDA v.s. M ORDA.
ues as compared to other algorithms with the
N
)
.
For
better
visualization,
we
do
rate O( log
?
eN
not include the comparison between M ORA and ORDA in Figure 2. Instead, we present the comparison separately in Figure 3. From Figure 3, the final objective values of both algorithms are very
close. An interesting observation is that, for M ORDA, each time when a new stage starts, it leads
to a sharp increase in the objective value following by a quick drop.
7
Conclusions and Future Works
In this paper, we propose a new dual averaging method which achieves the optimal rates for solving
stochastic regularized problems with both convex and strongly convex loss functions. We further
propose a multi-stage extension to achieve the uniformly-optimal convergence rate for strongly convex loss.
Although we study stochastic optimization problems in this paper, our algorithms can be easily
converted into online optimization approaches, where a sequence of decisions {xt }N
t=1 are generated
according to Algorithm 1 or 2. We often measure the quality of an online learning algorithm via the
PN
so-called regret, defined as RN (x? ) = t=1 (F (xt , ?t ) + h(xt )) ? (F (x? , ?t ) + h(x? )) . Given
the expected convergence rate in Corollary 1 and 2, the expected regret can be easily derived. For
PN
PN
1
?
example, for strongly convex f (x): ERN (x? ) ?
t=1 (E(?(xt )) ? ?(x )) ?
t=1 O( t ) =
O(ln N ). However, it would be a challenging future work to derive the regret bound for ORDA
instead of the expected regret. It would also be interesting to develop the parallel extensions of
ORDA (e.g., combining the distributed mini-batch strategy in [21] with ORDA) and apply them to
some large-scale real problems.
8
References
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gradient methods. In Advances in Neural Information Processing Systems (NIPS), 2011.
[2] O. Dekel, R. Gilad-Bachrach, O. Shamir, and L. Xiao. Optimal distributed online prediction
using mini-batches. Technical report, Microsoft Research, 2011.
[3] J. Duchi, P. L. Bartlett, and M. Wainwright. Randomized smoothing for stochastic optimization. arXiv:1103.4296v1, 2011.
[4] J. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and
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[5] J. Duchi, S. Shalev-Shwartz, Y. Singer, and A. Tewari. Composite objective mirror descent. In
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[9] A. Juditsky and Y. Nesterov. Primal-dual subgradient methods for minimizing uniformly convex functions. August 2010.
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Wiley New York, 1983.
[16] Y. Nesterov. Introductory lectures on convex optimization: a basic course. Kluwer Academic
Pub, 2003.
[17] Y. Nesterov. Primal-dual subgradient methods for convex problems. Mathematical Programming, 120:221?259, 2009.
[18] A. Rakhlin, O. Shamir, and K. Sridharan. To average or not to average? making stochastic gradient descent optimal for strongly convex problems. In International Conference on Machine
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1996.
[20] P. Tseng. On accelerated proximal gradient methods for convex-concave optimization. SIAM
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[22] H. Zou and T. Hastie. Regularization and variable selection via the elastic net. J. R. Statist.
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9
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3,915 | 4,544 | Learning the Architecture of Sum-Product Networks
Using Clustering on Variables
Dan Ventura
Department of Computer Science
Brigham Young University
Provo, UT 84602
[email protected]
Aaron Dennis
Department of Computer Science
Brigham Young University
Provo, UT 84602
[email protected]
Abstract
The sum-product network (SPN) is a recently-proposed deep model consisting of
a network of sum and product nodes, and has been shown to be competitive with
state-of-the-art deep models on certain difficult tasks such as image completion.
Designing an SPN network architecture that is suitable for the task at hand is an
open question. We propose an algorithm for learning the SPN architecture from
data. The idea is to cluster variables (as opposed to data instances) in order to
identify variable subsets that strongly interact with one another. Nodes in the SPN
network are then allocated towards explaining these interactions. Experimental
evidence shows that learning the SPN architecture significantly improves its performance compared to using a previously-proposed static architecture.
1
Introduction
The number of parameters in a textbook probabilistic graphical model (PGM) is an exponential
function of the number of parents of the nodes in the graph. Latent variables can often be introduced
such that the number of parents is reduced while still allowing the probability distribution to be
represented. Figure 1 shows an example of modeling the relationship between symptoms of a set of
diseases. The PGM at the left has no latent variables and the PGM at the right has an appropriately
added ?disease? variable. The model is able to be simplified because the symptoms are statistically
independent of one another given the disease. The middle PGM shows a model in which the latent
variable is introduced to no simplifying effect, demonstrating the need to be intelligent about what
latent variables are added and how they are added.
S1
S2
S3
(a)
S1
D
S2
S3
(b)
D
S1
S2
S3
(c)
Figure 1: Introducing a latent variable. The PGM in
(a) has no latent variables. The PGM in (b) has a latent
variable introduced to no beneficial effect. The PGM
in (c) has a latent variable that simplifies the model.
1
Deep models can be interpreted as PGMs
that introduce multiple layers of latent
variables over a layer of observed variables [1]. The architecture of these latent
variables (the size of the layers, the number of variables, the connections between
variables) can dramatically affect the performance of these models. Selecting a reasonable architecture is often done by hand.
This paper proposes an algorithm that automatically learns a deep architecture from
data for a sum-product network (SPN), a
recently-proposed deep model that takes
advantage of the simplifying effect of latent variables [2]. Learning the appropri-
+
x
+
x
+
+
+
+
?a
?a
?b
?b
A
B
x
+
Figure 2: A simple SPN over two binary variables A
and B. The leaf node ?a takes value 1 if A = 0 and
0 otherwise while leaf node ?a takes value 1 if A = 1
and 0 otherwise. If the value of A is not known then
both leaf nodes take value 1. Leaf nodes ?b and ?b behave similarly. Weights on the edges connecting sum
nodes with their children are not shown. The shortdashed edge causes the SPN to be incomplete. The
long-dashed edge causes the SPN to be inconsistent.
x
+
+
x
+
+
+
Figure 3: The Poon architecture with m = 1
sum nodes per region. Three product nodes
are introduced because the 2?3-pixel image
patch can be split vertically and horizontally
in three different ways. In general the Poon
architecture has number-of-splits times m2
product nodes per region.
ate architecture for a traditional deep model can be challenging [3, 4], but the nature of SPNs lend
themselves to a remarkably simple, fast, and effective architecture-learning algorithm.
In proposing SPNs, Poon & Domingos introduce a general scheme for building an initial SPN architecture; the experiments they run all use one particular instantiation of this scheme to build an
initial ?fixed? architecture that is suitable for image data. We will refer to this architecture as the
Poon architecture. Training is done by learning the parameters of an initial SPN; after training is
complete, parts of the SPN may be pruned to produce a final SPN architecture. In this way both the
weights and architecture are learned from data.
We take this a step further by also learning the initial SPN architecture from data. Our algorithm
works by finding subsets of variables (and sets of subsets of variables) that are highly dependent and
then effectively combining these together under a set of latent variables. This encourages the latent
variables to act as mediators between the variables, capturing and representing the dependencies
between them. Our experiments show that learning the initial SPN architecture in this way improves
its performance.
2
Sum-Product Networks
Sum-product networks are rooted, directed acyclic graphs (DAGs) of sum, product, and leaf nodes.
Edges connecting sum nodes to their children are weighted using non-negative weights. The value
of a sum node is computed as the dot product of its weights with the values of it child nodes. The
value of a product node is computed by multiplying the values of its child nodes. A simple SPN is
shown in Figure 2.
Leaf node values are determined by the input to the SPN. Each input variable has an associated set
of leaf nodes, one for each value the variable can take. For example, a binary variable would have
two associated leaf nodes. The leaf nodes act as indicator functions, taking the value 1 when the
variable takes on the value that the leaf node is responsible for and 0 otherwise.
An SPN can be constructed such that it is a representation of some probability distribution, with
the value of its root node and certain partial derivatives with respect to the root node having probabilistic meaning. In particular, all marginal probabilities and many conditional probabilities can be
computed [5]. Consequently an SPN can perform exact inference and does so efficiently when the
size of the SPN is polynomial in the number of variables.
2
If an SPN does represent a probability distribution then we call it a valid SPN; of course, not all
SPNs are valid, nor do they all facilitate efficient, exact inference. However, Poon & Domingos
proved that if the architecture of an SPN follows two simple rules then it will be valid. (Note that
this relationship does not go both ways; an SPN may be valid and violate one or both of these rules.)
This, along with showing that SPNs can represent a broader class of distributions than other models
that allow for efficient and exact inference are the key contributions made by Poon & Domingos.
To understand these rules it will help to know what the ?scope of an SPN node? means. The scope
of an SPN node n is a subset of the input variables. This subset can be determined by looking at the
leaf nodes of the subgraph rooted at n. All input variables that have one or more of their associated
leaf nodes in this subgraph are included in the scope of the node. We will denote the scope of n as
scope(n).
The first rule is that all children of a sum node must have the same scope. Such an SPN is called
complete. The second rule is that for every pair of children, (ci , cj ), of a product node, there must
not be contradictory leaf nodes in the subgraphs rooted at ci and cj . For example, if the leaf node
corresponding to the variable X taking on value x is in the subgraph rooted at ci , then the leaf nodes
corresponding to the variable X taking on any other value may not appear in the subgraph rooted at
cj . An SPN following this rule is called consistent. The SPN in Figure 2 violates completeness (due
to the short-dashed arrow) and it violates consistency (due to the long-dashed arrow).
An SPN may also be decomposable, which is a property similar to, but somewhat more restrictive
than consistency. A decomposable SPN is one in which the scopes of the children of each product
node are disjoint. All of the architectures described in this paper are decomposable.
Very deep SPNs can be built using these rules as a guide. The number of layers in an SPN can be
on the order of tens of layers, whereas the typical deep model has three to five layers. Recently it
was shown that deep SPNs can compute some functions using exponentially fewer resources than
shallow SPNs would need [6].
The Poon architecture is suited for modeling probability distributions over images, or other domains
with local dependencies among variables. It is constructed as follows. For every possible axisaligned rectangular region in the image, the Poon architecture includes a set of m sum nodes, all
of whose scope is the set of variables associated with the pixels in that region. Each of these (nonsingle-pixel) regions are conceptually split vertically and horizontally in all possible ways to form
pairs of rectangular subregions. For each pair of subregions, and for every possible pairing of sum
nodes (one taken from each subregion), a product node is introduced and made the parent of the
pair of sum nodes. The product node is also added as a child to all of the top region?s sum nodes.
Figure 3 shows a fragment of a Poon architecture SPN modeling a 2 ? 3 image patch.
3
Cluster Architecture
As mentioned earlier, care needs to be taken when introducing latent variables into a model. Since
the effect of a latent variable is to help explain the interactions between its child variables [7], it
makes little sense to add a latent variable as the parent of two statistically independent variables.
In the example in Figure 4, variables W and
X strongly interact and variables Y and Z
do as well. But the relationship between all
other pairs of variables is weak. The PGM
in (a), therefore, allows latent variable A to
take account of the interaction between W
and X. On the other hand, variable A does
little in the PGM in (b) since W and Y are
nearly independent. A similar argument can
be made about variable B. Consequently,
variable C in the PGM in (a) can be used to
explain the weak interactions between variables, whereas in the PGM in (b), variable
C essentially has the task of explaining the
interaction between all the variables.
C
C
A
W
B
X
Y
(a)
Z
W
A
B
X
Y
Z
(b)
Figure 4: Latent variables explain the interaction between child variables, causing the children to be independent given the latent variable parent. If variable
pairs (W, X) and (Y, Z) strongly interact and other
variable pairs do not, then the PGM in (a) is a more
suitable model than the PGM in (b).
3
In the probabilistic interpretation of an SPN, sum nodes are associated with latent variables. (The
evaluation of a sum node is equivalent to summing out its associated latent variable.) Each latent
variable helps the SPN explain interactions between variables in the scope of the sum nodes. Just as
in the example, then, we would like to place sum nodes over sets of variables with strong interactions.
The Poon architecture takes this principle into account. Images exhibit strong interactions between
pixels in local spatial neighborhoods. Taking advantage of this prior knowledge, the Poon architecture chooses to place sum nodes over local spatial neighborhoods that are rectangular in shape.
There are a few potential problems with this approach, however. One is that the Poon architecture
includes many rectangular regions that are long and skinny. This means that the pixels at each
end of these regions are grouped together even though they probably have only weak interactions.
Some grouping of weakly-interacting pixels is inevitable, but the Poon architecture probably does
this more than is needed. Another problem is that the Poon architecture has no way of explaining
strongly-interacting, non-rectangular local spatial regions. This is a major problem because such
regions are very common in images. Additionally, if the data does not exhibit strong spatially-local
interactions then the Poon architecture could perform poorly.
Our proposed architecture (we will call it the cluster architecture) avoids these problems. Large
regions containing non-interacting pixels are avoided. Sum nodes can be placed over spatially-local,
non-rectangular regions; we are not restricted to rectangular regions, but can explain arbitrarilyshaped blob-like regions. In fact, the regions found by the cluster architecture are not required to
exhibit spatial locality. This makes our architecture suitable for modeling data that does not exhibit
strong spatially-local interactions between variables.
3.1
Building a Cluster Architecture
As was described earlier, a sum node s in an SPN has the task of explaining the interactions between
all the variables in its scope. Let scope(s) = {V1 , ? ? ? , Vn }. If n is large, then this task will likely
be very difficult. SPNs have a mechanism for making it easier, however. Essentially, s delegates
part of its responsibilities to another set of sum nodes. This is done by first
S forming a partition
of scope(s), where {S1 , ? ? ? , Sk } is a partition of scope(s) if and only if i Si = scope(s) and
?i, j(Si ? Sj = ?). Then, for each subset Si in the partition, an additional sum node si is introduced
into the SPN and is given the task of explaining the interactions between all the variables in Si . The
original sum node s is then given a new child product node p and the product node becomes the
parent of each sum node si .
In this example the node s is analogous to the variable C in Figure 4 and the nodes si are analogous
to the variables A and B. So this partitioning process allows s to focus on explaining the interactions
between the nodes si and frees it from needing to explain everything about the interactions between
the variables {V1 , ? ? ? , Vn }. And, of course, the partitioning process can be repeated recursively,
with any of the nodes si taking the place of s.
This is the main idea behind the algorithm for building a cluster architecture (see Algorithm 1 and
Algorithm 2). However, due to the architectural flexibility of an SPN, discussing this algorithm in
terms of sum and product nodes quickly becomes tedious and confusing. The following definition
will help in this regard.
Definition 1. A region graph is a rooted DAG consisting of region nodes and partition nodes. The
root node is a region node. Partition nodes are restricted to being the children of region nodes and
vice versa. Region and partition nodes have scopes just like nodes in an SPN. The scope of a node
n in a region graph is denoted scope(n).
Region nodes can be thought of as playing the role of sum nodes (explaining interactions among
variables) and partition nodes can be thought of as playing the role of product nodes (delegating
responsibilities). Using the definition of the region graph may not appear to have made things any
simpler, but its benefits will become more clear when discussing the conversion of region graphs to
SPNs (see Figure 5).
At a high level the algorithm for building a cluster architecture is simple: build a region graph
(Algorithm 1 and Algorithm 2), then convert it to an SPN (Algorithm 3). These steps are described
below.
4
R1
Algorithm 1 BuildRegionGraph
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
Input: training data D
C 0 ? Cluster(D, 1)
for k = 2 to ? do
C ? Cluster(D, k)
r ? Quality(C)/Quality(C 0 )
if r < 1 + ? then
break
else
C0 ? C
G ? CreateRegionGraph()
n ? AddRegionNodeTo(G)
for i = 1 to k do
ExpandRegionGraph(G, n, Ci )
x
P1
x
+
+
x
x
x
P2
x
x
x
R1
R2
R2
R3
R4
(a)
R5
x
+
+
... x
R3
x
+
+
+
... x
x
+
+
R4
... x
x
+
R5
... x
(b)
Figure 5: Subfigure (a) shows a region graph fragment consisting of region nodes R1 , R2 , R3 , R4 , and R5 . R1 has
two parition nodes (the smaller, filled-in nodes). Subfigure
(b) shows the region graph converted to an SPN. In the SPN
each region is allotted two sum nodes. The product nodes
in R1 are surrounded by two rectangles labeled P1 and P2 ;
they correspond to the partition nodes in the region graph.
Algorithm 1 builds a region graph using training data to guide the construction. In lines 2 through 9
the algorithm clusters the training instances into k clusters C = {C1 , ? ? ? , Ck }. Our implementation
uses the scikit-learn [8] implementation of k-means to cluster the data instances, but any clustering
method could be used. The value for k is chosen automatically; larger values of k are tried until
increasing the value does not substantially improve a cluster-quality score. The remainder of the
algorithm creates a single-node region graph G and then adds nodes and edges to G using k calls to
Algorithm 2 (ExpandRegionGraph). To encourage the expansion of G in different ways, a different
subset of the training data, Ci , is passed to ExpandRegionGraph on each call.
At a high level, Algorithm 2 partitions scopes into sub-scopes recursively, adding region and partition nodes to G along the way. The initial call to ExpandRegionGraph partitions the scope of
the root region node. A corresponding partition node is added as a child of the root node. Two
sub-region nodes (whose scopes form the partition) are then added as children to the partition node.
Algorithm 2 is then called recursively with each of these sub-region nodes as arguments (unless the
scope of the sub-region node is too small).
In line 3 of Algorithm 2 the PartitionScope function in our implementation uses the k-means algorithm in an unusual way. Instead of partitioning the instances of the training dataset D into k
instance-clusters, it partitions variables into k variable-clusters as follows. D is encoded as a matrix,
each row being a data instance and each column corresponding to a variable. Then k-means is run
on DT , causing it to partition the variables into k clusters. Actually, the PartitionScope function
is only supposed to partition the variables in scope(n), not all the variables (note its input parameter). So before calling k-means we build a new matrix Dn by removing columns from D, keeping
only those columns that correspond to variables in scope(n). Then k-means is run on DnT and the
resulting variable partition is returned. The k-means algorithm serves the purpose of detecting subsets of variables that strongly interact with one another. Other methods (including other clustering
algorithms) could be used in its place.
After the scope Sn of a node n has been partitioned into S1 and S2 , Algorithm 2 (lines 4 through 11)
looks for region nodes in G whose scope is similar to S1 or S2 ; if region node r with scope Sr is
such a node, then S1 and S2 are adjusted so that S1 = Sr and {S1 , S2 } is still a partition of Sn .
Lines 12 through 18 expand the region graph based on the partition of Sn . If node n does not already
have a child partition node representing the partition {S1 , S2 } then one is created (p in line 15); p is
then connected to child region nodes n1 and n2 , whose scopes are S1 and S2 , respectively.
Note that n1 and n2 may be newly-created region nodes or they may be nodes that were created during a previous call to Algorithm 2. We recursively call ExpandRegionGraph only on newly-created
nodes; the recursive call is also not made if the node is a leaf node (|Si | = 1) since partitioning a
leaf node is not helpful (see lines 19 through 22).
5
Algorithm 2 ExpandRegionGraph
Algorithm 3 BuildSPN
Input: region graph G, sums per region m
Output: SPN S
R ? RegionNodesIn(G)
for all r ? R do
if IsRootNode(r) then
N ? AddSumNodesToSPN(S, 1)
else
N ? AddSumNodesToSPN(S, m)
P ? ChildPartitionNodesOf(r)
for all p ? P do
C ? ChildrenOf(p)
O ? AddProductNodesToSPN(S, m|C| )
for all n ? N do
AddChildrenToSumNode(n, O)
Q ? empty list
for all c ? C do
//We assume the sum nodes associated
//with c have already been created.
U ? SumNodesAssociatedWith(c)
AppendToList(Q, U )
ConnectProductsToSums(O, Q)
return S
1: Input: region graph G,
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
region node n in G, training data D
Sn ? scope(n)
{S1 , S2 } ? PartitionScope(Sn , D)
S ? ScopesOfAllRegionNodesIn(G)
for all Sr ? S s.t. Sr ? Sn do
p1 ? |S1 ? Sr |/|S1 ? Sr |
p2 ? |S2 ? Sr |/|S2 ? Sr |
if max{p1 , p2 } > threshold then
S1 ? Sr
S2 ? Sn \ Sr
break
n1 ? GetOrCreateRegionNode(G, S1 )
n2 ? GetOrCreateRegionNode(G, S2 )
if PartitionDoesNotExist(G, n, n1 , n2 ) then
p ? NewPartitionNode()
AddChildToRegionNode(n, p)
AddChildToPartitionNode(p, n1 )
AddChildToPartitionNode(p, n2 )
if S1 ?
/ S ? |S1 | > 1 then
ExpandRegionGraph(G, n1 )
if S2 ?
/ S ? |S2 | > 1 then
ExpandRegionGraph(G, n2 )
After the k calls to Algorithm 2 have been made, the resulting region graph must be converted to
an SPN. Figure 5 shows a small subgraph from a region graph and its conversion into an SPN;
this example demonstrates the basic pattern that can be applied to all region nodes in G in order
to generate an SPN. A more precise description of this conversion is given in Algorithm 3. In
this algorithm the assumption is made (noted in the comments) that certain sum nodes are inserted
before others. This assumption can be guaranteed if the algorithm performs a postorder traversal of
the region nodes in G in the outermost loop. Also note that the ConnectProductsToSums method
connects product nodes of the current region with sum nodes from its subregions; the children of a
product node consist of a single node drawn from each subregion, and there is a product node for
every possible combination of such sum nodes.
4
Experiments and Results
Poon showed that SPNs can outperform deep belief networks (DBNs), deep Boltzman machines
(DBMs), principle component analysis (PCA), and a nearest- neighbors algorithm (NN) on a difficult
image completion task. The task is the following: given the right/top half of an image, paint in
the left/bottom half of it. The completion results of these models were compared qualitatively by
inspection and quantitatively using mean squared error (MSE). SPNs produced the best results; our
experiments show that the cluster architecture significantly improves SPN performance.
We matched the experimental set-up reported in [2] in order to isolate the effect of changing the
initial SPN architecture and to make their reported results directly comparable to several of our
results. They add 20 sum nodes for each non-unit and non-root region. The root region has one
sum node and the unit regions have four sum nodes, each of which function as a Gaussian over pixel
values. The Gaussians means are calculated using the training data for each pixel, with one Gaussian
covering each quartile of the pixel-values histogram. Each training image is normalized such that
its mean pixel value is zero with a standard deviation of one. Hard expectation maximization (EM)
is used to train the SPNs; mini-batches of 50 training instances are used to calculate each weight
update. All sum node weights are initialized to zero; weight values are decreased after each training
epoch using an L0 prior; add-one smoothing on sum node weights is used during network evaluation.
6
Table 1: Results of experiments on the Olivetti, Caltech 101 Faces, artificial, and shuffled-Olivetti datasets
comparing the Poon and cluster architectures. Negative
log-likelihood (LLH) of the training set and test set is reported along with the MSE for the image completion results (both left-half and bottom-half completion results).
Dataset
Olivetti
Caltech
Faces
Artificial
Shuffled
Measurement
Train LLH
Test LLH
MSE (left)
MSE (bottom)
Train LLH
Test LLH
MSE (left)
MSE (bottom)
Train LLH
Test LLH
MSE (left)
MSE (bottom)
Train LLH
Test LLH
MSE (left)
MSE (bottom)
Poon
318 ? 1
863 ? 9
996 ? 42
963 ? 42
289 ? 4
674 ? 15
1968 ? 89
1925 ? 82
195 ? 0
266 ? 4
842 ? 51
877 ? 85
793 ? 3
1193 ? 3
811 ? 11
817 ? 17
Cluster
433 ? 17
715 ? 31
814 ? 35
820 ? 38
379 ? 8
557 ? 11
1746 ? 87
1561 ? 44
169 ? 0
223 ? 6
558 ? 27
561 ? 29
442 ? 14
703 ? 14
402 ? 16
403 ? 17
Figure 6: A cluster-architecture SPN
completed the images in the left column and a Poon-architecture SPN completed the images in the right column.
All images shown are left-half completions. The top row is the best results as
measured by MSE and the bottom row
is the worst results. Note the smooth
edges in the cluster completions and the
jagged edges in the Poon completions.
We test the cluster and Poon architectures by learning on the Olivetti dataset [9], the faces from the
Caltech-101 dataset [10], an artificial dataset that we generated, and the shuffled-Olivetti dataset,
which the Olivetti dataset with the pixels randomly shuffled (all images are shuffled in the same
way). The Caltech-101 faces were preprocessed as described by Poon & Domingos. The cluster
architecture is compared to the Poon architectures using the negative log-likelihood (LLH) of the
training and test sets as well as the MSE of the image completion results for the left half and bottom
half of the images. We train ten cluster architecture SPNs and ten Poon architecture SPNs. Average
results across the ten SPNs along with the standard deviation are given for each measurement.
On the Olivetti and Caltech-101 Faces datasets the Poon architecture resulted in better training set
LLH, but the cluster architecture generalized better, getting a better test set LLH (see Table 1). The
cluster architecture was also clearly better at the image completion tasks as measured by MSE.
The difference between the two architectures is most pronounced on the artificial dataset. The
images in this dataset are created by pasting randomly-shaded circle- and diamond-shaped image
patches on top of one another (see Figure 6), ensuring that various pixel patches are statistically
independent. The cluster architecture outperforms the Poon architecture across all measures on this
dataset (see Table 1); this is due to its ability to focus resources on non-rectangular regions.
To demonstrate that the cluster architecture does not rely on the presence of spatially-local, strong
interactions between the variables, we repeated the Olivetti experiment with the pixels in the images
having been shuffled. In this experiment (see Table 1) the cluster architecture was, as expected,
relatively unaffected by the pixel shuffling. The LLH measures remained basically unchanged from
the Olivetti to the Olivetti-shuffled datasets. (The MSE results did not stay the same because the
image completions happened over different subsets of the pixels.) On the other hand, the performance of the Poon architecture dropped considerably due to the fact that it was no longer able to
take advantage of strong correlations between neighboring pixels.
Figure 7 visually demonstrates the difference between the rectangular-regions Poon architecture and
the arbitrarily-shaped-regions cluster architecture. Artifacts of the different region shapes can be
seen in subfigure (a), where some regions are shaded lighter or darker, revealing region boundaries.
Subfigure (b) compares the best of both architectures, showing image completion results on which
both architectures did well, qualitatively speaking. Note how the Poon architecture produces results
that look ?blocky?, whereas the cluster architecture produces results that are smoother-looking.
7
(a)
(b)
Figure 7: The completion results in subfigure (a) highlight the difference between the rectangularshaped regions of the Poon architecture (top image) and the blob-like regions of the cluster architecture (bottom image), artifacts of which can be seen in the completions. Subfigure (b) shows
ground truth images, cluster-architecture SPN completions, and Poon-architecture SPN completions
in the left, middle, and right columns respectively. Left-half completions are in the top row and
bottom-half completions are in the bottom row.
Table 2: Test set LLH values for the Olivetti, Olivetti45, and Olivetti4590 datasets for different
values of k. For each dataset the best LLH value is marked in bold.
Dataset / k
Olivetti
Olivetti45
Olivetti4590
1
650
523
579
2
653
495
576
3
671
508
550
4
685
529
554
5
711
541
577
6
716
528
595
7
717
544
608
8
741
532
592
Algorithm 1 expands a region graph k times (lines 12 and 13). The value of k can significantly affect
test set LLH, as shown in Table 2. A value that is too low leads to an insufficiently powerful model
and a value that is too high leads to a model that overfits the training data and generalizes poorly.
A singly-expanded model (k = 1) is optimal for the Olivetti dataset. This may be due in part to the
Olivetti dataset having only one distinct class of images (faces in a particular pose). Datasets with
more image classes may benefit from additional expansions. To experiment with this hypothesis
we create two new datasets: Olivetti45 and Olivetti4590. Olivetti45 is created by augmenting the
Olivetti dataset with Olivetti images that are rotated by ?45 degrees. Olivetti4590 is built similarly
but with rotations by ?45 degrees and by ?90 degrees. The Olivetti45 dataset, then, has two distinct
classes of images: rotated and non-rotated. Similarly, Olivetti4590 has three distinct image classes.
Table 2 shows that, as expected, the optimal value of k for the Olivetti45 and Olivetti4590 datasets
is two and three, respectively.
Note that the Olivetti test set LLH with k = 1 in Table 2 is better than the test set LLH reported in
Table 1. This shows that the algorithm for automatically selecting k in Algorithm 1 is not optimal.
Another option is to use a hold-out set to select k, although this method may not not be appropriate
for small datasets.
5
Conclusion
The algorithm for learning a cluster architecture is simple, fast, and effective. It allows the SPN
to focus its resources on explaining the interactions between arbitrary subsets of input variables.
And, being driven by data, the algorithm guides the allocation of SPN resources such that it is able
to model the data more efficiently. Future work includes experimenting with alternative clustering algorithms, experimenting with methods for selecting the value of k, and experimenting with
variations of Algorithm 2 such as generalizing it to handle partitions of size greater than two.
8
References
[1] Geoffrey E. Hinton, Simon Osindero, and Yee-Whye Teh. A fast learning algorithm for deep
belief nets. Neural Computation, 18:1527?1554, July 2006.
[2] Hoifung Poon and Pedro Domingos. Sum-product networks: A new deep architecture. In
Proceedings of the Twenty-Seventh Annual Conference on Uncertainty in Artificial Intelligence
(UAI-11), pages 337?346, Corvallis, Oregon, 2011. AUAI Press.
[3] Ryan Prescott Adams, Hanna M. Wallach, and Zoubin Ghahramani. Learning the structure
of deep sparse graphical models. In Proceedings of the 13th International Conference on
Artificial Intelligence and Statistics, 2010.
[4] Nevin L. Zhang. Hierarchical latent class models for cluster analysis. Journal of Machine
Learning Research, 5:697?723, December 2004.
[5] Adnan Darwiche. A differential approach to inference in bayesian networks. Journal of the
ACM, 50:280?305, May 2003.
[6] Olivier Delalleau and Yoshua Bengio. Shallow vs. deep sum-product networks. In Advances
in Neural Information Processing Systems 24, pages 666?674. 2011.
[7] Daphne Koller and Nir Friedman. Probabilistic Graphical Models: Principles and Techniques.
MIT Press, 2009.
[8] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in python. Journal of Machine Learning
Research, 12:2825?2830, 2011.
[9] F.S. Samaria and A.C. Harter. Parameterisation of a stochastic model for human face identification. In Proceedings of the Second IEEE Workshop on Applications of Computer Vision,
pages 138 ?142, Dec 1994.
[10] Li Fei-Fei, R. Fergus, and P. Perona. Learning generative visual models from few training
examples: An incremental bayesian approach tested on 101 object categories. In IEEE CVPR
2004, Workshop on Generative-Model Based Vision, 2004.
9
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3,916 | 4,545 | Imitation Learning by Coaching
Jason Eisner
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21218
[email protected]
He He Hal Daum? III
Department of Computer Science
University of Maryland
College Park, MD 20740
{hhe,hal}@cs.umd.edu
Abstract
Imitation Learning has been shown to be successful in solving many challenging
real-world problems. Some recent approaches give strong performance guarantees by training the policy iteratively. However, it is important to note that these
guarantees depend on how well the policy we found can imitate the oracle on the
training data. When there is a substantial difference between the oracle?s ability and the learner?s policy space, we may fail to find a policy that has low error
on the training set. In such cases, we propose to use a coach that demonstrates
easy-to-learn actions for the learner and gradually approaches the oracle. By a
reduction of learning by demonstration to online learning, we prove that coaching can yield a lower regret bound than using the oracle. We apply our algorithm
to cost-sensitive dynamic feature selection, a hard decision problem that considers a user-specified accuracy-cost trade-off. Experimental results on UCI datasets
show that our method outperforms state-of-the-art imitation learning methods in
dynamic feature selection and two static feature selection methods.
1
Introduction
Imitation learning has been successfully applied to a variety of applications [1, 2]. The standard
approach is to use supervised learning algorithms and minimize a surrogate loss with respect to
an oracle. However, this method ignores the difference between distributions of states induced by
executing the oracle?s policy and the learner?s, thus has a quadratic loss in the task horizon T . A
recent approach called Dataset Aggregation [3] (DAgger) yields a loss linear in T by iteratively
training the policy in states induced by all previously learned policies. Its theoretical guarantees
are relative to performance of the policy that best mimics the oracle on the training data. In difficult
decision-making problems, however, it can be hard to find a good policy that has a low training error,
since the oracle?s policy may resides in a space that is not imitable in the learner?s policy space. For
instance, the task loss function can be highly non-convex in the learner?s parameter space and very
different from the surrogate loss.
When the optimal action is hard to achieve, we propose to coach the learner with easy-to-learn
actions and let it gradually approach the oracle (Section 3). A coach trains the learner iteratively in a
fashion similar to DAgger. At each iteration it demonstrates actions that the learner?s current policy
prefers and have a small task loss. The coach becomes harsher by showing more oracle actions
as the learner makes progress. Intuitively, this allows the learner to move towards a better action
without much effort. Thus our algorithm achieves the best action gradually instead of aiming at an
impractical goal from the beginning. We analyze our algorithm by a reduction to online learning
and show that our approach achieves a lower regret bound than DAgger that uses the oracle action
(Section 3.1). Our method is also related to direct loss minimization [4] for structured prediction
and methods of selecting oracle translations in machine translation [5, 6] (Section 5).
1
Our approach is motivated by a formulation of budgeted learning as a sequential decision-making
problem [7, 8] (Section 4). In this setting, features are acquired at a cost, such as computation
time and experiment expense. In dynamic feature selection, we would like to sequentially select a
subset of features for each instance at test time according to a user-specified accuracy-cost trade-off.
Experimental results show that coaching has a more stable training curve and achieves lower task
loss than state-of-the-art imitation learning algorithms.
Our major contribution is a meta-algorithm for hard imitation learning tasks where the available
policy space is not adequate for imitating the oracle. Our main theoretical result is Theorem 4 which
states that coaching as a smooth transition from the learner to the oracle have a lower regret bound
than only using the oracle.
2
Background
In a sequential decision-making problem, we have a set of states S, a set of actions A and a policy
space ?. An agent follows a policy ? : S ? A that determines which action to take in a given
state. After taking action a in state s, the environment responds by some immediate loss L(s, a).
We assume L(s, a) is bounded in [0, 1]. The agent is then taken to the next state s0 according to the
transition probability P (s0 |s, a). We denote dt? the state distribution at time t after executing ? from
time 1 to t ? 1, and d? the average state distribution of states over T steps. Then the T -step expected
PT
loss of ? is J(?) =
t=1 Es?dt? [L(s, ?(s)] = T Es?d? [L(s, ?(s))]. A trajectory is a complete
sequence of hs, a, L(s, a)i tuples from the starting state to a goal state. Our goal is to learn a policy
? ? ? that minimizes the task loss J(?). We assume that ? is a closed, bounded and non-empty
convex set in Euclidean space; a policy ? can be parameterized by a vector w ? Rd .
In imitation learning, we define an oracle that executes policy ? ? and demonstrates actions a?s =
arg min L(s, a) in state s. The learner only attempts to imitate the oracle?s behavior without any
a?A
notion of the task loss function. Thus minimizing the task loss is reduced to minimizing a surrogate
loss with respect to the oracle?s policy.
2.1
Imitation by Classification
A typical approach to imitation learning is to use the oracle?s trajectories as supervised data and learn
a policy (multiclass classifier) that predicts the oracle action under distribution of states induced by
running the oracle?s policy. At each step t, we collect a training example (st , ? ? (st )), where ? ? (st )
is the oracle?s action (class label) in state st . Let `(s, ?, ? ? (s)) denote the surrogate loss of executing
? in state s with respect to ? ? (s). This can be any convex loss function used for training the classifier,
for example, hinge loss in SVM. Using any standard supervised learning algorithm, we can learn a
policy
?
? = arg min Es?d?? [`(s, ?, ? ? (s))].
(1)
???
We then bound J(?
? ) based on how well the learner imitates the oracle. Assuming `(s, ?, ? ? (s)) is
an upper bound on the 0-1 loss and L(s, a) is bounded in [0,1], Ross and Bagnell [9] have shown
that:
Theorem 1. Let Es? d?? [`(s, ?
? , ? ? (s))] = , then J(?
? ) ? J(? ? ) + T 2 .
One drawback of the supervised approach is that it ignores the fact that the state distribution is
different for the oracle and the learner. When the learner cannot mimic the oracle perfectly (i.e.
classification error occurs), the wrong action will change the following state distribution. Thus the
learned policy is not able to handle situations where the learner follows a wrong path that is never
chosen by the oracle, hence the quadratically increasing loss. In fact in the worst case, performance
can approach random guessing, even for arbitrarily small [10].
Ross et al. [3] generalized Theorem 1 to any policy that has surrogate loss under its own state
0
distribution, i.e. Es?d? [`(s, ?, ? ? (s))] = . Let Q?t (s, ?) denote the t-step loss of executing ? in
0
the initial state and then running ? . We have the following:
?
?
Theorem 2. If Q?T ?t+1 (s, ?) ? Q?T ?t+1 (s, ? ? ) ? u for all action a, t ? {1, 2, . . . , T }, then
J(?) ? J(? ? ) + uT .
2
It basically says that when ? chooses a different action from ? ? at time step t, if the cumulative cost
due to this error is bounded by u, then the relative task loss is O(uT ).
2.2
Dataset Aggregation
The above problem of insufficient exploration can be alleviated by iteratively learning a policy
trained under states visited by both the oracle and the learner. For example, during training one
can use a ?mixture oracle? that at times takes an action given by the previous learned policy [11].
Alternatively, at each iteration one can learn a policy from trajectories generated by all previous
policies [3].
In its simplest form, the Dataset Aggregation (DAgger) algorithm [3] works as follows. Let
s? denote a state visited by executing ?. In the first iteration, we collect a training set D1 =
{(s?? , ? ? (s?? ))} from the oracle (?1 = ? ? ) and learn a policy ?2 . This is the same as the supervised approach to imitation. In iteration i, we collect trajectories by executing the previous policy
?i andS
form the training set Di by labeling s?i with the oracle action ? ? (s?i ); ?i+1 is then learned
on D1 . . . Di . Intuitively, this enables the learner to make up for past failures to mimic the oracle.
Thus we can obtain a policy that performs well under its own induced state distribution.
2.3
Reduction to Online Learning
Let `i (?) = Es?d?i [`(s, ?, ? ? (s))] denote the expected surrogate loss of executing ? in states distributed according to d?i . In an online learning setting, in iteration i an algorithm executes policy ?i
and observes loss `i (?i ). It then provides a different policy ?i+1 in the next iteration and observes
`i+1 (?i+1 ). A no-regret algorithm guarantees that in N iterations
N
N
1 X
1 X
`i (?i ) ? min
`i (?) ? ?N
??? N
N i=1
i=1
(2)
and limN ?? ?N = 0.
Assuming a strongly convex loss function, Follow-The-Leader is a simple no-regret algorithm. In
Pi
each iteration it picks the policy that works best so far: ?i+1 = arg min
j=1 `j (?). Similarly,
???
in DAgger at iteration i we choose the policy that has the minimum surrogate loss on all previous
data. Thus it can be interpreted as Follow-The-Leader where trajectories collected in each iteration
are treated as one online-learning example.
Assume that `(s, ?, ? ? (s)) is a strongly convex loss in ? upper bounding the 0-1 loss.
We denote the sequence of learned policies ?1 , ?2 , . . . , ?N by ?1:N .
Let N =
PN
min??? N1 i=1 Es?d?i [`(s, ?, ? ? (s))] be the minimum loss we can achieve in the policy space
?. In the infinite sample per iteration case, following proofs in [3] we have:
?
?
Theorem 3. For DAgger, if N is O(uT log T ) and Q?T ?t+1 (s, ?)?Q?T ?t+1 (s, ? ? ) ? u, there exists
a policy ? ? ?1:N s.t. J(?) ? J(? ? ) + uT N + O(1).
This theorem holds for any no-regret online learning algorithm and can be generalized to the finite
sample case as well.
3
Imitation by Coaching
An oracle can be hard to imitate in two ways. First, the learning policy space is far from the space
that the oracle policy lies in, meaning that the learner only has limited learning ability. Second,
the environment information known by the oracle cannot be sufficiently inferred from the state,
meaning that the learner does not have access to good learning resources. In the online learning
setting, a too-good oracle may result in adversarially varying loss functions over iterations from the
learner?s perspective. This may cause violent changes during policy updating. These difficulties
result in a substantial gap between the oracle?s performance and the best performance achievable in
the policy space ? (i.e. a large N in Theorem 3).
3
Algorithm 1 DAgger by Coaching
Initialize D ? ?
Initialize ?1 ? ? ?
for i = 1 to N do
Sample T -step trajectories using ?i
Collect coaching dataset Di = {(s?i , arg max ?i ? score?i (s?i , a) ? L(s?i , a))}
a?A
S
Aggregate datasets D ? D Di
Train policy ?i+1 on D
end for
Return best ?i evaluated on validation set
To address this problem, we define a coach in place of the oracle. To better instruct the learner, a
coach should demonstrate actions that are not much worse than the oracle action but are easier to
achieve within the learner?s ability. The lower an action?s task loss is, the closer it is to the oracle
action. The higher an action is ranked by the learner?s current policy, the more it is preferred by the
learner, thus easier to learn. Therefore, similar to [6], we define a hope action that combines the task
loss and the score of the learner?s current policy. Let score?i (s, a) be a measure of how likely ?i
chooses action a in state s. We define ?
?i by
?
?i (s) = arg max ?i ? score?i (s, a) ? L(s, a)
(3)
a?A
where ?i is a nonnegative parameter specifying how close the coach is to the oracle. In the first
iteration, we set ?1 = 0 as the learner has not learned any model yet. Algorithm 1 shows the
training process. Our intuition is that when the learner has difficulty performing the optimal action,
the coach should lower the goal properly and let the learner gradually achieving the original goal in
a more stable way.
3.1
Theoretical Analysis
Let `?i (?) = Es?d?i [`(s, ?, ?
?i (s))] denote the expected surrogate loss with respect to ?
?i . We denote
PN ?
1
?N = N min??? i=1 `i (?) the minimum loss of the best policy in hindsight with respect to hope
actions. The main result of this paper is the following theorem:
?
?
Theorem 4. For DAgger with coaching, if N is O(uT log T ) and Q?T ?t+1 (s, ?)?Q?T ?t+1 (s, ? ? ) ?
u, there exists a policy ? ? ?1:N s.t. J(?) ? J(? ? ) + uT ?N + O(1).
It is important to note that both the DAgger theorem and the coaching theorem provide a relative
guarantee. They depend on whether we can find a policy that has small training error in each FollowThe-Leader step. However, in practice, for hard learning tasks DAgger may fail to find such a good
policy. Through coaching, we can always adjust ? to create a more learnable oracle policy space,
thus get a relatively good policy that has small training error, at the price of running a few more
iterations.
To prove this theorem, we first derive a regret bound for coaching, and then follows the proofs of
DAgger.
We consider a policy ? parameterized by a vector w ? Rd . Let ? : S ? A ? Rd be a feature map
describing the state. The predicted action is
a
??,s = arg max wT ?(s, a)
(4)
a?A
and the hope action is
a
??,s = arg max ? ? wT ?(s, a) ? L(s, a).
(5)
a?A
We assume that the loss function ` : Rd ? R is a convex upper bound of the 0-1 loss. Further, it
can be written as `(s, ?, ? ? (s)) = f (wT ?(s, ?(s)), ? ? (s)) for a function f : R ? R and a feature
vector k?(s, a)k2 ? R. We assume that f is twice differentiable and convex in wT ?(s, ?(s)), which
is common for most loss functions used by supervised classification methods.
4
It has been shown that given a strongly convex loss function `, Follow-The-Leader has O(log N )
regret [12, 13]. More specifically, given the above assumptions we have:
Theorem 5. Let D = maxw1 ,w2 ?Rd kw1 ? w2 k2 be the diameter of the convex set Rd . For some
b, m > 0, assume that for all w ? Rd , we have |f 0 (wT ?(s, a))| ? b and |f 00 (wT ?(s, a))| ? m.
Then Follow-The-Leader on functions ` have the following regret:
N
X
i=1
`i (?i ) ? min
???
N
X
i=1
N
X
`i (?) ?
i=1
`i (?i ) ?
N
X
`i (?i+1 )
i=1
DRmN
2nb2
log
+1
m
b
?
To analyze the regret using surrogate loss with respect to hope actions, we use the following lemma:
PN
PN ?
PN
PN ?
Lemma 1.
`i (?i ) ? min???
`i (?) ?
`i (?i ) ?
`i (?i+1 ).
i=1
i=1
Proof. We prove inductively that
i=1
i=1
PN ?
PN ?
i=1 `i (?i+1 ) ? min???
i=1 `i (?).
When N = 1, by Follow-The-Leader we have ?2 = arg min `?1 (?), thus `?1 (?2 ) = min??? `?1 (?).
???
Assume correctness for N ? 1, then
N
X
i=1
`?i (?i+1 ) ?
?
min
???
N
?1
X
N
?1
X
`?i (?) + `?N (?N +1 ) (inductive assumption)
i=1
`?i (?N +1 ) + `?N (?N +1 ) = min
???
i=1
The last equality is due to the fact that ?N +1
N
X
`?i (?)
i=1
PN
= arg min i=1 `?i (?).
???
To see how learning from ?
? allows us to approaching ? ? , we derive the regret bound of
PN
PN ?i
i=1 `i (?i ) ? min???
i=1 `i (?).
Theorem 6. Assume that wi is upper bounded by C, i.e. for all i kwi k2 ? C, k?(s, a)k2 ? R and
|L(s, a) ? L(s, a0 )| ? for some action a, a0 ? A. Assume ?i is non-increasing and define n? as
. Let `max be an upper bound on the loss, i.e. for all i,
the largest n < N such that ?n? ?
2RC
`i (s, ?i , ? ? (s)) ? `max . We have
N
N
X
X
2nb2
DRmN
`i (?i ) ? min
`?i (?) ? 2`max n? +
log
+1
???
m
b
i=1
i=1
Proof. Given Lemma 1, we only need to bound the RHS, which can be written as
!
!
N
N
X
X
`i (?i ) ? `?i (?i ) +
`?i (?i ) ? `?i (?i+1 ) .
i=1
(6)
i=1
To bound the first term, we consider a binary action space A = {1, ?1} for clarity. The proof can
be extended to the general case in a straightforward manner.
Note that in states where a?s = a
??,s , `(s, ?, ? ? (s)) = `(s, ?, ?
? (s)). Thus we only need to consider
?
situations where as 6= a
??,s :
=
`i (?i ) ? `?i (?i )
h
i
Es?d?i (`i (s, ?i , ?1) ? `i (s, ?i , 1)) 1{s : a??i ,s =1,a?s =?1}
h
i
+Es?d?i (`i (s, ?i , 1) ? `i (s, ?i , ?1)) 1{s:?a?i ,s =?1,a?s =1}
5
In the binary case, we define ?L(s) = L(s, 1) ? L(s, ?1) and ??(s) = ?(s, 1) ? ?(s, ?1).
Case 1
a
??i ,s = 1 and a?s = ?1.
a
??i ,s = 1 implies ?i wTi ??(s) ? ?L(s) and a?s = ?1 implies ?L(s) > 0. Together we have
?L(s) ? (0, ?i wTi ??(s)]. From this we know that wTi ??(s) ? 0 since ?i > 0, which implies
a
??i = 1. Therefore we have
p(a?s = ?1, a
??i ,s = 1, a
??i ,s = 1)
= p(?
a?i ,s = 1|a?s = ?1, a
??i ,s = 1)p(?
a?i , s = 1)p(a?s = ?1)
?L(s)
= p ?i ? T
? p(wTi ??(s) ? 0) ? p(?L(s) > 0)
wi ??(s)
? 1 ? 1 = p ?i ?
? p ?i ?
2RC
2RC
, we have
Let n? be the largest n < N such that ?i ?
2RC
N
X
i=1
h
i
Es?d?i (`i (s, ?i , ?1) ? `i (s, ?i , 1)) 1{s : a??i ,s =1,a?s =?1} ? `max n?
eN
For example, let ?i decrease exponentially, e.g., ?i = ?0 e?i . If ?0 <
, Then n? =
2RC
2?0 RC
dlog
e.
Case 2 a
??i ,s = ?1 and a?s = 1. This is symmetrical to Case 1. Similar arguments yield the same
bound.
In the online learning setting, imitating the coach is to obsearve the loss `?i (?i ) and learn a policy
Pi ?
?i+1 = arg min
`j (?) at iteration i. This is indeed equivalent to Follow-The-Leader except
???
j=1
that we replaced the loss function. Thus Theorem 5 gives the bound of the second term.
DRmN
2nb2
log
+1 .
Equation 6 is then bounded by 2`max n? +
m
b
Now we can prove Theorem 4. Consider the best policy in ?1:N , we have
N
1 X
Es?d?i [`(s, ?i , ? ? (s))]
N i=1
2`max n?
2nb2
DRmN
? ?N +
+
log
+1
N
mN
b
min Es?d? [`(s, ?, ? ? (s))] ?
???1:N
When N is ?(T log T ), the regret is O(1/T ). Applying Theorem 2 completes the proof.
4
Experiments
We apply imitation learning to a novel dynamic feature selection problem. We consider the setting
where a pretrained model (data classifier) on a complete feature set is given and each feature has a
known cost. At test time, we would like to dynamically select a subset of features for each instance
and be able to explicitly specify the accuracy-cost trade-off. This can be naturally framed as a
sequential decision-making problem. The state includes all features selected so far. The action space
includes a set of non-selected features and the stop action. At each time step, the policy decides
whether to stop acquiring features and make a prediction; if not, which feature(s) to purchase next.
Achieving an accuracy-cost trade-off corresponds to finding the optimal policy minimizing a loss
function. We define the loss function as a combination of accuracy and cost:
L(s, a) = ? ? cost(s) ? margin(a)
6
(7)
1.00
0.95
0.90
reward
accuracy
0.55
0.50
0.45
0.85
0.80
0.75
|w|/cost
Forward
DAgger
Coaching
Oracle
0.70
DAgger
Coaching
0.40
0.26
0.28
0.30
0.32
0.34
0.36
average cost per example
0.65
0.60
0.0
0.38
0.2
(a) Reward of DAgger and Coaching.
0.4
0.6
average cost per example
0.8
1.0
(b) Radar dataset.
0.9
0.90
0.85
accuracy
accuracy
0.8
0.7
|w|/cost
Forward
DAgger
Coaching
Oracle
0.6
0.5
0.4
0.0
0.2
0.4
0.6
average cost per example
0.8
0.80
0.75
|w|/cost
Forward
DAgger
Coaching
Oracle
0.70
0.65
0.60
0.0
1.0
(c) Digit dataset.
0.2
0.4
0.6
average cost per example
0.8
1.0
(d) Segmentation dataset.
Figure 1: 1(a) shows reward versus cost of DAgger and Coaching over 15 iterations on the digit
dataset with ? = 0.5. 1(b) to 1(d) show accuracy versus cost on the three datasets. For DAgger and
Coaching, we show results when ? = 0, 0.1, 0.25, 0.5, 1.0, 1.5, 2.
where margin(a) denote the margin of classifying the instance after action a; cost(s) denote the
user-defined cost of all selected features in the current state s; and ? is a user-specified trade-off
parameter. Since we consider feature selection for each single instance here, the average margin
reflects accuracy on the whole datasets.
4.1
Dynamic Feature Selection by Imitation Learning
Ideally, an oracle should lead to a subset of features having the maximum reward. However, we
have too large a state space to exhaustedly search for the optimal subset of features. In addition,
the oracle action may not be unique since the optimal subset of features do not have to be selected
in a fixed order. We address this problem by using a forward-selection oracle. Given a state s, the
oracle iterates through the action space and calculates each action?s loss; it then chooses the action
that leads to the minimum immediate loss in the current state. We define ?(st , a) as a concatenation
of the current feature vector and a meta-feature vector that provides information about previous
classification results and cost.
In most cases, our oracle can achieve high accuracy with rather small cost. Considering a linear
classifier, as the oracle already knows the correct class label of an instance, it can simply choose,
for example, a positive feature that has a positive weight to correctly classify a positive instance. In
addition, at the start state even when ?(s0 , a) are almost the same for all instances, the oracle may
tend to choose features that favor the instance?s class. This makes the oracle?s behavior very hard to
imitate. In the next section we show that in this case coaching achieves better results than using an
oracle.
7
4.2
Experimental Results
We perform experiments on three UCI datasets (radar signal, digit recognition, image segmentation).
Random costs are assigned to features. We first compare the learning curve of DAgger and Coaching
over 15 iterations on the digit dataset with ? = 0.5 in Figure 1(a). We can see that DAgger makes
a big improvement in the second iteration, while Coaching takes smaller steps but achieves higher
reward gradually. In addition, the reward of Coaching changes smoothly and grows stably, which
means coaching avoids drastic change of the policy.
To test the effect of dynamic selection, we compare our results with DAgger and two static feature selection baselines that sequentially add features according to a ranked list. The first baseline
(denoted by Forward) ranks features according to the standard forward feature selection algorithm
without any notion of the cost. The second baseline (denoted by |w|/cost) uses a cost-sensitive
ranking scheme based on |w|/cost, the weight of a feature divided by its cost. Therefore, features
having high scores are expected to be cost-efficient. We give the results in Figure 1(b) to 1(d). To
get results of our dynamic feature selection algorithm at different costs, we set ? in the loss function
to be 0.0, 0.1, 0.25, 0.5, 1.0, 1.5, 2.0 and use the best policy evaluated on the development set for
each ?. For coaching, we set ?2 = 1 and decrease it by e?1 in each iteration. First, we can see that
dynamically selecting features for each instance significantly improves the accuracy at a small cost.
Sometimes, it even achieves higher accuracy than using all features. Second, we notice that there is
a substantial gap between the learned policy?s performance and the oracle?s, however, in almost all
settings Coaching achieves higher reward, i.e. higher accuracy at a lower cost as shown in the figures. Through coaching, we can reduce the gap by taking small steps towards the oracle. However,
the learned policy is still much worse compared to the oracle?s policy. This is because coaching
is still inherently limited by the insufficient policy space, which can be fixed by using expensive
kernels and nonlinear policies.
5
Related Work
The idea of using hope action is similar to what Chiang et al. [6] and Liang et al. [5] have used
for selecting oracle translations in machine translation. They maximized a linear combination of the
BLEU score (similar to negative task loss in our case) and the model score to find good translations
that are easier to train against. More recently, McAllester et al. [4] defined the direct label that
combines model score and task loss from a different view: they showed that using a perceptron-like
training methods and update towards the direct label is equivalent to perform gradient descent on
the task loss.
Coaching is also similar to proximal methods in online learning [14, 15]. They avoid large changes
during updating and achieve the original goal gradually. In proximal regularization, we want the
updated parameter vector to stay close to the previous one. Do et al. [14] showed that solving the
original learning problem through a sequence of modified optimization tasks whose objectives have
greater curvature can achieve a lower regret bound.
6
Conclusion and Future Work
In this paper, we consider the situation in imitation learning where an oracle?s performance is far
from what is achievable in the learning space. We propose a coaching algorithm that lets the learner
target at easier goals first and gradually approaches the oracle. We show that coaching has a lower
regret bound both theoretically and empirically. In the future, we are interested in formally defining
the hardness of a problem so that we know exactly in which cases coaching is more suitable than
DAgger. Another direction is to develop a similar coaching process in online convex optimization by
optimizing a sequence of approximating functions. We are also interested in applying coaching to
more complex structured prediction problems in natural language processing and computer vision.
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9
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3,917 | 4,546 | A Divide-and-Conquer Procedure for Sparse Inverse
Covariance Estimation
Inderjit S. Dhillon
Dept. of Computer Science
University of Texas, Austin
[email protected]
Cho-Jui Hsieh
Dept. of Computer Science
University of Texas, Austin
[email protected]
Pradeep Ravikumar
Dept. of Computer Science
University of Texas
[email protected]
Arindam Banerjee
Dept. of Computer Science & Engineering
University of Minnesota, Twin Cities
[email protected]
Abstract
We consider the composite log-determinant optimization problem, arising from
the `1 regularized Gaussian maximum likelihood estimator of a sparse inverse
covariance matrix, in a high-dimensional setting with a very large number of variables. Recent work has shown this estimator to have strong statistical guarantees
in recovering the true structure of the sparse inverse covariance matrix, or alternatively the underlying graph structure of the corresponding Gaussian Markov
Random Field, even in very high-dimensional regimes with a limited number of
samples. In this paper, we are concerned with the computational cost in solving
the above optimization problem. Our proposed algorithm partitions the problem
into smaller sub-problems, and uses the solutions of the sub-problems to build a
good approximation for the original problem. Our key idea for the divide step to
obtain a sub-problem partition is as follows: we first derive a tractable bound on
the quality of the approximate solution obtained from solving the corresponding
sub-divided problems. Based on this bound, we propose a clustering algorithm
that attempts to minimize this bound, in order to find effective partitions of the
variables. For the conquer step, we use the approximate solution, i.e., solution
resulting from solving the sub-problems, as an initial point to solve the original
problem, and thereby achieve a much faster computational procedure.
1
Introduction
Let {x1 , x2 , . . . , xn } be n sample points drawn from a p-dimensional Gaussian distribution
N (?, ?), also known as a Gaussian Markov Random Field (GMRF), where each xi is a pdimensional vector. An important problem is that of recovering the covariance matrix, or its inverse,
given the samples in a high-dimensional regime where n p, and p could number in the tens of
thousands. In such settings, the computational efficiency of any estimator becomes very important.
A popular approach for such high-dimensional inverse covariance matrix estimation is to impose the
structure of sparsity on the inverse covariance matrix (which can be shown to encourage conditional
independences among the Gaussian variables), and to solve the following `1 regularized maximum
likelihood problem:
arg min{? log det ? + tr(S?) + ?k?k1 } = arg min f (?),
(1)
? 0
? 0
Pn
Pn
where S = n1 i=1 (xi ? ?
?)(xi ? ?
?)T is the sample covariance matrix and ?
? = n1 i=1 xi is the
sample mean. The key focus in this paper is on developing computationally efficient methods to
solve this composite log-determinant optimization problem.
1
Due in part to its importance, many optimization methods [4, 1, 9, 7, 6] have been developed in
recent years for solving (1). However, these methods have a computational complexity of at least
O(p3 ) (typically this is the complexity per iteration). It is therefore hard to scale these procedures
to problems with tens of thousands of variables. For instance, in a climate application, if we are
modeling a GMRF over random variables corresponding to each Earth grid point, the number of
nodes can easily number in the tens of thousands. For this data, a recently proposed state-of-the-art
method QUIC [6], that uses a Newton-like method to solve (1), for instance takes more than 10
hours to converge.
A natural strategy when the computational complexity of a procedure scales poorly with the problem
size is a divide and conquer strategy: Given a partition of the set of nodes, we can first solve the
`1 regularized MLE over the sub-problems invidually, and than in the second step, aggregate the
? But how do we come up with a suitable partition? The main contribution
solutions together to get ?.
of this paper is to provide a principled answer to this question. As we show, our resulting divide and
conquer procedure produces overwhelming improvements in computational efficiency.
Interestingly, [8] recently proposed a decomposition-based method for GMRFs. They first observe
the following useful property of the composite log-determinant optimization problem in (1): if we
threshold the off-diagonal elements of the sample covariance matrix S, and the resulting thresholded
matrix is block-diagonal, then the corresponding inverse covariance matrix has the same blockdiagonal sparsity structure as well. Using this property, they decomposed the problem along these
block-diagonal components and solved these separately, thus achieving a sharp computational gain.
A major drawback to this approach of [8] however is that often the decomposition of the thresholded
sample covariance matrix can be very unbalanced ? indeed, in many of our real-life examples, we
found that the decomposition resulted in one giant component and several very small components.
In these cases, the approach in [8] is only a bit faster than directly solving the entire problem.
In this paper, we propose a different strategy based on the following simple idea. Suppose we are
given a particular partitioning, and solve the sub-problems specified by the partition components.
? clearly need not be equal to `1 regularized MLE (1). HowThe resulting decomposed estimator ?
ever, can we use bounds on the deviation to propose a clustering criterion? We first derive a bound
? ? ?? kF based on the off-diagonal error of the partition. Based on this bound, we propose
on k?
a normalized-cut spectral clustering algorithm to minimize the off-diagonal error, which is able to
? is very close to ?? . Interestingly, we show that this clustering
find a balanced partition such that ?
criterion can also be motivated as leveraging a property more general than that in [8] of the `1 reg? to initialize an iterative solver for the
ularized MLE (1). In the ?conquering? step, we then use ?
original problem (1). As we show, the resulting algorithm is much faster than other state-of-the-art
methods. For example, our algorithm can achieve an accurate solution for the climate data problem
in 1 hour, whereas directly solving it takes 10 hours.
In section 2, we outline the standard skeleton of a divide and conquer framework for GMRF estimation. The key step in such a framework is to come up with a suitable and efficient clustering
criterion. In the next section 3, we then outline our clustering criteria. Finally, in Section 4 we show
that in practice, our method achieves impressive improvements in computational efficiency.
2
The Proposed Divide and Conquer Framework
We first set up some notation. In this paper, we will consider each p ? p matrix X as an adjacency
matrix, where V = {1, . . . , p} is the node set, Xij is the weighted link between node i and node j.
We will use {Vc }kc=1 to denote a disjoint partitioning of the node set V, and each Vc will be called a
partition or a cluster.
Given a partition {Vc }kc=1 , our divide and conquer algorithm first solves GMRF for all node partitions to get the inverse covariance matrices {?(c) }kc=1 , and then uses the following matrix
? (1)
?
?
0
...
0
? 0
?(2) . . .
0 ?
?
? =?
?
(2)
? ..
..
..
.. ? ,
? .
.
.
. ?
0
0
0
?(k)
to initialize the solver for the whole GMRF. In this paper we use X (c) to denote the submatrix
XVc ,Vc for any matrix X. Notice that in our framework any sparse inverse covariance solver can
2
be used, however, in this paper we will focus on using the state-of-the-art method QUIC [6] as the
base solver, which was shown to have super-linear convergence when close to the solution. Using a
better starting point enables QUIC to more quickly reach this region of super-linear convergence, as
we will show later in our experiments.
The skeleton of the divide and conquer framework is quite simple and is summarized in Algorithm 1.
? defined in (2) should be close to the optimal
In order that Algorithm 1 be efficient, we require that ?
? F . Based
solution of the original problem ?? . In the following, we will derive a bound for k?? ? ?k
on this bound, we propose a spectral clustering algorithm to find an effective partitioning of the
nodes.
1
2
3
4
5
6
Algorithm 1: Divide and Conquer method for Sparse Inverse Covariance Estimation
Input : Empirical covariance matrix S, scalar ?
Output: ?? , the solution of (1)
Obtain a partition of the nodes {Vc }kc=1 ;
for c = 1, . . . , k do
Solve (1) on S (c) and subset of variables in Vc to get ?(c) ;
end
? by ?(1) , ?(2) , . . . , ?(k) as in (2) ;
Form ?
?
Use ? as an initial point to solve the whole problem (1) ;
2.1
Hierarchical Divide and Conquer Algorithm
Assume we conduct a k-way clustering, then the initial time for solving sub-problems is at least
O(k(p/k)3 ) = O(p3 /k 2 ) where p denotes the dimensionality, When we consider k = 2, the divide
and conquer algorithm can be at most 4 times faster than the original one. One can increase k,
however, a larger k entails a worse initial point for training the whole problem.
Based on this observation, we consider the hierarchical version of our divide-and-conquer algorithm.
For solving subproblems we can again apply a divide and conquer algorithm. In this way, the initial
time can be much less than O(p3 /k 2 ) if we use divide and conquer algorithm hierarchically for
each level. In the experiments, we will see that this hierarchical method can further improve the
performance of the divide-and-conquer algorithm.
3
Main Results: Clustering Criteria for GMRF
This section outlines the main contribution of this paper; in coming up with suitable efficient clustering criteria for use within the divide and framework structure in the previous section.
3.1
?
Bounding the distance between ?? and ?
To start, we discuss the following result from [8], which we reproduce using the notation in this
paper for convenience. Specifically, [8] shows that when all the between cluster edges in S have
absolute values smaller than ?, ?? will have a block-diagonal structure.
Theorem 1 ([8]). For any ? > 0 and a given partition {Vc }kc=1 , if |Sij | ? ? for all i, j in different
? where ?? is the optimal solution of (1) and ?
? is as defined in (2).
partitions, then ?? = ?,
? and ?? will be
As a consequence, if a partition {Vc }kc=1 satisfies the assumption of Theorem 1, ?
the same, and the last step of Algorithm 1 is not needed anymore. Therefore the result in [8] may be
viewed as a special case of our Divide-and-Conquer Algorithm 1.
However, in most real examples, a perfect partitioning as in Theorem 1 does not exist, which motivates a divide and conquer framework that does not need as stringent assumptions as in Theorem 1.
? we first prove a similar property for the
To allow a more general relationship between ?? and ?,
following generalized inverse covariance problem:
X
?? = arg min{? log det ? + tr(S?) +
?ij |?ij |} = arg min f? (?).
(3)
? 0
i,j
3
? 0
In the following, we use 1? to denote a matrix with all elements equal to ?. Therefore (1) is a special
case of (3) with ? = 1? . In (3), the regularization parameter ? is a p?p matrix, where each element
corresponds to a weighted regularization of each element of ?. We can then prove the following
theorem, as a generalization of Theorem 1.
Theorem 2. For any matrix regularization parameter ? (?ij > 0 ?i, j) and a given partition
{Vc }kc=1 , if |Sij | ? ?ij for all i, j in different partitions, then the solution of (3) will be the block
? defined in (2), where ?(c) is the solution for (3) with sample covariance S (c) and
diagonal matrix ?
regularization parameter ?(c) .
Proof. Consider the dual problem of (3):
max log det W s.t. |Wij ? Sij | ? ?ij ?i, j,
(4)
W 0
? = ?
? ?1 is a feasible sobased on the condition stated in the theorem, we can easily verify W
Pk
(c)
?
? is the oplution of (4) with the objective function value c=1 log det W . To show that W
? . From Fischer?s inequaltimal solution of (4), we consider an arbitrary feasible solution W
? ? Qk det W
? (c) for W
? 0. Since W
? (c) is the optimizer of the c-th block,
ity [2], det W
c=1
? (c) ? det W
? (c) for all c, which implies log det W
? ? log det W
? . Therefore ?
? is the primal
det W
optimal solution.
Next we apply Theorem 2 to develop a decomposition method. Assume our goal is to solve (1) and
we have clusters {Vc }kc=1 which may not satisfy the assumption in Theorem 1. We start by choosing
? such that
a matrix regularization weight ?
if i, j are in the same cluster,
? ij = ?
?
(5)
max(|Sij |, ?) if i, j are in different clusters.
? By construction,
Now consider the generalized inverse covariance problem (3) with this specified ?.
?
the assumption in Theorem 2 holds for ?, so we can decompose this problem into k sub-problems;
for each cluster c ? {1, . . . , k}, the subproblem has the following form:
?(c) = arg min{? log det ? + tr(S (c) ?) + ?k?k1 },
? 0
? is the optimal solution of
where S (c) is the sample covariance matrix of cluster c. Therefore, ?
? as the regularization parameter.
problem (3) with ?
Based on this observation, we will now provide another view of our divide and conquer algorithm as
follows. Considering the dual problem of the sparse inverse covariance estimation with the weighted
? defined in (5) to get
regularization defined in (4), Algorithm 1 can be seen to solve (4) with ? = ?
?
the initial point W , and then solve (4) with ? = 1? for all elements. Therefore we initially solve
the problem with looser bounded constraints to get an initial guess, and then solve the problem with
? are close to the real constraint 1? , the
tighter constraints. Intuitively, if the relaxed constraints ?
? and W ? will be close to each other. So in the following we derive a bound based on
solutions W
this observation.
?
For convenience, we use P ? to denote the original dual problem (4) with ? = 1? , and P ? to denote
the relaxed dual problem with different edge weights across edges as defined in (5). Based on the
? =?
? ?1 is the solution of P ?? . We
above discussions, W ? = (?? )?1 is the solution of P ? and W
define E as the following matrix:
0
if i, j are in the same cluster,
Eij =
(6)
max(|Sij | ? ?, 0) otherwise.
? by Theorem 2. In
If E = 0, all the off-diagonal elements are below the threshold ?, so W ? = W
the following we consider a more interesting case where E 6= 0. In this case kEkF measures how
much the off-diagonal elements exceed the threshold ?, and a good clustering algorithm should be
? kF can
able to find a partition to minimize kEkF . In the following theorem we show that kW ? ? W
? F can also be bounded by kEkF :
be bounded by kEkF , therefore k?? ? ?k
4
1
Theorem 3. If there exists a ? > 0 such that kEk2 ? (1 ? ?) kW
? k2 , then
?
?
? kF < p max(?max (W ), ?max (W )) kEkF ,
kW ? ? W
?)
??min (W
? 2
?
?
? F ? p max(?max (?), ?max (? )) ?max (?) kEkF ,
k?? ? ?k
?
? min(?min (?? ), ?min (?))
(7)
(8)
where ?min (?), ?max (?) denote the minimum/maximum singular values.
Proof. To prove Theorem 3, we need the following Lemma, which is proved in the Appendix:
Lemma 1. If A is a positive definite matrix and there exists a ? > 0 such that kA?1 Bk2 ? 1 ? ?,
then
log det(A + B) ? log det A ? p/(??min (A))kBkF .
(9)
?
? may not be a feasible solution of P ? .
Since P ? has a relaxed bounded constraint than P ? , W
?
?
However, we can construct a feasible solution W = W ? G ? E, where Gij = sign(Wij ) and
? indicates the entrywise product of two matrices. The assumption of this theorem implies that
? k2 , so kW
? ?1 (G ? E)k ? (1 ? ?). From Lemma 1 we have log det W
? ?
kG ? Ek2 ? (1 ? ?)/kW
p
?
?
?
?
log det W ? ??min (W
? ) kEkF . Since W is the optimal solution of P and W is a feasible solution
p
?
?
?
? ? log det W
? ?
of P , log det W ? log det W
? kEkF . Also, since W is the optimal
??min (W )
?
?
? . Therefore,
solution of P ? and W ? is a feasible solution of P ? , we have log det W ? < log det W
p
?
?
| log det W ? log det W | < ??min (W
? ) kEkF .
? )|
By the mean value theorem and some calculations, we have |f (W ? ) ? f (W
? ?W ? kF
kW
? ),?max (W ? )) , which implies (7).
max(?max (W
>
To establish the bound on ?, we use the mean value theorem again with g(W ) = W ?1 = ?,
?g(W ) = ? ? ? where ? is kronecker product. Moreover, ?max (? ? ?) = (?max (?))2 , so we
can combine with (7) to prove (8).
3.2
Clustering algorithm
In order to obtain computational savings, the clustering algorithm for the divide-and-conquer algorithm (Algorithm 1) should satisfy three conditions: (1) minimize the distance between the approx? ? ?? kF , (2) be cheap to compute, and (3) partition the nodes into
imate and the true solution k?
balanced clusters.
Assume the real inverse covariance matrix ?? is block-diagonal, then it is easy to show that W ?
is also block-diagonal. This is the case considered in [8]. Now let us assume ?? has almost a
block-diagonal structure but a few off-diagonal entries are not zero. Assume ?? = ?bd + vei eTj
where ?bd is the block-diagonal part of ?? and ei denotes the i-th standard basis vector, then from
Sherman-Morrison formula,
v
W ? = (?? )?1 = (?bd )?1 ?
?bd (?bd )T ,
1 + v(?bd )ij i j
where ?ibd is the ith column vector of ?bd . Therefore adding one off-diagonal element to ?bd will
introduce at most one nonzero off-diagonal block in W . Moreover, if block (i, j) of W is already
nonzero, adding more elements in block (i, j) of ? will not introduce any more nonzero blocks in
W . As long as just a few entries in off-diagonal blocks of ?? are nonzero, W will be block-diagonal
with a few nonzero off-diagonal blocks. Since kW ? ?S ? k? ? ?, we are able to use the thresholding
matrix S ? to guess the clustering structure of ?? .
In the following, we show this observation is consistent withP
the bound we get in Theorem 3. From
(8), ideally we want to find a partition to minimize kEk? = i ?
|?i (E)|. Since it is computationally
difficult to optimize this directly, we can use the bound kEk? ? pkEkF , so that minimizing kEkF
? ? ?? kF .
can be cast as a relaxation of the problem of minimizing k?
5
To find a partition minimizing kEkF , we want to find a partition {Vc }kc=1 such that the sum of
off-diagonal block entries of S ? is minimized, where S ? is defined as
?
(S ? )ij = max(|Sij | ? ?, 0)2 ? i 6= j and Sij
= 0 ?i = j.
(10)
At the same time, we want to have balanced clusters. Therefore, we minimize the following normalized cut objective value [10]:
p
k P
?
X
XX
i?Vc ,j ?V
/ c Sij
?
k
?
N Cut(S , {Vc }c=1 ) =
Sij
where d(Vc ) =
.
(11)
d(V
)
c
c=1
j=1
i?Vc
In (11), d(Vc ) is the volume of the vertex set Vc for balancing cluster sizes, and the numerator is
the sum of off-diagonal entries, which corresponds to kEk2F . As shown in [10, 3], minimizing the
normalized cut is equivalent to finding cluster indicators x1 , . . . , xc to maximize
min
x
k
X
xT (D ? S ? )xc
c
c=1
xTc Dx
= trace(Y T (I ? D?1/2 S ? D?1/2 )Y ),
(12)
Pp
?
where D is a diagonal matrix with Dii = j=1 Sij
, Y = D1/2 X and X = [x1 . . . xc ]. Therefore, a common way for getting cluster indicators is to compute the leading k eigenvectors of
D?1/2 S ? D?1/2 and then conduct kmeans on these eigenvectors.
The time complexity of normalized cut on S ? is mainly from computing the leading k eigenvectors
of D?1/2 S ? D?1/2 , which is at most O(p3 ). Since most state-of-the-art methods for solving (1)
require O(p3 ) per iteration, the cost for clustering is no more than one iteration for the original
solver. If S ? is sparse, as is common in real situations, we could speed up the clustering phase by
using the Graclus multilevel algorithm, which is a faster heuristic to minimize normalized cut [3].
4
Experimental Results
In this section, we first show that the normalized cut criterion for the thresholded matrix S ? in (10)
can capture the block diagonal structure of the inverse covariance matrix before solving (1). Using
the clustering results, we show that our divide and conquer algorithm significantly reduces the time
needed for solving the sparse inverse covariance estimation problem.
We use the following datasets:
1. Leukemia: Gene expression data ? originally provided by [5], we use the data after the
pre-processing done in [7].
2. Climate: This dataset is generated from NCEP/NCAR Reanalysis data 1 , with focus on
the daily temperature at several grid points on earth. We treat each grid point as a random
variable, and use daily temperature in year 2001 as features.
3. Stock: Financial dataset downloaded from Yahoo Finance 2 . We collected 3724 stocks,
each with daily closing price recorded in latest 300 days before May 15, 2012.
4. Synthetic: We generated synthetic data containing 20, 000 nodes with 100 randomly generated group centers ?1 , . . . , ?100 , each of dimension 200, such that each group c has half
of its nodes with feature ?c and the other half with features ??c . We then add Gaussian
noise to the features.
The data statistics are summarized in Table 1.
4.1
Clustering quality on real datasets
Given a clustering partition {Vc }kc=1 , we use the following ?within-cluster ratio? to determine its
performance on ?? :
Pk P
? 2
c=1
i,j:i6=j and i,j?Vc (?ij )
k
P
.
(13)
R({Vc }c=1 ) =
? 2
i6=j (?ij )
1
2
www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.surface.html
http://finance.yahoo.com/
6
p
n
Table 1: Dataset Statistics
Leukemia Climate Stock Synthetic
1255
10512
3724
20000
72
1464
300
200
Table 2: Within-cluster ratios (see (13)) on real datasets. We can see that our proposed clustering
? = ?? ? ?? , which we cannot see
method Spectral S ? is very close to the clustering based on ?
before solving (1).
Leukemia
Climate
Stock
Synthetic
? = 0.5 ? = 0.3 ? = 0.005 ? = 0.001 ? = 0.0005 ? = 0.0001 ? = 0.005 ? = 0.001
random clustering 0.26
0.24
0.24
0.25
0.24
0.24
0.25
0.24
spectral on S ?
0.91
0.84
0.87
0.65
0.96
0.87
0.98
0.93
?
spectral on ?
0.93
0.84
0.90
0.71
0.97
0.85
0.99
0.93
Higher values of R({Vc }kc=1 ) are indicative of better performance of the clustering algorithm.
In section 3.1, we presented theoretical justification for using normalized cut on the thresholded
matrix S ? . Here we show that this strategy shows great promise on real datasets. Table 2 shows the
within-cluster ratios (13) of the inverse covariance matrix using different clustering methods. We
include the following methods in our comparison:
? Random partition: partition the nodes randomly into k clusters. We use this as a baseline.
? Spectral clustering on thresholded matrix S ? : Our proposed method.
? = ?? ? ?? , which is the element-wise square of ?? : This is the
? Spectral clustering on ?
best clustering method we can conduct, which directly minimizes within-cluster ratio of
the ?? matrix. However, practically we cannot use this method as we do not know ?? .
We can observe in Table 2 that our proposed spectral clustering on S ? achieves almost the same
performance as spectral clustering on ?? ? ?? even though we do not know ?? .
Also, Figure 1 gives a pictorial view of how our clustering results help in recovering the sparse
inverse covariance matrix at different levels. We run a hierarchical 2-way clustering on the Leukemia
? with 1-level clustering and ?
? with 2-level
dataset, and plot the original ?? (solution of (1)), ?
?
clustering. We can see that although our clustering method does not look at ? , the clustering result
matches the nonzero pattern of ?? pretty well.
4.2
The performance of our divide and conquer algorithm
Next, we investigate the time taken by our divide and conquer algorithm on large real and synthetic
datasets. We include the following methods in our comparisons:
? DC-QUIC-1: Divide and Conquer framework with QUIC and with 1 level clustering.
(a) The inverse covariance matrix ?? .
? from level-1
(b) The recovered ?
clusters.
? from level 2
(c) The recovered ?
clusters.
Figure 1: The clustering results and the nonzero patterns of inverse covariance matrix ?? on
Leukemia dataset. Although our clustering method does not look at ?? , the clustering results
match the nonzero pattern in ?? pretty well.
7
(a) Leukemia
(b) Stock
(c) Climate
(d) Synthetic
Figure 2: Comparison of algorithms on real datasets. The results show that DC-QUIC is much faster
than other state-of-the-art solvers.
? DC-QUIC-3: Divide and Conquer QUIC with 3 levels of hierarchical clustering.
? QUIC: The original QUIC, which is a state-of-the-art second order solver for sparse inverse
estimation [6].
? QUIC-conn: Using the decomposition method described in [8] and using QUIC to solve
each smaller sub-problem.
? Glasso: The block coordinate descent algorithm proposed in [4].
? ALM: The alternating linearization algorithm proposed and implemented by [9].
All of our experiments are run on an Intel Xeon E5440 2.83GHz CPU with 32GB main memory.
Figure 2 shows the results. For DC-QUIC and QUIC-conn, we show the run time of the whole
process, including the preprocessing time. We can see that in the largest synthetic dataset, DCQUIC is more than 10 times faster than QUIC, and thus also faster than Glasso and ALM. For the
largest real dataset: Climate with more than 10,000 points, QUIC takes more than 10 hours to get a
reasonable solution (relative error=0), while DC-QUIC-3 converges in 1 hour. Moreover, on these
4 datasets QUIC-conn using the decomposition method of [8] provides limited savings, in part
because the connected components for the thresholded covariance matrix for each dataset turned
out to have a giant component, and multiple smaller components. DC-QUIC however was able to
leverage a reasonably good clustered decomposition, which dramatically reduced the inference time.
Acknowledgements
We would like to thank Soumyadeep Chatterjee and Puja Das for help with the climate and stock
data. C.-J.H., I.S.D and P.R. acknowledge the support of NSF under grant IIS-1018426. P.R. also
acknowledges support from NSF IIS-1149803. A.B. acknowledges support from NSF grants IIS0916750, IIS-0953274, and IIS-1029711.
8
References
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[4] J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3):432?441, July 2008.
[5] T. R. Golub, D. K. Slonim, P. Tamayo, C. Huard, M. Gaasenbeek, J. P. Mesirov, H. Coller,
M. L. Loh, J. R. Downing, M. A. Caligiuri, and C. D. Bloomfield. Molecular classication of
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531?537, 1999.
[6] C.-J. Hsieh, M. Sustik, I. S. Dhillon, and P. Ravikumar. Sparse inverse covariance matrix
estimation using quadratic approximation. In NIPS, 2011.
[7] L. Li and K.-C. Toh. An inexact interior point method for l1-reguarlized sparse covariance
selection. Mathematical Programming Computation, 2:291?315, 2010.
[8] R. Mazumder and T. Hastie. Exact covariance thresholding into connected components for
large-scale graphical lasso. Journal of Machine Learning Research, 13:723?736, 2012.
[9] K. Scheinberg, S. Ma, and D. Glodfarb. Sparse inverse covariance selection via alternating
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9
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3,918 | 4,547 | Non-parametric Approximate Dynamic
Programming via the Kernel Method
Nikhil Bhat
Graduate School of Business
Columbia University
New York, NY 10027
[email protected]
Vivek F. Farias
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA 02142
[email protected]
Ciamac C. Moallemi
Graduate School of Business
Columbia University
New York, NY 10027
[email protected]
Abstract
This paper presents a novel non-parametric approximate dynamic programming
(ADP) algorithm that enjoys graceful approximation and sample complexity guarantees. In particular, we establish both theoretically and computationally that our
proposal can serve as a viable alternative to state-of-the-art parametric ADP algorithms, freeing the designer from carefully specifying an approximation architecture. We accomplish this by developing a kernel-based mathematical program for
ADP. Via a computational study on a controlled queueing network, we show that
our procedure is competitive with parametric ADP approaches.
1
Introduction
Problems of dynamic optimization in the face of uncertainty are frequently posed as Markov decision processes (MDPs). The central computational problem is then reduced to the computation
of an optimal ?cost-to-go? function that encodes the cost incurred under an optimal policy starting
from any given MDP state. Many MDPs of practical interest suffer from the curse of dimensionality, where intractably large state spaces precluding exact computation of the cost-to-go function.
Approximate dynamic programming (ADP) is an umbrella term for algorithms designed to produce
good approximation to this function, yielding a natural ?greedy? control policy.
ADP algorithms are, in large part, parametric in nature; requiring the user to provide an ?approximation architecture? (i.e., a set of basis functions). The algorithm then produces an approximation in
the span of this basis. The strongest theoretical results available for such algorithms typically share
two features: (1) the quality of the approximation produced is comparable with the best possible
within the basis specified, and (2) the computational effort required for doing so typically scales as
the dimension of the basis specified.
These results highlight the importance of selecting a ?good? approximation architecture, and remain
somewhat dissatisfying in that additional sampling or computational effort cannot remedy a bad approximation architecture. On the other hand, a non-parametric approach would, in principle, permit
the user to select a rich, potentially full-dimensional architecture (e.g., the Haar basis). One would
then expect to compute increasingly accurate approximations with increasing computational effort.
The present work presents a practical algorithm of this type. Before describing our contributions,
we begin with summarizing the existing body of research on non-parametric ADP algorithms.
1
The key computational step in approximate policy iteration methods is approximate policy evaluation. This step involves solving the projected Bellman equation, a linear stochastic fixed point equation. A numerically stable approach to this is to perform regression with a certain ?2 -regularization,
where the loss is the ?2 -norm of the Bellman error. By substituting this step with a suitable nonparametric regression procedure, [2, 3, 4] come up with a corresponding non-parametric algorithm.
Unfortunately schemes such approximate policy iteration have no convergence guarantees in parametric settings, and these difficulties remain in non-parametric variations. Another idea has been to
use kernel-based local averaging ideas to approximate the solution of an MDP with that of a simpler
variation on a sampled state space [5, 6, 7]. However, convergence rates for local averaging methods are exponential in the dimension of the problem state space. As in our setting, [8] constructs
kernel-based cost-to-go function approximations. These are subsequently plugged into various ad
hoc optimization-based ADP formulations, without theoretical justification.
Closely related to our work, [9, 10] consider modifying the approximate linear program with an
?1 regularization term to encourage sparse approximations in the span of a large, but necessarily
tractable set of features. Along these lines, [11] discuss a non-parametric method that explicitly
restricts the smoothness of the value function. However, sample complexity results for this method
are not provided and it appears unsuitable for high-dimensional problems (such as, for instance, the
problem we consider in our experiments). In contrast to this line of work, our approach will allow
for approximations in a potentially infinite dimensional approximation architecture with a constraint
on an appropriate ?2 -norm of the weight vector.
The non-parametric ADP algorithm we develop enjoys non-trivial approximation and sample complexity guarantees. We show that our approach complements state-of-the-art parametric ADP algorithms by allowing the algorithm designer to compute what is essentially the best possible ?simple?
approximation1 in a full-dimensional approximation architecture as opposed to restricting attention
to some a-priori fixed low dimensional architecture. In greater detail, we make the following contributions:
A new mathematical programming formulation. We rigorously develop a kernel-based variation of
the ?smoothed? approximate LP (SALP) approach to ADP proposed by [12]. The resulting mathematical program, which we dub the regularized smoothed approximate LP (RSALP), is distinct from
simply substituting a kernel-based approximation in the SALP formulation. We develop a companion active set method that is capable of solving this mathematical program rapidly and with limited
memory requirements.
Theoretical guarantees. 2 We establish a graceful approximation guarantee for our algorithm. Our
algorithm can be interpreted as solving an approximate linear program in an appropriate Hilbert
space. We provide, with high probability, an upper bound on the approximation error of the algorithm relative to the best possible approximation subject to a regularization constraint. The sampling requirements for our method are, in fact, independent of the dimension of the approximation
architecture. Instead, we show that the number of samples grows polynomially as a function of a
regularization parameter. Hence, the sampling requirements are a function of the complexity of the
approximation, not of the dimension of the approximating architecture. This result can be seen as
the ?right? generalization of the prior parametric approximate LP approaches [13, 14, 12], where, in
contrast, sample complexity grows with the dimension of the approximating architecture.
A computational study. To study the efficacy of RSALP, we consider an MDP arising from a challenging queueing network scheduling problem. We demonstrate that our RSALP method yields
significant improvements over known heuristics and standard parametric ADP methods.
In what follows, proofs and a detailed discussion of our numerical procedure are deferred to the
Online Supplement to this paper.
1
In the sense that the ?2 norm of the weight vector can grow at most polynomially with a certain measure
of computational budget.
2
These guarantees come under assumption of being able to sample from a certain idealized distribution.
This is a common in the ADP literature.
2
2
Formulation
Consider a discrete time Markov decision process with finite state space S and finite action space A.
We denote by xt and at respectively, the state and action at time t. We assume time-homogeneous
Markovian dynamics: conditioned on being at state x and taking action a, the system transitions to
state x? with probability p(x, x? , a) independent of the past. A policy is a map ? : S ? A, so that
??
?
?
?
t
J (x) ? Ex,?
? gxt ,at
t=0
represents the expected (discounted, infinite horizon) cost-to-go under policy ? starting at state x.
Letting ? denote the set of all policies our goal is to find an optimal policy ?? such that ?? ?
argmax??? J ? (x) for all x ? S (it is well known that such a policy exists). We denote the optimal
?
cost-to-go function by J ? ? J ? . An optimal policy ?? can be recovered as a ?greedy? policy with
respect to J ? ,
?? (x) ? argmin gx,a + ?Ex,a [J ? (X ? )],
a?A
?
where we define Ex,a [f (X ? )] as x? ?S p(x, x? , a)f (x? ), for all f : S ? R.
Since in practical applications S is often intractably large, exact computation of J ? is untenable.
ADP algorithms are principally tasked with computing approximations to J ? of the form J ? (x) ?
?
z ? ?(x) ? J(x),
where ? : S ? Rm is referred to as an ?approximation architecture? or a basis and
must be provided as input to the ADP algorithm. The ADP algorithm computes a ?weight? vector z;
?
one then employs a policy that is greedy with respect to the corresponding approximation J.
2.1
Primal Formulation
Motivated by the LP for exact dynamic programming, a series of ADP algorithms [15, 13, 12] have
been proposed that compute a weight vector z by solving an appropriate modification of the exact LP
for dynamic programming. In particular, [12] propose solving the following optimization problem
where ? ? RS+ is a strictly positive probability distribution and ? > 0 is a penalty parameter:
?
?
max
?x z ? ?(x) ? ?
?x s x
x?S
s. t.
x?S
?
z ?(x) ? ga,x + ?Ex,a [z ? ?(X ? )] + sx ,
z ? Rm , s ? RS+ .
? x ? S, a ? A,
(1)
In parsing the above program notice that if one insisted that the slack variables s were precisely 0,
one is left with the ALP proposed by [15]. [13] provided a pioneering analysis that loosely showed
2
?J ? ? z ? ? ??1,? ?
inf ?J ? ? z ? ??? ,
1?? z
for an optimal solution z ? to the ALP; [12] showed that these bounds could be improved upon
substantially by ?smoothing? the constraints of the ALP, i.e., permitting positive slacks. In both
cases, one must solve a ?sampled? version of the above program.
Now, consider allowing ? to map from S to a general (potentially infinite dimensional) Hilbert space
H. We use bold letters to denote elements in the Hilbert space H, e.g., the weight vector is denoted
by z ? H. We further suppress the dependence on ? and denote the elements H corresponding to
their counterparts in S by bold letters. Hence, for example, x ? ?(x) and X ? ?(X). Further, we
denote X ? ?(S); X ? H. The value function approximation in this case would be given by
J?z,b (x) ? ?x, z? + b = ??(x), z? + b,
(2)
where b is a scalar offset corresponding to a constant basis function. The following generalization
of (1) ? which we dub the regularized SALP (RSALP) ? then essentially suggests itself:
?
?
?
max
?x ?x, z? + b ? ?
?x sx ? ?z, z?
2
x?S
x?S
(3)
s. t. ?x, z? + b ? ga,x + ?Ex,a [?X? , z? + b] + sx , ? x ? S, a ? A,
z ? H, b ? R, s ? RS+ .
3
The only ?new? ingredient in the
? program above is the fact that we regularize z using the parameter
? > 0. Constraining ?z?H ? ?z, z? to lie within some ?2 -ball anticipates that we will eventually
resort to sampling in solving this program and we cannot hope for a reasonable number of samples
to provide a good solution to a problem where z was unconstrained. This regularization, which plays
a crucial role both in theory and practice, is easily missed if one directly ?plugs in? a local averaging
approximation in place of z ? ?(x) as is the case in the earlier work of [5, 6, 7, 8] and others.
Since the RSALP, i.e., program (3), can be interpreted as a regularized stochastic optimization problem, one may hope to solve it via its sample average approximation. To this end, define the likelihood ratio wx ? ?x /?x , and let S? ? S be a set of N states sampled independently according to the
distribution ?. The sample average approximation of (3) is then
1 ?
? ?
?
max
wx ?x, z? + b ?
sx ? ?z, z?
N
N
2
x?S?
x?S?
(4)
? a ? A,
s. t. ?x, z? + b ? ga,x + ?Ex,a [?X? , z? + b] + sx , ? x ? S,
?
z ? H, b ? R, s ? RS+ .
? were small, it is still not clear that this program
We call this program the sampled RSALP. Even if |S|
can be solved effectively. We will, in fact, solve the dual to this problem.
2.2
Dual Formulation
We begin by establishing some notation. Let Nx,a ? {x} ? {x? ? S|p(x, x? , a) > 0}. Now, define
?
?
the symmetric positive semi-definite matrix Q ? R(S?A)?(S?A) according to
??
?
?
? ?
? ?
? ?
?
?
Q(x, a, x , a ) ?
1{x=y} ? ?p(x, y, a) 1{x =y } ? ?p(x , y , a) ?y, y? ?, (5)
y?Nx,a y ? ?Nx? ,a?
?
and the vector R ? RS?A according to
?
?
1 ? ?
?
R(x, a) ? ?gx,a ?
wx 1{x=y} ? ?p(x, y, a) ?y, x? ?.
N ?
(6)
x ?S? y?Nx,a
Notice that Q and R depend only on inner products in X (and other, easily computable quantities).
The dual to (4) is then given by:
min
s. t.
1 ?
2 ? Q?
+ R? ?
?
?
?x,a ? ,
N
a?A
??
?x,a =
x?S? a?A
?
? x ? S,
1
,
1??
??
(7)
?
RS?A
.
+
Assuming that Q and R can be easily computed, this finite dimensional quadratic program, is
tractable ? its size is polynomial in the number of sampled states. We may recover a primal solution (i.e., the weight vector z? ) from an optimal dual solution:
Proposition 1. The optimal solution to (7) is attained at some ?? , then optimal solution to (4) is
attained at some (z ? , s? , b? ) with
?
?
?
?
?
?
1 1
z? = ?
wx x ?
??x,a x ? ?Ex,a [X? ] ? .
(8)
? N
x?S?
?
x?S,a?A
Having solved this program, we may, using Proposition 1, recover our approximate cost-to-go func?
tion J(x)
= ?z? , x? + b? as
?
?
??
?
?
1
1
?
?
?
J(x)
=
wy ?y, x? ?
?y,a ?y, x? ? ?Ey,a [?X , x?]
+ b? .
(9)
? N
y?S?
?
y?S,a?A
4
A policy greedy with respect to J? is not affected by constant translations, hence in (9), the value of
b? can be set to be zero arbitrarily. Again note that given ?? , J? only involves the inner products.
At this point, we use the ?kernel? trick: instead of explicitly specifying H or the mapping ?, we
take the approach of specifying inner products. In particular, given any positive definite kernel
K : S ? S ? R, it is well known (Mercer?s theorem) that there exists a Hilbert space H and
? : S ? H such that K(x, y) = ??(x), ?(y)?. Consequently, given a positive definite kernel,
we simply replace every inner product ?x, x? ? in the defining of the program (7) with the quantity
K(x, x? ) and similarly in the approximation (9). In particular, this is equivalent to using a Hilbert
space, H and mapping ? corresponding to that kernel.
Solving (7) directly is costly. In particular, it is computationally expensive to pre-compute and store
the matrix Q. An alternative to this is to employ the following broad strategy, as recognized by
[16] and [17] in the context of solving SVM classification problems, referred to as an active set
method: At every point in time, one attempts to (a) change only a small number of variables while
not impacting other variables (b) maintain feasibility. It turns out that this results in a method that
requires memory and per-step computation that scales only linearly with the sample size. We defer
the details of the procedure as well as the theoretical analysis to the Online Supplement
3
Approximation Guarantees
Recall that we are employing an approximation J?z,b of the form (2), parameterized by the weight
vector z and the offset parameter b. Now denoting by C the feasible region of the RSALP projected
onto the z and b co-ordinates, the best possible approximation one may hope for among those permitted by the RSALP will have ?? -approximation error inf (z,b)?C ?J ? ? J?z,b ?? . Provided the
Gram matrix given by the kernel restricted to S is positive definite, this quantity can be made arbitrarily small by making ? small. The rate at which this happens would reflect the quality of the
kernel in use. Here we focus on asking the following question: for a fixed choice of regularization
parameters (i.e., with C fixed) what approximation guarantee can be obtained for a solution to the
RSALP? This section will show that one can achieve a guarantee that is, in essence, within a certain
constant multiple of the optimal approximation error using a number of samples that is independent
of the size of the state space and the dimension of the approximation architecture.
3.1
The Guarantee
Define the Bellman operator, T : RS ? RS according to
(T J)(x) ? min gx,a + ?Ex,a [J(X ? )].
a?A
Let S? be a set of N states drawn independently at random from S under the distribution ? over S.
Given the definition of J?z,b in (2), we consider the following sampled version of RSALP,
2 1 ?
max ? ? J?z,b ?
sx
1??N
x?S?
(10)
? a ? A,
s. t. ?x, z? + b ? ga,x + ?Ex,a [?X? , z? + b] + sx , ? x ? S,
?
?z?H ? C, |b| ? B,
z ? H, b ? R, s ? RS+ .
We will assume that states are sampled according to an idealized distribution. In particular, ? ?
??? ,? where
?
?
???? ,? ? (1 ? ?)
?t ? ? P?t ? .
(11)
t=0
Here, P?? is the transition matrix under the optimal policy ?? . This idealized assumption is also
common to the work of [14] and [12]. In addition, this program is somewhat distinct from the
program presented earlier, (4): (1) As opposed to a ?soft? regularization term in the objective, we
have a ?hard? regularization constraint, ?z?H ? C. It is easy to see that given a ?, we can choose
a radius C(?) that yields an equivalent optimization problem. (2) We bound the magnitude of the
offset b. This is for theoretical convenience; our sample complexity bound will be parameterized
5
by B. (3) We fix ? = 2/(1 ? ?). Our analysis reveals this to be the ?right? penalty weight on the
Bellman inequality violations.
Before stating our bound we establish a few bits of notation. We let (z? , b? ) denote an optimal
solution to (10). We let K ? maxx?X ?x?H , and finally, we define the quantity
?
??
?
?
1
?(C, B, K, ?) ?
1+
ln(1/?)
4CK(1 + ?) + 4B(1 ? ?) + 2?g?? .
2
We have the following theorem:
Theorem 1. For any ? > 0 and ? > 0, let N ? ?(C, B, K, ?)2 /?2 . If (10) is solved by sampling N
states from S with distribution ??? ,? , then with probability at least 1 ? ? ? ? 4 ,
?J ? ? J?z? ,b? ?1,? ?
inf
?z?H ?C,|b|?B
3+? ?
4?
?J ? J?z,b ?? +
.
1??
1??
(12)
Ignoring the ?-dependent error terms, we see that the quality of approximation provided by (z? , b? )
is essentially within a constant multiple of the optimal (in the sense of ?? -error) approximation to
J ? possible using a weight vector z and offsets b permitted by the regularization constraints. This
is a ?structural? error term that will persist even if one were permitted to draw an arbitrarily large
number of samples. It is analogous to the approximation results produced in parametric settings
with the important distinction that one allows comparisons to approximations in potentially fulldimensional basis sets which might be substantially superior.
In addition to the structural?error above, one incurs an additional additive ?sampling? error that scales
like O(N ?1/2 (CK + B) ln 1/?). This quantity has no explicit dependence on the dimension of
the approximation architecture. In contrast, comparable sample complexity results (eg. [14, 12])
typically scale with the dimension of the approximation architecture. Here, this space may be full
dimensional, so that such a dependence would yield a vacuous guarantee. The error depends on the
user specified quantities C and B, and K, which is bounded for many kernels. The result allows
for arbitrary ?simple? (i.e. with ?z?H small) approximations in a rich feature space as opposed to
restricting us to some a-priori fixed, low dimensional feature space. This yields some intuition for
why we expect the approach to perform well even with a relatively general choice of kernel.
As C and B grow large, the structural error will decrease to zero provided K restricted to S is
positive definite. In order to maintain the sampling error constant, one would then need to increase
N (at a rate that is ?((CK + B)2 ). In summary, increased sampling yields approximations of
increasing quality, approaching an exact approximation. If J ? admits a good approximation with
?z?H small, one can expect a good approximation with a reasonable number of samples.
3.2
Proof Sketch
A detailed proof of a stronger result is in the Online Supplement. Here, we provide a proof sketch.
The first step of the proof involves providing a guarantee for the exact (non-sampled) RSALP with
hard regularization. Assuming (z? , b? ) is the ?learned? parameter pair, we first establish the guarantee:
3+?
?J ? ? J?z? ,b? ?1,? ?
inf
?J ? ? J?z,b ?? .
1 ? ? ?z?H ?C,b?R
Geometrically, the proof works loosely by translating the ?best? approximation given the regularization constraints to one that is guaranteed to yield an approximation error no worse that that produced
by the RSALP.
To establish a guarantee for the sampled RSALP, we first pose the RSALP as a stochastic optimization problem by setting s(z, b) ? (J?z,b ? T J?z,b )+ . We must ensure that with high probability,
the sample averages in the sampled program are close to the exact expectations, uniformly for all
possible values of (z, b) with high accuracy. In order to establish such a guarantee, we bound the
Rademacher complexity of the class of functions given by
?
?
+
?
?
?
FS,? ? x ?? (Jz,b (x) ? T? Jz,b (x)) : ?z?H ? C, |b| ? B ,
6
queue 2
queue 4
?2 = 0.12
?4 = 0.08
?4 = 0.28
server 1
server 2
?3 = 0.28
?1 = 0.08
?1 = 0.12
queue 1
queue 3
Figure 1: The queueing network example.
(where T? is the Bellman operator associated with policy ?), This yields the appropriate uniform
large deviations bound. Using this guarantee we show that the optimal solution to the sampled
RSALP yields similar approximation guarantees as that with the exact RSALP; this proof is somewhat delicate as it appears difficult to directly show that the optimal solutions themselves are close.
4
Case Study: A Queueing Network
This section considers the problem of controlling the queuing network illustrated in Figure 1, with
the objective of minimizing long run average delay. There are two ?flows? in this network: the first
through server 1 followed by server 2 (with buffering at queues 1 and 2, respectively), and the second
through server 2 followed by server 1 (with buffering at queues 4 and 3, respectively). Here, all interarrival and service times are exponential with rate parameters summarized in Figure 1. This specific
network has been studied [13, 18] and is considered to be a challenging control problem. Our goal
in this section will be two-fold. First, we will show that the RSALP can surpass the performance
of both heuristic as well as established ADP-based approaches, when used ?out-of-the-box? with a
generic kernel. Second, we will show that the RSALP can be solved efficiently.
4.1
MDP Formulation
Although the control problem at hand is nominally a continuous time problem, it is routinely converted into a discrete time problem via a standard uniformization device; see [19], for instance, for
an explicit such example. In the equivalent discrete time problem, at most a single event can occur
in a given epoch, corresponding either to the arrival of a job at queues 1 or 4, or the arrival of a service token for one of the four queues with probability proportional to the corresponding rates. The
state of the system is described by the number of jobs is each of the four queues, so that S ? Z4+ ,
whereas the action space A consists of four potential actions each corresponding to a matching between servers and queues. We take the single period cost as the total number of jobs in the system,
so that gx,a = ?x?1 ; note that minimizing the average number of jobs in the system is equivalent to
minimizing average delay by Little?s law. Finally, we take ? = 0.9 as our discount factor.
4.2
Approaches
RSALP (this paper). We solve (7) using the active set method outlined in the Online Supplement,
taking as? our kernel the standard Gaussian radial basis function kernel K(x, y) ?
?
exp ??x ? y?22 /h , with the bandwidth parameter h ? 100. (The sensitivity of our results to this
bandwidth parameter appears minimal.) Note that this implicitly corresponds to a full-dimensional
basis function architecture. Since the idealized sampling distribution, ??? ,? is unavailable to us, we
use in its place the geometric distribution ?(x) ? (1 ? ?)4 ? ?x?1 , with the sampling parameter ? set
at 0.9, as in [13]. The regularization parameter ? was chosen via a line-search; we report results
for ? ? 10?8 . (Again performance does not appear to be very sensitive to ?, so that a crude linesearch appears to suffice.) In accordance with the theory we set the constraint violation parameter
? ? 2/(1 ? ?), as suggested by the analysis of Section 3.1, as well as by [12],
7
policy
performance
Longest Queue
Max-Weight
8.09
6.55
sample size
SALP, cubic basis
RSALP, Gaussian kernel
1000
7.19
6.72
(1.76)
(0.39)
3000
7.89
6.31
(1.76)
(0.11)
5000
6.94
6.13
(1.15)
(0.08)
10000
6.63
6.04
(0.92)
(0.05)
Table 1: Performance results in the queueing example. For the SALP and RSALP methods, the
number in the parenthesis gives the standard deviation across sample sets.
SALP [12]. The SALP formulation (1), is, as discussed earlier, the parametric counterpart to the
RSALP. It may be viewed as a generalization of the ALP approach proposed by [13] and has been
demonstrated to provide substantial performance benefits relative to the ALP approach. Our choice
of parameters for the SALP mirrors those for the RSALP to the extent possible, so as to allow for an
?apples-to-apples? comparison. Thus, we solve the sample average approximation of this program
using the same geometric sampling distribution and parameter ?. Approximation architectures in
which the basis functions are monomials of the queue lengths appear to be a popular choice for
queueing control problems [13]. We use all monomials with degree at most 3, which we will call
the cubic basis, as our approximation architectures.
Longest Queue (generic). This is a simple heuristic approach: at any given time, a server chooses
to work on the longest queue from among those it can service.
Max-Weight [20]. Max-Weight is a well known scheduling heuristic for queueing networks. The
policy is obtained as the greedy policy with respect to a value function approximation of the form
?4
J?M W (x) ? i=1 |xi |1+? , given a parameter ? > 0. This policy has been extensively studied and
shown to have a number of good properties, for example, being throughput optimal and offering
good performance for critically loaded settings [21]. Via a line-search, we chose to ? ? 1.5 as the
exponent for our experiments.
4.3
Results
Policies were evaluated using a common set of arrival process sample paths. The performance metric we report for each control policy is the long run average number of jobs in the system under that
?T
policy, t=1 ?xt ?1 /T , where we set T ? 10000. We further average this random quantity over
an ensemble of 300 sample paths. Further, in order to generate SALP and RSALP policies, state
sampling is required. To understand the effect of the sample size on the resulting policy performance, the different sample sizes listed in Table 1 were used. Since the policies generated involve
randomness to the sampled states, we further average performance over 10 sets of sampled states.
The results are reported in Table 1 and have the following salient features:
1. RSALP outperforms established policies: Approaches such as the Max-Weight or ?parametric? ADP with basis spanning polynomials have been previously shown to work well
for the problem of interest. We see that RSALP with 10000 samples achieves performance
that is superior to these extant schemes.
2. Sampling improves performance: This is expected from the theory in Section 3. Ideally, as
the sample size is increased one should relax the regularization. However, for our experiments we noticed that the performance is quite insensitive to the parameter ?. Nonetheless,
it is clear that larger sample sets yield a significant performance improvement.
3. RSALP in less sensitive to state sampling: We notice from the standard deviation values in
Table 1 that our approach gives policies whose performance varies significantly less across
different sample sets of the same size.
In summary we view these results as indicative of the possibility that the RSALP may serve as a
practical and viable alternative to state-of-the-art parametric ADP techniques.
8
References
[1] D. P. Bertsekas. Dynamic Programming and Optimal Control, Vol. II. Athena Scientific, 2007.
[2] B. Bethke, J. P. How, and A. Ozdaglar. Kernel-based reinforcement learning using Bellman
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[3] Y. Engel, S. Mannor, and R. Meir. Bayes meets Bellman: The Gaussian process approach to
temporal difference learning. In Proceedings of the 20th International Conference on Machine
Learning, pages 154?161. AAAI Press, 2003.
[4] X. Xu, D. Hu, and X. Lu. Kernel-based least squares policy iteration for reinforcement learning. IEEE Transactions on Neural Networks, 18(4):973?992, 2007.
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9
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3,919 | 4,548 | A Simple and Practical Algorithm
for Differentially Private Data Release
Moritz Hardt
IBM Almaden Research
San Jose, CA
[email protected]
Katrina Ligett?
Caltech
[email protected]
Frank McSherry
Microsoft Research SVC
[email protected]
Abstract
We present a new algorithm for differentially private data release, based on a simple combination of the Multiplicative Weights update rule with the Exponential
Mechanism. Our MWEM algorithm achieves what are the best known and nearly
optimal theoretical guarantees, while at the same time being simple to implement
and experimentally more accurate on actual data sets than existing techniques.
1
Introduction
Sensitive statistical data on individuals are ubiquitous, and publishable analysis of such private data
is an important objective. When releasing statistics or synthetic data based on sensitive data sets, one
must balance the inherent tradeoff between the usefulness of the released information and the privacy of the affected individuals. Against this backdrop, differential privacy [1, 2, 3] has emerged as a
compelling privacy definition that allows one to understand this tradeoff via formal, provable guarantees. In recent years, the theoretical literature on differential privacy has provided a large repertoire
of techniques for achieving the definition in a variety of settings (see, e.g., [4, 5]). However, data analysts have found that several algorithms for achieving differential privacy add unacceptable levels
of noise.
In this work we develop a broadly applicable, simple, and easy-to-implement algorithm, capable of
substantially improving the performance of linear queries on many realistic datasets. Linear queries
are equivalent to statistical queries (in the sense of [6]) and can serve as the basis of a wide range of
data analysis and learning algorithms (see [7] for some examples).
Our algorithm is a combination of the Multiplicative Weights approach of [8, 9], maintaining and
correcting an approximating distribution through queries on which the approximate and true datasets
differ, and the Exponential Mechanism [10], which selects the queries most informative to the Multiplicative Weights algorithm (specifically, those most incorrect vis-a-vis the current approximation).
One can view our approach as combining expert learning techniques (multiplicative weights) with
an active learning component (via the exponential mechanism).
We present experimental results for differentially private data release for a variety of problems studied in prior work: range queries as studied by [11, 12], contingency table release across a collection
of statistical benchmarks as in [13], and datacube release as studied by [14]. We empirically evaluate the accuracy of the differentially private data produced by MWEM using the same query class
and accuracy metric proposed by each of the corresponding prior works, improving on all. Beyond empirical improvements in these settings, MWEM matches the best known and nearly optimal
theoretical accuracy guarantees for differentially private data analysis with linear queries.
?
Computer Science Department, Cornell University. Work supported in part by an NSF Computing Innovation Fellowship (NSF Award CNF-0937060) and an NSF Mathematical Sciences Postdoctoral Fellowship
(NSF Award DMS-1004416).
1
Finally, we describe a scalable implementation of MWEM capable of processing datasets of substantial complexity. Producing synthetic data for the classes of queries we consider is known to be
computationally hard in the worst-case [15, 16]. Indeed, almost all prior work performs computation proportional to the size of the data domain, which limits them to datasets with relatively few
attributes. In contrast, we are able to process datasets with thousands of attributes, corresponding to
domains of size 21000 . Our implementation integrates a scalable parallel implementation of Multiplicative Weights, and a representation of the approximating distribution in a factored form that only
exhibits complexity when the model requires it.
2
Our Approach
The MWEM algorithm (Figure 1) maintains an approximating distribution over the domain D of
data records, scaled up by the number of records. We repeatedly improve the accuracy of this approximation with respect to the private dataset and the desired query set by selecting and posing a
query poorly served by our approximation and improving the approximation to better reflect the true
answer to this query. We select and pose queries using the Exponential [10] and Laplace Mechanisms [3], whose definitions and privacy properties we review in Subsection 2.1. We improve our
approximation using the Multiplicative Weights update rule [8], reviewed in Subsection 2.2.
2.1
Differential Privacy and Mechanisms
Differential privacy is a constraint on a randomized computation that the computation should not
reveal specifics of individual records present in the input. It places this constraint by requiring the
mechanism to behave almost identically on any two datasets that are sufficiently close.
Imagine a dataset A whose records are drawn from some abstract domain D, and which is described
as a function from D to the natural numbers N, with A(x) indicating the frequency (number of
occurrences) of x in the dataset. We use kA Bk to indicate the sum of the absolute values of
difference in frequencies (how many records would have to be added or removed to change A to B).
Definition 2.1 (Differential Privacy). A mechanism M mapping datasets to distributions over an
output space R provides (", )-differential privacy if for every S ? R and for all data sets A, B
where kA Bk ? 1,
P r[M (A) 2 S] ? e" Pr[M (B) 2 S] + .
If = 0 we say that M provides "-differential privacy.
The Exponential Mechanism [10] is an "-differentially private mechanism that can be used to select
among the best of a discrete set of alternatives, where ?best? is defined by a function relating each
alternative to the underlying secret data. Formally, for a set of alternative results R, we require
a quality scoring function s : dataset ? R ! R, where s(B, r) is interpreted as the quality of
the result r for the dataset B. To guarantee "-differential privacy, the quality function is required
to satisfy a stability property: that for each result r the difference |s(A, r) s(B, r)| is at most
kA Bk. The Exponential Mechanism E simply selects a result r from the distribution satisfying
Pr[E(B) = r] / exp(" ? s(B, r)/2).
Intuitively, the mechanism selects result r biased exponentially by its quality score. The Exponential
Mechanism takes time linear in the number of possible results, evaluating s(B, r) once for each r.
A linear query (also referred to as counting query or statistical query) is specified by a function q
mapping data records
P to the interval [ 1, +1]. The answer of a linear query on a data set D, denoted
q(B), is the sum x2D q(x) ? B(x).
The Laplace Mechanism is an "-differentially private mechanism which reports approximate sums
of bounded functions across a dataset. If q is a linear query, the Laplace Mechanism L obeys
Pr[L(B) = r] / exp ( " ? |r
q(B)|)
Although the Laplace Mechanism is an instance of the Exponential Mechanism, it can be implemented much more efficiently, by adding Laplace noise with parameter 1/" to the value q(B). As
the Laplace distribution is exponentially concentrated, the Laplace Mechanism provides an excellent
approximation to the true sum.
2
Inputs: Data set B over a universe D; Set Q of linear queries; Number of iterations T 2 N; Privacy
parameter " > 0; Number of records n.
Let A0 denote n times the uniform distribution over D.
For iteration i = 1, ..., T :
1. Exponential Mechanism: Select a query qi 2 Q using the Exponential Mechanism parameterized with epsilon value "/2T and the score function
si (B, q) = |q(Ai
1)
q(B)| .
2. Laplace Mechanism: Let measurement mi = qi (B) + Lap(2T /").
3. Multiplicative Weights: Let Ai be n times the distribution whose entries satisfy
Ai (x) / Ai
1 (x)
? exp(qi (x) ? (mi
qi (Ai
1 ))/2n)
.
Output: A = avgi<T Ai .
Figure 1: The MWEM algorithm.
2.2
Multiplicative Weights Update Rule
The Multiplicative Weights approach has seen application in many areas of computer science. Here
we will use it as proposed in Hardt and Rothblum [8], to repeatedly improve an approximate distribution to better reflect some true distribution. The intuition behind Multiplicative Weights is that
should we find a query whose answer on the true data is much larger than its answer or the approximate data, we should scale up the approximating weights on records contributing positively and
scale down the weights on records contributing negatively. If the true answer is much less than the
approximate answer, we should do the opposite.
More formally, let q be a linear query. If A and B are distributions over the domain D of records,
where A is a synthetic distribution intended to approximate a true distribution B with respect to
query q, then the Multiplicative Weights update rule recommends updating the weight A places on
each record x by:
Anew (x) / A(x) ? exp(q(x) ? (q(B) q(A))/2) .
The proportionality sign indicates that the approximation should be renormalized after scaling.
Hardt and Rothblum show that each time this rule is applied, the relative entropy between A and B
decreases by an additive (q(A) q(B))2 . As long as we can continue to find queries on which the
two disagree, we can continue to improve the approximation.
2.3
Formal Guarantees
As indicated in the introduction, the formal guarantees of MWEM represent the best known theoretical results on differentially private synthetic data release. We first describe the privacy properties.
Theorem 2.1. MWEM satisfies "-differential privacy.
Proof. The composition rules for differential privacy state that " values accumulate additively. We
make T calls to the Exponential Mechanism with parameter ("/2T ) and T calls to the Laplace
Mechanism with parameter ("/2T ), resulting in "-differential privacy.
We now bound the worst-case performance of the algorithm, in terms of the maximum error between
A and B across all q 2 Q. The natural range for q(A) is [ n, +n], and we see that by increasing T
beyond 4 log |D| we can bring the error asymptotically smaller than n.
Theorem 2.2. For any dataset B, set of linear queries Q, T 2 N, and " > 0, with probability at
least 1 2T /|Q|, MWEM produces A such that
r
log |D| 10T log |Q|
max |q(A) q(B)| ? 2n
+
.
q2Q
T
"
3
Proof. The proof of this theorem is an integration of pre-existing analyses of both the Exponential
Mechanism and the Multiplicative Weights update rule, omitted for reasons of space.
Note that these bounds are worst-case bounds, over adversarially chosen data and query sets. We
will see in Section 3 that MWEM works very well in more realistic settings.
2.3.1
Running time
The running time of our basic algorithm as described in Figure 1 is O(n|Q| + T |D||Q|)). The algorithm is embarrassingly parallel: query evaluation can be conducted independently, implemented
using modern database technology; the only required serialization is that the T steps must proceed
in sequence, but within each step essentially all work is parallelizable.
Results of Dwork et al. [17] show that for worst case data, producing differentially private synthetic
data for a set of counting queries requires time |D|0.99 under reasonable cryptographic hardness
assumptions. Moreover, Ullman and Vadhan [16] showed that similar lower bounds also hold for
more basic query classes such as we consider in Section 3.2. Despite these hardness results, we
provide an alternate implementation of our algorithm in Section 4 and demonstrate that its running
time is acceptable on real-world data even in cases where |D| is as large as 277 , and on simple
synthetic input datasets where |D| is as large as 21000 .
2.3.2
Improvements and Variations
There are several ways to improve the empirical performance of MWEM at the expense of the
theoretical guarantees. First, rather than use the average of the distributions Ai we use only the
final distribution. Second, in each iteration we apply the multiplicative weights update rule for all
measuments taken, multiple times; as long as any measurements do not agree with the approximating
distribution (within error) we can improve the result. Finally, it is occasionally helpful to initialize
A0 by performing a noisy count for each element of the domain; this consumes from the privacy
budget and lessens the accuracy of subsequent queries, but is often a good trade-off.
2.4
Related Work
The study of differentially private synthetic data release mechanisms for arbitrary counting queries
began with the work of Blum, Ligett, and Roth [18], who gave a computationally inefficient (superpolynomial in |D|) "-differentially private algorithm that achieves error that scales only logarithmically with the number of queries. The dependence on n and |Q| achieved by their algorithm is
O(n2/3 log1/3 |Q|) (which is the same dependence achieved by optimizing the choice of T in Theorem 2.2). Since [18], subsequent work [17, 19, 20, 8] has focused on computationally more efficient
algorithms (i.e., polynomial in |D|) as well as algorithms that work in the interactive query setting.
The latest of these results is the private
p Multiplicative Weights method of Hardt and Rothblum [8]
which achieves error rates of O( n log(|Q|)) for (", )-differential privacy (which is the same
dependence achieved by applying k-fold adaptive composition [19] and optimizing T in our Theorem 2.2). While their algorithm works in the interactive setting, it can also be used non-interactively
to produce synthetic data, albeit at a computational overhead of O(n). MWEM can also be cast as
an instance of a more general Multiplicative-Weights based framework of Gupta et al. [9], though
our specific instantiation and its practical appeal were not anticipated in their work.
Prior work on linear queries includes Fienberg et al. [13] and Barak et al. [21] on contingency tables;
Li et al. [22] on range queries (and substantial related work [23, 24, 22, 11, 12, 25] which Li and
Miklau [11, 25] show can all be seen as instances of the matrix mechanism of [22]); and Ding et
al. [14] on data cubes. In each case, MWEM?s theoretical guarantees and experimental performance
improve on prior work. We compare further in Section 3.
3
Experimental Evaluation
We evaluate MWEM across a variety of query classes, datasets, and metrics as explored by prior
work, demonstrating improvement in the quality of approximation (often significant) in each case.
The problems we consider are: (1) range queries under the total squared error metric, (2) binary
4
contingency table release under the relative entropy metric, and (3) datacube release under the average absolute error metric. Although contingency table release and datacube release are very similar,
prior work on the two have had different focuses: small datasets over many binary attributes vs. large
datasets over few categorical attributes, low-order marginals vs. all cuboids as queries, and relative
entropy vs. the average error within a cuboid as metrics.
Our general conclusion is that intelligently selecting the queries to measure can result in significant
accuracy improvements, in settings where accuracy is a scare resource. When the privacy parameters
are very lax, or the query set very simple, direct measurement of all queries yields better results than
expending some fraction of the privacy budget determining what to measure. On the other hand, in
the more challenging case of restrictions on privacy for complex data and query sets, MWEM can
substantially out-perform previous algorithms.
3.1
Range Queries
A range query over a domain D = {1, . . . , N } is a counting query specified by the indicator function
of an interval I ? D. Over a multi-dimensional domain D = D1 ? . . . Dd a range query is
defined by the product of indicator functions. Differentially private algorithms for range queries
were specifically considered by [18, 23, 24, 22, 11, 12, 25]. As noted in [11, 25], all previously
implemented algorithms for range queries can be seen as instances of the matrix mechanism of [22].
Moreover, [11, 25] show a lower bound on the total squared error achieved by the matrix mechanism
in terms of the singular values of a matrix associated with the set of queries. We refer to this bound
as the SVD bound.
Transfusion: monetary
Transfusion: recency x frequency
1.00E+10
1.00E+11
1.00E+10
1.00E+09
1.00E+09
1.00E+08
1.00E+08
1.00E+07
1.00E+07
1.00E+06
1.00E+06
0.0125
0.025
MWEM (T = 10)
0.5
0.1
0.0125
SVD Lower Bound
0.025
MWEM (T = 10)
Adult: capital loss
0.5
0.1
SVD Lower Bound
Adult: age x hours
1.00E+11
1.00E+11
1.00E+10
1.00E+10
1.00E+09
1.00E+09
1.00E+08
1.00E+08
1.00E+07
1.00E+07
1.00E+06
1.00E+06
0.0125
0.025
MWEM (T = 10)
0.5
0.1
0.0125
SVD Lower Bound
0.025
MWEM (T = 10)
0.5
0.1
SVD Lower Bound
Figure 2: Comparison of MWEM with the SVD lower bound on four data sets. The y-axis measures
the average squared error per query, averaged over 5 independent repetitions of the experiment,
as epsilon varies. The improvement is most significant for small epsilon, diminishing as epsilon
increases.
We empirically evaluate MWEM for range queries on restrictions of the Adult data set [26] to (a)
the ?capital loss? attribute, and (b) the ?age? and ?hours? attributes, as well as the restriction of
the Blood Transfusion data set [26, 27] to (c) the ?recency? and ?frequency? attributes, and (d) the
?monetary? attribute. We chose these data sets as they feature numerical attributes of suitable size.
In Figure 2, we compare the performance of MWEM on sets of randomly chosen range queries
against the SVD lower bound proved by [11, 25], varying " while keeping the number of queries
fixed. The SVD lower bound holds for algorithms achieving the strictly weaker guarantee of (", )differential privacy with > 0, permitting some probability of unbounded disclosure. The SVD
5
50
5
0.5
45
4.5
0.45
40
4
0.4
35
3.5
0.35
30
3
0.3
25
2.5
0.25
20
2
0.2
15
1.5
0.15
10
1
0.1
5
0.5
0.05
0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3: Relative entropy (y-axis) as a function of epsilon (x-axis) for the mildew, rochdale, and
czech datasets, respectively. The lines represent averages across 100 runs, and the corresponding
shaded areas one standard deviation in each direction. Red (dashed) represents the modified Barak
et al. [21] algorithm, green (dot-dashed) represents unoptimized MWEM, and blue (solid) represents
the optimized version thereof. The solid black horizontal line is the stated relative entropy values
from Fienberg et al. [13].
bound depends on ; in our experiments we fixed = 1/n when instantiating the SVD bound, as
any larger value of permits mechanisms capable of exact release of individual records.
3.2
Contingency Tables
A contingency table can be thought of as a table of records over d binary attributes, and the k-way
marginals of a contingency table correspond to the kd possible choices of k attributes, where each
marginal is represented by the 2k counts of the records with each possible setting of attributes. In
previous work, Barak et al. [21] describe an approach to differentially private contingency table release using linear queries defined by the Hadamard matrix. Importantly, all k-dimensional marginals
can be exactly recovered by examination of relatively few such queries: roughly kd out of the possible 2d , improving over direct measurement of the marginals by a factor of 2k . This algorithm is
evaluated by Fienberg et al. [13], and was found to do poorly on several benchmark datasets.
We evaluate our approximate dataset following Fienberg et al. [13] using relative entropy, also
known as the Kullback-Leibler (or KL) divergence. Formally, the relative entropy between our two
distributions (A/n and B/n) is
X
RE(B||A) =
B(x) log(B(x)/A(x))/n .
x2D
We use several statistical datasets from Fienberg et al. [13], and evaluate two variants of MWEM
(both with and without initialization of A0 ) against a modification of Barak et al. [21] which combines its observations using multiplicative weights (we find that without this modification, [21] is
terrible with respect to relative entropy). These experiments are therefore largely assessing the selective choice of measurements to take, rather than the efficacy of multiplicative weights.
Figure 3 presents the evaluation of MWEM on several small datasets in common use by statisticians.
Our findings here are fairly uniform across the datasets: the ability to measure only those queries
that are informative about the dataset results in substantial savings over taking all possible measurements. In many cases MWEM approaches the good non-private values of [13], indicating that we
can approach levels of accuracy at the limit of statistical validity.
We also consider a larger dataset, the National Long-Term Care Study (NLTCS), in Figure 4. This
dataset contains orders of magnitudes more records, and has 16 binary attributes. For our initial settings, maintaining all three-way marginals, we see similar behavior as above: the ability to choose
the measurements that are important allows substantially higher accuracy on those that matter. However, we see that the algorithm of Barak et al. [21] is substantially more competitive in the regime
where we are interested in querying all two-dimensional marginals, rather than the default three we
have been using. In this case, for values of epsilon at least 0.1, it seems that there is enough signal
present to simply measure all corresponding entries of the Hadamard transform; each is sufficiently
informative that measuring substantially fewer at higher accuracy imparts less information, rather
than more.
6
5
2
2
4.5
1.8
1.8
4
1.6
1.6
3.5
1.4
1.4
3
1.2
1.2
2.5
1
1
2
0.8
0.8
1.5
0.6
0.6
1
0.4
0.4
0.5
0.2
0.01
0.03
0.05
0.07
0.1
0
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4: Curves comparing our approach with that of Barak et al. on the National Long Term Care
Survey. The red (dashed) curve represents Barak et al, and the multiple blue (solid) curves represent
MWEM, with 20, 30, and 40 queries (top to bottom, respectively). From left to right, the first two
figures correspond to degree 2 marginals, and the third to degree 3 marginals. As before, the xaxis is the value of epsilon guaranteed, and the y-axis is the relative entropy between the produced
distribution and actual dataset. The lines represent averages across only 10 runs, owing to the high
complexity of Barak et al. on this many-attributed dataset, and the corresponding shaded areas one
standard deviation in each direction.
3.3
Data Cubes
We now change our terminology and objectives, shifting our view of contingency tables to one of
datacubes. The two concepts are interchangeable, a contingency table corresponding to the datacube,
and a marginal corresponding to its cuboids. However, the datasets studied and the metrics applied
are different. We focus on the restriction of the Adult dataset [26] to its eight categorical attributes,
as done in [14], and evaluate our approximations using average error within a cuboid, also as in [14].
Although MWEM is defined with respect to a single query at a time, it generalizes to sets of counting
queries, as reflected in a cuboid. The Exponential Mechanism can select a cuboid to measure using
a quality score function summing the absolute values of the errors within the cells of the cuboid. We
also (heuristically) subtract the number of cells from the score of a cuboid to bias the selection away
from cuboids with many cells, which would collect Laplace error in each cell. This subtraction
does not affect privacy properties. An entire cuboid can be measured with a single differentially
private query, as any record contributes to at most one cell (this is a generalization of the Laplace
Mechanism to multiple dimensions, from [3]). Finally, Multiplicative Weights works unmodified,
increasing and decreasing weights based on the over- or under-estimation of the count to which the
record contributes.
Average Average Error
Maximum Average Error
250
800
700
200
600
500
150
400
100
300
200
50
100
0
0
0.25
0.5
PMostC
1
1.5
2
0.25
MWEM (T = 10)
0.5
BMaxC
1
1.5
2
MWEM (T = 10)
Figure 5: Comparison of MWEM with the custom approaches from [14], varying epsilon through
the reported values from [14]. Each cuboid (marginal) is assessed by its average error, and either the
average or maximum over all 256 marginals is taken to evaluate the technique.
We compare MWEM with the work of [14] in Figure 5. The average average error improves noticeably, by approximately a factor of four. The maximum average error is less clear; experimentally
we have found we can bring the numbers lower using different heuristic variants of MWEM, but
without principled guidance we report only the default behavior. Of note, our results are achieved
7
by a single algorithm, whereas the best results for maximum and average error in [14] are achieved
by two different algorithms, each designed to optimize one specific metric.
4
A Scalable Implementation
The implementation of MWEM used in the previous experiments quite literally maintains a distribution Ai over the elements of the universe D. As the number of attributes grows, the universe D
grows exponentially, and it can quickly become infeasible to track the distribution explicitly. In
this section, we consider a scalable implementation with essentially no memory footprint, whose
running time is in the worst case proportional to |D|, but which for many classes of simple datasets
remains linear in the number of attributes.
Recall that the heart of MWEM maintains a distribution Ai over D that is then used in the Exponential Mechanism to select queries poorly approximated by the current distribution. From the
definition of the Multiplicative Weights distribution, we see that the weight Ai (x) can be determined
from the history Hi = {(qj , mj ) : j ? i}:
0
1
X
Ai (x) / exp @
qj (x) ? (mj qj (Aj 1 ))/2nA .
j?i
We explicitly record the scaling factors lj = mj
qj (Aj
{(qj , mj , lj ) : j ? i}, to remove the dependence on prior Aj .
1)
as part of the history Hi =
The domain D is often the product of many attributes. If we partition these attributes into disjoint
parts D1 , D2 , . . . Dk so that no query in Hi involves attributes from more than one part, then the
distribution produced by Multiplicative Weights is a product distribution over D1 ?D2 ?. . . Dk . For
query classes that factorize over the attributes of the domain (for example, range queries, marginal
queries, and cuboid queries) we can rewrite and efficiently perform the integration over D using
0
1
X
Y
X
@
q(x) ? Ai (x) =
q(xj ) ? Aji (xj )A .
x2D1 ?D2 ?...Dk
1?j?k
xj 2Dj
where Aji
from Hi .
is a mini Multiplicative Weights over attributes in part Dj , using only the relevant queries
So long as the measurements taken reflect modest groups of independent attributes, the
integration can be efficiently performed. As the measurements overlap more and more, additional
computation or approximation is required. The memory footprint is only the combined size of the
data, query, and history sets.
Experimentally, we are able to process a binarized form of the Adult dataset with 27 attributes efficiently (taking 80 seconds to process completely), and the addition of 50 new independent binary
attributes, corresponding to a domain of size 277 , results in neglible performance impact. For a simple synthetic dataset with up to 1,000 independent binary attributes, the factorized implementation
of MWEM takes only 19 seconds to for a complete execution.
5
Conclusions
We introduced MWEM, a simple algorithm for releasing data maintaining a high fidelity to the
protected source data, as well as differential privacy with respect to the records. The approach builds
upon the Multiplicative Weights approach of [8, 9], by introducing the Exponential Mechanism [10]
as a more judicious approach to determining which measurements to take. The theoretical analysis
matches previous work in the area, and experimentally we have evidence that for many interesting
settings, MWEM represents a substantial improvement over existing techniques.
As well as improving on experimental error, the algorithm is both simple to implement and simple
to use. An analyst does not require a complicated mathematical understanding of the nature of the
queries (as the community has for linear algebra [11] and the Hadamard transform [21]), but rather
only needs to enumerate those measurements that should be preserved. We hope that this generality
leads to a broader class of high-fidelity differentially-private data releases across a variety of data
domains.
8
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sequence. Expert Systems with Applications, 36(3), 2008.
9
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3,920 | 4,549 | Multiple Choice Learning:
Learning to Produce Multiple Structured Outputs
Abner Guzman-Rivera
University of Illinois
[email protected]
Dhruv Batra
Virginia Tech
[email protected]
Pushmeet Kohli
Microsoft Research Cambridge
[email protected]
Abstract
We address the problem of generating multiple hypotheses for structured prediction tasks that involve interaction with users or successive components in a cascaded architecture. Given a set of multiple hypotheses, such components/users
typically have the ability to retrieve the best (or approximately the best) solution
in this set. The standard approach for handling such a scenario is to first learn
a single-output model and then produce M -Best Maximum a Posteriori (MAP)
hypotheses from this model. In contrast, we learn to produce multiple outputs
by formulating this task as a multiple-output structured-output prediction problem with a loss-function that effectively captures the setup of the problem. We
present a max-margin formulation that minimizes an upper-bound on this lossfunction. Experimental results on image segmentation and protein side-chain prediction show that our method outperforms conventional approaches used for this
type of scenario and leads to substantial improvements in prediction accuracy.
1
Introduction
A number of problems in Computer Vision, Natural Language Processing and Computational
Biology involve predictions over complex but structured interdependent outputs, also known as
structured-output prediction. Formulations such as Conditional Random Fields (CRFs) [18], MaxMargin Markov Networks (M3 N) [27], and Structured Support Vector Machines (SSVMs) [28] have
provided principled techniques for learning such models.
In all these (supervised) settings, the learning algorithm typically has access to input-output pairs:
{(xi , yi ) | xi ? X , yi ? Y} and the goal is to learn a mapping from the input space to the output
space f : X ? Y that minimizes a (regularized) task-dependent loss function ` : Y ? Y ? R+ ,
where `(yi , y?i ) denotes the cost of predicting y?i when the correct label is yi .
Notice that the algorithm always makes a single prediction y?i and pays a penalty `(yi , y?i ) for that
prediction. However, in a number of settings, it might be beneficial (even necessary) to make multiple predictions:
1. Interactive Intelligent Systems. The goal of interactive machine-learning algorithms is to
produce an output for an expert or a user in the loop. Popular examples include tools for
interactive image segmentation (where the system produces a cutout of an object from a
picture [5, 25]), systems for image processing/manipulation tasks such as image denoising
and deblurring (e.g., Photoshop), or machine translation services (e.g., Google Translate).
These problems are typically modeled using structured probabilistic models and involve
computing the Maximum a Posteriori (MAP) solution. In order to minimize user interactions, the interface could show not just a single prediction but a small set of diverse
predictions, and simply let the user pick the best one.
2. Generating M -Best Hypotheses. Machine learning algorithms are often cascaded, with
the output of one model being fed into another. In such a setting, at the initial stages it is
1
not necessary to make the perfect prediction, rather the goal is to make a set of plausible
predictions, which may then be re-ranked or combined by a secondary mechanism. For
instance, in Computer Vision, this is the case for state-of-the-art methods for human-pose
estimation which produce multiple predictions that are then refined by employing a temporal model [23, 3]. In Natural Language Processing, this is the case for sentence parsing [8]
and machine translation [26], where an initial system produces a list of M -Best hypotheses [12, 24] (also called k-best lists in the NLP literature), which are then re-ranked.
The common principle in both scenarios is that we need to generate a set of plausible hypotheses
for an algorithm/expert downstream to evaluate. Traditionally, this is accomplished by learning a
single-output model and then producing M -Best hypotheses from it (also called the M -Best MAP
problem [20, 11, 29] or the Diverse M -Best problem [3] in the context of graphical models).
Notice that the single-output model is typically trained in the standard way, i.e., either to match the
data distribution (max-likelihood) or to score ground-truth the highest by a margin (max-margin).
Thus, there is a disparity between the way this model is trained and the way it is actually used. The
key motivating question for this paper is ? can we learn to produce a set of plausible hypotheses?
We refer to such a setting as Multiple Choice Learning (MCL) because the learner must learn to
produce multiple choices for an expert or other algorithm.
Overview. This paper presents an algorithm for MCL, formulated as multiple-output structuredoutput learning, where given an input sample xi the algorithm produces a set of M hypotheses
{?
yi1 , . . . , y?iM }. We first present a meaningful loss function for this task that effectively captures the
setup of the problem. Next, we present a max-margin formulation for training this M -tuple predictor
that minimizes an upper-bound on the loss-function. Despite the popularity of M -Best approaches,
to the best our knowledge, this is the first attempt to directly model the M -Best prediction problem.
Our approach has natural connections to SSVMs with latent variables, and resembles a structuredoutput version of k-means clustering. Experimental results on the problems of image segmentation
and protein side-chain prediction show that our method outperforms conventional M -Best prediction
approaches used for this scenario and leads to substantial improvements in prediction accuracy.
The outline for the rest of this paper is as follows: Section 2 provides the notation and discusses
classical (single-output) structured-output learning; Section 3 introduces the natural task loss for
multiple-output prediction and presents our learning algorithm; Section 4 discusses related work;
Section 5 compares our algorithm to other approaches experimentally and; we conclude in Section
6 with a summary and ideas for future work.
2
Preliminaries: (Single-Output) Structured-Output Prediction
We begin by reviewing classical (single-output) structured-output prediction and establishing the
notation used in the paper.
Notation. For any positive integer n, let [n] be shorthand for the set {1, 2, . . . , n}. Given a training
dataset of input-output pairs {(xi , yi ) | i ? [n], xi ? X , yi ? Y}, we are interested in learning a
mapping f : X ? Y from an input space X to a structured output space Y that is finite but typically
exponentially large (e.g., the set of all segmentations of an image, or all English translations of a
Chinese sentence).
Structured Support Vector Machines (SSVMs). In an SSVM setting, the mapping is defined as
f (x) = argmaxy?Y wT ?(x, y), where ?(x, y) is a joint feature map: ? : X ? Y ? Rd . The
quality of the prediction y?i = f (xi ) is measured by a task-specific loss function ` : Y ? Y ?
R+ , where `(yi , y?i ) denotes the cost of predicting y?i when the correct label is yi . Some examples
of loss functions are the intersection/union criteria used by the PASCAL Visual Object Category
Segmentation Challenge [10], and the BLEU score used to evaluate machine translations [22].
The task-loss is typically non-convex and non-continuous in w. Tsochantaridis et al. [28] proposed
to optimize a regularized surrogate loss function:
min
w
X
1
2
||w||2 + C
}i (w)
2
i?[n]
2
(1)
where C is a positive multiplier and }i (?) is the structured hinge-loss:
}i (w) = max `(yi , y) + wT ?(xi , y) ? wT ?(xi , yi ).
y
(2)
It can be shown [28] that the hinge-loss is an upper-bound on the task loss, i.e., }i (w) ? `(yi , f (xi )).
Moreover, }i (w) is a non-smooth convex function, and can be equivalently expressed with a set of
constraints:
X
1
2
min
||w||2 + C
?i
(3a)
w,?i
2
i?[n]
T
s.t. w ?(xi , yi ) ? wT ?(xi , y) ? `(yi , y) ? ?i
?i ? 0
?y ? Y \ yi
(3b)
(3c)
This formulation is known as the margin-rescaled n-slack SSVM [28]. Intuitively, we can see that it
minimizes the squared-norm of w subject to constraints that enforce a soft-margin between the score
of the ground-truth yi and the score of all other predictions. The above problem (3) is a Quadratic
Program (QP) with n|Y| constraints, which is typically exponentially large. If an efficient separation oracle for identifying the most violated constraint is available, then a cutting-plane approach
can be used to solve the QP. A cutting-plane algorithm maintains a working set of constraints and
incrementally adds the most violated constraint to this working set while solving for the optimum
solution under the working set. Tsochantaridis et al. [28] showed that such a procedure converges
in a polynomial number of steps.
3
Multiple-Output Structured-Output Prediction
We now describe our proposed formulation for multiple-output structured-output prediction.
Model. Our model is a generalization of the single-output SSVM. A multiple-output SSVM is a
mapping from the input space X to an M -tuple1 of structured outputs Yi = {?
yi1 , . . . , y?iM | y?i ? Y},
M
T
given by g : X ? Y , where g(x) = argmaxY ?Y M W ?(x, Y ). Notice that the joint feature map is now a function of the input and the entire set of predicted structured-outputs, i.e.,
? : X ? Y M ? Rd . Without further assumptions, optimizing over the output space |Y|M
would be intractable. We make a mean-field-like simplifying assumption that the set score factors into independent predictor scores, i.e., ?(xi , Y ) = [ ?1 (xi , y 1 )T , . . . , ?M (xi , y M )T ]T . Thus,
g is composed of M single-output predictors: g(x) = f 1 (x), . . . , f M (x) , where f m (x) =
T m
argmaxy?Y wm
? (x, y). Hence, the multiple-output SSVM is parameterized by an M -tuple of
T T
] .
weight vectors: W = [w1T , . . . , wM
3.1
Multiple-Output Loss
Let Y?i = {?
yi1 , . . . , y?iM } be the set of predicted outputs for input xi , i.e., y?im = f m (xi ). In the singleoutput SSVM, there typically exists a ground-truth output yi for each datapoint, and the quality of
y?i w.r.t. yi is given by `(yi , y?i ).
How good is a set of outputs? For our multiple-output predictor, we need to define a task-specific
loss function that can measure the quality of any set of predictions Y?i ? Y M . Ideally, the quality of
these predictions should be evaluated by the secondary mechanism that uses these predictions. For
instance, in an interactive setting where they are shown to a user, the quality of Y?i could be measured
by how much it reduces the user-interaction time. In the M-best hypotheses re-ranking scenario, the
accuracy of the top single output after re-ranking could be used as the quality measure for Y?i . While
multiple options exist, in order to provide a general formulation and to isolate our approach, we
propose the ?oracle? or ?hindsight? set-loss as a surrogate:
L(Y?i ) = min `(yi , y?i )
y?i ?Y?i
(4)
1
Our formulation is described with a nominal ordering of the predictions. However, both the proposed
objective function and optimization algorithm are invariant to permutations of this ordering.
3
i.e., the set of predictions Y?i only pays a loss for the most accurate prediction contained in this set
(e.g., the best segmentation of an image, or the best translation of a sentence). This loss has the
desirable behaviour that predicting a set that contains even a single accurate output is better than
predicting a set that has none. Moreover, only being penalized for the most accurate prediction
allows an ensemble to hedge its bets without having to pay for being too diverse (this is opposite
to the effect that replacing min with max or avg. would have). However, this also makes the setloss rather poorly conditioned ? if even a single prediction in the ensemble is the ground-truth, the
set-loss is 0, no matter what else is predicted.
Hinge-like Upper-Bound. The set-loss L(Y?i (W)) is a non-continuous non-convex function of W
and is thus difficult to optimize. If unique ground-truth sets Yi were available, we could set up a
standard hinge-loss approximation:
Hi (W) = max L(Y ) + WT ?(xi , Y ) ? WT ?(xi , Yi )
(5)
Y ?Y M
where ?(xi , Y ) = [ ?1 (xi , y 1 )T , . . . , ?M (xi , y M )T ]T are stacked joint feature maps.
However, no such natural choice for Yi exists. We propose a hinge-like upper-bound on the set-loss,
that we refer to as min-hinge:
? i (W) = min }i (wm ),
H
(6)
m?[M ]
i.e., we take the min over the hinge-losses (2) corresponding to each of the M predictors. Since
each hinge-loss is an upper-bound on the corresponding task-loss, i.e., }i (wm ) ? `(yi , f m (xi )), it
? i (W) ? L(Y?i ).
is straightforward to see that the min-hinge is an upper-bound on the set-loss, i.e., H
Notice that min-hinge is a min of convex functions, and thus not guaranteed to be convex.
3.2
Coordinate Descent for Learning Multiple Predictors
We now present our algorithm for learning a multiple-output SSVM by minimizing the regularized
min-hinge loss:
X
1
2
? i (W)
min
H
(7)
||W||2 + C
W
2
i?[n]
We begin by rewriting the min-hinge loss in terms of indicator ?flag? variables, i.e.,
X X
1
2
min
||W||2 + C
?i,m }i (wm )
2
W,{?i,m }
i?[n] m?[M ]
X
s.t.
?i,m = 1
?i ? [n]
(8a)
(8b)
m?[M ]
?i,m ? {0, 1}
?i ? [n], m ? [M ]
(8c)
where ?i,m is a flag variable that indicates which predictor produces the smallest hinge-loss.
Optimization problem 8 is a mixed-integer quadratic programming problem (MIQP), which is NPhard in general. However, we can exploit the structure of the problem via a block-coordinate descent
algorithm where W and {?i,m } are optimized iteratively:
1. Fix W; Optimize all {?i,m }.
Given
W, the optimization over {?i,m } reduces to the minimization of
P
P
i?[n]
m?[M ] ?i,m }i (wm ) subject to the ?pick-one-predictor? constraints (8b,
8c). This decomposes into n independent problems, which simply identify the best
predictor for each datapoint according to the current hinge-losses, i.e.:
1 if m = argmin } (w )
i
m
m?[M ]
?i,m =
(9)
0 else.
2. Fix {?i,m }; Optimize W.
Given {?i,m }, optimization over W decomposes into M independent problems, one for
each predictor, which are equivalent to single-output SSVM learning problems:
4
min
W
X X
1
2
||W||2 + C
?i,m }i (wm )
2
i?[n] m?[M ]
?
?
?
X
X ?1
2
?i,m }i (wm )
= min
||wm ||2 + C
W
?
?2
i?[n]
m?[M ]
?
?
?1
?
X
X
2
}i (wm )
=
min
||wm ||2 + C
wm ? 2
?
(10a)
(10b)
(10c)
i:?i,m 6=0
m?[M ]
Thus, each subproblem in 10c can be optimized using using any standard technique for
training SSVMs. We use the 1-slack algorithm of [14].
Convergence. Overall, the block-coordinate descent algorithm above iteratively assigns each datapoint to a particular predictor (Step 1) and then independently trains each predictor with just the
points that were assigned to it (Step 2). This is fairly reminiscent of k-means, where step 1 can be
thought of as the member re-assignment step (or the M-step in EM) and step 2 can be thought of
as the cluster-fitting step (or the E-step in EM). Since the flag variables take on discrete values and
the objective function is non-increasing with iterations, the algorithm is guaranteed to converge in a
finite number of steps.
Generalization. Formulation (8) canPbe generalized by replacing the ?pick-one-predictor? constraint with ?pick-K-predictors?, i.e., m?[M ] ?i,m = K, where K is a robustness parameter that
allows training data overlap between predictors. The M-step (cluster reassignment) is still simple,
and involves assigning a data-point to the top K best predictors. The E-step is unchanged. Notice that
at K = M , all predictors learn the same mapping. We analyze the effect of K in our experiments.
4
Related Work
At first glance, our work seems related to the multi-label classification literature, where the goal is
to predict multiple labels for each input instance (e.g., text tags for images on Flickr). However, the
motivation and context of our work is fundamentally different. Specifically, in multi-label classification there are multiple possible labels for each instance and the goal is to predict as many of them
as possible. On the other hand, in our setting there is a single ground-truth label for each instance
and the learner makes multiple guesses, all of which are evaluated against that single ground-truth.
For the unstructured setting (i.e. when |Y| is polynomial), Dey et al. [9] proposed an algorithm that
learns a multi-class classifier for each ?slot? in a M -Best list, and provide a formal regret reduction
from submodular sequence optimization.
To the best of our knowledge, the only other work that explicitly addresses the task of predicting
multiple structured outputs is multi-label structured prediction (MLSP) [19]. This work may be seen
as a technique to output predictions in the power-set of Y (2Y ) with a learning cost comparable to
algorithms for prediction over Y. Most critically, MLSP requires gold-standard sets of labels (one
set for each training example). In contrast, MCL neither needs nor has access to gold-standard sets.
At a high-level, MCL and MLSP are orthogonal approaches, e.g., we could introduce MLSP within
MCL to create an algorithm that predicts multiple (diverse) sets of structured-outputs (e.g., multiple
guesses by the algorithm where each guess is a set of bounding boxes of objects in an image).
A form of min-set-loss has received some attention in the context of ambiguously or incompletely
annotated data. For instance, [4] trains an SSVM for object detection implicitly defining a taskadapted loss, Lmin (Y, y?) = miny?Y `(y, y?). Note that in this case there is a set of ground-truth
labels and the model?s prediction is a single label (evaluated against the closest ground-truth).
Our formulation is also reminiscent of a Latent-SSVM with the indicator flags {?i,m | m ? [M ]}
taking a role similar to latent variables. However, the two play very different roles. Latent variable
models typically maximize or marginalize the model score across the latent variables, while MCL
uses the flag variables as a representation of the oracle loss.
At a high-level, our ideas are also related to ensemble methods [21] like boosting. However, the key
difference is that ensemble methods attempt to combine outputs from multiple weak predictors to
ultimately make a single prediction. We are interested in making multiple predictions which will all
5
(a) Image
(b) GT
(c) 3.44%
(d) 40.79% (e) 14.78% (f) 26.98% (g) 53.78% (h) 19.54%
(c) 16.59%
(d) 2.12%
(e) 11.54% (f) 11.34% (g) 77.91% (h) 65.34%
Figure 1: Each row shows the: (a) input image (b) ground-truth segmentation and (c-h) the set of predictions
produced by MCL (M = 6). Red border indicates the most accurate segmentation (i.e., lowest error). We can
see that the predictors produce different plausible foreground hypotheses, e.g., predictor (g) thinks foliage-like
things are foreground.
be handed to an expert or secondary mechanism that has access to more complex (e.g., higher-order)
features.
5
Experiments
Setup. We tested algorithm MCL on two problems: i) foreground-background segmentation in
image collections and ii) protein side-chain prediction. In both problems making a single perfect
prediction is difficult due to inherent ambiguity in the tasks. Moreover, inference-time computing
limitations force us to learn restricted models (e.g., pairwise attractive CRFs) that may never be able
to capture the true solution with a single prediction. The goal of our experiments is to study how
much predicting a set of plausible hypotheses helps. Our experiments will show that MCL is able
to produce sets of hypotheses which contain more accurate predictions than other algorithms and
baselines aimed at producing multiple hypotheses.
5.1
Foreground-Background Segmentation
Dataset. We used the co-segmentation dataset, iCoseg, of Batra et al. [2]. iCoseg consists of 37
groups of related images mimicking typical consumer photograph collections. Each group may be
thought of as an ?event? (e.g., images from a baseball game, a safari, etc.). The dataset provides
pixel-level ground-truth foreground-background segmentations for each image. We used 9 difficult
groups from iCoseg containing 166 images in total. These images were then split into train, validation and test sets of roughly equal size. See Fig. 1, 2 for some example images and segmentations.
Model and Features. The segmentation task is modeled as a binary pairwise MRF where each
node corresponds to a superpixel [1] in the image. We extracted 12-dim color features at each superpixel (mean RGB; mean HSV; 5 bin Hue histogram; Hue histogram entropy). The edge features,
computed for each pair of adjacent superpixels, correspond to a standard Potts model and a contrast
sensitive Potts model. The weights at each edge were constrained to be positive so that the resulting
supermodular potentials could be maximized via graph-cuts [6, 17].
Baselines and Evaluation. We compare our algorithm against three alternatives for producing
multiple predictions: i) Single SSVM + M -Best MAP [29], ii) Single SSVM + Diverse M -Best
MAP [3] and iii) Clustering + Multiple SSVMs.
For the first two baselines, we used all training images to learn a single SSVM and then produced
multiple segmentations via M -Best MAP and Diverse M -Best MAP [3]. The M -Best MAP baseline
was implemented via the BMMF algorithm [29] using dynamic graph-cuts [15] for computing maxmarginals efficiently. For the Diverse M -Best MAP baseline we implemented the D IV MB EST
algorithm of Batra et al. [3] using dynamic graph-cuts. The third baseline, Clustering + Multiple
SSVM (C-SSVM), involves first clustering the training images into M clusters and then training M
SSVMs independently on each cluster. For clustering, we used k-means with `2 distance on color
features (same as above) computed on foreground pixels.
For each algorithm we varied the number of predictors M ? {1, 2, . . . , 6} and tuned the regularization parameter C on validation. Since MCL involves non-convex optimization, a good initialization
is important. We used the output of k-means clustering as the initial assignment of images to predictors, so MCL?s first coordinate descent iteration produces the same results as C-SSVM. The task-loss
6
(a) 45.91%
(b) 76.84%
(c) 8.30%
(d) 40.55%
(e) 30.01%
(f) 29.16%
(g) 19.54%
(a) 37.00%
(b) 36.06%
(c) 9.14%
(d) 28.54%
(e) 17.43%
(f) 26.91%
(g) 11.09%
(a) 14.70%
(b) 1.17%
(c) 5.69%
(d) 5.86%
(e) 1.18%
(f) 13.32%
(g) 3.44%
Figure 2: In each column: first row shows input images; second shows ground-truth; third shows segmentation
produced by the single SSVM baseline; and the last two rows show the best MCL predictions (M = 6) at the
end of the first and last coordinate descent iteration.
in this experiment (`) is the percentage of incorrectly labeled pixels, and the evaluation metric is the
set-loss, L = miny?i ?Y?i `(yi , y?i ), i.e., the pixel error of the best segmentation among all predictions.
Comparison against Baselines. Fig. 3a show the performance of various algorithms as a function of
the number of predictors M . We observed that M -Best MAP produces nearly identical predictions
and thus the error drops negligibly as M is increased. On the other hand, the diverse M -Best
predictions output by D IV MB EST [3] lead to a substantial drop in the set-loss. MCL outperforms
both D IV MB EST and C-SSVM, confirming our hypothesis that it is beneficial to learn a collection
of predictors, rather than learning a single predictor and making diverse predictions from it.
Behaviour of Coordinate Descent. Fig. 3b shows the MCL objective and train/test errors as a
function of the coordinate descent steps. We verify that the objective function is improved at every
iteration and notice a nice correlation between the objective and the train/test errors.
Effect of C. Fig. 3c compares performance for different values of regularization parameter C. We
observe a fairly stable trend with MCL consistently outperforming baselines.
Effect of K. Fig. 3d shows the performance of MCL as robustness parameter K is increased from 1
to M . We observe a monotonic reduction in error as K decreases, which suggests there is a natural
clustering of the data and thus learning a single SSVM is detrimental.
Qualitative Results. Fig. 1 shows example images, ground-truth segmentations, and the predictions
made by M = 6 predictors. We observe that the M hypotheses are both diverse and plausible. The
evolution of the best prediction with coordinate descent iterations can be seen in Fig. 2.
5.2
Protein Side-Chain Prediction
Model and Dataset. Given a protein backbone structure, the task here is to predict the amino
acid side-chain configurations. This problem has been traditionally formulated as a pairwise MRF
with node labels corresponding to (discretized) side-chain configurations (rotamers). These models
include pairwise interactions between nearby side-chains, and between side-chains and backbone.
We use the dataset of [7] which consists of 276 proteins (up to 700 residues long) split into train and
test sets of sizes 55 and 221 respectively.2 The energy function is defined as a weighted sum of eight
2
Dataset available from: http://cyanover.fhcrc.org/recomb-2007/
7
14
Test error 13
Train error
12
4
15
11
3.5
10
3
3
4
5
6
20
25
15
10
8
2.5
2
25
MCL
MCL (train)
20
15
9
10
1
Task?Loss vs K; M = 6
MCL
MCL (train)
C?SSVM
M?Best
DivM?Best
Pixel Error %
Objective
Pixel Error %
20
Task?Loss vs C; M = 6
M = 6; C = 0.8
4.5
Objective
25
Pixel Error %
4
x 10
MCL
MCL (train)
C?SSVM
M?Best
DivM?Best
Pixel Error %
Task?Loss vs M
7
M
1
2
3
4
5
Coordinate?Descent Iteration
(a) Error vs. M .
(b) Error vs. Iterations.
10
0.1
0.4
1.6
C
6.4
25.6
1
2
3
4
5
6
K
(c) Error vs. C.
(d) Error vs. K.
Figure 3: Experiments on foreground-background segmentation.
Task?Loss vs M
Task?Loss vs C; M = 4
M = 4; C = 1
30
0.85
MCL
CRF
HCRF
Boost
29.5
29
30
28
MCL
CRF
HCRF
Boost
Objective
29.5
Test error
29
0.8
27
27.5
27
26.5
Error %
28
Error %
Objective
Error %
27.5
28.5
28.5
28
27.5
27
1
2
3
4
M
(a) Error vs. M .
0.75
1
2
3
4
Coordinate?Descent Iteration
(b) Error vs. Iterations.
26.5
26.5
1
10
C
100
(c) Error vs. C.
Figure 4: Experiments on protein side-chain prediction.
known energy terms where the weights are to be learned. We used TRW-S [16] (early iterations)
and ILP (CPLEX [13]) for inference.
Baselines and Evaluation. For this application there is no natural analogue to the C-SSVM baseline
and thus we used a boosting-like baseline where we first train an SSVM on the entire training data;
use the training instances with high error to train a second SSVM, and so on. For comparison,
we also report results from the CRF and HCRF models proposed in [7]. Following [7], we report
average error rates for the first two angles (?1 and ?2 ) on all test proteins.
Results. Fig. 4 shows the results. Overall, we observe behavior similar to the previous set of experiments. Fig. 4a confirms that multiple predictors are beneficial, and that MCL is able to outperform
the boosting-like baseline. Fig. 4b shows the progress of the MCL objective and test loss with coordinate descent iterations; we again observe a positive correlation between the objective and the loss.
Fig. 4c shows that MCL outperforms baselines across a range of values of C.
6
Discussion and Conclusions
We presented an algorithm for producing a set of structured outputs and argued that in a number
of problems it is beneficial to generate a set of plausible and diverse hypotheses. Typically, this
is accomplished by learning a single-output model and then producing M -best hypotheses from it.
This causes a disparity between the way the model is trained (to produce a single output) and the
way it is used (to produce multiple outputs). Our proposed algorithm (MCL) provides a principled
way to directly optimize the multiple prediction min-set-loss.
There are a number of directions to extend this work. While we evaluated performance of all algorithms in terms of oracle set-loss, it would be interesting to measure the impact of MCL and other
baselines on user experience or final stage performance in cascaded algorithms.
Further, our model assumes a modular scoring function S(Y ) = WT ?(x, Y ) =
P
T m
m
m?[M ] wm ? (x, y ), i.e., the score of a set is the sum of the scores of its members. In a number
of situations, the score S(Y ) might be a submodular function. Such scoring functions often arise
when we want the model to explicitly reward diverse subsets. We plan to make connections with
greedy-algorithms for submodular maximization for such cases.
8
Acknowledgments: We thank David Sontag for his assistance with the protein data. AGR was
supported by the C2S2 Focus Center (under the SRC?s Focus Center Research Program).
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3,921 | 455 | Temporal Adaptation
?
In a
Silicon Auditory Nerve
John Lazzaro
CS Division
UC Berkeley
571 Evans Hall
Berkeley, CA 94720
Abstract
Many auditory theorists consider the temporal adaptation of the
auditory nerve a key aspect of speech coding in the auditory periphery. Experiments with models of auditory localization and pitch
perception also suggest temporal adaptation is an important element of practical auditory processing. I have designed, fabricated,
and successfully tested an analog integrated circuit that models
many aspects of auditory nerve response, including temporal adaptation.
1. INTRODUCTION
We are modeling known and proposed auditory structures in the brain using analog
VLSI circuits, with the goal of making contributions both to engineering practice and biological understanding. Computational neuroscience involves modeling
biology at many levels of abstraction. The first silicon auditory models were constructed at a fairly high level of abstraction (Lyon and Mead, 1988; Lazzaro and
Mead, 1989ab; Mead et al., 1991; Lyon, 1991). The functional limitations of these
silicon systems have prompted a new generation of auditory neural circuits designed
at a lower level of abstraction (Watts et al., 1991; Liu et -al., 1991).
813
814
Lazzaro
The silicon model of auditory nerve response models sensory transduction and spike
generation in the auditory periphery at a high level of abstraction (Lazzaro and
Mead, 1989c); this circuit is a component in silicon models of auditory localization,
pitch perception, and spectral shape enhancement (Lazzaro and Mead, 1989ab;
Lazzaro, 1991a). Among other limitations, this circuit does not model the shortterm temporal adaptation of the auditory nerve. Many auditory theorists consider
the temporal adaptation of the auditory nerve a key aspect of speech coding in the
auditory periphery (Delgutte and Kiang, 1984). From the engineering perspective,
the pitch perception and auditory localization chips perform well with sustained
sounds as input; temporal adaptation in the silicon auditory nerve should improve
performance for transient sounds.
I have designed, fabricated, and tested an integrated circuit that models the temporal adaptation of spiral ganglion neurons in the auditory periphery. The circuit
receives an analog voltage input, corresponding to the signal at an output tap of a
silicon cochlea, and produces fixed-width, fixed-height pulses that are correlates to
the action potentials of an auditory nerve fiber. I have also fabricated and tested
an integrated circuit that combines an array of these neurons with a silicon cochlea
(Lyon and Mead, 1988); this design is a silicon model of auditory nerve response.
Both circuits were fabricated using the Orbit double polysilicon n-well 2/.l1n process.
2. TEMPORAL ADAPTATION
Figure 1 shows data from the temporal adaptation circuit; the data in this figure was
taken by connecting signals directly to the inner hair cell circuit input, bypassing
silicon cochlea processing. In (a), we apply a 1 kHz pure tone burst of 20ms in
duration to the input of the hair cell circuit (top trace), and see an adapting sequence
of spikes as the output (middle trace). If this tone burst in repeated at 80ms
intervals, each response in unique; by averaging the responses to 64 consecutive tone
bursts (bottom trace), we see the envelope ofthe temporal adaptation superimposed
on the cycle-by-cycle phase-locking of the spike train. These behaviors qualitatively
match biological experiments (Kiang et al., 1965).
In biological auditory nerve fibers, cycle-by-cycle phase locking ceases for auditory
fibers tuned to sufficiently high frequencies, but the temporal adaptation property
remains. In the silicon spiral ganglion neuron, a 10kHz pure tone burst fails to elicit
phae;e-Iocking (Figure 1(b), trace identities ae; in (a)). Temporal adaptation remains,
however, qualitatively matching biological experiments (Kiang et aI., 1965).
To compare this data with the previous generation of silicon auditory nerve circuits,
we set the control parameters of the new spiral ganglion model to eliminate temporal
adaptation. Figure 1(c) shows the 1 kHz tone burst response (trace identities as
in (a)). Phase locking occurs without temporal adaptation. The uneven response
of the averaged spike outputs is due to beat frequencies between the input tone
frequency and the output spike rate; in practice, the circuit noise of the silicon
cochleae; adds random variation to the auditory input and smooths this response
(Lazzaro and Mead, 1989c).
Temporal Adaptation in a Silicon Auditory Nerve
U I ,I
___UJ
j
I
1.
_,. -.......-.I.LW-..-J~_. . _
(a)
(b)
(c)
Figure 1. Responses of test chip to pure tone bursts. Horizontal axis is time
for all plots, all horizontal rules measure 5 ms. (a) Chip response to a 1 kHz,
20 ms tone burst. Top trace shows tone burst input, middle trace shows a sample
response from the chip, bottom trace shows averaged output of 64 responses to tone
bursts. Averaged response shows both temporal adaptation and phage locking. (b)
Chip response to a 10 kHz, 20 ms tone burst. Trace identifications identical to (a).
Response shows temporal adaptation without phase locking. (c) Chip response to
a 1 kHz, 20 ms tone burst, with adaptation circuitry disabled. Trace identifications
identical to (a). Response shows phase locking without temporal adaptation.
815
816
Lazzaro
3. CIRCUIT DESIGN
Figure 2 shows a block diagram of the model. The circuits modeling inner hair cell
transduction remain unchanged from the original model (Lazzaro and Mead, 1989c),
and are shown as a single box. This box performs time differentiation, nonlinear
compression and half-wave rectification on the input waveform Vi, producing a
unidirectional current waveform as output. The dependent current source represents
this processed signal.
The axon hillock circuit (Mead, 1989), drawn as a box marked with a pulse, converts this current signal into a series of fixed-width, fixed height spikes; Vo is the
output of the model. The current signal is connected to the pulse generator using a
novel current mirror circuit, that serves as the control element to regulate temporal
adaptation. This current mirror circuit has an additional high impedance input,
Va, that exponentially scales the current entering the axon hillock circuit (the current mirror operates in the subthreshold region) . The adaptation capacitor C a is
associated with the control voltage Va.
IHC
Figure 2. Circuit schematic of the enhanced silicon model of auditory nerve response. The circuit converts the analog voltage input Vi into the pulse train Vo;
control voltages VI and Vp control the temporal adaptation of state variable Va on
capacitor Ca. See text for details.
Temporal Adaptation in a Silicon Auditory Nerve
C a is constantly charged by the PFET transistor associated with control voltage
Vi, and is discharged during every pulse output of the axon hillock circuit, by an
amount set by the control voltage Vp. During periods with no input signal, Va is
charged to Vdd, and the current mirror is set to deliver maximum current with the
onset of an input signal. If an input signal occurs and neuron activity begins, the
capacitor Va is discharged with every spike, degrading the output of the current
mirror. In this way, temporal adaptation occurs, with characteristics determined
by Vp and Vi.
The nonlinear differential equations for this adaptation circuit are similar to the
equations governing the adaptive baroreceptor circuit (Lazzaro et al., 1991); the
publication describing this circuit includes an analysis deriving a recurrence relation
that describes the pulse output of the circuit given a step input.
26
40
Sp/sec
600
36
400
200
0.0
14 _ _ _ _ _ _ _ _-
20
10
30
(ms)
Figure 3. Instantaneous firing rate of the adaptive neuron, as a function of time;
tone burst begins at 0 ms. Each curve is marked with the amplitude of presented
tone burst, in dB. Tone burst frequency is 1Khz.
817
818
Lazzaro
4. DATA ANALYSIS
The experiment shown in Figure l(a) was repeated for tone bursts of different
amplitudes; this data set was used to produce several standard measures of adaptive
response (Hewitt and Meddis, 1991). The integrated auditory nerve circuit was used
for this set of experiments. Data was taken from an adaptive auditory nerve output
that had a best frequency of 1 Khz.; the frequency of all tone bursts was also 1 Khz.
Figure 3 shows the instantaneous firing rate of the auditory nerve output as a
function of time, for tone bursts of different amplitudes. Adaptation was more
pronounced for more intense sounds. This difference is also seen in Figure 4. In
this figure, instantaneous firing rate is plotted as a function of amplitude, both at
response onset and after full adaptation.
700
Spikes/sec
500
300
100
10
30
20
40
dB
Figure 4. Instantaneous firing rate of the adaptive neuron, as a function of amplitude (in dB). Top curve is firing rate at onset of response, bottom curve is firing
rate after adaptation. Tone burst frequency is 1Khz.
Temporal Adaptation in a Silicon Auditory Nerve
Figure 4 shows that the instantaneous spike rate saturates at moderate intensity
after full adaptation; at these moderate intensities, however, the onset instantaneous
spike rate continues to encode intensity. Figure 4 shows a non-monotonicity at high
intensities in the onset response; this undesired non-monotonicity is a result of the
undesired saturation of the silicon cochlea circuit (Lazzaro, 1991b).
5. CONCLUSION
This circuit improves the silicon model of auditory response, by adding temporal
adaptation. We expect this improvement to enhance existing architectures for auditory localization and pitch perception, and aid the creation of new circuits for
speech processing.
A cknow ledgeluen t s
Thanks to K. Johnson of CU Boulder and J. Wawrzynek of UC Berkeley for hosting this research in their laboratories. I also thank the Caltech auditory research
community, specifically C. l\-1ead, D. Lyon, M. Konishi, L. Watts, M. Godfrey, and
X. Arreguit. This work was funded by the National Science Foundation.
References
Delgutte, B., and Kiang, Y. S. (1984). Speech coding in the auditory nerve I-V. 1.
Acoust. Soc. Am 75:3,866-918.
Hewitt, M. J. and TvIeddis, R. (1991). An evaluation of eight computer models of
mammalian inner hair-cell function. J. Acoust. Soc. Am 90:2, 904.
Kiang, N. Y.-s, 'Watenabe, T., Thomas, E.C., and Clark, L.F. (1965). Discharge
Patterns of Single Fibers in the Cat's Auditory Nerve. Cambridge, MA: M.LT
Press.
Lazzaro, J. and Mead, C. (1989a). A silicon model of auditory localization. Neural
Computation 1: 41-70.
Lazzaro, J. and Mead, C. (1989b). Silicon modeling of pitch perception. Proceedings
National Academy of Sciences 86: 9597-960l.
Lazzaro, J. and Mead, C. (1989c). Circuit models of sensory transduction in the
cochlea. In Mead, C. and Ismail, M. (eds), Analog VLSI Implementations of Neural
Net'works. Nonvell, MA: Klmver Academic Publishers, pp. 85-1Ol.
Lazzaro, J. P. (1991a). A silicon model of an auditory neural representation of
spectral shape. IEEE Jour1lal Solid State Circuits 26: 772-777 .
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VLSL IEEE Asilomar COllference on Signals, System,s, and Computers .
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vanced Research. in VLSI, Proceedings of the 1991 Santa Cruz Conference, Cam-
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Liu, VV., Andreou, A., and Goldstein, M. (1991). Analog VLSI implementation of
an auditory periphery model. 25 Annual Conference on Information Sciences and
Systems, Baltimore, MD, 1991.
Lyon, R. and Mead, C. (1988). An analog electronic cochlea. IEEE Trans. Acoust.,
Speech, Signal Processing 36: 1119-1134.
Lyon, R. (1991). CCD correlators for auditory models. IEEE Asilomar Conference
on Signals, Systems, and Computers.
Mead, C. A., Arreguit, X., Lazzaro, J. P. (1991) Analog VLSI models of binaural
hearing. IEEE Journal of Neural Networks, 2: 230-236.
Mead, C. A. (1989). Analog VLSI and Neural Systems. Reading, MA: AddisonWesley.
Watts, L., Lyon, R., and Mead, C. (1991). A bidirectional analog VLSI cochlear
model. In Sequin, C. (ed), Advanced Research in VLSI, Proceedings of tlte 1991
Santa Cruz Conference, Cambridge, MA: MIT Press, pp. 153-163.
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3,922 | 4,550 | Finite Sample Convergence Rates of Zero-Order
Stochastic Optimization Methods
John C. Duchi1
Michael I. Jordan1,2
Martin J. Wainwright1,2
Andre Wibisono1
1
2
Department of Electrical Engineering and Computer Science and Department of Statistics
University of California, Berkeley
Berkeley, CA USA 94720
{jduchi,jordan,wainwrig,wibisono}@eecs.berkeley.edu
Abstract
We consider derivative-free algorithms for stochastic optimization problems that
use only noisy function values rather than gradients, analyzing their finite-sample
convergence rates. We show that if pairs of function values are available, algorithms that ?
use gradient estimates based on random perturbations suffer a factor
of at most d in convergence rate over traditional stochastic gradient methods,
where d is the problem dimension. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of
such problems, which show that our bounds are sharp with respect to all problemdependent quantities: they cannot be improved by more than constant factors.
1
Introduction
Derivative-free optimization schemes have a long history in optimization (see, for example, the
book by Spall [21]), and they have the clearly desirable property of never requiring explicit gradient
calculations. Classical techniques in stochastic and non-stochastic optimization, including KieferWolfowitz-type procedures [e.g. 17], use function difference information to approximate gradients
of the function to be minimized rather than calculating gradients. Researchers in machine learning
and statistics have studied online convex optimization problems in the bandit setting, where a player
and adversary compete, with the player choosing points ? in some domain ? and an adversary
choosing a point x, forcing the player to suffer a loss F (?; x), where F (?; x) : ? ? R is a convex
function [13, 5, 1]. The goal is to choose optimal ? based only on observations of function values
F (?; x). Applications including online auctions and advertisement selection in search engine results.
Additionally, the field of simulation-based optimization provides many examples of problems in
which optimization is performed based only on function values [21, 10], and problems in which
the objective is defined variationally (as the maximum of a family of functions), such as certain
graphical model and structured-prediction problems, are also natural because explicit differentiation
may be difficult [23].
Despite the long history and recent renewed interest in such procedures, an understanding of their
finite-sample convergence rates remains elusive. In this paper, we study algorithms for solving
stochastic convex optimization problems of the form
!
F (?; x)dP (x),
(1)
min f (?) := EP [F (?; X)] =
???
X
d
where ? ? R is a compact convex set, P is a distribution over the space X , and for P -almost every
x ? X , the function F (?; x) is closed convex. Our focus is on the convergence rates of algorithms
that observe only stochastic realizations of the function values f (?).
Work on this problem includes Nemirovski and Yudin [18, Chapter 9.3], who develop a randomized
sampling strategy that estimates ?F (?; x) using samples from the surface of the "2 -sphere, and
1
Flaxman et al. [13], who build on this approach, applying it to bandit convex optimization problems.
The convergence rates in these works are (retrospectively) sub-optimal [20, 2]: Agarwal et al.
?[2]
provide algorithms that achieve convergence rates (ignoring logarithmic factors) of O(poly(d)/ k),
where poly(d) is a polynomial in the dimension d, for stochastic algorithms receiving only single
function values, but (as the authors themselves note) the algorithms are quite complicated.
Some of the difficulties inherent in optimization using only a single function evaluation can be alleviated when the function F (?; x) can be evaluated at two points, as noted independently by Agarwal
et al. [1] and Nesterov [20]. The insight is that for small u, the quantity (F (? + uZ; x) ? F (?; x))/u
approximates a directional derivative of F (?; x) and can thus be used in first-order optimization
schemes. Such two-sample-based gradient estimators allow simpler analyses, with sharper convergence rates [1, 20], than algorithms that have access to only a single function evaluation in each
iteration. In the current paper, we take this line of work further, finding the optimal rate of convergence for procedures that are only able to obtain function evaluations, F (?; X), for samples X.
Moreover, adopting the two-point perspective, we present simple randomization-based algorithms
that achieve these optimal rates.
More formally, we study algorithms that receive paired observations Y (?, ? ) ? R2 , where ? and ?
are points the algorithm selects, and the tth sample is
"
#
F (?t ; X t )
t t t
Y (? , ? ) :=
(2)
F (? t ; X t )
where X t is a sample drawn from the distribution P . After k iterations, the algorithm returns a
$
vector ?(k)
? ?. In this setting, we analyze stochastic gradient and mirror-descent procedures [27,
18, 6, 19] that construct gradient estimators using the two-point observations Y t . By a careful
analysis of the dimension dependence of certain random perturbation schemes, we?show that the
convergence rate attained by our stochastic gradient methods is roughly a factor of d worse than
that attained by stochastic methods that observe the
? full gradient ?F (?; X). Under appropriate
conditions, our convergence rates are a factor of d better than those attained by Agarwal et al.
[1] and Nesterov [20]. In addition, though we present our results in the framework of stochastic
optimization, our analysis applies to (two-point) bandit online convex optimization problems [13,
5, 1], and we consequently obtain the sharpest rates for such problems. Finally, we show that
the convergence rates we provide are tight?meaning sharp to within constant factors?by using
information-theoretic techniques for constructing lower bounds on statistical estimators.
2
Algorithms
Stochastic mirror descent methods are a class of stochastic gradient methods for solving the problem
min??? f (?). They are based on a proximal function ?, which is a differentiable convex function
defined over ? that is assumed (w.l.o.g. by scaling) to be 1-strongly convex with respect to the norm
'?' over ?. The proximal function defines a Bregman divergence D? : ? ? ? ? R+ via
D? (?, ? ) := ?(?) ? ?(? ) ? )??(? ), ? ? ? * ?
1
2
'? ? ? ' ,
2
(3)
where the inequality follows from the strong convexity of ? over ?. The mirror descent (MD)
method proceeds in a sequence of iterations that we index by t, updating the parameter vector ?t ?
? using stochastic gradient information to form ?t+1 . At iteration t the MD method receives a
(subgradient) vector g t ? Rd , which it uses to update ?t via
%
(
& t '
1
t+1
t
?
= argmin g , ? +
D? (?, ? ) ,
(4)
?(t)
???
where {?(t)} is a non-increasing sequence of positive stepsizes.
We make two standard assumptions throughout the paper. Let ?? denote a minimizer of the problem (1). The first assumption [18, 6, 19] describes the properties of ? and the domain.
Assumption A. The proximal function ? is strongly convex with respect to the norm '?'. The
domain ? is compact, and there exists R < ? such that D? (?? , ?) ? 12 R2 for ? ? ?.
2
Our second assumption is standard for almost all first-order stochastic gradient methods [19, 24, 20],
and it holds whenever the functions F (?; x) are G-Lipschitz with respect to the norm '?'. We use
'?'? to denote the dual norm to '?', and let g : ? ? X ? Rd denote a measurable subgradient
selection for the functions F ; that is, g(?; x) ? ?F (?; x) with E[g(?; X)] ? ?f (?).
Assumption B. There is a constant G < ? such that the (sub)gradient selection g satisfies
2
E['g(?; X)'? ] ? G2 for ? ? ?.
When Assumptions A and B hold, the convergence rate of stochastic mirror descent methods is
well understood [6, 19, Section 2.3]. Indeed, let the variables X t ? X be sampled i.i.d. according
t
to P , set g?
= g(?t ; X t ), and let ?t be generated by the mirror descent iteration (4) with stepsize
?(t) = ?/ t. Then one obtains
$
E[f (?(k))]
? f (?? ) ?
?
1
? R2 + ? G2 .
2? k
k
(5)
For the remainder of this section, we explore the use of function difference information to obtain
subgradient estimates that can be used in mirror descent methods to achieve statements similar to
the convergence guarantee (5).
2.1
Two-point gradient estimates and general convergence rates
In this section, we show?under a reasonable additional assumption?how to use two samples of
the random function values F (?; X) to construct nearly unbiased estimators of the gradient ?f (?)
of the expected function f . Our analytic techniques are somewhat different than methods employed
in past work [1, 20]; as a consequence, we are able to achieve optimal dimension dependence.
Our method is based on an estimator of ?f (?). Our algorithm uses a non-increasing sequence
of positive smoothing parameters {ut } and a distribution ? on Rd (which we specify) satisfying
E? [ZZ # ] = I. Upon receiving the point X t ? X , we sample an independent vector Z t and set
gt =
F (?t + ut Z t ; X t ) ? F (?t ; X t ) t
Z .
ut
(6)
We then apply the mirror descent update (4) to the quantity g t .
The intuition for the estimator (6) of ?f (?) follows from an understanding of the directional derivatives of the random function realizations F (?; X). The directional derivative f $ (?, z) of the function
(?)
f at the point ? in the direction z is f $ (?, z) := limu?0 f (?+uz)?f
. The limit always exists when
u
f is convex [15, Chapter VI], and if f is differentiable at ?, then f $ (?, z) = )?f (?), z*. In addition,
we have the following key insight (see also Nesterov [20, Eq. (32)]): whenever ?f (?) exists,
E[f $ (?, Z)Z] = E[)?f (?), Z* Z] = E[ZZ # ?f (?)] = ?f (?)
if the random vector Z ? Rd has E[ZZ # ] = I. Intuitively, for ut small enough in the construction (6), the vector g t should be a nearly unbiased estimator of the gradient ?f (?).
To formalize our intuition, we make the following assumption.
Assumption C. There is a function L : X ? R+ such that for (P -almost every) x ? X , the function F (?; x) has L(x)-Lipschitz continuous gradient with respect to the norm '?', and the quantity
L(P )2 := E[L(X)2 ] < ?.
With Assumption C, we can show that g t is (nearly) an unbiased estimator of ?f (?t ). Furthermore,
for appropriate random vectors Z, we can also show that g t has small norm, which yields better
convergence rates for mirror descent-type methods. (See the proof of Theorem 1.) In order to study
the convergence of mirror descent methods using the estimator (6), we make the following additional
assumption on the distribution ?.
Assumption D. Let Z be sampled according to the distribution ?, where E[ZZ # ] = I. The quantity
4
2
M (?)2 := E['Z' 'Z'? ] < ?, and there is a constant s(d) such that for any vector g ? Rd ,
2
2
E[')g, Z* Z'? ] ? s(d) 'g'? .
3
As the next theorem shows, Assumption D is somewhat innocuous, the constant M (?) not even
appearing in the final bound. The dimension (and norm) dependent term s(d), however, is important for our results. In Section 2.2 we give explicit constructions of random variables that satisfy
Assumption D. For now, we present the following result.
Theorem 1. Let {ut } ? R+ be a non-increasing sequence of positive numbers, and let ?t be
generated according to the mirror descent update (4) using the gradient estimator (6). Under Assumptions A, B, C, and D, if we set the step and perturbation sizes
)
G s(d)
R
1
?(t) = ? )
? ,
? and ut = u
L(P )M (?) t
2G s(d) t
then
)
)
)
+
,
RG s(d)
RG s(d) log k
?
?1
2 RG s(d)
$
?
+u
,
E f (?(k)) ? f (? ) ? 2
max ?, ?
+ ?u
k
k
k
$ = 1 .k ?t , and the expectation is taken with respect to the samples X and Z.
where ?(k)
t=1
k
*
The proof of Theorem 1 requires some technical care?we never truly receive unbiased gradients?
and it builds on convergence proofs developed in the analysis of online and stochastic convex optimization [27, 19, 1, 12, 20] to achieve bounds of the form (5). Though we defer proof to Appendix A.1, at a very high level, the argument is as follows. By using Assumption C, we see that
for small enough ut , the gradient estimator g t from (6) is close (in expectation with respect to X t )
to f $ (?t , Z t )Z t , which is an unbiased estimate of ?f (?t ). Assumption C allows us to bound the
moments of the gradient estimator g t . By carefully showing that taking care to make sure that the
errors in g t as an estimator of ?f (?t ) scale with ut , we given an analysis similar to that used to
derive the bound (5) to obtain Theorem 1.
Before continuing, we make a few remarks. First, the method is reasonably robust to the selection
of the step-size multiplier
? ? (as noted by Nemirovski et al. [19] for gradient-based MD methods).
So long as ?(t) ? 1/ t, mis-specifying the multiplier ? results in a scaling at worst linear in
max{?, ??1 }. Perhaps more interestingly, our setting of ut was chosen mostly for convenience
and elegance of the final bound. In a sense, we can simply take u to be extremely close to zero (in
practice, we must avoid numerical precision issues, and the stochasticity in the method makes such
choices somewhat unnecessary). In addition, the convergence rate of the method is independent
of the Lipschitz continuity constant L(P ) of the instantaneous gradients ?F (?; X); the penalty for
nearly non-smooth objective functions comes into the bound only as a second-order term. This
suggests similar results should hold for non-differentiable functions; we have been able to show that
in some cases this is true, but a fully general result has proved elusive thus far. We are currently
investigating strategies for the non-differentiable case.
Using similar arguments based on Azuma-Hoeffding-type inequalities, it is possible to give highprobability convergence guarantees [cf. 9, 19] under additional tail conditions on g, for example,
2
that E[exp('g(?; X)'? /G2 )] ? exp(1). Additionally, though we have presented our results as
convergence guarantees for stochastic optimization problems, an inspection of our analysis in Appendix A.1 shows that we obtain (expected) regret bounds for bandit online convex optimization
problems [e.g. 13, 5, 1].
2.2
Examples and corollaries
In this section, we provide examples of random sampling strategies that give direct convergence
rate estimates for the mirror descent algorithm with subgradient samples (6). For each corollary,
we specify the norm '?', proximal function ?, and distribution ?, verify that Assumptions A, B, C,
and D hold, and then apply Theorem 1 to obtain a convergence rate.
We begin with a corollary that describes the convergence rate of our algorithm when the expected
function f is Lipschitz continuous with respect to the Euclidean norm '?'2 .
2
2
Corollary 1. Given the proximal function ?(?) := 21 '?'2 , suppose that E['g(?; X)'2 ] ? G2 and
?
that ? is uniform on the surface of the "2 -ball of radius d. With the step size choices in Theorem 1,
4
we have
?
?
?
+
RG d
RG d log k
?1
2 RG d
?
$
?
max{?, ? } + ?u
E f (?(k)) ? f (? ) ? 2
+u
.
k
k
k
*
?
6
Proof Note that 'Z'2 = d, which implies M (?)2 = E['Z'2 ] = d3 . Furthermore, it is easy to
see that E[ZZ # ] = I. Thus, for g ? Rd we have
2
2
2
E[')g, Z* Z'2 ] = dE[)g, Z* ] = dE[g # ZZ # g] = d 'g'2 ,
which gives us s(d) = d.
The rate provided by Corollary 1 is the fastest derived to date for zero-order stochastic optimization using two function evaluations.
Both Agarwal et al. [1] and Nesterov [20] achieve rates of
?
convergence of order RGd/ k. Admittedly, neither requires that the random functions F (?; X) be
continuously differentiable. Nonetheless, Assumption C does not require a uniform bound on the
Lipschitz constant L(X) of the gradients ?F (?; X); moreover, the convergence rate of the method
is essentially independent of L(P ).
In high-dimensional scenarios, appropriate choices for the proximal function ? yield better scaling
on the norm of the gradients [18, 14, 19, 12]. In online learning and stochastic optimization settings
where one observes gradients g(?; X), if the domain ? is.
the simplex, then exponentiated gradient
algorithms [16, 6] using the proximal function ?(?) =
j ?j log ?j obtain rates of convergence
dependent on the "? -norm of the gradients 'g(?; X)'? . This scaling is more palatable?than dependence on Euclidean norms applied to the gradient vectors, which may be a factor of d larger.
Similar results apply [7, 6] when using proximal functions based on "p -norms. Indeed, making the
2
1
'?'p , we obtain the following corollary.
choice p = 1 + 1/ log d and ?(?) = 2(p?1)
2
2
d
Corollary 2. Assume that E['g(?; X)'? ] ? G
? and that ? ? {? ? R : '?'1 ? R}. Set ? to be
uniform on the surface of the "2 -ball of radius d. Use the step sizes specified in Theorem 1. There
are universal constants C1 < 20e and C2 < 10e such that
?
?
*
+
,
0
RG d log d
RG d log d / 2
?
?1
$
?
E f (?(k)) ? f (? ) ? C1
?u + u log k .
max ?, ?
+ C2
k
k
Proof The proof of this corollary is somewhat involved. The main argument involves showing
that the constants M (?) and s(d) may be taken as M (?) ? d6 and s(d) ? 24d log d.
First, we recall [18, 7, Appendix 1] that our choice of ? is strongly convex with respect to the norm
'?'p . In addition, if we define q = 1 + log d, then we have 1/p + 1/q = 1, and 'v'q ? e 'v'? for
any v ? Rd and any d. As a consequence, we see that we may take the norm '?' = '?'1 and the dual
2
2
norm '?'? = '?'? , and E[')g, Z* Z'q ] ? e2 E[')g, Z* Z'? ]. To apply Theorem 1 with appropriate
2
values from Assumption D, we now bound E[')g, Z* Z'? ]; see Appendix A.3 for a proof.
?
Lemma 3. Let Z be distributed uniformly on the "2 -sphere of radius d. For any g ? Rd ,
2
2
E[')g, Z* Z'? ] ? C ? d log d 'g'? ,
where C ? 24 is a universal constant.
As a consequence of Lemma 3, the constant s(d) of Assumption D satisfies s(d) ? Cd log d.
4
2
Finally, we have the essentially trivial bound M (?)2 = E['Z'1 'Z'? ] ? d6 (we only need the
quantity M (?) to be finite to apply Theorem 1). Recalling that the set ? ? {? ? Rd : '?'1 ? R},
our choice of ? yields [e.g., 14, Lemma 3]
(p ? 1)D? (?, ? ) ?
1
1
2
2
'?'p + '? 'p + '?'p '? 'p .
2
2
We thus find that D? (?, ? ) ? 2R2 log d for any ?, ? ? ?, and using the step and perturbation size
choices of Theorem 1 gives the result.
5
?
Corollary 2 attains a convergence rate that scales with dimension as d log d. This dependence
on dimension is much worse than that of (stochastic) mirror descent using full gradient information [18, 19]. The additional dependence on d suggests that while O(1/'2 ) iterations are required to
achieve '-optimization accuracy for mirror descent methods (ignoring logarithmic factors), the twopoint method requires O(d/'2 ) iterations to obtain the same accuracy. A similar statement holds for
the results of Corollary 1. In the next section, we show that this dependence is sharp: except for logarithmic factors, no algorithm can attain better convergence rates, including the problem-dependent
constants R and G.
3
Lower bounds on zero-order optimization
We turn to providing lower bounds on the rate of convergence for any method that receives random
function values. For our lower bounds, we fix a norm '?' on Rd and as usual let '?'? denote its
dual norm. We assume that ? = {? ? Rd : '?' ? R} is the norm ball of radius R. We study
all optimization methods that receive function values of random convex functions, building on the
analysis of stochastic gradient methods by Agarwal et al. [3].
More formally, let Ak denote the collection of all methods that observe a sequence of data points
$ ? ?. The classes
(Y 1 , . . . , Y k ) ? R2 with Y t = [F (?t , X t ) F (? t , X t )] and return an estimate ?(k)
of functions over which we prove our lower bounds consist of those satisfying Assumption B, that is,
for a given Lipschitz constant G > 0, optimization problems over the set FG . The set FG consists of
pairs (F, P ) as described in the objective (1), and for (F, P ) ? FG we assume there is a measurable
2
subgradient selection g(?; X) ? ?F (?; X) satisfying EP ['g(?; X)'? ] ? G2 for ? ? ?.
Given an algorithm A ? Ak and a pair (F, P ) ? FG , we define the optimality gap
1
2
$
$
'k (A, F, P, ?) := f (?(k))
? inf f (?) = EP F (?(k);
X) ? inf EP [F (?; X)] ,
???
???
(7)
$
where ?(k)
is the output of A on the sequence of observed function values. The quantity (7) is a
random variable, since the Y t are random and A may use additional randomness. We we are thus
interested in its expected value, and we define the minimax error
'?k (FG , ?) := inf
sup
A?Ak (F,P )?FG
E['k (A, F, P, ?)],
(8)
where the expectation is over the observations (Y 1 , . . . , Y k ) and randomness in A.
3.1
Lower bounds and optimality
In this section, we give lower bounds on the minimax rate of optimization for a few specific settings.
We present our main results, then recall Corollaries 1 and 2, which imply we have attained the minimax rates of convergence for zero-order (stochastic) optimization schemes. The following sections
contain proof sketches; we defer technical arguments to appendices.
We begin by providing minimax lower bounds when the expected function f (?) = E[F (?; X)] is
Lipschitz continuous with respect to the "2 -norm. We have the following proposition.
,
Proposition 1. Let ? = ? ? Rd : '?'2 ? R and FG consist of pairs (F, P ) for which the sub2
gradient mapping g satisfies EP ['g(?; X)'2 ] ? G2 for ? ? ?. There exists a universal constant
c > 0 such that for k ? d,
?
GR d
?
'k (FG , ?) ? c ? .
k
Combining the lower bounds provided by Proposition 1 with our algorithmic scheme in Section 2
shows that our analysis is ?essentially
sharp, since Corollary 1 provides an upper bound for the
?
minimax optimality of RG d/ k. The stochastic gradient descent algorithm (4) coupled with the
sampling strategy (6) is thus optimal for stochastic problems with two-point feedback.
Now we investigate the minimax rates at which it is possible to solve stochastic convex optimization
problems whose objectives are Lipschitz continuous with respect to the "1 -norm. As noted earlier,
such scenarios are suitable for high-dimensional problems [e.g. 19].
6
Proposition 2. Let ? = {? ? Rd : '?'1 ? R} and FG consist of pairs (F, P ) for which the
2
subgradient mapping g satisfies EP ['g(?; X)'? ] ? G2 for ? ? ?. There exists a universal constant
c > 0 such that for k ? d,
?
GR d
?
'k (FG , ?) ? c ? .
k
We may again consider the optimality of our mirror descent algorithms, recalling Corollary 2. In this
2
1
case, the MD algorithm (4) with the choice ?(?) = 2(p?1)
'?'p , where p = 1 + 1/ log d, implies
that there exist universal constants c and C such that
?
?
GR d
GR d log d
?
c ?
? '?k (FG , ?) ? C
k
k
for the problem class described in Proposition 2. Here the upper bound is again attained by our
two-point mirror-descent procedure. Thus, to within logarithmic factors, our mirror-descent based
algorithm is optimal for these zero-order optimization problems.
When full gradient
information is available, that is, one has access to the subgradient selection
?
g(?; X),
the
d
factors
appearing in the lower bounds in Proposition 1 and 2 are not present [3].
?
The d factors similarly disappear from the convergence rates in Corollaries 1 and 2 when one
uses g t = g(?; X) in the mirror descent updates (4); said differently, the constant s(d) = 1 in
Theorem 1 [19, 6]. As noted in Section 2, our lower bounds consequently show that in addition to
dependence on the radius R and second moment
G2 in the case when gradients are available [3],
?
all algorithms must suffer an additional O( d) penalty in convergence rate. This suggests that for
high-dimensional problems, it is preferable to use full gradient information if possible, even when
the cost of obtaining the gradients is somewhat high.
3.2
Proofs of lower bounds
We sketch proofs for our lower bounds on the minimax error (8), which are based on the framework
introduced by Agarwal et al. [3]. The strategy is to reduce the optimization problem to a testing
problem: we choose a finite set of (well-separated) functions, show that optimizing well implies that
one can identify the function being optimized, and then, as in statistical minimax theory [26, 25],
apply information-theoretic lower bounds on the probability of error in hypothesis testing problems.
We begin with a finite set V ? Rd , to be chosen depending on the characteristics of the function
class FG , and a collection of functions and distributions G = {(Fv , Pv ) : v ? V} ? FG indexed by
V. Define fv (?) = EPv [Fv (?; X)], and let ?v? ? argmin??? fv (?). We also let ? > 0 be an accuracy
parameter upon which Pv and the following quantities implicitly depend. Following Agarwal et al.
[3], we define the separation between two functions as
1
2
?
?(fv , fw ) := inf fv (?) + fw (?) ? fv (?v? ) ? fw (?w
),
???
and the minimal separation of the set V (this may depend on the accuracy parameter ?) as
?? (V) := min{?(fv , fw ) : v, w ? V, v 0= w}.
For any algorithm A ? Ak , there exists a hypothesis test v$ : (Y 1 , . . . , Y k ) ? V such that for V
sampled uniformly from V (see [3, Lemma 2]),
P($
v (Y 1 , . . . , Y k ) 0= V ) ?
2
2
E['k (A, FV , PV , ?)] ? ?
max E['k (A, Fv , Pv , ?)], (9)
?? (V)
? (V) v?V
where the expectation is taken over the observations (Y 1 , . . . , Y k ). By Fano?s inequality [11],
P($
v 0= V ) ? 1 ?
I(Y 1 , . . . , Y k ; V ) + log 2
.
log |V|
(10)
We thus must upper bound the mutual information I(Y 1 , . . . , Y k ; V ), which leads us to the following. (See Appendix B.3 for the somewhat technical proof of the lemma.)
7
Lemma 4. Let X | V = v be distributed as N (?v, ? 2 I), and let F (?; x) = )?, x*. Let V be a
uniform random variable on V ? Rd , and assume that Cov(V ) 1 ?I for some ? ? 0. Then
?k? 2
I(Y 1 , Y 2 , . . . , Y k ; V ) ?
.
?2
Using Lemma 4, we can obtain a lower bound on the minimax optimization error whenever the
instantaneous objective functions are of the form F (?; x) = )?, x*. Combining inequalities (9),
(10), and Lemma 4, we find that if we choose the accuracy parameter
3
41/2
log |V|
?
? log 2
,
(11)
?=?
2
k?
(we assume that |V| > 4) we find that there exist a pair (F, P ) ? FG such that
E['k (A, F, P, ?)] ? ?? (V)/4.
(12)
The inequality (12) can give concrete lower bounds on the minimax optimization error. In our lower
bounds, we use Fv (?; x) = )?, x* and set Pv to be the N (?v, ? 2 I) distribution, which allows us to
apply Lemma 4. Proving Propositions 1 and 2 thus requires three steps:
1. Choose the set V with the property that Cov(V ) 1 ?I when V ? Uniform(V).
2. Choose the variance parameter ? 2 such that for each v ? V, the pair (Fv , Pv ) ? FG .
3. Calculate the separation value ?? (V) as a function of the accuracy parameter ?.
2
Enforcing (Fv , Pv ) ? FG amounts to choosing ? 2 so that E['X'? ] ? G2 for X ? N (?v, ? 2 I).
By construction fv (?) = ? )?, v*, which allows us to give lower bounds on the minimal separation
?? (V) for the choices of the norm constraint '?' ? R in Propositions 1 and 2. We defer formal
proofs to Appendices B.1 and B.2, providing sketches here.
For Proposition 1, an argument using the probabilistic method implies that there are universal
constants c1 , c2 > 0 for which there is a 12 packing V of the "2 -sphere of radius 1 with size
.
at least |V| ? exp(c1 d) and such that (1/|V|) v?V vv # 1 c2 Id?d /d. By the linearity of fv ,
we find ?(fv , fw ) ? ?R/16, and setting ? 2 = G2 /(2d) and ? as in the choice (11) implies that
2
E['X'2 ] ? G2 . Substituting ? and ?? (V) into the bound (12) proves Proposition 1.
For Proposition 2, we use the packing set V = {?ei : i = 1, . . . , d}. Standard bounds [8] on
2
the normal distribution imply that for Z ? N (0, I), we have E['Z'? ] = O(log d). Thus we find
2
that for ? 2 = O(G2 / log(d)) and suitably small ?, we have E['X'? ] = O(G2 ); linearity yields
?(fv , fw ) ? ?R for v 0= w ? V. Setting ? as in the expression (11) yields Proposition 2.
4
Discussion
We have analyzed algorithms for stochastic optimization problems that use only random function
values?as opposed to gradient computations?to minimize an objective function. As our development of minimax lower bounds shows, the algorithms we present, which build on those proposed by
Agarwal et al. [1] and Nesterov [20], are optimal: their convergence rates cannot be improved (in a
minimax sense) by more than numerical constant factors. As a consequence of our results, we have
attained sharp rates for bandit online convex optimization problems with multi-point feedback. We
have also shown that there is a necessary sharp transition in convergence rates between stochastic
gradient algorithms and algorithms that compute only function values. This result highlights the
advantages of using gradient information when it is available, but we recall that there are many
applications in which gradients are not available.
Finally, one question that this work leaves open, and which we are actively attempting to address,
is whether our convergence rates extend to non-smooth optimization problems. We conjecture that
they do, though it will be interesting to understand the differences between smooth and non-smooth
problems when only zeroth-order feedback is available.
Acknowledgments This material supported in part by ONR MURI grant N00014-11-1-0688 and
the U.S. Army Research Laboratory and the U.S. Army Research Office under grant no. W911NF11-1-0391. JCD was also supported by an NDSEG fellowship and a Facebook PhD fellowship.
8
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9
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3,923 | 4,551 | Inverse Reinforcement Learning
through Structured Classification
Edouard Klein1,2
LORIA ? team ABC
Nancy, France
[email protected]
2
1
Matthieu Geist2
Sup?lec ? IMS-MaLIS Research Group
Metz, France
[email protected]
Bilal Piot2,3 , Olivier Pietquin2,3
UMI 2958 (GeorgiaTech-CNRS)
Metz, France
{bilal.piot,olivier.pietquin}@supelec.fr
3
Abstract
This paper adresses the inverse reinforcement learning (IRL) problem, that is inferring a reward for which a demonstrated expert behavior is optimal. We introduce a new algorithm, SCIRL, whose principle is to use the so-called feature
expectation of the expert as the parameterization of the score function of a multiclass classifier. This approach produces a reward function for which the expert
policy is provably near-optimal. Contrary to most of existing IRL algorithms,
SCIRL does not require solving the direct RL problem. Moreover, with an appropriate heuristic, it can succeed with only trajectories sampled according to the
expert behavior. This is illustrated on a car driving simulator.
1
Introduction
Inverse reinforcement learning (IRL) [14] consists in finding a reward function such that a demonstrated expert behavior is optimal. Many IRL algorithms (to be briefly reviewed in Sec. 5) search
for a reward function such that the associated optimal policy induces a distribution over trajectories
(or some measure of this distribution) which matches the one induced by the expert. Often, this
distribution is characterized by the so-called feature expectation (see Sec. 2.1): given a reward function linearly parameterized by some feature vector, it is the expected discounted cumulative feature
vector for starting in a given state, applying a given action and following the related policy.
In this paper, we take a different route. The expert behavior could be mimicked by a supervised
learning algorithm generalizing the mapping from states to actions. Here, we consider generally
multi-class classifiers which compute from a training set the parameters of a linearly parameterized
score function; the decision rule for a given state is the argument (the action) which maximizes the
score function for this state (see Sec. 2.2). The basic idea of our SCIRL (Structured Classificationbased IRL) algorithm is simply to take an estimate of the expert feature expectation as the parameterization of the score function (see Sec. 3.1). The computed parameter vector actually defines a
reward function for which we show the expert policy to be near-optimal (Sec. 3.2).
Contrary to most existing IRL algorithms, a clear advantage of SCIRL is that it does not require
solving repeatedly the direct reinforcement learning (RL) problem. It requires estimating the expert
feature expectation, but this is roughly a policy evaluation problem (for an observed policy, so is less
involved than repeated policy optimization problems), see Sec. 4. Moreover, up to the use of some
heuristic, SCIRL may be trained solely from transitions sampled from the expert policy (no need to
sample the whole dynamic). We illustrate this on a car driving simulator in Sec. 6.
1
2
Background and Notations
2.1
(Inverse) Reinforcement Learning
A Markov Decision process (MDP) [12] is a tuple {S, A, P, R, ?} where S is the finite state space1 ,
A the finite actions space, P = {Pa = (p(s0 |s, a))1?s,s0 ?|S| , a ? A} the set of Markovian transition
probabilities, R ? RS the state-dependent reward function and ? the discount factor. A deterministic
policy ? ? S A defines the behavior of an agent. The quality of this control is quantified by the
?
value function vR
? RS , associating to each state the cumulative
P discounted reward for starting in
?
this state and following the policy ? afterwards: vR
(s) = E[ t?0 ? t R(St )|S0 = s, ?]. An optimal
?
?
policy ?R
(according to the reward function R) is a policy of associated value function vR
satisfying
?
?
vR ? vR , for any policy ? and componentwise.
Let P? be the stochastic matrix P? = (p(s0 |s, ?(s)))1?s,s0 ?|S| . With a slight abuse of notation,
we may write a the policy which associates the action a to each state s. The Bellman evaluation
?
?
?
(resp. optimality) operators TR
(resp. TR
) : RS ? RS are defined as TR
v = R + ?P? v and
?
?
?
?
TR v = max? TR v. These operators are contractions and vR and vR are their respective fixed?
? ?
?
? ?
points: vR
= TR
vR and vR
= TR
vR . The action-value function Q? ? RS?A adds a degree of
a ?
vR ](s). We also
freedom on the choice of the first action, it is formally defined as Q?R (s, a) = [TR
>
>
write ?? the stationary distribution of the policy ? (satisfying ?? P? = ?? ).
Reinforcement learning and approximate dynamic programming aim at estimating the optimal con?
when the model (transition probabilities and the reward function) is unknown (but
trol policy ?R
observed through interactions with the system to be controlled) and when the state space is too large
to allow exact representations of the objects of interest (as value functions or policies) [2, 15, 17].
We refer to this as the direct problem. On the contrary, (approximate) inverse reinforcement learning [11] aim at estimating a reward function for which an observed policy is (nearly) optimal. Let
us call this policy the expert policy, denoted ?E . We may assume that it optimizes some unknown
? such that the expert policy is
reward function RE . The aim of IRL is to compute some reward R
?E
?
(close to be) optimal, that is such that vR
?
v
.
We
refer
to
this
as
the inverse problem.
?
?
R
Similarly to the direct problem, the state space may be too large for the reward function to admit a
practical exact representation. Therefore, we restrict our search of a good reward among linearly parameterized functions. Let ?(s) = (?1 (s) . . . ?p (s))> be a feature vector composed ofPp basis funcp
tion ?i ? RS , we define the parameterized reward functions as R? (s) = ?> ?(s) = i=1 ?i ?i (s).
p
Searching a good reward thus reduces to searching a good parameter vector ? ? R . Notice that we
?
will use interchangeably R? and ? as subscripts (e.g., v?? for vR
). Parameterizing the reward this
?
way implies a related parameterization for the action-value function:
X
Q?? (s, a) = ?> ?? (s, a) with ?? (s, a) = E[
? t ?(St )|S0 = s, A0 = a, ?].
(1)
t?0
Therefore, the action-value function shares the parameter vector of the reward function, with an associated feature vector ?? called the feature expectation. This notion will be of primary importance
for the contribution of this paper. Notice that each component ??i of this feature vector is actually the
action-value function of the policy ? assuming the reward is ?i : ??i (s, a) = Q??i (s, a). Therefore,
any algorithm designed for estimating an action-value function may be used to estimate the feature
expectation, such as Monte-Carlo rollouts or temporal difference learning [7].
2.2
Classification with Linearly Parameterized Score Functions
Let X be a compact or a finite set (of inputs to be classified) and let Y be a finite set (of labels).
Assume that inputs x ? X are drawn according to some unknown distribution P(x) and that there
exists some oracle which associates to each of these inputs a label y ? Y drawn according to the
unknown conditional distribution P(y|x). Generally speaking, the goal of multi-class classification
is, given a training set {(xi , yi )1?i?N } drawn according to P(x, y), to produce a decision rule
g ? Y X which aims at minimizing the classification error E[?{g(x)6=y} ] = P(g(x) 6= y), where ?
denotes the indicator function.
1
This work can be extended to compact state spaces, up to some technical aspects.
2
Here, we consider a more restrictive set of classification algorithms. We assume that the decision
rule associates to an input the argument which maximizes a related score function, this score function being linearly parameterized and the associated parameters being learnt by the algorithm. More
formally, let ?(s, a) = (?1 (x, y) . . . ?d (x, y))> ? Rd be a feature vector whose components are d
basis functions ?i ? RX ?Y . The linearly parameterized score function sw ? RX ?Y of parameter
vector w ? Rd is defined as sw (x, y) = w> ?(x, y). The associated decision rule gw ? Y X is defined as gw (x) ? argmaxy?Y sw (x, y). Using a training set {(xi , yi )1?i?N }, a linearly parameterized score function-based multi-class classification (MC2 for short) algorithm computes a parameter
vector ?c . The quality of the solution is quantified by the classification error c = P(g?c (x) 6= y).
We do not consider a specific MC2 algorithm, as long as it classifies inputs by maximizing the
argument of a linearly parameterized score function. For example, one may choose a multi-class
support vector machine [6] (taking the kernel induced by the feature vector) or a structured large
margin approach [18]. Other choices may be possible, one can choose its preferred algorithm.
3
3.1
Structured Classification for Inverse Reinforcement Learning
General Algorithm
Consider the classification framework of Sec. 2.2. The input x may be seen as a state and the label y
as an action. Then, the decision rule gw (x) can be interpreted as a policy which is greedy according
to the score function w> ?(x, y), which may itself be seen as an action-value function. Making the
parallel with Eq. (1), if ?(x, y) is the feature expectation of some policy ? which produces labels
of the training set, and if the classification error is small, then w will be the parameter vector of a
reward function for which we may hope the policy ? to be near optimal. Based on these remarks,
we?re ready to present the proposed Structured Classification-based IRL (SCIRL) algorithm.
Let ?E be the expert policy from which we would like to recover a reward function. Assume that
we have a training set D = {(si , ai = ?E (si ))1?i?N } where states are sampled according to the
expert stationary distribution2 ?E = ??E . Assume also that we have an estimate ?
??E of the expert
?E
feature expectation ? defined in Eq. (1). How to practically estimate this quantity is postponed to
Sec. 4.1; however, recall that estimating ??E is simply a policy evaluation problem (estimating the
action-value function of a given policy), as noted in Sec. 2.1. Assume also that an MC2 algorithm
has been chosen. The proposed algorithm simply consists in choosing ?> ?
??E (s, a) as the linearly
parameterized score function, training the classifier on D which produces a parameter vector ?c , and
outputting the reward function R?c (s) = ?c> ?(s).
Algorithm 1: SCIRL algorithm
Given a training set D = {(si , ai = ?E (si ))1?i?N }, an estimate ?
??E of the expert feature
?E
2
expectation ? and an MC algorithm;
Compute the parameter vector ?c using the MC2 algorithm fed with the training set D and
considering the parameterized score function ?> ?
??E (s, a);
Output the reward function R?c (s) = ?c> ?(s) ;
The proposed approach is summarized in Alg. 1. We call this Structured Classification-based IRL
because using the (estimated) expert feature expectation as the feature vector for the classifier somehow implies taking into account the MDP structure into the classification problem and allows outputting a reward vector. Notice that contrary to most of existing IRL algorithms, SCIRL does not
require solving the direct problem. If it possibly requires estimating the expert feature expectation,
it is just a policy evaluation problem, less difficult than the policy optimization issue involved by the
direct problem. This is further discussed in Sec. 5.
2
For example, if the Markov chain induced by the expert policy is fast-mixing, sampling a trajectory will
quickly lead to sample states according to this distribution.
3
3.2
Analysis
In this section, we show that the expert policy ?E is close to be optimal according to the reward
function R?c , more precisely that Es??E [v??c (s) ? v??cE (s)] is small. Before stating our main result,
we need to introduce some notations and to define some objects.
We will use the first order discounted future state distribution concentration coefficient Cf [9]:
Cf = (1 ? ?)
X
? t c(t) with c(t) =
t?0
max
?1 ,...,?t ,s?S
(?>
E P?1 . . . P?t )(s)
.
?E (s)
We note ?c the decision rule of the classifier: ?c (s) ? argmaxa?A ?c> ?
??E (s, a). The classifica? ?E = ?c> ?
tion error is therefore c = Es??E [?{?c (s)6=?E (s)} ] ? [0, 1]. We write Q
??E the score
?c
function computed from the training set D (which can be interpreted as an approximate action-value
function). Let also ? = ?
??E ? ??E : S ? A ? Rp be the feature expectation error. Conse? ?E ? Q?E = ?c> (?
quently, we define the action-value function error as Q = Q
??E ? ??E ) =
?c
?c
?c> ? : S ? A ? R. We finally define the mean delta-max action-value function error as
?Q = Es??E [maxa?A Q (s, a) ? mina?A Q (s, a)] ? 0.
Theorem 1. Let R?c be the reward function outputted by Alg. 1. Let also the quantities Cf , c and
?Q be defined as above. We have
Cf
2?kR?c k?
?E
?
0 ? Es??E [vR?c ? vR?c ] ?
?Q + c
.
1??
1??
Proof. As the proof only relies on the reward R?c , we omit the related subscripts to keep the nota?
or R for R?c ). First, we link the error Es??E [v ? (s) ? v ?E (s)]
tions simple (e.g., v ? for v??c = vR
?c
to the Bellman residual Es??E [[T ? v ?E ](s) ? v ?E (s)]. Componentwise, we have that:
?
?
v ? ? v ?E = T ? v ? ? T ? v ?E + T ? v ?E ? T ? v ?E + T ? v ?E ? v ?E
(a)
(b)
? ?P?? (v ? ? v ?E ) + T ? v ?E ? v ?E ? (I ? ?P?? )?1 (T ? v ?E ? v ?E ).
?
Inequality (a) holds because T ? v ?E ? T ? v ?E and inequality (b) holds thanks to [9, Lemma 4.2].
Moreover, v ? being optimal we have that v ? ? v ?E ? 0 and T ? being the Bellman optimality
operP
ator, we have T ? v ?E ? T ?E v ?E = v ?E . Additionally, remark that (I ? ?P?? )?1 = t?0 ? t P?t ? .
Therefore, using the definition of the concentration coefficient Cf , we have that:
0 ? Es??E [v ? (s) ? v ?E (s)] ?
Cf
Es??E [[T ? v ?E ](s) ? v ?E (s)] .
1??
(2)
This results actually follows closely the one of [9, Theorem 4.2]. There remains to bound the
Bellman residual Es??E [[T ? v ?E ](s) ? v ?E (s)]. Considering the following decomposition,
T ? v ?E ? v ?E = T ? v ?E ? T ?c v ?E + T ?c v ?E ? v ?E ,
we will bound Es??E [[T ? v ?E ](s) ? [T ?c v ?E ](s)] and Es??E [[T ?c v ?E ](s) ? v ?E (s)].
? ?E = ?c> ?
The policy ?c (the decision rule of the classifier) is greedy with respect to Q
??E . Therefore,
for any state-action couple (s, a) ? S ? A we have:
? ?E (s, ?c (s)) ? Q
? ?E (s, a) ? Q?E (s, a) ? Q?E (s, ?c (s)) + Q (s, ?c (s)) ? Q (s, a).
Q
By definition, Q?E (s, a) = [T a v ?E ](s) and Q?E (s, ?c (s)) = [T ?c v ?E ](s). Therefore, for s ? S:
?a ? A, [T a v ?E ](s) ? [T ?c v ?E ](s) + Q (s, ?c (s)) ? Q (s, a)
? [T ? v ?E ](s) ? [T ?c v ?E ](s) + max Q (s, a) ? min Q (s, a).
a?A
a?A
? ?E
Taking the expectation according to ?E and noticing that T v
?v
?E
, we bound the first term:
0 ? Es??E [[T ? v ?E ](s) ? [T ?c v ?E ](s)] ? ?Q .
There finally remains to bound the term Es??E [[T
4
?c ?E
v
](s) ? v
?E
(s)].
(3)
Let us write M ? R|S|?|S| the diagonal matrix defined as M = diag(?{?c (s)6=?E (s)} ). Using this,
the Bellman operator T ?c may be written as, for any v ? RS :
T ?c v = R + ?M P?c v + ?(I ? M )P?E v = R + ?P?E v + ?M (P?c ? P?E )v.
Applying this operator to v ?E and recalling that R + ?P?E v ?E = T ?E v ?E = v ?E , we get:
?E
?c ?E
|.
? v ?E )| = ?|?>
T ?c v ?E ? v ?E = ?M (P?c ? P?E )v ?E ? |?>
E M (P?c ? P?E )v
E (T v
One can easily see that k(P?c ? P?E )v ?E k? ?
2
1?? kRk? ,
which allows bounding the last term:
|Es??E [[T ?c v ?E ](s) ? v ?E (s)]| ? c
2?
kRk? .
1??
(4)
Injecting bounds of Eqs. (3) and (4) into Eq. (2) gives the stated result.
This result shows that if the expert feature expectation is well estimated (in the sense that the estimation error ? is small for states sampled according to the expert stationary policy and for all actions)
and if the classification error c is small, then the proposed generic algorithm outputs a reward function R?c for which the expert policy will be near optimal. A direct corollary of Th. 1 is that given
the true expert feature expectation ??E and a perfect classifier (c = 0), ?E is the unique optimal
policy for R?c .
One may argue that this bounds trivially holds for the null reward function (a reward often exhibited
to show that IRL is an ill-posed problem), obtained if ?c = 0. However, recall that the parameter
vector ?c is computed by the classifier. With ?c = 0, the decision rule would be a random policy
and we would have c = |A|?1
|A| , the worst possible classification error. This case is really unlikely.
Therefore, we advocate that the proposed approach somehow allows disambiguating the IRL problem (at least, it does not output trivial reward functions such as the null vector). Also, this bound is
scale-invariant: one could impose k?c k = 1 or normalize (action-) value functions by kR?c k?1
?.
One should notice that there is a hidden dependency of the classification error c to the estimated
expert feature expectation ?
??E . Indeed, the minimum classification error depends on the hypothesis
space spanned by the chosen score function basis functions for the MC2 algorithm (here ?
??E ).
Nevertheless, provided a good representation for the reward function (that is a good choice of basis
functions ?i ) and a small estimation error, this should not be a practical problem.
Finally, if our bound relies on the generalization errors c and ?Q , the classifier will only use
(?
??E (si , a))1?i?N,a?A in the training phase, where si are the states from the set D. It outputs ?c , seen as a reward function, thus the estimated feature expectation ?
??E is no longer required. Therefore, practically it should be sufficient to estimate well ?
??E on state-action couples
(si , a)1?i?N,a?A , which allows envisioning Monte-Carlo rollouts for example.
4
4.1
A Practical Approach
Estimating the Expert Feature Expectation
SCIRL relies on an estimate ?
??E of the expert feature expectation. Basically, this is a policy evaluation problem. An already made key observation is that each component of ??E is the action-value
function of ?E for a reward function ?i : ??i E (s, a) = Q??Ei (s, a) = [T?ai v??iE ](s). We briefly review
its exact computation and possible estimation approaches, and consider possible heuristics.
If the model is known, the feature expectation can be computed explicitly. Let ? ? R|S|?p be the
feature matrix whose rows contain the feature vectors ?(s)> for all s ? S. For a fixed a ? A,
let ?a?E ? R|S|?p be the feature expectation matrix whose rows are the expert feature vectors, that
is (??E (s, a))> for any s ? S. With these notations, we have ??a E = ? + ?Pa (I ? ?P?E )?1 ?.
Moreover, the related computational cost is the same order of magnitude as evaluating a single policy
(as the costly part, computing (I ? ?P?E )?1 , is shared by all components).
If the model is unknown, any temporal difference learning algorithm can be used to estimate the
expert feature expectation [7], as LSTD (Least-Squares Temporal Differences) [4]. Let ? : S ?A ?
Rd be a feature vector composed of d basis functions ?i ? RS?A . Each component ??i E of the
5
expert feature expectation is parameterized by a vector ?i ? Rd : ??i E (s, a) ? ?i> ?(s, a). Assume
that we have a training set {(si , ai , s0i , a0i = ?E (s0i ))1?i?M } with actions ai not necessarily sampled
according to policy ?E (e.g., this may be obtained by sampling trajectories according to an expertbased -greedy policy), the aim being to have a better variability of tuples (non-expert actions should
? ? RM ?d (resp. ?
? 0 ) be the feature matrix whose rows are the feature vectors
be tried). Let ?
>
0
0 >
? ? RM ?p be the feature matrix whose rows are the
?(si , ai ) (resp. ?(si , ai ) ). Let also ?
>
reward?s feature vectors ?(si ) . Finally, let ? = [?1 . . . ?p ] ? Rd?p be the matrix of all
parameter vectors. Applying LSTD to each component of the feature expectation gives the LSTD-?
? > (?
? ? ??
? 0 ))?1 ?
? >?
? and ?
algorithm [7]: ? = (?
??E (s, a) = ?> ?(s, a). As for the exact case,
the costly part (computing the inverse matrix) is shared by all feature expectation components, the
computational cost is reasonable (same order as LSTD).
Provided a simulator and the ability to sample according to the expert policy, the expert feature
expectation may also be estimated using Monte-Carlo rollouts for a given state-action pair (as noted
in Sec. 3.2, ?
??E need only be known on (si , a)1?i?N,a?A ). Assuming that K trajectories are
sampled for each required state-action pair, this method would require KN |A| rollouts.
In order to have a small error ?Q , one may learn using transitions whose starting state is sampled
according to ?E and whose actions are uniformly distributed. However, it may happen that only
transitions of the expert are available: T = {(si , ai = ?E (si ), s0i )1?i?N }. If the state-action couples (si , ai ) may be used to feed the classifier, the transitions (si , ai , s0i ) are not enough to provide
an accurate estimate of the feature expectation. In this case, we can still expect an accurate estimate
of ??E (s, ?E (s)), but there is little hope for ??E (s, a 6= ?E (s)). However, one can still rely on
some heuristic; this does not fit the analysis of Sec. 3.2, but it can still provide good experimental
results, as illustrated in Sec. 6.
We propose such a heuristic. Assume that only data T is available and that we use it to provide an
(accurate) estimate ?
??E (s, ?E (s)) (this basically means estimating a value function instead of an
action-value function as described above). We may adopt an optimistic point of view by assuming
that applying a non-expert action just delays the effect of the expert action. More formally, we
associate to each state s a virtual state sv for which p(.|sv , a) = p(.|s, ?E (s)) for any action a
and for which the reward feature expectation is the null vector, ?(sv ) = 0. In this case, we have
??E (s, a 6= ?E (s)) = ???E (s, ?E (s)). Applying this idea to the available estimate (recalling that
the classifiers only requires evaluating ?
??E on (si , a)1?i?N,a?A ) provides the proposed heuristic:
for 1 ? i ? N , ?
??E (si , a 6= ai ) = ? ?
??E (si , ai ).
We may even push this idea further, to get the simpler estimate of the expert feature expectation (but
with the weakest guarantees). Assume that the set T consists of one long trajectory, that is s0i = si+1
(thus T = {s1 , a1 , s2 , . . . , sN ?1 , aN ?1 , sN , aN }). We may estimate ??E (si , ai ) using the single
rollout available in the training set and use the proposed heuristic for other actions:
?1 ? i ? N, ?
??E (si , ai ) =
N
X
? j?i ?(sj ) and ?
??E (si , a 6= ai ) = ? ?
??E (si , ai ).
(5)
j=i
To sum up, the expert feature expectation may be seen as a vector of action-value functions (for
the same policy ?E and different reward functions ?i ). Consequently, any action-value function
evaluation algorithm may be used to estimate ?? (s, a). Depending on the available data, one may
have to rely on some heuristic to assess the feature expectation for a unexperienced (non-expert)
action. Also, this expert feature expectation estimate is only required for training the classifier, so
it is sufficient to estimate on state-action couples (si , a)1?i?N,a?A . In any case, estimating ??E is
not harder than estimating the action-value function of a given policy in the on-policy case, which is
much easier than computing an optimal policy for an arbitrary reward function (as required by most
of existing IRL algorithms, see Sec. 5).
4.2
An Instantiation
As stated before, any MC2 algorithm may be used. Here, we choose the structured large margin
approach [18]. Let L : S ? A ? R+ be a user-defined margin function satisfying L(s, ?E (s)) ?
6
L(s, a) (here, L(si , ai ) = 0 and L(si , a 6= ai ) = 1). The MC2 algorithm solves:
N
1
? X
min k?k2 +
?i
?,? 2
N i=1
s.t. ?i, ?> ?
??E (si , ai ) + ?i ? max ?> ?
??E (si , a) + L(si , a).
a
Following [13], we express the equivalent hinge-loss form (noting that the slack variables ?i are
tight, which allows moving the constraints in the objective function):
J(?) =
N
1 X
?
??E (si , ai ) + k?k2 .
max ?> ?
??E (si , a) + L(si , a) ? ?> ?
N i=1 a
2
This objective function is minimized using a subgradient descent. The expert feature expectation is
estimated using the scheme described in Eq. (5).
5
Related Works
The notion of IRL has first been introduced in [14] and first been formalized in [11]. A classic approach to IRL, initiated in [1], consists in finding a policy (through some reward function) such that
its feature expectation (or more generally some measure of the underlying trajectories? distribution)
matches the one of the expert policy. See [10] for a review. Notice that related algorithms are not
always able to output a reward function, even if they may make use of IRL as an intermediate step.
In such case, they are usually refereed to as apprenticeship learning algorithms.
Closer to our contribution, some approaches also somehow introduce a structure in a classification
procedure [8][13]. In [8], a metric induced by the MDP is used to build a kernel which is used in
a classification algorithm, showing improvements compared to a non-structured kernel. However,
this approach is not an IRL algorithm, and more important assessing the metric of an MDP is a
quite involved problem. In [13], a classification algorithm is also used to produce a reward function.
However, instead of associating actions to states, as we do, it associates optimal policies (labels) to
MDPs (inputs), which is how the structure is incorporated. This involves solving many MDPs.
As far as we know, all IRL algorithms require solving the direct RL problem repeatedly, except [5, 3].
[5] applies to linearly-solvable MDPs (where the control is done by imposing any dynamic to the
system). In [3], based on a relative entropy argument, some utility function is maximized using a
subgradient ascent. Estimating the subgradient requires sampling trajectories according to the policy
being optimal for the current estimated reward. This is avoided thanks to the use of importance
sampling. Still, this requires sampling trajectories according to a non-expert policy and the direct
problem remains at the core of the approach (even if solving it is avoided).
SCIRL does not require solving the direct problem, just estimating the feature expectation of the
expert policy. In other words, instead of solving multiple policy optimization problems, we only
solve one policy evaluation problem. This comes with theoretical guarantees (which is not the case
of all IRL algorithms, e.g. [3]). Moreover, using heuristics which go beyond our analysis, SCIRL
may rely solely on data provided by expert trajectories. We demonstrate this empirically in the next
section. To the best of our knowledge, no other IRL algorithm can work in such a restrictive case.
6
Experiments
We illustrate the proposed approach on a car driving simulator, similar to [1, 16]. The goal si to drive
a car on a busy three-lane highway with randomly generated traffic (driving off-road is allowed on
both sides). The car can move left and right, accelerate, decelerate and keep a constant speed. The
expert optimizes a handcrafted reward RE which favours speed, punish off-road, punish collisions
even more and is neutral otherwise.
We compare SCIRL as instantiated in Sec. 4.2 to the unstructured classifier (using the same classification algorithm) and to the algorithm of [1] (called here PIRL for Projection IRL). We also consider
the optimal behavior according to a randomly sampled reward function as a baseline (using the same
reward feature vector as SCIRL and PIRL, the associated parameter vector is randomly sampled).
For SCIRL and PIRL we use a discretization of the state space as the reward feature vector, ? ?
R729 : 9 horizontal positions for the user?s car, 3 horizontal and 9 vertical positions for the closest
7
10
8
8
6
6
4
4
Es?U [VR?E (s)]
Es?U [VR?E (s)]
10
2
2
0
0
?2
?2
?4
?4
50
100
150
200
250
300
Number of samples from the expert
350
400
50
100
150
200
250
300
Number of samples from the expert
350
400
Figure 1: Highway problem. The highest line is the expert value. For each curves, we show the
mean (plain line), the standard deviation (dark color) and the min-max values (light color). The
policy corresponding to the random reward is in blue, the policy outputted by the classifier is in
yellow and the optimal policy according the SCIRL?s reward is in red. PIRL is the dark blue line.
traffic?s car and 3 speeds. Notice that these features are much less informative than the ones used
in [1, 16]. Actually, in [16] features are so informative that sampling a random positive parameter
vector ? already gives an acceptable behavior. The discount factor is ? = 0.9. The classifier uses
the same feature vector reproduced for each action.
SCIRL is fed with n trajectories of length n (started in a random state) with n varying from 3 to
20 (so fed with 9 to 400 transitions). Each experiment is repeated 50 times. The classifier uses the
same data. PIRL is an iterative algorithm, each iteration requiring to solve the MDP for some reward
function. It is run for 70 iterations, all required objects (a feature expectations for a non-expert policy
and an optimal policy according to some reward function at each iteration) are computed exactly
?
using the model. We measure the performance of each approach with Es?U [vR
(s)], where U is
E
the uniform distribution (this allows measuring the generalization capability of each approach for
states infrequently encountered), RE is the expert reward and ? is one of the following polices: the
optimal policy for RE (upper baseline), the optimal policy for a random reward (lower baseline),
the optimal policy for R?c (SCIRL), the policy produced by PIRL and the classifier decision rule.
Fig. 1 shows the performance of each approach as a number of used expert transitions (except PIRL
which uses the model). We can see that the classifier does not work well on this example. Increasing
the number of samples would improve its performance, but after 400 transitions it does not work as
well as SCIRL with only a ten of transitions. SCIRL works pretty well here: after only a hundred
of transitions it reaches the performance of PIRL, both being close to the expert value. We do not
report exact computational times, but running SCIRL one time with 400 transitions is approximately
hundred time faster than running PIRL for 70 iteration.
7
Conclusion
We have introduced a new way to perform IRL by structuring a linearly parameterized score
function-based multi-class classification algorithm with an estimate of the expert feature expectation. This outputs a reward function for which we have shown the expert to be near optimal, provided
a small classification error and a good expert feature expectation estimate. How to practically estimate this quantity has been discussed and we have introduced a heuristic for the case where only
transitions from the expert are available, along with a specific instantiation of the SCIRL algorithm.
We have shown on a car driving simulator benchmark that the proposed approach works well (even
combined with the introduced heuristic), much better than the unstructured classifier and as well as
a state-of-the-art algorithm making use of the model (and with a much lower computational time).
In the future, we plan to deepen the theoretical properties of SCIRL (notably regarding possible
heuristics) and to apply it to real-world robotic problems.
Acknowledgments. This research was partly funded by the EU FP7 project ILHAIRE (grant
n? 270780), by the EU INTERREG IVa project ALLEGRO and by the R?gion Lorraine (France).
8
References
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9
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3,924 | 4,552 | Bayesian active learning with localized priors
for fast receptive field characterization
Jonathan W. Pillow
Center For Perceptual Systems
The University of Texas at Austin
[email protected]
Mijung Park
Electrical and Computer Engineering
The University of Texas at Austin
[email protected]
Abstract
Active learning methods can dramatically improve the yield of neurophysiology
experiments by adaptively selecting stimuli to probe a neuron?s receptive field
(RF). Bayesian active learning methods specify a posterior distribution over the
RF given the data collected so far in the experiment, and select a stimulus on
each time step that maximally reduces posterior uncertainty. However, existing
methods tend to employ simple Gaussian priors over the RF and do not exploit
uncertainty at the level of hyperparameters. Incorporating this uncertainty can
substantially speed up active learning, particularly when RFs are smooth, sparse,
or local in space and time. Here we describe a novel framework for active learning
under hierarchical, conditionally Gaussian priors. Our algorithm uses sequential
Markov Chain Monte Carlo sampling (?particle filtering? with MCMC) to construct a mixture-of-Gaussians representation of the RF posterior, and selects optimal stimuli using an approximate infomax criterion. The core elements of this
algorithm are parallelizable, making it computationally efficient for real-time experiments. We apply our algorithm to simulated and real neural data, and show
that it can provide highly accurate receptive field estimates from very limited data,
even with a small number of hyperparameter samples.
1
Introduction
Neurophysiology experiments are costly and time-consuming. Data are limited by an animal?s willingness to perform a task (in awake experiments) and the difficulty of maintaining stable neural
recordings. This motivates the use of active learning, known in statistics as ?optimal experimental design?, to improve experiments using adaptive stimulus selection in closed-loop experiments.
These methods are especially powerful for models with many parameters, where traditional methods
typically require large amounts of data.
In Bayesian active learning, the basic idea is to define a statistical model of the neural response,
then carry out experiments to efficiently characterize the model parameters [1?6]. (See Fig. 1A).
Typically, this begins with a (weakly- or non-informative) prior distribution, which expresses our
uncertainty about these parameters before the start of the experiment. Then, recorded data (i.e.,
stimulus-response pairs) provide likelihood terms that we combine with the prior to obtain a posterior distribution. This posterior reflects our beliefs about the parameters given the data collected so
far in the experiment. We then select a stimulus for the next trial that maximizes some measure of
utility (e.g., expected reduction in entropy, mean-squared error, classification error, etc.), integrated
with respect to the current posterior.
In this paper, we focus on the problem of receptive field (RF) characterization from extracellularly
recorded spike train data. The receptive field is a linear filter that describes how the neuron integrates
its input (e.g., light) over space and time; it can be equated with the linear term in a generalized linear
1
model (GLM) of the neural response [7]. Typically, RFs are high-dimensional (with 10s to 100s of
parameters, depending on the choice of input domain), making them an attractive target for active
learning methods. Our paper builds on prior work from Lewi et al [6], a seminal paper that describes
active learning for RFs under a conditionally Poisson point process model.
Here we show that a sophisticated choice of prior distribution can lead to substantial improvements
in active learning. Specifically, we develop a method for learning under a class of hierarchical,
conditionally Gaussian priors that have been recently developed for RF estimation [8, 9]. These priors flexibly encode a preference for smooth, sparse, and/or localized structure, which are common
features of real neural RFs. In fixed datasets (?passive learning?), the associated estimators give substantial improvements over both maximum likelihood and standard lasso/ridge-regression shrinkage
estimators, but they have not yet been incorporated into frameworks for active learning.
Active learning with a non-Gaussian prior poses several major challenges, however, since the posterior is non-Gaussian, and requisite posterior expectations are much harder to compute. We address
these challenges by exploiting a conditionally Gaussian representation of the prior (and posterior)
using sampling at the level of the hyperparameters. We demonstrate our method using the Automatic
Locality Determination (ALD) prior introduced in [9], where hyperparameters control the locality
of the RF in space-time and frequency. The resulting algorithm outperforms previous active learning
methods on real and simulated neural data, even under various forms of model mismatch.
The paper is organized as follows. In Sec. 2, we formally define the Bayesian active learning problem and review the algorithm of [6], to which we will compare our results. In Sec. 3, we describe
a hierarchical response model, and in Sec. 4 describe the localized RF prior that we will employ
for active learning. In Sec. 5, we describe a new active learning method for conditionally Gaussian
priors. In Sec. 6, we show results of simulated experiments with simulated and real neural data.
2
Bayesian active learning
Bayesian active learning (or ?experimental design?) provides a model-based framework for selecting
optimal stimuli or experiments. A Bayesian active learning method has three basic ingredients:
(1) an observation model (likelihood) p(y|x, k), specifying the conditional probability of a scalar
response y given vector stimulus x and parameter vector k; (2) a prior p(k) over the parameters
of interest; and (3) a loss or utility function U , which characterizes the desirability of a stimulusresponse pair (x, y) under the current posterior over k. The optimal stimulus x is the one that
maximizes the expected utility Ey|x [U (x, y)], meaning the utility averaged over the distribution of
(as yet) unobserved y|x.
One popular choice of utility function is the mutual information between (x, y) and the parameters
k. This is commonly known as information-theoretic or infomax learning [10]. It is equivalent to
picking the stimulus on each trial that minimizes the expected posterior entropy.
Let Dt = {xi , yi }ti=1 denote the data collected up to time step t in the experiment. Under infomax
learning, the optimal stimulus at time step t + 1 is:
xt+1
=
arg max Ey|x,Dt [I(y, k|x, Dt )] = arg min Ey|x,Dt , [H(k|x, y, Dt )],
x
(1)
x
R
where H(k|x, y, D
R t ) = ? p(k|x, y, Dt ) log p(k|x, y, Dt )dk denotes the posterior entropy of k,
and p(y|x, Dt ) = p(y|x, k)p(k|Dt )dk is the predictive distribution over response y given stimulus
x and data Dt . The mutual information provided by (y, x) about k, denoted by I(y, k|x, Dt ), is
simply the difference between the prior and posterior entropy.
2.1
Method of Lewi, Butera & Paninski 2009
Lewi et al [6] developed a Bayesian active learning framework for RF characterization in closed-loop
neurophysiology experiments, which we henceforth refer to as ?Lewi-09?. This method employs a
conditionally Poisson generalized linear model (GLM) of the neural spike response:
?t
yt
= g(k> xt )
? Poiss(?t ),
2
(2)
A
select stimulus
B
experiment
RF model
(Lewi et al 09)
C
hierarchical RF model
hyperparameters
spike
count
stimulus
parameters
(RF)
update posterior
parameters
(RF)
Figure 1: (A) Schematic of Bayesian active learning for neurophysiology experiments. For each
presented stimulus x and recorded response y (upper right), we update the posterior over receptive
field k (bottom), then select the stimulus that maximizes expected information gain (upper left).
(B) Graphical model for the non-hierarchical RF model used by Lewi-09. It assumes a Gaussian
prior p(k) and Poisson likelihood p(yt |xt , k). (C) Graphical model for the hierarchical RF model
used here, with a hyper-prior p? (?) over hyper-parameters and conditionally Gaussian prior p(k|?)
over the RF. For simplicity and speed, we assume a Gaussian likelihood for p(yt |xt , k), though all
examples in the manuscript involved real neural data or simulations from a Poisson GLM.
where g is a nonlinear function that ensures non-negative spike rate ?t .
The Lewi-09 method assumes a Gaussian prior over k, which leads to a (non-Gaussian) posterior
given by the product of Poisson likelihood and Gaussian prior. (See Fig. 1B). Neither the predictive
distribution p(y|x, Dt ) nor the posterior entropy H(k|x, y, Dt ) can be computed in closed form.
However, the log-concavity of the posterior (guaranteed for suitable choice of g [11]) motivates a
tractable and accurate Gaussian approximation to the posterior, which provides a concise analytic
formula for posterior entropy [12, 13].
The key contributions of Lewi-09 include fast methods for updating the Gaussian approximation to
the posterior and for selecting the stimulus (subject to a maximum-power constraint) that maximizes
expected information gain. The Lewi-09 algorithm yields substantial improvement in characterization performance relative to randomized iid (e.g., ?white noise?) stimulus selection. Below, we will
benchmark the performance of our method against this algorithm.
3
Hierarchical RF models
Here we seek to extend the work of Lewi et al to incorporate non-Gaussian priors in a hierarchical
receptive field model. (See Fig. 1C). Intuitively, a good prior can improve active learning by reducing
the prior entropy, i.e., the effective size of the parameter space to be searched. The drawback of
more sophisticated priors is that they may complicate the problem of computing and optimizing the
posterior expectations needed for active learning.
To focus more straightforwardly on the role of the prior distribution, we employ a simple linearGaussian model of the neural response:
yt = k> xt + t ,
t ? N (0, ? 2 ),
(3)
2
where t is iid zero-mean Gaussian noise with variance ? . We then place a hierarchical, conditionally Gaussian prior on k:
k|?
?
? N (0, C? )
? p? ,
(4)
(5)
where C? is a prior covariance matrix that depends on hyperparameters ?. These hyperparameters
in turn have a hyper-prior p? . We will specify the functional form of C? in the next section.
In this setup, the effective prior over k is a mixture-of-Gaussians, obtained by marginalizing over ?:
Z
Z
p(k) = p(k|?)p(?)d? = N (0, C? ) p? (?)d?.
(6)
3
Given data X = (x1 , . . . , xt )> and Y = (y1 , . . . , yt )> , the posterior also takes the form of a
mixture-of-Gaussians:
Z
p(k|X, Y ) = p(k|X, Y, ?)p(?|X, Y )d?
(7)
where the conditional posterior given ? is the Gaussian
p(k|X, Y, ?) = N (?? , ?? ),
?? =
>
1
? 2 ?? X Y,
?? = ( ?12 X > X + C??1 )?1 ,
(8)
and the mixing weights are given by the marginal posterior,
p(?|X, Y ) ? p(Y |X, ?)p? (?),
(9)
which we will only need up to a constant of proportionality. The marginal likelihood or evidence
p(Y |X, ?) is the marginal probability of the data given the hyperparameters, and has a closed form
for the linear Gaussian model:
1
p(Y |X, ?) =
|2??? | 2
1
1
|2?? 2 I| 2 |2?C? | 2
where L = ? 2 (X > X)?1 and m =
1
>
? 2 LX Y
exp
1
2
?1
> ?1
?>
m ,
? ?? ?? ? m L
(10)
.
Several authors have pointed out that active learning confers no benefit over fixed-design experiments in linear-Gaussian models with Gaussian priors, due to the fact that the posterior covariance
is response-independent [1, 6]. That is, an optimal design (one that minimizes the final posterior
entropy) can be planned out entirely in advance of the experiment. However, this does not hold
for linear-Gaussian models with non-Gaussian priors, such as those considered here. The posterior
distribution in such models is data-dependent via the marginal posterior?s dependence on Y (eq. 9).
Thus, active learning is warranted even for linear-Gaussian responses, as we will demonstrate empirically below.
4
Automatic Locality Determination (ALD) prior
In this paper, we employ a flexible RF model underlying the so-called automatic locality determination (ALD) estimator [9].1 The key justification for the ALD prior is the observation that most neural
RFs tend to be localized in both space-time and spatio-temporal frequency. Locality in space-time
refers to the fact that (e.g., visual) neurons integrate input over a limited domain in time and space;
locality in frequency refers to the band-pass (or smooth / low pass) character of most neural RFs.
The ALD prior encodes these tendencies in the parametric form of the covariance matrix C? , where
hyperparameters ? control the support of both the RF and its Fourier transform.
The hyperparameters for the ALD prior are ? = (?, ?s , ?f , Ms , Mf )> , where ? is a ?ridge? parameter that determines the overall amplitude of the covariance; ?s and ?f are length-D vectors that
specify the center of the RF support in space-time and frequency, respectively (where D is the degree
of the RF tensor2 ); and Ms and Mf are D ? D positive definite matrices that describe an elliptical
(Gaussian) region of support for the RF in space-time and frequency, respectively. In practice, we
will also include the additive noise variance ? 2 (eq. 3) as a hyperparameter, since it plays a similar
role to C in determining the posterior and evidence. Thus, for the (D = 2) examples considered
here, there are 12 hyperparameters, including scalars ? 2 and ?, two hyperparameters each for ?s and
?f , and three each for symmetric matrices Ms and Mf .
Note that although the conditional ALD prior over k|? assigns high prior probability to smooth
and sparse RFs for some settings of ?, for other settings (i.e., where Ms and Mf describe elliptical
regions large enough to cover the entire RF) the conditional prior corresponds to a simple ridge
prior and imposes no such structure. We place a flat prior over ? so that no strong prior beliefs about
spatial locality or bandpass frequency characteristics are imposed a priori. However, as data from a
neuron with a truly localized RF accumulates, the support of the marginal posterior p(?|Dt ) shrinks
down on regions that favor a localized RF, shrinking the posterior entropy over k far more quickly
than is achievable with methods based on Gaussian priors.
1
?Automatic? refers to the fact that in [9], the model was used for empirical Bayes inference, i.e., MAP
inference after maximizing the evidence for ?. Here, we consider perform fully Bayesian inference under the
associated model.
2
e.g., a space?space?time RF has degree D = 3.
4
5
Bayesian active learning with ALD
To perform active learning under the ALD model, we need two basic ingredients: (1) an efficient
method for representing and updating the posterior p(k|Dt ) as data come in during the experiment;
and (2) an efficient algorithm for computing and maximizing the expected information gain given a
stimulus x. We will describe each of these in turn below.
5.1
Posterior updating via sequential Markov Chain Monte Carlo
To represent the ALD posterior over k given data, we will rely on the conditionally Gaussian representation of the posterior (eq. 7) using particles {?i }i=1,...,N sampled from the marginal posterior,
?i ? P (?|Dt ) (eq. 9). The posterior will then be approximated as:
p(k|Dt ) ?
1 X
p(k|Dt , ?i ),
N i
(11)
where each distribution p(k|Dt , ?i ) is Gaussian with ?i -dependent mean and covariance (eq. 8).
Markov Chain Monte Carlo (MCMC) is a popular method for sampling from distributions known
only up to a normalizing constant. In cases where the target distribution evolves over time by accumulating more data, however, MCMC samplers are often impractical due to the time required for
convergence (i.e., ?burning in?). To reduce the computational burden, we use a sequential sampling
algorithm to update the samples of the hyperparameters at each time step, based on the samples
drawn at the previous time step. The main idea of our algorithm is adopted from the resample-move
particle filter, which involves generating initial particles; resampling particles according to incoming data; then performing MCMC moves to avoid degeneracy in particles [14]. The details are as
follows.
Initialization: On the first time step, generate initial hyperparameter samples {?i } from the hyperprior p? , which we take to be flat over a broad range in ?.
Resampling: Given a new stimulus/response pair {x, y} at time t, resample the existing particles
according to the importance weights:
(t)
p(yt |?i , Dt?1 , xt ) = N (yt |?i > xt , xt > ?i xt + ?i2 ),
(12)
where (?i , ?i ) denote the mean and covariance of the Gaussian component attached to particle ?i ,
This ensures the posterior evolves according to:
(t)
(t)
(t)
p(?i |Dt ) ? p(yt |?i , Dt?1 , xt )p(?i |Dt?1 ).
(13)
MCMC Move: Propagate particles via Metropolis Hastings (MH), with multivariate Gaussian proposals centered on the current particle ?i of the Markov chain: ?? ? N (?i , ?), where ? is a diagonal
matrix with diagonal entries given by the variance of the particles at the end of time step t?1. Accept
?
)
the proposal with probability min(1, ?), where ? = q(?
q(?i ) , with q(?i ) = p(?i |Dt ). Repeat MCMC
moves until computational or time budget has expired.
The main bottleneck of this scheme is the updating of conditional posterior mean ?i and covariance
?i for each particle ?i , since this requires inversion of a d ? d matrix. (Note that, unlike Lewi09, these are not rank-one updates due to the fact that C?i changes after each ?i move). This cost
is independent of the amount of data, linear in the number of particles, and scales as O(d3 ) in
RF dimensionality d. However, particle updates can be performed efficiently in parallel on GPUs or
machines with multi-core processors, since the particles do not interact except for stimulus selection,
which we describe below.
5.2
Optimal Stimulus Selection
Given the posterior over k at time t, represented by a mixture of Gaussians attached to particles {?i }
sampled from the marginal posterior, our task is to determine the maximally informative stimulus to
present at time t + 1. Although the entropy of a mixture-of-Gaussians has no analytic form, we can
5
A
20
angle difference in degree
70
Passive-ALD
B
true filter
1
1
10
20
Lewi-09
ALD10
40
200
400
62.82
51.54
44.94
57.29
40.69
36.65
43.34
35.90
28.98
1000
trials
ALD100
0
ALD100
400
trials
50
30
ALD10
200
trials
10
60
Lewi-09
600
# trials
800
1000
Figure 2: Simulated experiment. (A) Angular error in estimates of a simulated RF (20 ? 20 pixels,
shown in inset) vs. number of stimuli, for Lewi-09 method (blue), the ALD-based active learning
method using 10 (pink) or 100 (red) particles, and the ALD-based passive learning method (black).
True responses were simulated from a Poisson-GLM neuron. Traces show average over 20 independent repetitions. (B) RF estimates obtained by each method after 200, 400, and 1000 trials. Red
numbers below indicate angular error (deg).
compute the exact posterior covariance via the formula:
N
1 X
?
?t =
?i + ?i ?i > ? ?
??
?> ,
N i=1
(14)
P
where ?
?t = N1
?i is the full posterior mean. This leads to an upper bound on posterior entropy, since a Gaussian is the maximum-entropy distribution for fixed covariance. We then take
the next stimulus to be the maximum-variance eigenvector of the posterior covariance, which is the
most informative stimulus under a Gaussian posterior and Gaussian noise model, subject to a power
constraint on stimuli [6].
Although this selection criterion is heuristic, since it is not guaranteed to maximize mutual information under the true posterior, it is intuitively reasonable since it selects the stimulus direction along
which the current posterior is maximally uncertain. Conceptually, directions of large posterior variance can arise in two different ways: (1) directions of large variance for all covariances ?i , meaning
that all particles assign high posterior uncertainty over k|Dt in the direction of x; or (2) directions in
which the means ?i are highly dispersed, meaning the particles disagree about the mean of k|Dt in
the direction of x. In either scenario, selecting a stimulus proportional to the dominant eigenvector
is heuristically justified by the fact that it will reduce collective uncertainty in particle covariances or
cause particle means to converge by narrowing of the marginal posterior. We show that the method
performs well in practice for both real and simulated data (Section 6). We summarize the complete
method in Algorithm 1.
Algorithm 1 Sequential active learning under conditionally Gaussian models
P
Given particles {?i } from p(?|Dt ), which define the posterior as P (k|Dt ) = i N (?i , ?i ),
? t from {(?i , ?i )} (eq. 14).
1. Compute the posterior covariance ?
?t
2. Select optimal stimulus xt+1 as the maximal eigenvector of ?
3. Measure response yt+1 .
4. Resample particles {?i } with the weights {N (yt+1 |?i > xt+1 , xt+1 > ?i xt+1 + ?i2 )}.
5. Perform MH sampling of p(?|Dt+1 ), starting from resampled particles.
repeat
6
true filter
Lewi-09
ALD10
Figure 3: Additional simulated examples comparing Lewi-09 and ALDbased active learning. Responses were
simulated from a GLM-Poisson model
with three different true 400-pixel RFs
(left column): (A) a Gabor filter
(shown previously in [6]); (B): a centersurround RF, typical in retinal ganglion
cells; (C): a relatively non-localized
grid-cell RF. Middle and right columns
show RF estimates after 400 trials of active learning under each method, with
average angular error (over independent
20 repeats) shown beneath in red.
(A)
angle difference: 60.68
37.82
62.82
42.57
60.32
50.73
(B)
(C)
6
Results
Simulated Data: We tested the performance of our algorithm using data simulated from a PoissonGLM neuron with a 20 ? 20 pixel Gabor filter and an exponential nonlinearity (See Fig. 2). This is
the response model assumed by the Lewi-09 method, and therefore substantially mismatched to the
linear-Gaussian model assumed by our method.
For the Lewi-09 method, we used a diagonal prior covariance with amplitude set by maximizing
marginal likelihood for a small dataset. We compared two versions of the ALD-based algorithm
(with 10 and 100 hyperparameter particles, respectively) to examine the relationship between performance and fidelity of the posterior representation. To quantify the performance, we used the
angular difference (in degrees) between the true and estimated RF.
Fig 2A shows the angular difference between the true RF and estimates obtained by Lewi-09 and
the ALD-based method, as a function of the number of trials. The ALD estimate exhibits more
rapid convergence, and performs noticeably better with 100 than with 10 particles (ALD100 vs.
ALD10 ), indicating that accurately preserving uncertainty over the hyperparameters is beneficial to
performance. We also show the performance of ALD inference under passive learning (iid random
stimulus selection), which indicates that the improvement in our method is not simply due to the use
of an improved RF estimator. Fig 2B shows the estimates obtained by each method after 200, 400,
and 1000 trials. Note that the estimate with 100 hyperparameter samples is almost indistinguishable
from the true filter after 200 trials, which is substantially lower than the dimensionality of the filter
itself (d = 400).
Fig. 3 shows a performance comparison using three additional 2-dimensional receptive fields, to
show that performance improves across a variety of different RF shapes. The filters included: (A)
a gabor filter similar to that used in [6]; (B) a retina-like center-surround receptive field; (C) a
grid-cell receptive field with multiple modes. As before, noisy responses were simulated from a
Poisson-GLM. For the grid-cell example, these filter is not strongly localized in space, yet the ALDbased estimate substantially outperforms Lewi-09 due to its sensitivity to localized components in
frequency. Thus, ALD-based method converges more quickly despite the mismatch between the
model used to simulate data and the model assumed for active learning.
Neural Data: We also tested our method with an off-line analysis of real neural data from a simple cell recorded in primate V1 (published in [15]). The stimulus consisted of 1D spatiotemporal
white noise (?flickering bars?), with 16 spatial bars on each frame, aligned with the cell?s preferred
orientation. We took the RF to have 16 time bins, resulting in a 256-dimensional parameter space
for the RF. We performed simulated active learning by extracting the raw stimuli from 46 minutes
of experimental data. On each trial, we then computed the expected information gain from presenting each of these stimuli (blind to neuron?s actual response to each stimulus). We used ALD-based
active learning with 10 hyperparameter particles, and examined performance of both algorithms for
960 trials (selecting from ? 276,000 possible stimuli on each trial).
7
B
70
avg angle difference
ml (46 min.)
16
60
8
1
40
0
1
8
16
Lewi-09
50
ALD
160
480
# of stimuli
960
480 stimuli 160 stimuli
A
C
Lewi-09
ALD
20
?20
angle: 55.0
45.1
?60
?100
47.2
42.5
?140
0
320
640
960
# of stimuli
Figure 4: Comparison of active learning methods in a simulated experiment with real neural data
from a primate V1 simple cell. (Original data recorded in response to white noise ?flickering bars?
stimuli, see [15]). (A): Average angular difference between the MLE (inset, computed from an entire
46-minute dataset) and the estimates obtained by active learning, as a function of the amount of data.
We simulated active learning via an offline analysis of the fixed dataset, where methods had access
to possible stimuli but not responses. (B): RF estimates after 10 and 30 seconds of data. Note that
the ALD-based estimate has smaller error with 10 seconds of data than Lewi-09 with 30 seconds of
data. (C): Average entropy of hyperparameter particles as a function of t, showing rapid narrowing
of marginal posterior.
Fig 4A shows the average angular difference between the maximum likelihood estimate (computed
with the entire dataset) and the estimate obtained by each active learning method, as a function of
the number of stimuli. The ALD-based method reduces the angular difference by 45 degrees with
only 160 stimuli, although the filter dimensionality of the RF for this example is 256. The Lewi-09
method requires four times more data to achieve the same accuracy. Fig 4B shows estimates after
160 and 480 stimuli. We also examined the average entropy of the hyperparameter particles as a
function of the amount of data used. Fig. 4C shows that the entropy of the marginal posterior over
hyperparameters falls rapidly during the first 150 trials of active learning.
? which
The main bottleneck of the algorithm is eigendecomposition of the posterior covariance ?,
took 30ms for a 256 ? 256 matrix on a 2 ? 2.66 GHz Quad-Core Intel Xeon Mac Pro. Updating
importance weights and resampling 10 particles took 4ms, and a single step of MH resampling for
each particle took 5ms. In total, it took <60 ms to compute the optimal stimulus in each trial using a
non-optimized implementation of our algorithm, indicating that our methods should be fast enough
for use in real-time neurophysiology experiments.
7
Discussion
We have developed a Bayesian active learning method for neural RFs under hierarchical response
models with conditionally Gaussian priors. To take account of uncertainty at the level of hyperparameters, we developed an approximate information-theoretic criterion for selecting optimal stimuli
under a mixture-of-Gaussians posterior. We applied this framework using a prior designed to capture
smooth and localized RF structure. The resulting method showed clear advantages over traditional
designs that do not exploit structured prior knowledge. We have contrasted our method with that
of Lewi et al [6], which employs a more flexible and accurate model of the neural response, but
a less flexible model of the RF prior. A natural future direction therefore will be to combine the
Poisson-GLM likelihood and ALD prior, which will combine the benefits of a more accurate neural
response model and a flexible (low-entropy) prior for neural receptive fields, while incurring only a
small increase in computational cost.
8
Acknowledgments
We thank N. C. Rust and J. A. Movshon for V1 data, and several anonymous reviewers for helpful advice on the original manuscript. This work was supported by a Sloan Research Fellowship,
McKnight Scholar?s Award, and NSF CAREER Award IIS-1150186 (JP).
References
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9
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3,925 | 4,553 | Learning High-Density Regions for a Generalized
Kolmogorov-Smirnov Test in High-Dimensional Data
Michael Lindenbaoum
Department of Computer Science
Technion ? Israel Institute of Technology
Haifa 32000, Israel
[email protected]
Assaf Glazer
Department of Computer Science
Technion ? Israel Institute of Technology
Haifa 32000, Israel
[email protected]
Shaul Markovitch
Department of Computer Science
Technion ? Israel Institute of Technology
Haifa 32000, Israel
Address
[email protected]
Abstract
We propose an efficient, generalized, nonparametric, statistical KolmogorovSmirnov test for detecting distributional change in high-dimensional data. To
implement the test, we introduce a novel, hierarchical, minimum-volume sets estimator to represent the distributions to be tested. Our work is motivated by the
need to detect changes in data streams, and the test is especially efficient in this
context. We provide the theoretical foundations of our test and show its superiority
over existing methods.
1
Introduction
The Kolmogorov-Smirnov (KS) test is efficient, simple, and often considered the choice method for
comparing distributions. Let X = {x1 , . . . , xn } and X 0 = {x01 , . . . , x0m } be two sets of feature
vectors sampled i.i.d. with respect to F and F 0 distributions. The goal of the KS test is to determine
whether F 6= F 0 . For one-dimensional distributions, the KS statistics are based on the maximal
difference between cumulative distribution functions (CDFs) of the two distributions. However,
nonparametric extensions of this test to high-dimensional data are hard to define since there are
2d?1 ways to represent a d-dimensional distribution by a CDF. Indeed, due to this limitation, several
extensions of the KS test to more than one dimension have been proposed [17, 9] but their practical
applications are mostly limited to a few dimensions.
One prominent approach of generalizing the KS test to beyond one-dimensional data is that
of Polonik [18]. It is based on a generalized quantile transform to a set of high-density hierarchical regions. The transform is used to construct two sets of plots, expected and empirical, which
serve as the two input CDFs for the KS test. Polonik?s transform is based on a density estimation
over X . It maps the input quantile in [0, 1] to a level-set of the estimated density such that the expected probability of feature vectors to lie within it is equal to its associated quantile. The expected
plots are the quantiles, and the empirical plots are fractions of examples in X 0 that lie within each
mapped region.
Polonik?s approach can handle multivariate data, but is hard to apply in high-dimensional or smallsample-sized settings where a reliable density estimation is hard. In this paper we introduce a generalized KS test, based on Polonik?s theory, to determine whether two samples are drawn from dif1
ferent distributions. However, instead of a density estimator, we use a novel hierarchical minimumvolume sets estimator to estimate the set of high-density regions directly. Because the estimation of
such regions is intrinsically simpler than density estimation, our test is more accurate than densityestimation approaches. In addition, whereas Polonik?s work was largely theoretical, we take a practical approach and empirically show the superiority of our test over existing nonparametric tests in
realistic, high-dimensional data.
To use Polonik?s generalization of the KS test, the high-density regions should be hierarchical.
Using classical minimum-volume set (MV-set) estimators, however, does not, in itself, guarantee
this property. We present here a novel method for approximate MV-sets estimation that guarantees
the hierarchy, thus allowing the KS test to be generalized to high dimensions. Our method uses
classical MV-set estimators as a basic component. We test our method with two types of estimators:
one-class SVMs (OCSVMs) and one-class neighbor machines (OCNMs).
While the statistical test introduced in this paper traces distributional changes in high dimensional
data in general, it is effective in particular for change detection in data streams. Many real-world
applications (e.g. process control) work in dynamic environments where streams of multivariate
data are collected over time, during which unanticipated distributional changes in data streams might
prevent the proper operation of these applications. Change-detection methods are thus required to
trace such changes (e.g. [6]). We extensively evaluate our test on a collection of change-detection
tasks. We also show that our proposed test can be used for the classical setting of the two-sample
problem using symmetric and asymmetric variations of our test.
2
Learning Hierarchical High-Density Regions
Our approach for generalizing the KS test is based on estimating a hierarchical set of MV-sets in
input space. In this section we introduce a method for finding such a set in high-dimensional data.
Following the notion of multivariate quantiles [8], let X = {x1 , . . . , xn } be a setof examples i.i.d.
with respect to a probability distribution F defined on a measurable space Rd , S . Let ? be a realvalued function defined on C ? S. Then, the minimum-volume set (MV-set) with respect to F , ?,
and C at level ? is
C (?) = argmin {?(C 0 ) : F (C 0 ) ? ?} .
(1)
C 0 ?C
If more than one
Pn set attains the minimum, one will be picked. Equivalently, if F (C) is replaced with
Fn (C) = n1 1 1C (xi ), then Cn (?) is one of the empirical MV-sets that attains the minimum. In
the following we think of ? as a Lebesgue measure on Rd .
Polonik introduced a new approach that uses a hierarchical set of MV-sets to generalize the KS test
beyond one dimension. Assume F has a density function f with respect to ?, and let Lf (c) =
{x : f (x) ? c} be the level set of f at level c. Sufficient regularity conditions on f are assumed.
Polonik observed that if Lf (c) ? C, then Lf (c) is an MV-set of F at level ? = F (Lf (c)). He thus
suggested that level-sets can be used as approximations of the MV-sets of a distribution. Hence, a
density estimator was used to define a family of MV-sets {C(?), ? ? [0, 1]} such that a hierarchy
constraint C(?) ? C(?) is satisfied for 0 ? ? < ? ? 1.
We also use hierarchical MV-sets to represent distributions in our research. However, since a density
estimation is hard to apply in high-dimensional data, a more practical solution is proposed. Instead
of basing our method on the products of a density estimation method, we introduce a novel nonparametric method, which uses MV-set estimators (OCSVM and OCNM) as a basic component, to
estimate hierarchical MV-sets without the need for a density estimation step.
2.1
Learning Minimum-Volume Sets with One-Class SVM Estimators
OCSVM is a nonparametric method for estimating a high-density region in a high-dimensional distribution [19]. Consider a function ? : Rd ? F mapping the feature vectors in X to a hypersphere in an infinite Hilbert space F. Let H be a hypothesis space of half-space decision functions
fC (x) = sgn ((w ? ?(x)) ? ?) such that fC (x) = +1 if x ? C, and ?1 otherwise. To separate X
2
from the origin, the learner is asked to solve this quadratic program:
1
1 X
minn
||w||2 ? ? +
?i , s.t. (w ? ? (xi )) ? ? ? ?i , ?i ? 0,
w?F ,??R ,??R 2
?n i
(2)
where ? is the vector of the slack variables, and 0 < ? < 1 is a regularization parameter related to the
proportion of outliers in the training data. All training examples xi for which (w ? ?(x)) ? ? ? 0 are
called support vectors (SVs). Outliers are referred to as examples that strictly satisfy (w ? ?(x)) ?
? < 0. Since the algorithm only depends on the dot product in F, ? never needs to be explicitly
computed, and a kernel function k (?, ?) is used instead such that k (xi , xj ) = (?(xi ) ? ?(xj ))F .
The following theorem draws the connection between the ? regularization parameter and the region
C provided by the solution of Equation 2:
Theorem 1 (Sch?olkopf et al. [19]). Assume the solution of Equation 2 satisfies ? 6= 0. The following
statements hold: (1) ? is an upper bound on the fraction of outliers. (2) ? is a lower bound on the
fraction of SVs. (3) Suppose X were generated i.i.d. from a distribution F which does not contain
discrete components. Suppose, moreover, that the kernel k is analytic and non-constant. Then, with
probability 1, asymptotically, ? is equal to both the fraction of SVs and to the fraction of outliers.
This theorem shows that we can use OCSVMs to estimate high-density regions in the input space
while bounding the number of examples in X lying outside these regions. Thus, by setting ? = 1??,
we can use OCSVMs to estimate regions approximating C(?). We use this estimation method with
its original quadratic optimization scheme to learn a family of MV-sets. However, a straightforward
approach of training a set of OCSVMs, each with different ? ? (0, 1), would not necessarily satisfy
the hierarchy requirement. In the following algorithm, we propose a modified construction of these
regions such that both the hierarchical constraint and the density assumption (Theorem 1) will hold
for each region.
Let 0 < ?1 < ?2 , . . . , < ?q < 1 be a sequence of quantiles. Given X and a kernel function k (?, ?),
our hierarchical MV-sets estimator iteratively trains a set of q OCSVMs, one for each quantile,
and returns a set of decision functions, f?C(?1 ) , . . . , f?C(?q ) that satisfy both hierarchy and density
requirements. Training starts from the largest quantile (?q ). Let Di be the training set of the OCSVM
trained for the ?i quantile. Let fC(?i ) , SVbi be the decision function and the calculated outliers
Sq
(bounded SVs) of the OCSVM trained for the i-th quantile. Let Oi = j=i SVbj . At each iteration,
Di contains examples in X that were not classified as outliers in previous iterations (not in Oi+1 ).
In addition, ? is set to the required fraction of outliers over Di that will keep the total fraction of
outliers over X equal to 1 ? ?i . After each iteration, f?C(?i ) corresponds to the intersection between
the region associated with the previous decision function and the half-space associated with the
current learned OCSVM. Thus f?C(?i ) corresponds to the region specified by an intersection of halfspaces. The outliers in Oi are points that lie strictly outside the constructed region. The pseudo-code
of our estimator is given in Algorithm 1.
Algorithm 1 Hierarchical MV-sets Estimator (HMVE)
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
Input: X , 0 < ?1 < ?2 , . . . , < ?q < 1, k (?, ?)
Output: f?C(?1 ) , . . . , f?C(?q )
Initialize: Dq ? X , Oq+1 ? ?
for i = q to 1 do
(1??i )|X |?|Oi+1 |
??
|Di |
fC(?i ) , SVbi ? OCSV M (Di , ?, k)
if i = q then
f?C(?i ) (x) ? fC(?i (x))
else
fC(?i (x)) : f?C(?i+1 ) (x)
f?C(?i ) (x) ?
?1 : otherwise
11:
Oi ? Oi+1 ? SVbi , Di?1 ? Di \ SVbi
12: return f?C(? ) , . . . , f?C(? )
1
q
The following theorem shows that the regions specified by the decision functions f?C(?1 ) , . . . , f?C(?q )
are: (a) approximations for the MV-sets in the same sense suggested by Sch?olkopf et al., and (b)
? i ) is denoted as the estimates of C(?i ) with respect to
hierarchically nested. In the following, C(?
?
fC(?i ) .
Theorem 2. Let f?C(?1 ) , . . . , f?eC(?q ) be the decision functions returned by Algorithm 1 with param? i ) be the region in the input space
eters {?1 , . . . , ?q }, X , k (?, ?). Assume X is separable. Let C(?
3
C1
C2
C2
C3
C3
C4
2D level?sets estimations: qcd ocsvm/ocnm Vs. kde
0.2
S
? ? x3 ?
x3
0.16
? ? x1 ?
symmetric difference
x2 C? ?? 3 ?
? ? x2 ?
C? ?? 2 ?
Fd
C? ??1 ?
HMVE (OCSVM)
HMVE (OCNM)
KDE2D
0.18
O
C? ?? 2 ?
x1
0.14
0.12
0.1
0.08
0.06
F1
0.04
0.02
10
? i ) in
Figure 1: Left: Estimated MV-sets C(?
the original input space,
q
=
3.
Right:
the
hj
? i ) in F.
projected hC(?
j ?1
15
20
25
30
35
# training points
40
45
50
Figure 2: Averaged symmetric differences
against the number of training points for the
OCSVM / OCNM versions of our estimator,
and the KDE2d density estimator
j
ptop
j ?1
ptop
associated with f?C(?i ) , and SVubi be the set of (unbounded) SVs lying on the separating hyperplane
? i ) ? C(?
? j ) for
j
in the region associated with fC(?ip(x))
. Then, the following statements hold:(1) C(?
sv
|SVj ?1 |+|O |
i
i|
psvubi
?i < ?j . (2) |O
. (3) Suppose X were i.i.d. drawn from a distribution
|X | ? 1 ? ?i ?
|X |
O
?j
F which does not contain discrete components, and k is analytic and non-constant. Then, 1 ? ?i is
wj
i| ?
asymptotically equal to |O
|X | .
j ?1
w j ?1
Proof. Statement (1) holds by definition of f?C(?i ) . Statements (2)-(3) are proved by induction
Hypersphere
with radius
on the number
of1 iterations. In the first iteration f?C(?q ) equals fC(?q ) . Thus, since Oq = SVbq
and ? = 1 ? ?q , statements (2)-(3) follow directly from Theorem 1 1 . Then, by the induction
hypothesis, statements (2)-(3) hold for the first n ? 1 iterations over the ?q , . . . , ?q?n+1 quantiles.
We Time
now prove that statements (2)-(3) hold for f?C(?q?n ) in the next iteration. Since f?C(?q?n+1 ) (x) =
x1 ,..., x100 ?x101 ,..., x150 . . . xi ,..., xi ? 49 . . . xm ? n ? 49 ,..., xm ? n
?1 implies
fC(?q?n ) (x) = ?1, Oq?n+1 are outliers with respect to f?C(?q?n ) . In addition, ? =
(1??q?n )|X |?|Oq?n+1 |
.
Training|D
set |
i
...
...
Hence,
Theorem 1, the total proportion of outliers with respect to
Testingfollowing
sets
X is |Oq?n | = |SVbq?n | + |Oq?n+1 | ? ?|Di | + |Oq?n+1 | = (1 ? ?q?n )|X |, and |SVubq?n | +
|O
|SVub
|
|+|Oq?n |
q?n
q?n
|Oq?n+1 | ? (1 ? ?q?n )|X |. Hence, |X
? 1 ? ?q?n ?
. In the same manner,
|
|X |
under the conditions of statement (3), |Oq?n | is asymptotically equal to (1 ? ?q?n )|X |, and hence,
|Oq?n |
asymptotically, 1 ? ?q?n = |X
| .
? i ) in both the original and the projected spaces. On
Figure 1 illustrates the estimated MV-sets C(?
? i ) regions in the original input space are colored with decreased gray levels. Note
the left, all C(?
? i ) is a subset of C(?
? j ) if i < j. On the right, the projected regions of all C(?
? i )s in F are
that C(?
marked with the same colors. Examples xi in the input space and their mapped vectors ?(xi ) in F
? i)
are contained in the same relative regions in both spaces. It can be seen that the projections of C(?
in F are the intersecting half-spaces learned by Algorithm 1.
2.2
Learning Minimum-Volume Sets with One-Class Neighbor Machine Estimators
OCNM [15] is as an alternative method to the OCSVM estimator for finding regions close to C(?).
Unlike OCSVM, the OCNM solution is proven to be asymptotically close to the MV-set specified 2 .
Degenerated structures in data that may damage the generalization of SVMs could be another reason
for choosing OCNM [24]. In practice, for finite sample size, it is not clear which estimator is more
accurate.
1
Note that the separability of the data implies that the solution of Equation 2 satisfies ? 6= 0.
Sch?olkopf et al. [19] proved that the set provided by OCSVM converges asymptotically to the correct
probability and not to the correct MV-set. Although this property should be sufficient for the correctness of our
test, Polonik observed that MV-sets are preferred.
2
4
OCNM uses either a sparsity or a concentration neighborhood measure. M (Rd , X ) ? R is a
sparsity measure if f (x) > f (y) implies lim|X |?? P (M (x, X ) < M (y, X )) = 1. An example
for a valid sparsity measure is the distance of x to its kth-nearest neighbor in X . When a sparsity
measure is used, the OCNM estimator solves the following linear problem
max
n
??R ,??R
?n? ?
n
X
?i , s.t. M (xi , X ) ? ? ? ?i , ?i ? 0,
(3)
i
such that the resulting decision function fC (x) = sgn (? ? M (x, X )) satisfies bounds and convergence properties similar to those mentioned in Theorem 1 (?-property).
OCNM can replace OCSVM in our hierarchical MV-sets estimator. In contrast to OCSVMs, when
OCNMs are iteratively trained on X using a growing sequence of ? values, outliers need not be
moved from previous iterations to ensure that the ?-property will hold for each decision function.
Hence, a simpler version of Algorithm 1 can be used, where X is used for training all OCNMs and
? = 1 ? ?i for each step 3 . Since Theorem 2 relies on the ?-property of the estimator, it can be
shown that similar statements to those of Theorem 2 also hold when OCNM is used.
As previously discussed, since the estimation of MV-sets is simpler than density estimation,
our test can achieve higher accuracy than approaches based on density estimation. To illustrate this hypothesis empirically, we conducted the following preliminary experiment. We sampled 10 to 50 i.i.d. points with respect to a two-dimensional, mixture of Gaussians, distribution
p = 12 N (? = (0.5, 0.5), ? = 0.1I) + 12 N (? = (?0.5, ?0.5), ? = 0.5I). We use the OCNM
and OCSVM versions of our estimator to approximate hierarchical MV-sets for q? = 9 quantiles:
? = 0.1, 0, 2, . . . , 0.9 (detailed setup parameters are discussed in Section 4). MV-sets estimated
with a KDE2d kernel-density estimation [2] were used for comparison. For each sample size, we
measured the error of each method according
P R to the mean weighted symmetric difference between
p(x)dx. Results, averaged over 50 simthe true MV-sets and their estimates, q1? ? x?C(?)?C(?)
?
ulations, are shows in Figure 2. The advantages of our approach can easily be seen: both versions
of our estimator preform notably better, especially for small sample sizes.
3
Generalized Kolmogorov-Smirnov Test
We now introduce a nonparametric, generalized Kolmogorov-Smirnov (GKS) statistical test for determining whether F 6= F 0 in high-dimensional data. Assume F, F 0 are one-dimensional continuous
0
distributions and Fn , Fm
are empirical distributions estimated from n and m examples i.i.d. drawn
0
from F, F . Then, the two-sample Kolmogorov-Smirnov (KS) statistic is
0
KSn,m = sup |Fn (x) ? Fm
(x)|
(4)
x?R
and
q
nm
n+m
KSn,m is asymptotically distributed, under the null hypothesis, as the distribution of
supx?R |B(F (x))| for a standard Brownian bridge B when F = F 0 . Under the null hypothesis,
assume F = F 0 and let F ?1 be a quantile transform of F , i.e., the inverse of F . Then we can
replace the supremum over x ? R with the supremum over ? ? [0, 1] as follows:
0
KSn,m = sup Fn (F ?1 (?)) ? Fm
(F ?1 (?)) .
(5)
??[0,1]
Note that in the one-dimensional setting, F ?1 (?) is the point x s.t. F (X ? x) ? ? where X is a
random variable drawn from F . Equivalently, F ?1 (?) can be identified with the interval [??, x].
In a high-dimensional space these intervals can be replaced by hierarchical MV-sets C(?) [18],
and hence, Equation 5 can be calculated regardless of the input space dimensionality. We suggest
replacing KSn,m with
0
Tn,m = sup |Fn (C(?)) ? Fm
(C(?))|.
(6)
??[0,1]
?
For estimating C(?) we use our nonparametric method from Section 2. C(?)
is learned with X
?
and marked as CX (?). In practice, when |X | is finite, the expected proportion of examples that lie
3
? i ).
Note that intersection is still needed (Algorithm 1, line 10) to ensure the hierarchical property on C(?
5
C1
C2
C2
C3
C3
C4
S
? ? x3 ?
C3
? ? x1 ?
? ? x2 ?
x
x2
within C?X (?i ) is not guaranteed
to be exactly3 ?i . Therefore, after learning the decision functions,
C2
we estimate Fn (C?X (?i )) by a k-folds
cross-validation procedure. Our final test statistic is
Fd
C1
O
? ?
C2?
?
(7)
Tn,m = sup Fn (CX (?i )) ? Fm (CX (?i )) ,
1?i?q
x1
where F?n (C?X (?i )) is the estimate of Fn (C?X (?i )). The two-sample KS statistical test is used over
F1
? n,m to calculate the resulting p-value.
T
The test defined above works only in one direction by predicting whether distributions of the samples
share the same ?concentrations? as regions estimated according to X , and not according to X 0 . We
may symmetrize it by running the non-symmetric
test twice, once in each direction, and return twice
hj
their minimum p-value (Bonferroni
correction).
Note that by doing so in the context of a change
h
j
detection task, we pay in runtimej ?1required for learning
MV-sets for each X 0 .
ptop
4
j ?1
ptop
Empirical Evaluation
We first evaluated our test on concept-drift detection problems in data-stream classification tasks.
Concept drifts are associated with distributional changes inj data streams that occur due to hidden
psv
context [22] ? changes of which the classifier is unaware.
j ?1 We used the 27 UCI datasets used
sv
in [6], and 6 additional high-dimensionality UCI datasets: parrhythmia,
madelon, semeion, internet
O musk. The average
?j
advertisement, hill-valley, and
number of features for all datasets is 123 4 .
w
? j ?1we generated, for each dataset, a sequence
Following the experimental setup used by j[11, 6],
hx1 , . . . , xn+m i, where the first n examples are associated
with the most frequent label, and the
w j ?1
following m examples with the second most frequent. Within each label the examples were shuffled
randomly. The first 100 examples hx1 , . . . , x100 i, associated, in all datasets, with the most common
label, were used as the baseline
dataset X . A sliding window of 50 consecutive examples over the
Hypersphere
following sequence of examples
used to define the most recent data X 0 at hand. Stawith was
radiusiteratively
1
tistical tests were evaluated with X and all possible X 0 windows. In total, for each dataset, the set
{hX , X 0 i i |X 0 i = {xi , . . . , xi+49 } , 101 ? i ? n + m ? 49} of pairs were used for evaluation. The
following figure illustrates this setup:
Time
x1 ,..., x100
x101 ,..., x150
...
xi ,..., xi ? 49
...
...
xm ? n ? 49 ,..., xm ? n
...
Training set
Testing sets
The pairs hX , X 0 i i , i ? n ? 49, where all examples in X 0 i have the same labels as in X , are
considered ?unchanged.? The remaining pairs are considered ?changed.? Performance is evaluated
using precision-recall values with respect to the change detection task.
We compare our one-directional (GKS1d ) and two-directional (GKS2d ) tests to the following 5 reference tests: kdq-tree test (KDQ) [4], Metavariable Wald-Wolfowitz test (WW) [10], Kernel change
detection (KCD) [5], Maximum mean discrepancy test (MMD) [12], and PAC-Bayesian margin test
(PBM) [6]. See section 5 for details. All tests, except of MMD, were implemented and parameters
were set with accordance to their suggested setting in their associate papers. The implementation of
MMD test provided by the authors 5 was used with default parameters (RBF kernels with automatic
kernel width detection) and Rademacher bounds. Similar results were also measured for asymptotic bounds. Note that we cannot compare our test to Polonik?s test since density estimations and
level-sets extractions are not practically feasible on high-dimensional data.
2
The LibSVM package [3] with a Gaussian kernel (? = #f eatures
) was used for the OCSVMs. A
distance from a point to its kth-nearest neighbor was used as a sparsity measure for the OCNMs. k
is set to 10% of the sample size 6 . ? = 0.1, 0.2, . . . , 0.9 were used for all experiments.
4
Nominal features were transformed into numeric ones using binary encoding; missing values were replaced
by their features? average values.
5
The code can be downloaded at http://people.kyb.tuebingen.mpg.de/arthur/mmd.htm.
6
Preliminary experiments show similar results obtained with k equal to 10, 20, . . . , 50% of |X |.
6
1
0.9
0.9
0.8
0.8
precision
precision
1
0.7
GKS1d (OCSVM)
GKS2d (OCSVM)
0.6
0.4
0.6
WW
MMD
PBM
KCD
KDQ
BEP
0.5
0
0.1
0.2
GKS1d (OCSVM)
GKS2d (OCSVM)
0.5
GKS1d (OCNM)
GKS2d (OCNM)
0.3
0.4
0.5
recall
0.6
0.7
0.8
0.9
0.4
1
0
0.1
0.2
0.3
0.4
0.5
recall
0.6
0.7
0.8
0.9
1
Figure 4: Precision-recall curves averaged over all 33 experiments for GKS1d
(OCSVMs), GKS2d (OCSVMs), GKS1d
(OSNMs), and GKS2d (OSNMs).
Figure 3: Precision-recall curves averaged over all 33 experiments for GKS1d
(OCSVMs), GKS2d (OCSVMs), and the 5
reference tests.
4.1
0.7
Results
For better visualization, results are shown in two separate figures: Figure 3 shows the precisionrecall plots averaged over the 33 experiments for the OCSVM version of our tests, and the 5 reference
tests. Figure 4 shows the precision-recall plots averaged over the 33 experiments for the OCSVM
and OCNM versions of our tests. In both versions, GKS1d and GKS2d provide the best precisionrecall compromise. For example, for the OCSVM version, at a recall of 0.86, GKS1d accurately
detects distributional changes with 0.90 precision and GKS2d with 0.88 precision, while the second
best competitor does so with 0.84 precision. In terms of their break even point (BEP) measures ?
the points at which precision equals recall ? GKS1d outperforms the other 5 reference tests with
a BEP of 0.89 while its second best competitor does so with BEP of 0.84. Mean precisions for
each dataset were compared using the Wilcoxon statistical test with ? = 0.05. Here, too, GKS1d
performs significantly better than all others for both OCSVM and OCNM versions, except for the
MMD with a p-value of 0.08 for GKS1d (OCSVM) and 0.12 for GKS1d (OCNM).
Although the plots for our GKS1d (OCSVM) test (Figure 4) look better than GKS2d , no significant
difference was found. This result is consistent with previous studies which claim that variants of
solutions whose goal is to make the tests more symmetric have empirically shown no conclusive
superiority [4]. We also found that the GKS1d (OCSVM) version of our test has the least runtime
and scales well with dimensionality, while the GKS1d (OSNM) version suffers from increased time
complexity, especially in high dimensions, due to its expensive neighborhood measure. However,
note that this observation is true only when off-line computational processing on X is not considered.
As opposed to the KCD, and, PBM, tests, our GKS1d test need not be retrained on each X 0 . Hence,
in the context where X is treated as a baseline dataset, GKS1d (OCSVM) is relatively cheap, and
estimated in O (nm) time (the total number of SVs used to calculate f 0 C(?1 ) , . . . , f 0 C(?q ) is O (n)).
In comparison to other tests, it is still the least computationally demanding 7 .
4.2
Topic Change Detection among Documents
We evaluated our test on an additional setup of high-dimensionality problems pertaining to the detection of topic changes in streams of documents. We used the 20-Newsgroup document corpus 8 .
1000 words were randomly picked to generate 1000 bag-of-words features. 12 categories were used
for the experiments 9 . Topic changes were simulated between all pairs of categories (66 pairs in total), using the same methodology as in the previous UCI experiments. Due to the excessive runtime
MMD and WW complexities are estimated in O (n + m)2 time where n, m are the sample sizes. KDQ
uses bootstrapping for p-value estimations, and hence, is more expensive.
8
The 20-Newsgroup corpus is at http://people.csail.mit.edu/jrennie/20Newsgroups/.
9
The selection of these categories is based on the train/test split defined in http://www.cad.zju.
edu.cn/home/dengcai/Data/TextData.html.
7
7
of some of the tests, especially with high-dimensional data, we evaluated only 4 of the 7 methods:
GKS1d (OCSVM), WW, MMD, and KDQ, whose expected runtime may be more reasonable.
Once again, our GKS1d test dominates the others with the best precision-recall compromise. With
regard to BEP values, GKS1d outperforms the other reference tests with a BEP of 0.67 (0.70 precision on average), while its second best competitor (MMD) does so with a BEP of 0.62 (0.64
precision on average). According to the Wilcoxon statistical test with ? = 0.05, GKS1d performs
significantly better than the others in terms of their average precision measures.
5
Related Work
Our proposed test belongs to a family of nonparametric tests for detecting change in multivariate
data that compare distributions without the intermediate density estimation step. Our reference tests
were thus taken from this family of studies. The kdq-tree test (KDQ) [4] uses a spatial scheme (called
kdq-tree) to partition the data into small cells. Then, the Kullback-Leibler (KL) distance is used to
measure the difference between data counts for the two samples in each cell. A permutation (bootstrapping) test [7] is used to calculate the significant difference (p-value). The metavariable WaldWolfowitz test (WW) [10] measures the differences between two samples according to the minimum
spanning tree in the graph of distances between all pairs in both samples. Then, the Wald-Wolfowitz
test statistics are computed over the number of components left in the graph after removing edges
between examples of different samples. The kernel change detection (KCD) [5] measures the distance between two samples according to a ?Fisher-like? distance between samples. This distance is
based on hypercircle characteristics of the resulting two OCSVMs, which were trained separately on
each sample. The maximum mean discrepancy test (MMD) [12] meausres discrepancy according to
a complete matrix of kernel-based dissimilarity measures between all examples, and test statistics
are then computed. (5) The PAC-Bayesian margin test (PBM) [6] measures the distance between
two samples according to the average margins of a linear SVM classifier between the samples, and
test statistics are computed.
As discussed in detail before, our test follows the general approach of Polonik but differs in three
important ways: (1) While Polonik uses a density estimator for specifying the MV-sets, we introduce
a simpler method that finds the MV-sets directly from the data. Our method is thus more practical
and accurate in high-dimensional or small-sample-sized settings. (2) Once the MV-sets are defined,
Polonik uses their hypothetical quantiles as the expected plots, and hence, runs the KS test in its onesample version (goodness-of-fit test). We take a more practically accurate approach for finite sample
size when approximations of MV-sets are not precise. Instead of using the hypothetical measures,
we estimate the expected plots of X empirically and use the two-sample KS test instead. (3) Unlike
Polonik?s work, ours was evaluated empirically and its superiority demonstrated over a wide range
of nonparametric tests. Moreover, since Polonik?s test relies on a density estimation and the ability
to extract its level-sets, it is not practically feasible in high-dimensional settings.
Other methods for estimating MV-sets exist in the literature [21, 1, 16, 13, 20, 23, 14]. Unfortunately, for problems beyond two dimensions and non-convex sets, there is often a gap between their
theoretical and practical estimates [20]. We chose here OCSVM and OSNM because they perform
well on small, high-dimensional samples.
6
Discussion and Summary
This paper makes two contributions. First, it proposes a new method that uses OCSVMs or OCNMs
to represent high-dimensional distributions as a hierarchy of high-density regions. This method
is used for statistical tests, but can also be used as a general, black-box, method for efficient and
practical representations of high-dimensional distributions. Second, it presents a nonparametric,
generalized, KS test that uses our representation method to detect distributional changes in highdimensional data. Our test was found superior to competing tests in the sense of average precision
and BEP measures, especially in the context of change-detection tasks.
An interesting and still open question is how we should set the input ? quantiles for our method.
The problem of determining the number of quantiles ? and the gaps between consecutive ones ? is
related to the problem of histogram design.
8
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9
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3,926 | 4,554 | Slice Normalized Dynamic Markov Logic Networks
Tivadar Papai
Henry Kautz
Daniel Stefankovic
Department of Computer Science
University of Rochester
Rochester, NY 14627
{papai,kautz,stefanko}@cs.rochester.edu
Abstract
Markov logic is a widely used tool in statistical relational learning, which uses
a weighted first-order logic knowledge base to specify a Markov random field
(MRF) or a conditional random field (CRF). In many applications, a Markov logic
network (MLN) is trained in one domain, but used in a different one. This paper focuses on dynamic Markov logic networks, where the size of the discretized
time-domain typically varies between training and testing. It has been previously
pointed out that the marginal probabilities of truth assignments to ground atoms
can change if one extends or reduces the domains of predicates in an MLN. We
show that in addition to this problem, the standard way of unrolling a Markov logic
theory into a MRF may result in time-inhomogeneity of the underlying Markov
chain. Furthermore, even if these representational problems are not significant for
a given domain, we show that the more practical problem of generating samples
in a sequential conditional random field for the next slice relying on the samples
from the previous slice has high computational cost in the general case, due to the
need to estimate a normalization factor for each sample. We propose a new discriminative model, slice normalized dynamic Markov logic networks (SN-DMLN),
that suffers from none of these issues. It supports efficient online inference, and
can directly model influences between variables within a time slice that do not
have a causal direction, in contrast with fully directed models (e.g., DBNs). Experimental results show an improvement in accuracy over previous approaches to
online inference in dynamic Markov logic networks.
1
Introduction
Markov logic [1] is a language for statistical relational learning, which employs weighted first-order
logic formulas to compactly represent a Markov random field (MRF) or a conditional random field
(CRF). A Markov logic theory where each predicate can take an argument representing a time point
is called a dynamic Markov logic network (DMLN). We will focus on two-slice dynamic Markov
logic networks, i.e., ones in which each quantified temporal argument is of the form t or t + 1, in
the conditional (CRF) setting. DMLNs are the undirected analogue of dynamic Bayesian networks
(DBN) [13] and akin to dynamic conditional random fields [19].
DMLNs have been shown useful for relational inference in complex dynamic domains; for example,
[17] employed DMLNs for reasoning about the movements and strategies of 14-player games of
Capture the Flag. The usual method for performing offline inference in a DMLN is to simply unroll
it into a CRF and employ a general MLN or CRF inference algorithm. We will show, however, that
the standard unrolling approach has a number of undesirable properties.
The first two negative properties derive from the fact that MLNs are in general sensitive to the
number of constants in each variable domain [6]; and so, in particular cases, unintuitive results can
occur when the length of training and testing sequences differ. First, as one increases the number
of time points in the domain, the marginals can fluctuate, even if the observations have little or no
influence on the hidden variables. Second, the model can become time-inhomogeneous, even if the
ground weighted formulas between the time slices originate from the same weighted first-order logic
formulas.
The third negative property is of greater practical concern. In domains where there are a large number of variables within each slice dynamic programming based exact inference cannot be used. When
1
the number of time steps is high and/or online inference is required, unrolling the entire sequence
(perhaps repeatedly) becomes prohibitively expensive. Kersting et al. [7] suggests reducing the cost
by exploiting symmetries while Nath & Domingos [14] propose reusing previously sent messages
while performing a loopy belief propagation. Both algorithms are restricted by the capabilities of
loopy belief propagation, which can fail to converge to the correct distribution in MLNs. Geier &
Biundo [2] provides a slice-by-slice approximate inference algorithm for DMLNs that can utilize
any inference algorithm as a black box, but assumes that projecting the distribution over the random
variables at every time step to the product of their marginal distributions does not introduce a large
degree of error ? an assumption that does not always hold. Sequential Monte Carlo methods, or
particle filters, are perhaps the most popular methods for online inference in high-dimensional sequential models. However, except for special cases such as, e.g., the Gaussian distributions used in
[11], sampling from a two-slice CRF model can become expensive, due to the need to evaluate a
partition function for each particle (see Sec. 3 for more details).
As a solution to all of these concerns, we propose a novel way of unrolling a Markov logic theory
such that in the resulting probabilistic model a smaller CRF is embedded into a larger CRF making the clique potentials between adjacent slices normalized. We call this model slice normalized
dynamic Markov logic network (SN-DMLN). Because of the embedded CRF and the undirected
components in our proposed model, the distribution represented by a SN-DMLN cannot be compactly captured by conventional chain graph [10], DBN or CRF graph representations, as we will
explain in Sec. 4. The SN-DMLN has none of the negative theoretical or practical properties outlined above, and for accuracy and/or speed of inference matches or outperforms unrolled CRFs and
the slice-by-slice approximate inference methods. Finally, because the maximum likelihood parameter learning for an SN-DMLN can be a non-convex optimization problem, we provide an effective
heuristic for weight learning, along with initial experimental results.
2
Background
Probabilistic graphical models compactly represent probability distributions using a graph structure that expresses conditional independences among the variables. Directed graphical models are
mainly used in the generative setting, i.e., they model the joint distribution of the hidden variables
and the observations, and during training the joint probability of the training data is maximized.
Hidden Markov models are the prototypical directed models used for sequential data with hidden
and observable parts. It has been demonstrated that for classification problems, discriminative models, which model the conditional probability of the hidden variables given the observations, can
outperform generative models [12]. The main justifications for their success are that complex dependencies between observed variables do not have to be modeled explicitly, and the conditional
probability of the training data (which is maximized during parameter learning) is a better objective
function if we eventually want to use our model for classification. Markov random fields (MRFs)
and conditional random fields (CRFs) belong to the class of undirected graphical models. MRFs
are generative models, while CRFs are their discriminative version. (For a more detailed discussion
of the relationships between these models see [8]). Markov logic [1] is a first-order probabilistic
language that allows one to define template features that apply to whole classes of objects at once.
A Markov logic network is a set of weighted first-order logic formulas and a finite set of constants
C = {c1 , c2 , . . . , c|C| } which together define a Markov network ML,C that contains a binary node
for each possible grounding of each predicate (ground atom) and a binary valued feature for each
grounding of each first-order logic formula. We will also call the ground atoms variables (since they
are random variables). In each truth assignment to the variables, each variable or feature (ground
formula) evaluates to 1 (true) or 0 (false). In this paper we assume function-free clauses and Herbrand interpretations. Using the knowledge base we can either create an MRF or a CRF. If we
instantiate the model as a CRF, the conditional probability of a truth assignment y to the hidden
ground atoms (query atoms) in an MLN, given truth assignment x to the observable ground atoms
(evidence atoms), is defined as:
P
P
exp( i wi j fi,j (x, y))
,
(1)
P r(Y = y|X = x) =
Z(x)
where fi,j (x, y) = 1 if the jth grounding of the ith formula is true under truth assignment {x, y},
and fi,j (x, y) = 0 otherwise. wi is the weight of the ith formula and Z(x) is the normalization
factor. Ground atoms share the same weight if they are groundings of
Pthe same weighted firstorder logic formula, and (1) could be expressed in terms of ni (x, y) = j fi,j (x, y). Instantiation
as an MRF can be done similarly, having an empty set of evidence atoms. Dynamic MLNs [7]
are MLNs with distinguished arguments in every predicate representing the flow of time or some
other sequential quantity. In our settings, Yt and Xt will denote the set of hidden and observable
random variables, respectively, at time t, and Y1:t and X1:t from time step 1 to t. Each set can
contain many variables, and we should note that their distribution will be represented compactly
by weighted first-order logic formulas. The formulas in the knowledge base can be partitioned into
2
two sets. The transitions part contains the formulas for which it is true that for any grounding of
each formula, there is a t such that the grounding shares variables only with Yt and Yt+1 . The
emission part represents the formulas which connect the hidden and observable variables, i.e. Yt and
Xt . We will use P? (Yt , Yt+1 ) (or P? (Yt:t+1 )) and P? (Yt , Xt ) to denote the product of the potentials
corresponding to weighted ground formulas at time t of the transition and the observation formulas,
respectively. Since some ground formulas may contain only variables from Yt ( i.e., defined over
hidden variables within the same slice), in order to count the corresponding potentials exactly once,
we always include their potentials P? (Yt , Yt?1 ), and for t = 1 we have a separate P? (Y1 ). Hence, the
distribution defined in (1) in sequential domains can be factorized as:
Qt
Qt
P?1 (Y1 = y1 ) i=2 P? (Yi?1:i = yi?1:i ) i=1 P? (Yi = yi , Xi = xi )
P r(Y1:t = y1:t |X1:t = x1:t ) =
Z(x1:t )
(2)
In the rest of the paper, we only allow the temporal domain to vary, and the rest of the domains are
fixed.
3
Unrolling MLNs into random fields in temporal domains
We now describe disadvantages of the standard definition of DMLNs, i.e., when the knowledge base
is unrolled into a CRF:
1. As one increases the number of time points the marginals can fluctuate, even if all the clique
potentials P? (Yi = yi , Xi = xi ) in (2) are uninformative.
2. The transition probability Pr(Yi+1 |Yi ) can be dependent on i, even if every P? (Yi =
yi , Xi = xi ) is uninformative and we use the same weighted first-order logic formula
responsible for the ground formulas covering the transitions between every i and i + 1.
3. Particle filtering is costly in general, i.e., if we have the marginal probabilities at time t, we
cannot compute them at time t + 1 using particle filtering unless certain special conditions
are satisfied.
Saying that P? (Yi = yi , Xi = xi ) is uninformative is equivalent to saying that P? (Yi = yi , Xi = xi )
is constant. (Note that, if Yi and Xi are independent, i.e., for some q and r P? (Yi = yi , Xi = xi ) =
r(yi )q(xi ) then q could be marginalized out and r(Yi ) could be snapped to P? (Yi , Yi?1 ) in (2).) To
demonstrate Property 1, consider an unrolled MRF with the temporal domain T = {1, . . . , T },
with only predicate P (t) (t ? T ) and with the weighted formulas (+?, P (t) ? P (t + 1))
(hard constraint) and (w, P (t)) (soft constraint). Because of the hard constraint, only the sequences ?t : P (t) and ?t : ?P (t) have non-zero probabilities. The soft weights imply that
Pr(P (t)) = exp(wT )Pr(?P (t)), i.e., Pr(P (t)) converges to 1, 0 or to 0.5 with exponential rate
depending on the sign of w. But we are not always fortunate to have converging marginals, e.g., if
we change the hard constraint to be P (t) ? ?P (t + 1) and w 6= 0 the marginals will diverge. If
T is even, then for every t ? T , Pr(P (t)) = Pr(?P (t)), since in both sequences P (t) has the same
number of true groundings. If T is odd then for every odd t ? T : Pr(P (t)) = exp(w)Pr(?P (t)).
Consequently, we have diverging marginals as T ? +?. This phenomenon not only makes the
inference unreliable, but a weight learning algorithm that maximizes the log-likelihood of the data
would produce different weights depending on whether T is even or odd. A similar effect arising from moving between different sized domains is discussed in more details in [6]. The akin
Property 2 (inhomogeneity) can be demonstrated similarly, consider, e.g., an MLN with a single
first-order logic formula P (t) ? P (t + 1) with weight w. For the sake of simplicity, assume T = 3.
1+exp(w)
The unrolled MRF defines a distribution where Pr(?P (3)|?P (2)) = 1+2exp(w)+exp(2w)
which is
not equal to Pr(?P (2)|?P (1)) =
1+exp(w)
1+exp(w)+2 exp(2w)
for an arbitrary choice of w.
The examples we just gave involved hard constraints. In fact, we can show that if there are no
hard hard constraints, as T increases the marginals converge and the system becomes homogeneous
(except for a finite number of transitions). Consider the matrix ? s.t. ?i,j = P? (Yt = aj , Yt?1 = ai ),
where ai , i = 1, . . . , N is an enumeration of the all the possible truth assignments within each
slice and N is the number of the possible truth assignments in the slice. Let PrT (Y1 = y1 ) =
P
QT ?1 ?
P
QT ?1 ?
1
y2 ,...,yT
y1 ,...,yT
i=1 P (Yi = yi , Yi+1 = yi+1 ), where Z(Y1:T ) =
i=1 P (Yi =
Z(Y1:T )
yi , Yi+1 = yi+1 ).
Proposition 1. limt?? Prt (Y1 = y) exists if ? is a positive matrix, i.e., ?i, j : ?i,j > 0.
3
Proof. Using ? and the notation ~1 for all one vector and e~i for a vector which has 1 at the ith
component and 0 everywhere else, we can express Prt (Y1 = y) as:
P ?
ei ?t?1~1
y2 P (Y1 = ai , Y2 = y2 )~
(3)
Prt (Y1 = y) =
~1T ?t~1
Since ? is positive we can apply theorem 8.2.8. from [5], i.e., if the spectral radius of ? is ?(?)
(which is always positive for positive matrices): limt?? (??1 (?)?)t = L, where L = xy T , ?x =
?(?)x, ?T y = ?(?)y, x > 0,y > 0 and xT y = 1. Dividing both the numerator and the denominator
by ?t (?) in (3) proves the convergence of Prt (Y1 = y).
The issue of diverging marginals and time-inhomogeneity has not been previously recognized as a
practical problem. However, the increasing interest in probabilistic models that contain large numbers of deterministic constraints (see, e.g. [4]) might bring this issues to the fore. This proposition
can serve as an explanation why in practice we do not encounter diverging marginals in linear chain
type CRFsand why except for a finite number of transitions the model becomes time-homogeneous.
A more significant practical challenge is described by Property 3, the problem of sampling from
Pr(Yt |X1:t = x1:t ) using the previously drawn samples from Pr(Yt?1 |X1:t?1 = x1:t?1 ). In a
directed graphical model (e.g., in a hidden Markov model), following standard particle filter design,
having sampled s1:t?1 ? Pr(Y1:t?1 = s1:t?1 |X1:t?1 = x1:t?1 ), and then using s1:t?1 one would
sample st ? Pr(Yt , Y1:t?1 = s1:t?1 |X1:t?1 ). Since
Pr(Y1:t = s1:t |X1:t?1 = x1:t?1 ) = Pr(Yt = st |Yt?1 = st?1 )Pr(Y1:t?1 = s1:t?1 |X1:t?1 = x1:t?1 )
(4)
we do not have any difficulty performing this sampling step, and all that is left is to re-sample
the collection of s1:t with importance weights Pr(Yt = st |Xt = xt ). The analogue of this process does not work in a CRF in general. If one first draws a sample s1:t?1 ? P? (Y1 , X1 =
Qt?1
x1 )P? (Y1 ) i=2 P? (Yi , Yi?1 )P? (Yi , Xi = xi ), and then draws st ? P? (Yt , Yt?1 = st?1 ), we end
up sampling from:
t
Y
1
s ? P? (Y1 , X1 = x1 )P? (Y1 )
P? (Yi , Yi?1 )P? (Yi , Xi = xi )
(5)
Z
(y
t?1 t?1 )
i=2
P ?
where Zt?1 (yt?1 ) =
yt P (Yt = yt , Yt?1 = yt?1 ). Unless Zt?1 (yt?1 ) is the same for every
yt?1 , it is necessary to approximate Zt?1 (st?1 ) for every st?1 . 1 Although several algorithms have
been proposed to estimate partition functions [16, 18], the partition function estimation can increase
both the running time of the sampling algorithm significantly and the error of the approximation of
the sampling algorithm. While there are restricted special cases where the normalization factor can
be ignored [11], in general ignoring the approximation of Zt?1 (yt?1 ) could cause a large error in
the computed marginals. Consider, e.g., when we have three weighted formulas in the previously
used toy domain, namely, w : ?P (Yt ) ? ?P (Yt+1 ), ?w : P (Yt ) ? ?P (Yt+1 ) and w? : P (Yt ) ?
?P (Yt+1 ), where w > 0 and w? < 0. It can be proved that in this setting using particle filtering in a
CRF without accounting for Zt?1 (yt?1 ) would result in limt?? Pr(P (Yt )) = 21 , while in the CRF
exp(w)
the correct marginal would be limt?? Pr(P (Yt )) = 1 ? 1+exp(w)
exp(w? ) + O(exp(2w? )), which
?
gets arbitrarily close to 1 as we decrease w .
4
Slice normalized DMLNs
As we demonstrated in Section 3, the root cause of the weaknesses
of an ordinarily unrolled CRF
P
lies in that P? (Yt = yt , Yt?1 = yt?1 ) is unnormalized, i.e., yt P? (Yt = yt , Yt?1 = yt?1 ) 6= 1 in
general. One approach to introduce normalization could be to use maximum entropy Markov models
(MEMM) [12]. In that case we would directly represent Pr(Yt |Xt , Yt?1 ), hence we could implement
a sequential Monte Carlo algorithm simply directly sampling st ? Pr(Yt |Xt = xt , Yt?1 = st?1 )
from slice to slice. However, in [9], it was pointed out that MEMMs suffer from the label-bias problem to which as a solution CRFs were invented. Chain graphs (see e.g. [10]) have also the advantage
of mixing directed and undirected components, and would be a tempting choice to use, but they could
only model the transition between slices by either representing (i) Pr(Yt |Xt = xt , Yt?1 = st?1 ),
1
Exploiting inner structure according to the graphical model within the slice would in worst case still result
in computation of the expensive partition function, or could result in a higher variance estimator the same way
as, e.g., using a uniform proposal distribution does.
4
in which case the model would again suffer from the label-bias problem, or (ii) Pr(Yt , Xt |Yt?1 )
or (iii) Pr(Xt |Yt ) and Pr(Yt |Yt?1 ). The defined distributions both in (ii) and (iii) do not give any
advantage performing the sampling step in (4), and similarly to CRFs would require the expensive
computation of a normalization factor. We propose a slice normalized dynamic Markov logic network (SN-DMLN) model, which consists of directed and undirected components on the high level,
and can be thought of as a smaller CRF nested into a larger CRF describing the transition probabilities constructed using weighted first-order logic formulas as templates. SN-DMLNs neither suffer
from the label bias problem, nor bear the disadvantageous properties presented in Section 3. The
distribution defined by an unrolled SN-DMLN is as follows:
Pr(Y1:t = y1:t |X1:t = x1:t ) =
t
Y
1
P1 (Y1 )
P? (Yi = yi , Xi = xi )
Z(x1:t )
i=1
t
Y
(6)
P (Yi = yi |Yi?1 = yi?1 ) ,
i=2
where
P? (Y1 = y1 )
P1 (Y1 = y1 ) = P
,
?
?
y ? P (Y1 = y1 )
1
P? (Yi = yi , Yi?1 = yi?1 )
P (Yi = yi |Yi?1 = yi?1 ) = P
,
?
?
y ? P (Yi = yi , Yi?1 = yi?1 )
i
and the partition function is defined by:
)
(
t
t
Y
Y
X
P (Yi = yi |Yi?1 = yi?1 ) .
P1 (Y1 = y1 )
P? (Yi = yi , Xi = xi )
Z(x1:t ) =
y1 ,...,yt
i=1
i=2
P (Yt = yt |Yt?1 = yt?1 ) is defined by a two-slice Markov logic network (CRF), which describes
the state transitions probabilities in a compact way. If we hide the details of this nested CRF component and treat it as one potential, we could represent the distribution in (6) by regular chain graphs or
CRFs; however we would lose then the compactness the nested CRF provides for describing the distribution. Similarly, we could collapse the variables at every time slice into one and could use a DBN
(or again a chain graph), but it would need exponentially more entries in its conditional probability
tables. If P? (Yi = yi , Xi = xi ) does not have any information content , the probability distribution
Qt
defined in (6) reduces to P1 (Y1 = y1 ) i=2 P (Yi = yi |Yi?1 = yi?1 ), which is a time-homogeneous
Markov chain 2 , hence this model clearly does not have Properties 1 and 2, no matter what formulas
are present in the knowledge base. Furthermore, we do not have to compute the partition function
between the slices, because equation (5) shows, drawing a sample yt ? P? (Yt , Yt?1 = yt?1 ) while
keeping the value yt?1 fixed is equivalent to sampling from P (Yt |Yt?1 = yt?1 ), the quantity present
in equation (6). This means that using our model one can avoid estimating Z(yt?1 ). To learn the
parameters of the model we will maximize the conditional log-likelihood (L) of the data. We use a
modified version of a hill climbing algorithm. The modification is needed, because in our case L is
not necessarily concave. We will partition the weights (parameters) of our model based on whether
they belong to transition or to emission part of the model. The gradient of the L of a data sequence
d = y1 , x1 , . . . , yt , xt w.r.t. an emission parameter we (to which feature ne belongs) is:
" t
#
t
X
X
?Ld
ne (yi , xi ) ? EP r(Y |X=x)
=
ne (Yi , xi ) ,
(7)
?we
i=1
i=1
which is analogous to what one would expect for CRFs. However, for a transition parameter wtr
(belonging to feature ntr ) we get something different:
t?1
t
X
X
?Ld
EP (Yi+1 |yi ) [ntr (Yi+1 , Yi = yi )]
ntr (yi+1 , yi ) ?
=
?wtr
i=1
i=1
?X
t?1
t?1
h
i?
X
?
? EP r(Y |X=x)
EP (Y?i+1 |Yi ) ntr (Yi+1 , Yi ) .
ntr (Yi+1 , Yi ) ?
(8)
i=1
i=1
(Note that, Ld is concave w.r.t. the emission parameters, i.e., when the transition parameters are
kept fixed, allowing the transition parameters to vary makes Ld no longer concave.) In (8) the first
2
Note that, in the SN-DMLN model the uniformity of P? (Yi = yi , Xi = xi ) is a stronger assumption than
the independence of Xi and Yi .
5
friendships reflect
people?s similarity in
smoking habits
symmetry and
reflexivity of
friendship
persistence of
smoking
people with different smoking
habits hang out separately
Smokes(p1 , t) ? ?Smokes(p2 , t) ? (p1 6= p2 ) ? ?F riends(p1 , p2 , t)
Smokes(p1 , t) ? Smokes(p2 , t) ? (p1 6= p2 ) ? F riends(p1 , p2 , t)
?Smokes(p1 , t) ? ?Smokes(p2 , t) ? (p1 6= p2 ) ? F riends(p1 , p2 , t)
?F riends(p1 , p2 , t) ? ?F riends(p2 , p1 , t)
F riends(p1 , p2 , t) ? F riends(p2 , p1 , t)
F riends(p, p, t)
Smokes(p, t) ? Smokes(p, t + 1)
?Smokes(p, t) ? ?Smokes(p, t + 1)
Hangout(p1 , g1 , t) ? Hangout(p2 , g2 , x) ? Smokes(p1 , t)?
(p1 6= p2 ) ? (g1 6= g2 ) ? ?Smokes(p2 , t)
Hangout(p1 , g1 , t) ? Hangout(p2 , g2 , t) ? ?Smokes(p1 , t)?
(p1 6= p2 ) ? (g1 6= g2 ) ? Smokes(p2 , t)
Table 1: Formulas in the knowledge base
two and the last two terms can be grouped together. The first group would represent the gradient
in the case of uninformative observations, i.e., when the model simplifies to a Markov chain with
a compactly represented transition probability distribution. The second group is the expected value
of the expression in the first group. The first three terms correspond to the gradient of a concave
function; while the fourth term corresponds to the gradient of a convex function, so the function as
a whole is not guaranteed to be maximized by convex optimization techniques alone. Therefore, we
chose a heuristic for our optimization algorithm which gradually increases the effects of the second
group in the gradient. More precisely, we always compute the gradient w.r.t. wo according to (7),
but w.r.t. wtr we use:
t?1
t
X
X
?Ld
EP (Yi+1 |yi ) [ntr (Yi+1 , yi )]
ntr (yi+1 , yi ) ?
=
?wtr
i=1
i=1
?X
t?1
t
h
i?
X
? ?EP r(Y |X=x)
EP (Y?i+1 |Yi ) ntr (Y?i+1 , Yi )
ntr (Yi+1 , Yi ) ?
(9)
i=1
i=1
where ? is kept at the value of 0 until convergence, and then gradually increased from 0 to 1 to
converge to the nearest local optimum. In Section 5, we experimentally demonstrate that this heuristic provides reasonably good results, hence we did not turn to more sophisticated algorithms. The
rationale behind our heuristic is that if P? (Yi = yi , Xi = xi ) had truly no information content, then
for ? = 0 we would find the global optimum, and as we increase ? we are taking into account that
the observations are correlated with the hidden variables with an increasing weight.
5
Experiments
For our experiments we extended the Probabilistic Consistency Engine (PCE) [3], a Markov logic
implementation that has been used effectively in different problem domains. For training, we
used 10000 samples for the unrolled CRF and 100 particles and 100 samples for approximating the conditional expectations in (9) for the SN-DMLN to estimate the gradients. For inference we used 10000 samples for the CRF and 10000 particles for the mixed model. The sampling algorithm we relied on was MC-SAT [15]. Our example training data set was a modified version of the dynamic social network example [7, 2]. The hidden predicates in our knowledge base were Smokes(person, time), F riends(person1 , person2 , time) and the observable
was Hangout(person, group, time). The goal of inference was to predict which people could
potentially be friends, based on the similarity in their smoking habits, which similarity could be inferred based on the groups the individuals hang out. We generated training and test data as follows:
there were two groups g1 , g2 , one for smokers and one for non-smokers. Initially 2 people were
randomly chosen to be smokers and 2 to be non-smokers. People with the same smoking habits
can become friends at any time step with probability 1 ? 0.05?, and a smoker and a non-smoker
can become friends with probability 0.05?. Every 5th time step (starting with t = 0) people hang
out in groups and for each person the probability of joining one of the groups is 1 ? 0.05?. With
probability 1? 0.05?, everyone spends time with the group reflecting their smoking habits, and with
probability 0.05? they go to hang out with the other group. The rest of the days people do not hang
out. The smoking habits persist, i.e., a smoker stays a smoker and a non-smoker stays a non-smoker
at the next time step with probability 1 ? 0.05?. In our two configurations we had ? = 0 (deterministic case) and ? = 1 (non-deterministic case). The weights of the clauses we learned using the
SN-DMLN and the CRF unrolled models are in Table 1.
We used chains with length 5, 10, 20 and 40 as training data, respectively. For each chain we had
40, 20, 10 and 5 examples both for the training and for testing, respectively. In our experiments
we compared three types of inference, and measured the prediction quality for the hidden predicate
F riends by assigning true to every ground atom the marginal probability of which was greater than
6
length
5
10
20
40
SN
1.0
1.0
1.0
1.0
?=0
accuracy
MAR MC-SAT SN
0.40
1.0
1.0
0.40
0.97
1.0
0.40
0.67
1.0
0.85
0.60
1.0
f1
MAR MC-SAT
0.14
1.0
0.14
0.95
0.14
0.49
0.72
0.43
SN
0.84
0.84
0.92
0.88
?=1
accuracy
MAR MC-SAT
SN
0.36
0.81
0.75
0.36
0.77
0.74
0.55
0.66
0.85
0.73
0.59
0.78
f1
MAR
0.10
0.11
0.32
0.55
MC-SAT
0.69
0.61
0.47
0.42
Table 2: Accuracy and F-score results when models were trained and tested on chains with the same
length
(a) ? = 0
(b) ? = 1
Figure 1: F-score of models trained and tested on the same length of data
0.55, and false if its probability was less than 0.45; otherwise we considered it as a misclassification.
Prediction of Smokes was impossible in the generated data set, because the data generation was
symmetric w.r.t to smoking and not smoking, and from the observations we could only tell that
certain pairs of people have similar or different smoking habits, but not who smokes and who does
not. The three methods we compared were (i) particle filtering in the SN-DMLN model (SN), (ii) the
approximate online inference algorithm of [2], which projects the inferred distribution of the random
variables at the previous slice to the product of their marginals, and incorporates this information
into a two slice MLN to infer the probabilities at the next slice (we re-implemented the algorithm
in PCE) (MAR), and (iii) using a general inference algorithm (MC-SAT [15]) for a CRF which is
always completely unrolled in every time step (UNR). In UNR and MAR the same CRF models
were used. The training of the SN-DMLN model took approximately for 120 minutes for all the test
cases, while for the CRF model, it took 120, 145, 175 and 240 minutes respectively. The inference
over the entire test set, took approximately 6 minutes for SN and MAR in every test case, while
UNR required 5, 8, 12 and 40 minutes for the different test cases. The accuracy and F-scores for the
different test cases are summarized in Table 2 and the F-scores are plotted in Fig. 1.
SN outperforms MAR, because as we see that in the knowledge base, MAR can only conclude that
people have the same or different smoking habits on the days when people hang out (every 5th time
step), and the marginal distributions of Smokes do not carry enough information about which pair
of people have different smoking habits, hence the quality of MAR?s prediction decreases on days
when people do not hang out. The performance of SN and MAR stays the same as we increase
the length of the chain while the performance of UNR degrades. This is most pronounced in the
deterministic case (? = 0). This can be explained by that MC-SAT requires more sampling steps to
maintain the same performance as the chain length increases.
To demonstrate that if we use the same number of particles in SN as number of samples in UNR,
the performance of SN stays approximately the same while the performance of UNR degrades over
time, we trained both the CRF and SN-DMLN on length 5 chains where both SN and UNR were
performing equally well and used test sets of different lengths up to length 150. F-scores are plotted
in Fig. 2.
We see from Fig. 2 that SN outperforms both UNR and MAR as the chain length increases. Moreover, UNR?s performance is clearly decreasing as the length of the chain increases.
6
Conclusion
In this paper, we explored the theoretical and practical questions of unrolling a sequential Markov
logic knowledge base into different probabilistic models. The theoretical issues arising in a CRF7
(a) ? = 0
(b) ? = 1
Figure 2: F-score of models trained and tested on different length of data
based MLN unrolling are a warning that unexpected results may occur if the observations are weakly
correlated with the hidden variables. We gave a qualitative justification why this phenomenon is
more of a theoretical concern in domains lacking deterministic constraints. We demonstrated that
the CRF based unrolling can be outperformed by a model that mixes directed and undirected components (the proposed model does not suffer from any of the theoretical weaknesses, nor from the
label-bias problem).
From a more practical point of view, we showed that our proposed model provides computational
savings, when the data has to be processed in a sequential manner. These savings are due to that
we do not have to unroll a new CRF at every time step, or estimate a partition function which is responsible for normalizing the product of clique potentials appearing in two consecutive slices. The
previously used approximate inference methods in dynamic MLNs either relied on belief propagation or assumed that approximating the distribution at every time step by the product of the marginals
would not cause any error. It is important to note that, although in our paper we focused on marginal
inference, finding the most likely state sequence could be done using the generated particles. Although the conditional log-likelihood of the training data in our model may be non-concave so that
hill climbing based approaches could fail to settle in a global maximum, we proposed a heuristic
for weight learning and demonstrated that it could train our model so that it performs as well as
conditional random fields. Although training the mixed model might have a higher computational
cost than training a conditional random field, but this cost is amortized over time, since in applications inference is performed many times, while weight learning only once. Designing more scalable
weight learning algorithms is among our future goals.
7
Acknowledgments
We thank Daniel Gildea for his insightful comments.
This research was supported by grants from ARO (W991NF-08-1-0242), ONR (N00014-11-10417),
NSF (IIS-1012017), DOD (N00014-12-C-0263), and a gift from Intel.
References
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tested:3 phenomenon:2 correlated:2 |
3,927 | 4,555 | Complex Inference in Neural Circuits with
Probabilistic Population Codes and Topic Models
Jeff Beck
Department of Brain and Cognitive Sciences
University of Rochester
[email protected]
Katherine Heller
Department of Statistical Science
Duke University
[email protected]
Alexandre Pouget
Department of Neuroscience
University of Geneva
[email protected]
Abstract
Recent experiments have demonstrated that humans and animals typically reason probabilistically about their environment. This ability requires a neural code
that represents probability distributions and neural circuits that are capable of
implementing the operations of probabilistic inference. The proposed probabilistic population coding (PPC) framework provides a statistically efficient neural
representation of probability distributions that is both broadly consistent with
physiological measurements and capable of implementing some of the basic operations of probabilistic inference in a biologically plausible way. However, these
experiments and the corresponding neural models have largely focused on simple
(tractable) probabilistic computations such as cue combination, coordinate transformations, and decision making. As a result it remains unclear how to generalize this
framework to more complex probabilistic computations. Here we address this short
coming by showing that a very general approximate inference algorithm known
as Variational Bayesian Expectation Maximization can be naturally implemented
within the linear PPC framework. We apply this approach to a generic problem
faced by any given layer of cortex, namely the identification of latent causes of
complex mixtures of spikes. We identify a formal equivalent between this spike
pattern demixing problem and topic models used for document classification, in
particular Latent Dirichlet Allocation (LDA). We then construct a neural network
implementation of variational inference and learning for LDA that utilizes a linear
PPC. This network relies critically on two non-linear operations: divisive normalization and super-linear facilitation, both of which are ubiquitously observed in
neural circuits. We also demonstrate how online learning can be achieved using a
variation of Hebb?s rule and describe an extension of this work which allows us to
deal with time varying and correlated latent causes.
1
Introduction to Probabilistic Inference in Cortex
Probabilistic (Bayesian) reasoning provides a coherent and, in many ways, optimal framework for
dealing with complex problems in an uncertain world. It is, therefore, somewhat reassuring that
behavioural experiments reliably demonstrate that humans and animals behave in a manner consistent
with optimal probabilistic reasoning when performing a wide variety of perceptual [1, 2, 3], motor
[4, 5, 6], and cognitive tasks[7]. This remarkable ability requires a neural code that represents
probability distribution functions of task relevant stimuli rather than just single values. While there
1
are many ways to represent functions, Bayes rule tells us that when it comes to probability distribution
functions, there is only one statistically optimal way to do it. More precisely, Bayes Rule states
that any pattern of activity, r, that efficiently represents a probability distribution over some task
relevant quantity s, must satisfy the relationship p(s|r) ? p(r|s)p(s), where p(r|s) is the stimulus
conditioned likelihood function that specifies the form of neural variability, p(s) gives the prior belief
regarding the stimulus, and p(s|r) gives the posterior distribution over values of the stimulus, s given
the representation r . Of course, it is unlikely that the nervous system consistently achieves this level
of optimality. None-the-less, Bayes rule suggests the existence of a link between neural variability as
characterized by the likelihood function p(r|s) and the state of belief of a mature statistical learning
machine such as the brain.
The so called Probabilistic Population Coding (or PPC) framework[8, 9, 10] takes this link seriously
by proposing that the function encoded by a pattern of neural activity r is, in fact, the likelihood
function p(r|s). When this is the case, the precise form of the neural variability informs the nature
of the neural code. For example, the exponential family of statistical models with linear sufficient
statistics has been shown to be flexible enough to model the first and second order statistics of in vivo
recordings in awake behaving monkeys[9, 11, 12] and anesthetized cats[13]. When the likelihood
function is modeled in this way, the log posterior probability over the stimulus is linearly encoded by
neural activity, i.e.
log p(s|r) = h(s) ? r ? log Z(r)
(1)
Here, the stimulus dependent kernel, h(s), is a vector of functions of s, the dot represents a standard
dot product, and Z(r) is the partition function which serves to normalize the posterior. This log
linear form for a posterior distribution is highly computationally convenient and allows for evidence
integration to be implemented via linear operations on neural activity[14, 8].
Proponents of this kind of linear PPC have demonstrated how to build biologically plausible neural networks capable of implementing the operations of probabilistic inference that are needed to
optimally perform the behavioural tasks listed above. This includes, linear PPC implementations
of cue combination[8], evidence integration over time, maximum likelihood and maximum a posterior estimation[9], coordinate transformation/auditory localization[10], object tracking/Kalman
filtering[10], explaining away[10], and visual search[15]. Moreover, each of these neural computations has required only a single recurrently connected layer of neurons that is capable of just two
non-linear operations: coincidence detection and divisive normalization, both of which are widely
observed in cortex[16, 17].
Unfortunately, this research program has been a piecemeal effort that has largely proceeded by
building neural networks designed deal with particular problems. As a result, there have been no
proposals for a general principle by which neural network implementations of linear PPCs might
be generated and no suggestions regarding how to deal with complex (intractable) problems of
probabilistic inference.
In this work, we will partially address this short coming by showing that Variation Bayesian Expectation Maximization (VBEM) algorithm provides a general scheme for approximate inference and
learning with linear PPCs. In section 2, we briefly review the VBEM algorithm and show how it
naturally leads to a linear PPC representation of the posterior as well as constraints on the neural
network dynamics which build that PPC representation. Because this section describes the VB-PPC
approach rather abstractly, the remainder of the paper is dedicated to concrete applications. As a
motivating example, we consider the problem of inferring the concentrations of odors in an olfactory
scene from a complex pattern of spikes in a population of olfactory receptor neurons (ORNs). In
section 3, we argue that this requires solving a spike pattern demixing problem which is indicative of
the generic problem faced by many layers of cortex. We then show that this demixing problem is
equivalent to the problem addressed by a class of models for text documents know as probabilistic
topic models, in particular Latent Dirichlet Allocation or LDA[18].
In section 4, we apply the VB-PPC approach to build a neural network implementation of probabilistic
inference and learning for LDA. This derivation shows that causal inference with linear PPC?s also
critically relies on divisive normalization. This result suggests that this particular non-linearity may
be involved in very general and fundamental probabilistic computation, rather than simply playing a
role in gain modulation. In this section, we also show how this formulation allows for a probabilistic
treatment of learning and show that a simple variation of Hebb?s rule can implement Bayesian
learning in neural circuits.
2
We conclude this work by generalizing this approach to time varying inputs by introducing the
Dynamic Document Model (DDM) which can infer short term fluctuations in the concentrations of
individual topics/odors and can be used to model foraging and other tracking tasks.
2
Variational Bayesian Inference with linear Probabilistic Population Codes
Variational Bayesian (VB) inference refers to a class of deterministic methods for approximating the
intractable integrals which arise in the context of probabilistic reasoning. Properly implemented it
can result a fast alternative to sampling based methods of inference such as MCMC[19] sampling.
Generically, the goal of any Bayesian inference algorithm is to infer a posterior distribution over
behaviourally relevant latent variables Z given observations X and a generative model which specifies
the joint distribution p(X, ?, Z). This task is confounded by the fact that the generative model
includes latent parameters ? which must be marginalized out, i.e. we wish to compute,
Z
p(Z|X) ?
p(X, ?, Z)d?
(2)
When the number of latent parameters is large this integral can be quite unwieldy. The VB algorithms
simplify this marginalization by approximating the complex joint distribution over behaviourally
relevant latents and parameters, p(?, Z|X), with a distribution q(?, Z) for which integrals of this
form are easier to deal with in some sense. There is some art to choosing the particular form for the
approximating distribution to make the above integral tractable, however, a factorized approximation
is common, i.e. q(?, Z) = q? (?)qZ (Z).
Regardless, for any given observation X, the approximate posterior is found by minimizing the
Kullback-Leibler divergence between q(?, Z) and p(?, Z|X). When a factorized posterior is
assumed, the Variational Bayesian Expectation Maximization (VBEM) algorithm finds a local
minimum of the KL divergence by iteratively updating, q? (?) and qZ (Z) according to the scheme
n
log q?
(?) ? hlog p(X, ?, Z)iqn (Z)
Z
and
n+1
log qZ
(Z) ? hlog p(X, ?, Z)iqn (?)
?
(3)
Here the brackets indicate an expected value taken with respect to the subscripted probability
distribution function and the tilde indicates equality up to a constant which is independent of ? and Z.
The key property to note here is that the approximate posterior which results from this procedure is in
an exponential family form and is therefore representable by a linear PPC (Eq. 1). This feature allows
for the straightforward construction of networks which implement the VBEM algorithm with linear
PPC?s in the following way. If rn? and rnZ are patterns of activity that use a linear PPC representation
of the relevant posteriors, then
n
log q?
(?) ? h? (?) ? rn?
and
n+1
log qZ
(Z) ? hZ (Z) ? rn+1
Z .
(4)
Here the stimulus dependent kernels hZ (Z) and h? (?) are chosen so that their outer product results
in a basis that spans the function space on Z ? ? given by log p(X, ?, Z) for every X. This choice
guarantees that there exist functions f? (X, rnZ ) and fZ (X, rn? ) such that
rn? = f? (X, rnZ )
and rn+1
= fZ (X, rn? )
Z
(5)
satisfy Eq. 3. When this is the case, simply iterating the discrete dynamical system described by Eq.
5 until convergence will find the VBEM approximation to the posterior. This is one way to build a
neural network implementation of the VB algorithm. However, its not the only way. In general, any
dynamical system which has stable fixed points in common with Eq. 5 can also be said to implement
the VBEM algorithm. In the example below we will take advantage of this flexibility in order to build
biologically plausible neural network implementations.
3
Response!
to Mixture !
of Odors!
Single
Odor
Response
Cause Intensity
Figure 1: (Left) Each cause (e.g. coffee) in isolation results in a pattern of neural activity (top). When
multiple causes contribute to a scene this results in an overall pattern of neural activity which is a
mixture of these patterns weighted by the intensities (bottom). (Right) The resulting pattern can be
represented by a raster, where each spike is colored by its corresponding latent cause.
3
Probabilistic Topic Models for Spike Train Demixing
Consider the problem of odor identification depicted in Fig. 1. A typical mammalian olfactory
system consists of a few hundred different types of olfactory receptor neurons (ORNs), each of which
responds to a wide range of volatile chemicals. This results in a highly distributed code for each
odor. Since, a typical olfactory scene consists of many different odors at different concentrations, the
pattern of ORN spike trains represents a complex mixture. Described in this way, it is easy to see that
the problem faced by early olfactory cortex can be described as the task of demixing spike trains to
infer latent causes (odor intensities).
In many ways this olfactory problem is a generic problem faced by each cortical layer as it tries to
make sense of the activity of the neurons in the layer below. The input patterns of activity consist of
spikes (or spike counts) labeled by the axons which deliver them and summarized by a histogram
which indicates how many spikes come from each input neuron. Of course, just because a spike came
from a particular neuron does not mean that it had a particular cause, just as any particular ORN
spike could have been caused by any one of a large number of volatile chemicals. Like olfactory
codes, cortical codes are often distributed and multiple latent causes can be present at the same time.
Regardless, this spike or histogram demixing problem is formally equivalent to a class of demixing
problems which arise in the context of probabilistic topic models used for document modeling. A
simple but successful example of this kind of topic model is called Latent Dirichlet Allocation (LDA)
[18]. LDA assumes that word order in documents is irrelevant and, therefore, models documents
as histograms of word counts. It also assumes that there are K topics and that each of these topics
appears in different proportions in each document, e.g. 80% of the words in a document might be
concerned with coffee and 20% with strawberries. Words from a given topic are themselves drawn
from a distribution over words associated with that topic, e.g. when talking about coffee you have a
5% chance of using the word ?bitter?. The goal of LDA is to infer both the distribution over topics
discussed in each document and the distribution of words associated with each topic. We can map
the generative model for LDA onto the task of spike demixing in cortex by letting topics become
latent causes or odors, words become neurons, word occurrences become spikes, word distributions
associated with each topic become patterns of neural activity associated with each cause, and different
documents become the observed patterns of neural activity on different trials. This equivalence is
made explicit in Fig. 2 which describes the standard generative model for LDA applied to documents
on the left and mixtures of spikes on the right.
4
LDA Inference and Network Implementation
In this section we will apply the VB-PPC formulation to build a biologically plausible network
capable of approximating probabilistic inference for spike pattern demixing. For simplicity, we will
use the equivalent Gamma-Poisson formulation of LDA which directly models word and topic counts
4
1. For each topic k = 1, . . . , K,
(a) Distribution over words
?k ? Dirichlet(?0 )
2. For document d = 1, . . . , D,
(a) Distribution over topics
?d ? Dirichlet(?0 )
(b) For word m = 1, . . . , ?d
i. Topic assignment
zd,m ? Multinomial(?d )
ii. Word assignment
?d,m ? Multinomial(?zm )
1. For latent cause k = 1, . . . , K,
(a) Pattern of neural activity
?k ? Dirichlet(?0 )
2. For scene d = 1, . . . , D,
(a) Relative intensity of each cause
?d ? Dirichlet(?0 )
(b) For spike m = 1, . . . , ?d
i. Cause assignment
zd,m ? Multinomial(?d )
ii. Neuron assignment
?d,m ? Multinomial(?zm )
Figure 2: (Left) The LDA generative model in the context of document modeling. (Right) The
corresponding LDA generative model mapped onto the problem of spike demixing. Text related
attributes on the left, in red, have been replaced with neural attributes on the right, in green.
rather than topic assignments. Specifically, we define, Rd,j to be the number of times neuron j fires
during trial d. Similarly, we let Nd,j,k to be the number of times a spike in neuron j comes from
cause k in trial d. These new variables play the roles of the cause and neuron assignment variables,
zd,m and ?d,m by simply
P counting them up. If we let cd,k be an un-normalized intensity of cause j
such that ?d,k = cd,k / k cd,k then the generative model,
Rd,j =
P
k
Nd,j,k
Nd,j,k ? Poisson(?j,k cd,k )
cd,k ? Gamma(?k0 , C ?1 ).
(6)
is equivalent to the topic models described above. Here the parameter C is a scale parameter
which sets the expected total number of spikes from the population on each trial. Note that, the
problem of inferring the wj,k and cd,k is a non-negative matrix factorization problem similar to
that considered by Lee and Seung[20]. The primary difference is that, here, we are attempting to
infer a probability distribution over these quantities rather than maximum likelihood estimates. See
supplement for details. Following the prescription laid out in section 2, we approximate the posterior
over latent variables given a set of input patterns, Rd , d = 1, . . . , D, with a factorized distribution of the form, qN (N)q
c (c)q? (?). This results in marginal posterior distributions q (?:,k |?:,k ),
?1
q cd,k |?d,k , C + 1 ), and q (Nd,j,: | log pd,j,: , Rd,i ) which are Dirichlet, Gamma, and Multinomial respectively. Here, the parameters ?:,k , ?d,k , and log pd,j,: are the natural parameters of these
distributions. The VBEM update algorithm yields update rules for these parameters which are
summarized in Fig. 3 Algorithm1.
Algorithm 1: Batch VB updates
1: while ?j,k not converged do
2:
for d = 1, ? ? ? , D do
3:
while pd,j,k , ?d,kP
not converged do
4:
?d,k ? ?0 + j Rd,j pd,j,k
5:
pd,j,k ?
Algorithm 2: Online VB updates
1: for d = 1, ? ? ? , D do
2:
reinitialize pj,k , ?k ?j, k
3:
while pj,k , ?k not
P converged do
4:
?k ? ?0 + j Rd,j pj,k
5:
pj,k ?
exp (?(?j,k )??(?
?k )) exp ?(?k )
P
?i )) exp ?(?i )
i exp (?(?j,i )??(?
exp (?(?j,k )??(?
?k )) exp ?(?d,k )
P
?i )) exp ?(?d,i )
i exp (?(?j,i )??(?
end while
?j,k ?
(1 ? dt)?j,k + dt(? 0 + Rd,j pj,k )
8: end for
6:
end while
7:
end for
P
8:
?j,k = ? 0 + d Rd,j pd,j,k
9: end while
6:
7:
P
Figure 3: Here ??k = j ?j,k and ?(x) is the digamma function so that exp ?(x) is a smoothed
threshold linear function.
Before we move on to the neural network implementation, note that this standard formulation of
variational inference for LDA utilizes a batch learning scheme that is not biologically plausible.
Fortunately, an online version of this variational algorithm was recently proposed and shown to give
5
superior results when compared to the batch learning algorithm[21]. This algorithm replaces the
sum over d in update equation for ?j,k with an incremental update based upon only the most recently
observed pattern of spikes. See Fig. 3 Algorithm 2.
4.1
Neural Network Implementation
Recall that the goal was to build a neural network that implements the VBEM algorithm for the
underlying latent causes of a mixture of spikes using a neural code that represents the posterior
distribution via a linear PPC. A linear PPC represents the natural parameters of a posterior distribution
via a linear operation on neural activity. Since the primary quantity of interest here is the posterior
distribution over odor concentrations, qc (c|?), this means that we need a pattern of activity r? which
is linearly related to the ?k ?s in the equations above. One way to accomplish this is to simply assume
that the firing rates of output neurons are equal to the positive valued ?k parameters.
Fig. 4 depicts the overall network architecture. Input patterns of activity, R, are transmitted to the
synapses of a population of output neurons which represent the ?k ?s. The output activity is pooled to
? j , given the output layer?s
form an un-normalized prediction of the activity of each input neuron, R
current state of belief about the latent causes of the Rj . The activity at each synapse targeted by input
neuron j is then inhibited divisively by this prediction. This results in a dendrite that reports to the
?j,k , which represents the fraction of unexplained spikes from input neuron j that
soma a quantity, N
could be explained by latent cause k. A continuous time dynamical system with this feature and the
property that it shares its fixed points with the LDA algorithm is given by
d ?
Nj,k
dt
d
?k
dt
?j N
?j,k
= wj,k Rj ? R
=
exp (? (?
?k )) (?0 ? ?k ) + exp (? (?k ))
(7)
X
?j,k
N
(8)
i
? j = P wj,k exp (? (?k )), and wj,k = exp (? (?j,k )). Note that, despite its form, it is
where R
k
Eq. 7 which implements the required divisive normalization operation since, in the steady state,
?j,k = wj,k Rj /R
?j .
N
Regardless, this network has a variety of interesting properties that align well with biology. It predicts
that a balance of excitation and inhibition is maintained in the dendrites via divisive normalization
and that the role of inhibitory neurons is to predict the input spikes which target individual dendrites.
It also predicts superlinear facilitation. Specifically, the final term on the right of Eq. 8 indicates
that more active cells will be more sensitive to their dendritic inputs. Alternatively, this could be
implemented via recurrent excitation at the population level. In either case, this is the mechanism by
which the network implements a sparse prior on topic concentrations and stands in stark contrast to the
winner take all mechanisms which rely on competitive mutual inhibition mechanisms. Additionally,
the ??j in Eq. 8 represents a cell wide ?leak? parameter that indicates that the total leak should be
roughly proportional to the sum total weight of the synapses which drive the neuron. This predicts
that cells that are highly sensitive to input should also decay back to baseline more quickly. This
implementation also predicts Hebbian learning of synaptic weights. To observe this fact, note that the
online update rule for the ?j,k parameters can be implemented by simply correlating the activity at
?j,k with activity at the soma ?j via the equation:
each synapse, N
?L
d
?j,k exp ? (?k )
wj,k = exp (? (?
?k )) (?0 ? 1/2 ? wj,k ) + N
dt
(9)
where ?L is a long time constant for learning and we have used the fact that exp (? (?jk )) ? ?jk ?1/2
for x > 1. For a detailed derivation see the supplementary material.
5
Dynamic Document Model
LDA is a rather simple generative model that makes several unrealistic assumptions about mixtures
of sensory and cortical spikes. In particular, it assumes both that there are no correlations between the
6
Targeted
Divisive Normalization
Targeted
Divisive Normalization
?j
Ri
Input
Neurons
Recurrent
Connections
?
?
-1
-1
? ?j
Nij
Ri
Synapses
Output
Neurons
Figure 4: The LDA network model. Dendritically targeted inhibition is pooled
from the activity of all neurons in the
output layer and acts divisively.
? jj'
Nij
Input
Neurons
Synapses
Output
Neurons
Figure 5: DDM network model also includes recurrent connections which target the soma with both
a linear excitatory signal and an inhibitory signal
that also takes the form of a divisive normalization.
intensities of latent causes and that there are no correlations between the intensities of latent causes
in temporally adjacent trials or scenes. This makes LDA a rather poor computational model for a
task like olfactory foraging which requires the animal to track the rise a fall of odor intensities as it
navigates its environment. We can model this more complicated task by replacing the static cause or
odor intensity parameters with dynamic odor intensity parameters whose behavior is governed by an
exponentiated Ornstein-Uhlenbeck process with drift and diffusion matrices given by (? and ?D ).
We call this variant of LDA the Dynamic Document Model (DDM) as it could be used to model
smooth changes in the distribution of topics over the course of a single document.
5.1
DDM Model
Thus the generative model for the DDM is as follows:
1. For latent cause k = 1, . . . , K,
(a) Cause distribution over spikes ?k ? Dirichlet(?0 )
2. For scene t = 1, . . . , T ,
(a) Log intensity of causes c(t) ? Normal(?ct?1 , ?D )
(b) Number of spikes in neuron j resulting from cause k,
Nj,k (t) ? Poisson(?j,k exp ck (t))
P
(c) Number of spikes in neuron j, Rj (t) = k Nj,k (t)
This model bears many similarities to the Correlated and Dynamic topic models[22], but models
dynamics over a short time scale, where the dynamic relationship (?, ?D ) is important.
5.2
Network Implementation
Once again the quantity of interest is the current distribution of latent causes, p(c(t)|R(? ), ? = 0..T ).
If no spikes occur then no evidence is presented and posterior inference over c(t) is simply given
by an undriven Kalman filter with parameters (?, ?D ). A recurrent neural network which uses a
linear PPC to encode a posterior that evolves according to a Kalman filter has the property that neural
responses are linearly related to the inverse covariance matrix of the posterior as well as that inverse
covariance matrix times the posterior mean. In the absence of evidence, it is easy to show that these
quantities must evolve according to recurrent dynamics which implement divisive normalization[10].
Thus, the patterns of neural activity which linearly encode them must do so as well. When a new spike
arrives, optimal inference is no longer possible and a variational approximation must be utilized. As
is shown in the supplement, this variational approximation is similar to the variational approximation
used for LDA. As a result, a network which can divisively inhibit its synapses is able to implement
approximate Bayesian inference. Curiously, this implies that the addition of spatial and temporal
correlations to the latent causes adds very little complexity to the VB-PPC network implementation
of probabilistic inference. All that is required is an additional inhibitory population which targets the
somata in the output population. See Fig. 5.
7
Natural Parameters
200
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Network Estimate
Network Estimate
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VBEM Estimate
VBEM Estimate
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0
Figure 6: (Left) Neural network approximation to the natural parameters of the posterior distribution
over topics (the ??s) as a function of the VBEM estimate of those same parameters for a variety of
?documents?. (Center) Same as left, but for the natural parameters of the DDM (i.e the entries of the
matrix ??1 (t) and ??1 ?(t) of the distribution over log topic intensities. (Right) Three example
traces for cause intensity in the DDM. Black shows true concentration, blue and red (indistinguishable)
show MAP estimates for the network and VBEM algorithms.
6
Experimental Results
We compared the PPC neural network implementations of the variational inference with the standard
VBEM algorithm. This comparison is necessary because the two algorithms are not guaranteed
to converge to the same solution due to the fact that we only required that the neural network
dynamics have the same fixed points as the standard VBEM algorithm. As a result, it is possible
for the two algorithms to converge to different local minima of the KL divergence. For the network
implementation of LDA we find good agreement between the neural network and VBEM estimates
of the natural parameters of the posterior. See Fig. 6(left) which shows the two algorithms estimates
of the shape parameter of the posterior distribution over topic (odor) concentrations (a quantity which
is proportional to the expected concentration). This agreement, however, is not perfect, especially
when posterior predicted concentrations are low. In part, this is due to the fact we are presenting the
network with difficult inference problems for which the true posterior distribution over topics (odors)
is highly correlated and multimodal. As a result, the objective function (KL divergence) is littered
with local minima. Additionally, the discrete iterations of the VBEM algorithm can take very large
steps in the space of natural parameters while the neural network implementation cannot. In contrast,
the network implementation of the DDM is in much better agreement with the VBEM estimation.
See Fig. 6(right). This is because the smooth temporal dynamics of the topics eliminate the need for
the VBEM algorithm to take large steps. As a result, the smooth network dynamics are better able to
accurately track the VBEM algorithms output. For simulation details please see the supplement.
7
Discussion and Conclusion
In this work we presented a general framework for inference and learning with linear Probabilistic
Population codes. This framework takes advantage of the fact that the Variational Bayesian Expectation Maximization algorithm generates approximate posterior distributions which are in an
exponential family form. This is precisely the form needed in order to make probability distributions
representable by a linear PPC. We then outlined a general means by which one can build a neural
network implementation of the VB algorithm using this kind of neural code. We applied this VB-PPC
framework to generate a biologically plausible neural network for spike train demixing. We chose
this problem because it has many of the features of the canonical problem faced by nearly every
layer of cortex, i.e. that of inferring the latent causes of complex mixtures of spike trains in the layer
below. Curiously, this very complicated problem of probabilistic inference and learning ended up
having a remarkably simple network solution, requiring only that neurons be capable of implementing
divisive normalization via dendritically targeted inhibition and superlinear facilitation. Moreover,
we showed that extending this approach to the more complex dynamic case in which latent causes
change in intensity over time does not substantially increase the complexity of the neural circuit.
Finally, we would like to note that, while we utilized a rate coding scheme for our linear PPC, the
basic equations would still apply to any spike based log probability codes such as that considered
Beorlin and Deneve[23].
8
References
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9
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3,928 | 4,556 | Learned Prioritization for
Trading Off Accuracy and Speed?
Jiarong Jiang?
Adam Teichert?
Hal Daum?e III?
Jason Eisner?
?
?
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21218
{teichert,eisner}@jhu.edu
Department of Computer Science
University of Maryland
College Park, MD 20742
{jiarong,hal}@umiacs.umd.edu
Abstract
Users want inference to be both fast and accurate, but quality often comes at
the cost of speed. The field has experimented with approximate inference algorithms that make different speed-accuracy tradeoffs (for particular problems and
datasets). We aim to explore this space automatically, focusing here on the case of
agenda-based syntactic parsing [12]. Unfortunately, off-the-shelf reinforcement
learning techniques fail to learn good policies: the state space is simply too large
to explore naively. An attempt to counteract this by applying imitation learning
algorithms also fails: the ?teacher? follows a far better policy than anything in our
learner?s policy space, free of the speed-accuracy tradeoff that arises when oracle information is unavailable, and thus largely insensitive to the known reward
functfion. We propose a hybrid reinforcement/apprenticeship learning algorithm
that learns to speed up an initial policy, trading off accuracy for speed according
to various settings of a speed term in the loss function.
1
Introduction
The nominal goal of predictive inference is to achieve high accuracy. Unfortunately, high accuracy
often comes at the price of slow computation. In practice one wants a ?reasonable? tradeoff between
accuracy and speed. But the definition of ?reasonable? varies with the application. Our goal is to
optimize a system with respect to a user-specified speed/accuracy tradeoff, on a user-specified data
distribution. We formalize our problem in terms of learning priority functions for generic inference
algorithms (Section 2).
Much research in natural language processing (NLP) has been dedicated to finding speedups for exact or approximate computation in a wide range of inference problems including sequence tagging,
constituent parsing, dependency parsing, and machine translation. Many of the speedup strategies in
the literature can be expressed as pruning or prioritization heuristics. Prioritization heuristics govern the order in which search actions are taken while pruning heuristics explicitly dictate whether
particular actions should be taken at all. Examples of prioritization include A? [13] and Hierarchical A? [19] heuristics, which, in the case of agenda-based parsing, prioritize parse actions so as
to reduce work while maintaining the guarantee that the most likely parse is found. Alternatively,
coarse-to-fine pruning [21], classifier-based pruning [23], [22] beam-width prediction [3], etc can
result in even faster inference if a small amount of search error can be tolerated.
Unfortunately, deciding which techniques to use for a specific setting can be difficult: it is impractical to ?try everything.? In the same way that statistical learning has dramatically improved
the accuracy of NLP applications, we seek to develop statistical learning technology that can dramatically improve their speed while maintaining tolerable accuracy. By combining reinforcement
learning and imitation learning methods, we develop an algorithm that can successfully learn such
a tradeoff in the context of constituency parsing. Although this paper focuses on parsing, we expect the approach to transfer to prioritization in other agenda-based algorithms, such as machine
translation and residual belief propagation. We give a broader discussion of this setting in [8].
?
This material is based upon work supported by the National Science Foundation under Grant No. 0964681.
1
2
Priority-based Inference
Inference algorithms in NLP (e.g. parsers, taggers, or translation systems) as well as more broadly
in artificial intelligence (e.g., planners) often rely on prioritized exploration. For concreteness, we
describe inference in the context of parsing, though it is well known that this setting captures all the
essential structure of a much larger family of ?deductive inference? problems [12, 9].
2.1
Prioritized Parsing
Given a probabilistic context-free grammar, one approach to inferring the best parse tree for a given
sentence is to build the tree from the bottom up by dynamic programming, as in CKY [29]. When a
prospective constituent such as ?NP from 3 to 8? is built, its Viterbi inside score is the log-probability
of the best known subparse that matches that description.1
A standard extension of the CKY algorithm [12] uses an agenda?a priority queue of constituents
built so far?to decide which constituent is most promising to extend next, as detailed in section 2.2
below. The success of the inference algorithm in terms of speed and accuracy hinge on its ability to
prioritize ?good? actions before ?bad? actions. In our context, a constituent is ?good? if it somehow
leads to a high accuracy solution, quickly.
Running Example 1. Either CKY or an agenda-based parser that prioritizes by Viterbi inside score will
find the highest-scoring parse. This achieves a percentage accuracy of 93.3, given the very large grammar
and experimental conditions described in Section 6. However, the agenda-based parser is over an order of
magnitude faster than CKY (wall clock time) because it stops as soon as it finds a parse, without building
further constituents. With mild pruning according to Viterbi inside score, the accuracy remains 93.3 and the
speed triples. With more aggressive pruning, the accuracy drops to 92.0 and the speed triples again.
Our goal is to learn a prioritization function that satisfies this condition. In order to operationalize
this approach, we need to define the test-time objective function we wish to optimize; we choose a
simple linear interpolation of accuracy and speed:
quality = accuracy ? ? ? time
(1)
where we can choose a ? that reflects our true preferences. The goal of ? is to encode ?how much
more time am I willing to spend to achieve an additional unit of accuracy?? In this paper, we consider
a very simple notion of time: the number of constituents popped from/pushed into the agenda during
inference, halting inference as soon as the parser pops its first complete parse.
When considering how to optimize the expectation of Eq (1) over test data, several challenges
present themselves. First, this is a sequential decision process: the parsing decisions made at a
given time may affect both the availability and goodness of future decisions. Second, the parser?s
total runtime and accuracy on a sentence are unknown until parsing is complete, making this an
instance of delayed reward. These considerations lead us to formulate this problem as a Markov
Decision Process (MDP), a well-studied model of decision processes.
2.2
Inference as a Markov Decision Process
A Markov Decision Process (MDP) is a formalization of a memoryless search process. An MDP
consists of a state space S, an action space A, and a transition function T . An agent in an MDP
observes the current state s ? S and chooses an action a ? A. The environment responds by
transitioning to a state s0 ? S, sampled from the transition distribution T (s0 | s, a). The agent then
observes its new state and chooses a new action. An agent?s policy ? describes how the (memoryless) agent chooses an action based on its current state, where ? is either a deterministic function of
the state (i.e., ?(s) 7? a) or a stochastic distribution over actions (i.e., ?(a | s)).
For parsing, the state is the full current chart and agenda (and is astronomically large: roughly 1017
states for average sentences). The agent controls which item (constituent) to ?pop? from the agenda.
The initial state has an agenda consisting of all single-word constituents, and an empty chart of
previously popped constituents. Possible actions correspond to items currently on the agenda. When
the agent chooses to pop item y, the environment deterministically adds y to the chart, combines y
as licensed by the grammar with adjacent items z in the chart, and places each resulting new item x
1
E.g., the maximum log-probability of generating some tree whose fringe is the substring spanning words
(3,8], given that NP (noun phrase) is the root nonterminal. This is the total log-probability of rules in the tree.
2
on the agenda. (Duplicates in the chart or agenda are merged: the one of highest Viterbi inside score
is kept.) The only stochasticity is the initial draw of a new sentence to be parsed.
We are interested in learning a deterministic policy that always pops the highest-priority available
action. Thus, learning a policy corresponds to learning a priority function. We define the priority
of action a in state s as the dot product of a feature vector ?(a, s) with the weight vector ?; our
features are described in Section 2.3. Formally, our policy is
?? (s) = arg max ? ? ?(a, s)
(2)
a
An admissible policy in the sense of A? search [13] would guarantee that we always return the parse
of highest Viterbi inside score?but we do not require this, instead aiming to optimize Eq (1).
2.3
Features for Prioritized Parsing
We use the following simple features to prioritize a possible constituent. (1) Viterbi inside score; (2)
constituent touches start of sentence; (3) constituent touches end of sentence; (4) constituent length;
length
(5) constituent
sentence length ; (6) log p(constituent label | prev. word POS tag) and log p(constituent label | next
word POS tag), where the part-of-speech (POS) tag of w is taken to be arg maxt p(w | t) under the
grammar; (7) 12 features indicating whether the constituent?s {preceding, following, initial} word
starts with an {uppercase, lowercase, number, symbol} character; (8) the 5 most positive and 5 most
negative punctuation features from [14], which consider the placement of punctuation marks within
the constituent.
The log-probability features (1), (6) are inspired by work on figures of merit for agenda-based parsing [4], while case and punctuation patterns (7), (8) are inspired by structure-free parsing [14].
3
Reinforcement Learning
Reinforcement learning (RL) provides a generic solution to solving learning problems with delayed
reward [25]. The reward function takes a state of the world s and an agent?s chosen action a and
returns a real value r that indicates the ?immediate reward? the agent receives for taking that action.
In general the reward function may be stochastic, but in our case, it is deterministic: r(s, a) ? R.
The reward function we consider is:
r(s, a) =
acc(a) ? ? ? time(s) if a is a full parse tree
0
otherwise
(3)
Here, acc(a) measures the accuracy of the full parse tree popped by the action a (against a gold
standard) and time(s) is a user-defined measure of time. In words, when the parser completes
parsing, it receives reward given by Eq (1); at all other times, it receives no reward.
3.1
Boltzmann Exploration
At test time, the transition between states is deterministic: our policy always chooses the action a
that has highest priority in the current state s. However, during training, we promote exploration of
policy space by running with stochastic policies ?? (a | s). Thus, there is some chance of popping a
lower-priority action, to find out if it is useful and should be given higher-priority. In particular, we
use Boltzmann exploration to construct a stochastic policy with a Gibbs distribution. Our policy is:
1
1
exp
? ? ?(a, s) with Z(s) as the appropriate normalizing constant (4)
?? (a | s) =
Z(s)
temp
That is, the log-likelihood of action a at state s is an affine function of its priority. The temperature
temp controls the amount of exploration. As temp ? 0, ?? approaches the deterministic policy
in Eq (2); as temp ? ?, ?? approaches the uniform distribution over available actions. During
training, temp can be decreased to shift from exploration to exploitation.
A trajectory ? is the complete sequence of state/action/reward triples from parsing a single sentence.
As is common, we denote ? = hs0 , a0 , r0 , s1 , a1 , r1 , . . . , sT , aT , rT i, where: s0 is the starting state;
at is chosen by the agent by ?? (at | st ); rt = r(st , at ); and st+1 is drawn by the environment from
3
T (st+1 | st , at ), deterministically in our case. At a given temperature, the weight vector ? gives rise
to a distribution over trajectories and hence to an expected total reward:
" T
#
X
R = E? ??? [R(? )] = E? ???
rt .
(5)
t=0
where ? is a random trajectory chosen by policy ?? , and rt is the reward at step t of ? .
3.2
Policy Gradient
Given our features, we wish to find parameters that yield the highest possible expected reward. We
carry out this optimization using a stochastic gradient ascent algorithm known as policy gradient
[27, 26]. This operates by taking steps in the direction of ?? R:
?? E? [R(? )] = E? [
T
h
i
h
i
X
?? p? (? )
R(? )] = E? R(? )?? log p? (? ) = E? R(? )
?? log ?(at | st )
p? (? )
t=0
(6)
The expectation can be approximated by sampling trajectories. It also requires computing the gradient of each policy decision, which, by Eq (4), is:
!
X
1
?(at , st ) ?
?? (a0 | st )?(a0 , st )
(7)
?? log ?? (at | st ) =
temp
0
a ?A
Combining Eq (6) and Eq (7) gives the form of the gradient with respect to a single trajectory. The
policy gradient algorithm samples one trajectory (or several) according to the current ?? , and then
takes a gradient step according to Eq (6). This increases the probability of actions on high-reward
trajectories more than actions on low-reward trajectories.
Running Example 2. The baseline system from Running Example 1 always returns the target parse (the
complete parse with maximum Viterbi inside score). This achieves an accuracy of 93.3 (percent recall) and
speed of 1.5 mpops (million pops) on training data. Unfortunately, running policy gradient from this starting
point degrades speed and accuracy. Training is not practically feasible: even the first pass over 100 training
sentences (sampling 5 trajectories per sentence) takes over a day.
3.3
Analysis
One might wonder why policy gradient performed so poorly on this problem. One hypothesis is that it is the
fault of stochastic gradient descent: the optimization problem was too hard or our step sizes were chosen poorly.
To address this, we attempted an experiment where we added a ?cheating? feature to the model, which had a
value of one for constituents that should be in the final parse, and zero otherwise. Under almost every condition,
policy gradient was able to learn a near-optimal policy by placing high weight on this cheating feature.
An alternative hypothesis is overfitting to the training data. However, we were unable to achieve significantly
higher accuracy even when evaluating on our training data?indeed, even for a single train/test sentence.
The main difficulty with policy gradient is credit assignment: it has no way to determine which actions were
?responsible? for a trajectory?s reward. Without causal reasoning, we need to sample many trajectories in order
to distinguish which actions are reliably associated with higher-reward. This is a significant problem for us,
since the average trajectory length of an A?0 parser on a 15 word sentence is about 30,000 steps, only about 40
of which (less than 0.15%) are actually needed to successfully complete the parse optimally.
3.4
Reward Shaping
A classic approach to attenuating the credit assignment problem when one has some knowledge about the
domain is reward shaping [10]. The goal of reward shaping is to heuristically associate portions of the total
reward with specific time steps, and to favor actions that are observed to be soon followed by a reward, on the
assumption that they caused that reward.
If speed is measured by the number of popped items and accuracy is measured by labeled constituent recall of
the first-popped complete parse (compared to the gold-standard parse), one natural way to shape rewards is to
give an immediate penalty for the time incurred in performing the action while giving an immediate positive
reward for actions that build constituents of the gold parse. Since only some of the correct constituents built
may actually make it into the returned tree, we can correct for having ?incorrectly? rewarded the others by
penalizing the final action. Thus, the shaped reward:
4
?
? 1 ? ?(s, a) ? ?
1??
r?(s, a) =
? ??
if a pops a complete parse (causing the parser to halt and return a)
if a pops a labeled constituent that appears in the gold parse
otherwise
(8)
? is from Eq (1), penalizing the runtime of each step. 1 rewards a correct constituent. The correction ?(s, a)
is the number of correct constituents popped into the chart of s that were not in the first-popped parse a. It is
easy to see that for any trajectory ending in a complete parse, the total shaped and unshaped rewards along a
trajectory are equal (i.e. r(? ) = r?(? )).
We now modify the total reward to use temporal discounting. Let 0 ? ? ? 1 be a discount factor. When
rewards are discounted over time, the policy gradient becomes
"
? ? (? )] = E? ??
E? ??? [R
?
T
X
T
X
t=0
t0 =t
?
t0 ?t
!
r?t0
#
?? log ?? (at | st )
(9)
where r?t0 = r?(st0 , at0 ). When ? = 1, the gradient of the above turns out to be equivalent to Eq (6) [20, section
3.1], and therefore following the gradient is equivalent to policy gradient. When ? = 0, the parser gets only
immediate reward?and in general, a small ? assigns the credit for a local reward r?t0 mainly to actions at at
closely preceding times.
This gradient step can now achieve some credit assignment. If an action is on a good trajectory but occurs after
most of the useful actions (pops of correct constituents), then it does not receive credit for those previously
occurring actions. However, if it occurs before useful actions, it still does receive credit because we do not
know (without additional simulation) whether it was a necessary step toward those actions.
Running Example 3. Reward shaping helps significantly, but not enough to be competitive. As the parser
speeds up, training is about 10 times faster than before. The best setting (? = 0, ? = 10?6 ) achieves an
accuracy in the mid-70?s with only about 0.2 mpops. No settings were able to achieve higher accuracy.
4
Apprenticeship Learning
In reinforcement learning, an agent interacts with an environment and attempts to learn to maximize its reward
by repeating actions that led to high reward in the past. In apprenticeship learning, we assume access to a
collection of trajectories taken by an optimal policy and attempt to learn to mimic those trajectories. The
learner?s only goal is to behave like the teacher at every step: it does not have any notion of reward. In contrast,
the related task of inverse reinforcement learning/optimal control [17, 11] attempts to infer a reward function
from the teacher?s optimal behavior.
Many algorithms exist for apprenticeship learning. Some of them work by first executing inverse reinforcement
learning [11, 17] to induce a reward function and then feeding this reward function into an off-the-shelf reinforcement learning algorithm like policy gradient to learn an approximately optimal agent [1]. Alternatively,
one can directly learn to mimic an optimal demonstrator, without going through the side task of trying to induce
its reward function [7, 24].
4.1
Oracle Actions
With a teacher to help guide the learning process, we would like to explore more intelligently than Boltzmann
exploration, in particular, focusing on high-reward regions of policy space. We introduce oracle actions as a
guidance for areas to explore.
Ideally, oracle actions should lead to a maximum-reward tree. In training, we will identify oracle actions to be
those that build items in the maximum likelihood parse consistent with the gold parse. When multiple oracle
actions are available on the agenda, we will break ties according to the priority assigned by the current policy
(i.e., choose the oracle action that it currently likes best).
4.2
Apprenticeship Learning via Classification
Given a notion of oracle actions, a straightforward approach to policy learning is to simply train a classifier to
follow the oracle?a popular approach in incremental parsing [6, 5]. Indeed, this serves as the initial iteration
of the state-of-the-art apprenticeship learning algorithm, DAGGER [24].
We train a classifier as follows. Trajectories are generated by following oracle actions, breaking ties using the
initial policy (Viterbi inside score) when multiple oracle actions are available. These trajectories are incredibly
5
short (roughly double the number of words in the sentence). At each step in the trajectory, (st , at ), a classification example is generated, where the action taken by the oracle (at ) is considered the correct class and all
other available actions are considered incorrect. The classifier that we train on these examples is a maximum
entropy classifier, so it has exactly the same form as the Boltzmann exploration model (Eq (4)) but without the
temperature control. In fact, the gradient of this classifier (Eq (10)) is nearly identical to the policy gradient
(Eq (6)) except that ? is distributed differently and the total reward R(? ) does not appear: instead of mimicking
high-reward trajectories we now try to mimic oracle trajectories.
" T
!#
X
X
0
0
E? ???
?(at , st ) ?
?? (a | st )?(a , st )
(10)
t=0
a0 ?A
where ? ? denotes the oracle policy so at is the oracle action. The potential benefit of the classifier-based
approach over policy gradient with shaped rewards is increased credit assignment. In policy gradient with
reward shaping, an action gets credit for all future reward (though no past reward). In the classifier-based
approach, it gets credit for exactly whether or not it builds an item that is in the true parse.
Running Example 4. The classifier-based approach performs only marginally better than policy gradient
with shaped rewards. The best accuracy we can obtain is 76.5 with 0.19 mpops.
To execute the DAGGER algorithm, we would continue in the next iteration by following the trajectories learned
by the classifier and generating new classification examples on those states. Unfortunately, this is not computationally feasible due to the poor quality of the policy learned in the first iteration. Attempting to follow
the learned policy essentially tries to build all possible constituents licensed by the grammar, which can be
prohibitively expensive. We will remedy this in section 5.
4.3
What?s Wrong With Apprenticeship Learning
An obvious practical issue with the classifier-based approach is that it trains the classifier only at states visited
by the oracle. This leads to the well-known problem that it is unable to learn to recover from past errors
[2, 28, 7, 24]. Even though our current feature set depends only on the action and not on the state, making
action scores independent of the current state, there is still an issue since the set of actions to choose from does
depend on the state. That is, the classifier is trained to discriminate only among the small set of agenda items
available on the oracle trajectory (which are always combinations of correct constituents). But the action sets
the parser faces at test time are much larger and more diverse.
An additional objection to classifiers is that not all errors are created equal. Some incorrect actions are more
expensive than others, if they create constituents that can be combined in many locally-attractive ways and
hence slow the parser down or result in errors. Our classification problem does not distinguish among incorrect
actions. The S EARN algorithm [7] would distinguish them by explicitly evaluating the future reward of each
possible action (instead of using a teacher) and incorporating this into the classification problem. But explicit
evaluation is computationally infeasible in our setting (at each time step, it must roll out a full future trajectory
for each possible action from the agenda). Policy gradient provides another approach by observing which
actions are good or bad across many random trajectories, but recall that we found it impractical as well. We do
not further address this problem in this paper, but in [8] we suggested explicit causality analysis.
A final issue has to do with the nature of the oracle. Recall that the oracle is ?supposed to? choose optimal
actions for the given reward. Also recall that our oracle always picks correct constituents. There seems to be
a contradiction here: our oracle action selector ignores ?, the tradeoff between accuracy and speed, and only
focuses on accuracy. This happens because for any reasonable setting of ?, the optimal thing to do is always to
just build the correct tree without building any extra constituents. Only for very large values of ? is it optimal to
do anything else, and for such values of ?, the learned model will have hugely negative reward. This means that
under the apprenticeship learning setting, we are actually never going to be able to learn to trade off accuracy
and speed: as far as the oracle is concerned, you can have both! The tradeoff only appears because our model
cannot come remotely close to mimicking the oracle.
5
Oracle-Infused Policy Gradient
The failure of both standard reinforcement learning algorithms and standard apprenticeship learning algorithms
on our problem leads us to develop a new approach. We start with the policy gradient algorithm (Section 3.2)
and use ideas from apprenticeship learning to improve it. Our formulation preserves the reinforcement learning
flavor of our overall setting, which involves delayed reward for a known reward function.
Our approach is specifically designed for the non-deterministic nature of the agenda-based parsing setting [8]:
once some action a becomes available (appears on the agenda), it never goes away until it is taken. This makes
the notion of ?interleaving? oracle actions with policy actions both feasible and sensible. Like policy gradient,
we draw trajectories from a policy and take gradient steps that favor actions with high reward under reward
shaping. Like S EARN and DAGGER, we begin by exploring the space around the optimal policy and slowly
explore out from there.
6
To achieve this, we define the notion of an oracle-infused policy. Let ? be an arbitrary policy and let ? ? [0, 1].
We define the oracle-infused policy ??+ as follows:
??+ (a | s) = ?? ? (a | s) + (1 ? ?)?(a | s)
(11)
In other words, when choosing an action, ??+ explores the policy space with probability 1 ? ? (according to its
current model), but with probability ?, we force it to take an oracle action.
Our algorithm takes policy gradient steps with reward shaping (Eqs (9) and (7)), but with respect to trajectories
drawn from ??+ rather than ?. If ? = 0, it reduces to policy gradient, with reward shaping if ? < 1 and
immediate reward if ? = 0. For ? = 1, the ? = 0 case reduces to the classifier-based approach with ? ? (which
in turn breaks ties by choosing the best action under ?).
Similar to DAGGER and S EARN, we do not stay at ? = 1, but wean our learner off the oracle supervision as
it starts to find a good policy ? that imitates the classifier reasonably well. We use ? = 0.8epoch , where epoch
is the total number of passes made through the training set at that point (so ? = 0.80 = 1 on the initial pass).
Over time, ? ? 0, so that eventually we are training the policy to do well on the same distribution of states
that it will pass through at test time (as in policy gradient). With intermediate values of ? (and ? ? 1), an
iteration behaves similarly to an iteration of S EARN, except that it ?rolls out? the consequences of an action
chosen randomly from (11) instead of evaluating all possible actions in parallel.
Running Example 5. Oracle-infusion gives a competitive speed and accuracy tradeoff. A typical result is 91.2
with 0.68 mpops.
6
Experiments
All of our experiments (including those discussed earlier) are based on the Wall Street Journal portion of
the Penn Treebank [15]. We use a probabilistic context-free grammar with 370,396 rules?enough to make
the baseline system accurate but slow. We obtained it as a latent-variable grammar [16] using 5 split-merge
iterations [21] on sections 2?20 of the Treebank, reserving section 22 for learning the parameters of our policy.
All approaches to trading off speed and accuracy are trained on section 22; in particular, for the running example
and Section 6.2, the same 100 sentences of at most 15 words from that section were used for training and
test. We measure accuracy in terms of labeled recall (including preterminals) and measure speed in terms of
the number of pops from on the agenda. The limitation to relatively short sentences is purely for improved
efficiency at training time.
6.1
Baseline Approaches
Our baseline approaches trade off speed and accuracy not by learning to prioritize, but by varying the pruning
level ?. A constituent is pruned if its Viterbi inside score is more than ? worse than that of some other
constituent that covers the same substring.
Our baselines are: (HA? ) a Hierarchical A? parser [18] with the same pruning threshold at each hierarchy
level; (A?0 ) an A? parser with a 0 heuristic function plus pruning; (IDA?0 ) an iterative deepening A? algorithm,
on which a failure to find any parse causes us to increase ? and try again with less aggressive pruning (note
that this is not the traditional meaning of IDA*); and (CTF) the default coarse-to-fine parser in the Berkeley
parser [21]. Several of these algorithms can make multiple passes, in which case the runtime (number of pops)
is assessed cumulatively.
6.2
Learned Prioritization Approaches
Model
# of pops Recall
F1
We explored four variants of our oracle-infused polA?0 (no pruning) 1496080
93.34 93.19
icy gradient with with ? = 10?6 . Figure 1 shows
D686641
56.35 58.74
the result on the 100 training sentences. The ?-? tests
I187403
76.48 76.92
are the degenerate case of ? = 1, or apprenticeship
D+
1275292
84.17 83.38
learning (section 4.2), while the ?+? tests use ? =
I+
682540
91.16 91.33
0.8epoch as recommended in section 5. Temperature
matters for the ?+? tests and we use temp = 1. We
Figure 1: Performance on 100 sentences.
performed stochastic gradient descent for 25 passes
over the data, sampling 5 trajectories in a row for each sentence (when ? < 1 so that trajectories are random).
We can see that the classifier-based approaches ?-? perform poorly: when training trajectories consist of only
oracle actions, learning is severely biased. Yet we saw in section 3.2 that without any help from the oracle
actions, we suffer from such large variance in the training trajectories that performance degrades rapidly and
learning does not converge even after days of training. Our ?oracle-infused? compromise ?+? uses some oracle
actions: after several passes through the data, the parser learns to make good decisions without help from the
oracle.
7
# of pops
3
x 10
Change of recall and # of pops
7
2
1
0
0.82
I+
A*
0
IDA*
0
CTF
HA*
0.84
0.86
0.88
0.9
Recall
0.92
0.94
0.96
Figure 2: Pareto frontiers: Our I+ parser at different values of ?, against the baselines at different
pruning levels.
The other axis of variation is that the ?D? tests (delayed reward) use ? = 1, while the ?I? tests (immediate
reward) use ? = 0. Note that I+ attempts a form of credit assignment and works better than D+.2 We were
not able to get better results with intermediate values of ?, presumably because this crudely assigns credit for
a reward (correct constituent) to the actions that closely preceded it, whereas in our agenda-based parser, the
causes of the reward (correct subconstituents) related actions may have happened much earlier [8].
6.3
Pareto Frontier
Our final evaluation is on the held-out test set (length-limited sentences from Section 23). A 5-split grammar
trained on section 2-21 is used. Given our previous results in Table 1, we only consider the I+ model: immediate reward with oracle infusion. To investigate trading off speed and accuracy, we learn and then evaluate a
policy for each of several settings of the tradeoff parameter: ?. We train our policy using sentences of at most
15 words from Section 22 and evaluate the learned policy on the held out data (from Section 23). We measure
accuracy as labeled constituent recall and evaluate speed in terms of the number of pops (or pushes) performed
on the agenda.
Figure 2 shows the baselines at different pruning thresholds as well as the performance of our policies trained
using I+ for ? ? {10?3 , 10?4 , . . . , 10?8 }, using agenda pops as the measure of time. I+ is about 3 times as
fast as unpruned A?0 at the cost of about 1% drop in accuracy (F-score from 94.58 to 93.56). Thus, I+ achieves
the same accuracy as the pruned version of A?0 while still being twice as fast. I+ also improves upon HA? and
IDA?0 with respect to speed at 60% of the pops. I+ always does better than the coarse-to-fine parser (CTF) in
terms of both speed and accuracy, though using the number of agenda pops as our measure of speed puts both
of our hierarchical baselines at a disadvantage.
We also ran experiments using the number of agenda pushes as a more accurate measure of time, again sweeping
over settings of ?. Since our reward shaping was crafted with agenda pops in mind, perhaps it is not surprising
that learning performs relatively poorly in this setting. Still, we do manage to learn to trade off speed and
accuracy. With a 1% drop in recall (F-score from 94.58 to 93.54), we speed up from A?0 by a factor of 4 (from
around 8 billion pushes to 2 billion). Note that known pruning methods could also be employed in conjunction
with learned prioritization.
7
Conclusions and Future Work
In this paper, we considered the application of both reinforcement learning and apprenticeship learning to
prioritize search in a way that is sensitive to a user-defined tradeoff between speed and accuracy. We found
that a novel oracle-infused variant of the policy gradient algorithm for reinforcement learning is effective for
learning a fast and accurate parser with only a simple set of features. In addition, we uncovered many properties
of this problem that separate it from more standard learning scenarios, and designed experiments to determine
the reasons off-the-shelf learning algorithms fail.
An important avenue for future work is to consider better credit assignment. We are also very interested in
designing richer feature sets, including ?dynamic? features that depend on both the action and the state of the
chart and agenda. One role for dynamic features is to decide when to halt. The parser might decide to continue
working past the first complete parse, or give up (returning a partial or default parse) before any complete parse
is found.
2
The D- and I- approaches are quite similar to each other. Both train on oracle trajectories where all actions
receive a reward of 1 ? ?, and simply try to make these oracle actions probable. However, D- trains more
aggressively on long trajectories, since (9) implies that it weights a given training action by T ? t + 1, the
number of future actions on that trajectory. The difference between D+ and I+ is more interesting because the
trajectory includes non-oracle actions as well.
8
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9
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3,929 | 4,557 | Putting Bayes to sleep
Wouter M. Koolen?
Dmitry Adamskiy?
Manfred K. Warmuth?
Abstract
We consider sequential prediction algorithms that are given the predictions from
a set of models as inputs. If the nature of the data is changing over time in that
different models predict well on different segments of the data, then adaptivity is
typically achieved by mixing into the weights in each round a bit of the initial prior
(kind of like a weak restart). However, what if the favored models in each segment
are from a small subset, i.e. the data is likely to be predicted well by models
that predicted well before? Curiously, fitting such ?sparse composite models? is
achieved by mixing in a bit of all the past posteriors. This self-referential updating
method is rather peculiar, but it is efficient and gives superior performance on
many natural data sets. Also it is important because it introduces a long-term
memory: any model that has done well in the past can be recovered quickly. While
Bayesian interpretations can be found for mixing in a bit of the initial prior, no
Bayesian interpretation is known for mixing in past posteriors.
We build atop the ?specialist? framework from the online learning literature to give
the Mixing Past Posteriors update a proper Bayesian foundation. We apply our
method to a well-studied multitask learning problem and obtain a new intriguing
efficient update that achieves a significantly better bound.
1
Introduction
We consider sequential prediction of outcomes y1 , y2 , . . . using a set of models m = 1, . . . , M for
this task. In practice m could range over a mix of human experts, parametric models, or even complex machine learning algorithms. In any case we denote the prediction of model m for outcome yt
given past observations y<t = (y1 , . . . , yt?1 ) by P (yt |y<t , m). The goal is to design a computationally efficient predictor P (yt |y<t ) that maximally leverages the predictive power of these models
as measured in log loss. The yardstick in this paper is a notion of regret defined w.r.t. a given comparator class of models or composite models: it is the additional loss of the predictor over the best
comparator. For example if the comparator class is the set of base models m = 1, . . . , M , then the
regret for a sequence of T outcomes y?T = (y1 , . . . , yT ) is
R :=
T
X
M
? ln P (yt |y<t ) ? min
m=1
t=1
T
X
? ln P (yt |y<t , m).
t=1
The Bayesian predictor (detailed below) with uniform model prior has regret at most ln M for all T .
Typically the nature of the data is changing with time: in an initial segment one model predicts
well, followed by a second segment in which another model has small loss and so forth. For this
scenario the natural comparator class is the set of partition models which divide the sequence of T
outcomes into B segments and specify the model that predicts in each segment. By running Bayes
on all exponentially
many partition models comprising the comparator class, we can guarantee regret
T ?1
ln B?1
+B ln M , which is optimal. The goal then is to find efficient algorithms with approximately
?
Supported by NWO Rubicon grant 680-50-1010.
Supported by Veterinary Laboratories Agency of Department for Environment, Food and Rural Affairs.
?
Supported by NSF grant IIS-0917397.
?
1
the same guarantee as full Bayes. In this case this is achieved by the Fixed Share [HW98] predictor.
It assigns a certain prior to all partition models for which the exponentially many posterior weights
collapse to M posterior weights that can be maintained efficiently. Modifications of this algorithm
achieve essentially the same bound for all T , B and M simultaneously [VW98, KdR08].
In an open problem Yoav Freund [BW02] asked whether there are algorithms that have small regret
against sparse partition models where the base models allocated to the segments are from a small
subset of N of the M models.
The Bayes algorithm when run on all such partition models achieves
T ?1
regret ln M
+
ln
+
B
ln N , but contrary to the non-sparse case, emulating this algorithm
N
B?1
is NP-hard. However in a breakthrough paper, Bousquet and Warmuth in 2001 [BW02] gave the
efficient MPP algorithm with only a slightly weaker regret bound. Like Fixed Share, MPP maintains
M ?posterior? weights, but it instead mixes in a bit of all past posteriors in each update. This causes
weights of previously good models to ?glow? a little bit, even if they perform bad locally. When
the data later favors one of those good models, its weight is pulled up quickly. However the term
?posterior? is a misnomer because no Bayesian interpretation for this curious self-referential update
was known. Understanding the MPP update is a very important problem because in many practical
applications [HLSS00, GWBA02]1 it significantly outperforms Fixed Share.
Our main philosophical contribution is finding a Bayesian interpretation for MPP. We employ the
specialist framework from online learning [FSSW97, CV09, CKZV10]. So-called specialist models
are either awake or asleep. When they are awake, they predict as usual. However when they are
asleep, they ?go with the rest?, i.e. they predict with the combined prediction of all awake models.
T outcomes
3
9
7 3
7 9 7
3
(a) A comparator partition model:
segmentation and model assignment
3
zz z
Instead of fully coordinated partition models, we construct partition
specialists consisting of a base model and a set of segments where
this base model is awake. The figure to the right shows how a comparator partition model is assembled from partition specialists. We
can emulate Bayes on all partition specialists; NP-completeness is
avoided by forgoing a-priori segment synchronization. By carefully choosing the prior, the exponentially many posterior weights
collapse to the small number of weights used by the efficient MPP
algorithm. Our analysis technique magically aggregates the contribution of the N partition specialists that constitute the comparator
partition, showing that we achieve regret close to the regret of Bayes
when run on all full partition models. Actually our new insights into
the nature of MPP result in slightly improved regret bounds.
3
7
9
7
ZZZZ....
zZ
3
7
9
(b) Decomposition into 3 partition
specialists, asleep at shaded times
We then apply our methods to an online multitask learning problem where a small subset of models
from a big set solve a large number of tasks. Again simulating Bayes on all sparse assignments
of models to tasks is NP-hard. We split an assignment into subset specialists that assign a single
base model to a subset of tasks. With the right prior, Bayes on these subset specialists again gently
collapses to an efficient algorithm with a regret bound not much larger than Bayes on all assignments.
This considerably improves the previous regret bound of [ABR07]. Our algorithm simply maintains
one weight per model/task pair and does not rely on sampling (often used for multitask learning).
Why is this line of research important? We found a new intuitive Bayesian method to quickly recover
information that was learned before, allowing us to exploit sparse composite models. Moreover,
it expressly avoids computational hardness by splitting composite models into smaller constituent
?specialists? that are asleep in time steps outside their jurisdiction. This method clearly beats Fixed
Share when few base models constitute a partition, i.e. the composite models are sparse.
We expect this methodology to become a main tool for making Bayesian prediction adapt to sparse
models. The goal is to develop general tools for adding this type of adaptivity to existing Bayesian
models without losing efficiency. It also lets us look again at the updates used in Nature in a new
light, where species/genes cannot dare adapt too quickly to the current environment and must guard
themselves against an environment that changes or fluctuates at a large scale. Surprisingly these
type of updates might now be amenable to a Bayesian analysis. For example, it might be possible
to interpret sex and the double stranded recessive/dominant gene device employed by Nature as a
Bayesian update of genes that are either awake or asleep.
1
The experiments reported in [HLSS00] are based on precursors of MPP. However MPP outperforms these
algorithms in later experiments we have done on natural data for the same problem (not shown).
2
2
Bayes and Specialists
We consider sequential prediction of outcomes y1 , y2 , . . . from a finite alphabet. Assume that
we have access to a collection of models m = 1, . . . , M with data likelihoods P (y1 , y2 , . . . |m).
We then design a prior P (m) with roughly two goals in mind: the Bayes algorithm should ?collapse? (become efficient) and have a good regret bound. After observing past outcomes y<t :=
(y1 , . . . , yt?1 ), the next outcome yt is predicted by the predictive distribution P (yt |y<t ), which
averages the model predictions P (yt |y<t , m) according to the posterior distribution P (m|y<t ):
P (yt |y<t ) =
M
X
P (yt |y<t , m)P (m|y<t ),
where P (m|y<t ) =
m=1
P (y<t |m)P (m)
.
P (y<t )
The latter is conveniently updated step-wise: P (m|yt , y<t ) = P (yt |y<t , m)P (m|y<t )/P (yt |y<t ).
The log loss of the Bayesian predictor on data y?T := (y1 , . . . , yT ) is the cumulative loss of the
predictive distributions and this readily relates to the cumulative loss of any model m:
?
M
X
? = ?ln
P (y?T |m)P (m) ? ?lnP (y?T |m)
? ? ?lnP (m).
?
?lnP (y?T ) ? ?lnP (y?T |m)
{z
}
|
{z
}
|
m=1
P
P
T
t=1
?lnP (yt |y<t )
T
t=1
?lnP (yt |y<t ,m)
?
That is, the additional loss (or regret) of Bayes w.r.t. model m
? is at most ? ln P (m).
? The uniform
prior P (m) = 1/M ensures regret at most ln M w.r.t. any model m.
? This is a so-called individual
sequence result, because no probabilistic assumptions were made on the data.
Our main results will make essential use of the following
fancier
weighted notion of regret.
Here U (m) is any distribution on the models and 4 U (m)
P (m) denotes the relative entropy
PM
U (m)
m=1 U (m) ln P (m) between the distributions U (m) and P (m):
M
X
U (m) ? ln P (y?T )?(? ln P (y?T |m)) = 4 U (m)
P (m) ?4 U (m)
P (m|y?T ) . (1)
m=1
By dropping the subtracted positive term we get an upper bound. The previous regret bound is now
the special case when U is concentrated on model m.
? However when multiple models are good we
achieve tighter regret bounds by letting U be the uniform distribution on all of them.
Specialists We now consider a complication of the prediction task, which was introduced in the
online learning literature under the name specialists [FSSW97]. The Bayesian algorithm, adapted
to this task, will serve as the foundation of our main results. The idea is that in practice the predictions P (yt |y<t , m) of some models may be unavailable. Human forecasters may be specialized,
unreachable or too expensive, algorithms may run out of memory or simply take too long. We call
models that may possibly abstain from prediction specialists. The question is how to produce quality
predictions from the predictions that are available.
We will denote by Wt the set of specialists whose predictions are available at time t, and call them
awake and the others asleep. The crucial idea, introduced in [CV09], is to assign to the sleeping
specialists the prediction P (yt |y<t ). But wait! That prediction P (yt |y<t ) is defined to average all
model predictions, including those of the sleeping specialists, which we just defined to be P (yt |y<t ):
X
X
P (yt |y<t ) =
P (yt |y<t , m) P (m|y<t ) +
P (yt |y<t ) P (m|y<t ).
m?Wt
m?W
/ t
Although this equation is self-referential, it does have a unique solution, namely
P
m?Wt P (yt |y<t , m)P (m|y<t )
P (yt |y<t ) :=
.
P (Wt |y<t )
Thus the sleeping specialists are assigned the average prediction of the awake ones. This completes
them to full models to which we can apply the unaltered Bayesian method as before. At first this
may seem like a kludge, but actually this phenomenon arises naturally wherever concentrations are
3
manipulated. For example, in a democracy abstaining essentially endorses the vote of the participating voters or in Nature unexpressed genes reproduce at rates determined by the active genes of the
organism. The effect of abstaining on the update of the posterior weights is also intuitive: weights
of asleep specialists are unaffected, whereas weights of awake models are updated with Bayes rule
and then renormalised to the original weight of the awake set:
?
? P (yt |y<t ,m)P (m|y<t ) = P P (yt |y<t ,m)P (m|y<t )
P (Wt |y<t ) if m ? Wt ,
P (yt |y<t )
m?Wt P (yt |y<t ,m)P (m|y<t )
)P (m|y<t )
P (m|y?t ) = P (y
(2)
|y
t
<t
?
=
P
(m|y
)
if
m
?
/
W
.
<t
t
P
(y
|y
)
t <t
To obtain regret bounds in the specialist setting, we use the fact that sleeping specialists m ?
/ Wt
are defined to predict P (yt |y<t , m) := P (yt |y<t ) like the Bayesian aggregate. Now (1) becomes:
Theorem 1 ([FSSW97, Theorem 1]). Let U (m) be any distribution on a set of specialists with wake
sets W1 , W2 , . . . Then for any T , Bayes guarantees
?
?
M
X
X
X
? ln P (yt |y<t , m)? ? 4 U (m)
P (m) .
? ln P (yt |y<t ) ?
U (m) ?
m=1
3
t?T : m?Wt
t?T : m?Wt
Sparse partition learning
We design efficient predictors with small regret compared to the best sparse partition model. We
do this by constructing partition specialists from the input models and obtain a proper Bayesian
predictor by averaging their predictions. We consider two priors. With the first prior we obtain the
Mixing Past Posteriors (MPP) algorithm, giving it a Bayesian interpretation and slightly improving
its regret bound. We then develop a new Markov chain prior. Bayes with this prior collapses to an
efficient algorithm for which we prove the best known regret bound compared to sparse partitions.
Construction Each partition specialist (?, m) is parameterized by a model index m and a circadian (wake/sleep pattern) ? = (?1 , ?2 , . . .) with ?t ? {w, s}. We use infinite circadians in order
to obtain algorithms that do not depend on a time horizon. The wake set Wt includes all partition
specialists that are awake at time t, i.e. Wt := {(?, m) | ?t = w}. An awake specialist (?, m)
in Wt predicts as the base model m, i.e. P(yt |y<t , (?, m)) := P (yt |y<t , m). The Bayesian joint
distribution P is completed2 by choosing a prior on partition specialists. In this paper we enforce
the independence P(?, m) := P(?)P(m) and define P(m) := 1/M uniform on the base models.
We now can apply Theorem 1 to bound the regret w.r.t. any partition model with time horizon T by
decomposing it into N partition specialists (?1?T , m
? 1 ), . . . , (?N
? N ) and choosing U (?) = 1/N
?T , m
uniform on these specialists:
N
R ? N ln
M X
+
? ln P(?n?T ).
N
n=1
(3)
The overhead of selecting N reference models from
the pool of size M closely approximates
M
the information-theoretic ideal N ln M
?
ln
This improves previous regret bounds
N
N .
[BW02, ABR07, CBGLS12] by an additive N ln N . Next we consider two choices for P(?): one
for which we retrieve MPP, and a natural one which leads to efficient algorithms and sharper bounds.
3.1
A circadian prior equivalent to Mixing Past Posteriors
The Mixing Past Posteriors algorithm is parameterized a so-called mixing scheme, which is a sequence ?1 , ?2 , . . . of distributions, each ?t with support {0, . . . , t ? 1}. MPP predicts outcome
PM
yt with Predt (yt ) :=
m=1 P (yt |y<t , m) vt (m), i.e. by averaging the model predictions with
weights vt (m) defined recursively by
vt (m) :=
t?1
X
v?s+1 (m) ?t (s) where v?1 (m) :=
s=0
2
1
M
and
v?t+1 (m) :=
P (yt |y<t , m)vt (m)
.
Predt (yt )
From here on we use the symbol P for the Bayesian joint to avoid a fundamental ambiguity: P(yt |y<t , m)
does not equal the prediction P (yt |y<t , m) of the input model m, since it averages over both asleep and awake
specialists (?, m). The predictions of base models are now recovered as P(yt |y<t , Wt , m) = P (yt |y<t , m).
4
The auxiliary distribution v?t+1 (m) is formally the (incremental) posterior from prior vt (m). The
predictive weights vt (m) are then the pre-specified ?t mixture of all such past posteriors.
To make the Bayesian predictor equal to MPP, we define from the MPP mixing scheme a circadian
prior measure P(?) that puts mass only on sequences with a finite nonzero number of w?s, by
P(?) :=
J
Y
1
sJ (sJ + 1) j=1
?sj (sj?1 ) where s?J are the indices of the w?s in ? and s0 = 0. (4)
We built the independence m ? ? into the prior P(?, m) and (4) ensures ?<t ? ?>t | ?t = w for
all t. Since the outcomes y?t are a stochastic function of m and ??t , the Bayesian joint satisfies
y?t , m ? ?>t | ?t = w
for all t.
(5)
Theorem 2. Let Predt (yt ) be the prediction of MPP for some mixing scheme ?1 , ?2 , . . . Let
P(yt |y<t ) be the prediction of Bayes with prior (4). Then for all outcomes y?t
Predt (yt ) = P(yt |y<t ).
Proof. Partition the event Wt = {?t = w} into Zqt := {?t = ?q = w and ?r = s for all q < r < t}
for all 0 ? q < t, with the convention that ?0 = w. We first establish that the Bayesian joint with
prior (4) satisfies y?t ? Wt for all t. Namely, by induction on t, for all q < t
(5)
Induction
P(y<t |Zqt ) = P(y<t |y?q )P(y?q |Zqt ) = P(y<t |y?q )P(y?q |Wq ) = P(y<t ),
Pt?1
and therefore P(y?t |Wt ) = P(yt |y<t ) q=0 P(y<t |Zqt )P(Zqt |Wt ) = P(y?t ), i.e. y?t ? Wt . The
theorem will be implied by the stronger claim vt (m) = P(m|y<t , Wt ), which we again prove by
induction on t. The case t = 1 is trivial. For t > 1, we expand the right-hand side, apply (5), use
the independence we just proved, and the fact that asleep specialist predict with the rest:
P(m|y<t , Wt ) =
t?1
X
q=0
=
P(m|y?q , Wq )
t
, Wq )P(Wq |H
m,
y?q
y?q
P(Zqt |
, m,
y?q
)
q
H) P(y<t |Z
P(y<t |y?q )
P(Wt |H
y<t
)
H
t?1
X
P (yq |y<q , m)P(m|y<q , Wq )
q=0
P(yq |y<q )
P(Zqt |Wt )
By (4) P(Zqt |Wt ) = ?t (q), and the proof is completed by applying the induction hypothesis.
The proof of the theorem provides a Bayesian interpretation of all the MPP weights: vt (m) =
P(m|y<t , Wt ) is the predictive distribution, v?t+1 (m) = P(m|y?t , Wt ) is the posterior, and ?t (q) =
P(Zqt |Wt ) is the conditional probability of the previous awake time.
3.2
A simple Markov chain circadian prior
In the previous section we recovered circadian priors corresponding to the MPP mixing schemes.
Here we design priors afresh from first principles. Our goal is efficiency and good regret bounds. A
simple and intuitive choice for prior P(?) is a Markov chain on states {w, s} with initial distribution
?(?) and transition probabilities ?(?|w) and ?(?|s), that is
P(??t ) := ?(?1 )
t
Y
?(?s |?s?1 ).
(6)
s=2
By choosing low transition probabilities we obtain a prior that favors temporal locality in that it
allocates high probability to circadians that are awake and asleep in contiguous segments. Thus if a
good sparse partition model exists for the data, our algorithm will pick up on this and predict well.
The resulting Bayesian strategy (aggregating infinitely many specialists) can be executed efficiently.
Theorem 3. The prediction P(yt |y<t ) of Bayes with Markov prior (6) equals the prediction
Predt (yt ) of Algorithm 1, which can be computed in O(M ) time per outcome using O(M ) space.
5
Proof. We prove by induction on t that vt (b, m) = P(?t = b, m|y<t ) for each model m and
b ? {w, s}. The base case t = 1 is automatic. For the induction step we expand
(6)
P(?t+1 = b, m|y?t ) = ?(b|w)P(?t = w, m|y?t ) + ?(b|s)P(?t = s, m|y?t )
P(?t = w, m|y<t )P (yt |y<t , m)
(2)
= ?(b|w) PM
+ ?(b|s)P(?t = s, m|y<t ).
i=1 P(i|?t = w, y<t )P (yt |y<t , i)
By applying the induction hypothesis we obtain the update rule for vt+1 (b, m).
Algorithm 1 Bayes with Markov circadian prior (6) (for Freund?s problem)
Input: Distributions ?(?), ?(?|w) and ?(?|s) on {w, s}.
Initialize v1 (b, m) := ?(b)/M for each model m and b ? {w, s}
for t = 1, 2, . . . do
Receive prediction P (yt |y<t , m) of each model m
PM
Predict with Predt (yt ) :=
m=1 P (yt |y<t , m)vt (m|w) where vt (m|w) =
PM
vt (w,m)
vt (w,m0 )
m0 =1
Observe outcome yt and suffer loss ? ln Predt (yt ).
(yt |y<t ,m)
Update vt+1 (b, m) := ?(b|w) P Pred
vt (w, m) + ?(b|s)vt (s, m).
t (yt )
end for
The previous theorem establishes that we can predict fast. Next we show that we predict well.
Theorem 4. Let m
? 1, . . . , m
? T be an N -sparse assignment of M models to T times with B segments. The regret of Bayes (Algorithm 1) with tuning ?(w) = 1/N , ?(s|w) = B?1
T ?1 and
B?1
?(w|s) = (N ?1)(T ?1) is at most
M
1
B?1
B?1
R ? N ln
+NH
+ (T ? 1) H
+ (N ? 1)(T ? 1) H
,
N
N
T ?1
(N ? 1)(T ? 1)
where H(p) := ?p ln(p) ? (1 ? p) ln(1 ? p) is the binary entropy function.
Proof. Without generality assume m
? t ? {1, . . . , N }. For each reference model n pick circadian
? t = n. Expanding the definition of the prior (6) we find
?n?T with ?nt = w iff m
N
Y
P(?n?T ) = ?(w)?(s)N ?1 ?(s|s)
(N ?1)(T ?1)?(B?1)
T ?B
?(w|w)
B?1
?(w|s)
?(s|w)
B?1
,
n=1
which is in fact maximized by the proposed tuning. The theorem follows from (3).
T ?1
The information-theoretic ideal regret is ln M
N + ln B?1 + B ln N . Theorem 4 is very close to
this except for a factor of 2 in front of the middle term; since n H(k/n) ? k ln(n/k) + k we have
M
T ?1
R ? N ln
+ 2 (B ? 1) ln
+ B ln N + 2B.
N
B?1
The origin of this factor remained a mystery in [BW02], but becomes clear in our analysis: it is the
price of coordination between the specialists that constitute the best partition model. To see this, let
us regard a circadian as a sequence of wake/sleep transition times. With this viewpoint (3) bounds
the regret by summing the prior costs of all the reference wake/sleep transition times. This means
that we incur overhead at each segment boundary of the comparator twice: once as the sleep time of
the preceding model, and once more as the wake time of the subsequent model.
In practice the comparator parameters T , N and B are unknown. This can be addressed by standard
orthogonal techniques. Of particular interest is the method inspired by [SM99, KdR08, Koo11] of
changing the Markov transition probabilities as a function of time. It can be shown that by setting
1
) we keep the update time and
?(w) = 1/2 and increasing ?(w|w) and ?(s|s) as exp(? t ln2 (t+1)
space of the algorithm at O(M ) and guarantee regret bounded for all T , N and B as
M
R ? N ln
+ 2N + 2(B ? 1) ln T + 4(B ? 1) ln ln(T + 1).
N
At no computational overhead, this bound is remarkably close to the fully tuned bound of the theorem above, especially when the number of segments B is modest as a function of T .
6
4
Sparse multitask learning
We transition to an extension of the sequential prediction setup called online multitask learning
[ABR07, RAB07, ARB08, LPS09, CCBG10, SRDV11]. The new ingredient is that before predicting outcome yt we are given its task number ?t ? {1, . . . , K}. The goal is to exploit similarities
between tasks. As before, we have access to M models that each issue a prediction each round. If a
single model predicts well on several tasks we want to figure this out quickly and exploit it. Simply
ignoring the task number would not result in an adaptive algorithm. Applying a separate Bayesian
predictor to each task independently would not result in any inter-task synergy. Nevertheless, it
would guarantee regret at most K ln M overall. Now suppose each task is predicted well by some
model from a small subset of models of size N M . Running Bayes on all N -sparse allocations
would achieve regret ln M
N + K ln N . However, emulating Bayes in this case is NP-hard [RAB07].
The goal is to design efficient algorithms with approximately the same regret bound.
In [ABR07] this multiclass problem is reduced to MPP, giving regret bound N ln M
N + B ln N . Here
B is the number of same-task segments in the task sequence ??T . When all outcomes with the same
task number are consecutive, i.e. B = K, then the desired bound is achieved. However the tasks
may be interleaved, making the number of segments B much larger than K. We now eliminate the
dependence on B, i.e. we solve a key open problem of [ABR07].
We apply the method of specialists to multitask learning, and obtain regret bounds close to the
information-theoretic ideal, which in particular do not depend on the task segment count B at all.
Construction We create a subset specialist (S, m) for each basic model index m and subset of
tasks S ? {1, . . . , K}. At time t, specialists with the current task ?t in their set S are awake,
i.e. Wt := {(S, m) | ?t ? S}, and issue the prediction P(yt |y<t , S, m) := P (yt |y<t , m) of
model m. We assign to subset specialist (S, m) prior probability P(S, m) := P(S)P(m) where
P(m) := 1/M is uniform, and P(S) includes each task independently with some fixed bias ?(w)
P(S) := ?(w)|S| ?(s)K?|S| .
(7)
This construction has the property that the product of prior weights of two loners ({?1 }, m)
? and
({?2 }, m)
? is dramatically lower than the single pair specialist ({?1 , ?2 }, m),
? especially so when the
number of models M is large or when we consider larger task clusters. By strongly favoring it in
the prior, any inter-task similarity present will be picked up fast.
The resulting Bayesian strategy involving M 2K subset specialists can be implemented efficiently.
Theorem 5. The predictions P(yt |y<t ) of Bayes with the set prior (7) equal the predictions
Predt (yt ) of Algorithm 2. They can be computed in O(M ) time per outcome using O(KM ) storage.
?
Of particular interest is Algorithm 2?s update rule for ft+1
(m). This would be a regular Bayesian
posterior calculation if vt (m) in Predt (yt ) were replaced by ft? (m). In fact, vt (m) is the communication channel by which knowledge about the performance of model m in other tasks is received.
Proof. The resource analysis follows from inspection, noting that the update is fast because only
the weights ft? (m) associated to the current task ? are changed. We prove by induction on t that
P(m|y<t , Wt ) = vt (m). In the base case t = 1 both equal 1/M . For the induction step we expand
P(m|y?t , Wt+1 ), which is by definition proportional to
?
??
?
X 1
Y
Y
?(w)|S| ?(s)K?|S| ?
P (yq |y<q , m)? ?
P(yq |y<q )? .
(8)
M
q?t : ?q ?S
S3?t+1
q?t : ?q ?S
/
The product form of both set prior and likelihood allows us to factor this exponential sum of products
into a product of binary sums. It follows from the induction hypothesis that
Y
?(w)
P (yq |y<q , m)
ftk (m) =
?(s)
P(yq |y<q )
q?t : ?q =k
K
Then we can divide (8) by P(y?t )?(s)
Y
1 ?t+1
P(m|y?t , Wt+1 ) ?
ft
(m)
M
and reorganize to
ftk (m)
k6=?t+1
7
+1
K
?
1 ft t+1 (m) Y k
=
ft (m) + 1
?t+1
M ft
(m) + 1 k=1
Since the algorithm maintains ?t (m) =
QK
k
k=1 (ft (m)
+ 1) this is proportional to vt+1 (m).
Algorithm 2 Bayes with set prior (7) (for online multitask learning)
Input: Number of tasks K ? 2, distribution ?(?) on {w, s}.
Q
w)
k
:= K
Initialize f1k (m) := ?(
k=1 (f1 (m) + 1).
?(s) for each task k and ?1 (m)
for t = 1, 2, . . . do
Observe task index ? = ?t .
f ? (m) ?t (m)/(ft? (m)+1)
.
Compute auxiliary vt (m) := PtM f ? (i)
? (i)/(f ? (i)+1)
i=1
t
t
t
Receive prediction P (yt |y<t , m) of each model m
PM
Issue prediction Predt (yt ) := m=1 P (yt |y<t , m)vt (m).
Observe outcome yt and suffer loss ? ln Predt (yt ).
(yt |y<t ,m) ?
?
k
Update ft+1
(m) := P Pred
ft (m) and keep ft+1
(m) := ftk (m) for all k 6= ?.
t (yt )
Update ?t+1 (m) :=
end for
?
ft+1
(m)+1
ft? (m)+1 ?t (m).
The Bayesian strategy is hence emulated fast by Algorithm 2. We now show it predicts well.
Theorem 6. Let m
? 1, . . . , m
? K be an N -sparse allocation of M models to K tasks. With tuned
inclusion rate ?(w) = 1/N , the regret of Bayes (Algorithm 2) is bounded by
R ? N ln ( M /N ) + KN H(1/N ).
Proof. Without loss of generality assume that m
? k ? {1, . . . , N }. Let Sn := {1 ? k ? K |
QN
m
? k = n}. The sets Sn for n = 1, . . . , N form a partition of the K tasks. By (7) n=1 P(Sn ) =
?(w)K ?(s)(N ?1)K , which is maximized by the proposed tuning. The theorem now follows from
(3).
We achieve the desired goal since KN H(1/N ) ? K ln N . In practice N is of course unavailable
for tuning, and we may tune ?(w) = 1/K pessimistically to get K ln K + N instead for all N
simultaneously. Or alternatively, we may sacrifice some time efficiency to externally mix over all
M possible values with decreasing prior, increasing the tuned regret by just ln N + O(ln ln N ).
If in addition the number of tasks is unknown or unbounded, we may (as done in Section 3.2)
decrease the membership rate ?(w) with each new task encountered and guarantee regret R ?
N ln(M/N ) + K ln K + 4N + 2K ln ln K where now K is the number of tasks actually received.
5
Discussion
We showed that Mixing Past Posteriors is not just a heuristic with an unusual regret bound: we
gave it a full Bayesian interpretation using specialist models. We then applied our method to a
multitask problem. Again an unusual algorithm resulted that exploits sparsity by pulling up the
weights of models that have done well before in other tasks. In other words, if all tasks are well
predicted by a small subset of base models, then this algorithm improves its prior over models as it
learns from previous tasks. Both algorithms closely circumvent NP-hardness. The deep question is
whether some of the common updates used in Nature can be brought into the Bayesian fold using
the specialist mechanism.
There are a large number of more immediate technical open problems (we just discuss a few). We
presented our results using probabilities and log loss. However the bounds should easily carry over to
the typical pseudo-likelihoods employed in online learning in connection with other loss functions.
Next, it would be worthwhile to investigate for which infinite sets of models we can still employ our
updates implicitly. It was already shown in [KvE10, Koo11] that MPP can be efficiently emulated
on all Bernoulli models. However, what about Gaussians, exponential families in general, or even
linear regression? Finally, is there a Bayesian method for modeling concurrent multitasking, i.e. can
the Bayesian analysis be generalized to the case where a small subset of models solve many tasks in
parallel?
8
References
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Jacob Ducan Abernethy, Peter Bartlett, and Alexander Rakhlin. Multitask learning
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Alekh Agarwal, Alexander Rakhlin, and Peter Bartlett. Matrix regularization techniques for online multitask learning, October 2008.
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Olivier Bousquet and Manfred K. Warmuth. Tracking a small set of experts by mixing
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[CBGLS12] Nicol`o Cesa-Bianchi, Pierre Gaillard, G?abor Lugosi, and Gilles Stoltz. A new look at
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Giovanni Cavallanti, Nicol`o Cesa-Bianchi, and Claudio Gentile. Linear algorithms for
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[FSSW97]
Y. Freund, R. E. Schapire, Y. Singer, and M. K. Warmuth. Using and combining predictors that specialize. In Proc. 29th Annual ACM Symposium on Theory of Computing,
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[KdR08]
[Koo11]
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9
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3,930 | 4,558 | A Polynomial-time Form of Robust Regression
?
Yaoliang Yu, Ozlem
Aslan and Dale Schuurmans
Department of Computing Science, University of Alberta, Edmonton AB T6G 2E8, Canada
{yaoliang,ozlem,dale}@cs.ualberta.ca
Abstract
Despite the variety of robust regression methods that have been developed, current regression formulations are either NP-hard, or allow unbounded response
to even a single leverage point. We present a general formulation for robust
regression?Variational M-estimation?that unifies a number of robust regression
methods while allowing a tractable approximation strategy. We develop an estimator that requires only polynomial-time, while achieving certain robustness and
consistency guarantees. An experimental evaluation demonstrates the effectiveness of the new estimation approach compared to standard methods.
1
Introduction
It is well known that outliers have a detrimental effect on standard regression estimators. Even a
single erroneous observation can arbitrarily affect the estimates produced by methods such as least
squares. Unfortunately, outliers are prevalent in modern data analysis, as large data sets are automatically gathered without the benefit of manual oversight. Thus the need for regression estimators
that are both scalable and robust is increasing.
Although the field of robust regression is well established, it has not considered computational complexity analysis to be one of its central concerns. Consequently, none of the standard regression
estimators in the literature are both robust and tractable, even in a weak sense: it has been shown
that standard robust regression formulations with non-zero breakdown are NP-hard [1, 2], while
any estimator based on minimizing a convex loss cannot guarantee bounded response to even a single leverage point [3] (definitions given below). Surprisingly, there remain no standard regression
formulations that guarantee both polynomial run-time with bounded response to even single outliers.
It is important to note that robustness and tractability can be achieved under restricted conditions. For
example, if the domain is bounded, then any estimator based on minimizing a convex and Lipschitzcontinuous loss achieves high breakdown [4]. Such results have been extended to kernel-based
regression under the analogous assumption of a bounded kernel [5, 6]. Unfortunately, these results
can no longer hold when the domain or kernel is unbounded: in such a case arbitrary leverage can
occur [4, 7] and no (non-constant) convex loss, even Lipschitz-continuous, can ensure robustness
against even a single outlier [3]. Our main motivation therefore is to extend these existing results
to the case of an unbounded domain. Unfortunately, the inapplicability of convex losses in this
situation means that computational tractability becomes a major challenge, and new computational
strategies are required to achieve tractable robust estimators.
The main contribution of this paper is to develop a new robust regression strategy that can guarantee
both polynomial run-time and bounded response to individual outliers, including leverage points.
Although such an achievement is modest, it is based on two developments of interest. The first
is a general formulation of adaptive M-estimation, Variational M-estimation, that unifies a number
of robust regression formulations, including convex and bounded M-estimators with certain subsetselection estimators such as Least Trimmed Loss [7]. By incorporating Tikhonov regularization,
these estimators can be extended to reproducing kernel Hilbert spaces (RKHSs). The second development is a convex relaxation scheme that ensures bounded outlier influence on the final estimator.
1
The overall estimation procedure is guaranteed to be tractable, robust to single outliers with unbounded leverage, and consistent under non-trivial conditions. An experimental evaluation of the
proposed estimator demonstrates effective performance compared to standard robust estimators.
The closest previous works are [8], which formulated variational representations of certain robust
losses, and [9], which formulated a convex relaxation of bounded loss minimization. Unfortunately,
[8] did not offer a general characterization, while [9] did not prove their final estimator was robust,
nor was any form of consistency established. The formulation we present in this paper generalizes
[8] while the convex relaxation scheme we propose is simpler and tighter than [9]; we are thus able
to establish non-trivial forms of both robustness and consistency while maintaining tractability.
There are many other notions of ?robust? estimation in the machine learning literature that do not
correspond to the specific notion being addressed in this paper. Work on ?robust optimization? [10?
12], for example, considers minimizing the worst case loss achieved given bounds on the maximum
data deviation that will be considered. Such results are not relevant to the present investigation because we explicitly do not bound the magnitude of the outliers. Another notion of robustness is
algorithmic stability under leave-one-out perturbation [13], which analyzes specific learning procedures rather than describing how a stable algorithm might be generally achieved.
2
Preliminaries
We start by considering the standard linear regression model
y
= xT ? ? + u
(1)
where x is an Rp -valued random variable, u is a real-valued random noise term, and ? ? ? ? ? Rp
is an unknown deterministic parameter vector. Assume we are given a sample of n independent
identically distributed (i.i.d.) observations represented by an n ? p matrix X and an n ? 1 vector
y, where each row Xi: is drawn from some unknown marginal probability measure Px , and yi are
generated according to (1). Our task is to estimate the unknown deterministic parameter ? ? ? ?.
Clearly, this is a well-studied problem in statistics and machine learning. If the noise distribution
has a known density p(?), then a standard estimator is given by maximum likelihood
Pn
Pn
??ML ? arg min n1 i=1 ? log p(yi ? Xi: ?) = arg min n1 i=1 ? log p(ri ),
(2)
???
???
where ri = yi ? Xi: ? is the ith residual. When the noise distribution is unknown, one can replace
the negative log-likelihood with a loss function ?(?) and use the estimator
??M
? arg min n1 1T ?(y ? X?),
(3)
???
where ?(r) denotes the
of losses obtained by applying the loss componentwise to each residPvector
n
ual, hence 1T ?(r) = i=1 ?(ri ). Such a procedure is known as M -estimation in the robust statistics literature, and empirical risk minimization in the machine learning literature.1
Although uncommon in robust regression, it is conventional in machine learning to include a regularizer. In particular we will use Tikhonov (?ridge?) regularization by adding a squared penalty
??MR
?
arg min n1 1T ?(y ? X?) + ?2 k?k22
???
for ? ? 0,
(4)
The significance of Tikhonov regularization is that it ensures ??MR = X T ? for some ? ? Rn
[14]. More generally, under Tikhonov regularization, the regression problem can be conveniently
expressed in a reproducing kernel Hilbert space (RKHS). If we let H denote the RKHS corresponding to positive semidefinite kernel ? : X ? X ? R, then f (x) = h?(x, ?), f iH for any f ? H by
the reproducing property [14, 15]. We consider the generalized regression model
y
=
f ? (x) + u
(5)
?
where x is an X -valued random variable, u is a real-valued random noise term as above, and f ? H
is an unknown deterministic function. Given a sample of n i.i.d. observations (x1 , y1 ), ..., (xn , yn ),
1
Generally one has to introduce an additional scale parameter ? and allow rescaling of the residuals via
ri /?, to preserve parameter equivariance [3, 4]. However, we will initially assume a known scale.
2
where each xi is drawn from some unknown marginal probability measure Px , and yi are generated
according to (5),2 the task is then to estimate the unknown deterministic function f ? ? H. To do so
we can express the estimator (4) more generally as
Pn
f?MR ? arg min 1
?(yi ? f (xi )) + ? kf k2 .
(6)
f ?H n
i=1
2
H
Pn
By the representer theorem [14], the solution to (6) can be expressed by f?MR (x) = i=1 ?i ?(xi , x)
for some ? ? Rn , and therefore (6) can be recovered by solving the finite dimensional problem
? MR ? arg min n1 1T ?(y ? K?) + ?2 ?T K? such that Kij = ?(xi , xj ).
(7)
?
?
Our interest is understanding the tractability, robustness and consistency aspects of such estimators.
Consistency: Much is known about the consistency properties of estimators expressed as regularized empirical risk minimizers. For example, the ML-estimator (2) and the M -estimator (3) are both
known to be parameter consistent under general conditions [16].3 The regularized M -estimator in
RKHSs (6), is loss consistent under some general assumptions on the kernel, loss and training distribution.4 Furthermore, a weak form of f -consistency has also established in [6]. For bounded
kernel and bounded Lipschitz losses, one can similarly prove the loss consistency of the regularized
M -estimator (6) (in RKHS). See Appendix C.1 of the supplement for more discussion.
Generally speaking, any estimator that can be expressed as a regularized empirical loss minimization
is consistent under ?reasonable? conditions. That is, one can consider regularized loss minimization
to be a (generally) sound principle for formulating regression estimators, at least from the perspective
of consistency. However, this is no longer the case when we consider robustness and tractability;
here sharp distinctions begin to arise within this class of estimators.
Robustness: Although robustness is an intuitive notion, it has not been given a unique technical
definition in the literature. Several definitions have been proposed, with distinct advantages and
disadvantages [4]. Some standard definitions consider the asymptotic invariance of estimators to
an infinitesimal but arbitrary perturbation of the underlying distribution, e.g. the influence function
[4, 17]. Although these analyses can be useful, we will focus on finite sample notions of robustness
since these are most related to concerns of computational tractability. In particular, we focus on the
following definition related to the finite sample breakdown point [18, 19].
Definition 1 (Bounded Response). Assuming the parameter set ? is metrizable, an estimator has
bounded response if for any finite data sample its output remains in a bounded interior subset of the
closed parameter set ? (or respectively H), no matter how a single observation pair is perturbed.
This is a much weaker definition than having a non-zero breakdown point: a breakdown of requires that bounded response be guaranteed when any fraction of the data is perturbed arbitrarily.
Bounded response is obviously a far more modest requirement. However, importantly, the definition
of bounded response allows the possibility of arbitrary leverage; that is, no bound is imposed on the
magnitude of a perturbed input (i.e. kx1 k ? ? or ?(x1 , x1 ) ? ?). Surprisingly, we find that even
such a weak robustness property is difficult to achieve while retaining computational tractability.
Computational Dilemma: The goals of robustness and computational tractability raise a dilemma:
it is easy to achieve robustness (i.e. bounded response) or tractability (i.e. polynomial run-time) in a
consistent estimator, but apparently not both.
Consider, for example, using a convex loss function. These are the best known class of functions
that admit computationally efficient polynomial-time minimization [20] (see also [21] ). It is sufficient that the objective be polynomial-time evaluable, along with its first and second derivatives,
2
We are obviously assuming X is equipped with an appropriate ?-algebra, and R with the standard Borel
?-algebra, such that the joint distribution
P over X ? R is well defined and ?(?, ?) is measurable.
P
3
T
T
In particular, let Mn (?) = n1 n
i=1 ?(yi ? xi ?), let M (?) = E(?(y1 ? x1 ?)), and equip the parameter
(n)
space ? with the uniform metric k ? k? . Then ??M ? ? ? , provided kMn ? M k? ? 0 in outer probability
(adopted to avoid measurability issues) and M (? ? ) > sup??G M (?) for every open set G that contains ? ? .
The latter assumption is satisfied in particular when M : ? 7? R is upper semicontinuous with a unique
maximum at ? ? . It is also possible to derive asymptotic convergence rates for general M -estimators [16].
P
4
?
?
Specifically, let ?? = inf f ?H E[?(y1 ? f (x1 ))]. Then [6] showed that n1 n
i=1 ?(yi ? fMR (xi )) ? ?
2
provided the regularization constant ?n ? 0 and ?n n ? ?, the loss ? is convex and Lipschitz-continuous,
and the RKHS H (induced by some bounded measurable kernel ?) is separable and dense in L1 (P) (the space
of P-integrable functions) for all distributions P on X . Also, Y ? R is required to be closed where y ? Y.
3
and that the objective be self-concordant [20].5 Since a Tikhonov regularizer is automatically selfconcordant, the minimization problems outlined above can all be solved in polynomial time with
Newton-type algorithms, provided ?(r), ?0 (r), and ?00 (r) can all be evaluated in polynomial time
for a self-concordant ? [22, Ch.9]. Standard loss functions, such as squared error or Huber?s loss
satisfy these conditions, hence the corresponding estimators are polynomial-time.
Unfortunately, loss minimization with a (non-constant) convex loss yields unbounded response to
even a single outlier [3, Ch.5]. We extend this result to also account for regularization and RKHSs.
Theorem 1. Empirical risk minimization based on a (non-constant) convex loss cannot have
bounded response if the domain (or kernel) is unbounded, even under Tikhonov regularization.
(Proof given in Appendix B of the supplement.)
By contrast, consider the case of a (non-constant) bounded loss function.6 Bounded loss functions
are a common choice in robust regression because they not only ensure bounded response, trivially,
they can also ensure a high breakdown point of (n ? p)/(2n) [3, Ch.5]. Unfortunately, estimators
based on bounded losses are inherently intractable.
Theorem 2. Bounded (non-constant) loss minimization is NP-hard. (Proof given in Appendix E.)
These difficulties with empirical risk minimization have led the field of robust statistics to develop
a variety of alternative estimators [4, Ch.7]. For example, [7] recommends subset-selection based
regression estimators, such as Least Trimmed Loss:
Pn0
??LTL ? arg min??? i=1 ?(r[i] ).
(8)
Here r[i] denotes sorted residuals r[1] ? ? ? ? ? r[n] and n0 < n is the number of terms to consider.
Traditionally ?(r) = r2 is used. These estimators are known to have high breakdown [7],7 and
obviously demonstrate bounded response to single outliers. Unfortunately, (8) is NP-hard [1].
3
Variational M-estimation
To address the dilemma, we first adopt a general form of adaptive M-estimator that allows flexibility
while allowing a general approximation strategy. The key construction is a variational representation
of M-estimation that can express a number of standard robust (and non-robust) methods in a common
framework. In particular, consider the following adaptive form of loss function
?(r)
=
min ?`(r) + ?(?).
(9)
0???1
where r is a residual value, ` is a closed convex base loss, ? is an adaptive weight on the base loss,
and ? is a convex auxiliary function. The weight can choose to ignore the base loss if `(r) is large,
but this is balanced against a prior penalty ?(?). Different choices of base loss and auxiliary function
will yield different results, and one can represent a wide variety of loss functions ? in this way [8].
For example, any convex loss ? can be trivially represented in the form (9) by setting ` = ?, and
?(?) = ?{1} (?).8 Bounded loss functions can also be represented in this way, for example
(Geman-McClure) [8]
?(r) =
(Geman-Reynolds) [8]
?(r) =
(LeClerc) [8]
(Clipped-loss) [9]
r2
1+r 2
|r|
1+|r|
`(r) = r2
?(r) = 1 ? exp(?`(r))
?(r) = max(1, `(r))
`(r) = |r|
`(?) convex
`(?) convex
?
?(?) = ( ? ? 1)2
?
?(?) = ( ? ? 1)2
?(?) = ? log ? ? ? + 1
?(?) = 1 ? ?.
(10)
(11)
(12)
(13)
Appendix D in the supplement demonstrates how one can represent general functions ? in the form
(9), not just specific examples, significantly extending [8] with a general characterization.
A function ? is self-concordant if |?000 (r)| ? 2?00 (r)3/2 ; see e.g. [22, Ch.9].
A bounded function obviously cannot be convex over an unbounded domain unless it is constant.
7
When n0 approaches n/2 the breakdown of (8) approaches 1/2 [7].
8
We use ?C (?) to denote the indicator for the point set C; i.e., ?C (?) = 0 if ? ? C, otherwise ?C (?) = ?.
5
6
4
Therefore, all of the previous forms of regularized empirical risk minimization, whether with a
convex or bounded loss ?, can be easily expressed using only convex base losses ` and convex
auxiliary functions ?, as follows
??VM
f?VM
? VM
?
? arg min min ? T `(y ? X?) + 1T ?(?) + ?2 k?k1 k?k22
??? 0???1
Pn
?
2
? arg min min
i=1 {?i `(yi ? f (xi )) + ?(?i )} + 2 k?k1 kf kH
f ?H 0???1
?
arg min min ? T `(y ? K?) + 1T ?(?) + ?2 k?k1 ?T K?.
? 0???1
(14)
(15)
(16)
Note that we have added a regularizer k?k1 /n, which increases robustness by encouraging ? weights
to prefer small values (but adaptively increase on indices with small loss). This particular form of
regularization has two advantages: (i) it is a smooth function of ? on 0 ? ? ? 1 (since k?k1 = 1T ?
in this case), and (ii) it enables a tight convex approximation strategy, as we will see below.
Note that other forms of robust regression can be expressed in a similar framework. For example,
generalized M-estimation (GM-estimation) can be formulated simply by forcing each ?i to take on
a specific value determined by kxi k or ri [7], ignoring the auxilary function ?. Least Trimmed Loss
(8) can be expressed in the form (9) provided only that we add a shared constraint over ?:
??LT L
? arg min
min
??? 0???1:1T ?=n0
? T `(r) + ?(?)
(17)
where ?(?i ) = 1 ? ?i and n0 < n specifies the number of terms to consider in the sum of losses.
Since ? ? {0, 1}n at a solution (see e.g. [9]), (17) is equivalent to (8) if ? is the clipped loss (13).
These formulations are all convex in the parameters given the auxiliary weights, and vice versa.
However, they are not jointly convex in the optimization variables (i.e. in ? and ?, or in ? and ?).
Therefore, one is not assured that the problems (14)?(16) have only global minima; in fact local
minima exist and global minima cannot be easily found (or even verified).
4
Computationally Efficient Approximation
We present a general approximation strategy for the variational regression estimators above that can
guarantee polynomial run-time while ensuring certain robustness and consistency properties. The
approximation is significantly tighter than the existing work [9], which allows us to achieve stronger
guarantees while providing better empirical performance. In developing our estimator we follow
standard methodology from combinatorial optimization: given an intractable optimization problem,
first formulate a (hopefully tight) convex relaxation that provides a lower bound on the objective,
then round the relaxed minimizer back to the feasible space, hopefully verifying that the rounded
solution preserves desirable properties, and finally re-optimize the rounded solution to refine the
result; see e.g. [23].
To maintain generality, we formulate the approximate estimator in the RKHS setting. Consider (16).
Although the problem is obviously convex in ? given ?, and vice versa, it is not jointly convex (recall
the assumption that ` and ? are both convex functions). This suggests that an obvious computational
strategy for computing the estimator (16) is to alternate between ? and ? optimizations (or use
heuristic methods [2]), but this cannot guarantee anything other than local solutions (and thus may
not even achieve any of the desired theoretical properties associated with the estimator).
Reformulation: We first need to reformulate the problem to allow a tight relaxation. Let ?(?)
denote putting a vector ? on the main diagonal of a square matrix, and let ? denote componentwise
multiplication. Since ` is closed and convex by assumption, we know that `(r) = sup? ?r ? ?`? (?),
where `? is the Fenchel conjugate of ` [22]. This allows (16) to be reformulated as follows.
Lemma 1. min min ? T `(y ? K?) + 1T ?(?) + ?2 k?k1 ?T K?
(18)
0???1 ?
1 T
T
= min sup 1T ?(?) ? ? T (`? (?) ? ?(y)?) ? 2?
? K ? (?k?k?1
(19)
1 ? ) ?,
0???1 ?
where the function evaluations are componentwise. (Proof given in Appendix A of the supplement.)
Although no relaxation has been introduced, the new form (25) has a more convenient structure.
5
T
Relaxation: Let N = ?k?k?1
1 ? and note that, since 0 ? ? ? 1, N must satisfy a number of
useful properties. We can summarize these by formulating a constraint set N ? N? given by:
N? = {N : N < 0, N 1 = ?, rank(N ) = 1}
(20)
M? = {M : M < 0, M 1 = ?, tr(M ) ? 1}.
(21)
Unfortunately, the set N? is not convex because of the rank constraint. However, relaxing this
constraint leads to a set M? ? N? which preserves much of the key structure, as we verify below.
?
1
(22)
Lemma 2. (25) = min min sup 1T ?(?) ? ? T (` (?) ? ?(y)?) ? 2? ? T (K ? N ) ?
0???1 N ?N?
?
min
?
min sup 1T ?(?) ? ? T (`? (?) ? ?(y)?) ?
0???1 M ?M?
?
1 T
2? ?
(K ? M ) ?. (23)
using the fact that N? ? M? . (Proof given in Appendix A of the supplement.)
Crucially, the constraint set {(?, M ) : 0 ? ? ? 1, M ? M? } is jointly convex in ? and M , thus
(35) is a convex-concave min-max problem. To see why, note that the inner objective function is
jointly convex in ? and M , and concave in ?. Since a pointwise maximum of convex functions is
convex, the problem is convex in (?, M ) [22, Ch.3]. We conclude that all local minima in (?, M )
are global. Therefore, (35) provides the foundation for an efficiently solvable relaxation.
Rounding: Unfortunately the solution to M in (35) does not allow direct recovery of an estimator
? achieving the same objective value in (24), unless M satisfies rank(M ) = 1. In general we first
need to round M to a rank 1 solution. Fortunately, a trivial rounding procedure is available: we
simply use ? (ignoring M ) and re-solve for ? in (24). This is equivalent to replacing M with the
? = ?k?k?1 ? T ? N? , which restores feasibility in the original problem. Of course,
rank 1 matrix N
1
such a rounding step will generally increase the objective value.
Reoptimization: Finally, the rounded solution can be locally improved by alternating between ?
?
and ? updates in (24) (or using any other local optimization method), yielding the final estimate ?.
5
Properties
Although a tight a priori bound on the size of the optimality gap is difficult to achieve, a rigorous
bound on the optimality gap can be recovered post hoc once the re-optimized estimator is computed.
Let R0 denote the minimum value of (24) (not efficiently computable); let R1 denote the minimum
value of (35) (the relaxed solution); let R2 denote the value of (24) achieved by freezing ? from
the relaxed solution but re-optimizing ? (the rounded solution); and finally let R3 denote the value
of (24) achieved by re-optimizing ? and ? from the rounded solution (the re-optimized solution).
Clearly we have the relationships R1 ? R0 ? R3 ? R2 . An upper bound on the relative optimality
gap of the final solution (R3 ) can be determined by (R3 ? R0 )/R3 ? (R3 ? R1 )/R3 , since R1 and
R3 are both known quantities.
Tractability: Under mild assumptions on ` and ?, computation of the approximate estimator (solving the relaxed problem, rounding, then re-optimizing) admits a polynomial-time solution; see Appendix E in the supplement. (Appendix E also provides details for an efficient implementation for
solving (35).) Once ? is recovered from the relaxed solution, the subsequent optimizations of (24)
can be solved efficiently under weak assumptions about ` and ?; namely that they both satisfy the
self-concordance and polynomial-time computation properties discussed in Section 2.
Robustness: Despite the approximation, the relaxation remains sufficiently tight to preserve some
of the robustness properties of bounded loss minimization. To establish the robustness (and consistency) properties, we will need to make use of a specific technical definition of outliers and inliers.
Definition 2 (Outliers and Inliers). For an L-Lipschitz loss `, an outlier is a point (xi , yi ) that
satisfies `(yi ) > L2 Kii /(2?) ? ? 0 (0), while an inlier satisfies `(yi ) + L2 Kii /(2?) < ?? 0 (1).
Theorem 3. Assume the loss ? is bounded and has a variational representation (9) such that `
is Lipschitz-continuous and ? 0 is bounded. Also assume there is at least one (unperturbed) inlier,
and consider the perturbation of a single data point (y1 , x1 ). Under the following conditions, the
rounded (re-optimized) estimator maintains bounded response:
(i) If either y1 remains bounded, or ?(x1 , x1 ) remains bounded.
(ii) If |y1 | ? ?, ?(x1 , x1 ) ? ? and `(y1 )/?(x1 , x1 ) ? ?.
(Proof given in Appendix B of the supplement.)
6
Methods
L2
L1
Huber
LTS
GemMc
[9]
AltBndL2
AltBndL1
CvxBndL2
CvxBndL1
Outlier Probability
p = 0.4
p = 0.2
p = 0.0
43.5 (13) 57.6 (21.21) 0.52 (0.01)
4.89 (2.81) 3.6
(2.04)
0.52 (0.01)
4.89 (2.81) 3.62 (2.02) 0.52 (0.01)
6.72 (7.37) 8.65 (14.11) 0.52 (0.01)
0.53 (0.03) 0.52 (0.02) 0.52 (0.01)
0.52 (0.01) 0.52 (0.01) 0.52 (0.01)
0.52 (0.01) 0.52 (0.01) 0.52 (0.02)
0.73 (0.12) 0.74 (0.16) 0.52 (0.01)
0.52 (0.01) 0.52 (0.01) 0.52 (0.01)
0.53 (0.02) 0.55 (0.05) 0.52 (0.01)
Table 1: RMSE on clean test data for an artificial data set with 5 features and 100 training points,
with outlier probability p, and 10000 test data points. Results are averaged over 10 repetitions.
Standard deviations are given in parentheses.
Note that the latter condition causes any convex loss ` to demonstrate unbounded response (see
proof of Theorem 5 in Appendix B). Therefore, the approximate estimator is strictly more robust (in
terms of bounded response) than regularized empirical risk minimization with a convex loss `.
Consistency: Finally, we can establish consistency of the approximate estimator in a limited albeit
non-trivial setting, although we have yet to establish it generally.
Theorem 4. Assume ` is Lipschitz-continuous and ?(?) = 1 ? ?. Assume that the data is generated
from a mixture of inliers and outliers, where P (inlier) > P (outlier). Then the estimate ?? produced
by the rounded (re-optimized) method is loss consistent.(Proof given in Appendix C.2.)
6
Experimental Evaluation
We conducted a set of experiments to evaluate the effectiveness of the proposed method compared
to standard methods from the literature. Our experimental evaluation was conducted in two parts:
first a synthetic experiment where we could control data generation, then an experiment on real data.
The first synthetic experiment was conducted as follows. A target weight vector ? was drawn from
N (0, I), with Xi: sampled uniformly from [0, 1]m , m = 5, and outputs yi computed as yi =
Xi: ? + i , i ? N (0, 12 ). We then seeded the data set with outliers by randomly re-sampling each
yi and Xi: from N (0, 108 ) and N (0, 104 ) respectively, governed by an outlier probability p. Then
we randomly sampled 100 points as the training set and another 10000 samples are used for testing.
We implemented the proposed method with two different base losses, L2 and L1 , respectively; referring to these as CvxBndL2 and CvxBndL1. We compared to standard L2 and L1 loss minimization,
as well as minimizing the Huber minimax loss (Huber) [4]. We also considered standard methods from the robust statistics literature, including the least trimmed square method (LTS) [7, 24],
and bounded loss minimization based on the Geman-McClure loss (GemMc) [8]. Finally we also
compared to the alternating minimization strategies outlined at the end of Section 3 (AltBndL2 and
AltBndL1 for L2 and L1 losses respectively), and implemented the strategy described in [9]. We
added the Tikhonov regularization to each method and the regularization parameter ? was selected
(optimally for each method) on a separate validation set. Note that LTS has an extra parameter n0 ,
which is the number of inliers. The ideal setting n0 = (1 ? p)n was granted to LTS. We also tried
30 random restarts for LTS and picked the best result.
All experiments are repeated 10 times and the average root mean square errors (RMSE) (with standard deviations) on the clean test data are reported in Table 1. For p = 0 (i.e. no outliers), all methods
perform well; their RMSEs are close to optimal (1/2, the standard deviation of i ). However, when
outliers start to appear, the result of least squares is significantly skewed, while the results of classic robust statistics methods, Huber, L1 and LTS, indeed turn out to be more robust than the least
squares, but nevertheless are still affected significantly. Both implementations of the new method
performs comparably to the the non-convex Geman-McClure loss while substantially improving the
alternating strategy under the L1 loss. Note that the latter improvement clearly demonstrates that
7
Methods
L2
L1
Huber
LTS
GemMc
[9]
AltBndL2
AltBndL1
CvxBndL2
CvxBndL1
Gap(Cvx2)
Gap(Cvx1)
cal-housing
1185 (124.59)
1303 (244.85)
1221 (119.18)
533 (398.92)
28 (88.45)
967 (522.40)
967 (522.40)
1005 (603.00)
9
(0.64)
8
(0.28)
2e-12 (3e-12)
0.005
(0.01)
Datasets
abalone
pumadyn
7.93
(0.67)
1.24 (0.42)
7.30
(0.40)
1.29 (0.42)
7.73
(0.49)
1.24 (0.42)
755.1 (126)
0.32 (0.41)
2.30
(0.01)
0.12 (0.12)
8.39
(0.54)
0.81 (0.77)
8.39
(0.54)
0.81 (0.77)
7.30
(0.40)
1.29 (0.42)
7.60
(0.86)
0.07 (0.07)
2.98
(0.08)
0.08 (0.07)
3e-9
(4e-9)
0.025 (0.052)
0.001 (0.001) 0.267 (0.269)
bank-8fh
18.21 (6.57)
6.54 (3.09)
7.37 (3.18)
10.96 (6.67)
0.93 (0.80)
3.91 (6.18)
7.74 (9.40)
1.61 (2.51)
0.20 (0.05)
0.10 (0.07)
0.001 (0.003)
0.011 (0.028)
Table 2: RMSE on clean test data for 108 training data points and 1000 test data points, with 10 repeats. Standard deviations shown parentheses. The mean gap values of CvxBndL2 and CvxBndL1,
Gap(Cvx2) and Gap(Cvx1) respectively, are given in the last two rows.
alternating can be trapped in poor local minima. The proposal from [9] was not effective in this
setting (which differed from the one investigated there).
Next, we conducted an experiment on four real datasets taken from the StatLib repository9 and
DELVE.10 For each data set, we randomly selected 108 points as the training set, and another random
1000 points as the test set. Here the regularization constant is tuned by 10-fold cross validation. To
seed outliers, 5% of the training set are randomly chosen and their X and y values are multiplied
by 100 and 10000, respectively. All of these data sets have 8 features, except pumadyn which has
32 features. We also estimated the scale factor on the training set by the mean absolute deviation
method, a common method in robust statistics [3]. Again, the ideal parameter n0 = (1 ? 5%)n is
granted to LTS and 30 random restarts are performed.
The RMSE on test set for all methods are reported in Table 2. It is clear that all methods based on
convex losses (L2, L1, Huber) suffer significantly from the added outliers. The method proposed in
this paper consistently outperform all other methods with a noticeable margin, except on the abalone
data set where GemMc performs slightly better.11 Again, we observe evidence that the alternating
strategy can be trapped in poor local minima, while the method from [9] was less effective. We also
measured the relative optimality gaps for the approximate CvxBnd procedures. The gaps were quite
small in most cases (the gaps were very close to zero in the synthetic case, and so are not shown),
demonstrating the tightness of the proposed approximation scheme.
7
Conclusion
We have developed a new robust regression method that can guarantee a form of robustness (bounded
response) while ensuring tractability (polynomial run-time). The estimator has been proved consistent under some restrictive but non-trivial conditions, although we have not established general consistency. Nevertheless, an empirical evaluation reveals that the method meets or surpasses the generalization ability of state-of-the-art robust regression methods in experimental studies. Although
the method is more computationally involved than standard approaches, it achieves reasonable scalability in real problems. We are investigating whether the proposed estimator achieves stronger
robustness properties, such as high breakdown or bounded influence. It would be interesting to extend the approach to also estimate scale in a robust and tractable manner. Finally, we continue to
investigate whether other techniques from the robust statistics and machine learning literatures can
be incorporated in the general framework while preserving desired properties.
Acknowledgements
Research supported by AICML and NSERC.
9
http://lib.stat.cmu.edu/datasets/
http://www.cs.utoronto.ca/ delve/data/summaryTable.html
11
Note that we obtain different results than [9] arising from a very different outlier process.
10
8
References
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on Influence Functions. Wiley, 1986.
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9
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3,931 | 4,559 | Multi-Task Averaging
Sergey Feldman, Maya R. Gupta, and Bela A. Frigyik
Department of Electrical Engineering
University of Washington
Seattle, WA 98103
Abstract
We present a multi-task learning approach to jointly estimate the means of multiple independent data sets. The proposed multi-task averaging (MTA) algorithm
results in a convex combination of the single-task averages. We derive the optimal
amount of regularization, and show that it can be effectively estimated. Simulations and real data experiments demonstrate that MTA outperforms both maximum likelihood and James-Stein estimators, and that our approach to estimating
the amount of regularization rivals cross-validation in performance but is more
computationally efficient.
1
Introduction
The motivating hypothesis behind multi-task learning (MTL) algorithms is that leveraging data from
related tasks can yield superior performance over learning from each task independently. Early
evidence for this hypothesis is Stein?s work on the estimation of the means of T distributions (tasks)
[1]. Stein showed that it is better (in a summed squared error sense) to estimate each of the means
of T Gaussian random variables using data sampled from all of them, even if they are independent
and have different means. That is, it is beneficial to consider samples from seemingly unrelated
distributions in the estimation of the tth mean. This surprising result is often referred to as Stein?s
paradox [2].
Estimating means is perhaps the most common of all estimation tasks, and often multiple means
need to be estimated. In this paper we consider a multi-task regularization approach to the problem
of estimating multiple means that we call multi-task averaging (MTA). We show that MTA has
provably nice theoretical properties, is effective in practice, and is computationally efficient. We
define the MTA objective in Section 2, and review related work in Section 3. We present some
key properties of MTA in Section 4 (proofs are omitted due to space constraints). In particular, we
state the optimal amount of regularization to be used, and show that this optimal amount can be
effectively estimated. Simulations in Section 5 verify the advantage of MTA over standard sample
means and James-Stein estimation if the true means are close compared to the sample variance. In
Section 6.1, two experiments estimating expected sales show that MTA can reduce real errors by
over 30% compared to the sample mean. MTA can be used anywhere multiple averages are needed;
we demonstrate this by applying it fruitfully to the averaging step of kernel density estimation in
Section 6.1.
2
Multi-Task Averaging
Consider the T -task problem of estimating the means of T random variables that have finite mean
t
and variance. Let {Yti }N
i=1 be Nt independent and identically-distributed random samples for t =
1, . . . , T . The MTA objective and many of the results in this paper generalize trivially to samples that
are vectors rather than scalars, but for notational simplicity we restrict our focus to scalar samples
Yti ? R. Key notation is given in Table 1.
1
Table 1: Key Notation
T
Nt
Yti ? R
Y?t ? R
Yt? ? R
?t2
?
A ? RT ?T
L=D?A
number of tasks
number of samples for tth task
ith random sample from
Ptth task
tth sample average N1t i Yti
MTA estimate of tth mean
variance of the tth task
?2
diagonal covariance matrix of Y? with ?tt = Ntt
pairwise task similarity matrix
PT
graph Laplacian of A, with diagonal D s.t. Dtt = r=1 Atr
In addition, assume that the T ? T matrix A describes the relatedness or similarity of any pair of the
T tasks, with Att = 0 for all t without loss of generality (because the diagonal self-similarity terms
are canceled in the objective below). The proposed MTA objective is
{Yt? }Tt=1 = arg min
{Y?t }T
t=1
T
T Nt
T
(Yti ? Y?t )2
? XX
1 XX
Ars (Y?r ? Y?s )2 .
+
T t=1 i=1
?t2
T 2 r=1 s=1
(1)
The first term minimizes the sum of the empirical losses, and the second term jointly regularizes
the estimates by regularizing their pairwise differences. The regularization parameter ? balances
the empirical risk and the multi-task regularizer. Note that if ? = 0, then (1) decomposes to T
separate minimization problems, producing the sample averages Y?t . The normalization of each
error term in (1) by its task-specific variance ?t2 (which may be estimated) scales the T empirical
loss terms relative to the variance of their distribution; this ensures that high-variance tasks do not
disproportionately dominate the loss term.
A more general formulation of MTA is
{Yt? }Tt=1
T Nt
1 XX
= arg min
L(Yti , Y?t ) + ?J {Y?t }Tt=1 ,
T t=1 i=1
{Y?t }T
t=1
where L is some loss function and J is a regularization function. If L is chosen to be any Bregman
loss, then setting ? = 0 will produce the T sample averages [3]. For the analysis and experiments
in this paper, we restrict our focus to the tractable squared-error formulation given in (1).
The task similarity matrix A can be specified as side information (e.g. from a domain expert), or
set in an optimal fashion. In Section 4 we derive two optimal choices of A for the T = 2 case: the
A that minimizes expected squared error, and a minimax A. We use the T = 2 analysis to propose
practical estimators of A for any number of tasks.
3
Related Work
MTA is an approach to the problem of estimating T means. We are not aware of other work in the
multi-task literature that addresses this problem; most MTL methods are designed for regression,
classification, or feature selection, e.g. [4, 5, 6]. The most closely related work is Stein estimation,
an empirical Bayes strategy for estimating multiple means simultaneously [7, 8, 2, 9]. James and
Stein [7] showed that the maximum likelihood estimate of the tth mean ?t can be dominated by
a shrinkage estimate given Gaussian assumptions. There have been a number of extensions to the
original James-Stein estimator. We compare to the positive-part residual James-Stein estimator for
multiple data points per task and independent unequal variances [8, 10], such that the estimated
mean for the tth task is
T ?3
(Y?t ? ?),
(2)
?+ 1? ?
(Y ? ?)T ??1 (Y? ? ?) +
2
where (x)+ = max(0, x); ? is a diagonal matrix of the estimated variances of each sample mean
?
?2
where ?tt = Ntt and the estimate is shrunk towards ?, which is usually set to be the mean of the
P
sample means (other choices are sometimes used) ? = Y? = T1 t Y?t . Bock?s formulation of (2)
uses the effective dimension (defined as the ratio of the trace of ? to the maximum eigenvalue of ?)
rather than the T in the numerator of (2) [8, 7, 10]. In preliminary practical experiments where ?
must be estimated from the data, we found that using the effective dimension significantly crippled
the performance of the James-Stein estimator. We hypothesize that this is due to the high variance
of the estimate of the maximum eigenvalue of ?.
MTA can be interpreted as estimating means of T Gaussians with an intrinsic Gaussian Markov
random field prior [11]. Unlike most work in graphical models, we do not assume any variables are
conditionally independent, and generally have non-sparse inverse covariance.
A key issue for MTA and many other multi-task learning methods is how to estimate the similarity
(or task relatedness) between tasks and/or samples if it is not provided. A common approach is to
estimate the similarity matrix jointly with the task parameters [12, 13, 5, 14, 15]. For example, Zhang
and Yeung [15] assumed that there exists a covariance matrix for the task relatedness, and proposed
a convex optimization approach to estimate the task covariance matrix and the task parameters in
a joint, alternating way. Applying such joint and alternating approaches to the MTA objective (1)
leads to a degenerate solution with zero similarity. However, the simplicity of MTA enables us to
specify the optimal task similarity matrix for T = 2 (see Sec. 4), which we generalize to obtain an
estimator for the general multi-task case.
4
MTA Theory
For symmetric A with non-negative components1 , the MTA objective given in (1) is continuous,
differentiable, and convex. It is straightforward to show that (1) has closed-form solution:
?1
?
Y ? = I + ?L
Y? ,
T
(3)
PNt
where Y? is the vector of sample averages with tth entry Y?t = N1t i=1
Yti , L is the graph Laplacian
of A, and ? is defined as before. With non-negative A and ?, the matrix inverse in (3) can be shown
to always exist using the Gershgorin Circle Theorem [16].
Note that the (r, s)th entry of T? ?L goes to 0 as Nt approaches infinity, and since matrix inversion
?1
is a continuous operation, I + T? ?L
? I in the norm. By the law of large numbers one can
conclude that Y ? asymptotically approaches the true means.
4.1
Convexity of MTA Solution
From inspection of (3), it is clear that each of the elements of Y ? is a linear combination of the
sample averages Y? . However, a stronger statement can be made:
?2
Theorem: If ? ? 0, 0 ? Ars < ? for all r, s and 0 < Ntt < ? for all t, then the MTA estimates
{Yt? } given in (3) are a convex combination of the task sample averages {Y?t }.
?1
exists and is
Proof Sketch: The theorem requires showing that the matrix W = I + T? ?L
right-stochastic. Using the Gershgorin Circle Theorem [16], we can show that the real part of every
eigenvalue of W ?1 is positive. The matrix W ?1 is a Z-matrix [17], and if the real part of each of
the eigenvalues of a Z-matrix is positive, then its inverse has all non-negative entries (See Chapter
6, Theorem 2.3, G20 , and N38 , [17]). Finally, to prove that W has rows that sum
to 1, first note that
by definition the rows of the graph Laplacian L sum to zero. Thus I + T? ?L 1 = 1, and because
?1
we established invertibility, this implies the desired right-stochasticity: 1 = I + T? ?L
1.
1
If an asymmetric A is provided, using it with MTA is equivalent to using the symmetric (AT + A)/2.
3
4.2
Optimal A for the Two Task Case
In this section we analyze the T = 2 task case, with N1 and N2 samples for tasks 1 and 2 respectively. Suppose {Y1i } are iid (independently and identically distributed) with finite mean ?1 and
finite variance ?12 , and {Y2i } are iid with finite mean ?2 = ?1 + ? and finite variance ?22 . Let the
task-relatedness matrix be A = [0 a; a 0], and without loss of generality, we fix ? = 1. Then the
closed-form solution (3) can be simplified:
?
?
?
?
?22
?12
T
+
a
a
N2
N1
? Y?1 + ?
? Y?2 .
Y1? = ?
(4)
?2
?2
?2
?2
T + N11 a + N22 a
T + N11 a + N22 a
It is straightforward to derive the mean squared error of Y1? :
?
?
?22
?4
?12 ?22 2
?24 2
2
2
T
+
2T
?2 N12 a2
a
+
a
+
a
2
?1 ?
N2
N1 N2
N2
?
1
?+
MSE[Y1 ] =
.
?12
?22
?12
?2
N1
2
(T + N1 a + N2 a)
(T + N1 a + N22 a)2
(5)
Comparing to the MSE of the sample average, one obtains the following relationship:
?2
?2
4
MSE[Y1? ] < MSE[Y?1 ] if ?2 ? 1 ? 2 < ,
N1
N2
a
(6)
Thus the MTA estimate of the first mean has lower MSE if the squared mean-separation ?2 is small
compared to the variances of the sample averages. Note that as a approaches 0 from above, the RHS
of (6) approaches infinity, which means that a small amount of regularization can be helpful even
when the difference between the task means ? is large. Summarizing, if the two task means are
close relative to each task?s sample variance, MTA will help.
The risk is the sum of the mean squared errors: MSE[Y1? ]+MSE[Y2? ], which is a convex, continuous,
and differentiable function of a, and therefore the first derivative can be used to specify the optimal
value a? , when all the other variables are fixed. Minimizing MSE[Y1? ] + MSE[Y2? ] w.r.t. a one
obtains the following solution:
2
(7)
a? = 2 ,
?
which is always non-negative.
Analysis of the second derivative shows that this minimizer always holds for the cases of interest
(that is, for N1 , N2 ? 1). In the limit case, when the difference in the task means ? goes to zero
(while ?t2 stay constant), the optimal task-relatedness a? goes to infinity, and the weights in (4) on
Y?1 and Y?2 become 1/2 each.
4.3
Estimating A from Data
Based on our analysis of the optimal A for the two-task case, we propose two methods to estimate
A from data for arbitrary T . The first method is designed to minimize the approximate risk using a
constant similarity matrix. The second method provides a minimax estimator. With both methods we
can use the Sherman-Morrison formula to avoid taking the matrix inverse in (3), and the computation
of Y ? is O(T ).
4.3.1
Constant MTA
Recalling that E[Y? Y? T ] = ??T + ?, the risk of estimator Y? = W Y? of unknown parameter vector
? for the squared loss is the sum of the mean squared errors:
R(?, W Y? ) = E[(W Y? ? ?)T (W Y? ? ?)] = tr(W ?W T ) + ?T (I ? W )T (I ? W )?.
(8)
One approach to generalizing the results of Section 4.2 to arbitrary T is to try to find a symmetric,
non-negative matrix A such that the (convex, differentiable) risk R(?, W Y? ) is minimized for W =
?1
I + T? ?L
(recall L is the graph Laplacian of A). The problem with this approach is two-fold:
(i) the solution is not analytically tractable for T > 2 and (ii) an arbitrary A has T (T ? 1) degrees
of freedom, which is considerably more than the number of means we are trying to estimate in
4
the first place. To avoid these problems, we generalize the two-task results by constraining A to
be a scaled constant matrix A = a11T , and find the optimal a? that minimizes the risk in (8). In
addition, w.l.o.g. we set ? to 1, and for analytic tractability we assume that all the tasks have the
same variance, estimating ? as tr(?)
T I. Then it remains to solve:
?1 !
1
tr(?)
a? = arg min R ?, I +
L(a11T )
Y? ,
T T
a
which has the solution
a? =
1
T (T ?1)
PT
r=1
2
PT
s=1 (?r
? ?s )2
,
which reduces to the optimal two task MTA solution (7) when T = 2. In practice, one of course
does not have {?r } as these are precisely the values one is trying to estimate. So, to estimate a? we
PT 2PT
use the sample means {?
yr }: a
?? =
. Using this estimated optimal constant
1
(?
y ??
y )2
T (T ?1)
r=1
s=1
r
s
? produces what we refer to as the constant MTA
similarity and an estimated covariance matrix ?
estimate
?1
1?
T
?
?
Y = I + ?L(?
a 11 )
Y? .
(9)
T
Note that we made the assumption that the entries of ? were the same in order to be able to derive
? used with a
the constant similarity a? , but we do not need nor suggest that assumption on the ?
?? in
(9).
4.4
Minimax MTA
Bock?s James-Stein estimator is minimax in that it minimizes the worst-case loss, not necessarily the
expected loss [10]. This leads to a more conservative use of regularization. In this section, we derive
a minimax version of MTA, that prescribes less regularization than the constant MTA. Formally, an
estimator Y M of ? is called minimax if it minimizes the maximum risk:
inf sup R(?, Y? ) = sup R(?, Y M ).
Y?
?
?
First, we will specify minimax MTA for the T = 2 case. To find a minimax estimator Y M it is
sufficient to show that (i) Y M is a Bayes estimator w.r.t. the least favorable prior (LFP) and (ii)
it has constant risk [10]. To find a LFP, we first need to specify a constraint set for ?t ; we use an
interval: ?t ? [bl , bu ], for all t, where bl ? R and bu ? R. With this constraint set the minimax
estimator is:
?1
2
T
M
Y = I+
?L(11 )
Y? ,
(10)
T (bu ? bl )2
which reduces to (7) when T = 2. This minimax analysis is only valid for the case when T = 2,
but we found that good practical results for larger T using (10) with the data-dependent interval
?bl = mint y?t and ?bu = maxt y?t .
5
Simulations
We first illustrate the performance of the proposed MTA using Gaussian and uniform simulations
so that comparisons to ground truth can be made. Simulation parameters are given in the table in
Figure 1, and were set so that the variances of the distribution of the true means were the same in
both types of simulations. Simulation results are reported in Figure 1 for different values of ??2 ,
which determines the variance of the distribution over the means.
We compared constant MTA and minimax MTA to single-task sample averages and to the JamesStein estimator given in (2). We also compared to a randomized 5-fold 50/50 cross-validated (CV)
version of constant MTA, and minimax MTA, and the James-Stein estimator (which is simply a convex regularization towards the average of the sample means: ??
yt +(1??)y?.). For the cross-validated
versions, we randomly subsampled Nt /2 samples and chose the value of ? for constant/minimax
5
Gaussian Simulations
?t ? N (0, ??2 )
?t2 ? Gamma(0.9, 1.0) + 0.1
Nt ? U {2, . . . , 100}
yti ? N (?t , ?t2 )
Uniform Simulations
q
q
?t ? U (? 3??2 , 3??2 )
?t2 ? U (0.1, 2.0)
Nt ? U {2, . . .p
, 100}
p
yti ? U [?t ? 3?t2 , ?t + 3?t2 ]
T=2
T=2
10
% change vs. single?task
% change vs. single?task
10
0
?10
?20
?30
?40
?50
0
0.5
Single?Task
James?Stein
MTA, constant
MTA, minimax
James?Stein (CV)
MTA, constant (CV)
MTA, minimax (CV)
1
1.5
2
2.5
3
2
?? (variance of the means)
0
?10
?20
?30
?40
?50
0
0.5
Single?Task
James?Stein
MTA, constant
MTA, minimax
James?Stein (CV)
MTA, constant (CV)
MTA, minimax (CV)
1
1.5
2
2.5
3
2
?? (variance of the means)
T=5
T=5
10
% change vs. single?task
% change vs. single?task
10
0
?10
?20
?30
?40
?50
0
0.5
1
1.5
2
2.5
?2 (variance of the means)
0
?10
?20
?30
?40
?50
0
3
0.5
?
T = 25
T = 25
10
% change vs. single?task
% change vs. single?task
3
?
10
0
?10
?20
?30
?40
?50
0
1
1.5
2
2.5
?2 (variance of the means)
0.5
1
1.5
2
2.5
?2? (variance of the means)
0
?10
?20
?30
?40
?50
0
3
0.5
1
1.5
2
2.5
?2? (variance of the means)
3
Figure 1: Average (over 10000 random draws) percent change in risk vs. single-task. Lower is
better.
MTA or ? for James-Stein that resulted in the lowest average left-out risk compared to the sample
mean estimated with all Nt samples. In the optimal versions of constant/minimax MTA, ? was set
to 1, as this was the case during derivation.
We used the following parameters for CV: ? ? {2?5 , 2?4 , . . . , 25 } for the MTA estimators and a
?
comparable set of ? spanning (0, 1) by the transformation ? = ?+1
. Even when cross-validating,
an advantage of using the proposed constant MTA or minimax MTA is that these estimators provide
?
a data-adaptive scale for ?, where ? = 1 sets the regularization parameter to be aT or T (bu1?bl )2 ,
respectively.
Some observations from Figure 1: further to the right on the x-axis, the means are more likely to be
further apart, and multi-task approaches help less on average. For T = 2, the James-Stein estimator
reduces to the single-task estimator, and is of no help. The MTA estimators provide a gain while
6
??2 < 1 but deteriorates quickly thereafter. For T = 5, constant MTA dominates in the Gaussian
case, but in the uniform case does worse than single-task when the means are far apart. Note that for
all T > 2 minimax MTA almost always outperforms James-Stein and always outperforms singletask, which is to be expected as it was designed conservatively. For T = 25, we see the trend that
all estimators benefit from an increase in the number of tasks.
For constant MTA, cross-validation is always worse than the estimated optimal regularization. Since
both constant MTA and minimax MTA use a similarity matrix of all ones scaled by a constant, crossvalidating over a set of possible ? may result in nearly identical performance, and this can be seen in
the Figure (i.e. the green and blue dotted lines are superimposed). To conclude, when the tasks are
close to each other compared to their variances, constant MTA is the best estimator to use by a wide
margin. When the tasks are farther apart, minimax MTA will provide a win over both James-Stein
and maximum likelihood.
6
Applications
We present two applications with real data. The first application parallels the simulations, estimating
expected values of sales of related products. The second application uses MTA for multi-task kernel
density estimation, highlighting the applicability of MTA to any algorithm that uses sample averages.
6.1
Application: Estimating Product Sales
We consider two multi-task problems using sales data over a certain time period supplied by Artifact
Puzzles, a company that sells jigsaw puzzles online. For both problems, we model the given samples
as being drawn iid from each task.
The first problem estimates the impact of a particular puzzle on repeat business: ?Estimate how
much a random customer will spend on an order on average, if on their last order they purchased
the tth puzzle, for each of T = 77 puzzles.? The samples were the amounts different customers had
spent on orders after buying each of the t puzzles, and ranged from 480 down to 0 for customers that
had not re-ordered. The number of samples for each puzzle ranged from Nt = 8 to Nt = 348.
The second problem estimates the expected order size of a particular customer: ?Estimate how much
the tth customer will spend on a order on average, for each of the T = 477 customers that ordered
at least twice during the data timeframe.? The samples were the order amounts for each of the T
customers. Order amounts varied from 15 to 480. The number of samples for each customer ranged
from Nt = 2 to Nt = 17.
There is no ground truth. As a metric to compare the estimates, we treat each task?s sample average
computed from all of the samples as the ground truth, and compare to estimates computed from a
uniformly randomly chosen 50% of the samples. Results in Table 2 are averaged over 1000 random
draws of the 50% used for estimation. We used 5-fold cross-validation with the same parameter
choices as in the simulations section.
Table 2: Percent change in average risk (for puzzle and buyer data, lower is better), and mean
reciprocal rank (for terrorist data, higher is better).
Estimator
Pooled Across Tasks
James-Stein
James-Stein (CV)
Constant MTA
Constant MTA (CV)
Minimax MTA
Minimax MTA (CV)
Expert MTA
Expert MTA (CV)
Puzzles
T = 77
181.67%
-6.87%
-21.18%
-17.48%
-21.65%
-8.41%
-19.83 %
-
Customers
T = 477
109.21%
-14.04%
-31.01%
-32.29%
-30.89%
-2.96%
-25.04%
7
Suicide Bombings
T =7
0.13
0.15
0.15
0.19
0.19
0.19
0.19
0.19
0.19
6.2
Density Estimation for Terrorism Risk Assessment
MTA can be used whenever multiple averages are taken. In this section we present multi-task
kernel density estimation, as an application of MTA. Recall that for standard single-task kernel
density estimation (KDE) [18], a set of random samples xi ? Rd , i ? {1, . . . , N } are assumed
to be iid from an unknown distribution pX , and the problem is to estimate the density for a query
sample, z ? Rd . Given a kernel function K(xi , xj ), the un-normalized single-task KDE estimate is
PN
p?(z) = N1 i=1 K(xi , z), which is just a sample average.
When multiple kernel densities {pt (z)}Tt=1 are estimated for the same domain, we replace the multiple sample averages with MTA estimates, which we refer to as multi-task kernel density estimation
(MT-KDE).
We compared KDE and MT-KDE on a problem of estimating the probability of terrorist events in
Jerusalem using the Naval Research Laboratory?s Adversarial Modeling and Exploitation Database
(NRL AMX-DB). The NRL AMX-DB combined multiple open primary sources2 to create a rich
representation of the geospatial features of urban Jerusalem and the surrounding region, and accurately geocoded locations of terrorist attacks. Density estimation models are used to analyze the
behavior of such violent agents, and to allocate security and medical resources. In related work,
[19] also used a Gaussian kernel density estimate to assess risk from past terrorism events.
The goal in this application is to estimate a risk density for 40,000 geographical locations (samples)
in a 20km ? 20km area of interest in Jerusalem. Each geographical location is represented by a
d = 76-dimensional feature vector. Each of the 76 features is the distance in kilometers to the
nearest instance of some geographic location of interest, such as the nearest market or bus stop.
Locations of past events are known for 17 suicide bombings. All the events are attributed to one of
seven terrorist groups. The density estimates for these seven groups are expected to be related, and
are treated as T = 7 tasks.
The kernel K was taken to be a Gaussian kernel with identity covariance. In addition to constant A
and minimax A, we also obtained a side-information A from terrorism expert Mohammed M. Hafez
of the Naval Postgraduate School; he assessed the similarity between the seven groups during the
Second Intifada (the time period of the data), providing similarities between 0 and 1.
We used leave-one-out cross validation to assess KDE and MT-KDE for this problem, as follows.
After computing the KDE and MT-KDE density estimates using all but one of the training examples
{xti } for each task, we sort the resulting 40,000 estimated probabilities for each of the seven tasks,
and extract the rank of the left-out known event. The mean reciprocal rank (MRR) metric is reported
in Table 2. Ideally, the MRR of the left-out events would be as close to 1 as possible, and indicating
that the location of the left-out event is at high-risk. The results show that the MRR for MT-KDE
are lower or not worse than those for KDE for both problems; there are, however, too few samples
to verify statistical significance of these results.
7
Summary
Though perhaps unintuitive, we showed that both in theory and in practice, estimating multiple unrelated means using an MTL approach can improve the overall risk, even more so than James-Stein
estimation. Averaging is common, and MTA has potentially broad applicability as a subcomponent in many algorithms, such as k-means clustering, kernel density estimation, or non-local means
denoising.
Acknowledgments
We thank Peter Sadowski, Mohammed Hafez, Carol Chang, Brian Sandberg and Ruth Wilis for help
with preliminary experiments and access to the terorrist dataset.
2
Primary sources included the NRL Israel Suicide Terrorism Database (ISD) cross referenced with open
sources (including the Israel Ministry of Foreign Affairs, BBC, CPOST, Daily Telegraph, Associated Press,
Ha?aretz Daily, Jerusalem Post, Israel National News), as well as the University of New Haven Institute for
the Study of Violent Groups, the University of Maryland Global Terrorism Database, and the National Counter
Terrorism Center Worldwide Incident Tracking System.
8
References
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9
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3,932 | 456 | Shooting Craps in Search of an Optimal Strategy for
Training Connectionist Pattern Classifiers
J. B. Hampshire IT
and B. V. K. Vijaya Kumar
Department of Electrical & Computer Engineering
Carnegie Mellon University
Pittsbwgh. PA 15213-3890
[email protected]
and
[email protected]
Abstract
We compare two strategies for training connectionist (as well as nonconnectionist) models for statistical pattern recognition. The probabilistic strategy is based on the notion that Bayesian discrimination (i.e .? optimal classification) is achieved when the classifier learns the a posteriori class distributions of
the random feature vector. The differential strategy is based on the notion that
the identity of the largest class a posteriori probability of the feature vector is
all that is needed to achieve Bayesian discrimination. Each strategy is directly
linked to a family of objective functions that can be used in the supervised training
procedure. We prove that the probabilistic strategy - linked with error measure
objective functions such as mean-squared-error and cross-entropy - typically
used to train classifiers necessarily requires larger training sets and more complex
classifier architectures than those needed to approximate the Bayesian discriminant function. In contrast. we prove that the differential strategy - linked
with classificationfigure-of-merit objective functions (CF'MmoIlO) [3] - requires
the minimum classifier functional complexity and the fewest training examples
necessary to approximate the Bayesian discriminant function with specified precision (measured in probability of error). We present our proofs in the context of
a game of chance in which an unfair C-sided die is tossed repeatedly. We show
that this rigged game of dice is a paradigm at the root of all statistical pattern
recognition tasks. and demonstrate how a simple extension of the concept leads
us to a general information-theoretic model of sample complexity for statistical
pattern recognition.
1125
1126
Hampshire and Kumar
1 Introduction
Creating a connectionist pattern classifier that generalizes well to novel test data has recently
focussed on the process of finding the network architecture with the minimum functional
complexity necessary to model the training data accurately (see, for example, the works of
Baum. Cover, Haussler, and Vapnik). Meanwhile, relatively little attention has been paid to
the effect on generalization of the objective function used to train the classifier. In fact, the
choice of objective function used to train the classifier is tantamount to a choice of training
strategy, as described in the abstract [2,3].
We formulate the proofs outlined in the abstract in the context of a rigged game of dice in
which an unfair C-sided die is tossed repeatedl y. Each face of the die has some probability of
turning up. We assume that one face is always more likely than all the others. As a result, all
the probabilities may be different, but at most C - 1 of them can be identical. The objective
of the game is to identify the most likely die face with specified high confidence. The
relationship between this rigged dice paradigm and statistical pattern recognition becomes
clear if one realizes that a single unfair die is analogous to a specific point on the domain
of the randomfeature vector being classified. Just as there are specific class probabilities
associated with each point in feature vector space, each die has specific probabilities
associated with each of its faces. The number of faces on the die equals the number of
classes associated with the analogous point in feature vector space. Identifying the most
likely die face is equivalent to identifying the maximum class a posteriori probability for
the analogous point in feature vector space - the requirement for Bayesian discrimination.
We formulate our proofs for the case of a single die, and conclude by showing how a simple
extension of the mathematics leads to general expressions for pattern recognition involving
both discrete and continuous random feature vectors.
Authors' Note: In the interest of brevity, our proofs are posed as answers to questions that
pertain to the rigged game of dice. It is hoped that the reader will find the relevance of
each question/answer to statistical pattern recognition clear. Owing to page limitations, we
cannot provide our proofs in full detail; the reader seeking such detail should refer to [1],
Definitions of symbols used in the following proofs are given in table 1.
1.1 A Fixed-Point Representation
The Mq-bit approximation qM[X] to the real number x E (-1, 1] is of the form
MSB (most significant bit) = sign [x]
MSB - 1 = 2- 1
!
LSB (least significant bit)
=
(1)
2-(M.-1)
with the specific value defined as the mid-point of the 2-(M.-1) -wide interval in which x
is located:
A { sign[x] . (L Ixl . 2(M.-l) J . 2-(M.-1) + 2- M.) ,
Ixl < 1
qM[X] =
(2)
sign [x] . (1 - 2- M.) ,
Ixl = 1
The lower and upper bounds on the quantization interval are
LM.[X]
< x <
UM.[X]
(3)
An Optimal Strategy for Training Connectionist Pattern Classifiers
Thble 1: Definitions of symbols used to describe die faces, probabilities, probabilistic
differences, and associated estimates.
Symbol
Wrj
P(Wrj)
kq
P(Wrj)
Definition
The lrUe jth most likely die face (w;j is the estimated jth most likely face).
The probability of the lrUe jth most likely die face.
The number of occurrences of the true jth most likely die face.
An empirical estimate of the probability of the true jth most likely die face:
=!c;.
P(wrj)
(note n denotes the sample size)
The probabilistic difference involving the true rankings and probabilities of the
C die faces:
..1ri = P(Wri) - SUPj,..i P(Wrj)
The probabilistic difference involving the true rankings but empirically estimated
probabilities of the C die faces:
"
..1ri
= P(Wri)
-
sUPhfj
?
P(wrj)
=
k,; - lupifi ~
II
where
(4)
and
(5)
The fixed-point representation described by (1) - (5) differs from standard fixed-point
representations in its choice of quantization interval. The choice of (2) - (5) represents zero
as a negative - more precisely, a non-positive - finite precision number. See [1] for the
motivation of this format choice.
1.2 A Mathematical Comparison of the Probabilistic and Differential Strategies
The probabilistic strategy for identifying the most likely face on a die with C faces involves
estimating the C face probabilities. In order for us to distinguish P(Wrl) from P(W r2) , we
must choose Mq (i.e. the number of bits in our fiXed-point representation of the estimated
probabilities) such that
(6)
The distinction between the differential and probabilistic strategies is made more clear if
one considers the way in which the Mq-bit approximation jrl is computed from a random
sample containing krl occurrences of die face Wr l and kl'2. occurrences of die face Wr2. For
the differential strategy
.1rl dijferelltUJl
=
qM [krl : kl'2.]
(7)
and for the probabilistic strategy
.1rl probabilistic
(8)
1127
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Hampshire and Kumar
where
.d;
6.
i = 1,2, ... C
P(Wi) - sup P(Wj)
iii
(9)
Note that when i = rl
(10)
and when i
::f r 1
(ll)
Note also
(12)
Since
rC
=
L P(Wj) -
i=1
(C - 2) P(W rl)
(13)
i..,.3
we can show that the C differences of (9) yield the C probabilities by
=
P(Wrj)
~C [1 -
t
.di ]
(14)
izr2
= .drj + P(Wrl)
Vj
>1
Thus, estimating the C differences of (9) is equivalent to estimating the C probabilities
P(Wl), P(W2) , ... ,P(wc).
Clearly, the sign of L1rl in (7) is modeled correctly (i.e., L1rl differentWl can correctly identify
the most likely face) when Mq = 1, while this is typically not the case for .dr l probabilistic
in (8). In the latter case, L1rl probabilistic is zero when Mq = 1 because qm[p(Wrl)] and
QM[P(Wr'2)] are indistinguishable for Mq below some minimal value implied by (6). That
minimal value of Mq can be found by recognizing that the number of bits necessary for (6)
to hold for asymptotically large n (Le., for the quantized difference in (8) to exceed one
LSB) is
1
+
r-log2 [.drd 1,
~"
sign bit
~ +
sign bit
magnit~de bits
-log2 [P(Wrj)] :3 Z+
j E {1,2}
J
J-log2 [.drd 1 + 1)
otherwise
magnit~de bits
(15)
where Z+ represents the set of all positive integers. Note that the conditional nature of
Mq min in (15) prevents the case in which lime-+o P(Wrl) - ? = LM. [P(Wrl)] or P(W r2) =
UM.[P(W r2)]; either case would require an infinitely large sample size before the variance
of the corresponding estimated probability became small enough to distinguish QM[P(Wrl)]
from QM[P(Wr'2)]. The sign bit in (15)is not required to estimate the probabilities themselves
in (8), but it is necessary to compute the difference between the two probabilities in that
equation - this difference being the ultimate computation by which we choose the most
likely die face.
An Optimal Strategy for Training Connectionist Pattern Classifiers
1.3 The Sample Complexity Product
We introduce the sample complexity product (SCP) as a measure of both the number of
samples and the functional complexity (measured in bits) required to identify the most
likely face of an unfair die with specified probability.
A
SCP = n . Mq
s.t.
P(most likely face correctly IO'd)
~
a
(16)
2 A Comparison of the Sample Complexity Requirements for the
Probabilistic and Differential Strategies
Axiom 1 We view the number of bits Mq in the finite-precision approximation qM[X] to
the real number x E (-1, 1] as a measure of the approximation's functional complexity.
That is, the functional complexity of an approximation is the number of bits with which it
represents a real number on (-1, 1].
Assumption 1 If P(Wrl) > P(W r2), then P(Wrl) will be greater than P(Wrj) Vj> 2 (see [1]
for an analysis of cases in which this assumption is invalid).
Question: What is the probability that the most likely face of an unfair die will be empirically identifiable after n tosses?
Answer for the probabilistic strategy:
P(qM[P(Wrl)]
~
n!
t
k.1=>',
>
qM[P(Wrj)] , Vj
P(Wrl)k.,
krl!
> 1)
[t P(Wr2)~
~=>'l
(1 - P(Wrl) - P(Wr2))(II-k., -~)] (17)
kr2! (n - krl - kr2)!
where
Al
= max ( B + 1, nC-_k~ + 1 )
=
A2 =
V].
=
B =
Vl
VC > 2
n
0
(18)
min (B , n - krl )
{BM9}
= kUN9 [P(Wr2)] = kt..vq [P(Wrl)]
-
1
There is a simple recursion in [1] by which every possible boundary for Mq-bit quantization
leads to itself and two additional boundaries in the set {BM9} for (Mq + I)-bit quantization.
Answer for the differential strategy:
1129
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Hampshire and Kumar
where
Al
VI
=
=
A2 =
Vl
max (
kL.v [Llrd, nC-k~
_ 1 +
t
1)
\lC > 2
n
(20)
max ( 0 , krl - kUJft [LlrlJ )
= min (krl - kr.Jft [Llrd , n - krl )
Since the multinomial distribution is positive semi-definite, it should be clear from a
comparisonof(17)-(18) and (19)-(20) thatP (LMt[Llrd
< Lirl < UMt[Llrtl)
islargest
(and larger than any possible P (qM[P(Wrl)] > qM[P(Wrj)] , \I j > 1) ) for a given sample
size n when the differential strategy is employed with Mq = 1 such that LMt [Llrtl = 0 and
UMt [Llrd = 1 (Le., lr.
[Llrtl = 1 and ku~
[Llrd = n). The converse is also true, to wit:
ALNt
-t
Theorem 1 For aftxed value ofn in (19), the l-bitapproximationto Llrl yields the highest
probability of identifying the most likely die face Wrl .
It can be shown that theorem 1 does not depend on the validity of assumption 1 [1]. Given
Axiom 1, the following corollary to theorem 1 holds:
Corollary 1 The differential strategy's minimum-complexity l-bit approximation of Llrl
yields the highest probability of identifying the most likely die face Wrl for a given number
of tosses n.
Corollary 2 The differential strategy's minimum-complexity l-bit approximation of Llrl
requires the smallest sample size necessary (nmi,,) to identify P(Wrl) -and thereby the most
likely die face Wrl - correctly with specified confidence. Thus, the differential strategy
requires the minimum SCP necessary to identify the most likely die face with specified
confidence.
2.1 Theoretical Predictions versus Empirical Results
Figures 1 and 2 compare theoretical predictions of the number of samples n and the number
of bits Mq necessary to identify the most likely face of a particular die versus the actual
requirements obtained from 1000 games (3000 tosses of the die in each game). The die has
five faces with probabilities P(W rl) = 0.37 ,P(Wr2) = 0.28, P(W r3) = 0.2, P(Wr4) = 0.1 ,and
P(W rl) = 0.05. The theoretical predictions for Mq and n (arrows with boxed labels based
on iterative searches employing equations (17) and (19)) that would with 0.95 confidence
correctly identify the most likely die face Wrl are shown to correspond with the empirical
results: in figure 1 the empirical 0.95 confidence interval is marked by the lower bound of
the dark gray and the upper bound of the light gray; in figure 2 the empirical 0.95 confidence
interval is marked by the lower bound of the P(W rl) distribution and the upper bound of the
An Optimal Strategy for Training Connectionist Pattern Classifiers
O~~f
__- - - - - - - - - - - -__- -__- -__
-G.l.'Ot;;:;;.-_ _ _ _ _ _ _ _ _ _ __
~
FigUre 1: Theoretical predictions of the
number of tosses needed to identify the
most likely face Wrl with 95% confidence
(Die 1): Differential strategy prediction superimposed on empirical results of 1000
games (3000 tosses each).
1000
1~
2000
2~
3000
Figure 2: Theoretical predictions of the
number of tosses needed to identify the
most likely face Wrl with 95% confidence
(Die 1): Probabilistic strategy prediction
superimposed on empirical results of 1000
games (3000 tosses each).
P(Wr2) distribution. These figures illustrate that the differential strategy's minimum SCP
is 227 (n = 227, Mq = 1) while the minimum SCP for the probabilistic strategy is 2720
(n = 544 , Mq = 5). A complete tabulation of SCP as a function of P(Wrl) , P(W r2) , and the
worst-case choice for C (the number of classes/die faces) is given in [1].
3 Conclusion
The sample complexity product (SCP) notion of functional complexity set forth herein is
closely aligned with the complexity measures of Kolmogorov and Rissanen [4, 6]. We have
used it to prove that the differential strategy for learning the Bayesian discriminant function
is optimal in terms of its minimum requirements for classifier functional complexity and
number of training examples when the classification task is identifying the most likely face
of an unfair die. It is relatively straightforward to extend theorem 1 and its corollaries to
the general pattern recognition case in order to show that the expected SCP for the I-bit
differential strategy
E [SCP]diffenntial
~
Ix
nmin
[p (Wrl I x) , P (wr21 x)] '~q min [p (Wrl I~) , P (wr21 x) tp{x)dx
=1
(21)
(or the discrete random vector analog of this equation) is minimal [1]. This is because nmin
is by corollary 2 the smallest sample size necessary to distinguish any and all P(Wrl) from
1131
1132
Hampshire and Kumar
lesser P(Wr2). The resulting analysis confinns that the classifier trained with the differential
strategy for statistical pattern recognition (Le., using a CFMmoM objective function) has
the highest probability of learning the Bayesian discriminant function when the functional
capacity of the classifier and the available training data are both limited.
The relevance of this work to the process of designing and training robust connectionist pattern classifiers is evident if one considers the practical meaning of the terms
nmilt [p (Wrl I x) , P (wr21 x)] and Mq mill [p (Wrl I x) , P (wr21 x)] in the sample complexity product of (21). Oi ven one's choice of connectionist model to employ as a classifier, the
M q milt term dictates the minimum necessary connectivity of that model. For example, (21)
can be used to prove that a partially connected radial basis function (RBF) with trainable
variance parameters and three hidden layer ''nodes'' has the minimum Mq necessary for
Bayesian discrimination in the 3-class task described by [5]. However, because both SCP
terms are functions of the probabilistic nature of the random feature vector being classified
and the learning strategy employed. that minimal RBF architecture will only yield Bayesian
discrimination if trained using the differential strategy. The probabilistic strategy requires
significantly more functional complexity in the RBF in order to meet the requirements
of the probabilistic strategy's SCP [1]. Philosophical arguments regarding the use of the
differential strategy in lieu of the more traditional probabilistic strategy are discussed at
length in [1].
Acknowledgement
This research was funded by the Air Force Office of Scientific Research under grant
AFOSR-89-0551. We gratefully acknowledge their support.
References
[1] J. B. Hampshire II. A Differential Theory of Statistical Pattern Recognition. PhD
thesis, Carnegie Mellon University, Department of ElectricaI & Computer Engineering,
Hammerschlag Hall, Pittsburgh. PA 15213-3890,1992. manuscript in progress.
[2] J. B. Hampshire II and B. A. Pearlmutter. Equivalence Proofs for Multi-Layer Perceptton Classifiers and the Bayesian Discriminant Function. In Touretzky, Elman.
Sejnowski, and Hinton, editors, Proceedings of the 1990 Connectionist Models Summer School. pages 159-172. San Mateo, CA, 1991. Morgan-Kaufmann.
[3] J. B. Hampshire II and A. H. Waibel. A Novel Objective Function for Improved
Phoneme Recognition Using Time-Delay Neural Networks. IEEE Transactions on
Neural Networks, 1(2):216-228, June 1990. A revised and extended version of work
first presented at the 1989 International Joint Conference on Neural Networks, vol. I.
pp.235-241.
[4] A. N. Kolmogorov. Three Approaches to the Quantitative Definition of Information.
Problems of Information Transmission. 1(1):1-7, Jan. - Mar. 1965. Faraday Press
ttanslation of Problemy Peredachi Informatsii.
[5] M. D. Richard and R. P. Lippmann. Neural Network Classifiers Estimate Bayesian a
posteriori Probabilities. Neural Computation, 3(4):461-483.1991.
[6] J. Rissanen. Modeling by shortest data description. Automatica, 14:465-471,1978.
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3,933 | 4,560 | Semi-supervised Eigenvectors
for Locally-biased Learning
Michael W. Mahoney
Department of Mathematics
Stanford University
Stanford, CA 94305
[email protected]
Toke Jansen Hansen
Section for Cognitive Systems
DTU Informatics
Technical University of Denmark
[email protected]
Abstract
In many applications, one has side information, e.g., labels that are provided
in a semi-supervised manner, about a specific target region of a large data set,
and one wants to perform machine learning and data analysis tasks ?nearby?
that pre-specified target region. Locally-biased problems of this sort are particularly challenging for popular eigenvector-based machine learning and data analysis tools. At root, the reason is that eigenvectors are inherently global quantities. In this paper, we address this issue by providing a methodology to construct
semi-supervised eigenvectors of a graph Laplacian, and we illustrate how these
locally-biased eigenvectors can be used to perform locally-biased machine learning. These semi-supervised eigenvectors capture successively-orthogonalized directions of maximum variance, conditioned on being well-correlated with an input
seed set of nodes that is assumed to be provided in a semi-supervised manner. We
also provide several empirical examples demonstrating how these semi-supervised
eigenvectors can be used to perform locally-biased learning.
1
Introduction
We consider the problem of finding a set of locally-biased vectors that inherit many of the ?nice?
properties that the leading nontrivial global eigenvectors of a graph Laplacian have?for example,
that capture ?slowly varying? modes in the data, that are fairly-efficiently computable, that can be
used for common machine learning and data analysis tasks such as kernel-based and semi-supervised
learning, etc.?so that we can perform what we will call locally-biased machine learning in a principled manner.
By locally-biased machine learning, we mean that we have a very large data set, e.g., represented as
a graph, and that we have information, e.g., given in a semi-supervised manner, that certain ?regions?
of the data graph are of particular interest. In this case, we may want to focus predominantly on those
regions and perform data analysis and machine learning, e.g., classification, clustering, ranking, etc.,
that is ?biased toward? those pre-specified regions. Examples of this include the following.
? Locally-biased community identification. In social and information network analysis, one
might have a small ?seed set? of nodes that belong to a cluster or community of interest [2,
13]; in this case, one might want to perform link or edge prediction, or one might want to
?refine? the seed set in order to find other nearby members.
? Locally-biased image segmentation. In computer vision, one might have a large corpus
of images along with a ?ground truth? set of pixels as provided by a face detection algorithm [7, 14, 15]; in this case, one might want to segment entire heads from the background
for all the images in the corpus in an automated manner.
1
? Locally-biased neural connectivity analysis. In functional magnetic resonance imaging applications, one might have small sets of neurons that ?fire? in response to some external
experimental stimulus [16]; in this case, one might want to analyze the subsequent temporal dynamics of stimulation of neurons that are ?nearby,? either in terms of connectivity
topology or functional response.
These examples present considerable challenges for spectral techniques and traditional eigenvectorbased methods. At root, the reason is that eigenvectors are inherently global quantities, thus limiting
their applicability in situations where one is interested in very local properties of the data.
In this paper, we provide a methodology to construct what we will call semi-supervised eigenvectors
of a graph Laplacian; and we illustrate how these locally-biased eigenvectors inherit many of the
properties that make the leading nontrivial global eigenvectors of the graph Laplacian so useful in
applications. To achieve this, we will formulate an optimization ansatz that is a variant of the usual
global spectral graph partitioning optimization problem that includes a natural locality constraint as
well as an orthogonality constraint, and we will iteratively solve this problem.
In more detail, assume that we are given as input a (possibly weighted) data graph G = (V, E), an
indicator vector s of a small ?seed set? of nodes, a correlation parameter ? ? [0, 1], and a positive
integer k. Then, informally, we would like to construct k vectors that satisfy the following bicriteria:
first, each of these k vectors is well-correlated with the input seed set; and second, those k vectors
describe successively-orthogonalized directions of maximum variance, in a manner analogous to the
leading k nontrivial global eigenvectors of the graph Laplacian. (We emphasize that the seed set s
of nodes, the integer k, and the correlation parameter ? are part of the input; and thus they should
be thought of as being available in a semi-supervised manner.) Somewhat more formally, our main
algorithm, Algorithm 1 in Section 3, returns as output k semi-supervised eigenvectors; each of these
is the solution to an optimization problem of the form of G ENERALIZED L OCAL S PECTRAL in Figure 1, and thus each ?captures? (say) ?/k of the correlation with the seed set. Our main theoretical
result states that these vectors define successively-orthogonalized directions of maximum variance,
conditioned on being ?/k-well-correlated with an input seed set s; and that each of these k semisupervised eigenvectors can be computed quickly as the solution to a system of linear equations.
From a technical perspective, the work most closely related to ours is that of Mahoney et al. [14].
The original algorithm of Mahoney et al. [14] introduced a methodology to construct a locally-biased
version of the leading nontrivial eigenvector of a graph Laplacian and showed (theoretically and empirically in a social network analysis application) that the resulting vector could be used to partition
a graph in a locally-biased manner. From this perspective, our extension incorporates a natural orthogonality constraint that successive vectors need to be orthogonal to previous vectors. Subsequent
to the work of [14], [15] applied the algorithm of [14] to the problem of finding locally-biased cuts
in a computer vision application. Similar ideas have also been applied somewhat differently. For
example, [2] use locally-biased random walks, e.g., short random walks starting from a small seed
set of nodes, to find clusters and communities in graphs arising in Internet advertising applications;
[13] used locally-biased random walks to characterize the local and global clustering structure of
a wide range of social and information networks; [11] developed the Spectral Graph Transducer
(SGT), that performs transductive learning via spectral graph partitioning. The objectives in both
[11] and [14] are considered constrained eigenvalue problems, that can be solved by finding the
smallest eigenvalue of an asymmetric generalized eigenvalue problem, but in practice this procedure
can be highly unstable [8]. The SGT reduces the instabilities by performing all calculations in a subspace spanned by the d smallest eigenvectors of the graph Laplacian, whereas [14] perform a binary
search, exploiting the monotonic relationship between a control parameter and the corresponding
Lagrange multiplier.
In parallel, [3] and a large body of subsequent work including [6] used eigenvectors of the graph
Laplacian to perform dimensionality reduction and data representation, in unsupervised and semisupervised settings. Many of these methods have a natural interpretation in terms of kernel-based
learning [18]. Many of these diffusion-based spectral methods also have a natural interpretation
in terms of spectral ranking [21]. ?Topic sensitive? and ?personalized? versions of these spectral
ranking methods have also been studied [9, 10]; and these were the motivation for diffusion-based
methods to find locally-biased clusters in large graphs [19, 1, 14]. Our optimization ansatz is a
generalization of the linear equation formulation of the PageRank procedure [17, 14, 21], and the
solution involves Laplacian-based linear equation solving, which has been suggested as a primitive
2
of more general interest in large-scale data analysis [20]. Finally, the form of our optimization
problem has similarities to other work in computer vision applications: e.g., [23] and [7] find good
conductance clusters subject to a set of linear constraints.
2
Background and Notation
Let G = (V, E, w) be a connected undirected graph with n = |V | vertices and m = |E| edges,
in which edge {i, j} has non-negative weight wij . In the following, AG ? RV ?V will denote the
V ?V
adjacency matrix of
will denote the diagonal degree matrix of G, i.e.,
P G, while DG ? R
DG (i, i) = di = {i,j}?E wij , the weighted degree of vertex i. Moreover, for a set of vertices
def P
S ? V in a graph, the volume of S is vol(S) =
i?S di . The Laplacian of G is defined as
def
LG = DG ? AG . (This is also called the combinatorial Laplacian, in which case the normalized
def
?1/2
?1/2
Laplacian of G is LG = DG LG DG .)
P
2
The Laplacian is the symmetric matrix having quadratic form xT LG x =
ij?E wij (xi ? xj ) ,
for x ? RV . This implies that LG is positive semidefinite and that the all-one vector 1 ? RV is
the eigenvector corresponding to the smallest eigenvalue 0. The generalized eigenvalues of LG x =
?i DG x are 0 = ?1 < ?2 ? ? ? ? ? ?N . We will use v2 to denote smallest non-trivial eigenvector,
i.e., the eigenvector corresponding to ?2 ; v3 to denote the next eigenvector; and so on. Finally, for
a matrix A, let A+ denote its (uniquely defined) Moore-Penrose pseudoinverse. For two vectors
x, y ? Rn , and the degree matrix DG for a graph G, we define the degree-weighted inner product
def Pn
as xT DG y = i=1 xi yi di . In particular, if a vector x has unit norm, then xT DG x = 1. Given a
subset of vertices S ? V , we denote by 1S the indicator vector of S in RV and by 1 the vector in
RV having all entries set equal to 1.
3
3.1
Optimization Approach to Semi-supervised Eigenvectors
Motivation for the Program
Recall the optimization perspective on how one computes the leading nontrivial global eigenvectors
of the normalized Laplacian LG . The first nontrivial eigenvector v2 is the solution to the problem
G LOBAL S PECTRAL that is presented on the left of Figure 1. Equivalently, although G LOBAL S PEC TRAL is a non-convex optimization problem, strong duality holds for it and it?s solution may be
computed as v2 , the leading nontrivial generalized eigenvector of LG . The next eigenvector v3 is
the solution to G LOBAL S PECTRAL, augmented with the constraint that xT DG v2 = 0; and in general the tth generalized eigenvector of LG is the solution to G LOBAL S PECTRAL, augmented with
the constraints that xT DG vi = 0, for i ? {2, . . . , t ? 1}. Clearly, this set of constraints and the
constraint xT DG 1 = 0 can be written as xT DG Q = 0, where 0 is a (t ? 1)-dimensional all-zeros
vector, and where Q is an n ? (t ? 1) orthogonal matrix whose ith column equals vi (where v1 = 1,
the all-ones vector, is the first column of Q).
Also presented in Figure 1 is L OCAL S PECTRAL, which includes a constraint requiring the solution
to be well-correlated with an input seed set. This L OCAL S PECTRAL optimization problem was introduced in [14], where it was shown that the solution to L OCAL S PECTRAL may be interpreted as
a locally-biased version of the second eigenvector of the Laplacian. In particular, although L OCAL S PECTRAL is not convex, it?s solution can be computed efficiently as the solution to a set of linear
equations that generalize the popular Personalized PageRank procedure; in addition, by performing
a sweep cut and appealing to a variant of Cheeger?s inequality, this locally-biased eigenvector can
be used to perform locally-biased spectral graph partitioning [14].
3.2
Our Main Algorithm
We will formulate the problem of computing semi-supervised vectors in terms of a primitive optimization problem of independent interest. Consider the G ENERALIZED L OCAL S PECTRAL optimization problem, as shown in Figure 1. For this problem, we are given a graph G = (V, E), with
associated Laplacian matrix LG and diagonal degree matrix DG ; an indicator vector s of a small
3
G LOBAL S PECTRAL
T
minimize x LG x
s.t xT DG x = 1
L OCAL S PECTRAL
T
minimize x LG x
s.t xT DG x = 1
xT DG 1 = 0
xT D G 1 = 0
?
xT D G s ? ?
G ENERALIZED L OCAL S PECTRAL
minimize xT LG x
s.t xT DG x = 1
xT DG Q = 0
?
xT DG s ? ?
Figure 1: Left: The usual G LOBAL S PECTRAL partitioning optimization problem; the vector achieving the optimal solution is v2 , the leading nontrivial generalized eigenvector of LG with respect
to DG . Middle: The L OCAL S PECTRAL optimization problem, which was originally introduced
in [14]; for ? = 0, this coincides with the usual global spectral objective, while for ? > 0, this
produces solutions that are biased toward the seed vector s. Right: The G ENERALIZED L OCAL S PECTRAL optimization problem we introduce that includes both the locality constraint and a more
general orthogonality constraint. Our main algorithm for computing semi-supervised eigenvectors
will iteratively compute the solution to G ENERALIZED L OCAL S PECTRAL for a sequence of Q matrices. In all three cases, the optimization variable is x ? Rn .
?seed set? of nodes; a correlation parameter ? ? [0, 1]; and an n?? constraint matrix Q that may be
assumed to be an orthogonal matrix. We will assume (without loss of generality) that s is properly
normalized and orthogonalized so that sT DG s = 1 and sT DG 1 = 0. While s can be a general unit
vector orthogonal to 1, it may be helpful to think of s as the indicator vector of one or more vertices
in V , corresponding to the target region of the graph.
In words, the problem G ENERALIZED L OCAL S PECTRAL asks us to find a vector x ? Rn that minimizes the variance xT LG x subject
? to several constraints: that x is unit length; that x is orthogonal
to the span of Q; and that x is ?-well-correlated with the input seed set vector s. In our application of G ENERALIZED L OCAL S PECTRAL to the computation of semi-supervised eigenvectors, we
will iteratively compute the solution to G ENERALIZED L OCAL S PECTRAL, updating Q to contain
the already-computed semi-supervised eigenvectors. That is, to compute the first semi-supervised
eigenvector, we let Q = 1, i.e., the n-dimensional all-ones vector, which is the trivial eigenvector of
LG , in which case Q is an n ? 1 matrix; and to compute each subsequent semi-supervised eigenvector, we let the columns of Q consist of 1 and the other semi-supervised eigenvectors found in each
of the previous iterations.
To show that G ENERALIZED L OCAL S PECTRAL is efficiently-solvable, note that it is a quadratic
program with only one quadratic constraint and one linear equality constraint. In order to remove the
equality constraint, which will simplify the problem, let?s change variables by defining the n?(n??)
matrix F as {x : QT DG x = 0} = {x : x = F y}. That is, F is a span for the null space of QT ;
and we will take F to be an orthogonal matrix. Then, with respect to the y variable, G ENERALIZED
L OCAL S PECTRAL becomes
minimize y T F T LG F y
y
(1)
subject to y T F T DG F y = 1,
?
y T F T DG s ? ?.
In terms of the variable x, the solution to this optimization problem is of the form
+
x? = cF F T (LG ? ?DG ) F F T DG s
+
= c F F T (LG ? ?DG ) F F T DG s,
(2)
?
for a normalization constant c ? (0, ?) and for some ? that depends on ?. The second line follows
from the first since F is an n ? (n ? ?) orthogonal matrix. This so-called ?S-procedure? is described
in greater detail in Chapter 5 and Appendix B of [4]. The significance of this is that, although it is
a non-convex optimization problem, the G ENERALIZED L OCAL S PECTRAL problem can be solved
by solving a linear equation, in the form given in Eqn. (2).
Returning to our problem of computing semi-supervised eigenvectors, recall that, in addition to the
input for the G ENERALIZED L OCAL S PECTRAL problem, we need to specify a positive integer k
that indicates the number of vectors to be computed. In the simplest case, we would assume that
4
we would like the correlation
p to be ?evenly distributed? across all k vectors, in which case we will
require that each vector is ?/k-well-correlated with the input seed set vector s; but this assumption
can easily be relaxed, and thus Algorithm 1 is formulated more generally as taking a k-dimensional
vector ? = [?1 , . . . , ?k ]T of correlation coefficients as input.
To compute the first semi-supervised eigenvector, we will let Q = 1, the all-ones vector, in which
case the first nontrivial semi-supervised eigenvector is
+
x?1 = c (LG ? ?1 DG ) DG s,
(3)
where ?1 is chosen to saturate the part of the correlation constraint along the first direction. (Note
that the projections F F T from Eqn. (2) are not present in Eqn. (3) since by design sT DG 1 = 0.)
That is, to find the correct setting of ?1 , it suffices to perform a binary search over the possible
values of ?1 in the interval (?vol(G), ?2 (G)) until the correlation constraint is satisfied, that is,
until (sT DG x)2 is sufficiently close to ?21 , see [8, 14].
To compute subsequent semi-supervised eigenvectors, i.e., at steps t = 2, . . . , k if one ultimately
wants a total of k semi-supervised eigenvectors, then one lets Q be the n ? (t ? 1) matrix with first
column equal to 1 and with j th column, for i = 2, . . . , t ? 1, equal to x?j?1 (where we emphasize
that x?j?1 is a vector not an element of a vector). That is, Q is of the form Q = [1, x?1 , . . . , x?t?1 ],
where x?i are successive semi-supervised eigenvectors, and the projection matrix F F T is of the
form F F T = I ? DG Q(QT DG DG Q)?1 QT DG , due to the degree-weighted inner norm. Then, by
Eqn. (2), the tth semi-supervised eigenvector takes the form
+
x?t = c F F T (LG ? ?t DG )F F T DG s.
(4)
Algorithm 1 Semi-supervised eigenvectors
Input: LG , DG , s, ? = [?1 , . . . , ?k ]T ,
Require: sT DG 1 = 0, sT DG s = 1, ?T 1 ? 1
1: Q = [1]
2: for t = 1 to k do
3:
F F T ? I ? DG Q(QT DG DG Q)?1 QT DG
4:
> ? ?2 where F F T LG F F T v2 = ?2 F F T DG F F T v2
5:
? ? ?vol(G)
6:
repeat
7:
?t ? (? + >)/2 (Binary search over ?t )
8:
xt ? (F F T (LG ? ?t DG )F F T )+ F F T DG s
9:
Normalize xt such that xTt DG xt = 1
10:
if (xTt DG s)2 > ?t then ? ? ?t else > ? ?t end if
11:
until k(xTt DG s)2 ? ?t k ? or k(? + >)/2 ? ?t k ?
12:
Augment Q with x?t by letting Q = [Q, x?t ].
13: end for
In more detail, Algorithm 1 presents pseudo-code for our main algorithm for computing semisupervised eigenvectors. Several things should be noted about our implementation. First, note
that we implicitly compute the projection matrix F F T . Second, a na??ve approach to Eqn. (2) does
not immediately lead to an efficient solution, since DG s will not be in the span of (F F T (LG ?
?DG )F F T ), thus leading to a large residual. By changing variables so that x = F F T y, the solution becomes x? ? F F T (F F T (LG ? ?DG )F F T )+ F F T DG s. Since F F T is a projection matrix,
this expression is equivalent to x? ? (F F T (LG ? ?DG )F F T )+ F F T DG s. Third, we exploit that
F F T (LG ? ?i DG )F F T is an SPSD matrix, and we apply the conjugate gradient method, rather
than computing the explicit pseudoinverse. That is, in the implementation we never represent the
dense matrix F F T , but instead we treat it as an operator and we simply evaluate the result of applying a vector to it on either side. Fourth, we use that ?2 can never decrease (here we refer to
?2 as the smallest non-zero eigenvalue of the modified matrix), so we only recalculate the upper
bound for the binary search when an iteration saturates without satisfying k(xTt DG s)2 ? ?t k ? .
In case of saturation one can for instance recalculate ?2 iteratively by using the inverse iteration
T
T +
T
T k
method, v2k+1 ? (F F T LG F F T ? ?est
2 F F DG F F ) F F DG F F v2 , and normalizing such
k+1 T k+1
that (v2 ) v2 = 1.
5
4
Illustrative Empirical Results
In this section, we will provide a detailed empirical evaluation of our method of semi-supervised
eigenvectors and how they can be used for locally-biased machine learning. Our goal will be twofold: first, to illustrate how the ?knobs? of our method work; and second, to illustrate the usefulness
of the method in a real application. To do so, we will consider:
? Toy data. In Section 4.1, we will consider one-dimensional examples of the popular ?small
world? model [22]. This is a parameterized family of models that interpolates between
low-dimensional grids and random graphs; and, as such, it will allow us to illustrate the
behavior of our method and it?s various parameters in a controlled setting.
? Handwritten image data. In Section 4.2, we will consider the data from the MNIST digit
data set [12]. These data have been widely-studied in machine learning and related areas
and they have substantial ?local heterogeneity?; and thus these data will allow us to illustrate how our method may be used to perform locally-biased versions of common machine
learning tasks such as smoothing, clustering, and kernel construction.
4.1
Small-world Data
To illustrate how the ?knobs? of our method work, and in particular how ? and ? interplay, we consider data constructed from the so-called small-world model. To demonstrate how semi-supervised
eigenvectors can focus on specific target regions of a data graph to capture slowest modes of local
variation, we plot semi-supervised eigenvectors around illustrations of (non-rewired and rewired)
realizations of the small-world graph; see Figure 2.
?2
?3
?4
?5
p = 0,
= 0.000011,
= 0.000011,
= 0.000046,
= 0.000046.
(a) Global eigenvectors
?2
?3
?4
?5
p = 0.01,
= 0.000149,
= 0.000274,
= 0.000315,
= 0.000489.
(b) Global eigenvectors
p = 0.01, ? = 0.005,
?1 = 0.000047,
?2 = 0.000052,
?3 = ?0.000000,
?4 = ?0.000000.
p = 0.01, ? = 0.05,
?1 = ?0.004367,
?2 = ?0.001778,
?3 = ?0.001665,
?4 = ?0.000822.
(c) Semi-supervised eigenvectors
(d) Semi-supervised eigenvectors
Figure 2: In each case, (a-d) the data consist of 3600 nodes, each connected to it?s 8 nearestneighbors. In the center of each subfigure, we show the nodes (blue) and edges (black and light
gray are the local edges, and blue are the randomly-rewired edges). In each subfigure, we wrap a
plot (black x-axis and gray background) visualizing the 4 smallest semi-supervised eigenvectors,
allowing us to see the effect of random edges (different values of rewiring probability p) and degree
of localization (different values of ?). Eigenvectors are color coded as blue, red, yellow, and green,
starting with the one having the smallest eigenvalue. See the main text for more details.
In Figure 2.a, we show a graph with no randomly-rewired edges (p = 0) and a locality parameter
? such that the global eigenvectors are obtained. This yields a symmetric graph with eigenvectors
corresponding to orthogonal sinusoids, i.e., for all eigenvectors, except the all-ones with eigenvalue
0, the algebraic multiplicity is 2, i.e., the first two capture the slowest mode of variation and correspond to a sine and cosine with equal random phase-shift (rotational ambiguity). In Figure 2.b,
random edges have been added with probability p = 0.01 and the locality parameter ? is still chosen such that the global eigenvectors of the rewired graph are obtained. In particular, note small
kinks in the eigenvectors at the location of the randomly added edges. Since the graph is no longer
symmetric, all of the visualized eigenvectors have algebraic multiplicity 1. Moreover, note that the
slow mode of variation in the interval on the top left; a normalized-cut based on the leading global
eigenvector would extract this region since the remainder of the ring is more well-connected due
to the degree of rewiring. In Figure 2.c, we see the same graph realization as in Figure 2.b, except
that the semi-supervised eigenvectors have a seed node at the top of the circle and the correlation
6
parameter ?t = 0.005. Note that, like the global eigenvectors, the local approach produces modes
of increasing variation. In addition, note that the neighborhood around ?11 o-clock? contains more
mass, when compared with Figure 2.b; the reason for this is that this region is well-connected with
the seed via a randomly added edge. Above the visualization we also show the ?t that saturates ?t ,
i.e., ?t is the Lagrange multiplier that defines the effective correlation ?t . Not shown is that if we
kept reducing ?, then ?t would tend towards ?t+1 , and the respective semi-supervised eigenvector
would tend towards the global eigenvector. Finally, in Figure 2.d, the desired correlation is increased
to ? = 0.05 (thus decreasing the value of ?t ), making the different modes of variation more localized in the neighborhood of the seed. It should be clear that, in addition to being determined by the
locality parameter, we can think of ? as a regularizer biasing the global eigenvectors towards the
region near the seed set.
4.2
MNIST Digit Data
We now demonstrate the semi-supervised eigenvectors as a feature extraction preprocessing step in
a machine learning setting. We consider the well-studied MNIST dataset containing 60000 training
digits and 10000 test digits ranging from 0 to 9. We construct the complete 70000 ? 70000 k-NN
graph with k = 10 and with edge weights given by wij = exp(? ?42 kxi ? xj k2 ), where ?i2 being
i
the Euclidean distance to it?s nearest neighbor, and we define the graph Laplacian in the usual way.
We evaluate the semi-supervised eigenvectors in a transductive learning setting by disregarding the
majority of labels in the entire training data. We then use a few samples from each class to seed
our semi-supervised eigenvectors, and a few others to train a downstream classification algorithm.
Here we choose to apply the SGT of [11] for two main reasons. First, the transductive classifier is
inherently designed to work on a subset of global eigenvectors of the graph Laplacian, making it
ideal for validating that our localized basis constructed by the semi-supervised eigenvectors can be
more informative when we are solely interested in the ?local heterogeneity? near a seed set. Second,
using the SGT based on global eigenvectors is a good point of comparison, because we are only
interested in the effect of our subspace representation. (If we used one type of classifier in the local
setting, and another in the global, the classification accuracy that we measure would obviously be
biased.) As in [11], we normalize the spectrum of both global and semi-supervised eigenvectors
2
by replacing the eigenvalues with some monotonically increasing function. We use ?i = ki 2 , i.e.,
focusing on ranking among smallest cuts; see [5]. Furthermore, we fix the regularization parameter
of the SGT to c = 3200, and for simplicity we fix ? = 0 for all semi-supervised eigenvectors,
implicitly defining the effective ? = [?1 , . . . , ?k ]T . Clearly, other correlation distributions and
values of ? may yield subspaces with even better discriminative properties1 .
Labeled points
1:1
1 : 10
5 : 50
10 : 100
50 : 500
#Semi-supervised eigenvectors for SGT
1
2
4
6
8
10
0.39
0.39
0.38
0.38
0.38
0.36
0.30
0.31
0.25
0.23
0.19
0.15
0.12
0.15
0.09
0.08
0.07
0.06
0.09
0.10
0.07
0.06
0.05
0.05
0.03
0.03
0.03
0.03
0.03
0.03
1
0.50
0.49
0.49
0.49
0.49
#Global eigenvectors for SGT
5
10
15
20
0.48
0.36
0.27
0.27
0.36
0.09
0.08
0.06
0.09
0.08
0.07
0.05
0.08
0.07
0.06
0.04
0.10
0.07
0.06
0.04
25
0.19
0.06
0.04
0.04
0.04
Table 1: Classification error for the SGT based on respectively semi-supervised and global eigenvectors. The first column from the left encodes the configuration, e.g., 1:10 interprets as 1 seed and 10
training samples from each class (total of 22 samples - for the global approach these are all used for
training). When the seed is well determined and the number of training samples moderate (50:500)
a single semi-supervised eigenvector is sufficient, where for less data we benefit from using multiple
semi-supervised eigenvectors. All experiments have been repeated 10 times.
Here, we consider the task of discriminating between fours and nines, as these two classes tend to
overlap more than other combinations. (A closed four usually resembles nine more than an ?open?
four.) Hence, we expect localization on low order global eigenvectors, meaning that class separation
will not be evident in the leading global eigenvector, but instead will be ?buried? further down the
spectrum. Thus, this will illustrate how semi-supervised eigenvectors can represent relevant heterogeneities in a local subspace of low dimensionality. Table 1 summarizes our classification results
based on respectively semi-supervised and global eigenvectors. Finally, Figure 3 and 4 illustrates
two realizations for the 1:10 configuration, where the training samples are fixed, but where we vary
1
A thorough analysis regarding the importance of this parameter will appear in the journal version.
7
the seed nodes, to demonstrate the influence of the seed. See the caption in these figures for further
details.
s+ = { }
}
s? = { }
l? = {
1 vs. 2
}
?
?????????
? Test data ?
?????????
?
l+ = {
0.6
1 vs. 3
1 vs. 4
1 vs. 5
2 vs. 3
2 vs. 4
2 vs. 5
3 vs. 4
3 vs. 5
Classification error
Unexplained correlation
0.5
0.4
0.3
4 vs. 5
0.2
0.1
0
0.08 0.07 0.06
0.05
1
2
3
4
0.03 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03
5
6
7
8
9 10 11 12
#Semi-supervised eigenvectors
13
14
15
Figure 3: Left: Shows a subset of the classification results for the SGT based on 5 semi-supervised
eigenvectors seeded in s+ and s? , and trained using samples l+ and l? . Misclassifications are
marked with black frames. Right: Visualizes all test data spanned by the first 5 semi-supervised
eigenvectors, by plotting each component as a function of the others. Red (blue) points correspond
to 4 (9), whereas green points correspond to remaining digits. As the seed nodes are good representatives, we note that the eigenvectors provide a good class separation. We also plot the error as
a function of local dimensionality, as well as the unexplained correlation, i.e., initial components
explain the majority of the correlation with the seed (effect of ? = 0). The particular realization
based on the leading 5 semi-supervised eigenvectors yields an error of ? 0.03 (dashed circle).
s+ = { }
?
?????????
? Test data ?
?????????
?
l+ = {
}
s? = { }
l? = {
1 vs. 2
}
0.6
0.5
0.31 0.30 0.30 0.30 0.29
0.3
0.2
0.1
0
1 vs. 4
1 vs. 5
2 vs. 3
2 vs. 4
2 vs. 5
3 vs. 4
3 vs. 5
Classification error
Unexplained correlation
0.48
0.4
1 vs. 3
0.27
0.24
0.20
4 vs. 5
0.15
0.10
0.04 0.04 0.04 0.04
1
2
3
4
5
6
7
8
9 10 11 12
#Semi-supervised eigenvectors
13
14
15
Figure 4: See the general description in Figure 3. Here we illustrate an instance where the s+ shares
many similarities with s? , i.e., s+ is on the boundary of the two classes. This particular realization
achieves a classification error of ? 0.30 (dashed circle). In this constellation we first discover
localization on low order semi-supervised eigenvectors (? 12 eigenvectors), which is comparable
to the error based on global eigenvectors (see Table 1), i.e., further down the spectrum we recover
from the bad seed and pickup the relevant mode of variation.
In summary: We introduced the concept of semi-supervised eigenvectors that are biased towards
local regions of interest in a large data graph. We demonstrated the feasibility on a well-studied
dataset and found that our approach leads to more compact subspace representations by extracting
desired local heterogeneities. Moreover, the algorithm is scalable as the eigenvectors are computed
by the solution to a sparse system of linear equations, preserving the low O(m) space complexity.
Finally, we foresee that the approach will prove useful in a wide range of data analysis fields, due to
the algorithm?s speed, simplicity, and stability.
8
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2?14, 2010.
[21] S. Vigna. Spectral ranking. Technical report. Preprint: arXiv:0912.0238 (2009).
[22] D.J. Watts and S.H. Strogatz. Collective dynamics of small-world networks. Nature, 393:440?
442, 1998.
[23] S. X. Yu and J. Shi. Grouping with bias. In Annual Advances in Neural Information Processing
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9
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3,934 | 4,561 | Feature-aware Label Space Dimension Reduction for
Multi-label Classification
Hsuan-Tien Lin
Department of Computer Science
& Information Engineering,
National Taiwan University
[email protected]
Yao-Nan Chen
Department of Computer Science
& Information Engineering,
National Taiwan University
[email protected]
Abstract
Label space dimension reduction (LSDR) is an efficient and effective paradigm
for multi-label classification with many classes. Existing approaches to LSDR,
such as compressive sensing and principal label space transformation, exploit only
the label part of the dataset, but not the feature part. In this paper, we propose
a novel approach to LSDR that considers both the label and the feature parts.
The approach, called conditional principal label space transformation, is based
on minimizing an upper bound of the popular Hamming loss. The minimization
step of the approach can be carried out efficiently by a simple use of singular
value decomposition. In addition, the approach can be extended to a kernelized
version that allows the use of sophisticated feature combinations to assist LSDR.
The experimental results verify that the proposed approach is more effective than
existing ones to LSDR across many real-world datasets.
1
Introduction
The multi-label classification problem is an extension of the traditional multiclass classification
problem. In contrast to the multiclass problem, which associates only a single label to each instance,
the multi-label classification problem allows multiple labels for each instance. General solutions
to this problem meet the demands of many real-world applications for classifying instances into
multiple concepts, including categorization of text [1], scene [2], genes [3] and so on. Given the
wide range of such applications, the multi-label classification problem has been attracting much
attention of researchers in machine learning [4, 5, 6].
Label space dimension reduction (LSDR) is a new paradigm in multi-label classification [4, 5].
By viewing the set of multiple labels as a high-dimensional vector in some label space, LSDR
approaches use certain assumed or observed properties of the vectors to ?compress? them. The
compression step transforms the original multi-label classification problem (with many labels) to a
small number of learning tasks. If the compression step, de-compression step, and learning steps
can be efficient and effective, LSDR approaches can be useful for multi-label classification because
of the appropriate use of joint information within the labels [5]. For instance, a representative LSDR
approach is the principal label space transformation [PLST; 5]. PLST takes advantage of the key
linear correlations between labels to build a small number of regression tasks.
LSDR approaches are homologous to the feature space dimension reduction (FSDR) approaches and
share similar advantages: saving computational power and storage without much loss of prediction
accuracy and improving performance by removing irrelevant, redundant, or noisy information [7].
There are two types of FSDR approaches: unsupervised and supervised. Unsupervised FSDR considers only feature information during reduction, while supervised FSDR considers the additional
label information. A typical instance of unsupervised FSDR is principal component analysis [PCA;
8]. PCA transforms the features into a small number of uncorrelated variables. On the other hand,
the supervised FSDR approaches include supervised principal component analysis [9], sliced inverse
regression [10], and kernel dimension reduction [11]. In particular, for multi-label classification, a
1
leading supervised FSDR approach is canonical correlation analysis [CCA; 6, 12] which is based
on linear projections in both the feature space and the label space. In general, well-tuned supervised FSDR approaches can perform better than unsupervised ones because of the additional label
information.
PLST can be viewed as the counterpart of PCA in the label space [5] and is feature-unaware. That is,
it considers only the label information during reduction. Motivated by the superiority of supervised
FSDR over unsupervised approaches, we are interested in studying feature-aware LSDR: LSDR that
considers feature information.
In this paper, we propose a novel feature-aware LSDR approach, conditional principal label space
transformation (CPLST). CPLST combines the concepts of PLST (LSDR) and CCA (supervised
FSDR) and can improve PLST through the addition of feature information. We derive CPLST by
minimizing an upper bound of the popular Hamming loss and show that CPLST can be accomplished
by a simple use of singular value decomposition. Moreover, CPLST can be flexibly extended by the
kernel trick with suitable regularization, thereby allowing the use of sophisticated feature information to assist LSDR. The experimental results on real-world datasets confirm that CPLST can reduce
the number of learning tasks without loss of prediction performance. In particular, CPLST is usually
better than PLST and other related LSDR approaches.
The rest of this paper is organized as follows. In Section 2, we define the multi-label classification
problem and review related works. Then, in Section 3, we derive the proposed CPLST approach.
Finally, we present the experimental results in Section 4 and conclude our study in Section 5.
2
Label Space Dimension Reduction
The multi-label classification problem aims at finding a classifier from the input vector x to a label
set Y, where x ? Rd , Y ? {1, 2, . . . , K} and K is the number of classes. The label set Y is often
conveniently represented as a label vector, y ? {0, 1}K , where y[k] = 1 if and only if k ? Y.
Given a dataset D = {(xn , yn )}N
n=1 , which contains N training examples (xn , yn ), the multi-label
classification algorithm uses D to find a classifier h: X ? 2{1,2,??? ,K} anticipating that h predicts y
well on any future (unseen) test example (x, y).
There are many existing algorithms for solving multi-label classification problems. The simplest and
most intuitive one is binary relevance [BR; 13]. BR decomposes the original dataset D into K binary
classification datasets, Dk = {(xn , yn [k])}N
n=1 , and learns K independent binary classifiers, each of
which is learned from Dk and is responsible for predicting whether the label set Y includes label k.
When K is small, BR is an efficient and effective baseline algorithm for multi-label classification.
However, when K is large, the algorithm can be costly in training, prediction, and storage.
Facing the above challenges, LSDR (Label Space Dimension Reduction) offers a potential solution
to these issues by compressing the K-dimensional label space before learning. LSDR transforms D
N
into M datasets, where Dm = {(xn , tn [m])}n=1 , m = 1, 2, . . . , M , and M K such that the
multi-label classification problem can be tackled efficiently without significant loss of prediction
performance. In particular, LSDR involves solving, predicting with, and storing the models for
only M , instead of K, learning tasks.
For instance, compressive sensing [CS; 4], a precursor of LSDR, is based on the assumption that the
label set vector y is sparse (i.e., contains few ones) to ?compressed? y to a shorter code vector t by
projecting y on M random directions v1 , ? ? ? , vM , where M K can be determined according
to the assumed sparsity level. CS transforms the original multi-label classification problem into M
T
regression tasks with Dm = {(xn , tn [m])}N
n=1 , where tn [m] = vm yn . After obtaining a multioutput regressor r(x) for predicting the code vector t, CS decodes r(x) to the optimal label set
vector by solving an optimization problem for each input instance x under the sparsity assumption,
which can be time-consuming.
2.1
Principal Label Space Transformation
Principal label space transformation [PLST; 5] is another approach to LSDR. PLST first shifts each
P
? , where y
? = N1 N
label set vector y to z = y ? y
n=1 yn is the estimated mean of the label set
vectors. Then, PLST takes a matrix V that linearly maps z to the code vector t by t = Vz. Unlike
CS, however, PLST takes principal directions vm (to be introduced next) rather than the random
ones, and does not need to solve an optimization problem during decoding.
2
In particular, PLST considers only a matrix V with orthogonal rows, and decodes r(x) to the pre? ), which is called round-based decoding. Tai and Lin [5]
dicted labels by h(x) = round(VT r(x)+ y
prove that when using round-based decoding and a linear transformation V that contains orthogonal
rows, the common Hamming loss for evaluating multi-label classifiers [14] is bounded by
2
2
T
T
,
(1)
Training Hamming Loss ? c
r(X) ? ZV
+
Z ? ZV V
F
T
F
zTn
where r(X) contains r(xn ) as rows, Z contains
as rows and c is a constant that depends on K
and N . The matrix ZVT then contains the code vector tTn as rows.
The bound can be divided into two parts. The first part is kr(X) ? ZVT k2F , which represents
the prediction error from the regressor r(xn ) to the desired code vectors tn . The second part is
kZ ? ZVT Vk2F , which stands for the encoding error for projecting zn into the closest vector in
span{v1 , ? ? ? , vM }, which is VT tn .
PLST is derived by minimizing the encoding error [5] and finds the optimal M by K matrix V
by applying the singular value decomposition on Z and take the M right-singular vectors vm that
correspond to the M largest singular values. The M right-singular vectors are called the principal
directions for representing zn .
PLST can be viewed as a linear case of the kernel dependency estimation (KDE) algorithm [15].
Nevertheless, the general nonlinear KDE must solve a computationally expensive pre-image problem for each test input x during the prediction phase. The linearity of PLST avoids the pre-image
problem and enjoys efficient round-based decoding. In this paper, we will focus on the linear case
in order to design efficient algorithms for LSDR during both the training and prediction phases.
2.2
Canonical Correlation Analysis
A related technique that we will consider in this paper is canonical correlation analysis [CCA;
6], a well-known statistical technique for analyzing the linear relationship between two multidimensional variables. Traditionally, CCA is regarded as a FSDR approach in multi-label classification [12]. In this subsection, we discuss whether CCA can also be viewed as an LSDR approach.
Formally, given an N by d matrix X with the n-th row being xTn (assumed to be zero mean) as
well as an N by K matrix Z with the n-th row being zTn (assumed to be zero mean), CCA aims at
(2)
(1)
(2)
(1)
finding two lists of basis vectors, (wx , wx , ? ? ? ) and (wz , wz , ? ? ? ), such that the correlation
(i)
(i)
(i)
(i)
coefficient between the canonical variables cx = Xwx and cz = Zwz is maximized, under the
(j)
(j)
(i)
constraint that cx is uncorrelated to all other cx and cz for 1 ? j < i. Kettenring [16] showed
that CCA is equivalent to simultaneously solving the following constrained optimization problem:
XWxT ? ZWzT
2
min
subject to Wx XT XWxT = Wz ZT ZWzT = I, (2)
F
Wx ,Wz
(i) T
(i) T
where Wx is the matrix with the i-th row (wx ) , and Wz is the matrix with the i-th row (wz ) .
When CCA is considered in the context of multi-label classification, X is the matrix that contains the
mean-shifted xTn as rows and Z is the shifted label matrix that contains the mean-shifted ynT as rows.
Traditionally, CCA is used as a supervised FSDR approach that discards Wz and uses only Wx to
project features onto a lower-dimension space before learning with binary relevance [12, 17].
On the other hand, due to the symmetry between X and Z, we can also view CCA as an approach to feature-aware LSDR. In particular, CCA is equivalent to first seeking projection directions Wz of Z, and then performing a multi-output linear regression from xn to Wz zn , under the
constraints Wx XT XWxT = I, to obtain Wx . However, it has not been seriously studied how to use
CCA for LSDR because Wz does not contain orthogonal rows. That is, unlike PLST, round-based
decoding cannot be used and it remains to be an ongoing research issue for designing a suitable
decoding scheme with CCA [18].
3
Proposed Algorithm
Inspired by CCA, we first design a variant that involves an appropriate decoding step. As suggested
in Section 2.2, CCA is equivalent to finding a projection that minimizes the squared prediction error
under the constraints Wx XT XWxT = Wz ZT ZWzT = I. If we drop the constraint on Wx in order
to further decrease the squared prediction error and change Wz ZT ZWzT = I to Wz WzT = I in
3
order to enable round-based decoding, we obtain
XWxT ? ZWzT
2
min
subject to Wz WzT = I
F
Wx ,Wz
(3)
Problem (3) preserves the original objective function of CCA and specifies that Wz must contain orthogonal rows for applying round-based decoding. We call this algorithm orthogonally
constrained CCA (OCCA). Then, using the Hamming loss bound (1), when V = Wz and
r(x) = XWzT , OCCA minimizes kr(x) ? ZWzT k in (1) with the hope that the Hamming loss
is also minimized. In other words, OCCA is employed for the orthogonal directions V that are
?easy to learn? (of low prediction error) in terms of linear regression.
For every fixed Wz = V in (3), the optimization problem for Wx is simply a linear regression from
X to ZVT . Then, the optimal Wx can be computed by a closed-form solution WxT = X? ZVT ,
where X? is the pseudo inverse of X. When the optimal Wx is inserted back into (3), the optimiza
XX? ZVT ? ZVT
2 which is equivalent to
tion problem becomes min
F
VVT =I
min tr VZT (I ? H) ZVT .
VVT =I
(4)
The matrix H = XX? is called the hat matrix for linear regression [19]. Similar to PLST, by EckartYoung theorem [20], we can solve problem (4) by considering the eigenvectors that correspond to
the largest eigenvalues of ZT (H ? I)Z.
3.1
Conditional Principal Label Space Transformation
From the previous discussions, OCCA captures the input-output relation to minimize the prediction
error in bound (1) with the ?easy? directions. In contrast, PLST minimizes the encoding error in
bound (1) with the ?principal? directions. Now, we combine the benefits of the two algorithms, and
minimize the two error terms simultaneously with the ?conditional principal? directions. We begin
by continuing our derivation of OCCA, which obtains r(x) by a linear regression from X to ZVT .
If we minimize both terms in (1) together with such a linear regression, the optimization problem
becomes
2
2
T
T
T
min c
XW ? ZV
+
Z ? ZV V
F
F
W,VVT =I
T
T
T
T
?
min tr VZ (I ? H) ZV ? V VZ Z ? ZT ZVT V + VT VZT ZVT V
(5)
VVT =I
?
max tr VZT HZVT
(6)
VVT =I
Problem (6) is derived by a cyclic permutation to eliminate a pair of V and VT and combine the
last three terms of (5). The problem can again be solved by taking the eigenvectors with the largest
eigenvalues of ZT HZ as the rows of V. Such a matrix V minimizes the prediction error term and
the encoding error term simultaneously. The resulting algorithm is called conditional principal label
space transformation (CPLST), as shown in Algorithm 1.
Algorithm 1 Conditional Principal Label Space Transformation
T
?.
1: Let Z = [z1 . . . zN ] with zn = yn ? y
2: Preform SVD on ZT HZ to obtain ZT HZ = A?B with ?1 ? ?2 ? ? ? ? ? ?N . Let VM
contain the top M rows of B.
N
3: Encode {(xn , yn )}N
n=1 to {(xn , tn )}n=1 , where tn = VM zn .
4: Learn a multi-dimension regressor r(x) from {(xn , tn )}N
n=1 .
T
? .
5: Predict the label-set of an instance x by h(x) = round VM
r(x) + y
CPLST balances the prediction error with the encoding error and is closely related with bound (1).
Moreover, in contrast with PLST, which uses the key unconditional correlations, CPLST is featureaware and allows the capture of conditional correlations [14].
We summarize the three algorithms in Table 1, and we will compare them empirically in Section 4.
The three algorithms are similar. They all operate with an SVD (or eigenvalue decomposition)
on a K by K matrix. PLST focuses on the encoding error and does not consider the features
during LSDR, i.e. it is feature-unaware. On the other hand, CPLST and OCCA are feature-aware
approaches, which consider features during LSDR. When using linear regression as the multi-output
4
Table 1: Summary of three LSDR algorithms
Algorithm
PLST
OCCA
CPLST
Matrix for SVD
ZT Z
ZT (H ? I)Z
ZT HZ
LSDR
feature-unaware
feature-aware
feature-aware
Relation to bound (1)
minimizes the encoding error
minimizes the prediction error
minimizes both
regressor, CPLST simultaneously minimizes the two terms in bound (1), while OCCA minimizes
only one term of the bound.
In contrast to PLST, the two feature-aware approaches OCCA and CPLST must calculate the matrix H and are thus slower than PLST if the dimension d of the input space is large.
3.2
Kernelization and Regularization
Kernelization?extending a linear model to a nonlinear one using the kernel trick [21]?and regularization are two important techniques in machine learning. The former expands the power of the
linear models while the latter regularizes the complexity of the learning model. In this subsection,
we show that kernelization and regularization can be applied to CPLST (and OCCA).
In Section 3.1, we derive CPLST by using linear regression as the underlying multi-output regression
method. Next, we replace linear regression by its kernelized form with `2 regularization, kernel ridge
regression [22], as the underlying regression algorithm. Kernel ridge regression considers a feature
mapping ? : X ? F before performing regularized linear regression. According to ?, the kernel
function k(x, x0 ) = ?(x)T ?(x0 ) is defined as the inner product in the space F. When applying
kernel ridge regression with a regularization parameter ? to map from X to ZV, if ?(x) can be
explicitly computed, it is known that the closed-form solution is [22]
?1
?1
W = ?T ?I + ??T
ZVT = ?T (?I + K) ZVT ,
(7)
where ? is the matrix containing ?(xn )T as rows, and K is the matrix with Kij = k(xi , xj ) =
?(xi )T ?(xj ). That is, K = ??T and is called the kernel matrix of X.
Now, we derive kernel-CPLST by inserting the optimal W into the Hamming loss bound (1). When
substituting (7) into minimizing the loss bound (1) with r(X) = ?W and letting Q = (?I + K)?1 ,
2
2
min c
??T QZVT ? ZVT
F +
Z ? ZVT V
F
VVT =I
2
KQZVT ? ZVT
2 +
? min
Z ? ZVT V
F
F
VVT =I
T
T
? max tr VZ (2KQ ? QKKQ ? I) ZV
(8)
VVT =I
Notice that in equation (8), kernel-CPLST do not need to explicitly compute the matrix ? and only
needs the kernel matrix K (that can be computed through the kernel function k). Therefore, a high or
even an infinite dimensional feature transform can be used to assist LSDR in kernel-CPLST through
a suitable kernel function. Problem (8) can again be solved by considering the eigenvectors with the
largest eigenvalues of ZT (2KQ ? QKKQ) Z as the rows of V.
4
Experiment
In this section, we conduct experiments on eight real-world datasets, downloaded from Mulan [23],
to validate the performance of CPLST and other LSDR approaches. Table 2 shows the number of labels of each dataset. Because kernel ridge regression itself, kernel-CPLST need to invert an N by N
matrix, we can only afford to conduct a fair comparison using mid-sized datasets. In each run of the
experiment, we randomly sample 80% of the dataset for training and reserve the rest for testing. All
the results are reported with the mean and the standard error over 100 different random runs.
Dataset
# Labels (K)
Table 2: The number of labels of each dataset
bib. cor. emo. enr. gen. med.
159
374
6
53
27
45
sce.
6
yea.
14
We take PLST, OCCA, CPLST, and kernel-CPLST in our comparison. We do not include Compressive Sensing [13] in the comparison because earlier work [24] has shown that the algorithm is
more sophisticated while being inferior to PLST. We conducted some side experiments on CCA [6]
for LSDR (see Subsection 2.2) and found that it is at best comparable to OCCA. Given the space
5
0.245
0.235
PBR
OCCA
PLST
CPLST
2000
0.23
|Z ? ZVTV|2
Hamning loss
2500
PBR
OCCA
PLST
CPLST
0.24
0.225
0.22
1500
1000
0.215
0.21
500
0.205
0.2
0
5
10
0
15
0
5
# of dimension
(a) Hamming loss
|XWT ? ZVT|2 + |Z ? ZVTV|2
2250
|XWT ? ZVT|2
2000
1500
1000
PBR
OCCA
PLST
CPLST
500
0
5
15
(b) encoding error
2500
0
10
# of dimension
10
2150
2100
2050
2000
1950
15
# of dimension
PBR
OCCA
PLST
CPLST
2200
0
5
10
15
# of dimension
(c) prediction error
(d) loss bound
Figure 1: yeast: test results of LSDR algorithm when coupled with linear regression
constraints, we decide to only report the results on OCCA. In addition to those LSDR approaches,
we also consider a simple baseline approach [24], partial binary relevance (PBR). PBR randomly
selects M labels from the original label set during training and only learns those M binary classifiers for prediction. For the other labels, PBR directly predicts ?1 without any training to match the
sparsity assumption as exploited by Compressive Sensing [13].
4.1
Label Space Dimension Reduction with Linear Regression
In this subsection, we couple PBR, OCCA, PLST and CPLST with linear regression. The yeast
dataset reveals clear differences between the four LSDR approaches and is hence taken for presentation here, while similar differences have been observed on other datasets as well. Figure 1(a) shows
the test Hamming loss with respect to the possible M (labels) used. It is clear that CPLST is better
than the other three approaches. PLST can reach similar performance to CPLST only at a larger M .
The other two algorithms, OCCA and PBR, are both significantly worse than CPLST.
To understand the cause of the different performance, we plot the (test) encoding error kZ ?
ZVT Vk2F , the prediction error kXWT ? ZVT k2F , and the loss bound (1) in Figure 1. Figure 1(b)
shows the encoding error on the test set, which matches the design of PLST. Regardless of the approaches used, the encoding error decreases to 0 when using all 14 dimensions because the {vm }?s
can span the whole label space. As expected, PLST achieves the lowest encoding error across every
number of dimensions. CPLST partially minimizes the encoding error in its objective function, and
hence also achieves a decent encoding error. On the other hand, OCCA is blind to and hence worst
at the encoding error. In particular, its encoding error is even worse than that of the baseline PBR.
Figure 1(c) shows the prediction error kXWT ? ZVT k2F on the test set, which matches the design
of OCCA. First, OCCA indeed achieves the lowest prediction error across all number of dimensions.
PLST, which is blind to the prediction error, reaches the highest prediction error, and is even worse
than PBR. The results further reveal the trade-off between the encoding error and the prediction
error: more efficient encoding of the label space are harder to predict. PLST takes the more efficient
encoding to the extreme, and results in worse prediction error; OCCA, on the other hand, is better
in terms of the prediction error, but leads to the least efficient encoding.
Figure 1(d) shows the scaled upper bound (1) of the Hamming loss, which equals the sum of the encoding error and the prediction error. CPLST is designed to knock down this bound, which explains
its behavior in Figure 1(d) and echoes its superior performance in Figure 1(a). In fact, Figure 1(d)
shows that the bound (1) is quite indicative of the performance differences in Figure 1(a). The results
6
Table 3: Test Hamming loss of PLST and CPLST with linear regression
Dataset
bibtex
corel5k
emotions
enron
genbase
medical
scene
yeast
Algorithm
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
M = 20%K
0.0129 ? 0.0000
0.0127 ? 0.0000
0.0094 ? 0.0000
0.0094 ? 0.0000
0.2207 ? 0.0020
0.2189 ? 0.0019
0.0728 ? 0.0004
0.0729 ? 0.0004
0.0169 ? 0.0004
0.0168 ? 0.0004
0.0346 ? 0.0004
0.0346 ? 0.0004
0.1809 ? 0.0004
0.1744 ? 0.0004
0.2150 ? 0.0008
0.2069 ? 0.0008
40%
0.0125 ? 0.0000
0.0124 ? 0.0000
0.0094 ? 0.0000
0.0094 ? 0.0000
0.2064 ? 0.0023
0.2059 ? 0.0022
0.0860 ? 0.0005
0.0864 ? 0.0005
0.0040 ? 0.0002
0.0041 ? 0.0002
0.0407 ? 0.0005
0.0406 ? 0.0005
0.1718 ? 0.0006
0.1532 ? 0.0005
0.2052 ? 0.0009
0.2041 ? 0.0009
60%
0.0124 ? 0.0000
0.0123 ? 0.0000
0.0094 ? 0.0000
0.0094 ? 0.0000
0.1982 ? 0.0022
0.1990 ? 0.0022
0.0946 ? 0.0006
0.0943 ? 0.0006
0.0012 ? 0.0001
0.0012 ? 0.0001
0.0472 ? 0.0005
0.0471 ? 0.0005
0.1566 ? 0.0007
0.1349 ? 0.0005
0.2033 ? 0.0009
0.2024 ? 0.0009
80%
0.0123 ? 0.0000
0.0123 ? 0.0000
0.0094 ? 0.0000
0.0094 ? 0.0000
0.2013 ? 0.0020
0.2015 ? 0.0021
0.1006 ? 0.0007
0.1006 ? 0.0007
0.0009 ? 0.0001
0.0008 ? 0.0001
0.0490 ? 0.0005
0.0490 ? 0.0005
0.1321 ? 0.0008
0.1209 ? 0.0007
0.2020 ? 0.0009
0.2020 ? 0.0009
100%
0.0123 ? 0.0000
0.0123 ? 0.0000
0.0094 ? 0.0000
0.0094 ? 0.0000
0.2040 ? 0.0022
0.2040 ? 0.0022
0.1028 ? 0.0007
0.1028 ? 0.0007
0.0007 ? 0.0001
0.0007 ? 0.0001
0.0497 ? 0.0006
0.0497 ? 0.0006
0.1106 ? 0.0008
0.1106 ? 0.0008
0.2022 ? 0.0009
0.2022 ? 0.0009
(those within one standard error of the lower one are in bold)
Table 4: Test Hamming loss of LSDR algorithm with M5P
Dataset
bibtex
corel5k
emotions
enron
genbase
medical
scene
yeast
Algorithm
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
PLST
CPLST
M = 20%K
0.0130 ? 0.0001
0.0129 ? 0.0001*
0.0094 ? 0.0000*
0.0094 ? 0.0000*
0.2213 ? 0.0030
0.2209 ? 0.0031*
0.0490 ? 0.0002
0.0489 ? 0.0003*
0.0215 ? 0.0004*
0.0215 ? 0.0004*
0.0127 ? 0.0002
0.0126 ? 0.0002*
0.1802 ? 0.0005
0.1674 ? 0.0005
0.2162 ? 0.0008
0.2083 ? 0.0009*
40%
0.0128 ? 0.0001*
0.0128 ? 0.0001*
0.0094 ? 0.0000*
0.0094 ? 0.0000*
0.2109 ? 0.0030
0.2085 ? 0.0032*
0.0488 ? 0.0002*
0.0489 ? 0.0003
0.0202 ? 0.0004*
0.0202 ? 0.0004*
0.0099 ? 0.0002*
0.0099 ? 0.0002*
0.1688 ? 0.0007
0.1538 ? 0.0006*
0.2082 ? 0.0009
0.2064 ? 0.0009*
60%
0.0128 ? 0.0001
0.0127 ? 0.0001*
0.0094 ? 0.0000*
0.0094 ? 0.0000*
0.2039 ? 0.0029
0.2004 ? 0.0031*
0.0489 ? 0.0002*
0.0490 ? 0.0003
0.0195 ? 0.0003*
0.0195 ? 0.0003*
0.0097 ? 0.0002
0.0096 ? 0.0002*
0.1540 ? 0.0008
0.1428 ? 0.0007*
0.2071 ? 0.0009
0.2063 ? 0.0009*
80%
0.0127 ? 0.0001*
0.0127 ? 0.0001*
0.0094 ? 0.0000*
0.0094 ? 0.0000*
0.2051 ? 0.0029
0.2020 ? 0.0031*
0.0490 ? 0.0002*
0.0490 ? 0.0003*
0.0194 ? 0.0003*
0.0195 ? 0.0003
0.0097 ? 0.0002
0.0096 ? 0.0002*
0.1396 ? 0.0011
0.1289 ? 0.0007*
0.2064 ? 0.0009*
0.2064 ? 0.0009*
100%
0.0127 ? 0.0001*
0.0127 ? 0.0001*
0.0094 ? 0.0000*
0.0094 ? 0.0000*
0.2063 ? 0.0030
0.2046 ? 0.0031*
0.0490 ? 0.0002*
0.0490 ? 0.0003*
0.0194 ? 0.0003*
0.0195 ? 0.0003
0.0097 ? 0.0002
0.0096 ? 0.0002*
0.1281 ? 0.0008
0.1268 ? 0.0008*
0.2067 ? 0.0009
0.2066 ? 0.0009*
(those with the lowest mean are marked with *; those within one standard error of the lowest one are in bold)
demonstrate that CPLST explores the trade-off between the encoding error and the prediction error
in an optimal manner to reach the best performance for label space dimension reduction.
The results of PBR and OCCA are consistently inferior to PLST and CPLST across most of the
datasets in our experiments [25] and are not reported here because of space constraints. The test
Hamming loss achieved by PLST and CPLST on other datasets with different percentage of used
labels are reported in Table 3. In most datasets, CPLST is at least as effective as PLST; in bibtex,
scene and yeast, CPLST performs significantly better than PLST.
Note that in the medical and enron datasets, both PLST and CPLST overfit when using many
dimensions. That is, the performance of both algorithms would be better when using fewer dimensions (than the full binary relevance, which is provably equivalent to either PLST or CPLST with
M = K when using linear regression). These results demonstrate that LSDR approaches, like their
feature space dimension reduction counterparts, can potentially help resolve the issue of overfitting.
4.2
Coupling Label Space Dimension Reduction with the M5P Decision Tree
CPLST is designed by assuming a specific regression method. Next, we demonstrate that the inputoutput relationship captured by CPLST is not restricted for coupling with linear regression, but can
be effective for other regression methods in the learning stage (step 4 of Algorithm 1). We do so
by coupling the LSDR approaches with the M5P decision tree [26]. M5P decision tree is a nonlinear regression method. We take the implementation from WEKA [27] for M5P with the default
parameter setting.
The experimental results are shown in Table 4. The relations between PLST and CPLST when
coupled with M5P are similar to the ones when coupled with linear regression. In particular, in
the yeast, scene, and emotions, CPLST outperforms PLST. The results demonstrate that the
captured input-output relation is also effective for regression methods other than linear regression.
4.3
Label Space Dimension Reduction with Kernel Ridge Regression
In this subsection, we conduct experiments for demonstrating the performance of kernelization and
regularization. For kernel-CPLST, we use the Gaussian kernel k(xi , xj ) = exp ??kxi ? xj k2
7
Table 5: Test Hamming loss of LSDR algorithm with kernel ridge regression
Dataset
bibtex
corel5k
emotions
enron
genbase
medical
scene
yeast
Algorithm
PLST
kernel-CPLST
PLST
kernel-CPLST
PLST
kernel-CPLST
PLST
kernel-CPLST
PLST
kernel-CPLST
PLST
kernel-CPLST
PLST
kernel-CPLST
PLST
kernel-CPLST
M = 20%K
0.0151 ? 0.0000
0.0127 ? 0.0000
0.0094 ? 0.0000
0.0092 ? 0.0000
0.2218 ? 0.0020
0.2231 ? 0.0020
0.0460 ? 0.0002
0.0453 ? 0.0002
0.0169 ? 0.0004
0.0170 ? 0.0004
0.0136 ? 0.0002
0.0131 ? 0.0002
0.1713 ? 0.0004
0.1733 ? 0.0004
0.2030 ? 0.0008
0.2018 ? 0.0008
40%
0.0151 ? 0.0000
0.0123 ? 0.0000
0.0094 ? 0.0000
0.0092 ? 0.0000
0.2074 ? 0.0023
0.2071 ? 0.0024
0.0462 ? 0.0002
0.0454 ? 0.0002
0.0039 ? 0.0002
0.0040 ? 0.0002
0.0106 ? 0.0002
0.0098 ? 0.0002
0.1468 ? 0.0006
0.1470 ? 0.0006
0.1913 ? 0.0009
0.1904 ? 0.0009
60%
0.0151 ? 0.0000
0.0121 ? 0.0000
0.0094 ? 0.0000
0.0092 ? 0.0000
0.1983 ? 0.0026
0.1981 ? 0.0025
0.0466 ? 0.0002
0.0455 ? 0.0002
0.0014 ? 0.0001
0.0013 ? 0.0001
0.0103 ? 0.0002
0.0096 ? 0.0002
0.1173 ? 0.0008
0.1179 ? 0.0007
0.1892 ? 0.0009
0.1875 ? 0.0009
80%
0.0151 ? 0.0000
0.0120 ? 0.0000
0.0094 ? 0.0000
0.0092 ? 0.0000
0.2000 ? 0.0025
0.1973 ? 0.0027
0.0468 ? 0.0002
0.0455 ? 0.0002
0.0010 ? 0.0001
0.0009 ? 0.0001
0.0102 ? 0.0002
0.0096 ? 0.0002
0.0932 ? 0.0011
0.0905 ? 0.0007
0.1882 ? 0.0009
0.1869 ? 0.0009
100%
0.0151 ? 0.0000
0.0120 ? 0.0000
0.0094 ? 0.0000
0.0092 ? 0.0000
0.2002 ? 0.0025
0.1988 ? 0.0027
0.0469 ? 0.0002
0.0456 ? 0.0002
0.0008 ? 0.0001
0.0008 ? 0.0001
0.0102 ? 0.0002
0.0096 ? 0.0002
0.0731 ? 0.0007
0.0717 ? 0.0007
0.1881 ? 0.0009
0.1868 ? 0.0009
(those within one standard error of the lower one are in bold)
during LSDR and take kernel ridge regression with the same kernel and the same regularization
parameter as the underlying multi-output regression method. We also couple PLST with kernel ridge
regression for a fair comparison. We select the Gaussian kernel parameter ? and the regularization
parameter ? with a grid search on (log2 ?, log2 ?) using a 5-fold cross validation using the sum of
the Hamming loss across all dimensions. The details of the grid search can be found in the Master?s
Thesis of the first author [25].
When coupled with kernel ridge regression, the comparison between PLST and kernel-CPLST in
terms of the Hamming loss is shown in Table 5. kernel-CPLST performs well for LSDR and outperforms the feature-unaware PLST in most cases. In particular, in five out of the eight datasets,
kernel-CPLST is significantly better than PLST regardless of the number of dimensions used. In
addition, in the medical and enron datasets, the overfitting problem is eliminated with regularization (and parameter selection), and hence kernel-CPLST not only performs better than PLST with
kernel ridge regression, but also is better than the (unregularized) linear regression results in Table 3.
From the previous comparison between CPLST and PLST, CPLST is at least as good as, and usually
better than, PLST. The difference between CPLST and PLST is small but consistent, and does suggest that CPLST is a better choice for label-space dimension reduction. The results provide practical
insights on the two types of label correlation [14]: unconditional correlation (feature-unaware) and
conditional correlation (feature-aware). The unconditional correlation, exploited by PLST and other
LSDR algorithms, readily leads to promising performance in practice. On the other hand, there is
room for some (albeit small) improvements when exploiting the conditional correlation properly like
CPLST.
5
Conclusion
In this paper, we studied feature-aware label space dimension reduction (LSDR) approaches, which
utilize the feature information during LSDR and can be viewed as the counterpart of supervised
feature space dimension reduction. We proposed a novel feature-aware LSDR algorithm, conditional principal label space transformation (CPLST) which utilizes the key conditional correlations
for dimension reduction. CPLST enjoys the theoretical guarantee in balancing between the prediction error and the encoding error in minimizing the Hamming loss bound. In addition, we extended
CPLST to a kernelized version for capturing more sophisticated relations between features and labels. We conducted experiments for comparing CPLST and its kernelized version with other LSDR
approaches. The experimental results demonstrated that CPLST is the best among the LSDR approaches when coupled with linear regression or kernel ridge regression. In particular, CPLST is
consistently better than its feature-unaware precursor, PLST. Moreover, the input-output relation
captured by CPLST can be utilized by regression method other than linear regression.
Acknowledgments
We thank the anonymous reviewers of the conference and members of the Computational Learning
Laboratory at National Taiwan University for valuable suggestions. This work is partially supported
by National Science Council of Taiwan via the grant NSC 101-2628-E-002-029-MY2.
8
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3,935 | 4,562 | 3D Object Detection and Viewpoint Estimation with a
Deformable 3D Cuboid Model
Sven Dickinson
University of Toronto
[email protected]
Sanja Fidler
TTI Chicago
[email protected]
Raquel Urtasun
TTI Chicago
[email protected]
Abstract
This paper addresses the problem of category-level 3D object detection. Given
a monocular image, our aim is to localize the objects in 3D by enclosing them
with tight oriented 3D bounding boxes. We propose a novel approach that extends
the well-acclaimed deformable part-based model [1] to reason in 3D. Our model
represents an object class as a deformable 3D cuboid composed of faces and parts,
which are both allowed to deform with respect to their anchors on the 3D box. We
model the appearance of each face in fronto-parallel coordinates, thus effectively
factoring out the appearance variation induced by viewpoint. Our model reasons
about face visibility patters called aspects. We train the cuboid model jointly and
discriminatively and share weights across all aspects to attain efficiency. Inference
then entails sliding and rotating the box in 3D and scoring object hypotheses.
While for inference we discretize the search space, the variables are continuous
in our model. We demonstrate the effectiveness of our approach in indoor and
outdoor scenarios, and show that our approach significantly outperforms the stateof-the-art in both 2D [1] and 3D object detection [2].
1
Introduction
Estimating semantic 3D information from monocular images is an important task in applications
such as autonomous driving and personal robotics. Let?s consider for example, the case of an autonomous agent driving around a city. In order to properly react to dynamic situations, such an agent
needs to reason about which objects are present in the scene, as well as their 3D location, orientation
and 3D extent. Likewise, a home robot requires accurate 3D information in order to navigate in
cluttered environments as well as grasp and manipulate objects.
While impressive performance has been achieved for instance-level 3D object recognition [3],
category-level 3D object detection has proven to be a much harder task, due to intra-class variation as well as appearance variation due to viewpoint changes. The most common approach to
3D detection is to discretize the viewing sphere into bins and train a 2D detector for each viewpoint [4, 5, 1, 6]. However, these approaches output rather weak 3D information, where typically a
2D bounding box around the object is returned along with an estimated discretized viewpoint.
In contrast, object-centered approaches represent and reason about objects using more sophisticated
3D models. The main idea is to index (or vote) into a parameterized pose space with local geometric [7] or appearance features, that bear only weak viewpoint dependencies [8, 9, 10, 11]. The main
advantage of this line of work is that it enables a continuous pose representation [10, 11, 12, 8], 3D
bounding box prediction [8], and potentially requires less training examples due to its more com1
Figure 1: Left: Our deformable 3D cuboid model. Right Viewpoint angle ?.
pact visual representation. Unfortunately, these approaches work with weaker appearance models
that cannot compete with current discriminative approaches [1, 6, 13]. Recently, Hedau et al. [2]
proposed to extend the 2D HOG-based template detector of [14] to predict 3D cuboids. However,
since the model represents object?s appearance as a rigid template in 3D, its performance has been
shown to be inferior to (2D) deformable part-based models (DPMs) [1].
In contrast, in this paper we extend DPM to reason in 3D. Our model represents an object class with
a deformable 3D cuboid composed of faces and parts, which are both allowed to deform with respect
to their anchors on the 3D box (see Fig 1). Towards this goal, we introduce the notion of stitching
point, which enables the deformation between the faces and the cuboid to be encoded efficiently.
We model the appearance of each face in fronto-parallel coordinates, thus effectively factoring out
the appearance variation due to viewpoint. We reason about different face visibility patterns called
aspects [15]. We train the cuboid model jointly and discriminatively and share weights across all
aspects to attain efficiency. In inference, our model outputs 2D along with oriented 3D bounding
boxes around the objects. This enables the estimation of object?s viewpoint which is a continuous
variable in our representation. We demonstrate the effectiveness of our approach in indoor [2] and
outdoor scenarios [16], and show that our approach significantly outperforms the state-of-the-art in
both 2D [1] and 3D object detection [2].
2
Related work
The most common way to tackle 3D detection is to represent a 3D object by a collection of independent 2D appearance models [4, 5, 1, 6, 13], one for each viewpoint. Several authors augmented
the multi-view representation with weak 3D information by linking the features or parts across
views [17, 18, 19, 20, 21]. This allows for a dense representation of the viewing sphere by morphing
related near-by views [12]. Since these methods usually require a significant amount of training
data, renderings of synthetic CAD models have been used to supplement under-represented views
or provide supervision for training object parts or object geometry [22, 13, 8].
Object-centered approaches, represent object classes with a 3D model typically equipped with viewinvariant geometry and appearance [7, 23, 24, 8, 9, 10, 11, 25]. While these types of models are
attractive as they enable continuous viewpoint representations, their detection performance has typically been inferior to 2D deformable models.
Deformable part-based models (DPMs) [1] are nowadays arguably the most successful approach
to category-level 2D detection. Towards 3D, DPMs have been extended to reason about object
viewpoint by training the mixture model with viewpoint supervision [6, 13]. Pepik et al. [13] took
a step further by incorporating supervision also at the part level. Consistency was enforced by
forcing the parts for different 2D viewpoint models to belong to the same set of 3D parts in the
physical space. However, all these approaches base their representation in 2D and thus output only
2D bounding boxes along with a discretized viewpoint.
The closest work to ours is [2], which models an object with a rigid 3D cuboid, composed of independently trained faces without deformations or parts. Our model shares certain similarities with
this work, but has a set of important differences. First, our model is hierarchical and deformable:
we allow deformations of the faces, while the faces themselves are composed of deformable parts.
We also explicitly reason about the visibility patterns of the cuboid model and train the model accordingly. Furthermore, all the parameters in our model are trained jointly using a latent SVM
formulation. These differences are important, as our approach outperforms [2] by a significant margin.
2
Figure 2: Aspects, together with the range of ? that they cover, for (left) cars and (right) beds.
Finally, in concurrent work, Xiang and Savarese [26] introduced a deformable 3D aspect model,
where an object is represented as a set of planar parts in 3D. This model shares many similarities
with our approach, however, unlike ours, it requires a collection of CAD models in training.
3
A Deformable 3D Cuboid Model
In this paper, we are interested in the problem of, given a single image, estimating the 3D location
and orientation of the objects present in the scene. We parameterize the problem as the one of
estimating a tight 3D bounding box around each object. Our 3D box is oriented, as we reason about
the correspondences between the faces in the estimated bounding box and the faces of our model
(i.e., which face is the top face, front face, etc). Towards this goal, we represent an object class as
a deformable 3D cuboid, which is composed of 6 deformable faces, i.e., their locations and scales
can deviate from their anchors on the cuboid. The model for each cuboid?s face is a 2D template
that represents the appearance of the object in view-rectified coordinates, i.e., where the face is
frontal. Additionally, we augment each face with parts, and employ a deformation model between
the locations of the parts and the anchor points on the face they belong to. We assume that any
viewpoint of an object in the image domain can be modeled by rotating our cuboid in 3D, followed
by perspective projection onto the image plane. Thus inference involves sliding and rotating the
deformable cuboid in 3D and scoring the hypotheses.
A necessary component of any 3D model is to properly reason about the face visibility of the object
(in our case, the cuboid). Assuming a perspective camera, for any given viewpoint, at most 3 faces
are visible in an image. Topologically different visibility patterns define different aspects [15] of
the object. Note that a cuboid can have up to 26 aspects, however, not all necessarily occur for
each object class. For example, for objects supported by the floor, the bottom face will never be
visible. For cars, typically the top face is not visible either. Our model only reasons about the
occurring aspects of the object class of interest, which we estimate from the training data. Note
that the visibility, and thus the aspect, is a function of the 3D orientation and position of a cuboid
hypothesis with respect to the camera. We define ? to be the angle between the outer normal to the
front face of the cuboid hypothesis, and the vector connecting the camera and the center of the 3D
box. We refer the reader to Fig. 1 for a visualization. Assuming a camera overlooking the center of
the cuboid, Fig. 2 shows the range of the cuboid orientation angle on the viewing sphere for which
each aspect occurs in the datasets of [2, 16], which we employ for our experiments. Note however,
that in inference we do not assume that the object?s center lies on the camera?s principal axis.
In order to make the cuboid deformable, we introduce the notion of stitching point, which is a point
on the box that is common to all visible faces for a particular aspect. We incorporate a quadratic
deformation cost between the locations of the faces and the stitching point to encourage the cuboid
to be as rigid as possible. We impose an additional deformation cost between the visible faces,
ensuring that their sizes match when we stitch them into a cuboid hypothesis. Our model represents each aspect with its own set of weights. To reduce the computational complexity and impose
regularization, we share the face and part templates across all aspects, as well as the deformations
between them. However, the deformations between the faces and the cuboid are aspect specific as
they depend on the stitching point.
We formally define the model by a (6 ? (n + 1) + 1)-tuple ({(Pi , Pi,1 , . . . , Pi,n )}i=1,..,6 , b) where Pi
models the i-th face, Pi,j is a model for the j-th part belonging to face i, and b is a real valued bias
term. For ease of exposition, we assume each face to have the same number of parts, n; however,
the framework is general and allows the numbers of parts to vary across faces. For each aspect a,
3
100
50
R?T
L?T
F?T F?R?T F?L?T
400
DPM: mixture statistics for bed
num of training examples
150
0
BBOX3D: face statistics for bed
num of training examples
num of training examples
BBOX3D: aspect statistics for bed
200
300
200
100
0
front
cuboid aspects
left
right
top
cuboid faces
400
300
200
100
0
1
2
3
4
5
6
mixture id
Figure 3: Dataset [2] statistics for training our cuboid model (left and middle) and DPM [1] (right).
we define each of its visible faces by a 3-tuple (Fi , ra,i , dstitch
, ba ), where Fi is a filter for the i-th
a,i
face, ra,i is a two-dimensional vector specifying the position of the i-th face relative to the position
of the stitching point in the rectified view, and di is a four-dimensional vector specifying coefficients
of a quadratic function defining a deformation cost for each possible placement of the face relative
to the position of the stitching point. Here, ba is a bias term that is aspect specific and allows us to
calibrate the scores across aspects with different number of visible faces. Note that Fi will be shared
across aspects and thus we omit index a.
The model representing each part is face-specific, and is defined by a 3-tuple (Fi,j , ri,j , di,j ), where
Fi,j is a filter for the j-th part of the i-th face, ri,j is a two-dimensional vector specifying an ?anchor? position for part j relative to the root position of face i, and di,j is a four dimensional vector
specifying coefficients of a quadratic function defining a deformation cost for each possible placement of the part relative to the anchor position on the face. Note that the parts are defined relative to
the face and are thus independent of the aspects. We thus share them across aspects.
The appearance templates as well as the deformation parameters in the model are defined for each
face in a canonical view where that face is frontal. We thus score a face hypothesis in the rectified
view that makes the hypothesis frontal. Each pair of parallel faces shares a homography, and thus
at most three rectifications are needed for each viewpoint hypothesis ?. In indoor scenarios, we
estimate the 3 orthogonal vanishing points and assume a Manhattan world. As a consequence only
3 rectifications are necessary altogether. In the outdoor scenario, we assume that at least the vertical
vanishing point is given, or equivalently, that the orientation (but not position) of the ground plane
is known. As a consequence, we only need to search for a 1-D angle ?, i.e., the azimuth, in order
to estimate the rotation of the 3D box. A sliding window approach is then used to score the cuboid
hypotheses, by scoring the parts, faces and their deformations in their own rectified view, as well as
the deformations of the faces with respect to the stitching point.
Following 2D deformable part-based models [1], we use a pyramid of HOG features to describe
each face-specific rectified view, H(i, ?), and score a template for a face as follows:
X
score(pi , ?) =
Fi (u0 , v 0 ) ? H[ui + u0 ; vi + v 0 ; i, ?]
(1)
u0 ,v 0
where pi = (ui , vi , li ) specifies the position (ui , vi ) and level li of the face filters in the face-rectified
feature pyramids. We score each part pi,j = (ui,j , vi,j , li,j ) in a similar fashion, but the pyramid is
indexed at twice the resolution of the face. We define the compatibility score between the parts and
the corresponding face, denoted as pi = {pi , {pi,j }j=1,...,n }, as the sum over the part scores and
their deformations with respect to the anchor positions on the face:
n
X
scoreparts (pi , ?) =
(score(pi,j , ?) ? dij ? ?d (pi , pi,j )) ,
(2)
j=1
We thus define the score of a 3D cuboid hypothesis to be the sum of the scores of each face and its
parts, as well as the deformation of each face with respect to the stitching point and the deformation
of the faces with respect to each other as follows
6
X
score(x, ?, s, p) =
V (i, a) score(pi , ?) ? dstitch
? ?stich
(pi , s, ?) ?
a,i
d
i=1
?
6
X
ace f ace
V (i, a) ? dfi,ref
?d (pi , pref , ?) +
6
X
i=1
i>ref
4
V (i, a) ? scoreparts (pi , ?) + ba
Figure 4: Learned models for (left) bed, (right) car.
where p = (p1 , ? ? ? , p6 ) and V (i, a) is a binary variable encoding whether face i is visible under
aspect a. Note that a = a(?, s) can be deterministically computed from the rotation angle ? and the
position of the stitching point s (which we assume to always be visible), which in turns determines
the face visibility V . We use ref to index the first visible face in the aspect model, and
?d (pi , pi,j , ?) = ?d (du, dv) = (du, dv, du2 , dv 2 )
(3)
are the part deformation features, computed in the rectified image of face i implied by the 3D angle
?. As in [1], we employ a quadratic deformation cost to model the relationships between the parts
and the anchor points on the face, and define (dui,j , dvi,j ) = (ui,j , vi,j ) ? (2 ? (ui , vi ) + ri,j ) as
the displacement of the j-th part with respect to its anchor (ui , vi ) in the rectified j-th face. The
deformation features ?stich
(pi , s, ?) between the face pi and the stitching point s are defined as
d
(dui , dvi ) = (ui , vi ) ? (u(s, i), v(s, i)) + ra,i ). Here, (u(s, i), v(s, i)) is the position of the stitching
point in the rectified coordinates corresponding to face i and level l.
We define the deformation cost between the faces to be a function of their relative dimensions:
(
i ,ek )
0, if max(e
f ace
min(ei ,ek ) < 1 +
?d (pi , pk , ?) =
(4)
? otherwise
with ei and ek the lengths of the common edge between faces i and k. We define the deformation of
a face with respect to the stitching point to also be quadratic. It is defined in the rectified view, and
thus depend on ?. We additionally incorporate a bias term for each aspect, ba , to make the scores of
multiple aspects comparable when we combine them into a full cuboid model.
Given an image x, the score of a hypothesized 3D cuboid can be obtained as the dot product between
the model?s parameters and a feature vector, i.e., score(x, ?, s, p) = wa ? ?(x, a(?, s), p), with
ace
ace
0
0
wa = (F10 , ? ? ? , F60 , F1,1
, ? ? ? , F6,n
, d1,1 , ? ? ? , d6,n , dstitch
, ? ? ? , dstitch
, df1,2
, ? ? ? , df5,6
, ba ), (5)
a,1
a,6
and the feature vector:
? 1 , i, ?), ? ? ? , H(p
? 1,1 , i, ?), ???d (p1 , p1,1 ), ? ? ? , ???d (p6 , p6,n ),
?(x, a(?, s), p) = H(p
? ??stitch (p1 , s, ?), ? ? ? , ???stitch (p6 , s, ?), ???f ace (p1 , p2 ), ? ? ? , 1
d
d
d
? ?) = V (i, a) ? ?(i, ?).
where ?? includes the visibility score in the feature vector, e.g., ?(i,
Inference: Inference in this model can be done by computing
fw (x) = max wa ? ?(x, a(?, s), p)
?,s,p
This can be solved exactly via dynamic programming, where the score is first computed for each ?,
i.e., maxs,p wa ? ?(x, a(?, s), p), and then a max is taken over the angles ?. We use a discretization
of 20 deg for the angles. To get the score for each ?, we first compute the feature responses for
the part and face templates (Eq. (1)) using a sliding window approach in the corresponding feature
pyramids. As in [1], distance transforms are used to compute the deformation scores of the parts
efficiently, that is, Eq. (2). The score for each face simply sums the response of the face template
and the scores of the parts. We again use distance transforms to compute the deformation scores
for each face and the stitching point, which is carried out in the rectified coordinates for each face.
We then compute the deformation scores between the faces in Eq. (4), which can be performed
efficiently due to the fact that sides of the same length along one dimension (horizontal or vertical)
in the coordinates of face i will also be constant along the corresponding line when projected to the
coordinate system of face j. Thus, computing the side length ratios of two faces is not quadratic in
the number of pixels but only in the number of horizontal or vertical lines. Finally, we reproject the
scores to the image coordinate system and sum them to get the score for each ?.
5
Hedau et al. [2]
ours
Detectors? performance
DPM [1]
3D det.
combined
54.2%
51.3%
59.6%
55.6%
59.4%
60.5%
DPM [1]
60.0%
Layout rescoring
3D det.
combined
62.8%
64.6%
63.8%
Table 1: Detection performance (measured in AP at 0.5 IOU overlap) for the bed dataset of [2]
3D measure DPM fit3D BBOX3D combined BBOX3D + layout comb. + layout
48.2%
53.9%
53.9%
57.8%
57.1%
convex hull
16.3%
33.0%
34.4%
33.5%
33.6%
face overlap
Table 2: 3D detection performance in AP (50% IOU overlap of convex hulls and faces)
bed: 3D perf.: conv hull overlap
DPM (AP = 0.556)
3D BBOX (AP = 0.594)
combined (AP = 0.605)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
recall
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
bed: 3D perf.: face overlap
DPM fit3D (AP = 0.482)
3D BBOX (AP = 0.539)
combined (AP = 0.539)
precision
precision
precision
bed: 2D Detection performance
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
recall
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
DPM fit3D (AP = 0.163)
3D BBOX (AP = 0.330)
combined (AP = 0.344)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
recall
Figure 5: Precision-recall curves for (left) 2D detection (middle) convex hull, (right) face overlap.
Learning: Given a set of training samples D = (hx1 , y1 , bb1 i, ? ? ? hxN , yN , bbN i), where x is an
image, yi ? {?1, 1}, and bb ? R8?2 are the eight coordinates of the 3D bounding box in the
image, our goal is to learn the weights w = [wa1 , ? ? ? , waP ] for all P aspects in Eq. (5). To train
our model using partially labeled data, we use a latent SVM formulation [1], however, frameworks
such as latent structural SVMs [27] are also possible. To initialize the full model, we first learn a
deformable face+parts model for each face independently, where the faces of the training examples
are rectified to be frontal prior to training. We estimate the different aspects of our 3D model from
the statistics of the training data, and compute for each training cuboid the relative positions va,i of
face i and the stitching point in the rectified view of each face. We then perform joint training of the
full model, treating the training cuboid and the stitching point as latent, however, requiring that each
face filter and the face annotation overlap more than 70%. Following [1], we utilize a stochastic
gradient descent approach which alternates between solving for the latent variables and updating the
weights w. Note that this algorithm is only guaranteed to converge to a local optimum, as the latent
variables make the problem non-convex.
4
Experiments
We evaluate our approach on two datasets, the dataset of [2] as well as KITTI [16], an autonomous
driving dataset. To our knowledge, these are the only datasets which have been labeled with 3D
bounding boxes. We begin our experimentation with the indoor scenario [2]. The bedroom dataset
contains 181 train and 128 test images. To enable a comparison with the DPM detector [1], we
trained a model with 6 mixtures and 8 parts using the same training instances but employing 2D
bounding boxes. Our 3D bed model was trained with two parts per face. Fig. 3 shows the statistics
of the dataset in terms of the number of training examples for each aspect (where L-R-T denotes an
aspect for which the front, right and the top face are visible), as well as per face. Note that the fact
that the dataset is unbalanced (fewer examples for aspects with two faces) does not affect too much
our approach, as only the face-stitching point deformation parameters are aspect specific. As we
share the weights among the aspects, the number of training instances for each face is significantly
higher (Fig. 3, middle). We compare this to DPM in Fig. 3, right. Our method can better exploit the
training data by factoring out the viewpoint dependance of the training examples.
We begin our quantitative evaluation by using our model to reason about 2D detection. The 2D
bounding boxes for our model are computed by fitting a 2D box around the convex hull of the
projection of the predicted 3D box. We report average precision (AP) where we require that the
output 2D boxes overlap with the ground-truth boxes at least 50% using the intersection-over-union
(IOU) criteria. The precision-recall curves are shown in Fig. 5. We compare our approach to the
deformable part model (DPM) [1] and the cuboid model of Hedau et al. [2]. As shown in Table 1
we outperform the cuboid model of [2] by 8.1% and DPM by 3.8%. This is notable, as to the best
6
Figure 6: Detection examples obtained with our model on the bed dataset [2].
Figure 7: Detections in 3D + layout
of our knowledge, this is the first time that a 3D approach outperforms the DPM.
detections of our model are shown in Fig. 6.
1
Examples of
A standard way to improve the detector?s performance has been to rescore object detections using
contextual information [1]. Following [2], we use two types of context. We first combined our
detector with the 2D-DPM [1] to see whether the two sources of information complement each
other. The second type of context is at the scene level, where we exploit the fact that the objects in
indoor environments do not penetrate the walls and usually respect certain size ratios in 3D.
We combine the 3D and 2D detectors using a two step process, where first the 2D detector is run
inside the bounding boxes produced by our cuboid model. A linear SVM that utilizes both scores
as input is then employed to produce a score for the combined detection. While we observe a slight
improvement in performance (1.1%), it seems that our cuboid model is already scoring the correct
boxes well. This is in contrast to the cuboid model of [2], where the increase in performance is more
significant due to the poorer accuracy of their 3D approach.
Following [2], we use an estimate of the room layout to rescore the object hypotheses at the scene
level. We use the approach by Schwing et al. [28] to estimate the layout. To train the re-scoring
classifier, we use the image-relative width and height features as in [1], footprint overlap between
the 3D box and the floor as in [2] as well as 3D statistics such as distance between the object 3D
box and the wall relative to the room height and the ratio between the object and room height in 3D.
This further increases our performance by 5.2% (Table 1). Examples of 3D reconstruction of the
room and our predicted 3D object hypotheses are shown in Fig. 7.
To evaluate the 3D performance of our detector we use the convex hull overlap measure as introduced in [2]. Here, instead of computing the overlap between the predicted boxes, we require that
the convex hulls of our 3D hypotheses projected to the image plane and groundtruth annotations
overlap at least 50% in IOU measure. Table 2 reports the results and shows that only little is lost in
performance due to a stricter overlap measure.
1
Note that the numbers for our and [2]?s version of DPM slightly differ. The difference is likely due to how
the negative examples are sampled during training (the dataset has a positive example in each training image).
7
Figure 8: KITTI: examples of car detections. (top) Ground truth, (bottom) Our 3D detections,
augmented with best fitting CAD models to visualize inferred 3D box orientations.
Since our model also predicts the locations of the dominant object faces (and thus the 3D object
orientation), we would like to quantify its accuracy. We introduce an even stricter measure where
we require that also the predicted cuboid faces overlap with the faces of the ground-truth cuboids. In
particular, a hypothesis is correct if the average of the overlaps between top faces and vertical faces
exceeds 50% IOU. We compare the results of our approach to DPM [1]. Note however, that [1]
returns only 2D boxes and hence a direct comparison is not possible. We thus augment the original
DPM with 3D information in the following way. Since the three dominant orientations of the room,
and thus the objects, are known (estimated via the vanishing points), we can find a 3D box whose
projection best overlaps with the output of the 2D detector. This can be done by sliding a cuboid
(whose dimensions match our cuboid model) in 3D to best fit the 2D bounding box. Our approach
outperforms the 3D augmented DPM by a significant margin of 16.7%. We attribute this to the fact
that our cuboid is deformable and thus the faces localize more accurately on the faces of the object.
We also conducted preliminary tests for our model on the autonomous driving dataset KITTI [16].
We trained our model with 8 aspects (estimated from the data) and 4 parts per face. An example of a
learned aspect model is shown in Fig. 4. Note that the rectangular patches on the faces represent the
parts, and color coding is used to depict the learned part and face deformation weights. We can observe that the model effectively and compactly factors out the appearance changes due to changes in
viewpoint. Examples of detections are shown in Fig.8. The top rows show groundtruth annotations,
while the bottom rows depict our predicted 3D boxes. To showcase also the viewpoint prediction
of our detector we insert a CAD model inside each estimated 3D box, matching its orientation in
3D. In particular, for each detection we automatically chose a CAD model out of a collection of 80
models whose 3D bounding box best matches the dimensions of the predicted box. One can see that
our 3D detector is able to predict the viewpoints of the objects well, as well as the type of car.
5
Conclusion
We proposed a novel approach to 3D object detection, which extends the well-acclaimed DPM to
reason in 3D by means of a deformable 3D cuboid. Our cuboid allows for deformations at the face
level via a stitching point as well as deformations between the faces and the parts. We demonstrated
the effectiveness of our approach in indoor and outdoor scenarios and showed that our approach
outperforms [1] and [2] in terms of 2D and 3D estimation. In future work, we plan to reason jointly
about the 3D scene layout and the objects in order to improve the performance in both tasks.
Acknowledgements. S.F. has been supported in part by DARPA, contract number W911NF-10-20060. The views and conclusions contained in this document are those of the authors and should not
be interpreted as representing the official policies, either express or implied, of the Army Research
Laboratory or the U.S. Government.
8
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9
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3,936 | 4,563 | Distributed Non-Stochastic Experts
Varun Kanade?
UC Berkeley
[email protected]
Zhenming Liu?
Princeton University
[email protected]
Bo?zidar Radunovi?c
Microsoft Research
[email protected]
Abstract
We consider the online distributed non-stochastic experts problem, where the distributed system consists of one coordinator node that is connected to k sites, and
the sites are required to communicate with each other via the coordinator. At each
time-step t, one of the k site nodes has to pick an expert from the set {1, . . . , n},
and the same site receives information about payoffs of all experts for that round.
The goal of the distributed system is to minimize regret at time horizon T , while
simultaneously keeping communication to a minimum. The two extreme solutions
to this problem are: (i) Full communication:
p This essentially simulates the nondistributed setting to obtain the optimal O( log(n)T ) regret bound at the cost of
T communication.
p (ii) No communication: Each site runs an independent copy ?
the regret is O( log(n)kT ) and the communication is 0. This paper shows
the
?
difficulty of simultaneously achieving regret asymptotically better than kT and
communication better than T . We give a novel algorithm
that for an oblivious
?
adversary achieves a non-trivial trade-off: regret O( k 5(1+)/6 T ) and communication O(T /k ), for any value of ? (0, 1/5). We also consider a variant of the
model, where the coordinator picks the expert. In this model, we show that the
label-efficient forecaster of Cesa-Bianchi et al. (2005) already gives us strategy
that is near optimal in regret vs communication trade-off.
1
Introduction
In this paper, we consider the well-studied non-stochastic expert problem in a distributed setting.
In the standard (non-distributed) setting, there are a total of n experts available for the decisionmaker to consult, and at each round t = 1, . . . , T , she must choose to follow the advice of one of
the experts, say at , from the set [n] = {1, . . . , n}. At the end of the round, she observes a payoff
vector pt ? [0, 1]n , where pt [a] denotes the payoff that would have been received by following
the advice of expert a. The payoff received by the decision-maker is pt [at ]. In the non-stochastic
setting, an adversary decides the payoff vectors at any time step. At the end of the T rounds, the
regret of the decision maker is the difference in the payoff that she would have received using the
single best expert at all times in hindsight, and the payoff that she actually received, i.e. R =
PT
PT
maxa?[n] t=1 pt [a] ? t=1 pt [at ]. The goal here is to minimize her regret; this general problem
?
This work was performed while the author was at Harvard University supported in part by grant NSF-CCF09-64401
?
This work was performed while the author was at Harvard University supported in part by grants NSF-IIS0964473 and NSF-CCF-0915922.
1
in the non-stochastic setting captures several applications of interest, such as experiment design,
online ad-selection, portfolio optimization, etc. (See [1, 2, 3, 4, 5] and references therein.)
Tight bounds on regret for the non-stochastic expert problem are obtained by the so-called follow
the regularized leader approaches; at time t, the decision-maker chooses a distribution, xt , over the
Pt?1
n experts. Here xt minimizes the quantity s=1 pt ? x + r(x), where r is a regularizer. Common
regularizers are the entropy function, which results in Hedge [1] or the exponentially weighted
forecaster (see chap. 2 in [2]), or as we consider in this paper r(x) = ?? ? x, where ?? ?R [0, ?]n is a
random vector, which gives the follow the perturbed leader (FPL) algorithm [6].
We consider the setting when the decision maker is a distributed system, where several different
nodes may select experts and/or observe payoffs at different time-steps. Such settings are common,
e.g. internet search companies, such as Google or Bing, may use several nodes to answer search
queries and the performance is revealed by user clicks. From the point of view of making better predictions, it is useful to pool all available data. However, this may involve significant communication
which may be quite costly. Thus, the question of interest is studying the trade-off between cost of
communication and cost of inaccuracy (because of not pooling together all data).
2
Models and Summary of Results
We consider a distributed computation model consisting of one central coordinator node connected
to k site nodes. The site nodes must communicate with each other using the coordinator node. At
each time step, the distributed system receives a query1 , which indicates that it must choose an
expert to follow. At the end of the round, the distributed system observes the payoff vector. We consider two different models described in detail below: the site prediction model where one of the k
sites receives a query at any given time-step, and the coordinator prediction model where the query
is always received at the coordinator node. In both these models, the payoff vector, pt , is always
observed at one of the k site nodes. Thus, some communication is required to share the information
about the payoff vectors among nodes. As we shall see, these two models yield different algorithms
and performance bounds. All missing proofs are provided in the long version [7]
Goal: The algorithm implemented on the distributed system may use randomness, both to decide
which expert to pick and to decide when to communicate with other nodes. We focus on simultaneously minimizing the expected regret and the expected communication used by the (distributed)
algorithm. Recall, that the expected regret is:
"
E[R] = E max
a?[n]
T
X
t
p [a] ?
t=1
T
X
#
t
t
p [a ],
(1)
t=1
where the expectation is over the random choices made by the algorithm. The expected communication is simply the expected number (over the random choices) of messages sent in the system.
As we show in this paper, this is a challenging problem and to keep the analysis simple we focus on
bounds in terms of the number of sites k and the time horizon T , which are often the most important
scaling parameters. In particular, our algorithms are variants of follow the perturbed leader (FPL)
and hence our bounds are not optimal in terms of the number of experts n. We believe that the
dependence on the number of experts in our algorithms (upper bounds) can be strengthened using
a different regularizer. Also, all our lower bounds are shown in terms of T and k, for n = 2. For
larger n, using techniques similar to Thm. 3.6 in [2] should give the appropriate dependence on n.
Adversaries: In the non-stochastic setting, we assume that an adversary may decide the payoff vectors, pt , at each time-step and also the site, st , that receives the payoff vector (and also the query in
the site-prediction model). An oblivious adversary cannot see any of the actions of the distributed
system, i.e. selection of expert, communication patterns or any random bits used. However, the
oblivious adversary may know the description of the algorithm. In addition to knowing the description of the algorithm, an adaptive adversary is stronger and can record all of the past actions of the
algorithm, and use these arbitrarily to decide the future payoff vectors and site allocations.
Communication: We do not explicitly account for message sizes, since we are primarily concerned
with scaling in terms of T and k. We require that message size not depend k or T , but only on the
1
We do not use the word query in the sense of explicitly giving some information or context, but merely as
indication of occurrence of an event that forces some site or coordinator to choose an expert
2
number of experts n. In other words, we assume that n is substantially smaller than T and k. All the
messages used in our algorithms contain at most n real numbers. As is standard in the distributed
systems literature, we assume that communication delay is 0, i.e. the updates sent by any node are
received by the recipients before any future query arrives. All our results still hold under the weaker
assumption that the number of queries received by the distributed system in the duration required to
complete a broadcast is negligible compared to k. 2
We now describe the two models in greater detail, state our main results and discuss related work:
1. S ITE P REDICTION M ODEL: At each time step t = 1, . . . , T , one of the k sites, say st , receives
a query and has to pick an expert, at , from the set, [n] = {1, . . . , n}. The payoff vector pt ? [0, 1]n ,
where pt [i] is the payoff of the ith expert is revealed only to the site st and the decision-maker
(distributed system) receives payoff pt [at ], corresponding to the expert actually chosen. The site
prediction model is commonly studied in distributed machine learning settings (see [8, 9, 10]). The
payoff vectors p1 , . . . , pT and also the choice of sites that receive the query, s1 , . . . , sT , are decided
by an adversary. There are two very simple algorithms in this model:
(i) Full communication: The coordinator always maintains the current cumulative payoff vector,
Pt?1 ?
Pt?1 ?
t
? =1 p . At time step t, s receives the current cumulative payoff vector
? =1 p from the
coordinator, chooses an expert at ? [n] using FPL, receives payoff vector pt and sends pt to the
coordinator, which updates its cumulative payoff vector. Note that the total communication
? is 2T
and the system simulates (non-distributed) FPL to achieve (optimal) regret guarantee O( nT ).
(ii) No communication: Each site maintains cumulative payoff vectors corresponding to the queries
received by them, thus implementing k independent versions of FPL. Suppose that the ?
ith site
?
Pk
Pk
receives a total of Ti queries ( i=1 Ti = T ), the regret is bounded by i=1 O( nTi ) = O( nkT )
and the total communication is 0. This upper bound is actually tight in the event that there is 0
communication (see the accompanying long version [7]).
?
Simultaneously achieving regret that is asymptotically lower than knT using communication
asymptotically lower than T turns out to be a significantly challenging question. Our main positive
result is the first distributed expert algorithm in the oblivious adversarial (non-stochastic) setting,
using sub-linear communication. Finding such an algorithm in the case of an adaptive adversary is
an interesting open problem.
Theorem 1. When T ? 2k 2.3 , there exists an algorithm ?
for the distributed experts problem that
against an oblivious adversary achieves regret O(log(n) k 5(1+)/6 T ) and uses communication
O(T /k ), giving non-trivial guarantees in the range ? (0, 1/5).
2. C OORDINATOR P REDICTION M ODEL: At every time step, the query is received by the coordinator node, which chooses an expert at ? [n]. However, at the end of the round, one of the
site nodes, say st , observes the payoff vector pt . The payoff vectors pt and choice of sites st are
decided by an adversary. This model is also a natural one and is explored in the distributed systems
and streaming literature (see [11, 12, 13] and references therein).
?
The full communication protocol is equally applicable here getting optimal regret bound, O( nT ) at
the cost of substantial (essentially T ) communication. But here, we do not have any straightforward
algorithms that achieve non-trivial regret without using any communication. This model is closely
related to the label-efficient prediction problem (see Chapter 6.1-3 in [2]), where the decision-maker
has a limited budget and has to spend part of its budget to observe any payoff information. The
optimal strategy is to request payoff information randomly with probability C/T at each time-step,
if C is the communication budget. We refer to this algorithm as LEF (label-efficient forecaster) [14].
Theorem 2.p[14] (Informal) The LEF algorithms using FPL with communication budget C achieves
regret O(T n/C) against both an adaptive and an oblivious adversary.
One of the crucial differences between this model and that of the label-efficient setting is that when
communication does occur, the site can send cumulative payoff vectors comprising all previous
updates to the coordinator rather than just the latest one. The other difference is that, unlike in
the label-efficient case, the sites have the knowledge of their local regrets and can use it to decide
2
This is because in regularized leader like approaches, if the cumulative payoff vector changes by a small
amount the distribution over experts does not change much because of the regularization effect.
3
when to communicate. However, our lower bounds for natural types of algorithms show that these
advantages probably do not help to get better guarantees.
Lower Bound Results: In the case of an adaptive adversary, we have an unconditional (for any
type of algorithm) lower bound in both the models:
Theorem 3. Let n?= 2 be the number of experts. Then any (distributed) algorithm that achieves
expected regret o( kT ) must use communication (T /k)(1 ? o(1)).
The proof appears in [7]. Notice that in the coordinator prediction model, when C = T /k, this
lower bound is matched by the upper bound of LEF.
In the case of an oblivious adversary, our results are weaker, but we can show that certain natural
types of algorithms are not applicable directly in this setting. The so called regularized leader
algorithms, maintain a cumulative payoff vector, Pt , and use only this and a regularizer to select an
expert at time t. We consider two variants in the distributed setting:
? t , which is an (approximate)
(i) Distributed Counter Algorithms: Here the forecaster only uses P
t
version of the cumulative payoff vector P . But we make no assumptions on how the forecaster will
? t. P
? t can be maintained while using sub-linear communication by applying techniques from
use P
distributed systems literature [12]. (ii) Delayed Regularized Leader: Here the regularized leaders
don?t try to explicitly maintain an approximate version of the cumulative payoff vector. Instead,
they may use an arbitrary communication protocol, but make prediction using the cumulative payoff
vector (using any past payoff vectors that they could have received) and some regularizer.
We show in Section 3.2 that the distributed counter approach does not yield any non-trivial guarantee
in the site-prediction model even against an oblivious adversary. It is possible to show a similar lower
bound the in the coordinator prediction model, but is omitted since it follows easily from the idea in
the site-prediction model combined with an explicit communication lower bound given in [12].
Section 4 shows that the delayed regularized leader approach is ineffective even against an oblivious
adversary for coordinator prediction model, suggesting LEF algorithm is near optimal.
Related Work: Recently there has been significant interest in distributed online learning questions
(see for example [8, 9, 10]). However, these works have focused mainly on stochastic optimization problems. Thus, the techniques used, such as reducing variance through mini-batching, are not
applicable to our setting. Questions such as network structure [9] and network delays [10] are interesting in our setting as well, however, at present our work focuses on establishing some non-trivial
regret guarantees in the distributed online non-stochastic experts setting. Study of communication
as a resource in distributed learning is also considered in [15, 16, 17]; however, this body of work
seems only applicable to offline learning.
The other related work is that of distributed functional monitoring [11] and in particular distributed
counting[12, 13], and sketching [18]. Some of these techniques have been successfully applied
in offline machine learning problems [19]. However, we are the first to analyze the performancecommunication trade-off of an online learning algorithm in the standard distributed functional monitoring framework [11]. An application of a distributed counter to an online Bayesian regression was
proposed in Liu et al. [13]. Our lower bounds discussed below, show that approximate distributed
counter techniques do not directly yield non-trivial algorithms.
3
3.1
Site-prediction model
Upper Bounds
We describe our algorithm that simultaneously achieves non-trivial bounds on expected regret and
expected communication. We begin by making two assumptions that simplify the exposition. First,
we assume that there are only 2 experts. The generalization from 2 experts to n is easy, as discussed
in the Remark 1 at the end of this section. Second, we assume that there exists a global query
counter, that is available to all sites and the co-ordinator, which keeps track of the total number of
queries received across the k sites. We discuss this assumption in Remark 2 at the end of the section.
As is often the case in online algorithms, we assume that the time horizon T is known. Otherwise,
the standard doubling trick may be employed. The notation used in this Section is defined in Table 1.
4
Symbol
pt
`
b
Pi
Qi
M (v)
FPi (?)
FRia (?)
FRi (?)
Definition
Payoff vector at time-step t, pt ? [0, 1]2
The length of block into which inputs are divided
Number of input blocks b = T /`
Pi`
Cumulative payoff vector within block i, Pi = t=(i?1)`+1 pt
Pi?1
Cumulative payoff vector until end of block (i ? 1), Qi = j=1 Pj
For vector v ? R2 , M (v) = 1 if v1 > v2 ; M (v) = 2 otherwise
Random variable denoting the payoff obtained by playing FPL(?) on block i
Random variable denoting the regret with respect to action a of playing FPL(?) on block i
FRia (?) = Pi [a] ? FPi (?)
Random variable denoting the regret of playing FPL(?) on payoff vectors in block i
FRi (?) = maxa=1,2 Pi [a] ? FPi (?) = maxa=1,2 FRia (?)
Table 1: Notation used in Algorithm DFPL (Fig. 1) and in Section 3.1.
DFPL(T , `, ?)
?
set b = T /`; ? 0 = `; q = 2`3 T 2 /? 5
for i = 1 . . . , b
let Yi = Bernoulli(q)
if Yi = 1 then #step phase
play FPL(? 0 ) for time-steps (i ? 1)` + 1, . . . , i`
else #block phase
ai = M (Qi + r) where r ?R [0, ?]2
play ai for time-steps (i ? 1)` + 1, . . . , i`
P
i
t
P = i`
t=(i?1)`+1 p
i+1
i
i
Q
=Q +P
FPL(T, n = 2, ?)
for t = 1, . .P
.,T
?
2
at = M ( t?1
? =1 p + r) where r ?R [0, ?]
t
follow expert a at time-step t
observe payoff vector pt
(a)
(b)
Figure 1: (a) DFPL: Distributed Follow the Perturbed Leader, (b) FPL: Follow the Perturbed Leader with
parameter ? for 2 experts (M (?) is defined in Table 1, r is a random vector)
Algorithm Description: Our algorithm DFPL is described in Figure 1(a). We make use of FPL
algorithm, described in Figure 1(b), which takes as a parameter the amount of added noise ?. DFPL
algorithm treats the T time steps as b(= T /`) blocks, each of length `. At a high level, with
probability q on any given block the algorithm is in the step phase, running a copy of FPL (with
noise parameter ? 0 ) across all time steps of the block, synchronizing after each time step. Otherwise
it is in a block phase, running a copy of FPL (with noise parameter ?) across blocks with the same
expert being followed for the entire block and synchronizing after each block. This effectively makes
Pi , the cumulative payoff over block i, the payoff vector for the block FPL. The block FPL has on
average (1 ? q)T /` total time steps. We begin by stating a (slightly stronger) guarantee for FPL.
Lemma 1. Consider the case n = 2. Let p1 , . . . , pT ? [0, 1]2 be a sequence of payoff vectors such
that maxt |pt |? ? B and let the number of experts be 2. Then FPL(?) has the following guarantee
PT
t
t
on expected regret, E[R] ? B
t=1 |p [1] ? p [2]| + ?.
?
The proof is a simple modification to the proof of the standard analysis [6] and is given in [7]. The
rest of this section is devoted to the proof of Lemma 2
Lemma 2. Consider the case n = 2. If T > 2k 2.3 , Algorithm DFPL (Fig. 1) when ?
run with
parameters `, T , ? = `5/12 T 1/2 and b, ? 0 , q as defined in Fig 1, has expected regret O( `5/6 T )
and expected communication O(T k/`). In particular for ` = k 1+?for 0 < < 1/5, the algorithm
simultaneously achieves regret that is asymptotically lower than kT and communication that is
asymptotically lower3 than T .
3
T
Note that here asymptotics is in terms
? of both parameters k and T . Getting communication of the form
f (k) for regret bound better than kT , seems to be a fairly difficult and interesting problem
1??
5
Since we are in the case of an oblivious adversary, we may assume that the payoff vectors p1 , . . . , pT
are fixed ahead of time. Without loss of generality let expert 1 (out of {1, 2}) be the one that has
greater payoff in hindsight. Recall that FRi1 (? 0 ) denotes the random variable that is the regret of
playing FPL(? 0 ) in a step phase on block i with respect to the first expert. In particular, this will
be negative if expert 2 is the best expert on block i, even though globally expert 1 is better. In fact,
this is exactly what our algorithm exploits: it gains on regret in the communication-expensive, step
phase while saving on communication in the block phase.
Pb
The regret can be written as R = i=1 Yi ? FRi1 (? 0 ) + (1 ? Yi )(Pi [1] ? Pi [ai ]) . Note that the
random variables Yi are independent of the random variables FRi1 (? 0 ) and the random variables ai .
As E[Yi ] = q, we can bound the expression for expected regret as follows:
E[R] ? q
b
X
E[FRi1 (? 0 )] + (1 ? q)
i=1
b
X
E[Pi [1] ? Pi [ai ]]
(2)
i=1
We first analyze the second term of the above equation. This is just the regret corresponding
to running FPL(?) at the block level, with T /` time steps. Using the fact that maxi |Pi |? ?
` maxt |pt |? ? `, Lemma 1 allows us to conclude that:
b
X
E[Pi [1] ? Pi [ai ]] ?
i=1
b
`X i
|P [1] ? Pi [2]| + ?
? i=1
(3)
?
Next, we also analyse the first term of the inequality
(2). We chose ? 0 = ` (see Fig. 1) and the
?
analysis of FPL guarantees that E[FRi (? 0 )] ? 2 `, where FRi (? 0 ) denotes the random variable
that is the actual regret of FPL(? 0 ), not the regret with respect to expert 1 (which is FRi1 (? 0 )). Now
i
i 0
i
0
0
either
1 (? ) (i.e. expert 1 was the better one on block i), in which case E[FR1 (? )] ?
? FR (? ) = FR
i 0
i
0
2 `; otherwise ?
FR (? ) = FR2 (? ) (i.e. expert 2 was the better one on block i), in which case
E[FRi1 (? 0 )] ? 2 ` + Pi [1] ? Pi [2]. Note that in this expression
Pi [1] ? Pi [2] is negative. Putting
?
i
0
i
everything together we can write that E[FR1 (? )] ? 2 ` ? (P [2] ? Pi [1])+ , where (x)+ = x if
x ? 0 and 0 otherwise. Thus, we get the main equation for regret.
b
b
X
?
`X i
(Pi [2] ? Pi [1])+ +
|P [1] ? Pi [2]| +?
E[R] ? 2qb ` ? q
?
i=1
i=1
|
{z
} |
{z
}
term 1
(4)
term 2
?
?
Note that the first (i.e. 2qb `) and last (i.e. ?) terms of inequality (4) are O( `5/6 T ) for the setting
of the parameters as in Lemma 2. The strategy is to show that when ?term 2? becomes large, then
?term 1? is also large in magnitude, but negative, compensating the effect of ?term 1?. We consider
a few cases:
Case 1: When the best expert is identified quickly and not changed thereafter. Let ? denote the
maximum index, i, such that Qi [1] ? Qi [2] ? ?. Note that after the block ? is processed, the
algorithm in the block phase will never follow expert 2.
Suppose that ? ? (?/`)2 . We note that the correct bound for ?term 2? is now actually
P?
(`/?) i=1 |Pi [1] ? Pi [2]| ? (`2 ?/?) ? ? since |Pi [1] ? Pi [2]| ? ` for all i.
Case 2 The best expert may not be identified quickly, furthermore |Pi [1] ? Pi [2]| is large often. In
this case, although ?term 2? may be large (when (Pi [1] ? Pi [2]) is large), this is compensated by
the negative regret in ?term 1? in expression (4). This is because if |Pi [1] ? Pi [2]| is large often,
but the best expert is not identified quickly, there must be enough blocks on which (Pi [2] ? Pi [1])
is positive and large.
Notice that ? ? (?/`)2 . Define ? = ? 2 /T and let S = {i ? ? | |Pi [1] ? Pi [2]| ? ?}.
P?
Let ? = |S|/?. We show that i=1 (Pi [2] ? Pi [1])+ ? (???)/2
P ? ?. To see this consider
S1 = {i ? S | Pi [1] > Pi [2]} and S2 = S \ S1 . First, observe that i?S |Pi [1] ? Pi [2]| ? ???.
P
P
Then, if i?S2 (Pi [2] ? Pi [1]) ? (???)/2, we are done. If not i?S1 (Pi [1] ? Pi [2]) ? (???)/2.
P?
P?
Now notice that i=1 Pi [1] ? Pi [2] ? ?, hence it must be the case that i=1 (Pi [2] ? Pi [1])+ ?
6
(???)/2 ? ?. Now for the value of q = 2`3 T 2 /? 5 and if ? ? ? 2 /(T `), the negative contribution of
?term 1? is at least q???/2 which greater than the maximum possible positive contribution of ?term
2? which is `2 ?/?. It is easy to see that these quantities are equal and hence the total contribution of
?term 1? and ?term 2? together is at most ?.
Case 3 When |Pi [1] ? Pi [2]| is ?small? most of the time. In this case the parameter ? is actually
well-tuned (which was not the case when |Pi [1] ? Pi [2]| ? `) and gives us a small overall regret.
(See Lemma 1.) We have ? < ? 2 /(T `). Note that ?` ? ? = ? 2 /T and that ? ? T /`. In this case
P?
?term 2? can be bounded easily as follows: ?` i=1 |Pi [1] ? Pi [2]| ? ?` (??` + (1 ? ?)??) ? 2?
The above three cases exhaust all possibilities and hence no matter what the nature of the payoff
sequence, the expected regret of DFPL is bounded by O(?) as required. The expected total communication is easily seen to be O(qT +T k/`) ? the q(T /`) blocks on which step FPL is used contribute
O(`) communication each, and the (1 ? q)(T /`) blocks where block FPL is used contributed O(k)
communication each.
Remark 1. Our algorithm can be generalized to n experts by recursively dividing the set of experts
in two and applying our algorithm to two meta-experts, to give the result of Theorem 1. Details are
provided in [7].
Remark 2. Instead of a global counter, it suffices for the co-ordinator to maintain an approximate
counter and notify all sites of beginning an end of blocks by broadcast. This only adds 2k communication per block. See [7] for more details.
3.2
Lower Bounds
In this section we give a lower bound on distributed counter algorithms in the site prediction model.
Distributed counters allow tight approximation guarantees,
i.e. for factor ? additive approximation,
?
the communication required
is
only
O(T
log(T
)
k/?)
[12].
We observe that the noise used by
?
FPL is quite large, O( T ), and so it is tempting to find a suitable ? and run FPL using approximate
cumulative payoffs. We consider the class of algorithms such that:
(i) Whenever each site receives a query, it has an (approximate) cumulative payoff of each expert to
additive accuracy ?. Furthermore, any communication is only used to maintain such a counter.
(ii) Any site only uses the (approximate) cumulative payoffs and any local information it may have
to choose an expert when queried.
However, our negative result shows that even with a highly accurate counter?? = O(k), the nonstochasticity of the payoff sequence may cause any such algorithm to have ?( kT ) regret. Furthermore, we show that any distributed algorithm that implements (approximate) counters to additive
error k/10 on all sites4 is at least ?(T ).
Theorem 4. At any time step t, suppose each site has an (approximate) cumulative payoff count,
? t [a], for every expert such that |Pt [a] ? P
? t [a]| ? ?. Then we have the following:
P
? t [a] and any local information at the
1. If ? ? k, any algorithm that uses the approximate counts P
?
site making the decision, cannot achieve expected regret asymptotically better than ?T .
2. Any protocol on the distributed system that guarantees that at each time step, each site has a
? = k/10 approximate cumulative payoff with probability ? 1/2, uses ?(T ) communication.
4
Coordinator-prediction model
In the co-ordinator prediction model, as mentioned earlier it is possible to use the label-efficient forecaster, LEF (Chap. 6 [2, 14]). Let C be an upper bound on the total amount of communication we are
allowed to use. The label-efficient predictor translates into the following simple protocol: Whenever
a site receives a payoff vector, it will forward that particular payoff to the coordinator with probability p ? C/T . The coordinator will always execute the exponentially weighted forecaster
p over the
sampled subset of payoffs to make new decisions.
Here, the expected regret is O(T log(n)/C).
?
In other words, if our regret needs to be O( T ), the communication needs to be linear in T .
4
The approximation guarantee is only required when a site receives a query and has to make a prediction.
7
We observe that in principle there is a possibility of better algorithms in this setting for mainly two
reasons: (i) when the sites send payoff vectors to the co-ordinator, they can send cumulative payoffs
rather than the latest ones, thus giving more information, and (ii) the sites may decided when to
communicate as a function of the payoff vectors instead of just randomly. However,
? we present a
lower-bound that shows that for a natural family of algorithms achieving regret O( T ) requires at
least ?(T 1? ) for every > 0, even when k = 1. The type of algorithms we consider may have an
arbitrary communication protocol, but it satisfies the following: (i) Whenever a site communicates
with the coordinator, the site will report?
its local cumulative payoff vector. (ii) When the
? coordinator
makes a decision, it will execute, FPL( T ), (follow the perturbed leader with noise T ) using the
latest cumulative payoff vector. The proof of Theorem 5 appears in [7] and the results could be
generalized to other regularizers.
Theorem 5. Consider the distributed non-stochastic expert problem in ?coordinator prediction
model. Any algorithm of the kind described above that achieves regret O( T ) must use ?(T 1? )
communication against an oblivious adversary for every constant .
Simulations
400
Cumulative regret
4
x 10
No?communication
Mini?batch, p=4.64e?002
All?communication
HYZ, p=2.24e?001
DFPL, ?=0.00e+000
DFPL, ?=1.48e?001
500
300
200
Worst?case communication
5
100
0
?100
0
0.5
1
?
1.5
6
4
2
0
0
2
4
DFPL
Mini?batches
HYZ
8
500
x 10
(a)
1000
1500
Worst?case regret
2000
(b)
Figure 2: (a) - Cumulative regret for the MC sequences as a function of correlation ?, (b) - Worst-case
cumulative regret vs. communication cost for the MC and zig-zag sequences.
In this section, we describe some simulation results comparing the efficacy of our algorithm DFPL
with some other techniques. We compare DFPL against simple algorithms ? full communication
and no communication, and two other algorithms which we refer to as mini-batch and HYZ. In the
mini-batch algorithm, the coordinator requests randomly, with some probability p at any time step,
all cumulative payoff vectors at all sites. It then broadcasts the sum (across all of the sites) back to
the sites, so that all sites have the latest cumulative payoff vector. Whenever such a communication
does occur, the cost is 2k. We refer to this as mini-batch because it is similar in spirit to the minibatch algorithms used in the stochastic optimization problems. In the HYZ algorithm, we use the
distributed counter technique of Huang et al. [12] to maintain the (approximate) cumulative payoff
for each expert. Whenever a counter update occurs, the coordinator must broadcast to all nodes to
make sure they have the most current update.
We consider two types of synthetic sequences. The first is a zig-zag sequence, with ? being the
length of one increase/decrease. For the first ? time steps the payoff vector is always (1, 0) (expert
1 being better), then for the next 2? time steps, the payoff vector is (0, 1) (expert 2 is better), and
then again for the next 2? time-steps, payoff vector is (1, 0) and so on. The zig-zag sequence is also
the sequence used in the proof of the lower bound in Theorem 5. The second is a two-state Markov
1
. While in state 1, the payoff vector
chain (MC) with states 1, 2 and Pr[1 ? 2] = Pr[2 ? 1] = 2?
is (1, 0) and when in state 2 it is (0, 1).
In our simulations we use T = 20000 predictions, and k = 20 sites. Fig. 2 (a) shows the performance of the above algorithms for the MC sequences, the results are averaged across 100 runs,
over both the randomness of the MC and the algorithms. Fig. 2 (b) shows the worst-case cumulative communication vs the worst-case cumulative regret trade-off for three algorithms: DFPL,
mini-batch and HYZ, over all the described sequences. While in general it is hard to compare algorithms on non-stochastic inputs, our results confirm that for non-stochastic sequences inspired by
the lower-bounds in the paper, our algorithm DFPL outperforms other related techniques.
8
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3,937 | 4,564 | On Triangular versus Edge Representations ?
Towards Scalable Modeling of Networks
Qirong Ho
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Junming Yin
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Eric P. Xing
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
In this paper, we argue for representing networks as a bag of triangular motifs,
particularly for important network problems that current model-based approaches
handle poorly due to computational bottlenecks incurred by using edge representations. Such approaches require both 1-edges and 0-edges (missing edges) to be
provided as input, and as a consequence, approximate inference algorithms for
these models usually require ?(N 2 ) time per iteration, precluding their application to larger real-world networks. In contrast, triangular modeling requires less
computation, while providing equivalent or better inference quality. A triangular
motif
P is a vertex triple containing 2 or 3 edges, and the number of such motifs is
?( i Di2 ) (where Di is the degree of vertex i), which is much smaller than N 2
for low-maximum-degree networks. Using this representation, we develop a novel
mixed-membership network model and approximate inference algorithm suitable
for large networks with low max-degree. For networks with high maximum degree, the triangular motifs can be naturally subsampled in a node-centric fashion,
allowing for much faster inference at a small cost in accuracy. Empirically, we
demonstrate that our approach, when compared to that of an edge-based model,
has faster runtime and improved accuracy for mixed-membership community detection. We conclude with a large-scale demonstration on an N ? 280, 000-node
network, which is infeasible for network models with ?(N 2 ) inference cost.
1
Introduction
Network analysis methods such as MMSB [1], ERGMs [20], spectral clustering [17] and latent
feature models [12] require the adjacency matrix A of the network as input, reflecting the natural assumption that networks are best represented as a set of edges taking on the values 0 (absent)
or 1 (present). This assumption is intuitive, reasonable, and often necessary for some tasks, such
as link prediction, but it comes at a cost (which is not always necessary, as we will discuss later)
for other tasks, such as community detection in both the single-membership or admixture (mixedmembership) settings. The fundamental difference between link prediction and community detection is that the first is concerned with link outcomes on pairs of vertices, for which providing links
as input is intuitive. However, the second task is about discovering the community memberships of
individual vertices, and links are in fact no longer the only sensible representation. By representing
the input network as a bag of triangular motifs ? by which we mean vertex triples with 2 or 3
edges ? one can design novel models for mixed-membership community detection that outperform
models based on the adjacency matrix representation.
The main advantage of the bag-of-triangles representation lies in its huge reduction of computational cost for certain network analysis problems, with little or no loss of outcome quality. In the
traditional edge representation, if N is the number of vertices, then the adjacency matrix has size
?(N 2 ) ? thus, any network analysis algorithm that touches every element must have ?(N 2 ) runtime complexity. For probabilistic network models, this statement applies to the cost of approximate
1
i
j
i
k
(a)
j
i
k
j
i
k
j
i
j
k
(b)
i
k
j
i
k
j
i
k
(c)
j
k
(d)
Figure 1: Four types of triangular motifs: (a) full-triangle; (b) 2-triangle; (c) 1-triangle; (d) empty-triangle.
For mixed-membership community detection, we only focus on full-triangles and 2-triangles.
inference. For example, the Mixed Membership Stochastic Blockmodel (MMSB) [1] has ?(N 2 )
latent variables, implying an inference cost of ?(N 2 ) per iteration. Looking beyond, the popular p?
or Exponential Random Graph models [20] are normally estimated via MCMC-MLE, which entails
drawing network samples (each of size ?(N 2 )) from some importance distribution. Finally, latent
factor models such as [12] only have ?(N ) latent variables, but the Markov blanket of each variable
depends on ?(N ) observed variables, resulting in ?(N 2 ) computation per sweep over all variables.
With an inference cost of ?(N 2 ), even modestly large networks with only ? 10, 000 vertices are
infeasible, to say nothing of modern social networks with millions of vertices or more.
On the other hand,
P it can be shown that the number of 2- and 3-edge triangular motifs is upperbounded by ?( i Di2 ), where Di is the degree of vertex i. For networks with low maximum degree,
this quantity is N 2 , allowing us to construct more parsimonious models with faster inference
algorithms. Moreover, for networks with high maximum degree, one can subsample ?(N ? 2 ) of
these triangular motifs in a node-centric fashion, where ? is a user-chosen parameter. Specifically,
we assign triangular motifs to nodes in a natural manner, and then subsample motifs only from nodes
with too many of them. In contrast, MMSB and latent factor models rely on distributions over 0/1edges (i.e. edge probabilities), and for real-world networks, these distributions cannot be preserved
with small (i.e. o(N 2 )) sample sizes because the 0-edges asymptotically outnumber the 1-edges.
As we will show, a triangular representation does not preserve all information found in an edge representation. Nevertheless, we argue that one should represent complex data objects in a task-dependent
manner, especially since computational cost is becoming a bottleneck for real-world problems like
analyzing web-scale network data. The idea of transforming the input representation (e.g. from
network to bag-of-triangles) for better task-specific performance is not new. A classic example is
the bag-of-words representation of a document, in which the ordering of words is discarded. This
representation has proven effective in natural language processing tasks such as topic modeling [2],
even though it eliminates practically all grammatical information. Another example from computer
vision is the use of superpixels to represent images [3, 4]. By grouping adjacent pixels into larger
superpixels, one obtains a more compact image representation, in turn leading to faster and more
meaningful algorithms. When it comes to networks, triangular motifs (Figure 1) are already of
significant interest in biology [13], social science [19, 9, 10, 16], and data mining [21, 18, 8]. In
particular, 2- and 3-edge triangular motifs are central to the notion of transitivity in the social sciences ? if we observe edges A-B and B-C, does A have an edge to C as well? Transitivity is of
special importance, because high transitivity (i.e. we frequently observe the third edge A-C) intuitively leads to stronger clusters with more within-cluster edges. In fact, the ratio of 3-edge triangles
to connected vertex triples (i.e. 2- and 3-edge triangular motifs) is precisely the definition of the
network clustering coefficient [16], which is a popular measure of cluster strength.
In the following sections, we begin by characterizing the triangular motifs, following which we develop a mixed-membership model and inference algorithm based on these motifs. Our model, which
we call MMTM or the Mixed-Membership Triangular Model, performs mixed-membership community detection, assigning each vertex i to a mixture of communities. This allows for better outlier detection and more informative visualization compared to single-membership modeling. In addition,
mixed-membership modeling has two key advantages: first, MM models such as MMSB, Latent
Dirichlet Allocation and our MMTM are easily modified for specialized tasks ? as evidenced by
the rich literature on topic models [2, 1, 14, 5]. Second, MM models over disparate data types (text,
network, etc.) can be combined by fusing their latent spaces, resulting in a multi-view model ? for
example, [14, 5] model both text and network data from the same mixed-membership vectors. Thus,
our MMTM can serve as a basic modeling component for massive real-world networks with copious
side information. After developing our model and inference algorithm, we present simulated experiments comparing them on a variety of network types to an adjacency-matrix-based model (MMSB)
and its inference algorithm. These experiments will show that triangular mixed-membership modeling results in both faster inference and more accurate mixed-membership recovery. We conclude
by demonstrating our model/algorithm on a network with N ? 280, 000 nodes and ? 2, 300, 000
edges, which is far too large for ?(N 2 ) inference algorithms such as variational MMSB [1] and the
Gibbs sampling MMSB inference algorithm we developed for our experiments.
2
2
Triangular Motif Representation of a Network
In this work, we consider undirected networks over N vertices, such as social networks. Most of
the ideas presented here also generalize to directed networks, though the analysis is more involved
since directed networks can generate more motifs than undirected ones. To prevent confusion, we
shall use the term ?1-edge? to refer to edges that exist between two vertices, and the term ?0edge? to refer to missing edges. Now, define a triangular motif Eijk involving vertices i < j < k
to be the type of subgraph over these 3 vertices. There are 4 basic classes of triangular motifs
(Figure 1), distinguished by their number of 1-edges: full-triangle ?3 (three 1-edges), 2-triangle ?2
(two 1-edges), 1-triangle ?1 (one 1-edge), and empty-triangle ?0 (no 1-edges). The total number of
triangles, over all 4 classes, is ?(N 3 ). However, our goal is not to account for all 4 classes; instead,
we will focus on ?3 and ?2 while ignoring ?1 and ?0 . We have three primary motivations for this:
1. In the network literature, the most commonly studied ?network motifs? [13], defined as
patterns of significantly recurring inter-connections in complex networks, are the threenode connected subgraphs (namely ?3 and ?2 ) [13, 19, 9, 10, 16].
2. Since the full-triangle and 2-triangle classes are regarded as the basic structural elements
of most networks [19, 13, 9, 10, 16], we naturally expect them to characterize most of the
community structure in networks (cf. network clustering coefficient, as explained in the
introduction). In particular, the ?3 and ?2 triangular motifs preserve almost all 1-edges
from the original network: every 1-edge appears in some triangular motif ?2 , ?3 , except
for isolated 1-edges (i.e. connected components of size 2), which are less interesting from
a large-scale community detection perspective.
3. For real networks, which have far more 0- than 1-edges, focusing only on ?3 and ?2
greatly reduces the number of triangular motifs, via the following lemma:
P 1
Lemma 1. The total number of ?3 ?s and ?2 ?s is upper bounded by
i 2 (Di )(Di ? 1) =
P 2
?( i Di ), where Di is the degree of vertex i.
Proof. Let Ni be the neighbor set of vertex i. For each vertex i, form the set Ti of tuples (i, j, k)
where j < k and j, k ? Ni , which represents the set of all pairs of neighbors of i. Because j and
k are neighbors of i, for every tuple (i, j, k) ? Ti , Eijk is either a ?3 or a ?2 . It is easy to see
that each ?2 is accounted for by exactly one Ti , where i is the center vertex of the ?2 , and that
each
accounted for by three sets Ti , Tj and Tk , one for each vertex in the full-triangle. Thus,
P ?3 isP
1
|T
|
=
i i
i 2 (Di )(Di ? 1) is an upper bound of the total number of ?3 ?s and ?2 ?s.
P
For networks with low maximum degree D, ?( i Di2 ) = ?(N D2 ) is typically much smaller than
?(N 2 ), allowing triangular models to scale to larger networks than edge-based models. As for networks with high maximum degree, we suggest the following node-centric subsampling procedure,
which we call ?-subsampling: for each vertex i with degree Di > ? for some threshold ?, sample
1
2 ?(? ? 1) triangles without replacement and uniformly at random from Ti ; intuitively, this is similar
to capping the network?s maximum degree at Ds = ?. A full-triangle ?3 associated with vertices
i, j and k shall appear in the final subsample only if it has been subsampled from at least one of
Ti , Tj and Tk . To obtain the set of all subsampled triangles ?2 and ?3 , we simply take the union of
subsampled triangles from each Ti , discarding those full-triangles duplicated in the subsamples.
Although this node-centric subsampling does not preserve all properties of a network, such as the
distribution of node degrees, it approximately preserves the local cluster properties of each vertex,
thus capturing most of the community structure in networks. Specifically, the ?local? clustering
coefficient (LCC) of each vertex i, defined as the ratio of #(?3 ) touching i to #(?3 , ?2 ) touching
i, is well-preserved. This follows from subsampling the ?3 and ?2 ?s at i uniformly at random,
though the LCC has a small upwards bias since each ?3 may also be sampled by the other two
vertices j and k. Hence, we expect community detection based on the subsampled triangles to be
nearly as accurate as with the original set of triangles ? which our experiments will show.
We note that other subsampling strategies [11, 22] preserve various network properties, such as
degree distribution, diameter, and inter-node random walk times. In our triangular model, the main
property of interest is the distribution over ?3 and ?2 , analogous to how latent factor models and
MMSB model distributions over 0- and 1-edges. Thus, subsampling strategies that preserve ?3 /?2
distributions (e.g. our ?-subsampling) would be appropriate for our model. In contrast, 0/1-edge
subsampling for MMSB and latent factor models is difficult: most networks have ?(N 2 ) 0-edges
but only o(N 2 ) 1-edges, thus sampling o(N 2 ) 0/1-edges leads to high variance in their distribution.
3
3
Mixed-Membership Triangular Model
Given a network, now represented by triangular motifs ?3 and ?2 , our goal is to perform community
detection for each network vertex i, in the same sense as what an MMSB model would enable. Under
an MMSB, each vertex i is assigned to a mixture over communities, as opposed to traditional singlemembership community detection, which assigns each vertex to exactly one community. By taking
a mixed-membership approach, one gains many benefits over single-membership models, such as
outlier detection, improved visualization, and better interpretability [2, 1].
Following a design principle similar to the one underly?
ing the MMSB models, we now present a new mixedmembership network model built on the more parsimonious triangular representation. For each triplet of ver?i
?j
?k
tices i, j, k ? {1, . . . , N } , i < j < k, if the subgraph on
i, j, k is a 2-triangle with i, j, or k at the center, then let
Eijk = 1, 2 or 3 respectively, and if the subgraph is a fullsk,ij
si,jk
sj,ik
triangle, then let Eijk = 4. Whenever i, j, k corresponds
to a 1- or an empty-triangle, we do not model Eijk . We
assume K latent communities, and that each vertex takes
Bxyz
Eijk
a distribution (i.e. mixed-membership) over them. The
?
observed bag-of-triangles {Eijk } is generated according
to (1) the distribution over community-memberships at
Figure 2: Graphical model representation
each vertex, and (2) a tensor of triangle generation proba- for MMTM, our mixed-membership model
bilities, containing different triangle probabilities for dif- over triangular motifs.
ferent combinations of communities.
More specifically, each vertex i is associated with a community mixed-membership vector ?i ?
?K?1 restricted to the (K ? 1)-simplex ?K?1 . This mixed-membership vector ?i is used to generate community indicators si,jk ? {1, . . . , K}, each of which represents the community chosen
by vertex i when it is forming a triangle with vertices j and k. The probability of observing a triangular motif Eijk depends on the community-triplet si,jk , sj,ik , sk,ij , and a tensor of multinomial
parameters B. Let x, y, z ? {1, . . . , K} be the values of si,jk , sj,ik , sk,ij , and assume WLOG that
x < y < z 1 . Then, Bxyz ? ?3 represents the probabilities of generating the 4 triangular motifs2
among vertices i, j and k. In detail, Bxyz,1 is the probability of the 2-triangle whose center vertex
has community x, and analogously for Bxyz,2 and community y, and for Bxyz,3 and community z;
Bxyz,4 is the probability of the full-triangle.
The MMTM generative model is summarized below; see Figure 2 for a graphical model illustration.
? Triangle tensor Bxyz ? Dirichlet (?) for all x, y, z ? {1, . . . , K}, where x < y < z
? Community mixed-membership vectors ?i ? Dirichlet (?) for all i ? {1, . . . , N }
? For each triplet (i, j, k) where i < j < k,
? Community indices si,jk ? Discrete (?i ), sj,ik ? Discrete (?j ), sk,ij ? Discrete (?k ).
? Generate the triangular motif Eijk based on Bxyz and the ordered values of
si,jk , sj,ik , sk,ij ; see Table 1 for the exact conditional probabilities. There are 6 entries
in Table 1, corresponding to the 6 possible orderings of si,jk , sj,ik , sk,ij .
4
Inference
We adopt a collapsed, blocked Gibbs sampling approach, where ? and B have been integrated out.
Thus, only the community indices s need to be sampled. For each triplet (i, j, k) where i < j < k,
P (si,jk , sj,ik , sk,ij | s?ijk , E, ?, ?) ? P (Eijk |E?ijk , s, ?) P (si,jk | si,?jk , ?)
P (sj,ik | sj,?ik , ?) P (sk,ij | sk,?ij , ?) ,
1
The cases x = y = z, x = y < z and x < y = z require special treatment, due to ambiguity cased by
having identical communities. In the interest of keeping our discussion at a high level, we shall refer the reader
to the appendix for these cases.
2
It is possible to generate a set of triangles that does not correspond to a network, e.g. a 2-triangle centered
on i for (i, j, k) followed by a 3-triangle for (j, k, `), which produces a mismatch on the edge (j, k). This is a
consequence of using a bag-of-triangles model, just as the bag-of-words model in Latent Dirichlet Allocation
can generate sets of words that do not correspond to grammatical sentences. In practice, this is not an issue for
either our model or LDA, as both models are used for mixed-membership recovery, rather than data simulation.
4
si,jk
si,jk
sj,ik
sj,ik
sk,ij
sk,ij
Order
< sj,ik < sk,ij
< sk,ij < sj,ik
< si,jk < sk,ij
< sk,ij < si,jk
< si,jk < sj,ik
< sj,ik < si,jk
Conditional probability of Eijk ? {1, 2, 3, 4}
Discrete([Bxyz,1 , Bxyz,2 , Bxyz,3 , Bxyz,4 ])
Discrete([Bxyz,1 , Bxyz,3 , Bxyz,2 , Bxyz,4 ])
Discrete([Bxyz,2 , Bxyz,1 , Bxyz,3 , Bxyz,4 ])
Discrete([Bxyz,3 , Bxyz,1 , Bxyz,2 , Bxyz,4 ])
Discrete([Bxyz,2 , Bxyz,3 , Bxyz,1 , Bxyz,4 ])
Discrete([Bxyz,3 , Bxyz,2 , Bxyz,1 , Bxyz,4 ])
Table 1: Conditional probabilities of Eijk given si,jk , sj,ik and sk,ij . We define x, y, z to be the ordered (i.e.
sorted) values of si,jk , sj,ik , sk,ij .
where s?ijk is the set of all community memberships except for si,jk , sj,ik , sk,ij , and si,?jk is the
set of all community memberships of vertex i except for si,jk . The last three terms are predictive
distributions of a multinomial-Dirichlet model, with the multinomial parameter ? marginalized out:
P (si,jk | si,?jk , ?)
# [si,?jk = si,jk ] + ?
.
# [si,?jk ] + K?
=
The first term is also a multinomial-Dirichlet predictive distribution (refer to appendix for details).
5
Comparing Mixed-Membership Network Models on Synthetic Networks
For a mixed-membership network model to be useful, it must recover some meaningful notion of
mixed community membership for each vertex. The precise definition of network community has
been a subject of much debate, and various notions of community [1, 15, 17, 12, 6] have been
proposed under different motivations. Our MMTM, too, conveys another notion of community
based on membership in full triangles ?3 and 2-triangles ?2 , which are key aspects of network
clustering coefficients. In our simulations, we shall compare our MMTM against an adjacencymatrix-based model (MMSB), in terms of how well they recover mixed-memberships from networks
generated under a range of assumptions. Note that some of these synthetic networks will not match
the generative assumptions of either our model or MMSB; this is intentional, as we want to compare
the performance of both models under model misspecification.
We shall also demonstrate that MMTM leads to faster inference, particularly when ?-subsampling
triangles (as described in Section 2). Intuitively, we expect the mixed-membership recovery of our
inference algorithm to depend on (a) the degree distribution of the network, and (b) the ?degree
limit? ? used in subsampling the network; performance should increase as the number of vertices i
having degree Di ? ? goes up. In particular, our experiments will demonstrate that subsampling
yields good performance even when the network contains a few vertices with very large degree Di
(a characteristic of many real-world networks).
Synthetic networks We compared our MMTM to MMSB3 [1] on multiple synthetic networks,
evaluating them according to how well their inference algorithms recovered the vertex mixedmembership vectors ?i . Each network was generated from N = 4, 000 mixed-membership vectors
?i of dimensionality K = 5 (i.e. 5 possible communities), according to one of several models:
1. The Mixed Membership Stochastic Blockmodel [1], an admixture generalization of the
stochastic blockmodel. The probability of a link from i to j is ?i B?j for some block matrix
B, and we convert all directed edges into undirected edges. In our experiments, we use a
B with on-diagonal elements Baa = 1/80, and off-diagonal elements Bab = 1/800. Our
values of B are lower than typically seen in the literature, because they are intended to
replicate the 1-edge density of real-world networks with size around N = 4, 000.
2. A simplex Latent position model, where the probability of a link between i, j is ?(1 ?
1
2 ||?i ? ?j ||1 ) for some scaling parameter ?. In other words, the closer that ?i and ?j are,
the higher the link probability. Note that 0 ? ||?i ? ?j ||1 ? 2, because ?i and ?j lie in the
simplex. We choose ? = 1/40, again to reproduce the 1-edge density seen in real networks.
3. A ?Biased? scale-free model that combines the preferred attachment model [7] with a
mixed-membership model. Specifically, we generated M = 60, 000 1-edges as follows: (a)
pick a vertex i with probability proportional to its degree; (b) randomly pick a destination
community k from ?i ; (c) find the set Vk of all vertices v such that ?vk is the largest
element of ?v (i.e. the vertices that mostly belong to community k); (d) within Vk , pick
the destination vertex j with probability proportional to its degree. The resulting network
3
MMSB is applicable to both directed and undirected networks; our experiments use the latter.
5
MMSB
Latent position
Biased scale-free
Pure membership
#0,1-edges
7,998,000
q
q
q
#1-edges
55,696
56,077
60,000
55,651
max(Di )
51
51
231
44
#?3 , ?2
1,541,085
1,562,710
3,176,927
1,533,365
? = 20
749,018
746,979
497,737
746,796
? = 15
418,764
418,448
304,866
418,222
? = 10
179,841
179,757
144,206
179,693
?=5
39,996
39,988
35,470
39,986
Table 2: Number of edges, maximum degree, and number of 3- and 2-edge triangles ?3 , ?2 for each N =
4, 000 synthetic network, as well as #triangles when subsampling at various degree thresholds ?. MMSB
inference is linear in #0,1-edges, while our MMTM?s inference is linear in #?3 , ?2 .
exhibits both a block diagonal structure, as well as a power-law degree distribution. In
contrast, the other two models have binomial (i.e. Gaussian-like) degree distributions.
To use these models, we must input mixed-memberships ?i . These were generated as follows:
1. Divide the N = 4, 000 vertices into 5 groups of size 800. Assign each group to a (different)
dominant community k ? {1, . . . , 5}.
2. Within each group:
(a) Pick 160 vertices to have mixed-membership in 3 communities: 0.8 in the dominant
community k, and 0.1 in two other randomly chosen communities.
(b) The remaining 640 vertices have mixed-membership in 2 communities: 0.8 in the
dominant community k, and 0.2 in one other randomly chosen community.
In other words, every vertex has a dominant community, and one or two other minor communities.
Using these ?i ?s, we generated one synthetic network for each of the three models described. In
addition, we generated a fourth ?pure membership? network under the MMSB model, using pure
?i ?s with full membership in the dominant community. This network represents the special case of
single-community membership. Statistics for all 4 networks can be found in Table 2.
Inference and Evaluation For our MMTM4 , we used our collapsed, blocked Gibbs sampler for
inference. The hyperparameters were fixed at ?, ? = 0.1 and K = 5, and we ran each experiment
for 2,000 iterations. For evaluation, we estimated all ?i ?s using the last sample, and scored the
P
estimates according to i ||??i ? ?i ||2 , the sum of `2 distances of each estimate ??i from its true value
?i . These results were taken under the most favorable permutation for the ??i ?s, in order to avoid the
permutation non-identifiability issue. We repeated every experiment 5 times.
To investigate the effect of ?-subsampling triangles (Section 2), we repeated every MMTM experiment under four different values of ?: 20, 15, 10 and 5. The triangles were subsampled prior to
running the Gibbs sampler, and they remained fixed during inference.
With MMSB, we opted not to use the variational inference algorithm of [1], because we wanted our
experiments to be, as far as possible, a comparison of models rather than inference techniques. To
accomplish this, we derived a collapsed, blocked Gibbs sampler for the MMSB model, with added
Beta hyperparameters ?1 , ?2 on each element of the block matrix B. The mixed-membership vectors
?i (?i in the original paper) and blockmatrix B were integrated out, and we Gibbs sampled each edge
(i, j)?s associated community indicators zi?j , zi?j in a block fashion. Hence, this MMSB sampler
uses the exact same techniques as our MMTM sampler, ensuring that we are comparing models
rather than inference strategies. Furthermore, its per-iteration runtime is still ?(N 2 ), equal to the
original MMSB variational algorithm. All experiments were conducted in exactly the same manner
as with MMTM, with the MMSB hyperparameters fixed at ?, ?1 , ?2 = 0.1 and K = 5.
Results Figure 3 plots the cumulative `2 error for each experiment, as well as the time taken per
trial. On all 4 networks, the full MMTM model performs better than MMSB ? even on the MMSBgenerated network! MMTM also requires less runtime for all but the biased scale-free network,
which has a much larger maximum degree than the others (Table 2). Furthermore, ?-subsampling
is effective: MMTM with ? = 20 runs faster than full MMTM, and still outperforms MMSB while
approaching full MMTM in accuracy. The runtime benefit is most noticable on the biased scale-free
network, underscoring the need to subsample real-world networks with high maximum degree.
We hypothesize MMSB?s poorer performance on networks of this size (N = 4, 000) results from
having ?(N 2 ) latent variables, while noting that the literature has only considered smaller N <
1, 000 networks [1]. Compared to MMTM, having many latent variables not only increases runtime
per iteration, but also the number of iterations required for convergence, since the latent variable state
space grows exponentially with the number of latent variables. In support of this, we have observed
4
As explained in Section 2, we first need to preprocess the network adjacency list into the ?3 , ?2 triangle
representation. The time required is linear in the number of ?3 , ?2 triangles, and is insignificant compared to
the actual cost of MMTM inference.
6
4
Mixed?membership community recovery: Accuracy
12
4500
4000
10
2500
2000
6
4
1500
MMSB
MMTM
MMTM ?=20
MMTM ?=15
MMTM ?=10
MMTM ?=5
1000
500
0
Mixed?membership community recovery: Total runtime
MMSB
MMTM
MMTM ?=20
MMTM ?=15
MMTM ?=10
MMTM ?=5
8
3000
Total runtime (s)
Cumulative L2 error
3500
x 10
MMSB
Latent position
Biased scale?free
2
0
Pure membership
MMSB
Latent position
Biased scale?free
Pure membership
Figure 3: Mixed-membership community recovery task: Cumulative `2 errors and runtime per trial for MMSB,
MMTM and MMTM with ?-subsampling, on N = 4, 000 synthetic networks.
that the MMSB sampler?s complete log-likelihood fluctuates greatly across all 2000 iterations; in
contrast, the MMTM sampler plateaus within 500 iterations, and remains stable.
Scalability Experiments Although the preceding N = 4, 000 experiments appear fairly small, in
actual fact, they are close to the feasible limit for adjacency-matrix-based models like MMSB. To
demonstrate this, we generated four networks with sizes N ? {1000, 4000, 10000, 40000} from the
MMSB generative model. The generative parameters for the N = 4, 000 network are identical to our
earlier experiment, while the parameters for the other three network sizes were adjusted to maintain
the same average degree5 . We then ran the MMSB, MMTM, and MMTM with ?-subsampling
inference algorithms on all 4 networks, and plotted the average per-iteration runtime in Figure 4.
The figure clearly exposes the scalability differences between MMSB and MMTM. The ?subsampled MMTM experiments show linear runtime dependence on N , which is expected since
the number of subsampled triangles is O(N ? 2 ). The full MMTM experiment is also roughly linear
? though we caution that this is not necessarily true for all networks, particularly high maximum
degree ones such as scale-free networks. Conversely, MMSB shows a clear quadratic dependence on
N . In fact, we had to omit the MMSB N = 40, 000 experiment because the latent variables would
not fit in memory, and even if they did, the extrapolated runtime would have been unreasonably long.
6
A Larger Network Demonstration
The MMTM model with ?-subsampling scales to even larger networks than the ones we have been
discussing. To demonstrate this, we ran the MMTM Gibbs sampler with ? = 20 on the SNAP
Stanford Web Graph6 , containing N = 281, 903 vertices (webpages), 2, 312, 497 1-edges, and approximately 4 billiion 2- and 3-edge triangles ?3 , ?2 , which we reduced to 11, 353, 778 via ? = 20subsampling. Note that the vast majority of triangles are associated with exceptionally high-degree
vertices, which make up a small fraction of the network. By using ?-subsampling, we limited the
number of triangles that come from such vertices, thus making the network feasible for MMTM.
We ran the MMTM sampler with settings identical to our synthetic experiments: 2,000 sampling
iterations, hyperparameters fixed to ?, ? = 0.1. The experiment took 74 hours, and we observed
log-likelihood convergence within 500 iterations.
The recovered mixed-membership vectors ?i are visualized in Figure 5. A key challenge is that
the ?i exist in the 4-simplex ?4 , which is difficult to visualize in two dimensions. To overcome
this, Figure 5 uses both position and color to communicate the values of ?i . Every vertex i is
displayed as a circle ci , whose size is proportional to the network degree of i. The position of ci is
equal to a convex combination of the 5 pentagon corners? (x, y) coordinates, where the coordinates
are weighted by the elements of ?i . In particular, circles ci at the pentagon?s corners represent
single-membership ?i ?s, while circles on the lines connecting the corners represent ?i ?s with mixedmembership in 2 communities. All other circles represent ?i ?s with mixed-membership in ? 3
communities. Furthermore, each circle ci ?s color is also a ?i -weighted convex combination, this
time of the RGB values of 5 colors: blue, green, red, cyan and purple. This use of color helps
distinguish between vertices with 2 versus 3 or more communities: for example, even though the
largest circle sits on the blue-red line (which initially suggets mixed-membership in 2 communities),
its dark green color actually comes from mixed-membership in 3 communities: green, red and cyan.
5
6
Note that the maximum degree still increases with N , because MMSB has a binomial degree distribution.
Available at http://snap.stanford.edu/data/web-Stanford.html
7
Per?iteration runtime for MMSB and MMTM Gibbs samplers
250
MMSB
MMTM
MMTM ?=20
MMTM ?=15
MMTM ?=10
MMTM ?=5
Time per iteration (s)
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
Number of vertices
4
4
x 10
Figure 4:
Per-iteration runtimes for MMSB,
MMTM and MMTM with ?-subsampling, on synthetic networks with N ranging from 1,000 to 40,000,
but with constant average degree.
Figure 5: N = 281, 903 Stanford web graph,
MMTM mixed-membership visualization.
Most high-degree vertices (large circles) are found at the pentagon?s corners, leading to the intuitive
conclusion that the five communities are centered on hub webpages with many links. Interestingly,
the highest-degree vertices are all mixed-membership, suggesting that these webpages (which are
mostly frontpages) lie on the boundaries between the communities. Finally, if we focus on the sets
of vertices near each corner, we see that the green and red sets have distinct degree (i.e. circle size)
distributions, suggesting that those communities may be functionally different from the other three.
7
Future Work and Conclusion
We have focused exclusively on triangular motifs because of their popularity in the literature, their
relationship to community structure through the network clustering coefficient, and the ability to
subsample them in a natural, node-centric fashion with minor impact on accuracy. However, the
bag-of-network-motifs idea extends beyond triangles ? one could easily consider subgraphs over 4
or more vertices, as in [13]. As with triangular motifs, it is algorithmically infeasible to consider all
possible subgraphs; rather, we must focus our attention on a meaningful subset of them. Nevertheless, higher order motifs could be more suited for particular tasks, thus meriting their investigation.
In modeling terms, we have applied triangular motifs to a generative mixed-membership setting,
which is suitable for visualization but not necessarily for attribute prediction. Recent developments
in constrained learning of generative models [23, 24] have yielded significant improvements in predictive accuracy, and these techniques are also applicable to mixed-membership triangular modeling.
Also, given how well ? = 20-subsampling works for MMTM at N = 4, 000, the next step would be
investigating how to adaptively choose ? as N increases, in order to achieve good performance.
To summarize, we have introduced the bag-of-triangles representation as a parsimonius alternative to
the network adjacency matrix, and developed a model (MMTM) and inference algorithm for mixedmembership community detection in networks. Compared to mixed-membership models that use
the adjacency matrix (exemplified by MMSB), our model features a much smaller latent variable
space, leading to faster inference and better performance at mixed-membership recovery. When
combined with triangle subsampling, our model and inference algorithm scale easily to networks
with 100,000s of vertices, which are completely infeasible for ?(N 2 ) adjacency-matrix-based models ? the adjacency matrix might not even fit in memory, to say nothing of runtime.
As a final note, we speculate that the local nature of the triangles lends itself better to parallel inference than the adjacency matrix representation; it may be possible to find good ?triangle separators?,
small subsets of triangles that divide the remaining triangles into large, non-vertex-overlapping subsets, which can then be inferred in parallel. This is similar to classical 1-edge separators that divide networks into non-overlapping subgraphs, which are unfortunately inapplicable to adjacencymatrix-based models, as they require separators over both the 0- and 1-edges. With triangle separators, we expect triangle models to scale to networks with millions of vertices and more.
Acknowledgments
This work was supported by AFOSR FA9550010247, NIH 1R01GM093156 to Eric P. Xing. Qirong
Ho is supported by an Agency for Science, Research and Technology, Singapore fellowship. Junming Yin is a Lane Fellow under the Ray and Stephanie Lane Center for Computational Biology.
8
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3,938 | 4,565 | Near-optimal Differentially Private Principal
Components
Kamalika Chaudhuri
UC San Diego
[email protected]
Anand D. Sarwate
TTI-Chicago
[email protected]
Kaushik Sinha
UC San Diego
[email protected]
Abstract
Principal components analysis (PCA) is a standard tool for identifying good lowdimensional approximations to data sets in high dimension. Many current data
sets of interest contain private or sensitive information about individuals. Algorithms which operate on such data should be sensitive to the privacy risks in publishing their outputs. Differential privacy is a framework for developing tradeoffs
between privacy and the utility of these outputs. In this paper we investigate the
theory and empirical performance of differentially private approximations to PCA
and propose a new method which explicitly optimizes the utility of the output.
We demonstrate that on real data, there is a large performance gap between the
existing method and our method. We show that the sample complexity for the two
procedures differs in the scaling with the data dimension, and that our method is
nearly optimal in terms of this scaling.
1
Introduction
Dimensionality reduction is a fundamental tool for understanding complex data sets that arise in
contemporary machine learning and data mining applications. Even though a single data point
can be represented by hundreds or even thousands of features, the phenomena of interest are often
intrinsically low-dimensional. By reducing the ?extrinsic? dimension of the data to its ?intrinsic? dimension, analysts can discover important structural relationships between features, more efficiently
use the transformed data for learning tasks such as classification or regression, and greatly reduce
the space required to store the data. One of the oldest and most classical methods for dimensionality
reduction is principal components analysis (PCA), which computes a low-rank approximation to the
second moment matrix of a set of points in Rd . The rank k of the approximation is chosen to be the
intrinsic dimension of the data. We view this procedure as specifying a k-dimensional subspace of
Rd .
Much of today?s machine-learning is performed on the vast amounts of personal information collected by private companies and government agencies about individuals, such as customers, users,
and subjects. These datasets contain sensitive information about individuals and typically involve
a large number of features. It is therefore important to design machine-learning algorithms which
discover important structural relationships in the data while taking into account its sensitive nature.
We study approximations to PCA which guarantee differential privacy, a cryptographically motivated definition of privacy [9] that has gained significant attention over the past few years in the
machine-learning and data-mining communities [19, 21, 20, 10, 23]. Differential privacy measures
privacy risk by a parameter ? that bounds the log-likelihood ratio of output of a (private) algorithm
under two databases differing in a single individual.
There are many general tools for providing differential privacy. The sensitivity method [9] computes
the desired algorithm (PCA) on the data and then adds noise proportional to the maximum change
than can be induced by changing a single point in the data set. The PCA algorithm is very sensitive
1
in this sense because the top eigenvector can change by 90 by changing one point in the data set.
Relaxations such as smoothed sensitivity [24] are difficult to compute in this setting as well. The
SULQ method of Blum et al. [2] adds noise to the second moment matrix and then runs PCA on
the noisy matrix. As our experiments show, the amount of noise required is often quite severe and
SULQ seems impractical for data sets of moderate size.
The general SULQ method does not take into account the quality of approximation to the nonprivate PCA output. We address this by proposing a new method, PPCA, that is an instance of the
exponential mechanism of McSherry and Talwar [22]. For any k < d, this differentially private
method outputs a k-dimensional subspace; the output is biased towards subspaces which are close
to the output of PCA. In our case, the method corresponds to sampling from the matrix Bingham
distribution. We implement this method using a Markov Chain Monte Carlo (MCMC) procedure
due to Hoff [15] and show that it achieves significantly better empirical performance.
In order to understand the performance gap, we prove sample complexity bounds in case of k = 1 for
SULQ and PPCA, as well as a general lower bound on the sample complexity for any differentially
p
private algorithm. We show that (up to log factors) the sample complexity scales as ?(d3/2 d)
for SULQ and as O(d) for PPCA. Furthermore, any differentially private algorithm requires ?(d)
samples, showing that PPCA is nearly optimal in terms of sample complexity as a function of data
dimension. These theoretical results suggest that our experiments exhibit the limit of how well ?differentially private algorithms can perform, and our experiments show that this gap should persist
for general k.
There are several interesting open questions suggested by this work. One set of issues is computational. Differentially privacy is a mathematical definition, but algorithms must be implemented
using finite precision machines. Privacy and computation interact in many places, including pseudorandomness, numerical stability, optimization, and in the MCMC procedure we use to implement
PPCA; investigating the impact of approximate sampling is an avenue for future work. A second set
of issues is theoretical ? while the privacy guarantees of PPCA hold for all k, our theoretical analysis of sample complexity applies only to k = 1 in which the distance and angles between vectors
are related. An interesting direction is to develop theoretical bounds for general k; challenges here
are providing the right notion of approximation of PCA, and extending the theory using packings of
Grassman or Stiefel manifolds.
2
Preliminaries
The data given to our algorithm is a set of n vectors D = {x1 , x2 , . . . , xn } where each xi corresponds to the private value of one individual, xi 2 Rd , and kxi k ? 1 for all i. Let X = [x1 , . . . , xn ]
be the matrix whose columns are the data vectors {xi }. Let A = n1 XX T denote the d ? d second
moment matrix of the data. The matrix A is positive semidefinite, and has Frobenius norm at most
1.
The problem of dimensionality reduction is to find a ?good? low-rank approximation to A. A popular
? F , where k is much
solution is to compute a rank-k matrix A? which minimizes the norm kA Ak
lower than the data dimension d. The Schmidt approximation theorem [25] shows that the minimizer
is given by the singular value decomposition, also known as the PCA algorithm in some areas of
computer science.
Definition 1. Suppose A is a positive semidefinite matrix whose first k eigenvalues are distinct. Let
the eigenvalues of A be 1 (A)
???
0 and let ? be a diagonal matrix with
2 (A)
d (A)
?ii = i (A). The matrix A decomposes as
A = V ?V T ,
(1)
where V is an orthonormal matrix of eigenvectors. The top-k subspace of A is the matrix
Vk (A) = [v1 v2 ? ? ? vk ] ,
(2)
where vi is the i-th column of V in (1).
Given the top-k subspace and the eigenvalue matrix ?, we can form an approximation A(k) =
Vk (A)?k Vk (A)T to A, where ?k contains the k largest eigenvalues in ?. In the special case k = 1
2
we have A(1) = 1 (A)v1 v1T , where v1 is the eigenvector corresponding to 1 (A). We refer to v1 as
the top eigenvector of the data. For a d ? k matrix V? with orthonormal columns, the quality of V? in
approximating A can be measured by
?
?
qF (V? ) = tr V? T AV? .
(3)
The V? which maximizes q(V? ) has columns equal to {vi : i 2 [k]}, corresponding to the top k
eigenvectors of A.
Our theoretical results apply to the special case k = 1. For these results, we measure the inner
product between the output vector v?1 and the true top eigenvector v1 :
(4)
qA (?
v1 ) = |h?
v1 , v1 i| .
This is related to (3). If we write v?1 in the basis spanned by {vi }, then
qF (?
v1 ) =
v1 )
1 qA (?
2
+
d
X
v 1 , vi i
i h?
i=2
2
.
Our proof techniques use the geometric properties of qA (?).
Definition 2. A randomized algorithm A(?) is an (?, ?)-close approximation to the top eigenvector
if for all data sets D of n points,
P (qA (A(D))
?)
1
?,
(5)
where the probability is taken over A(?).
We study approximations to A? that preserve the privacy of the underlying data. The notion of
privacy that we use is differential privacy, which quantifies the privacy guaranteed by a randomized
algorithm P applied to a data set D.
Definition 3. An algorithm A(B) taking values in a set T provides ?-differential privacy if
sup sup
S D,D 0
? (S | B = D)
? e? ,
? (S | B = D0 )
(6)
where the first supremum is over all measurable S ? T , the second is over all data sets D and
D0 differing in a single entry, and ?(?|B) is the conditional distribution (measure) on T induced by
the output A(B) given a data set B. The ratio is interpreted to be 1 whenever the numerator and
denominator are both 0.
Definition 4. An algorithm A(B) taking values in a set T provides (?, )-differential privacy if
P (A(D) 2 S) ? e? P (A(D0 ) 2 S) + ,
(7)
for all all measurable S ? T and all data sets D and D differing in a single entry.
0
Here ? and are privacy parameters, where low ? and ensure more privacy. For more details about
these definitions, see [9, 26, 8]. The second privacy guarantee is weaker; the parameter bounds the
probability of failure, and is typically chosen to be quite small.
In this paper we are interested in proving results on the sample complexity of differentially private algorithms that approximate PCA. That is, for a given ? and ?, how large must the number of
individuals n in the data set be such that it is ?-differentially private and also a (?, ?)-close approximation to PCA? It is well known that as the number of individuals n grows, it is easier to guarantee
the same level of privacy with relatively less noise or perturbation, and therefore the utility of the
approximation also improves. Our results characterize how privacy and utility scale with n and the
tradeoff between them for fixed n.
Related Work Differential privacy was proposed by Dwork et al. [9], and has spawned an extensive literature of general methods and applications [1, 21, 27, 6, 24, 3, 22, 10] Differential privacy
has been shown to have strong semantic guarantees [9, 17] and is resistant to many attacks [12] that
succeed against some other definitions of privacy. There are several standard approaches for designing differentially-private data-mining algorithms, including input perturbation [2], output perturbation [9], the exponential mechanism [22], and objective perturbation [6]. To our knowledge, other
3
than SULQ method [2], which provides a general differentially-private input perturbation algorithm, this is the first work on differentially-private PCA. Independently, [14] consider the problem
of differentially-private low-rank matrix reconstruction for applications to sparse matrices; provided
certain coherence conditions hold, they provide an algorithm for constructing a rank 2k approximation B to a matrix A such that kA BkF is O(kA Ak k) plus some additional terms which
depend on d, k and n; here Ak is the best rank k approximation to A. Because of their additional
assumptions, their bounds are generally incomparable to ours, and our bounds are superior for dense
matrices.
The data-mining community has also considered many different models for privacy-preserving computation ? see Fung et al. for a survey with more references [11]. Many of the models used have
been shown to be susceptible to composition attacks, when the adversary has some amount of prior
knowledge [12]. An alternative line of privacy-preserving data-mining work [28] is in the Secure
Multiparty Computation setting; one work [13] studies privacy-preserving singular value decomposition in this model. Finally, dimension reduction through random projection has been considered
as a technique for sanitizing data prior to publication [18]; our work differs from this line of work
in that we offer differential privacy guarantees, and we only release the PCA subspace, not actual
data. Independently, Kapralov and Talwar [16] have proposed a dynamic programming algorithm
for differentially private low rank matrix approximation which involves sampling from a distribution
induced by the exponential mechanism. The running time of their algorithm is O(d6 ), where d is
the data dimension.
3
Algorithms and results
In this section we describe differentially private techniques for approximating (2). The first is a modified version of the SULQ method [2]. Our new algorithm for differentially-private PCA, PPCA,
is an instantiation of the exponential mechanism due to McSherry and Talwar [22]. Both procedures provide differentially private approximations to the top-k subspace: SULQ provides (?, )differential privacy and PPCA provides ?-differential privacy.
Input perturbation. The only differentially-private approximation to PCA prior to this work is
the SULQ method [2]. The SULQ method perturbs each entry of the empirical second moment matrix A to ensure differential privacy and releases the top k eigenvectors of this perturbed matrix. In
2
2
(d/ )
particular, SULQ recommends adding a matrix N of i.i.d. Gaussian noise of variance 8d log
n2 ? 2
and applies the PCA algorithm to A + N . This guarantees a weaker privacy definition known as
(?, )-differential privacy. One problem with this approach is that with probability 1 the matrix
A + N is not symmetric, so the largest eigenvalue may not be real and the entries of the corresponding eigenvector may be complex. Thus the SULQ algorithm is not a good candidate for practical
privacy-preserving dimensionality reduction.
However, a simple modification to the basic SULQ approach does guarantee (?, ) differential
privacy. Instead of adding a asymmetric Gaussian matrix, the algorithm can add the a symmetric
matrix with i.i.d. Gaussian entries N . That is, for 1 ? i ? j ? d, the variable Nij is an independent
Gaussian random variable with variance 2 . Note that this matrix is symmetric but not necessarily
positive semidefinite, so some eigenvalues may be negative but the eigenvectors are all real. A
derivation for the noise variance is given in Theorem 1.
Algorithm 1: Algorithm MOD-SULQ (input pertubation)
inputs: d ? n data matrix X, privacy parameter ?, parameter
outputs: d ? k matrix V?k = [?
v1 v?2 ? ? ? v?k ] with orthonormal columns
1 Set A = n1 XX T .;
r
?
?
2
d+1
2 Set = n? 2 log d p+d + p1?n . Generate a d ? d symmetric random matrix N whose
2 2?
entries are i.i.d. drawn from N 0, 2 . ;
3 Compute V?k = Vk (A + N ) according to (2). ;
4
Exponential mechanism. Our new method, PPCA, randomly samples a k-dimensional subspace
from a distribution that ensures differential privacy and is biased towards high utility. The distribution from which our released subspace is sampled is known in the statistics literature as the matrix
Bingham distribution [7], which we denote by BMFk (B). The algorithm is in terms of general
k < d but our theoretical results focus on the special case k = 1 where we wish to release a onedimensional approximation to the data covariance matrix. The matrix Bingham distribution takes
values on the set of all k-dimensional subspaces of Rd and has a density equal to
1
f (V ) =
exp(tr(V T BV )),
(8)
1
1
F
k,
d,
B
1 1 2
2
where V is a d ? k matrix whose columns are orthonormal and 1 F1
hypergeometric function [7, p.33].
1
1
2 k, 2 d, B
is a confluent
Algorithm 2: Algorithm PPCA (exponential mechanism)
inputs: d ? n data matrix X, privacy parameter ?, dimension k
outputs: d ? k matrix V?k = [?
v1 v?2 ? ? ? v?k ] with orthonormal columns
1 Set A = n1 XX T ;
2 Sample V?k = BMF n ?
2A ;
By combining results on the exponential mechanism [22] along with properties of PCA algorithm,
we can show that this procedure is differentially private. In many cases, sampling from the distribution specified by the exponential mechanism distribution may be difficult computationally, especially
for continuous-valued outputs. We implement PPCA using a recently-proposed Gibbs sampler due
to Hoff [15]. Gibbs sampling is a popular Markov Chain Monte Carlo (MCMC) technique in which
samples are generated according to a Markov chain whose stationary distribution is the density in
(8). Assessing the ?burn-in time? and other factors for this procedure is an interesting question in its
own right; further details are in Section E.3.
Other approaches. There are other general algorithmic strategies for guaranteeing differential
privacy. The sensitivity method [9] adds noise proportional to the maximum change that can be
induced by changing a single point in the data set. Consider a data set D with m + 1 copies of a unit
vector u and m copies of a unit vector u0 with u ? u0 and let D0 havep
m copies of u and m+1 copies
of u0 . Then v1 (D) = u but v1 (D0 ) = u0 , so kv1 (D) v1 (D0 )k = 2. Thus the global sensitivity
does not scale with the number of data points, so as n increases the variance of the noise required
by the Laplace mechanism [9] will not decrease. An alternative to global sensitivity is smooth
sensitivity [24]; except for special cases, such as the sample median, smooth sensitivity is difficult
to compute for general functions. A third method for computing private, approximate solutions
to high-dimensional optimization problems is objective perturbation [6]; to apply this method, we
require the optimization problems to have certain properties (namely, strong convexity and bounded
norms of gradients), which do not apply to PCA.
Main results. Our theoretical results are sample complexity bounds for PPCA and MOD-SULQ
as well as a general lower bound on the sample complexity for any ?-differentially private algorithm.
These results show that the PPCA is nearly optimal in terms the scaling of the sample complexity
with respect to the data dimension d, privacy parameter ?, and eigengap . We further show that
MOD-SULQ requires more samples as a function of d, despite having a slightly weaker privacy
guarantee. Proofs are deferred to the supplementary material.
Even though both algorithms can output the top-k PCA subspace for general k ? d, we prove results
for the case k = 1. Finding the scaling behavior of the sample complexity with k is an interesting
open problem that we leave for future work; challenges here are finding the right notion of approximation of the PCA, and extending the theory using packings of Grassman or Stiefel manifolds.
Theorem 1. For the in Algorithm 1, the MOD-SULQ algorithm is (?, ) differentially private.
Theorem 2. Algorithm PPCA is ?-differentially private.
The fact that these two algorithms are differentially private follows from some simple calculations.
Our first sample complexity result provides an upper bound on the number of samples required by
5
PPCA to guarantee a certain level of privacy and accuracy. The sample complexity of PPCA n
grows linearly with the dimension d, inversely with ?, and inversely with the correlation gap (1 ?)
and eigenvalue gap 1 (A)
2 (A).
?
?
4 1
Theorem 3 (Sample complexity of PPCA). If n > ?(1 ?)(d 1 2 ) log(1/?)
+
log
,
2
d
(1 ? )( 1
2)
then PPCA is a (?, ?)-close approximation to PCA.
Our second result shows a lower bound on the number of samples required by any ?-differentiallyprivate algorithm to guarantee a certain level of accuracy for a large class of datasets, and uses proof
techniques in [4, 5].
Theorem
4 (Sample complexity
lower bound). Fix d, ?,
? 12 and let 1
=
?
?
ln 8+ln(1+exp(d))
1
exp
2?
. For any ?
1
d 2
16 , no ?-differentially private algorithm A can
approximate PCA with expected utility greater
than
? on all databases with n points in dimension d
?
q
having eigenvalue gap , where n < max d? , 180 ? ?pd1 ? .
Theorem 3 shows that if n scales like ? (1d ?) log 1 1?2 then PPCA produces an approximation v?1
pd
that has correlation ? with v1 , whereas Theorem 4 shows that n must scale like
for any
?
(1 ?)
?-differentially private algorithm. In terms of scaling with d, ? and , the upper and lower bounds
match, and they also match up to square-root factors with respect to the correlation. By contrast, the
following lower bound on the number of samples required by MOD-SULQ to ensure a certain level
of accuracy shows that MOD-SULQ has a less favorable scaling with dimension.
0
Theorem 5 (Sample
p complexity lower bound for MOD-SULQ). There are constants c and c such
d3/2
log(d/ )
that if n < c
(1 c0 (1 ?)), then there is a dataset of size n in dimension d such that
?
the top PCA direction v and the output v? of MOD-SULQ satisfy E[|h?
v1 , v1 i|] ? ?.
Notice that the dependence on n grows as d3/2 in SULQ as opposed to d in PPCA. Dimensionality
reduction via PCA is often used in applications where the data points occupy a low dimensional
space but are presented in high dimensions. These bounds suggest that PPCA is better suited to
such applications than MOD-SULQ. We next turn to validating this intuition on real data.
4
Experiments
We chose four datasets from four different domains ? kddcup99, which includes features
of 494,021 network connections, census, a demographic data set on 199, 523 individuals,
localization, a medical dataset with 164,860 instances of sensor readings on individuals engaged in different activities, and insurance, a dataset on product usage and demographics of
9,822 individuals. After preprocessing, the dimensions of these datasets are 116, 513, 44 and 150
respectively. We chose k to be 4, 8, 10, and 11 such that the top-k PCA subspace had qF (Vk ) at least
80% of kAkF . More details are in Appendix E in the supplementary material.
We ran three algorithms on these data sets : standard (non-private) PCA, MOD-SULQ with ? = 0.1
and = 0.01, and PPCA with ? = 0.1. As a sanity check, we also tried a uniformly generated
random projection ? since this projection is data-independent we would expect it to have low utility.
Standard PCA is non-private; changing a single data point will change the output, and hence violate
differential privacy. We measured the utility qF (U ), where U is the k-dimensional subspace output
by the algorithm; kU k is maximized when U is the top-k PCA subspace, and thus this reflects how
close the output subspace is to the true PCA subspace in terms of representing the data. Although
our theoretical results hold for qA (?), the ?energy? qF (?) is more relevant in practice for larger k.
Figures 1(a), 1(b), 1(c), and 1(d) show qF (U ) as a function of sample size for the k-dimensional
subspace output by PPCA, MOD-SULQ, non-private PCA, and random projections. Each value
in the figure is an average over 5 random permutations of the data, as well as 10 random starting
points of the Gibbs sampler per permutation (for PPCA), and 100 random runs per permutation (for
MOD-SULQ and random projections).
6
0.7
0.6
0.6
0.5
0.5
0.4
0.2
Algorithm
0.4
Nonprivate
PPCA
Random
SULQ
0.3
Utility
Utility
Algorithm
Nonprivate
PPCA
Random
SULQ
0.3
0.2
0.1
0.1
50000
100000
150000
2e+04
4e+04
n
6e+04
8e+04
1e+05
n
(a) census
(b) kddcup
0.5
0.5
0.4
Algorithm
Utility
0.3
Utility
Nonprivate
PPCA
Random
SULQ
0.4
Algorithm
0.3
Nonprivate
PPCA
Random
SULQ
0.2
0.2
0.1
2e+04
4e+04
6e+04
8e+04
1e+05
2000
4000
n
6000
8000
10000
n
(c) localization
(d) insurance
Figure 1: Utility qF (U ) for the four data sets
KDDCUP
LOCALIZATION
Non-private PCA
98.97 ? 0.05
100 ? 0
PPCA
98.95 ? 0.05
100 ? 0
Table 1:
MOD-SULQ
98.18 ? 0.65
97.06 ? 2.17
Random projections
98.23 ? 0.49
96.28 ? 2.34
Classification accuracy in the k-dimensional subspaces for kddcup99(k =
localization(k = 10) in the k-dimensional subspaces reported by the different algorithms.
4), and
The plots show that PPCA always outperforms MOD-SULQ, and approaches the performance of
non-private PCA with increasing sample size. By contrast, for most of the problems and sample
sizes considered by our experiments, MOD-SULQ does not perform much better than random projections. The only exception is localization, which has much lower dimension (44). This
confirms that MOD-SULQ does not scale very well with the data dimension d. The performance of
both MOD-SULQ and PPCA improve as the sample size increases; the improvement is faster for
PPCA than for MOD-SULQ. However, to be fair, MOD-SULQ is simpler and hence runs faster
than PPCA. At the sample sizes in our experiments, the performance of non-private PCA does not
improve much with a further increase in samples. Our theoretical results suggest that the performance of differentially private PCA cannot be significantly improved over these experiments.
Effect of privacy on classification. A common use of a dimension reduction algorithm is as a
precursor to classification or clustering; to evaluate the effectiveness of the different algorithms,
we projected the data onto the subspace output by the algorithms, and measured the classification
accuracy using the projected data. The classification results are summarized in Table 4. We chose
the normal vs. all classification task in kddcup99, and the falling vs. all classification task in
localization. 1 We used a linear SVM for all classification experiments.
For the classification experiments, we used half of the data as a holdout set for computing a projection subspace. We projected the classification data onto the subspace computed based on the holdout
set; 10% of this data was used for training and parameter-tuning, and the rest for testing. We repeated the classification process 5 times for 5 different (random) projections for each algorithm, and
then ran the entire procedure over 5 random permutations of the data. Each value in the figure is
thus an average over 5 ? 5 = 25 rounds of classification.
1
For the other two datasets, census and insurance, the classification accuracy of linear SVM after
(non-private) PCAs is as low as always predicting the majority label.
7
Utility versus privacy parameter
0.7
?? ?
?
?
?
?
?
?
?
Utility q(U)
0.6
Algorithm
?
0.5
Non?Private
SULQ
PPCA 1000
?
0.4
0.3
0.2
??
?
0.5
1.0
1.5
2.0
Privacy parameter alpha
Figure 2: Plot of qF (U ) versus ? for a synthetic data set with n = 5,000, d = 10, and k = 2.
The classification results show that our algorithm performs almost as well as non-private PCA for
classification in the top k PCA subspace, while the performance of MOD-SULQ and random projections are a little worse. The classification accuracy while using MOD-SULQ and random projections
also appears to have higher variance compared to our algorithm and non-private PCA; this can be
explained by the fact that these projections tend to be farther from the PCA subspace, in which the
data has higher classification accuracy.
Effect of the privacy requirement. To check the effect of the privacy requirement,
we generated a synthetic data set of n = 5,000 points drawn from a Gaussian distribution in d = 10 with mean 0 and whose covariance matrix had eigenvalues
{0.5, 0.30, 0.04, 0.03, 0.02, 0.01, 0.004, 0.003, 0.001, 0.001}. In this case the space spanned by the
top two eigenvectors has most of the energy, so we chose k = 2 and plotted the utility qF (?) for nonprivate PCA, MOD-SULQ with = 0.05, and PPCA. We drew 100 samples from each privacypreserving algorithm and the plot of the average utility versus ? is shown in Figure 2. As ? increases,
the privacy requirement is relaxed and both MOD-SULQ and PPCA approach the utility of PCA
without privacy constraints. However, for moderate ? the PPCA still captures most of the utility,
whereas the gap between MOD-SULQ and PPCA becomes quite large.
5
Conclusion
In this paper we investigated the theoretical and empirical performance of differentially private approximations to PCA. Empirically, we showed that MOD-SULQ and PPCA differ markedly in how
well they approximate the top-k subspace of the data.
p The reason for this, theoretically, is that the
sample complexity of MOD-SULQ scales with d3/2 log d whereas PPCA scales with d. Because
PPCA uses the exponential mechanism with qF (?) as the utility function, it is not surprising that
it performs well. However, MOD-SULQ often had a performance comparable to random projections, indicating that the real data sets we used were too small for it to be effective. We furthermore
showed that PPCA is nearly optimal, in that any differentially private approximation to PCA must
use ?(d) samples.
Our investigation brought up many interesting issues to consider for future work. The description of
differentially private algorithms assume an ideal model of computation : real systems require additional security assumptions that have to be verified. The difference between truly random noise and
pseudorandomness and the effects of finite precision can lead to a gap between the theoretical ideal
and practice. Numerical optimization methods used in objective perturbation [6] can only produce
approximate solutions, and have complex termination conditions unaccounted for in the theoretical
analysis. Our MCMC sampling has this flavor : we cannot sample exactly from the Bingham distribution because we must determine the Gibbs sampler?s convergence empirically. Accounting for
these effects is an interesting avenue for future work that can bring theory and practice together.
Finally, more germane to the work on PCA here is to prove sample complexity results for general k
rather than the case k = 1 here. For k = 1 the utility functions qF (?) and qA (?) are related, but for
general k it is not immediately clear what metric best captures the idea of ?approximating? PCA.
Developing a framework for such approximations is of interest more generally in machine learning.
8
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9
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3,939 | 4,566 | Communication/Computation Tradeoffs in
Consensus-Based Distributed Optimization
Konstantinos I. Tsianos, Sean Lawlor, and Michael G. Rabbat
Department of Electrical and Computer Engineering
McGill University, Montr?eal, Canada
{konstantinos.tsianos, sean.lawlor}@mail.mcgill.ca
[email protected]
Abstract
We study the scalability of consensus-based distributed optimization algorithms
by considering two questions: How many processors should we use for a given
problem, and how often should they communicate when communication is not
free? Central to our analysis is a problem-specific value r which quantifies the
communication/computation tradeoff. We show that organizing the communication among nodes as a k-regular expander graph [1] yields speedups, while when
all pairs of nodes communicate (as in a complete graph), there is an optimal number of processors that depends on r. Surprisingly, a speedup can be obtained,
in terms of the time to reach a fixed level of accuracy, by communicating less
and less frequently as the computation progresses. Experiments on a real cluster
solving metric learning and non-smooth convex minimization tasks demonstrate
strong agreement between theory and practice.
1
Introduction
How many processors should we use and how often should they communicate for large-scale distributed optimization? We address these questions by studying the performance and limitations of a
class of distributed algorithms that solve the general optimization problem
m
minimize F (x) =
x?X
1 X
lj (x)
m j=1
(1)
where each function lj (x) is convex over a convex set X ? Rd . This formulation applies widely in
machine learning scenarios, where lj (x) measures the loss of model x with respect to data point j,
and F (x) is the cumulative loss over all m data points.
Although efficient serial algorithms exist [2], the increasing size of available data and problem dimensionality are pushing computers to their limits and the need for parallelization arises [3]. Among
many proposed distributed approaches for solving (1), we focus on consensus-based distributed optimization [4, 5, 6, 7] where each component function in (1) is assigned to a different node in a
network (i.e., the data is partitioned among the nodes), and the nodes interleave local gradient-based
optimization updates with communication using a consensus protocol to collectively converge to a
minimizer of F (x).
Consensus-based algorithms are attractive because they make distributed optimization possible without requiring centralized coordination or significant network infrastructure (as opposed to, e.g., hierarchical schemes [8]). In addition, they combine simplicity of implementation with robustness to
node failures and are resilient to communication delays [9]. These qualities are important in clusters,
which are typically shared among many users, and algorithms need to be immune to slow nodes that
1
use part of their computation and communication resources for unrelated tasks. The main drawback
of consensus-based optimization algorithms comes from the potentially high communication cost
associated with distributed consensus. At the same time, existing convergence bounds in terms of iterations (e.g., (7) below) suggest that increasing the number of processors slows down convergence,
which contradicts the intuition that more computing resources are better.
This paper focuses on understanding the limitations and potential for scalability of consensus-based
optimization. We build on the distributed dual averaging framework [4]. The key to our analysis is
to attach to each iteration a cost that involves two competing terms: a computation cost per iteration which decreases as we add more processors, and a communication cost which depends on the
network. Our cost expression quantifies the communication/computation tradeoff by a parameter r
that is easy to estimate for a given problem and platform. The role of r is essential; for example,
when nodes communicate at every iteration, we show that in complete graph topologies, there exists
an optimal number of processors nopt = ?1r , while for k-regular expander graphs [1], increasing
the network size yields a diminishing speedup. Similar results are obtained when nodes communicate every h > 1 iterations and even when h increases with time. We validate our analysis with
experiments on a cluster. Our results show a remarkable agreement between theory and practice.
In Section 2 we formalize the distributed optimization problem and summarize the distributed dual
averaging algorithm. Section 3 introduces the communication/computation tradeoff and contains the
basic analysis where nodes communicate at every iteration. The general case of sparsifying communication is treated in Section 4. Section 5 tests our theorical results on a real cluster implementation
and Section 6 discusses some future extensions.
2
Distributed Convex Optimization
Assume we have at our disposal a cluster with n processors to solve (1), and suppose without loss
of generality that m is divisible by n. In the absence of any other information, we partition the data
evenly among the processors and our objective becomes to solve the optimization problem,
?
?
m
m
n
n
n
1 X
1 X? n X
1X
?
minimize F (x) =
lj (x) =
lj|i (x) =
fi (x)
(2)
x?X
m j=1
n i=1 m j=1
n i=1
where we use the notation lj|i to denote loss associated with the jth local data point at processor
i (i.e., j|i = (i ? 1) m
n + j). The local objective functions fi (x) at each node are assumed to be
L-Lipschitz and convex. The recent distributed optimization literature contains multiple consensusbased algorithms with similar rates of convergence for solving this type of problem. We adopt
the distributed dual averaging (DDA) framework [4] because its analysis admits a clear separation
between the standard (centralized) optimization error and the error due to distributing computation
over a network, facilitating our investigation of the communication/computation tradeoff.
2.1
Distributed Dual Averaging (DDA)
In DDA, nodes iteratively communicate and update optimization variables to solve (2). Nodes only
communicate if they are neighbors in a communication graph G = (V, E), with the |V | = n vertices
being the processors. The communication graph is user-defined (application layer) and does not
necessarily correspond to the physical interconnections between processors. DDA requires three
additional quantities: a 1-strongly convex proximal function ? : Rd ? R satisfying ?(x) ? 0 and
?(0) = 0 (e.g., ?(x) = 12 xT x); a positive step size sequence a(t) = O( ?1t ); and a n ? n doubly
stochastic consensus matrix P with entries pij > 0 only if either i = j or (j, i) ? E and pij = 0
otherwise. The algorithm repeats for each node i in discrete steps t, the following updates:
n
X
zi (t) =
pij zj (t ? 1) + gi (t ? 1)
(3)
j=1
1
xi (t) =argmin hzi (t), xi +
?(x)
a(t)
x?X
1
x
?i (t) = (t ? 1) ? x
?i (t ? 1) + xi (t)
t
2
(4)
(5)
where gi (t ? 1) ? ?fi (xi (t ? 1)) is a subgradient of fi (x) evaluated at xi (t ? 1). In (3), the variable
zi (t) ? Rd maintains an accumulated subgradient up to time t and represents node i?s belief of
the direction of the optimum. To update zi (t) in (3), each node must communicate to exchange the
variables zj (t) with its neighbors in G. If ?(x? ) ? R2 , for the local running averages x
?i (t) defined
in (5), the error from a minimizer x? of F (x) after T iterations is bounded by (Theorem 1, [4])
Erri (T ) = F (?
xi (T )) ? F (x? ) ?
T
R2
L2 X
+
a(t ? 1)
T a(T ) 2T t=1
?
?
T
n
X
LX
2
+
a(t) ?
k?
z (t) ? zj (t)k? + k?
z (t) ? zi (t)k? ? (6)
T t=1
n j=1
Pn
where L is the Lipschitz constant, k?k? indicates the dual norm, z?(t) = n1 i=1 zi (t), and
k?
z (t) ? zi (t)k? quantifies the network error as a disagreement between the direction to the optimum at node i and the consensus direction z?(t) at time t. Furthermore, from Theorem 2 in [4], with
a(t) = ?At , after optimizing for A we have a bound on the error,
s
?
12
log (T n)
?
? ,
Erri (T ) ? C1
,
C1 = 2LR 19 +
(7)
1
?
?2
T
where ?2 is the second largest eigenvalue of P . The dependence on the communication topology
is reflected through ?2 , since the sparsity structure of P is determined by G. According to (7),
increasing n slows down the rate of convergence even if ?2 does not depend on n.
3
Communication/Computation Tradeoff
In consensus-based distributed optimization algorithms such as DDA, the communication graph G
and the cost of transmitting a message have an important influence on convergence speed, especially
when communicating one message requires a non-trivial amount of time (e.g., if the dimension d of
the problem is very high).
We are interested in the shortest time to obtain an -accurate solution (i.e., Erri (T ) ? ). From (7),
convergence
is faster for topologies with good expansion properties; i.e., when the spectral gap
?
1 ? ?2 does not shrink too quickly as n grows. In addition, it is preferable to have a balanced
network, where each node has the same number of neighbors so that all nodes spend roughly the
same amount of time communicating per iteration. Below we focus on two particular cases and take
G to be either a complete graph (i.e., all pairs of nodes communicate) or a k-regular expander [1].
By using more processors, the total amount of communication inevitably increases. At the same
time, more data can be processed in parallel in the same amount of time. We focus on the scenario
where the size m of the dataset is fixed but possibly very large. To understand whether there is room
for speedup, we move away from measuring iterations and employ a time model that explicitly accounts for communication cost. This will allow us to study the communication/computation tradeoff
and draw conclusions based on the total amount of time to reach an accuracy solution.
3.1
Time model
At each iteration, in step (3), processor i computes a local subgradient on its subset of the data:
m
n
?lj|i (x)
?fi (x)
n X
gi (x) =
=
.
?x
m j=1 ?x
(8)
The cost of this computation increases linearly with the subset size. Let us normalize time so that
one processor compute a subgradient on the full dataset of size m in 1 time unit. Then, using n cpus,
each local gradient will take n1 time units to compute. We ignore the time required to compute the
projection in step (4); often this can be done very efficiently and requires negligible time when m is
large compared to n and d.
3
We account for the cost of communication as follows. In the consensus update (3), each pair of
neighbors in G transmits and receives one variable zj (t ? 1). Since the message size depends only
on the problem dimension d and does not change with m or n, we denote by r the time required to
transmit and receive one message, relative to the 1 time unit required to compute the full gradient
on all the data. If every node has k neighbors, the cost of one iteration in a network of n nodes is
1
+ kr time units / iteration.
(9)
n
Using this time model, we study the convergence rate bound (7) after attaching an appropriate time
unit cost per iteration. To obtain a speedup by increasing the number of processors n for a given
problem, we must ensure that -accuracy is achieved in fewer time units.
3.2
Simple Case: Communicate at every Iteration
In the original DDA description (3)-(5), nodes communicate at every iteration. According to our
time model, T iterations will cost ? = T ( n1 + kr) time units. From (7), the time ? () to reach error
is found by substituting for T and solving for ? (). Ignoring the log factor in (7), we get
1
C2 1
C1 q
= =? ? () = 21
+ kr time units.
(10)
? ()
n
1
n +kr
This simple manipulation reveals some important facts. If communication is free, then r = 0. If in
addition the network G is a k-regular expander, then ?2 is fixed [10], C1 is independent of n and
? () = C12 /(2 n). Thus, in the ideal situation, we obtain a linear speedup by increasing the number
of processors, as one would expect. In reality, of course, communication is not free.
Complete graph. Suppose that G is the complete graph, where k = n ? 1 and ?2 = 0. In this
scenario we cannot keep increasing the network size without eventually harming performance due
to the excessive communication cost. For a problem with a communication/computation tradeoff r,
the optimal number of processors is calculated by minimizing ? () for n:
?? ()
1
= 0 =? nopt = ? .
(11)
?n
r
Again, in accordance with intuition, if the communication cost is too high (i.e., r ? 1) and it takes
more time to transmit and receive a gradient than it takes to compute it, using a complete graph
cannot speedup the optimization. We reiterate that r is a quantity that can be easily measured for
a given hardware and a given optimization problem. As we report in Section 5, the optimal value
predicted by our theory agrees very well with experimental performance on a real cluster.
Expander. For the case where G is a k-regular expander, the communication cost per node remains
constant as n increases. From (10) and the expression for C1 in (7), we see that n can be increased
without losing performance, although the benefit diminishes (relative to kr) as n grows.
4
General Case: Sparse Communication
The previous section analyzes the case where processors communicate at every iteration. Next we
investigate the more general situation where we adjust the frequency of communication.
4.1
Bounded Intercommunication Intervals
Suppose that a consensus step takes place once every h + 1 iterations. That is, the algorithm repeats
h ? 1 cheap iterations (no communication) of cost n1 time units followed by an expensive iteration
(with communication) with cost n1 + kr. This strategy clearly reduces the overall average cost per
iteration. The caveat is that the network error k?
z (t) ? zi (t)k? is higher because of having executed
fewer consensus steps.
In a cheap iteration we replace the update (3) by zi (t) = zi (t ? 1) + gi (t ? 1). After some straightforward algebra we can show that [for (12), (16) please consult the supplementary material]:
Q
H
n
t ?1
t ?1 h?1
X
X
XX
Ht ?w
zi (t) =
P
g
(wh
+
k)
+
gi (t ? Qt + k).
(12)
ij j
w=0 k=0 j=1
k=0
4
where Ht = b t?1
h c counts the number of communication steps in t iterations, and Qt = mod(t, h)
if mod(t, h) > 0 and Qt = h otherwise. Using the fact that P 1 = 1, we obtain
z?(t) ? zi (t) =
H
n
h?1
n
t ?1 X
X
1X
1 Ht ?w X
zs (t) ? zi (t) =
? P
gj (wh + k)
ij
n s=1
n
w=0 j=1
(13)
k=0
+
n Qt ?1
1X X
gs (t ? Qt + k) ? gi (t ? Qt + k) .
n s=1
(14)
k=0
Taking norms, recalling that the fi are convex and Lipschitz, and since Qt ? h, we arrive at
H
t ?1
X
1 T H ?w
hL + 2hL
1 ? P t
k?
z (t) ? zi (t)k? ?
n
i,:
1
w=0
(15)
Using a technique similar to that in [4] to bound the `1 distance of row i of P Ht ?w to its stationary
distribution as t grows, we can show that
?
log(T n)
?
k?
z (t) ? zi (t)k? ? 2hL
+ 3hL
(16)
1 ? ?2
for all t ? T . Comparing (16) to equation (29) in [4], the network error within t iterations is no more
than h times larger when a consensus step is only performed once every h + 1 ?
iterations. Finally,
PT
we substitute the network error in (6). For a(t) = ?At , we have t=1 a(t) ? 2A T , and
2
?
?
R
log (T n)
log (T n)
12h
2
? + 18h
?
?
= Ch
.
(17)
Erri (T ) ?
+ AL 1 +
A
1 ? ?2
T
T
We minimize the leading term Ch over A to obtain
s
s
!?1
R
12h
12h
?
? .
A=
and Ch = 2RL 1 + 18h +
1 + 18h +
L
1 ? ?2
1 ? ?2
Of the T iterations, only HT = b T h?1 c involve communication. So, T iterations will take
1
T
1
? = (T ? HT ) + HT
+ kr = + HT kr time units.
n
n
n
(18)
(19)
C2
To achieve -accuracy, ignoring again the logarithmic factor, we need T = 2h iterations, or
T
T ?1
C2 1
kr
? () =
+
kr ? 2h
+
time units.
n
h
n
h
(20)
From the last expression, for a fixed number of processors n, there exists an optimal value for h that
depends on the network size and communication graph G:
s
nkr
hopt =
.
(21)
?
18 + 1?12
?
2
If the network is a complete graph, using hopt yields ? () = O(n); i.e., using more processors
hurts performance when not communicating every iteration. On the other hand, if the network is a
k-regular expander then ? () = ?c1n + c2 for constants c1 , c2 , and we obtain a diminishing speedup.
4.2
Increasingly Sparse Communication
Next, we consider progressively increasing the intercommunication intervals. This captures the
intuition that as the optimization moves closer to the solution, progress slows down and a processor
should have ?something significantly new to say? before it communicates. Let hj ? 1 denote the
number of cheap iterations performed between the (j ? 1)st and jth expensive iteration; i.e., the first
communication is at iteration h1 , the second at iteration h1 + h2 , and so on. We consider schemes
5
where hj = j p for p ? 0. The number of iterations that nodes communicate out of the first T total
PH
iterations is given by HT = max{H :
j=1 hj ? T }. We have
Z
Z HT
H
T
HT
X
HTp+1 ? 1
H p+1 + p
y p dy =?
jp ? 1 +
y p dy ?
?T ? T
,
(22)
p+1
p+1
y=1
y=1
j=1
1
which means that HT = ?(T p+1 ) as T ? ?. Similar to (15), the network error is bounded as
w ?1
H
H
t ?1
t ?1
X
X
1 T H ?w
hX
t
L
+
2h
L
=
L
1
?
P
k?k1 hw + 2ht L. (23)
k?
z (t) ? zi (t)k? ?
t
n
i,:
1
w=0
w=0
k=0
We split the sum into?two terms based on whether or not the powers of P have converged. Using the
n)
?
, the `1 term is bounded by 2 when w is large and by T1 when w is small:
split point t? = log(T
1? ?
2
k?
z (t) ? zi (t)k? ?L
HtX
?1?t?
k?k1 hw + L
w=0
?
L
T
HtX
?1?t?
H
t ?1
X
k?k1 hw + 2ht L
(24)
w=Ht ?t?
H
t ?1
X
wp + 2L
w=0
wp + 2tp L
(25)
w=Ht ?t?
1
L (Ht ? t? ? 1) p+1 + p
+ 2Lt?(Ht ? 1)p + 2tp L
T
p+1
Lp
L
+
+ 2Lt?Htp + 2tp L
?
p + 1 T (p + 1)
?
(26)
(27)
since T > Ht ? t? ? 1. Substituting this bound into (6) and taking the step size sequence to be
a(t) = tAq with A and q to be determined, we get
Erri (T ) ?
R2
L2 A
3L2 A
3L2 pA
+
+
+
1?q
q
q
AT
2(1 ? q)T
(p + 1)(1 ? q)T
(p + 1)(1 ? q)T 1+q
T
+
T
6L2 A X p?q
6L2 t?A X Htp
+
t .
q
T t=1 t
T t=1
(28)
1
The first four summands converge to zero when 0 < q < 1. Since Ht = ?(t p+1 ),
!
p
1
T
T
p
1 X O(t p+1 )p
1 X Htp
T p+1 ?q+1
p+1 ?q
?
?
O
=
O
T
(29)
T t=1 tq
T t=1
tq
T
PT
p
T p?q
which converges to zero if p+1
< q. To bound the last term, note that T1 t=1 tp?q ? p?q+1
,
so the term
goes
to
zero
as
T
?
?
if
p
<
q.
In
conclusion,
Err
(T
)
converges
no
slower
than
i
?
n)
1
O( logT(T
) since q?1 p < T q?p
. If we choose q = 12 to balance the first three summands, for
q?p
T
p+1
?
(T
small p > 0, the rate of convergence is arbitrarily close to O( log ?
T
increasingly infrequently as T ? ?.
n)
), while nodes communicate
p
Out of T total iterations, DDA executes HT = ?(T p+1 ) expensive iterations involving communication and T ? HT cheap iterations without communication, so
p
T
1
kr
? () = O
+ T p+1 kr = O T
+
.
(30)
1
n
n T p+1
In this case, the communication cost kr becomes a less and less significant proportion of ? () as T
increases. So for any 0 < p < 12 , if k is fixed, we approach a linear speedup behaviour ?( Tn ). To
get Erri (T ) ? , ignoring the logarithmic factor, we need
s
2
1?2p
Cp
12p + 12
12
?
T =
iterations, with Cp = 2LR 7 +
.
(31)
+
(3p + 1)(1 ? ?2 ) 2p + 1
From this last equation we see that for 0 < p < 12 we have Cp < C1 , so using increasingly sparse
communication should, in fact, be faster than communicating at every iteration.
6
5
Experimental Evaluation
To verify our theoretical findings, we implement DDA on a cluster of 14 nodes with 3.2 GHz Pentium 4HT processors and 1 GB of memory each, connected via ethernet that allows for roughly
11 MB/sec throughput per node. Our implementation is in C++ using the send and receive functions
of OpenMPI v1.4.4 for communication. The Armadillo v2.3.91 library, linked to LAPACK and
BLAS, is used for efficient numerical computations.
5.1
Application to Metric Learning
Metric learning [11, 12, 13] is a computationally intensive problem where the goal is to find a
distance metric D(u, v) such that points that are related have a very small distance under D while
for unrelated points D is large. Following the formulation in [14], we have a data set {uj , vj , sj }m
j=1
with uj , vj ? Rd and sj = {?1, 1} signifying whether or not uj is similar to vj (e.g., similar if
they are from the same class). Our goal is to find a symmetric
positive semi-definite matrix A 0
p
to define a pseudo-metric of the form DA (u, v) = (u ? v)T A(u ? v). To that end, we use a
hinge-type loss function lj (A, b) = max{0, sj DA (uj , vj )2 ? b + 1} where b ? 1 is a threshold
that determines whether two points are dissimilar according to DA (?, ?). In the batch setting, we
formulate the convex optimization problem
m
X
minimize F (A, b) =
lj (A, b) subject to A 0, b ? 1.
(32)
A,b
j=1
The subgradient of lj at (A, b) is zero if sj (DA (uj , vj )2 ? b) ? ?1. Otherwise
?lj (A, b)
?lj (A, b)
= sj (uj ? vj )T (uj ? vj ), and
= ?sj .
(33)
?A
?b
Since DDA uses vectors xi (t) and zi (t), we represent each pair (Ai (t), bi (t)) as a d2 +1 dimensional
vector. The communication cost is thus quadratic in the dimension. In step (3) of DDA, we use the
proximal function ?(x) = 21 xT x, in which case (4) simplifies to taking xi (t) = ?a(t ? 1)zi (t),
followed by projecting xi (t) to the constraint set by setting bi (t) ? max{1, bi (t)} and projecting
Ai (t) to the set of positive semi-definite matrices by first taking its eigenvalue decomposition and
reconstructing Ai (t) after forcing any negative eigenvalues to zero.
We use the MNIST digits dataset which consists of 28 ? 28 pixel images of handwritten digits 0
through 9. Representing images as vectors, we have d = 282 = 784 and a problem with d2 + 1 =
614657 dimensions trying to learn a 784 ? 784 matrix A. With double precision arithmetic, each
DDA message has a size approximately 4.7 MB. We construct a dataset by randomly selecting 5000
pairs from the full MNIST data. One node needs 29 seconds to compute a gradient on this dataset,
and sending and receiving 4.7 MB takes 0.85 seconds. The communication/computation tradeoff
value is estimated as r = 0.85
29 ? 0.0293. According to (11), when G is a complete graph, we
expect to have optimal performance when using nopt = ?1r = 5.8 nodes. Figure 1(left) shows the
P
xi (t)) for 1 to 14 processors connected as
evolution of the average function value F? (t) = n1 i F (?
a complete graph, where x
?i (t) is as defined in (5). There is a very good match between theory and
practice since the fastest convergence is achieved with n = 6 nodes.
In the second experiment, to make r closer to 0, we apply PCA to the original data and keep the top
87 principal components, containing 90% of the energy. The dimension of the problem is reduced
dramatically to 87 ? 87 + 1 = 7570 and the message size to 59 KB. Using 60000 random pairs of
MNIST data, the time to compute one gradient on the entire dataset with one node is 2.1 seconds,
while the time to transmit and receive 59 KB is only 0.0104 seconds. Again, for a complete graph,
Figure 1(right) illustrates the evolution of F? (t) for 1 to 14 nodes. As we see, increasing n speeds up
the computation. The speedup we get is close to linear at first, but diminishes since communication
is not entirely free. In this case r = 0.0104
2.1 = 0.005 and nopt = 14.15.
5.2
Nonsmooth Convex Minimization
Next we create an artificial problem where the minima of the components fi (x) at each node are
very different, so that communication is essential in order to obtain an accurate optimizer of F (x).
7
4
2
1.2
1
0.8
1.5
0.6
1
0.4
0.5
0
n=1
n=2
n=4
n=6
n=8
n = 10
n = 12
n = 14
1.4
F? (t)
3
2.5
F? (t)
1.6
n=1
n=2
n=4
n=6
n=8
n = 10
n = 12
n = 14
3.5
0.2
50
100
150
200
250
Time (sec)
300
350
400
450
0
10
20
30
Time (sec)
40
50
60
Figure 1: (Left) In a subset of the Full MNIST data for our specific hardware, nopt = ?1r = 5.8. The
fastest convergence is achieved on a complete graph of 6 nodes. (Right) In the reduced MNIST data
using PCA, the communication cost drops and a speedup is achieved by scaling up to 14 processors.
We define fi (x) as a sum of high dimensional quadratics,
fi (x) =
M
X
1
2
max lj|i
(x), lj|i
(x) ,
?
lj|i
(x) = (x ? c?j|i )T (x ? c?j|i ),
? ? {1, 2},
(34)
j=1
where x ? R10,000 , M = 15, 000 and c1j|i , c2j|i are the centers of the quadratics. Figure 2 illustrates
again the average function value F? (t) for 10 nodes in a complete graph topology. The baseline performance is when nodes communicate at every iteration (h = 1). For this problem r = 0.00089 and,
from (21), hopt = 1. Naturally communicating every 2 iterations (h = 2) slows down convergence.
Over the duration of the experiment, with h = 2, each node communicates with its peers 55 times.
We selected p = 0.3 for increasingly sparse communication, and got HT = 53 communications
per node. As we see, even though nodes communicate as much as the h = 2 case, convergence is
even faster than communicating at every iteration. This verifies our intuition that communication is
more important in the beginning. Finally, the case where p = 1 is shown. This value is out of the
permissible range, and as expected DDA does not converge to the right solution.
5
x 10
h=1
h=2
0.3
h=t
h=t
2.4
2.2
F? (t)
2
1.8
1.6
1.4
1.2
20
40
60
80
100
Time (sec)
120
140
160
Figure 2: Sparsifying communication to minimize (34) with 10 nodes in a complete graph topology.
When waiting t0.3 iterations between consensus steps, convergence is faster than communicating
at every iteration (h = 1), even though the total number of consensus steps performed over the
duration of the experiment is equal to communicating every 2 iterations (h = 2). When waiting a
linear number of iterations between consensus steps (h = t) DDA does not converge to the right
solution. Note: all methods are initialized from the same value; the x-axis starts at 5 sec.
6
Conclusions and Future Work
The analysis and experimental evaluation in this paper focus on distributed dual averaging and reveal the capability of distributed dual averaging to scale with the network size. We expect that
similar results hold for other consensus-based algorithms such as [5] as well as various distributed
averaging-type algorithms (e.g., [15, 16, 17]). In the future we will extend the analysis to the case of
stochastic optimization, where ht = tp could correspond to using increasingly larger mini-batches.
8
References
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[2] Y. Nesterov, ?Primal-dual subgradient methods for convex problems,? Mathematical Programming Series
B, vol. 120, pp. 221?259, 2009.
[3] R. Bekkerman, M. Bilenko, and J. Langford, Scaling up Machine Learning, Parallel and Distributed
Approaches. Cambridge University Press, 2011.
[4] J. Duchi, A. Agarwal, and M. Wainwright, ?Dual averaging for distributed optimization: Convergence
analysis and network scaling,? IEEE Transactions on Automatic Control, vol. 57, no. 3, pp. 592?606,
2011.
[5] A. Nedic and A. Ozdaglar, ?Distributed subgradient methods for multi-agent optimization,? IEEE Transactions on Automatic Control, vol. 54, no. 1, January 2009.
[6] B. Johansson, M. Rabi, and M. Johansson, ?A randomized incremental subgradient method for distributed
optimization in networked systems,? SIAM Journal on Control and Optimization, vol. 20, no. 3, 2009.
[7] S. S. Ram, A. Nedic, and V. V. Veeravalli, ?Distributed stochastic subgradient projection algorithms for
convex optimization,? Journal of Optimization Theory and Applications, vol. 147, no. 3, pp. 516?545,
2011.
[8] A. Agarwal and J. C. Duchi, ?Distributed delayed stochastic optimization,? in Neural Information Processing Systems, 2011.
[9] K. I. Tsianos and M. G. Rabbat, ?Distributed dual averaging for convex optimization under communication delays,? in American Control Conference (ACC), 2012.
[10] F. Chung, Spectral Graph Theory.
AMS, 1998.
[11] E. P. Xing, A. Y. Ng, M. I. Jordan, and S. Russell, ?Distance metric learning, with application to clustering
with side-information,? in Neural Information Processing Systems, 2003.
[12] K. Q. Weinberger and L. K. Saul, ?Distance metric learning for large margin nearest neighbor classification,? Journal of Optimization Theory and Applications, vol. 10, pp. 207?244, 2009.
[13] K. Q. Weinberger, F. Sha, and L. K. Saul, ?Convex optimizations for distance metric learning and pattern
classification,? IEEE Signal Processing Magazine, 2010.
[14] S. Shalev-Shwartz, Y. Singer, and A. Y. Ng, ?Online and batch learning of pseudo-metrics,? in ICML,
2004, pp. 743?750.
[15] M. A. Zinkevich, M. Weimer, A. Smola, and L. Li, ?Parallelized stochastic gradient descent,? in Neural
Information Processing Systems, 2010.
[16] R. McDonald, K. Hall, and G. Mann, ?Distributed training strategies for the structured perceptron,? in
Annual Conference of the North American Chapter of the Association for Computational Linguistics,
2012, pp. 456?464.
[17] G. Mann, R. McDonald, M. Mohri, N. Silberman, and D. D. Walker, ?Efficient large-scale distributed
training of conditional maximum entropy models,? in Neural Information Processing Systems, 2009, pp.
1231?1239.
9
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3,940 | 4,567 | Fully Bayesian inference for neural models with
negative-binomial spiking
Jonathan W. Pillow
Center for Perceptual Systems
Department of Psychology
The University of Texas at Austin
[email protected]
James G. Scott
Division of Statistics and Scientific Computation
McCombs School of Business
The University of Texas at Austin
[email protected]
Abstract
Characterizing the information carried by neural populations in the brain requires
accurate statistical models of neural spike responses. The negative-binomial distribution provides a convenient model for over-dispersed spike counts, that is,
responses with greater-than-Poisson variability. Here we describe a powerful
data-augmentation framework for fully Bayesian inference in neural models with
negative-binomial spiking. Our approach relies on a recently described latentvariable representation of the negative-binomial distribution, which equates it to
a Polya-gamma mixture of normals. This framework provides a tractable, conditionally Gaussian representation of the posterior that can be used to design efficient EM and Gibbs sampling based algorithms for inference in regression and
dynamic factor models. We apply the model to neural data from primate retina
and show that it substantially outperforms Poisson regression on held-out data,
and reveals latent structure underlying spike count correlations in simultaneously
recorded spike trains.
1
Introduction
A central problem in systems neuroscience is to understand the probabilistic representation of information by neurons and neural populations. Statistical models play a critical role in this endeavor, as
they provide essential tools for quantifying the stochasticity of neural responses and the information
they carry about various sensory and behavioral quantities of interest.
Poisson and conditionally Poisson models feature prominently in systems neuroscience, as they
provide a convenient and tractable description of spike counts governed by an underlying spike rate.
However, Poisson models are limited by the fact that they constrain the ratio between the spike count
mean and variance to one. This assumption does not hold in many brain areas, particularly cortex,
where responses are often over-dispersed relative to Poisson [1].
A second limitation of Poisson models in regression analyses (for relating spike responses to stimuli)
or latent factor analyses (for finding common sources of underlying variability) is the difficulty of
performing fully Bayesian inference. The posterior formed under Poisson likelihood and Gaussian
prior has no tractable representation, so most theorists resort to either fast, approximate methods
based on Gaussians, [2?9] or slower, sampling-based methods that may scale poorly with data or
dimensionality [10?15].
The negative-binomial (NB) distribution generalizes the Poisson with a shape parameter that controls the tradeoff between mean and variance, providing an attractive alternative for over-dispersed
spike count data. Although well-known in statistics, it has only recently been applied for neural
data [16?18]. Here we describe fully Bayesian inference methods for the neural spike count data
based on a recently developed representation of the NB as a Gaussian mixture model [19]. In the
1
weights
B
shape
stimulus
C
300
response
variance
A
200
100
on
Poiss
latent
0
0
50
mean
100
Figure 1: Representations of the negative-binomial (NB) regression model. (A) Graphical model for
standard gamma-Poisson mixture representation of the NB. The linearly projected stimulus t =
T
xt defines the scale parameter for a gamma r.v. with shape parameter ?, giving t ? Ga(e t , ?),
which is in turn the rate for a Poisson spike count: yt ? Poiss( t ). (B) Graphical model illustrating
novel representation as a Polya-Gamma (PG) mixture of normals. Spike counts are represented as
NB distributed with shape ? and rate pt = 1/(1 + e t ). The latent variable !t is conditionally PG,
while (and |x) are normal given (!t , ?), which facilitates efficient inference. (C) Relationship
between spike-count mean and variance for different settings of shape parameter ?, illustrating superPoisson variability of the NB model.
following, we review the conditionally Gaussian representation for the negative-binomial (Sec. 2),
describe batch-EM, online-EM and Gibbs-sampling based inference methods for NB regression
(Sec. 3), sampling-based methods for dynamic latent factor models (Sec. 4), and show applications
to spiking data from primate retina.
2
The negative-binomial model
Begin with the single-variable case where the data Y = {yt } are scalar counts observed at times
t = 1, . . . , N . A standard Poisson generalized linear model (GLM) assumes that yt ? Pois(e t ),
where the log rate parameter t may depend upon the stimulus. One difficulty with this model is
that the variance of the Poisson distribution is equal to its mean, an assumption that is violated in
many data sets [20?22].
To relax this assumption, we can consider the negative binomial model, which can be described as a
doubly-stochastic or hierarchical Poisson model [18]. Suppose that yt arises according to:
(
(yt |
t
?
t)
| ?,
?
t)
Pois( t )
Ga ?, e
t
,
where we have parametrized the Gamma distribution in terms of its shape and scale parameters. By
marginalizing over the top-level model for t , we recover a negative-binomial distribution for yt :
where pt is related to
p(yt | ?,
t
t)
/ (1
pt )? pyt t ,
via the logistic transformation:
e t
.
1+e t
The extra parameter ? therefore allows for over-dispersion compared to the Poisson, with the count
yt having expected value ?e t and variance ?e t (1 + e t ). (See Fig. 1).
pt =
Bayesian inference for models of this form has long been recognized as a challenging problem, due
to the analytically inconvenient form of the likelihood function. To see the difficulty, suppose that
T
is a linear function of known inputs xt = (xt1 , . . . , xtP )T . Then the conditional posterior
t = xt
distribution for , up to a multiplicative constant, is
p( | ?, Y ) / p( ) ?
N
Y
{exp(xTt )}yt
,
{1 + exp(xTt )}?+yt
t=1
(1)
where p( ) is the prior distribution, and where we have assumed for the moment that ? is fixed.
The two major issues are the same as those that arise in Bayesian logistic regression: the response
2
depends non-linearly upon the parameters, and there is no natural conjugate prior p( ) to facilitate
posterior computation.
One traditional approach for Bayesian inference in logistic models is to work directly with the
discrete-data likelihood. A variety of tactics along these lines have been proposed, including numerical integration [23], analytic approximations to the likelihood [24?26], or Metropolis-Hastings [27].
A second approach is to assume that the discrete outcome is some function of an unobserved continuous quantity or latent variable. This is most familiar in the case of Bayesian inference for the probit
or dichotomized-Gaussian model [28, 29], where binary outcomes yi are assumed to be thresholded
versions of a latent Gaussian quantity zi . The same approach has also been applied to logistic
and Poisson regression [30, e.g.]. Unfortunately, none of these schemes lead to a fully automatic
approach to posterior inference, as they require either approximations (whose quality must be validated) or the careful selection of tuning constants (as is typically required when using, for example,
the Metropolis?Hastings sampler in very high dimensions).
To proceed with Bayesian inference in the negative-binomial model, we appeal to a recent latentvariable construction (depicted in Fig. 1B) from [19] based on the theory of Polya-Gamma random
variables. The basic result we exploit is that the negative binomial likelihood can be represented as
a mixture of normals with Polya-Gamma mixing distribution. The algorithms that result from this
scheme are both exact (in the sense of avoiding analytic approximations) and fully automatic.
Definition 1. A random variable X has a Polya-Gamma distribution with parameters b > 0 and
c 2 R, denoted X ? PG(b, c), if
1
1 X
X=
2? 2
(k
gk
,
1/2)2 + c2 /(4? 2 )
D
k=1
(2)
D
where each gk ? Ga(b, 1) is an independent gamma random variable, and where = denotes equality
in distribution.
We make use of four important facts about Polya-Gamma variables from [19]. First, suppose that
p(!) denotes the density of the random variable ! ? PG(b, 0), for b > 0. Then for any choice of a,
Z 1
2
(e )a
b ?
=2 e
e ! /2 p(!) d! ,
(3)
(1 + e )b
0
where ? = a b/2. This integral identity allows us to rewrite each term in the negative binomial
likelihood (eq. 1) as
Z 1
2
{exp( t )}yt
? yt
?t t
(1 pt ) pt =
/e
e !t /2 p(! | ? + yt , 0) d! ,
(4)
{1 + exp( t )}h+yt
0
where ?t = (yt ?)/2, and where the mixing distribution is Polya-Gamma. Conditional upon !t ,
we have a likelihood proportional to e Q( t ) for some quadratic form Q, which will be conditionally
conjugate to any Gaussian or mixture-of-Gaussians prior for t . This conditional Gaussianity can
be exploited to great effect in MCMC, EM, and sequential Monte Carlo algorithms, as described in
the next section.
A second important fact is that the conditional distribution
p(! | ) = R 1
0
e
e
!
2
!
2 /2
/2
p(!)
p(!) d!
is also in the Polya-Gamma class: (! | ) ? PG(b, ). In this sense, the Polya-Gamma distribution
is conditionally conjugate to the NB likelihood, which is very useful for Gibbs sampling.
Third, although the density of a Polya-Gamma random variable can be expressed only as an infinite
series, its expected value is known in closed form: if ! ? PG(b, c), then
b
tanh(c/2) .
(5)
2c
As we show in the next section, this expression comes up repeatedly when fitting negative-binomial
models via expectation-maximization, where these moments of !t form a set of sufficient statistics
for the complete-data log posterior distribution in .
E(!) =
3
Finally, despite the awkward form of the density function, it is still relatively easy to simulate random
Polya-Gamma draws, avoiding entirely the need to truncate the infinite sum in Equation 2. As the
authors of [19] show, this can be accomplished via a highly efficient accept-reject algorithm using
ideas from [31]. The proposal distribution requires only exponential, uniform, and normal random
variates; and the algorithm?s acceptance probability is uniformly bounded below at 0.9992 (implying
roughly 8 rejected draws out of every 10,000 proposals).
As we now describe, these four facts are sufficient to allow straightforward Bayesian inference for
negative-binomial models. We focus first on regression models, for which we derive simple Gibbs
sampling and EM algorithms. We then turn to negative-binomial dynamic factor models, which can
be fit using a variant of the forward-filter, backwards-sample (FFBS) algorithm [32].
3
3.1
Negative-binomial regression
Fully Bayes inference via MCMC
Suppose that t = xTt for some p-vector of regressors xt . Then, conditional upon !t , the contribution of observation t to the likelihood is
Lt ( )
/
/
exp{?t xTt
!t (xTt )2 /2}
(
?
?2 )
! t yt ?
T
exp
xt
.
2
2!t
Let ? = diag(!1 , . . . , !n ); let zt = (yt ?)/(2!t ); and let z denote the stacked vector of zt terms.
Combining all terms in the likelihood leads to a Gaussian linear-regression model where
1
(z | , ?) ? N (X , ?
).
It is usually reasonable to assume a conditionally Gaussian prior, ? N (c, C). Note that C itself
may be random, as in, for example, a Bayesian lasso or horseshoe prior [33?35]. Gibbs sampling
proceeds in two simple steps:
(!t | ?, )
( | ?, z)
?
?
PG(yt + ?, xTt )
N (m, V ) ,
where PG denotes a Polya-Gamma draw, and where
V
=
(X T ?X + C
m
=
V (X T ?z + C
1
)
1
1
c) .
One may update the dispersion parameter ? via Gibbs sampling, using the method described in [36].
3.2
Batch EM for MAP estimation
We may also use the same data-augmentation trick in an expectation-maximization (EM) algorithm
to compute the maximum a-posteriori (MAP) estimate ?. Returning to the likelihood in (4) and
ignoring constants of proportionality, we may write the complete-data log posterior distribution,
given !1 , . . . , !N , as
N ?
X
yt ?
(xT )2
Q( ) = log p( | Y, !1 , . . . , !N ) =
(xTt ) ?
!t t
+ log p( )
2
2
t=1
for some prior p( ). This expression is linear in !t . Therefore we may compute E{Q( )} by
substituting !
? t = E(!t | ), given the current value of , into the above expression. Appealing to
(5), these conditional expectations are available in closed form:
?
?
?t
E(!t | ) =
tanh(xTt /2) ,
xTt
where ?t = (yt
?)/2. In the M step, we re-express E{Q( )} as
1 T
E{Q( )} =
S + T d + log p( ) ,
2
4
where the complete-data sufficient statistics are
S
d
?
X T ?X
T
X ?
=
=
? = diag(!?1 , . . . , !
for ?
? N ) and ? = (?1 , . . . , ?N )T . Thus the M step is a penalized weighted least
squares problem, which can be solved using standard methods. In fact, it is typically unnecessary
to maximize E{Q( )} exactly at each iteration. As is well established in the literature on the EM
algorithm, it is sufficient to move to a value of that merely improves that observed-data objective
function. We have found that it is much faster to take a single step of the conjugate conjugategradient algorithm (in which case in will be important to check for improvement over the previous
iteration); see, e.g. [37] for details.
3.3
Online EM
For very large data sets, the above batch algorithm may be too slow. In such cases, we recommend
computing the MAP estimate via an online EM algorithm [38], as follows. Suppose that our current
estimate of the parameter is (t 1) , and that the current estimate of the complete-data log posterior
is
1 T (t 1)
Q( ) =
S
+ T d(t 1) + log p( ) ,
(6)
2
where
S (t
1)
t 1
X
=
!
? i xi xTi
i=1
d(t
1)
t 1
X
=
?i x i ,
i=1
recalling that ?i = (yi
value of !t as
?)/2. After observing new data (yt , xt ), we first compute the expected
? ?
?t
(t 1)
!
? t = E(!t | yt ,
)=
tanh( t /2) ,
t
with t =
denoting the linear predictor evaluated at the current estimate. We then update
the sufficient statistics recursively as
xTt
(t 1)
S (t)
=
(1
d(t)
=
(1
(t 1)
+ t!
? t xt xTt
t )S
(t 1)
+ t ?t x t ,
t )d
where t is the learning rate. We then plug these updated sufficient statistics into (6), and solve the
M step to move to a new value of . The data can also be processed in batches of size larger than 1,
(t)
(t)
with
p obvious modifications to the updates for S and d ; we have found that batch sizes of order
p tend to work well, although we are unaware of any theory to support this choice.
In high-dimensional problems, the usual practice is to impose sparsity via an `1 penalty on the
regression coefficients, leading to a lasso-type prior. In this case, the M-step in the online algorithm
can be solved very efficiently using the modified shooting algorithm, a coordinate-descent method
described in a different context by [39] and [40].
This online EM is guaranteed to converge
P1to a stationary point
P1 of the log posterior distribution if the
learning rate decays in time such that t=1 t = 1 and t=1 t2 < 1. (If the penalty function is
concave and ? is fixed, then this stationary point will be the global maximum.) A simple choice for
the learning rate is t = 1/ta for a 2 (0.5, 1), with a = 0.7 being our default choice.
4
Factor analysis for negative-binomial spiking
Let t = ( t1 , . . . , tK ) denote a vector of K linear predictors at time t, corresponding to K
different neurons with observed counts Yt = (yt1 , . . . , ytK )T . We propose a dynamic negative5
binomial factor model for Yt , with a vector autoregressive (VAR) structure for the latent factors:
NB(?, e tk ) for k = 1, . . . K
? + Bft
ft 1 + ?t , ?t ? N(0, ? 2 I) .
?
=
=
ytk
t
ft
Here ft denotes an L-vector of latent factors, with L typically much smaller than P . The K ? L
factor-loadings matrix B is restricted to have zeroes above the diagonal, and to have positive diagonal entries. These restrictions are traditional in Bayesian factor analysis [41], and ensure that B
is formally identified. We also assume that is a diagonal matrix, and impose conjugate inversegamma priors on ? 2 to ensure that, marginally over the latent factors ft , the entries of t have
approximately unit variance. Although we do not pursue the point here, the mean term ? can incorporate the effect of known predictors with no additional complication to the analysis.
By exploiting the Polya-Gamma data-augmentation scheme, posterior inference in this model may
proceed via straightforward Gibbs sampling?something not previously possible for count-data factor models. Prior work on latent variable modeling of spike data has relied on either Gaussian
approximations [2?6, 8] or variants of particle filtering [10?13].
Gibbs sampling proceeds as follows. Conditional upon B and ft , we update the latent variables as
!tk ? PG(ytk + ?, Bk ft ), where Bk denotes the kth row of the loadings matrix. The mean vector
? and factor-loadings matrix B can both be updated in closed-form via a Gaussian draw using the
full conditional distributions given in, for example, [42] or [43].
Given all latent variables and other parameters of the model, the factors ft can be updated in a single
block using the forward-filter, backwards-sample (FFBS) algorithm from [32]. First, pass forwards
through the data from y1 to yN , recursively computing the filtered moments of ft as
Mt
=
mt
=
(Vt
1
+ B T ?t B)
1
T
1
Mt (B ?t zt + Vt
mt
1) ,
where
Mt
T
+ ? 2I
Vt
=
zt
=
(zt1 , . . . , ztK )T
?t
=
diag(!t1 , . . . , !tK ) .
1
,
ztk =
ytk ?
2!tk
?k
Then draw fN ? N(mN , MN ) from its conditional distribution. Finally, pass backwards through
the data, sampling ft as (ft | mt , Mt , ft+1 ) ? N(at , At ), where
At
1
at
=
=
Mt
1
2
+?
1
1
I
At (Mt mt + ?
2
ft+1 ) .
This will result in a block draw of all N ? L factors from their joint conditional distribution.
5
Experiments
To demonstrate our methods, we performed regression and dynamic factor analyses on a dataset of
27 neurons recorded from primate retina (published in [44] and re-used with authors? permission).
Briefly, these data consist of spike responses from a simultaneously-recorded population of ON and
OFF parasol retinal ganglion cells, stimulated with a flickering, 120-Hz binary white noise stimulus.
5.1
Regression
Figure 2 shows a comparison of a Poisson model versus a negative-binomial model for each of the
27 neurons in the retinal dataset. We binned spike counts in 8 ms bins, and regressed against a
temporally lagged stimulus, resulting in a 100-element (10 ? 10 pixel) spatial receptive field for
each neuron. To benchmark the two methods, we created 50 random train/test splits from a full
dataset of 30,000 points, with 7,500 points held out for validation. Using each training set, we used
6
120
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Increase in Held-Out Log Likelihood
140
Neuron
Figure 2: Boxplots of improvement in held-out log likelihoods (NB versus Poisson regression) for
50 train/test splits on each of the 27 neurons in the primate retinal data.
our online maximum-likelihood method to fit an NB model to each of the 27 neurons, and then used
these models to compute held-out log-likelihoods on the test set versus a standard Poisson GLM.
As Figure 2 shows, the NB model has a higher average held-out log-likelihood than the Poisson
model. In some cases it is dozens of orders of magnitude better (as in neurons 12?14 and 22?27),
suggesting that there is substantial over-dispersion in the data that is not faithfully captured by the
Poisson model. We emphasize that this is a ?weak-signal? regime, and that overdispersion is likely
to be less when the signal is stronger. Yet these results suggest, at the very least, that many of these
neurons have marginal distributions that are quite far from Poisson. Moreover, regardless of the
underlying signal strength, the regression problem can be handled quite straightforwardly using our
online method, even in high dimensions, without settling for the restrictive Poisson assumption.
5.2
Dynamic factor analysis
To study the factor-modeling framework, we conducted parallel experiments on both simulated and
real data. First, we simulated two different data sets comprising 1000 time points and 11 neurons,
each from a two-factor model: one with high factor autocorrelation ( = 0.98), and one with low
factor autocorrelation ( = 0.5). The two questions of interest here are: how well does the fully
Bayesian method reconstruct the correlation structure among the unobserved rate parameters tk ;
and how well does it distinguish between a high-autocorrelation and low-autocorrelation regime in
the underlying low-dimensional representation?
The results in Figure 3 suggest that the results, on both counts, are highly accurate. It is especially
interesting to compare the left-most column of Figure 3 with the actual cross-sectional correlation
of t , the systematic component of variation, in the second column. The correlation of the raw
counts yt show a dramatic attenuation effect, compared to the real latent states. Yet this structure is
uncovered easily by the model, with together with a full assessment of posterior uncertainty. The
approach behaves much like a model-based version of principal-components analysis, appropriate
for non-Gaussian data.
Finally, Figure 4 shows the results of fitting a two-factor model to the primate retinal data. We
are able to uncover latent structure in the data in a completely unsupervised fashion. As with the
simulated data, it is interesting to compare the correlation of the raw counts yt with the estimated
correlation structure of the latent states. There is also strong support for a low-autocorrelation regime
in the factors, in light of the posterior mean factor scores depicted in the right-most pane.
6
Discussion
Negative-binomial models have only recently been explored in systems neuroscience, despite their
favorable properties for handling data with larger-than-Poisson variation. Likewise, Bayesian inference for the negative binomial model has traditionally been a difficult problem, with the existence
of a fully automatic Gibbs sampler only recently discovered [19]. Our paper has made three specific contributions to this literature. First, we have shown that negative-binomial models can lead to
7
1
Neuron
2
3
Index
4
Index
5
Index
6
Index
7
Index
8
Index
9
Index
10
Index
11
Index
Correlation Among Spike Counts
Actual Correlation Among Latent States
Estimated Correlation Among Latent States
1
Neuron
2
3
Index
4
Index
5
Index
6
Index
7
Index
8
Index
9
Index
10
Index
11
Index
Correlation Among Spike Counts
Actual Correlation Among Latent States
Estimated Correlation Among Latent States
Neuron
Figure 3: Results for two simulated data sets with high factor autocorrelation (top row) and low
factor autocorrelation (bottom row). The three left-most columns show the raw correlation among
the counts yt ; the actual correlation, E( t tT ), of the latent states; and the posterior mean estimator
for the correlation of the latent states. The right-most column shows the simulated spike trains for
the 11 neurons, along with the factors ft in blue (with 75% credible intervals), plotted over time.
1
2
3
4
5
6
7
8
9
10
11
Index
Index
Index
Index
Index
Index
Index
Index
Index
Index
Correlation among spike counts
Estimated correlation of latent states
Spike counts
Posterior mean factor scores
Figure 4: Results for factor analysis of the primate retinal data.
substantial improvements in fit, compared to the Poisson, for neural data exhibiting over-dispersion.
Such models can be fit straightforwardly via MCMC for a wide class of prior distributions over
model parameters (including sparsity-inducing choices, such as the lasso). Second, we have proposed a novel online-EM algorithm for sparse NB regression. This algorithm inherits all the convergence properties of EM, but is scalable to extremely large data sets. Finally, we have embedded
a dynamic factor model inside a negative-binomial likelihood. This latter approach can be extended
quite easily to spatial interactions, more general state-space models, or mixed models incorporating
both regressors and latent variables. All of these extensions, as well as the model-selection question
(how many factors?) form promising areas for future research.
Acknowledgments
We thank E. J. Chichilnisky, A. M. Litke, A. Sher and J. Shlens for retinal data, J. Windle for
PG sampling code, and J. H. Macke for helpful comments. This work was supported by a Sloan
Research Fellowship, McKnight Scholar?s Award, and NSF CAREER Award IIS-1150186 (JP).
8
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9
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3,941 | 4,568 | Dynamic Pruning of Factor Graphs
for Maximum Marginal Prediction
Christoph H. Lampert
IST Austria (Institute of Science and Technology Austria)
Am Campus 1, 3400 Klosterneuburg, Austria
http://www.ist.ac.at/?chl
[email protected]
Abstract
We study the problem of maximum marginal prediction (MMP) in probabilistic
graphical models, a task that occurs, for example, as the Bayes optimal decision
rule under a Hamming loss. MMP is typically performed as a two-stage procedure: one estimates each variable?s marginal probability and then forms a prediction from the states of maximal probability.
In this work we propose a simple yet effective technique for accelerating MMP
when inference is sampling-based: instead of the above two-stage procedure we
directly estimate the posterior probability of each decision variable. This allows us
to identify the point of time when we are sufficiently certain about any individual
decision. Whenever this is the case, we dynamically prune the variables we are
confident about from the underlying factor graph. Consequently, at any time only
samples of variables whose decision is still uncertain need to be created.
Experiments in two prototypical scenarios, multi-label classification and image
inpainting, show that adaptive sampling can drastically accelerate MMP without
sacrificing prediction accuracy.
1
Introduction
Probabilistic graphical models (PGMs) have become useful tools for classical machine learning
tasks, such as multi-label classification [1] or semi-supervised learning [2], as well for many realworld applications, for example image processing [3], natural language processing [4], bioinformatics [5], and computational neuroscience [6]. Despite their popularity, the question of how to
best perform (approximate) inference in any given graphical models is still far from solved. While
variational approximations and related message passing algorithms have been proven useful for certain classes of models (see [7] for an overview), there is still a large number of cases for which
sampling-based approaches are the safest choice. Unfortunately, inference by sampling is often
computationally costly: many samples are required to reach a confident result, and generating the
individual samples can be a complex task in itself, in particular if the underlying graphical model is
large and highly connected.
In this work we study a particular inference problem: maximum marginal prediction (MMP) in
binary-valued PGMs, i.e. the task of determining for each variable in the graphical model which of
its states has highest marginal probability. MMP occurs naturally as the Bayes optimal decision rule
under Hamming loss [8], and it has also found use as a building block for more complex prediction
tasks, such as M -best MAP prediction [9]. The standard approach to sampling-based MMP is
to estimate each variable?s marginal probability distribution from a set of samples from the joint
probability, and for each variable pick the state of highest estimated marginal probability. In this
work, we propose an almost as simple, but more efficient way. We introduce one binary indicator
variable for each decision we need to make, and keep estimates of the posterior probabilities of
each of these during the process of sampling. As soon as we are confident enough about any of
1
the decisions, we remove it from the factor graph that underlies the sampling process, so no more
samples are generated for it. Consequently, the factor graph shrinks over time, and later steps in the
sampling procedure are accelerated, often drastically so.
Our main contribution lies in the combination of two relatively elementary components that we
will introduce in the following section: an estimate for the posterior distributions of the decision
variables, and a mean field-like construction for removing individual variables from a factor graph.
2
Adaptive Sampling for Maximum Marginal Prediction
Let p(x) be a fixed probability distribution over the set X = {0, 1}V of binary labelings of a vertex
set V = {1, . . . , n}. We assume that p is given to us by means of a factor graph, G = (V, F), with
factor set F = {F1 . . . , Fk }. Each factor, Fj ? V , has an associated log-potential, ?j , which is a
real-valued function of only the variables occurring in Fj . Writing xFj = (xi )i?Fj we have
p(x) ? exp ? E(x)
with E(x) =
X
F ?F
?F (xF ).
(1)
for any x ? {0, 1}V . Our goal is maximum marginal prediction, i.e. to infer the values of decision
variables (zi )i?V that are defined by zi := 0 if ?i ? 0.5, and zi := 1 otherwise, where ?i := p(xi =
1) is the marginal probability of the ith variable taking the value 1. Computing the marginals ?i in a
loopy graphical model is in general #P-complete [10], so one has to settle for approximate marginals
and approximate predictions. In this work, we assume access to a suitable constructed sampler
based on the Markov chain Monte Carlo (MCMC) principle [11, 12], e.g. a Gibbs sampler [3] It
produces a chain of states Sm = {x(1) , . . . , x(N ) }, where each x(i) is a random sample from the
Pm (j)
1
joint distribution p(x). From the set of sample we can compute an estimate, ?
?i = m
j=1 xi of
the true marginal, ?i , and make approximate decisions: z?i := 1 if and only if ?
?i ? 0.5. Under mild
conditions on the sampling procedure the law of large number guarantees that limN ?? ?
?i = ?i ,
and the decisions will become correct almost surely.
The main problem with sampling-based inference is when to stop sampling [13]. The more samples
we have, the lower the variance on the estimates, so the more confident we can be about our decisions. However, each sample we generate increases the computational cost at least proportionally to
the numbers of factors and variables involved. At the same time, the variance of the estimators ?
?i
is reduced only proportionally to the square root of the sample size. In combination, this means that
often, one spends a large amount of computational resources on a small win in predictive accuracy.
In the rest of this section, we explain our proposed idea of adaptive sampling in graphical models,
which reduces the number of variables and factors during the course of the sampling procedure.
As an illustrative example we start by the classical situation of adaptive sampling in the case of a
single binary variable. This is a special case of Bayesian hypothesis selection, and ?for the case of
i.i.d. data? has recently also been rediscovered in the pattern recognition literature, for example for
evaluating decision trees [14]. We then introduce our proposed extensions to correlated samples, and
show how the per-variable decisions can be applied in the graphical model situation with potentially
many variables and dependencies between them.
2.1
Adaptive Sampling of Binary Variables
Let x be a single binary variable, for which we have a set of samples, S = {x(1) , . . . , x(N ) },
available. The main insight lies in the fact that even though samples are used to empirically estimate
the (marginal) probability ?, the latter is not the actual quantity of interest to us. Ultimately, we are
only interested in the value of the associated decision variable z.
Independent samples. Assuming for the moment that the samples are independent (i.i.d.), we can
derive an analytic expression for the posterior probability of z given the observed samples,
Z
1
2
p(q|S)dq
p(z = 0|S) =
(2)
0
2
where p(q|S) is the conditional probability density for ? having the value q. Applying Bayes? rule
with likelihood p(x|q) = q x (1 ? q)1?x and uniform prior, p(q) = 1, results in
Z 12
1
=
q m (1?q)N?m dq = I 21 (m+1, N ?m+1), (3)
B(m+ 1, N ?m+1) 0
PN
?(?)?(?)
(j)
where m =
j=1 x . The normalization factor B(?, ?) = ?(?+?) is the beta function; the
integral is called the incomplete beta function (here evaluated at 12 ). In combination, they form the
regularized incomplete beta function Ix (?, ?) [15].
From the above derivation we obtain a stopping criterion of -confidence: given any number of
samples we compute p(z = 0|S) using Equation (3). If its value is above 1 ? , we are -confident
that the correct decision is z = 0. If it is below , we are equally confident that the correct decision
is z = 1. Only if it lies inbetween we need to continue sampling. An analogue derivation to the
above leads to a confidence bound for estimates of the marginal probability, ?
? = m/N , itself:
p(|?
? ? ?| ? ?|S) = I??+? (m+1, N ?m+1) ? I???? (m+1, N ?m+1).
(4)
Note that both tests are computable fast enough to be done after each sample, or small batches of
samples. Evaluating the regularized incomplete beta function does not require numeric integration,
and for fixed parameter the values N and m that bound the regions of confidence can also be
tabulated [16]. A figure illustrating the difference between confidence in the MMP, and confidence
in the estimated marginals can be found in the supplemental material. It shows that only relatively
few independent samples (tens to hundreds) are sufficient to get a very confident MMP decision,
if the actual marginals are close to 0 or 1. Intuitively, this makes sense, since in this situation a
even coarse estimate of the marginal is sufficient to make of a decision with low error probability.
Only if the true marginal lies inside of a relatively narrow interval around 0.5, the MMP decision
becomes hard, and a large number of samples will be necessary to make a confident decision. Our
experiments in Section 4 will show that in practical problem where the probability distribution is
learned from data, the regions close to 0 and 1 are in fact the most relevant ones.
Dependent samples. Practical sampling procedures, such as MCMC, do not create i.i.d. samples,
but dependent ones. Using the above bounds directly with these would make the tests overconfident.
We overcome this problem, approximately, by borrowing the concept of effective sample size (ESS)
from the statistics literature. Intuitively, the ESS reflects how many independent samples, N 0 , a set of
N correlated sample is equivalent to. In first order 1 , one estimates the effective sample size as N 0 =
PN ?1 (x(j) ???)(x(j+1) ???)
1?r
1
, and
j=1
1+r N , where r is the first order autocorrelation coefficient, r = N ?1
?2
? 2 is the estimated variance of the sample sequence. Consequently, we can adjust the confidence
tests defined above to correlated data: we first collecting a small number of samples, N0 , which we
use to estimate initial values of ? 2 and r. Subsequently, we estimate the confidence of a decision by
p(z = 0|S) = I 12 (?
?N 0 + 1, (1 ? ?
?)N 0 + 1),
(5)
i.e. we replace the sample size N by the effective sample size N 0 and the raw count m by its adjusted
value ?
?N 0 .
2.2
Adaptive Sampling in Graphical Models
In this section we extend the above confidence criterion from single binary decisions to the situation
of joint sampling from the joint probability of multiple binary variables. Note that we are only inter(j)
ested in per-variable decisions, so we can treat the value of each variable xi in a joint sample x(j) as
a separate sample from the marginal probability p(xi ). We will have to take the dependence between
(j)
(k)
different samples xi and xi into account, but between variable dependencies within a sample do
not pose problems. Consequently, estimate the confidence of any decision variable zi is straight
(1)
(N )
forward from Equation (5), applied separately to the binary sample set Si = {xi , . . . , xi }. Note
that all quantities defined above for the single variable case need to be computed separately for each
decision. For example, each variable has its own autocorrelation estimate and effective sample size.
1
Many more involved methods for estimating the effective sample size exist, see, for example, [17], but in
our experiments the first-order method proved sufficient for our purposes.
3
The difference to the binary situation lies in what we do when we are confident enough about the
decision of some subset of variables, V c ? V . Simply stopping all sampling would be too risky,
since we are still uncertain about the decisions of V u := V \ V c . Continuing to sample until we are
certain about all decision will be wasteful, since we know that variables with marginal close to 0.5
require many more samples than others for a confident decision. We therefore propose to continue
sampling, but only for the variables about which we are still uncertain. This requires us to derive an
expression for p(xu ), the marginal probability of all variables that we are still uncertain about.
P
Computing p(xu ) =
xc , xu ) exactly is almost always infeasible, otherwise, we
x
?c ?{0,1}V c p(?
would not have needed to resort to sampling based inference in the first place. An alternative idea
would be to continue using the original factor graph, but to clamp all variables we are certain about
to their MMP values. This is computationally feasible, but it results in samples from a conditional
distribution, p(xu |xc = zc ), not from the desired marginal one. The new construction that we introduce combines advantages of both previous ideas: it is computationally as efficient as the value
clamping, but it uses a distribution that approximates the marginal distribution as closely as possible.
Similar as in mean-field methods [7], the main step consists of finding distributions q and q 0 such
0
that p(x) ? q(xu )q
P (xc ). Subsequently,Pq(xu ) can be used as approximate replacement to p(xu ),
because p(xu ) = x?c ?{0,1}V c p(x) ? x?c ?{0,1}V c q 0 (?
xc )q(xu ) = q(xu ). The main difference to
mean-field inference lies in the fact that q and q 0 have different role in our construction. For q 0 we
prefer a distribution that factorizes over the variables that we are confident about. Because we want
q also to respect the marginal probabilities,
?
?i for i ? V c , as estimated them from the sampling
Q
xi
0
process so far, we obtain q (xc ) = i?V c ?
?i (1 ? ?
?i )xi . The distribution q contain all variables
that we are not yet confident about, so we want to avoid making any limiting assumptions about its
potential values or structure. Instead, we define it as the solution of minimizing KL(p|qq 0 ) over all
distributions q, which yields the solution
q(xu ) ? exp( ?Ex?c ?q0 (xc ) {E(?
xc , xu )} ).
0
(6)
0
What remains is to define factors F and log-potentials ? , such that q(xu ) ? exp ?
P
0
F ?F 0 ?F (xF ) while also allowing for efficient sampling from q. For this we partition the original factor set into three disjoint sets, F = F c ? F u ? F0 , with F c := {F ? F : F ? V c },
F u := {F ? F : F ? V u }, and F0 := F \ (F c ? F u ). Each factor F0 ? F0 we split further into
its certain and uncertain components, F0c ? V c and F0u ? V u , respectively.
With this we obtain a decomposition of the exponent in Equation (6):
X X
X
X X
Ex?c ?q0 {E(?
xc , xu )} =
q 0 (?
xc )?F c (xF c ) +
?Fu (xFu )+
q 0 (?
xF0c )?F (?
xF0c , xF0u )
?F c
F c ?F c x
Fu ?F u
?F c
F0 ?F0 x
0
The first sum is a constant with respect to xu , so we can disregard it in the construction of F 0 . The
factors and log-potentials in the second sum already depend only on V u , so we can re-use them
in unmodified form for F 0 , we set ?F0 = ?F for every F ? F u . The third sum we rewrite as
P
0
u
{F u =F ?V u :F ?F0 } ?F u (xF ), with
X Y
?F0 u (xu ) :=
?
?xi?i (1 ? ?
?i )1??xi ?F (?
xc , xu ).
(7)
x
?c ?{0,1}Fc
i?Fc
for any F ? F0 , where we have made use of the explicit form of q?. If factors with identical variable
set occur during this construction, we merge them by summing their log-potentials. Ultimately, we
obtain a new factor set F 0 := F u ? {F ? V u : F ? F0 }, and probability distribution
X
u
q(xu ) ? exp
?F0 (xF )
for xu ? {0, 1}V .
(8)
F ?F 0
Note that during the process, not only the number of variables is reduced, but also the number of
factors and the size of each factor can never grow. Consequently, if sampling was feasible for the
original distribution p, it will also be feasible for q, and potentially more efficient.
3
Related Work
Sequential sampling with the option of early stopping has a long tradition in Bayesian statistics. First
introduced by Wald in 1945 [18], the ability to continuously accumulate information until a decision
can be made with sufficient confidence was one of the key factors that contributed to the success of
4
Bayesian reasoning for decision making. Today, it has been a standard technique in areas as diverse
as clinical medicine (e.g. for early stop of drug trials [19]), social sciences (e.g. for designing and
evaluating experiments [20]), and economics (e.g. in modelling stock market behavior [21]).
In current machine learning research, sequential sampling is used less frequently for making individual decisions, but in the form of MCMC it has become one of the most successful techniques for
statistical inference of probability distributions with many dependent variables [12, 22]. Nevertheless, to the best of our knowledge, the method we propose is the first one that performs early stopping
of subsets of variables in this context. Many other approaches to reduce the complexity of sampling
iterations exist, however, for example to approximate complex graphical models by simpler ones,
such as trees [23], or loopy models of low treewidth [24]. These fall into a different category than the
proposed method, though, as they are usually performed statically and prior to the actual inference
step, so they cannot dynamically assign computational resources where they are needed most. Beam
search [25] and related techniques take an orthogonal approach to ours. They dynamically exclude
low-likelihood label combinations from the inference process, but they keep the size and topology of
the factor graph fixed. Select and sample [26] disregards a data-dependent subset of variables during each sampling iterations. It is not directly applicable in our situation, though, since it requires
that the underlying graphical model is bipartite, such that the individual variables are conditionally
independent of each other. Given their complementary nature, we believe that the idea of combining
adaptive MMP with beam search and/or select and sample could be a promising direction for future
work.
4
Experimental Evaluation
To demonstrate the effect of adaptive MMP compared to naive MMP, we performed experiments in
two prototypical applications: multi-label classification and binary image inpainting. In both tasks,
performance is typically measured by the Hamming loss, so MMP is the preferred method of test
time prediction.
4.1
Multi-Label Classification
In multi-label classification, the task is to predict for each input y ? Y, which labels out of a label
set L = {1, . . . , K} are correct. The difference to multi-class classification is that several labels can
be correct simultaneously, or potentially none at all. Multi-label classification can be formulated
as simultaneous prediction of K binary labels (xi )i=1,...K , where xi = 1 indicates that the label i
is part of the prediction, and xi = 0 indicates that it is not. Even though multi-label classification
can in principle be solved by training K independent predictors, several studies have shown that
by making use of dependencies between label, the accuracy of the individual predictions can be
improved [1, 27, 28].
For our experiments we follow [1] in using a fully-connected conditional random field model.
Given an input y, each label variable i has a unary factor Fi = {i} with log-linear potential
?i (xi ) = hwi , yixi , where wi is a label-specific weight vector that was learned from training
data. Additionally there are K(K ? 1)/2 pairwise factors, Fij = {i, j}, with log-potentials
?ij (xi , xj ) = ?ij xi xj . Its free parameter ?ij is learned as well. The resulting conditional joint
distribution has the form of a Boltzmann machine, p(x|y) ? exp(?Ey (x)), with energy function
PL PL
PK
Ey (x) = i=1 ?i xi + i=1 j=i+1 ?ij xi xj in minimal representation, where ?i and ?ij depend
on y. We downloaded several standard datasets and trained the CRF on each of them using a stochastic gradient descent procedure based on the sgd2 package. The necessary gradients are computing
using a junction tree algorithms for problems with 20 variables or less, and by Gibbs sampling
otherwise. For model selection, when required, we used 10-fold cross-validation on the training set.
Note that our goal in this experiment is not to advocate a new model multi-label classification, but to
create probability distributions as they would appear in real problems. Nevertheless, we also report
classification accuracy in Table 1 to show that a) the learned models have similar characteristics as
earlier work, in particular to [29], where the an identical model was trained using structured SVM
learning, and b) adaptive MMP can achieve as high prediction accuracy as ordinary Gibbs sampling,
as long as the confidence parameter is not chosen overly optimistically. In fact, in many cases even
2
http://leon.bottou.org/projects/sgd
5
Dataset
S YNTH 1 [29]
S YNTH 2 [29]
S CENE
R CV 1-10 [29]
M EDIAMILL -10 [29]
Y EAST
T MC 2007
AWA [30]
M EDIAMILL
R CV 1
#Labels
6
10
6
10
10
14
22
85
101
103
#Train
471
1000
1211
2916
29415
1500
21519
24295
29415
3000
#Test
5045
10000
1196
2914
12168
917
7077
6180
12168
3000
[29]
[28]
6.9
?
7.0
?
10.1 9.5 ? 2.1
5.6
?
18.8
?
20.2 20.2 ? 1.3
? 3.3 ? 2.7
?
?
? 3.6 ? 0.5
?
?
Exact Gibbs
Proposed
5.2
5.3 5.2 / 5.2 / 5.2
10.0 10.0 10.0/10.0/10.0
10.4 10.3 10.2/10.2/10.2
4.2
4.2 4.6 / 4.4 / 4.2
18.4 18.6 19.0/18.6/18.4
20.0 20.2 23.4/21.4/20.5
5.3
5.3 5.3 / 5.3 / 5.3
?
32.2 32.7/32.7/32.7
?
3.7 3.6 / 3.5 / 3.6
?
1.5 1.7 / 1.6 / 1.5
Variables Iterations
Table 1: Multi-label classification. Dataset characteristics (number of labels, number of training
examples, number of test examples) and classification error rate in percent. [29] used the same model
as we do, but trained it using a structured SVM framework and predicted using MAP. [28] compared
12 different multi-label classification techniques, we report their mean and standard deviation. The
remaining columns give MMP prediction accuracy of the trained CRF models: Exact computes the
exact marginal values by a junction tree, Gibbs and Proposed performs ordinary Gibbs sampling, or
the proposed adaptive version with = 10?2 /10?5 /10?8 , both run for up to 500 iterations.
1.0
1.0
1.0
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.0
Factors
0.1
1
10
100
1000
10000
0.0
0.01
0.1
1
10
100
1000
10000
0.5
1.0
1.0
0.4
0.8
0.8
0.3
0.6
0.6
0.2
0.4
0.4
0.1
0.2
0.2
0.0
0.0
0.01
0.1
1
10
100
1000
10000
0.1
1
10
100
1000
10000
0.5
1.0
0.20
0.4
0.8
0.15
0.3
0.6
0.10
0.2
0.4
0.05
0.1
0.2
0.00
0.0
0.1
1
10
100
1000
10000
0.1
1
10
100
1000
10000
1.0
1.0
0.4
0.8
0.8
0.3
0.6
0.6
0.2
0.4
0.4
0.1
0.2
0.2
0.0
0.0
0.1
1
10
100
1000
10000
0.1
1
10
100
1000
10000
0.01
0.1
1
10
100
1000
10000
0.01
0.1
1
10
100
1000
10000
0.01
0.1
1
10
100
1000
10000
0.0
0.01
0.5
0.01
0.01
0.0
0.01
0.25
0.01
Runtime
0.2
0.0
0.01
0.0
0.01
0.1
1
10
100
1000
10000
Figure 1: Results of adaptive pruning on RCV 1 dataset for = 10?2 , 10?5 , 10?8 (left to right).
x-axis: regularization parameter C used for training, y-axis: ratio of iterations/variables/factors/
runtime used by adaptive sampling relative to Gibbs sampling.
a relative large value, such as = 0.01 results in a smaller loss of accuracy than the potential 1%,
but overall, a value of 10?5 or less seems advisable.
Figures 1 and 2 show in more detail how the adaptive sampling behaves on two exemplary datasets
with respect to four aspects: the number of iterations, the number of variables, the number of factors, and the overall runtime. For each aspect we show a box plot of the corresponding relative
quantity compared to the Gibbs sampler. For example, a value of 0.5 in iterations means that the
adaptive sample terminated after 250 iterations instead of the maximum of 500, because it was confident about all decisions. Values of 0.2 in variables and factors means that the number of variable
states samples by the adaptive sampler was 20%, and the number of factors in the corresponding
factor graphs was 10% of the corresponding quantities for the Gibbs sampler. Within each plot, we
reported results for the complete range of regularization parameters in order to illustrate the effect
that regularization has on the distribution of marginals.
6
Variables Iterations
1.0
1.0
1.0
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.0
Factors
0.1
1
10
100
1000
10000
0.0
0.01
0.1
1
10
100
1000
10000
0.5
1.0
1.0
0.4
0.8
0.8
0.3
0.6
0.6
0.2
0.4
0.4
0.1
0.2
0.2
0.0
0.0
0.01
0.1
1
10
100
1000
10000
0.1
1
10
100
1000
10000
0.5
1.0
0.20
0.4
0.8
0.15
0.3
0.6
0.10
0.2
0.4
0.05
0.1
0.2
0.00
0.0
0.1
1
10
100
1000
10000
0.1
1
10
100
1000
10000
1.0
1.0
0.4
0.8
0.8
0.3
0.6
0.6
0.2
0.4
0.4
0.1
0.2
0.1
1
10
100
1000
10000
1
10
100
1000
10000
0.01
0.1
1
10
100
1000
10000
0.01
0.1
1
10
100
1000
10000
0.01
0.1
1
10
100
1000
10000
0.2
0.0
0.01
0.1
0.0
0.01
0.5
0.0
0.01
0.0
0.01
0.25
0.01
Runtime
0.2
0.0
0.01
0.0
0.01
0.1
1
10
100
1000
10000
Figure 2: Results of adaptive pruning on YEAST dataset for = 10?2 , 10?5 , 10?8 (left to right).
x-axis: regularization parameter C used for training, y-axis: ratio of iterations/variables/factors/
runtime used by adaptive sampling relative to Gibbs sampling. Note that the scaling of the y-axis
differs between columns.
Figure 1 shows results for the relatively simple RCV 1 dataset. As one can see, a large number of
variables and factors are removed quickly from the factor graph, leading to a large speedup compared
to the ordinary Gibbs sampler. In fact, as the first row shows, it was possible to make a confident
decision for all variables far before the 500th iteration, such that the adaptive method terminated
early. As a general trend, the weaker the regularization (larger C value in the plot), the earlier the
adaptive sampler is able to remove variables and factors, presumably because more extreme values
of the energy function result in more marginal probabilities close to 0 or 1. A second insight is that
despite the exponential scaling of the confidence parameter between the columns, the runtime grows
only roughly linearly. This indicates that we can choose conservatively without taking a large
performance hit. On the hard YEAST dataset (Figure 2) in the majority of cases the adaptive sampling
does not terminate early, indicating that some of the variables have marginal probabilities close to
0.5. Nevertheless, a clear gain in speed can be observed, in particular in the weakly regularized case,
indicating that nevertheless, many tests for confidence are successful early during the sampling.
4.2
Binary Image Inpainting
Inpainting is a classical image processing task: given an image (in our case black-and-white) in
which some of the pixels are occluded or have missing values, the goal is to predict a completed
image in which the missing pixels are set to their correct value, or at least in a visually pleasing
way. Image inpainting has been tackled successfully by grid-shaped Markov random field models,
where each pixel is represented by a random variable, unary factors encode local evidence extracted
from the image, and pairwise terms encode the cooccurrence of pixel value. For our experiment, we
use the Hard Energies from Chinese Characters (HECC) dataset [31], for which the authors provide
pre-computed energy functions. The dataset has 100 images, each with between 4992 and 17856
pixels, i.e. binary variables. Each variable has one unary and up to 64 pairwise factors, leading to an
overall factor count of 146224 to 553726. Because many of the pairwise factors act repulsively, the
underlying energy function is highly non-submodular, and sampling has proven a more successful
mean of inference than, for example, message passing [31].
Figure 3 shows exemplary results of the task. The complete set can be found in the supplemental
material. In each case, we ran an ordinary Gibbs sampler and the adaptive sampler for 30 seconds,
7
input
Gibbs
= 10?2
= 10?5
= 10?8
Figure 3: Example results of binary image inpainting on HECC dataset. From left to right: image to
be inpainted, result of Gibbs sampling, result of adaptive sampling, where each method was run for
up to 30 seconds per image. The left plot of each result shows the marginal probabilities, the right
plot shows how often each pixel was sampled on a log scale from 10 (dark blue) to 100000 (bright
red). Gibbs sampling treats all pixels uniformly, reaching around 100 sampling sweeps within the
given time budget. Adaptive sampling stops early for parts of the image that it is certain about, and
concentrates its samples in the uncertain regions, i.e. pixels with marginal probability close to 0.5.
The larger , the more pronounced this effect it.
and we visualize the resulting marginal probabilities as well as the number of samples created for
each of the pixels. One can see that adaptive sampling comes to a more confident prediction within
the given time budget. The larger the parameter, the earlier to stops sampling the ?easy? pixels,
spending more time on the difficult cases, i.e. pixel with marginal probability close to 0.5.
5
Summary and Outlook
In this paper we derived an analytic expression for how confident one can be about the maximum
marginal predictions (MMPs) of a binary graphical model after a certain number of samples, and
we presented a method for pruning factor graphs when we want to stop sampling for a subset of
the variables. In combination, this allowed us to more efficiently infer the MMPs: starting from
the whole factor graph, we sample sequentially, and whenever we are sufficiently certain about a
prediction, we prune it from the factor graph before continuing to sample. Experiments on multilabel classification and image inpainting show a clear increase in performance at virtually no loss in
accuracy, unless the confidence is chosen too optimistically.
Despite the promising results there are two main limitations that we plan to address. On the one
hand, the multi-label experiments showed that sometimes, a conservative estimate of the confidence
is required to achieve highest accuracy. This is likely a consequence of the fact that our pruning
uses the estimated marginal to build a new factor graph, and even if the decision confidence is high,
the marginals can still vary considerately. We plan to tackle this problem by also integrating bounds
on the marginals with data-dependent confidence into our framework. A second limitation is that
we can currently only handle binary-valued labelings. This is sufficient for multi-label classification
and many problems in image processing, but ultimately, one would hope to derive similar early
stopping criteria also for graphical models with larger label set. Our pruning method would be
readily applicable to this situation, but an open challenge lies in finding a suitable criterion when
to prune variables. This will require a deeper understanding of tail probabilities of multinomial
decision variables, but we are confident it will be achievable, for example based on existing prior
works from the case of i.i.d. samples [14, 32].
8
References
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9
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3,942 | 4,569 | Efficient Monte Carlo Counterfactual Regret
Minimization in Games with Many Player Actions
Richard Gibson, Neil Burch, Marc Lanctot, and Duane Szafron
Department of Computing Science, University of Alberta
Edmonton, Alberta, T6G 2E8, Canada
{rggibson | nburch | lanctot | dszafron}@ualberta.ca
Abstract
Counterfactual Regret Minimization (CFR) is a popular, iterative algorithm for
computing strategies in extensive-form games. The Monte Carlo CFR (MCCFR)
variants reduce the per iteration time cost of CFR by traversing a smaller, sampled
portion of the tree. The previous most effective instances of MCCFR can still be
very slow in games with many player actions since they sample every action for a
given player. In this paper, we present a new MCCFR algorithm, Average Strategy Sampling (AS), that samples a subset of the player?s actions according to the
player?s average strategy. Our new algorithm is inspired by a new, tighter bound on
the number of iterations required by CFR to converge to a given solution quality.
In addition, we prove a similar, tighter bound for AS and other popular MCCFR
variants. Finally, we validate our work by demonstrating that AS converges faster
than previous MCCFR algorithms in both no-limit poker and Bluff.
1
Introduction
An extensive-form game is a common formalism used to model sequential decision making problems containing multiple agents, imperfect information, and chance events. A typical solution concept in games is a Nash equilibrium profile. Counterfactual Regret Minimization (CFR) [12] is an
iterative algorithm that, in 2-player zero-sum extensive-form games, converges to a Nash equilibrium. Other techniques for computing Nash equilibria of 2-player zero-sum games include linear
programming [8] and the Excessive Gap Technique [6]. Theoretical results indicate that for a fixed
solution quality, CFR takes a number of iterations at most quadratic in the size of the game [12, Theorem 4]. Thus, as we consider larger games, more iterations are required to obtain a fixed solution
quality. Nonetheless, CFR?s versatility and memory efficiency make it a popular choice.
Monte Carlo CFR (MCCFR) [9] can be used to reduce the traversal time per iteration by considering
only a sampled portion of the game tree. For example, Chance Sampling (CS) [12] is an instance of
MCCFR that only traverses the portion of the game tree corresponding to a single, sampled sequence
of chance?s actions. However, in games where a player has many possible actions, such as no-limit
poker, iterations of CS are still very time consuming. This is because CS considers all possible
player actions, even if many actions are poor or only factor little into the algorithm?s computation.
Our main contribution in this paper is a new MCCFR algorithm that samples player actions and is
suitable for games involving many player choices. Firstly, we provide tighter theoretical bounds on
the number of iterations required by CFR and previous MCCFR algorithms to reach a fixed solution
quality. Secondly, we use these new bounds to propel our new MCCFR sampling algorithm. By
using a player?s average strategy to sample actions, convergence time is significantly reduced in
large games with many player actions. We prove convergence and show that our new algorithm
approaches equilibrium faster than previous sampling schemes in both no-limit poker and Bluff.
1
2
Background
A finite extensive game contains a game tree with nodes corresponding to histories of actions h ? H
and edges corresponding to actions a ? A(h) available to player P (h) ? N ? {c} (where N is the
set of players and c denotes chance). When P (h) = c, ?c (h, a) is the (fixed) probability of chance
generating action a at h. Each terminal history z ? Z has associated utilities ui (z) for each player
i. We define ?i = maxz,z0 ?Z ui (z) ? ui (z 0 ) to be the range of utilities for player i. Non-terminal
histories are partitioned into information sets I ? Ii representing the different game states that
player i cannot distinguish between. For example, in poker, player i does not see the private cards
dealt to the opponents, and thus all histories differing only in the private cards of the opponents are
in the same information set for player i. The action sets A(h) must be identical for all h ? I, and we
denote this set by A(I). We define |Ai | = maxI?Ii |A(I)| to be the maximum number of actions
available to player i at any information set. We assume perfect recall that guarantees players always
remember information that was revealed to them and the order in which it was revealed.
A (behavioral) strategy for player i, ?i ? ?i , is a function that maps each information set I ? Ii to
a probability distribution over A(I). A strategy profile is a vector of strategies ? = (?1 , ..., ?|N | ) ?
?, one for each player. Let ui (?) denote the expected utility for player i, given that all players play
according to ?. We let ??i refer to the strategies in ? excluding ?i . Let ? ? (h) be the probability
?
of
Q history h ?occurring if all? players choose actions according to ?. We can decompose ? (h) =
i?N ?{c} ?i (h), where ?i (h) is the contribution to this probability from player i when playing
?
according to ?i (or from chance when i = c). Let ??i
(h) be the product of all players? contributions
?
(including chance) except that of player i. Let ? (h, h0 ) be the probability of history h0 occurring
after h, given h has occurred. Furthermore, for I ? Ii , the probability of player i playing to reach I
is ?i? (I) = ?i? (h) for any h ? I, which is well-defined due to perfect recall.
A best response to ??i is a strategy that maximizes player i?s expected payoff against ??i . The
best response value for player i is the value of that strategy, bi (??i ) = max?i0 ??i ui (?i0 , ??i ). A
strategy profile ? is an -Nash equilibrium if no player can unilaterally deviate from ? and gain
more than ; i.e., ui (?) + ? bi (??i ) for all i ? N . A game is two-player zero-sum if N = {1, 2}
and u1 (z) = ?u2 (z) for all z ? Z. In this case, the exploitability of ?, e(?) = (b1 (?2 )+b2 (?1 ))/2,
measures how much ? loses to a worst case opponent when players alternate positions. A 0-Nash
equilibrium (or simply a Nash equilibrium) has zero exploitability.
Counterfactual Regret Minimization (CFR) [12] is an iterative algorithm that, for two-player zero
sum games, computes an -Nash equilibrium profile with ? 0. CFR has also been shown to work
well in games with more than two players [1, 3]. On each iteration t, the base algorithm, ?vanilla?
CFR, traverses the entire game tree once per player, computing the expected utility for player i at
each information set I ? Ii under the current profile ? t , assuming
to reach I. This
P player i plays
?
expectation is the counterfactual value for player i, vi (I, ?) = z?ZI ui (z)??i
(z[I])? ? (z[I], z),
where ZI is the set of terminal histories passing through I and z[I] is that history along z contained
in I. For each action a ? A(I), these values determine the counterfactual regret at iteration t,
t
rit (I, a) = vi (I, ?(I?a)
) ? vi (I, ? t ),
where ?(I?a) is the profile ? except that at I, action a is always taken. The regret rit (I, a) measures
how much player i would rather play action a at I than play ? t . These regrets are accumulated to
PT
obtain the cumulative counterfactual regret, RiT (I, a) = t=1 rit (I, a), and are used to update
the current strategy profile via regret matching [5, 12],
? T +1 (I, a) = P
RiT,+ (I, a)
b?A(I)
RiT,+ (I, b)
,
(1)
where x+ = max{x, 0} and actions are chosen uniformly at random when the denominator is zero.
It is well-known that in a two-player zero-sum game, if both players? average (external) regret,
T
1X
RiT
t
t
= max
ui (?i0 , ??i
) ? ui (?it , ??i
) ,
0
?i ??i T
T
t=1
is at most /2, then the average profile ?
? T is an -Nash equilibrium. During computation, CFR
PT
T
?t
t
stores a cumulative profile si (I, a) =
t=1 ?i (I)?i (I, a) and outputs the average profile
2
?
?iT (I, a) = sTi (I, a)/
P
T
b?A(I) si (I, b).
The original CFR analysis shows that player i?s regret
is bounded by the sum of the positive parts of the cumulative counterfactual regrets RiT,+ (I, a):
Theorem 1 (Zinkevich et al. [12])
RiT ?
X
I?I
max RiT,+ (I, a).
a?A(I)
Regret matching minimizes the average of the cumulative counterfactual regrets, and so player i?s
average regret is also minimized by Theorem 1. For each player i, let Bi be the partition of Ii such
that two information sets I, I 0 are in the same part B ? Bi if and only if player i?s sequence of
actions leading to I is the same as the sequence of actions leading to I 0 . Bi is well-defined
p due to
P
perfect recall. Next, define the M -value of the game to player i to be Mi = B?Bi |B|. The
best known bound on player i?s average regret is:
Theorem 2 (Lanctot et al. [9]) When using vanilla CFR, average regret is bounded by
p
?i Mi |Ai |
RiT
?
?
.
T
T
We prove a tighter bound in Section 3. For large games, CFR?s full game tree traversals can be very
expensive. Alternatively, one can traverse a smaller, sampled portion of the tree on each iteration
using Monte Carlo CFR (MCCFR) [9]. Let Q = {Q1 , ..., QK } be a set of subsets, or blocks, of
the terminal histories Z such that the union of Q spans Z. For example, Chance Sampling (CS)
[12] is an instance of MCCFR that partitions Z into blocks such that two histories are in the same
block if and only if no two chance actions differ. On each iteration, a block Qj is sampled with
PK
probability qj , where k=1 qk = 1. In CS, we generate a block by sampling a single action a at
each history h ? H with P (h) = c according to its likelihood of occurring, ?c (h, a). In general, the
sampled counterfactual value for player i is
X
?
v?i (I, ?) =
ui (z)??i
(z[I])? ? (z[I], z)/q(z),
z?ZI ?Qj
P
where q(z) =
k:z?Qk qk is the probability that z was sampled. For example, in CS, q(z) =
?
t
?c (z). Define the sampled counterfactual regret for action a at I to be r?it (I, a) = v?i (I, ?(I?a)
)?
P
T
t
T
t
?
v?i (I, ? ). Strategies are then generated by applying regret matching to R (I, a) =
r? (I, a).
i
t=1 i
CS has been shown to significantly reduce computing time in poker games [11, Appendix A.5.2].
Other instances of MCCFR include External Sampling (ES) and Outcome Sampling (OS) [9].
ES takes CS one step further by considering only a single action for not only chance, but also for
t
the opponents, where opponent actions are sampled according to the current profile ??i
. OS is the
most extreme version of MCCFR that samples a single action at every history, walking just a single
trajectory through the tree on each traversal (Qj = {z}). ES and OS converge to equilibrium faster
than vanilla CFR in a number of different domains [9, Figure 1].
ES and OS yield a probabilistic bound on the average regret, and thus provide a probabilistic guarantee that ?
? T converges to a Nash equilibrium. Since
Q both algorithms generate blocks by sampling
actions independently, we can decompose q(z) = i?N ?{c} qi (z) so that qi (z) is the probability
contributed to q(z) by sampling player i?s actions.
Theorem 3 (Lanctot et al. [9]) 1 Let X be one of ES or OS (assuming OS also samples opponent
actions according to ??i ), let p ? (0, 1], and let ? = minz?Z qi (z) > 0 over all 1 ? t ? T . When
using X, with probability 1 ? p, average regret is bounded by
! p
p
2|Ii ||Bi |
1 ?i |Ai |
RiT
?
? Mi +
.
?
T
p
?
T
1
The bound
p presented by Lanctot et al. appears slightly different, but the last step of their proof mistakenly
used Mi ? |Ii ||Bi |, which is actually incorrect. The bound we present here is correct.
3
3
New CFR Bounds
While Zinkevich et al. [12] bound a player?s regret by a sum of cumulative counterfactual regrets (Theorem 1), we can actually equate a player?s regret to a weighted sum of counterfactual
set I ? Ii , define RiT (I, ?i ) =
P regrets. For aT strategy ?i ? ?i and an information
?
In addition, let ?i ? ?i be a player i strategy such that
a?A(I) ?i (I, a)Ri (I, a).
PT
PT
t
?
t
?
0
). Note that in a two-player game,
?i = arg max?i ??i t=1 ui (?i0 , ??i
t=1 ui (?i , ??i ) =
?
T
?
T ui (?i , ?
??i ), and thus ?i is a best response to the opponent?s average strategy after T iterations.
Theorem 4
RiT =
X
?
?i? (I)RiT (I, ?i? ).
I?Ii
All proofs in this paper are provided in full as supplementary material. Theorem 4 leads to a tighter
bound on the average regret when
p using CFR. For a strategy ?i ? ?i , define the M -value of ?i to
P
be Mi (?i ) = B?Bi ?i? (B) |B|, where ?i? (B) = maxI?B ?i? (I). Clearly, Mi (?i ) ? Mi for all
?i ? ?i since ?i? (B) ? 1. For vanilla CFR, we can simply replace Mi in Theorem 2 with Mi (?i? ):
Theorem 5 When using vanilla CFR, average regret is bounded by
p
?i Mi (?i? ) |Ai |
RiT
?
.
?
T
T
For MCCFR, we can show a similar improvement to Theorem 3. Our proof includes a bound for CS
that appears to have been omitted in previous work. Details are in the supplementary material.
Theorem 6 Let X be one of CS, ES, or OS (assuming OS samples opponent actions according to
??i ), let p ? (0, 1], and let ? = minz?Z qi (z) > 0 over all 1 ? t ? T . When using X, with
probability 1 ? p, average regret is bounded by
! p
p
2|Ii ||Bi |
1 ?i |Ai |
RiT
?
?
? Mi (?i ) +
.
?
T
p
?
T
Theorem 4 states that player i?s regret is equal to the weighted sum of player i?s counterfactual
regrets at each I ? Ii , where the weights are equal to player i?s probability of reaching I under ?i? .
Since our goal is to minimize average regret, this means that we only need to minimize the average
cumulative counterfactual regret at each I ? Ii that ?i? plays to reach. Therefore, when using
MCCFR, we may want to sample more often those information sets that ?i? plays to reach, and less
often those information sets that ?i? avoids. This inspires our new MCCFR sampling algorithm.
4
Average Strategy Sampling
Leveraging the theory developed in the previous section, we now introduce a new MCCFR sampling algorithm that can minimize average regret at a faster rate than CS, ES, and OS. As we just
described, we want our algorithm to sample more often the information sets that ?i? plays to reach.
Unfortunately, we do not have the exact strategy ?i? on hand. Recall that in a two-player game, ?i? is
T
a best response to the opponent?s average strategy, ?
??i
. However, for two-player zero-sum games,
T
we do know that the average profile ?
? converges to a Nash equilibrium. This means that player i?s
T
average strategy, ?
?iT , converges to a best response of ?
??i
. While the average strategy is not an exact
best response, it can be used as a heuristic to guide sampling within MCCFR. Our new sampling algorithm, Average Strategy Sampling (AS), selects actions for player i according to the cumulative
profile and three predefined parameters. AS can be seen as a sampling scheme between OS and ES
where a subset of player i?s actions are sampled at each information set I, as opposed to sampling
one action (OS) or sampling every action (ES). Given the cumulative profile sTi (I, ?) on iteration
T , an exploration parameter ? (0, 1], a threshold parameter ? ? [1, ?), and a bonus parameter
? ? [0, ?), each of player i?s actions a ? A(I) are sampled independently with probability
(
)
? + ? sTi (I, a)
P
?(I, a) = max ,
,
(2)
? + b?A(I) sTi (I, b)
4
Algorithm 1 Average Strategy Sampling (Two-player version)
1: Require: Parameters , ?, ?
2: Initialize regret and cumulative profile: ?I, a : r(I, a) ? 0, s(I, a) ? 0
3:
4: WalkTree(history h, player i, sample prob q):
5:
if h ? Z then return ui (h)/q end if
6:
if h ? P (c) then Sample action a ? ?c (h, ?), return WalkTree(ha, i, q) end if
7:
I ? Information set containing h , ?(I, ?) ? RegretMatching(r(I, ?))
8:
if h ?
/ P (i) then
9:
for a ? A(I) do s(I, a) ? s(I, a) + (?(I, a)/q) end for
10:
Sample action a ? ?(I, ?), return WalkTree(ha, i, q)
11:
end if
12:
for a ? A(I) ndo
o
13:
?+? s(I,a)
? ? max , ?+P
?(a) ? 0
s(I,b) , v
b?A(I)
14:
15:
16:
17:
if Random(0, 1) < ? then v?(a) ? WalkTree(ha, i, q ? min{1, ?}) end if
end for
P
for a ? A(I) do r(I, a) ? r(I, a) + v?(a) ? a?A(I) ?(I, a)?
v (a) end for
P
return a?A(I) ?(I, a)?
v (a)
P
or with probability 1 if either ?(I, a) > 1 or ? + b?A(I) sTi (I, b) = 0. As in ES, at opponent and
T
chance nodes, a single action is sampled on-policy according to the current opponent profile ??i
and the fixed chance probabilities ?c respectively.
If ? = 1P
and ? = 0, then ?(I, a) is equal to the probability that the average strategy ?
?iT =
T
T
si (I, a)/ b?A(I) si (I, b) plays a at I, except that each action is sampled with probability at least
. For choices greater than 1, ? acts as a threshold so that any action taken with probability at least
1/? by the average strategy is always sampled by AS. Furthermore, ??s purpose is to increase the
rate of exploration during early AS iterations. When ? > 0, we effectively add ? as a bonus to the
cumulative value sTi (I, a) before normalizing. Since player i?s average strategy ?
?iT is not a good
?
approximation of ?i for small T , we include ? to avoid making ill-informed choices early-on. As
the cumulative profile sTi (I, ?) grows over time, ? eventually becomes negligible. In Section 5, we
present a set of values for , ? , and ? that work well across all of our test games.
Pseudocode for a two-player version of AS is presented in Algorithm 1. In Algorithm 1, the recursive
function WalkTree considers four different cases. Firstly, if we have reached a terminal node, we
return the utility scaled by 1/q (line 5), where q = qi (z) is the probability of sampling z contributed
from player i?s actions. Secondly, when at a chance node, we sample a single action according to ?c
and recurse down that action (line 6). Thirdly, at an opponent?s choice node (lines 8 to 11), we again
sample a single action and recurse, this time according to the opponent?s current strategy obtained
via regret matching (equation (1)). At opponent nodes, we also update the cumulative profile (line
9) for reasons that we describe in a previous paper [2, Algorithm 1]. For games with more than two
players, a second tree walk is required and we omit these details.
The final case in Algorithm 1 handles choice nodes for player i (lines 7 to 17). For each action a, we
compute the probability ? of sampling a and stochastically decide whether to sample a or not, where
Random(0,1) returns a random real number in [0, 1). If we do sample a, then we recurse to obtain
t
the sampled counterfactual value v?(a) = v?i (I, ?(I?a)
) (line 14). Finally, we update the regrets at I
P
(line 16) and return the sampled counterfactual value at I, a?A(I) ?(I, a)?
v (a) = v?i (I, ? t ).
Repeatedly running WalkTree(?, i, 1) ?i ? N provides a probabilistic guarantee that all players?
average regret will be minimized. In the supplementary material, we prove that AS exhibits the same
regret bound as CS, ES, and OS provided in Theorem 6. Note that ? in Theorem 6 is guaranteed
to be positive for AS by the inclusion of in equation (2). However, for CS and ES, ? = 1 since
all of player i?s actions are sampled, whereas ? ? 1 for OS and AS. While this suggests that fewer
iterations of CS or ES are required to achieve the same regret bound compared to OS and AS,
iterations for OS and AS are faster as they traverse less of the game tree. Just as CS, ES, and OS
5
have been shown to benefit from this trade-off over vanilla CFR, we will show that in practice, AS
can likewise benefit over CS and ES and that AS is a better choice than OS.
5
Experiments
In this section, we compare the convergence rates of AS to those of CS, ES, and OS. While AS can
be applied to any extensive game, the aim of AS is to provide faster convergence rates in games
involving many player actions. Thus, we consider two domains, no-limit poker and Bluff, where we
can easily scale the number of actions available to the players.
No-limit poker. The two-player poker game we consider here, which we call 2-NL Hold?em(k),
is inspired by no-limit Texas Hold?em. 2-NL Hold?em(k) is played over two betting rounds. Each
player starts with a stack of k chips. To begin play, the player denoted as the dealer posts a small
blind of one chip and the other player posts a big blind of two chips. Each player is then dealt two
private cards from a standard 52-card deck and the first betting round begins. During each betting
round, players can either fold (forfeit the game), call (match the previous bet), or raise by any
number of chips in their remaining stack (increase the previous bet), as long as the raise is at least as
big as the previous bet. After the first betting round, three public community cards are revealed (the
flop) and a second and final betting round begins. If a player has no more chips left after a call or a
raise, that player is said to be all-in. At the end of the second betting round, if neither player folded,
then the player with the highest ranked five-card poker hand wins all of the chips played. Note that
the number of player actions in 2-NL Hold?em(k) at one information set is at most the starting stack
size, k. Increasing k adds more betting options and allows for more actions before being all-in.
Bluff. Bluff(D1 , D2 ) [7], also known as Liar?s Dice, Perduo, and Dudo, is a two-player dice-bidding
game played with six-sided dice over a number of rounds. Each player i starts with Di dice. In each
round, players roll their dice and look at the result without showing their opponent. Then, players
alternate by bidding a quantity q of a face value f of all dice in play until one player claims that
the other is bluffing (i.e., claims that the bid does not hold). To place a new bid, a player must
increase q or f of the current bid. A face value of six is considered ?wild? and counts as any other
face value. The player calling bluff wins the round if the opponent?s last bid is incorrect, and loses
otherwise. The losing player removes one of their dice from the game and a new round begins.
Once a player has no more dice left, that player loses the game and receives a utility of ?1, while
the winning player earns +1 utility. The maximum number of player actions at an information set
is 6(D1 + D2 ) + 1 as increasing Di allows both players to bid higher quantities q.
Preliminary tests. Before comparing AS to CS, ES, and OS, we first run some preliminary experiments to find a good set of parameter values for , ? , and ? to use with AS. All of our preliminary
experiments are in two-player 2-NL Hold?em(k). In poker, a common approach is to create an abstract game by merging similar card dealings together into a single chance action or ?bucket? [4]. To
keep the size of our games manageable, we employ a five-bucket abstraction that reduces the branching factor at each chance node down to five, where dealings are grouped according to expected hand
strength squared as described by Zinkevich et al. [12].
Firstly, we fix ? = 1000 and test different values for and ? in 2-NL Hold?em(30). Recall that
? = 1000 implies actions taken by the average strategy with probability at least 0.001 are always
sampled by AS. Figure 1a shows the exploitability in the five-bucket abstract game, measured in
milli-big-blinds per game (mbb/g), of the profile produced by AS after 1012 nodes visited. Recall
that lower exploitability implies a closer approximation to equilibrium. Each data point is averaged
over five runs of AS. The = 0.05 and ? = 105 or 106 profiles are the least exploitable profiles
within statistical noise (not shown).
Next, we fix = 0.05 and ? = 106 and test different values for ? . Figure 1b shows the abstract
game exploitability over the number of nodes visited by AS in 2-NL Hold?em(30), where again each
data point is averaged over five runs. Here, the least exploitable strategies after 1012 nodes visited
are obtained with ? = 100 and ? = 1000 (again within statistical noise). Similar results to Figure
1b hold in 2-NL Hold?em(40) and are not shown. Throughout the remainder of our experiments, we
use the fixed set of parameters = 0.05, ? = 106 , and ? = 1000 for AS.
6
Exploitability (mbb/g)
1
0.5
0.4
0.3
? 0.2
0.1
0.05
0.01
0.8
0.6
0.4
0.2
0
100 101 102 103 104 105 106 107 108 109
?
Abstract game exploitability (mbb/g)
102
1
10
0
100
?=101
?=10
?=1023
?=104
?=10
5
?=106
?=10
10-1 10
10
11
10
Nodes Visited
12
10
(b) = 0.05, ? = 106
(a) ? = 1000
Figure 1: (a) Abstract game exploitability of AS profiles for ? = 1000 after 1012 nodes visited
in 2-NL Hold?em(30). (b) Log-log plot of abstract game exploitability over the number of nodes
visited by AS with = 0.05 and ? = 106 in 2-NL Hold?em(30). For both figures, units are in
milli-big-blinds per hand (mbb/g) and data points are averaged over five runs with different random
seeds. Error bars in (b) indicate 95% confidence intervals.
Main results. We now compare AS to CS, ES, and OS in both 2-NL Hold?em(k) and Bluff(D1 , D2 ).
Similar to Lanctot et al. [9], our OS implementation is -greedy so that the current player i samples
a single action at random with probability = 0.5, and otherwise samples a single action according
to the current strategy ?i .
Firstly, we consider two-player 2-NL Hold?em(k) with starting stacks of k = 20, 22, 24, ..., 38,
and 40 chips, for a total of eleven different 2-NL Hold?em(k) games. Again, we apply the same
five-bucket card abstraction as before to keep the games reasonably sized. For each game, we ran
each of CS, ES, OS, and AS five times, measured the abstract game exploitability at a number of
checkpoints, and averaged the results. Figure 2a displays the results for 2-NL Hold?em(36), a game
with approximately 68 million information sets and 5 billion histories (nodes). Here, AS achieved
an improvement of 54% over ES at the final data points. In addition, Figure 2b shows the average
exploitability in each of the eleven games after approximately 3.16 ? 1012 nodes visited by CS, ES,
and AS. OS performed much worse and is not shown. Since one can lose more as the starting stacks
are increased (i.e., ?i becomes larger), we ?normalized? exploitability across each game by dividing
the units on the y-axis by k. While there is little difference between the algorithms for the smaller
20 and 22 chip games, we see a significant benefit to using AS over CS and ES for the larger games
that contain many player actions. For the most part, the margins between AS, CS, and ES increase
with the game size.
Figure 3 displays similar results for Bluff(1, 1) and Bluff(2, 1), which contain over 24 thousand and
3.5 million information sets, and 294 thousand and 66 million histories (nodes) respectively. Again,
AS converged faster than CS, ES, and OS in both Bluff games tested. Note that the same choices
of parameters ( = 0.05, ? = 106 , ? = 1000) that worked well in 2-NL Hold?em(30) also worked
well in other 2-NL Hold?em(k) games and in Bluff(D1 , D2 ).
6
Conclusion
This work has established a number of improvements for computing strategies in extensive-form
games with CFR, both theoretically and empirically. We have provided new, tighter bounds on the
average regret when using vanilla CFR or one of several different MCCFR sampling algorithms.
These bounds were derived by showing that a player?s regret is equal to a weighted sum of the
player?s cumulative counterfactual regrets (Theorem 4), where the weights are given by a best response to the opponents? previous sequence of strategies. We then used this bound as inspiration for
our new MCCFR algorithm, AS. By sampling a subset of a player?s actions, AS can provide faster
7
3
10
102
101
100
-1
10
CS
ES
OS
AS
1010
1011
Nodes Visited
0.16
Abstract game exploitability (mbb/g) / k
Abstract game exploitability (mbb/g)
104
1012
0.14
0.12
CS
ES
AS
k=40
0.1
0.08
0.06
0.04
k=30
0.02 k=20
0
106
107
108
Game size (# information sets)
(b) 2-NL Hold?em(k), k ? {20, 22, ..., 40}
(a) 2-NL Hold?em(36)
100
100
10-1
10-1
10-2
10-2
Exploitability
Exploitability
Figure 2: (a) Log-log plot of abstract game exploitability over the number of nodes visited by CS,
ES, OS, and AS in 2-NL Hold?em(36). The initial uniform random profile is exploitable for 6793
mbb/g, as indicated by the black dashed line. (b) Abstract game exploitability after approximately
3.16 ? 1012 nodes visited over the game size for 2-NL Hold?em(k) with even-sized starting stacks
k between 20 and 40 chips. For both graphs, units are in milli-big-blinds per hand (mbb/g) and data
points are averaged over five runs with different random seeds. Error bars indicate 95% confidence
intervals. For (b), units on the y-axis are normalized by dividing by the starting chip stacks.
-3
10
10-4
CS
ES
OS
AS
10-5 7
10
108
109
1010 1011
Nodes Visited
1012
1013
(a) Bluff(1, 1)
-3
10
10-4
CS
ES
OS
AS
10-5 7
10
108
109
1010 1011
Nodes Visited
1012
1013
(b) Bluff(2, 1)
Figure 3: Log-log plots of exploitability over number of nodes visited by CS, ES, OS, and AS in
Bluff(1, 1) and Bluff(2, 1). The initial uniform random profile is exploitable for 0.780 and 0.784
in Bluff(1, 1) and Bluff(2, 1) respectively, as indicated by the black dashed lines. Data points are
averaged over five runs with different random seeds and error bars indicate 95% confidence intervals.
convergence rates in games containing many player actions. AS converged faster than previous MCCFR algorithms in all of our test games. For future work, we would like to apply AS to games with
many player actions and with more than two players. All of our theory still applies, except that
player i?s average strategy is no longer guaranteed to converge to ?i? . Nonetheless, AS may still find
strong strategies faster than CS and ES when it is too expensive to sample all of a player?s actions.
Acknowledgments
We thank the members of the Computer Poker Research Group at the University of Alberta for helpful conversations pertaining to this work. This research was supported by NSERC, Alberta Innovates
? Technology Futures, and computing resources provided by WestGrid and Compute Canada.
8
References
[1] Nick Abou Risk and Duane Szafron. Using counterfactual regret minimization to create competitive multiplayer poker agents. In Ninth International Conference on Autonomous Agents
and Multiagent Systems (AAMAS), pages 159?166, 2010.
[2] Richard Gibson, Marc Lanctot, Neil Burch, Duane Szafron, and Michael Bowling. Generalized
sampling and variance in counterfactual regret minimization. In Twenty-Sixth Conference on
Artificial Intelligence (AAAI), pages 1355?1361, 2012.
[3] Richard Gibson and Duane Szafron. On strategy stitching in large extensive form multiplayer
games. In Advances in Neural Information Processing Systems 24 (NIPS), pages 100?108,
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[4] Andrew Gilpin and Tuomas Sandholm. A competitive Texas Hold?em poker player via automated abstraction and real-time equilibrium computation. In Twenty-First Conference on
Artificial Intelligence (AAAI), pages 1007?1013, 2006.
[5] Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68:1127?1150, 2000.
[6] Samid Hoda, Andrew Gilpin, Javier Pe?na, and Tuomas Sandholm. Smoothing techniques
for computing Nash equilibria of sequential games. Mathematics of Operations Research,
35(2):494?512, 2010.
[7] Reiner Knizia. Dice Games Properly Explained. Blue Terrier Press, 2010.
[8] Daphne Koller, Nimrod Megiddo, and Bernhard von Stengel. Fast algorithms for finding
randomized strategies in game trees. In Annual ACM Symposium on Theory of Computing
(STOC?94), pages 750?759, 1994.
[9] Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte Carlo sampling
for regret minimization in extensive games. In Advances in Neural Information Processing
Systems 22 (NIPS), pages 1078?1086, 2009.
[10] Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte Carlo sampling
for regret minimization in extensive games. Technical Report TR09-15, University of Alberta,
2009.
[11] Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. Technical Report TR07-14, University of
Alberta, 2007.
[12] Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In Advances in Neural Information Processing
Systems 20 (NIPS), pages 905?912, 2008.
9
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3,943 | 457 | Constant-Time Loading of Shallow 1-Dimensional
Networks
Stephen Judd
Siemens Corporate Research,
755 College Rd. E.,
Princeton, NJ 08540
[email protected]
Abstract
The complexity of learning in shallow I-Dimensional neural networks has
been shown elsewhere to be linear in the size of the network. However,
when the network has a huge number of units (as cortex has) even linear
time might be unacceptable. Furthermore, the algorithm that was given to
achieve this time was based on a single serial processor and was biologically
implausible.
In this work we consider the more natural parallel model of processing
and demonstrate an expected-time complexity that is constant (i.e. independent of the size of the network). This holds even when inter-node
communication channels are short and local, thus adhering to more biological and VLSI constraints.
1
Introduction
Shallow neural networks are defined in [J ud90]; the definition effectively limits the
depth of networks while allowing the width to grow arbitrarily, and it is used as a
model of neurological tissue like cortex where neurons are arranged in arrays tens
of millions of neurons wide but only tens of neurons deep. Figure I exemplifies
a family of networks which are not only shallow but "I-dimensional" as well-we
allow the network to be extended as far as one liked in width (i.e. to the right) by
repeating the design segments shown. The question we address is how learning time
scales with the width. In [Jud88], it was proved that the worst case time complexity
863
864
Judd
of training this family is linear in the width. But the proof involved an algorithm
that was biologically very implausible and it is this objection that will be somewhat
redressed in this paper.
The problem with the given algorithm is that it operates only a monolithic serial
computer; the single-CPU model of computing has no overt constraints on communication capacities and therefore is too liberal a model to be relevant to our neural
machinery. Furthermore, the algorithm reveals very little about how to do the
processing in a parallel and distributed fashion. In this paper we alter the model
of computing to attain a degree of biological plausibility. We allow a linear number processors and put explicit constraints on the time required to communicate
between processors. Both of these changes make the model much more biological
(and also closer to the connectionist sty Ie of processing).
This change alone, however, does not alter the time complexity-the worst case
training time is still linear. But when we change the complexity question being
asked, a different answer is obtained. We define a class of tasks (viz. training data)
that are drawn at random and then ask for the expected time to load these tasks,
rather than the worst-case time. This alteration makes the question much more
environmentally relevant. It also leads us into a different domain of algorithms and
yields fast loading times.
2
2.1
Shallow I-D Loading
Loading
A family of the example shallow I-dimensional architectures that we shall examine
is characterized solely by an integer, d, which defines the depth of each architecture
in the family. An example is shown in figure 1 for d 3. The example also happens
to have a fixed fan-in of 2 and a very regular structure, but this is not essential. A
member of the family is specified by giving the width n, which we will take to be
the number of output nodes.
=
A task is a set of pairs of binary vectors, each specifying an stimulus to a net and
its desired response. A random task of size t is a set of t pairs of independently
drawn random strings; there is no guarantee it is a function.
Our primary question has to do with the following problem, which is parameterized
by some fixed depth d, and by a node function set (which is the collection of different
transfer functions that a node can be tuned to perform):
Shallow 1-D Loading:
Instance: An integer n, and a task.
Objective: Find a function (from the node function set) for each node in the
network in the shallow I-D architecture defined by d and n such that the
resulting circuit maps all the stimuli in the task to their associated responses.
Constant-Time Loading of Shallow I-Dimensional Networks
Figure 1: A Example Shallow 1-D Architecture
2.2
Model of Computation
Our machine model for solving this question is the following: For an instance of
shallow 1-D loading of width n, we allow n processors. Each one has access to
a piece of the task, namely processor i has access to bits i through i + d of each
stimulus, and to bit i of each response. Each processor i has a communication link
only to its two neighbours, namely processors i - I and i + 1. (The first and nth
processors have only one neighbour.) It takes one time step to communicate a fixed
amount of data between neighbours. There is no charge for computation, but this is
not an unreasonable cheat because we can show that a matrix multiply is sufficient
for this problem, and the size of the matrix is a function only of d (which is fixed).
This definition accepts the usual connectionist ideal of having the processor closely
identified with the network nodes for which it is "finding the weights", and data
available at the processor is restricted to the same "local" data that connectionist
machines have.
This sort of computation sets the stage for a complexity question,
2.3
Question and Approach
We wish to demonstrate that
Claim 1 This parallel machine solves shallow J-D loading where each processor is
finished in constant expected time The constant is dependent on the depth of the
architecture and on the size of the task, but not on the width. The expectation is
over the tasks.
865
866
Judd
For simplicity we shall focus on one particular processor-the one at the leftmost
end-and we shall further restrict our at tention to finding a node function for one
particular node.
To operate in parallel, it is necessary and sufficient for each processor to make its
local decisions in a "safe" manner-that is, it must make choices for its nodes in
such a way as to facilitate a global solution. Constant-time loading precludes being
able to see all the data; and if only local data is accessible to a processor, then
its plight is essentially to find an assignment that is compatible with all nonlocal
satisfying assignments.
Theorem 2 The expected communication complexity of finding a "safe" node function assignment for a particular node in a shallow l-D architecture is a constant
dependent on d and t, but not on n.
If decisions about assignments to single nodes can be made easily and essentially
without having to communicate with most of the network, then the induced partitioning of the problem admits of fast parallel computation. There are some complications to the details because all these decisions must be made in a coordinated
fashion, but we omit these details here and claim they are secondary issues that do
not affect the gross complexity measurements.
The proof of the theorem comes in two pieces. First, we define a computational
problem called path finding and the graph-theoretic notion of domination which
is its fundamental core. Then we argue that the loading problem can be reduced
to path finding in constant parallel time and give an upper bound for determining
domination.
3
Path Finding
The following problem is parameterized by an integer I<, which is fixed.
Path finding :
Instance: An integer n defining the number of parts in a partite graph, and a
series of I<xI< adjacency matrices, M I , M 2 , ??. M n - I . Mj indicates connections
between the K nodes of part i and the I< nodes of part i + 1.
Objective: Find a path of n nodes, one from each part of the n-partite graph.
Define Xh to be the binary matrix representing connectivity between the first part of
the graph and the ith part: Xl = MI and Xh(j, k) = 1 iff 3m such that Xh(j, m) = 1
and Mh(m, k) = 1. We say "i includes j at h" if every bit in the ith row of Xh is 1
whenever the corresponding bit in the jth row of X h is 1. We say "i dominates at
h" or "i is a dominator' if for all rows j, i includes j at h.
Lemma 3 Before an algorithm can select a node i from the first part of the graph
to be on the path, it is necessary and sufficient for i to have been proven to be a
dominator at some h.
0
Constant-Time Loading of Shallow l-Dimensional Networks
The minimum h required to prove domination stands as our measure of "communication complexity" .
Lemma 4 Shallow J-D Loading can be reduced to path finding in constant parallel
time.
Proof: Each output node in a shallow architecture has a set of nodes leading into it
called a support cone (or "receptive field"), and the collection of functions assigned
to those nodes will determine whether or not the output bit is correct in each
response. Nodes A,B,C,D,E,G in Figure 1 are the support cone for the first output
node (node C), and D,E,F,G,H,J are the cone for the second. Construct each part
of the graph as a set of points each corresponding to an assignment over the whole
support cone that makes its output bit always correct. This can be done for each
cone ih parallel, and since the depth (and the fan-in) is fixed, the set of all possible
assignments for the support cone can be enumerated in constant time. Now insert
edges between adjacent parts wherever two points correspond to assignments that
are mutually compatible. (Note that since the support cones overlap one another,
we need to ensure that assignments are consistent with each other.) This also can
be done in constant parallel time. We call this construction a compatibility graph.
A solution to the loading problem corresponds exactly to a path in the compatibility
graph.
0
A dominator in this path-finding graph is exactly what was meant above by a "safe"
assignment in the loading problem.
4
Proof of Theorem
Since it is possible that there is no assignments to certain cones that correctly
map the stimuli it is trivial to prove the theorem, but as a practical matter we are
interested in the case where the architecture is actually capable of performing the
task. We will prove the theorem using a somewhat more satisfying event.
Proof of theorem 2: For each support cone there is 1 output bit per response and
there are t such responses. Given the way they are generated, these responses could
all be the same with probability .5 t - 1 . The probability of two adjacent cones both
having to perform such a constant mapping is .5 2(t-l).
Imagine the labelling in Figure 1 to be such that there were many support cones
to the left (and right) of the piece shown. Any path through the left side of the
compatibility graph that arrived at some point in the part for the cone to the left
of C would imply an assignment for nodes A, B, and D. Any path through the
right side of the compatibility graph that arrived at some point in the part for the
cone of I would imply an assignment for nodes G, H, and J. If cones C and F were
both required to merely perform constant mappings, then any and all assignments
to A, B, and D would be compatible with any and all assignments to G, H, and J
(because nodes C and F could be assigned constant functions themselves, thereby
making the others irrelevant). This insures that any point on a path to the left will
dominate at the part for I.
867
868
Judd
Thus 2 2(t-l) (the inverse of the probability of this happening) is an upper bound
on the domination distance, i.e. the communication complexity, i.e. the loading
time.
0
More accurately, the complexity is min(c(d, t), f(t), n), where c and f are some
unknown functions. But the operative term here is usually c because d is unlikely
to get so large as to bring f into play (and of course n is unbounded).
The analysis in the proof is sufficient, but it is a far cry from complete. The actual
Markovian process in the sequence of X's is much richer; there are so many events
in the compatibility graph that cause domination to occur that is takes a lot of
careful effort to construct a task that will avoid it!
5
Measuring the Constants
Unfortunately, the very complications that give rise to the pleasant robustness of
the domination event also make it fiendishly difficult to analyze quantitatively. So
to get estimates for the actual constants involved we ran Monte Carlo experiments.
We ran experiments for 4 different cases. The first experiment was to measure
the distance one would have to explore before finding a dominating assignment for
the node labeled A in figure 1. The node function set used was the set of linearly
separable functions. In all experiments, if domination occurred for the degenerate
reason that there were no solutions (paths) at all, then that datum was thrown out
and the run was restarted with a different seed.
Figure 2 reports the constants for the four cases. There is one curve for each
experiment. The abscissa represents t, the size of the task. The ordinate is the
number of support cones that must be consulted before domination can be expected
to occur. All points given are the average of at least 500 trials. Since t is an integer
the data should not have been interpolated between points, but they are easier to
see as connected lines. The solid line (labeled LSA) is for the case just described.
It has a bell shape, reflecting three facts:
? when the task is very small almost every choice of node function for one node
is compatible with choices for the neighbouring nodes.
? when the task is very large, there so many constraints on what a node must
compute that it is easy to resolve what that should be without going far afield.
? when the task is intermediate-sized, the problem is harder.
Note the very low distances involved-even the peak of the curve is well below 2,
so nowhere would you expect to have to pass data more than 2 support cones away.
Although this worst-expected-case would surely be larger for deeper nets, current
work is attempting to see how badly this would scale with depth (larger d).
The curve labeled LUA is for the case where all Boolean functions are used as the
node function set. Note that it is significantly higher in the region 6 < t < 12. The
implication is that although the node function set being used here is a superset of
the linearly separable functions, it takes more computation at loading time to be
able to exploit that extra power.
Constant-Time Loading of Shallow I-Dimensional Networks
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Figure 2: Measured Domination Distances.
The curve labeled LSB shows the expected distance one has to explore before finding
a dominating assignment for the node labelled B in figure 1. The node function
set used was the set of linearly separable functions, Note that it is everywhere
higher that the LSA curve, indicating that the difficulty of settling on a correct
node function for a second-layer node is somewhat higher than finding one for a
first-layer node.
Finally, there is a curve for node B when all Boolean functions are used (LUB), It
is generally higher than when just linearly separable functions are used, but not so
markedly so as in the case of node A.
6
Conclusions
The model of computation used here is much more biologically relevant than the
ones previously used for complexity results, but the algorithm used here runs in an
off-line "batch mode" (i.e. it has all the data before it starts processing). This has
an unbiological nature, but no more so than the customary connectionist habit of
repeating the data many times.
869
870
Judd
A weakness of our analysis is that (as formulated here) it is only for discrete node
functions, exact answers, and noise-free data. Extensions for any of these additional
difficulties may be possible, and the bell shape of the curves should survive.
The peculiarities of the regular 3-layer network examined here may appear restrictive, but it was taken as an example only; what is really implied by the term "l-D"
is only that the bandwidth of the SCI graph for the architecture be bounded (see
[J ud90] for definitions). This constraint allows several degrees of freedom in choosing the architecture, but domination is such a robust combinatoric event that the
essential observation about bell-shaped curves made in this paper will persist even
in the face of large changes from these examples.
We suggest that whatever architectures and node function sets a designer cares to
use, the notion of domination distance will help reveal important computational
characteristics of the design.
Acknowledgements
Thanks go to Siemens and CalTech for wads of computer time.
References
[Jud88] J. S. Judd. On the complexity ofloading shallow neural networks. Journal
of Complexity, September 1988. Special issue on Neural Computation, in
press.
[Jud90] J. Stephen Judd. Neural Network Design and the Complexity of Learning.
MIT Press, Cambridge, Massachusetts, 1990.
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3,944 | 4,570 | Memorability of Image Regions
Aditya Khosla
Jianxiong Xiao
Antonio Torralba
Aude Oliva
Massachusetts Institute of Technology
{khosla,xiao,torralba,oliva}@csail.mit.edu
Abstract
While long term human visual memory can store a remarkable amount of visual
information, it tends to degrade over time. Recent works have shown that image
memorability is an intrinsic property of an image that can be reliably estimated
using state-of-the-art image features and machine learning algorithms. However,
the class of features and image information that is forgotten has not been explored
yet. In this work, we propose a probabilistic framework that models how and
which local regions from an image may be forgotten using a data-driven approach
that combines local and global images features. The model automatically discovers memorability maps of individual images without any human annotation. We
incorporate multiple image region attributes in our algorithm, leading to improved
memorability prediction of images as compared to previous works.
1
Introduction
Human long-term memory can store a remarkable amount of visual information and remember thousands of different pictures even after seeing each of them only once [25, 1]. However, it appears to
be the fate of visual memories that they degrade [13, 30]. While most of the work in visual cognition
has examined how people forget for general classes of visual or verbal stimuli [30], little work has
looked at which image information is forgotten and which is retained. Does all visual information
fade alike? Are there some features, image regions or objects that are forgotten more easily than
others? Inspired by work in visual cognition showing that humans selectively forget some objects
and regions from an image while retaining others [22], we propose a novel probabilistic framework
for modeling image memorability, based on the fading of local image information.
Recent work on image memorability [6, 7, 12] has shown that there are large differences between
the memorabilities of different images, and these differences are consistent across context and observers, suggesting that memory differences are intrinsic to the images themselves. Using machine
learning tools such as support vector regression and a fully annotated dataset of images with human memorability scores, Isola et al [7] show that an automatic image ranking algorithm matches
individual image memory scores quite well: with dynamic scenes with people interacting as most
memorable, static indoor environments and human-scale objects as somewhat less memorable, and
outdoor vistas as forgettable. In addition, using manual annotation, Isola et al. quantified the contribution of segmented regions to the image memorability score, creating a memorability map for each
individual image that identifies objects that are correlated with high or low memorability scores.
However, this previous work did not attempt to discover in an automatic fashion which part of the
image is memorable and which regions are forgettable.
In this paper, we introduce a novel framework for predicting image memorability that is able to
account for how memorability of image regions and different types of features fade over time, offering memorability maps that are more interpretable than [7]. The current work offers three original
contributions: (1) a probabilistic model that simulates the forgetting local image regions, (2) the
automatic discovery of memorability maps of individual images that reveal which regions are memorable/forgettable, and (3) an improved overall image memorability prediction from [7], using an
automatic, data-driven approach combining local and global images features.
1
Original Image!
External!
Representation!
Internal!
Representation!
+"
#"
+"
?,"?"
#"
#"
+"
Internal Image!
Memory !
+" Noisy
+"
Process!
+"
#"
~
vj!
vj!!
Figure 1: Overview of our probabilistic framework. This figure illustrates a possible external or
?observed? representation of an image. The conversion to an internal representation in memory can
be thought of as a noisy process where some elements of the image are changed probabilistically as
described by ? and ? (Sec. 3.1). The image on the right illustrates a possible internal representation:
the green and blue regions remain unchanged, while the red region is forgotten and the pink region
is hallucinated. Note that the internal representation cannot be observed and is only shown here for
illustrating the framework.
2
Related work
Large scale visual memory experiments [26, 25, 1, 13, 14, 28] have shown that humans can remember specific images they have seen among thousands of images, hours to days later, even after
being exposed to each picture only once. In addition, humans seem to have a massive capacity in
long term memory to store specific details about these images, like remembering whether the glass
of orange juice they saw thousands of images earlier was full or half full [1] or which specific door
picture they saw after being exposed to hundreds of pictures of doors [28].
However, not all images are equally memorable as shown by the Memory Game experiment described in [7, 12], and importantly, not all kinds of local information are equally retained from an
image: on average, observers will more likely remember visual details attached to objects that have
a specific semantic label or a distinctive interpretation (for example observers will remember different types of cars by tagging each car with a different brand name, but would more likely confuse
different types of apples, which only differ by their color [14]). This suggests that different features,
objects and regions in an image may have themselves different memorability status: indeed, works
by Isola et al [7, 6] have shown that different individual features, objects, local regions and attributes
are correlated with image that are highly memorable or forgettable. For instance, indoor spaces,
pictures containing people, particularly if their face is visible, close up views on objects, animals,
are more memorable than buildings, pictures of natural landscapes, and natural surfaces in general
(like mountains, grass, field). However, to date, there is no work which has attempted to predict
which local information from an image is memorable or forgettable, in an automatic manner.
3
Modeling memorability using image regions
We propose to predict memorability using a noisy memory process of encoding images in our memory, illustrated in Fig. 1. In our setting, an image consists of different types of image regions and
features. After a delay between the first and second presentation of an image, people are likely to
remember some image regions and objects more than others. For example, as shown in [7], people
and close up views on objects tend to be more memorable than natural objects and regions of landscapes, suggesting for instance that an image region containing a person is less likely to be forgotten
than an image region containing a tree. It is well established that stored visual information decays
over time [30, 31, 14], which can be represented in a model by a novel image vector with missing
global and local information. We postulate that the farther the stored representation of the image is
from its veridical representation, the less likely it is to be remembered.
Here, we propose to model this noisy memorability process in a probabilistic framework. We assume
that the representation of an image is composed of image regions where different regions of an
2
image correspond to different sets of objects. These regions have different probabilities of being
forgotten and some regions have a probability of being imagined or hallucinated. We postulate that
the likelihood of an image to be remembered depends on the distance between the initial image
representation and its internal degraded version. An image with a larger distance to the internal
representation is more likely to be forgotten, thereby the image should have a lower memorability
score. In our algorithm, we model this probabilistic process and show its effectiveness at predicting
image memorability and at producing interpretable memorability maps.
3.1
Formulation
Given some image Ij , we define its representation vj and v?j as the external and internal representation of the image respectively. The external representation refers to the original image which is
observed, while internal representation refers to the noisy representation of the same image that is
stored in the observer?s memory. Assume that there are N types of regions or objects an image can
contain. We define vj ? {0, 1}N as a binary vector of size N containing a 1 at index n when the
corresponding region is present in image Ij and 0 otherwise. Similarly, the internal representation
consists of the same set of region types, but has different presence and absence values as memory is
noisy.
In this setting, one of two things can happen when the external representation of an image is observed: (1) An image region that was shown is forgotten i.e. v?j (i) = 0 when vj (i) = 1, where vj (i)
refers to the ith element of vj , or (2) An image region is hallucinated i.e. an image region that did
not exist in the image is believed to be present. We expect this to happen with different probabilities
for different types of image regions. Therefore, we define two probability vectors ?
~ , ?~ ? [0, 1]N ,
where ?i corresponds to the probability of region type i being forgotten while ?i corresponds to the
probability of hallucinating a region of type i.
Using this representation, we define the distance between the internal and external representation as
Dj = D(vj , v?j ) = ||vj ? v?j ||1 . Dj is inversely proportional to the memorability score of an image
sj ; the higher the distance of an image in the brain from its true representation, the less likely it is
to be remembered, i.e. when D increases, s decreases. Thus, we can compute the expected distance
E(Dj |vj ) of an image as:
E(Dj |vj ) =
N
X
v (i)
?i j
1?vj (i)
? ?i
= vjT ?
~ + (?vj )T ?~
(1)
i=1
This represents the expected number of modifications in v from 1 to 0 (?) or from 0 to 1 (?). Thus,
over all images, we can define the expected distance E(D|v) as
?
?
v1T
?v1T
?
~
?
?
.
.
..
..
E(D|v) = ?
(2)
? ? ~ ?rank ?~s
?
T
T
vM
?vM
where ?i , ?i ? [0, 1] and ?rank represents that the proportionality is only related to the relative
ranking of the image memorability scores, and M is the total number of images. We do not explicitly
predict a memorability score, rather the ranking of scores between images.
The above equation represents a typical ordinal rank regression setting with additional constraints
~ Since we are only interested in the rank, we can rescale the
on the learning parameters ?
~ and ?.
learned parameters to lie between [0, 1], allowing us to use standard solvers such as SVM-Rank [9].
~ cannot be uniquely determined when considering ranking of images alone, and thus
We note that ?
we focus our attention on ?
~ for the rest of this paper.
Implementation details: To generate the region types automatically, we randomly sample rectangular regions of arbitrary width and height from the training images. The regions can be overlapping
with each other. For each region, we compute a particular feature (described in Sec. 4.2), ensuring
the same dimension for all regions of different shapes and sizes (using Bag-of-Words like representations). Then we perform k-means clustering to learn the dictionary of region types as cluster
centroids. The region type is determined by the closest cluster centroid. This method allows us to
3
feature !
memorability maps!
gradient!
?gradient!
overall!
memorability map!
?gradient!
color!
?color!
+"
?color!
pooling!
texture!
?texture!
?texture!
Figure 2: Illustration of multiple feature integration. Refer to Sec. 3.2 for details.
bypass the need for human annotation as done in [7]. The details of the dictionary size and feature
types used are provided in Sec. 4. As we sample overlapping regions, we only encode the presence of a region type by 1 or 0. There may be more than one sampled region that corresponds to a
particular region type.
We evaluate our algorithm on test images by applying a similar method as that on the train images.
In this case, we assume the dictionary of region types is given, and we simply assign the randomly
~ to compute a score.
sampled image regions to region types, and use the learned parameters (~
?, ?)
3.2
Multiple feature integration
We incorporate multiple attributes of each region type such as color, texture and gradient in the form
of image features into our algorithm. Our method is illustrated in Fig. 2. For each attribute, we learn
a separate dictionary of region types. An image region is encoded using each feature dictionary
independently, and the ?
~ , ?~ parameters are learned jointly in our learning algorithm. Subsequently,
we use each set of ?
~ and ?~ for individual features to construct memorability maps that are later
combined using weighted pooling1 to produce an overall memorability map as shown in Fig. 2.
We demonstrate experimentally (Sec. 4) that multiple feature integration helps to improve both the
memorability score prediction and produce visually more consistent memorability maps.
4
Experiments
In this section, we describe the experimental setup and dataset used (Sec. 4.1), provide details about
the region attributes used in our experiments (Sec. 4.2) and describe the experimental results on
the image memorability dataset (Sec. 4.3). Experimental results show that our method outperforms
state-of-the-art methods on this dataset while providing automatic memorability maps of images that
compare favorably to when ground truth segmentation is used.
4.1
Setup
Dataset: We use the dataset proposed by Isola et al. [7] consisting of 2222 images from the SUN
dataset [32]. The images are fully annotated with segmented object regions and randomly sampled
from different scene categories. The images are cropped and resized to 256?256 and a memorability
score corresponding to each image is provided. The memorability score is defined as the percentage
of correct detections by participants in their study.
Performance evaluation: The performance is evaluated using Spearman?s rank correlation(?). We
evaluate our performance on 25 different training/testing splits of the data (same splits as [7]) with
1
We weight the importance of individual features by summing the ?
~ corresponding to the particular feature.
4
an equal number of images for training and testing (1111). The train splits are scored by one half
of the participants and the test splits are scored by the other half of the participants with a human
consistency of ? = 0.75. This can be thought of as an upper bound in the performance of automatic
methods.
Algorithmic details: We sample 2000 patches per image with size 0.2 ? 0.2 to 0.7 ? 0.7 with random
aspect ratios in normalized image coordinates. To speed up convergence of SVM-Rank, we do not
include rank constraints for memorability scores that lie within 0.001 of each other. We find that
this does not affect the performance significantly. The hyperparameter of the SVM-Rank algorithm
is set using 5-fold cross-validation.
4.2
Image region attributes
Our goal is to choose various features as attributes that human likely use to represent image regions.
In this work, we consider six common attributes, namely gradient, color, texture, shape, saliency
and semantic meaning of the images. The attributes are extracted for each region and assigned to a
region type as described in Sec. 3.2 with a dictionary size of 1024 for each feature. For each of the
attributes, we describe our motivation and the method used for extraction.
Gradient: In human vision system, much evidence suggests that retinal ganglion cells and receptive
fields of cells in the visual cortex V1 are essentially gradient-based features. Furthermore, recent
success of many computer vision algorithms [2, 4] also demonstrated the power of such features. In
this work, we use the powerful Histogram of Oriented Gradients (HOG) features for our task. We
densely sample HOG [2] with a cell size of 2x2 at a grid spacing of 4 and learn a dictionary of size
256. The descriptors for a given image region are max-pooled at 2 spatial pyramid levels[15] using
Locality-Constrained Linear Coding (LLC) [29].
Color: Color is an important part of human vision. Color usually has large variations caused by
changes in illumination, shadows, etc, and these variations make the task of robust color description
difficult. Isola et al. [7] show that simple image color features, such as mean hue, saturation and
intensity, only exhibits very weak correlation with memorability. In contrast to this, color has been
shown to yield excellent results in combination with shape features for image classification [11].
Furthermore, many studies show that color names are actually linguistic labels that humans assign
to color spectrum space. In this paper, we use the color names feature [27] to better exploit the
color information. We densely sample the feature at multiple scales (12, 16, 24 and 32) with a grid
spacing of 4. Then we learn a dictionary of size 100 and apply LLC at 2-level spatial pyramid to
obtain the color descriptor for each region.
Texture: We interact with a variety of materials on a daily basis and we constantly assess their texture properties by visual means and tactile touch. To encode visual texture perception information,
we make use of the popular texture features ? Local Binary Pattern [21] (LBP). We use a 2-level
spatial pyramid of non-uniform LBP descriptor.
Saliency: Image saliency is a biologically inspired model to capture the regions that attract more
visual attention and fixation focus [8]. Inspired by this, we extract a saliency value for each pixel
using natural statistics [10]. Then we perform average pooling at 3-level spatial pyramid to obtain
the descriptor for each region.
Shape: Humans constantly use geometric patterns to determine the similarity between visual entities, and the layout of shapes is directly relevant to mid level representations of the image. We denote
shape as a histogram of local Self-Similarity geometric pattens (SSIM [23]). We densely sample the
SSIM descriptor with a grid spacing of 4 and learn a dictionary of sie 256. The descriptors for a
given image region are max-pooled at 2 spatial pyramid levels using LLC.
Semantic: High-level semantic meaning contained in images has been shown to be strongly correlated to image memorability [7], where manual annotation of object labels lead to great performance
in predicting image memorability. Here, our goal is to design a fully automatic approach to predict
image memorability, while still exploiting the semantic information. Thus, we use the automatic
Object Bank [17] feature to model the presence/absence of various objects in the images. We reduce
the feature dimension by using simple max pooling instead of spatial pyramid pooling.
5
Table 1: Images are sorted into sets according to predictions made on the basis of a variety of
features (denoted by column headings). Average measured memorabilities are reported for each set.
e.g. The Top 20 row reports average measured memorability of the 20 highest predicted images. ?
is the Spearman rank correlation between predictions and measurements.
Top 20
Top 100
Bottom 100
Bottom 20
?
?!
Multiple global features [7]
83%
80%
57%
55%
0.46
Our Global
84%
80%
56%
53%
0.48
Our Local
83%
80%
57%
54%
0.45
Our Full Model
85%
81%
55%
52%
0.50
Gradient (HOG)!
Semantic (ObjectBank)!
?!
0.048!
0.107!
0.931!
0.909!
Figure 3: Visualization of region types and corresponding ? learned by our algorithm for gradient
and semantic features. The histograms represent the distribution of memorability scores corresponding to the particular region type. We observe that high-scoring images tend to have a small value of
? while low scoring regions have a high value. This corresponds well with the proposed framework.
The color of the bounding boxes corresponds to the memorability score of the image shown (using
a jet color scheme).
4.3
Results
In this section, we evaluate the performance of our model with single and multiple features, and later
explore what the model has learned using memorability maps and the ranking of different types of
image regions.
Single + multiple features: Fig. 6(a) and Tbl. 1 summarize the performance of our algorithm when
using single and multiple features. We compare our results with [7], and find that our algorithm
outperforms the automatic methods from [7] by 4%, and achieve comparable performance to when
ground truth annotation is used. This shows the effectiveness of our method at predicting memorability. Further, we note that our model provides complementary information to global features as
it focuses on local image regions, increasing performance by 2% when combined with our global
features. We use the same set of attributes described in Sec. 4.2 as global features in our model.
The global features are learned independently using SVM-Rank and the predicted score is combined with the predicted scores of our local model in SVM-Rank algorithm. Despite using the same
set of features, we are able to obtain performance gain suggesting that our algorithm is effective at
capturing local information in the image that was overlooked by the global features.
Memorability maps: We obtain memorability maps using max-pooling of the ? from different
image regions. Fig. 4 shows the memorability maps obtained when using different features and
the overall memorability map when combining multiple features. Despite using no annotation, the
learned maps are similar to those obtained using ground truth objects and segments. From the
images shown, we observe that there is no single attribute that is always effective at producing
memorability maps, but the combination of the attributes leads to a significantly improved version.
We show additional results in Fig. 5.
6
Memory! Original !
Score!
Image!
Overall !
memorability map!
Feature !
Ground truth!
memorability maps! segments!
0.900!
1"
2"
3"
4"
5"
6"
high!
0.811!
0.561!
0.406!
1"
2"
3"
4"
5"
6"
1"
2"
3"
1
2
3
4
5
6
4"
5"
6"
low!
1"
2"
3"
4"
5"
6"
Gradient#
Saliency#
Color#
Texture#
Shape#
Semantic!
Figure 4: Visualization of the memorability maps obtained using different features, and the overall
memorability map. Additionally, we also include the memorability map obtained when using ground
truth segmentation on the right. We observe that it resembles our automatically generated maps.
Figure 5: Additional examples of memorability maps generated by our algorithm.
Image region types: In Fig. 3, we rank the image region types by their ? value and visualize the
regions for the corresponding region type when ? is close to 0 or 1. We observe that the region types
are consistent with our intuition of what is memorable from [7]. People often exist in image regions
with low ? (i.e. low probability of being forgotten) while natural scenes and plain backgrounds are
observed in high ?.
Further, we analyze the image region types by computing the standard deviation of the memorability
scores of the image regions that correspond to the particular type. Fig. 6(b) and 6(c) show the
results. The results are encouraging as regions that have high standard deviation tend to have a value
of ? close to 0.5, which means they are not very informative for prediction. The same behavior is
observed for multiple feature types, and we find that the overall performance for individual features
(shown in Fig. 6(a)) corresponds well with the distance of the peaks in Fig. 6(b) from ? = 0.5. This
suggests that our algorithm is effective at learning the regions with high and low probability of being
forgotten as proposed in our framework.
7
80
75
Gradient)
Color)
Texture)
Shape)
Seman4c)
Saliency)
70
0
100
200
300
400
500
600
700
Image rank (N)
800
900
1000
(a) Comparison of results averaged
across the 25 splits. Images are
ranked by predicted memorability
and plotted against the cumulative
average of measured memorability
scores.
?=0"
Standard Deviation"
85
Standard Deviation"
Average memorability for top N ranked images (%)
Other Human [0.75]
Objects and Scenes [0.50]
Our Final Model [0.50]
Global Only [0.48]
Isola et al. [0.46]
Local Only [0.45]
Gradient [0.40]
Shape [0.38]
Semantic [0.37]
Texture [0.34]
Color [0.29]
Saliency [0.28]
?=0.5"
?=1"
(b) Standard deviation of memorability score of all region types averaged across the 25 splits for all
features, sorted by ?. Graphs are
smoothed using a median filter.
?=0"
?=0.5"
?=1"
(c) Standard deviation of region types for Gradient feature
averaged across the 25 splits.
No smoothing is applied in this
case.
Figure 6: Plot of various results and analysis of our method. Fig. 6(b) and Fig. 6(c) are explained in
greater detail in Sec. 4.3
5
Conclusion
With the emergence of large scale photo collections and growing demands in storing, organizing,
interpreting, and summarizing large amount of digital information, it becomes essential to be able
to automatically annotate images on various novel dimensions that are interpretable to human users.
Recently, learning algorithms have been proposed to automatically interpret whether an image is
aesthetically pleasant or not [20, 3], memorable or forgettable [7, 6], and the role that other high
level photographic properties plays in image interpretation (photo quality [19], attractiveness [16],
composition [5, 18], and object importance [24]). Here, we propose a novel probabilistic framework for automatically constructing memorability maps, discovering regions in the image that are
more likely to be memorable or forgettable by human observers. We demonstrate an effective yet
interpretable framework to model the process of forgetting. Future development of such automatic
algorithms of image memorability could have many exciting and far-reaching applications in computer science, graphics, media, designs, gaming and entertainment industries in general.
Acknowledgements
We thank Phillip Isola and the reviewers for helpful discussions. This work is funded by NSF grant
(1016862) to A.O, Google research awards to A.O and A.T, ONR MURI N000141010933 and NSF
Career Award (0747120) to A.T. J.X. is supported by Google U.S./Canada Ph.D. Fellowship in
Computer Vision.
References
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9
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3,945 | 4,571 | Online `1-Dictionary Learning with Application to
Novel Document Detection
Huahua Wang?
University of Minnesota
[email protected]
Shiva Prasad Kasiviswanathan?
General Electric Global Research
[email protected]
Arindam Banerjee?
University of Minnesota
[email protected]
Prem Melville
IBM T.J. Watson Research Center
[email protected]
Abstract
Given their pervasive use, social media, such as Twitter, have become a leading
source of breaking news. A key task in the automated identification of such news
is the detection of novel documents from a voluminous stream of text documents
in a scalable manner. Motivated by this challenge, we introduce the problem of
online `1 -dictionary learning where unlike traditional dictionary learning, which
uses squared loss, the `1 -penalty is used for measuring the reconstruction error.
We present an efficient online algorithm for this problem based on alternating
directions method of multipliers, and establish a sublinear regret bound for this
algorithm. Empirical results on news-stream and Twitter data, shows that this
online `1 -dictionary learning algorithm for novel document detection gives more
than an order of magnitude speedup over the previously known batch algorithm,
without any significant loss in quality of results.
1
Introduction
The high volume and velocity of social media, such as blogs and Twitter, have propelled them to
the forefront as sources of breaking news. On Twitter, it is possible to find the latest updates on
diverse topics, from natural disasters to celebrity deaths; and identifying such emerging topics has
many practical applications, such as in marketing, disease control, and national security [14]. The
key challenge in automatic detection of breaking news, is being able to detect novel documents in
a stream of text; where a document is considered novel if it is ?unlike? documents seen in the past.
Recently, this has been made possible by dictionary learning, which has emerged as a powerful data
representation framework. In dictionary learning each data point y is represented as a sparse linear
combination Ax of dictionary atoms, where A is the dictionary and x is a sparse vector [1, 12]. A
dictionary learning approach can be easily converted into a novel document detection method: let A
be a dictionary representing all documents till time t ? 1, for a new data document y arriving at time
t, if one does not find a sparse combination x of the dictionary atoms, and the best reconstruction
Ax yields a large loss, then y clearly is not well represented by the dictionary A, and is hence novel
compared to documents in the past. At the end of timestep t, the dictionary is updated to represent
all the documents till time t.
Kasiviswanathan et al. [10] presented such a (batch) dictionary learning approach for detecting novel
documents/topics. They used an `1 -penalty on the reconstruction error (instead of squared loss com?
Part of this wok was done while the author was a postdoc at the IBM T.J. Watson Research Center.
H. Wang and A. Banerjee was supported in part by NSF CAREER grant IIS-0953274, NSF grants IIS0916750, 1029711, IIS-0812183, and NASA grant NNX12AQ39A.
?
1
monly used in the dictionary learning literature) as the `1 -penalty has been found to be more effective
for text analysis (see Section 3). They also showed this approach outperforms other techniques, such
as a nearest-neighbor approach popular in the related area of First Story Detection [16]. We build
upon this work, by proposing an efficient algorithm for online dictionary learning with `1 -penalty.
Our online dictionary learning algorithm is based on the online alternating directions method which
was recently proposed by Wang and Banerjee [19] to solve online composite optimization problems
with additional linear equality constraints. Traditional online convex optimization methods such
as [25, 8, 5, 6, 22] require explicit computation of the subgradient making them computationally
expensive to be applied in our high volume text setting, whereas in our algorithm the subgradients
are computed implicitly. The algorithm has simple closed form updates for all steps yielding a fast
and scalable algorithm for updating the dictionary. Under suitable assumptions
(to cope with the
?
non-convexity of the dictionary learning problem), we establish an O( T ) regret bound for the objective, matching the regret bounds of existing methods [25, 5, 6, 22]. Using this online algorithm
for `1 -dictionary learning, we obtain an online algorithm for novel document detection, which we
empirically validate on traditional news-streams as well as streaming data from Twitter. Experimental results show a substantial speedup over the batch `1 -dictionary learning based approach of
Kasiviswanathan et al. [10], without a loss of performance in detecting novel documents.
Related Work. Online convex optimization is an area of active research and for a detailed survey
on the literature we refer the reader to [18]. Online dictionary learning was recently introduced
by Mairal et al. [12] who showed that it provides a scalable approach for handling large dynamic
datasets. They considered an `2 -penalty and showed that their online algorithm converges to the
minimum objective value in the stochastic case (i.e., with distributional assumptions on the data).
However, the ideas proposed in [12] do not translate to the `1 -penalty. The problem of novel document/topics detection was also addressed by a recent work of Saha et al. [17], where they proposed a
non-negative matrix factorization based approach for capturing evolving and novel topics. However,
their algorithm operates over a sliding time window (does not have online regret guarantees) and
works only for `2 -penalty.
2
Preliminaries
Notation. Vectors P
are always column vectors
P and2are denoted by boldface letters. For a matrix Z
its norm, kZk1 = i,j |zij | and kZk2F = ij zij
. For arbitrary real matrices the standard inner
product is defined as hY, Zi = Tr(Y > Z). We use ?max (Z) to denote the largest eigenvalue of
Z > Z. For a scalar r ? R, let sign(r) = 1 if r > 0, ?1 if r < 0, and 0 if r = 0. Define
soft(r, T ) = sign(r) ? max{|r| ? T, 0}. The operators sign and soft are extended to a matrix by
applying it to every entry in the matrix. 0m?n denotes a matrix of all zeros of size m ? n and the
subscript is omitted when the dimension of the represented matrix is clear from the context.
Dictionary Learning Background. Dictionary learning is the problem of estimating a collection
of basis vectors over which a given data collection can be accurately reconstructed, often with sparse
encodings. It falls into a general category of techniques known as matrix factorization. Classic dictionary learning techniques for sparse representation (see [1, 15, 12] and references therein) consider
a finite training set of signalsPP = [p1 , . . . , pn ] ? Rm?n and optimize the empirical cost function
n
which is defined as f (A) = i=1 l(pi , A), where l(?, ?) is a loss function such that l(pi , A) should
be small if A is ?good? at representing the signal pi in a sparse fashion. Here, A ? Rm?k is referred
to as the dictionary. In this paper, we use a `1 -loss function with an `1 -regularization term, and our
l(pi , A) = min kpi ? Axk1 + ?kxk1 , where ? is the regularization parameter.
x
We define the problem of dictionary learning as that of minimizing the empirical cost f (A). In other
words, the dictionary learning is the following optimization problem
n
X
def
min f (A) = f (A, X) = min
l(pi , A) = min kP ? AXk1 + ?kXk1 .
A
A,X
A,X
i=1
For maintaining interpretability of the results, we would additionally require that the A and X matrices be non-negative. To prevent A from being arbitrarily large (which would lead to arbitrarily
small values of X), we add a scaling constant on A as follows. Let A be the convex set of matrices
defined as
A = {A ? Rm?k : A ? 0m?k ?j = 1, . . . , k , kAj k1 ? 1}, where Aj is the jth column in A.
2
We use ?A to denote the Euclidean projection onto the nearest point in the convex set A. The
resulting optimization problem can be written as
min
A?A,X?0
kP ? AXk1 + ?kXk1
(1)
The optimization problem (1) is in general non-convex. But if one of the variables, either A or X is
known, the objective function with respect to the other variable becomes a convex function (in fact,
can be transformed into a linear program).
3
Novel Document Detection Using Dictionary Learning
In this section, we describe the problem of novel document detection and explain how dictionary
learning could be used to tackle this problem. Our problem setup is similar to [10].
Novel Document Detection Task. We assume documents arrive in streams. Let {Pt : Pt ?
Rmt ?nt , t = 1, 2, 3, . . . } denote a sequence of streaming matrices where each column of Pt represents a document arriving at time t. Here, Pt represents the term-document matrix observed at time
t. Each document is represented is some conventional vector space model such as TF-IDF [13].
The t could be at any granularity, e.g., it could be the day that the document arrives. We use nt to
represent the number of documents arriving at time t. We normalize Pt such that each column (document) in Pt has a unit `1 -norm. For simplicity in exposition, we will assume that mt = m for all
t.1 We use the notation P[t] to denote the term-document matrix obtained by vertically concatenating
the matrices P1 , . . . , Pt , i.e., P[t] = [P1 |P2 | . . . |Pt ]. Let Nt be the number of documents arriving at
time ? t, then P[t] ? Rm?Nt . Under this setup, the goal of novel document detection is to identify
documents in Pt that are ?dissimilar? to the documents in P[t?1] .
Sparse Coding to Detect Novel Documents. Let At ? Rm?k represent the dictionary matrix after
time t ? 1; where dictionary At is a good basis to represent of all the documents in P[t?1] . The exact
construction of the dictionary is described later. Now, consider a document y ? Rm appearing at
time t. We say that it admits a sparse representation over At , if y could be ?well? approximated as
a linear combination of few columns from At . Modeling a vector with such a sparse decomposition
is known as sparse coding. In most practical situations it may not be possible to represent y as At x,
e.g., if y has new words which are absent in At . In such cases, one could represent y = At x + e
where e is an unknown noise vector. We consider the following sparse coding formulation
l(y, At ) = min ky ? At xk1 + ?kxk1 .
x?0
(2)
The formulation (2) naturally takes into account both the reconstruction error (with the ky ? At xk1
term) and the complexity of the sparse decomposition (with the kxk1 term). It is quite easy to
transform (2) into a linear program. Hence, it can be solved using a variety of methods. In our
experiments, we use the alternating directions method of multipliers (ADMM) [2] to solve (2).
ADMM has recently gathered significant attention in the machine learning community due to its
wide applicability to a range of learning problems with complex objective functions [2].
We can use sparse coding to detect novel documents as follows. For each document y arriving at
time t, we do the following. First, we solve (2) to check whether y could be well approximated as a
sparse linear combination of the atoms of At . If the objective value l(y, At ) is ?big? then we mark
the document as novel, otherwise we mark the document as non-novel. Since, we have normalized
all documents in Pt to unit `1 -length, the objective values are in the same scale.
Choice of the Error Function. A very common choice of reconstruction error is the `2 -penalty. In
fact, in the presence of isotopic Gaussian noise the `2 -penalty on e = y ? At x gives the maximum
likelihood estimate of x [21, 23]. However, for text documents, the noise vector e rarely satisfies the
Gaussian assumption, as some of its coefficients contain large, impulsive values. For example, in
fields such as politics and sports, a certain term may become suddenly dominant in a discussion [10].
In such cases imposing an `1 -penalty on the error is a better choice than imposing an `2 -penalty
(e.g., recent research [21, 24, 20] have successfully shown the superiority of `1 over `2 penalty for a
1
As new documents come in and new terms are identified, we expand the vocabulary and zero-pad the
previous matrices so that at the current time t, all previous and current documents have a representation over
the same vocabulary space.
3
different but related application domain of face recognition). We empirically validate the superiority
of using the `1 -penalty for novel document detection in Section 5.
Size of the Dictionary. Ideally, in our application setting, changing the size of the dictionary (k)
dynamically with t would lead to a more efficient and effective sparse coding. However, in our
theoretical analysis, we make the simplifying assumption that k is a constant independent of t. In
our experiments, we allow for small increases in the size of the dictionary over time when required.
Batch Algorithm for Novel Document Detection. We now describe a simple batch algorithm
(slightly modified from [10]) for detecting novel documents. The Algorithm BATCH alternates between a novel document detection and a batch dictionary learning step.
Algorithm 1 : BATCH
Input: P[t?1] ? Rm?Nt?1 , Pt = [p1 , . . . , pnt ] ? Rm?nt , At ? Rm?k , ? ? 0, ? ? 0
Novel Document Detection Step:
for j = 1 to nt do
Solve: xj = argminx?0 kpj ? At xk1 + ?kxk1
if kpj ? At xj k1 + ?kxj k1 > ?
Mark pj as novel
Batch Dictionary Learning Step:
Set P[t] ? [P[t?1] | p1 , . . . , pnt ]
Solve: [At+1 , X[t] ] = argminA?A,X?0 kP[t] ? AXk1 + ?kXk1
Batch Dictionary Learning. We now describe the batch dictionary learning step. At time t, the
dictionary learning step is2
[At+1 , X[t] ] = argminA?A,X?0 kP[t] ? AXk1 + ?kXk1 .
(3)
Even though conceptually simple, Algorithm BATCH is computationally inefficient. The bottleneck
comes in the dictionary learning step. As t increases, so does the size of P[t] , so solving (3) becomes
prohibitive even with efficient optimization techniques. To achieve computational efficiency, in [10],
the authors solved an approximation of (3) where in the dictionary learning step they only update
the A?s and not the X?s.3 This leads to faster running times, but because of the approximation, the
quality of the dictionary degrades over time and the performance of the algorithm decreases. In this
paper, we propose an online learning algorithm for (3) and show that this online algorithm is both
computationally efficient and generates good quality dictionaries under reasonable assumptions.
Online `1 -Dictionary Learning
4
In this section, we introduce the online `1 -dictionary learning problem and propose an efficient algorithm for it. The standard goal of online learning is to design algorithms whose regret is sublinear
in time T , since this implies that ?on the average? the algorithm performs as well as the best fixed
strategy in hindsight [18]. Now consider the `1 -dictionary learning problem defined in (3). Since
this problem is non-convex, it may not be possible to design efficient (i.e., polynomial running time)
algorithms that solves it without making any assumptions on either the dictionary (A) or the sparse
code (X). This also means that it may not be possible to design efficient online algorithm with
sublinear regret without making any assumptions on either A or X because an efficient online algorithm with sublinear regret would imply an efficient algorithm for solving (1) in the offline case.
Therefore, we focus on obtaining regret bounds for the dictionary update, assuming that the at each
timestep the sparse codes given to the batch and online algorithms are ?close?. This motivates the
following problem.
Definition 4.1 (Online `1 -Dictionary Learning Problem). At time t, the online algorithm picks
? t+1 ) with Pt+1 ? Rm?n and X
? t+1 ?
A?t+1 ? A. Then, the nature (adversary) reveals (Pt+1 , X
2
In our algorithms, it is quite straightforward to replace the condition A ? A by some other condition
A ? C, where C is some closed non-empty convex set.
3
e[t] = [X
e[t?1] | x1 , . . . , xnt ] where xj ?s are coming from the novel
In particular, define (recursively) X
e[t] k1 .
document detection step at time t. In [10], the dictionary learning step is At+1 = argminA?A kP[t] ? AX
4
Rk?n . The problem is to pick the A?t+1 sequence such that the following regret function is minimized4
T
T
X
X
? t k1 ? min
R(T ) =
kPt ? A?t X
kPt ? AXt k1 ,
A?A
t=1
t=1
? t = Xt + Et and Et is an error matrix dependent on t.
where X
The regret defined above admits the discrepancy between the sparse coding matrices supplied to the
batch and online algorithms through the error matrix. The reason for this generality is because in
our application setting, the sparse coding matrices used for updating the dictionaries of the batch
and online algorithms could be different. We will later establish the conditions on Et ?s under which
we can achieve sublinear regret. All missing proofs and details appear in the full version of the
paper [11].
Online `1 -Dictionary Algorithm
4.1
In this section, we design an algorithm for the online `1 -dictionary learning problem, which we
call Online Inexact ADMM (OIADMM)5 and bound its regret. Firstly note that because of the
non-smooth `1 -norms involved it is computationally expensive to apply standard online learning
algorithms like online gradient descent [25, 8], COMID [6], FOBOS [5], and RDA [22], as they
require computing a costly subgradient at every iteration. The subgradient of kP ? AXk1 at A = A?
is (X ? sign(X > A?> ? P > ))> .
Our algorithm for online `1 -dictionary learning is based on the online alternating direction method
which was recently proposed by Wang et al. [19]. Our algorithm first performs a simple variable
substitution by introducing an equality constraint. The update for each of the resulting variable has
a closed-form solution without the need of estimating the subgradients explicitly.
Algorithm 2 : OIADMM
? t ? Rk?n , ?t ? 0, ?t ? 0
Input: Pt ? Rm?n , A?t ? Rm?k , ?t ? Rm?n , X
e t ?? Pt ? A?t X
?t
?
e t ? ?i + (?t /2)k?
e t ? ?k2
?t+1 = argmin? k?k1 + h?t , ?
F
e t + ?t /?t , 1/?t ))
(? ?t+1 = soft(?
>
e
?
Gt+1 ?? ?(?t /?t + ?t ? ?t+1 )Xt
A?t+1 = argminA?A ?t (hGt+1 , A ? A?t i + (1/2?t )kA ? A?t k2F )
(? A?t+1 = ?A (max{0, A?t ? ?t Gt+1 }))
?
?
?t+1 = ?t + ?t (Pt ? At+1 Xt ? ?t+1 )
Return A?t+1 and ?t+1
The Algorithm OIADMM is simple. Consider the following minimization problem at time t
? t k1 .
min kPt ? AX
A?A
We can rewrite this above minimization problem as:
? t = ?.
min k?k1 such that Pt ? AX
A?A,?
(4)
The augmented Lagrangian of (4) is:
L(A, ?, ?)
=
min
A?A,?
? t ? ?i +
k?k1 + h?, Pt ? AX
2
?t
? t ? ?
Pt ? AX
, (5)
2
F
where ? ? Rm?n is a multiplier and ?t > 0 is a penalty parameter.
4
For ease of presentation and analysis, we will assume that m and n don?t vary with time. One could allow
for changing m and n by carefully adjusting the size of the matrices by zero-padding.
5
The reason for naming it OIADMM is because the algorithm is based on alternating directions method of
multipliers (ADMM) procedure.
5
OIADMM is summarized in Algorithm 2. The algorithm generates a sequence of iterates
{?t , At , ?t }?
t=1 . At each time t, instead of solving (4) completely, it only runs one step ADMM
update of the variables (?t , At , ?t ). The complete analysis of Algorithm 2 is presented in the full
version of the paper [11]. Here, we just summarize the main result in the following theorem.
Theorem 4.2. Let {?t , A?t , ?t } be the sequences generated by the OIADMM procedure and R(T )
be the regret as defined above. Assume the following conditions hold: (i) ?t, the Frobenius norm
of ?k?t k1 is upper bounded by ?, (ii) A?1 = 0m?k , kAopt kF ? D, (iii) ?1 = 0m?n , and (iv) ?t,
?
? t ). Setting ?t, ?t = ? ?m T where ?m = maxt {1/?t }, we have
1/?t ? 2?max (X
D
?
T
?D T X opt
R(T ) ? ?
+
kA Et k1 .
?m
t=1
In the above theorem one could replace ?m by any upper bound on it (i.e., we don?t need to know
?m exactly).
? t ) made
Condition on Et ?s for Sublinear Regret. In a standard online learning setting, the (Pt , X
available to the online learning algorithm will be the same as (Pt , Xt ) made available ?
to the batch
? t = Xt ? Et = 0, yielding a O( T ) regret.
dictionary learning algorithm in hindsight, so that X
PT
More generally, as long as t=1 kEt kp = o(T ) for some suitable p-norm, we get a sublinear regret
bound.6 For example, if {Zt } is a sequence of matrices such that for all t, kZt kp = O(1), then
setting Et = t? Zt , > 0 yields a sublinear regret. This gives a sufficient condition for sublinear
regret, and it is an interesting open problem to extend the analysis to other cases.
Running Time. For the ith column in the dictionary matrix the projection onto A can be done in
O(si log m) time where si is the number of non-zero elements in the ith column using the projection onto `1 -ball algorithm of Duchi et al. [4]. The simplest implementation of OIADMM takes
O(mnk) time at each timestep because of the matrix multiplications involved.
5
Experimental Results
In this section, we present experiments to compare and contrast the performance of `1 -batch and
`1 -online dictionary learning algorithms for the task of novel document detection. We also present
results highlighting the superiority of using an `1 - over an `2 -penalty on the reconstruction error for
this task (validating the discussion in Section 3).
Implementation of BATCH. In our implementation, we grow the dictionary size by ? in each
timestep. Growing the dictionary size is essential for the batch algorithm because as t increases the
number of columns of P[t] also increases, and therefore, a larger dictionary is required to compactly
represent all the documents in P[t] . For solving (3), we use alternative minimization over the variables. The pseudo-code description is given in the full version of the paper [11]. The optimization
problems arising in the sparse coding and dictionary learning steps are solved using ADMM?s.
Online Algorithm for Novel Document Detection. Our online algorithm7 uses the same novel
document detection step as Algorithm BATCH, but dictionary learning is done using OIADMM. For
a pseudo-code description, see full version of the paper [11]. Notice that the sparse coding matrices
?1, . . . , X
? t . If these sequence of
of the Algorithm BATCH, X1 , . . . , Xt could be different from X
matrices are close to each other, then we have a sublinear regret on the objective function.8
Evaluation of Novel Document Detection. For performance evaluation, we assume that documents
in the corpus have been manually identified with a set of topics. For simplicity, we assume that each
document is tagged with the single, most dominant topic that it associates with, which we call the
true topic of that document. We call a document y arriving at time t novel if the true topic of
y has not appeared before the time t. So at time t, given a set of documents, the task of novel
P
P
This follows from H?older?s inequality which gives Tt=1 kAopt Et k1 ? kAopt kq ( Tt=1 kEt kp ) for 1 ?
opt
p, q ? ? and 1/p + 1/q = 1, and by the assuming kA kq is bounded. Here, k ? kp denotes Schatten p-norm.
7
In our experiments, the number of documents introduced in each timestep is almost of the same order, and
hence there is no need to change the size of the dictionary across timesteps for the online algorithm.
8
As noted earlier, we can not do a comparison without making any assumptions.
6
6
document detection is to classify each document as either novel (positive) or non-novel (negative).
For evaluating this classification task, we use the standard Area Under the ROC Curve (AUC) [13].
Performance Evaluation for `1 -Dictionary Learning. We use a simple reconstruction error measure for comparing the dictionaries produced by our `1 -batch and `1 -online algorithms. We want the
dictionary at time t to be a good basis to represent all the documents in P[t] ? Rm?Nt . This leads
us to define the sparse reconstruction error (SRE) of a dictionary A at time t as
def 1
SRE(A) =
min kP[t] ? AXk1 + ?kXk1 .
Nt X?0
A dictionary with a smaller SRE is better on average at sparsely representing the documents in P[t] .
Novel Document Detection using `2 -dictionary learning. To justify the choice of using an `1 penalty (on the reconstruction error) for novel document detection, we performed experiments comparing `1 - vs. `2 -penalty for this task. In the `2 -setting, for the sparse coding step we used a fast
implementation of the LARS algorithm with positivity constraints [7] and the dictionary learning
was done by solving a non-negative matrix factorization problem with additional sparsity constraints
(also known as the non-negative sparse coding problem [9]). A complete pseudo-code description
is given in the full version of the paper [11].9
Experimental Setup. All reported results are based on a Matlab implementation running on a quadcore 2.33 GHz Intel processor with 32GB RAM. The regularization parameter ? is set to 0.1 which
? t ))
yields reasonable sparsities in our experiments. OIADMM parameters ?t is set 1/(2?max (X
(chosen according to Theorem 4.2) and ?t is fixed to 5 (obtained through tuning). The ADMM
parameters for the sparse coding and batch dictionary learning steps are set as suggested in [10]
(refer to the full version [11]). In the batch algorithms, we grow the dictionary sizes by ? = 10 in
each timestep. The threshold value ? is treated as a tunable parameter.
5.1
Experiments on News Streams
Our first dataset is drawn from the NIST Topic Detection and Tracking (TDT2) corpus which consists of news stories in the first half of 1998. In our evaluation, we used a set of 9000 documents
represented over 19528 terms and distributed into the top 30 TDT2 human-labeled topics over a
period of 27 weeks. We introduce the documents in groups. At timestep 0, we introduce the first
1000 documents and these documents are used for initializing the dictionary. We use an alternative
minimization procedure over the variables of (1) to initialize the dictionary. In these experiments
the size of the initial dictionary k = 200. In each subsequent timestep t ? {1, . . . , 8}, we provide
the batch and online algorithms the same set of 1000 documents. In Figure 1, we present novel
document detection results for those timesteps where at least one novel document was introduced.
Table 1 shows the corresponding AUC numbers. The results show that using an `1 -penalty on the
reconstruction error is better for novel document detection than using an `2 -penalty.
0.5
False Positive Rate
1
0.5
0
0
ONLINE
BATCH?IMPL
L2?BATCH
0.5
False Positive Rate
1
0.5
0
0
ONLINE
BATCH?IMPL
L2?BATCH
0.5
False Positive Rate
1
Timestep 8
1
0.5
0
0
ONLINE
BATCH?IMPL
L2?BATCH
0.5
False Positive Rate
1
True Positive Rate
0
0
ONLINE
BATCH?IMPL
L2?BATCH
Timestep 6
1
True Positive Rate
0.5
Timestep 5
1
True Positive Rate
Timestep 2
True Positive Rate
True Positive Rate
Timestep 1
1
1
0.5
0
0
ONLINE
BATCH?IMPL
L2?BATCH
0.5
False Positive Rate
Figure 1: ROC curves for TDT2 for timesteps where novel documents were introduced.
Comparison of the `1 -online and `1 -batch Algorithms. The `1 -online and `1 -batch algorithms
have almost identical performance in terms of detecting novel documents (see Table 1). However,
the online algorithm is much more computationally efficient. In Figure 2(a), we compare the running
times of these algorithms. As noted earlier, the running time of the batch algorithm goes up as
t increases (as it has to optimize over the entire past). However, the running time of the online
algorithm is independent of the past and only depends on the number of documents introduced
in each timestep (which in this case is always 1000). Therefore, the running time of the online
9
We used the SPAMS package http://spams-devel.gforge.inria.fr/ in our implementation.
7
1
Timestep
1
2
5
6
8
Avg.
No. of Novel Docs.
19
53
116
66
65
AUC `1 -online
0.791
0.694
0.732
0.881
0.757
0.771
No. of Nonnovel Docs.
981
947
884
934
935
AUC `1 -batch
0.815
0.704
0.764
0.898
0.760
0.788
AUC `2 -batch
0.674
0.586
0.601
0.816
0.701
0.676
Table 1: AUC Numbers for ROC Plots in Figure 1.
algorithm is almost the same across different timesteps. As expected the run-time gap between the
`1 -batch and `1 -online algorithms widen as t increases ? in the first timestep the online algorithm is
5.4 times faster, and this rapidly increases to a factor of 11.5 in just 7 timesteps.
200
100
0
0
2
4
Timestep
6
8
Sparse Reconstruction Error Plot for TDT2
1
ONLINE
BATCH?IMPL
0.9
0.8
0.7
0.6
0
2
4
Timestep
6
8
Run Time Plot for Twitter
400
ONLINE
BATCH?IMPL
300
200
100
0
0
5
Timestep
10
Sparse Reconstruction Error (SRE)
300
ONLINE
BATCH?IMPL
L2?BATCH
CPU Running Time (in mins)
CPU Running Time (in mins)
Running Time Plot for TDT2
400
Sparse Reconstruction Error (SRE)
In Figure 2(b), we compare the dictionaries produced by the `1 -batch and `1 -online algorithms
under the SRE metric. In the first few timesteps, the SRE of the dictionaries produced by the online
algorithm is slightly lower than that of the batch algorithm. However, this gets corrected after a few
timesteps and as expected later on the batch algorithm produces better dictionaries.
Sparse Reconstruction Error Plot for Twitter
1
0.9
0.8
0.7
0.6
0.5
0
ONLINE
BATCH?IMPL
5
Timestep
10
(b)
(c)
(d)
(a)
Figure 2: Running time and SRE plots for TDT2 and Twitter datasets.
5.2
Experiments on Twitter
Our second dataset is from an application of monitoring Twitter for Marketing and PR for smartphone and wireless providers. We used the Twitter Decahose to collect a 10% sample of all tweets
(posts) from Sept 15 to Oct 05, 2011. From this, we filtered the tweets relevant to ?Smartphones?
using a scheme presented in [3] which utilizes the Wikipedia ontology to do the filtering. Our dataset
comprises of 127760 tweets over these 21 days and the vocabulary size is 6237 words. We used the
tweets from Sept 15 to 21 (34292 in number) to initialize the dictionaries. Subsequently, at each
timestep, we give as input to both the algorithms all the tweets from a given day (for a period of 14
days between Sept 22 to Oct 05). Since this dataset is unlabeled, we do a quantitative evaluation of
`1 -batch vs. `1 -online algorithms (in terms of SRE) and do a qualitative evaluation of the `1 -online
algorithm for the novel document detection task. Here, the size of the initial dictionary k = 100.
Figure 2(c) shows the running times on the Twitter dataset. At first timestep the online algorithm is
already 10.8 times faster, and this speedup escalates to 18.2 by the 14th timestep. Figure 2(d) shows
the SRE of the dictionaries produced by these algorithms. In this case, the SRE of the dictionaries
produced by the batch algorithm is consistently better than that of the online algorithm, but as the
running time plots suggests this improvement comes at a very steep price.
Date
2011-09-26
2011-09-29
2011-10-03
2011-10-04
2011-10-05
Sample Novel Tweets Detected Using our Online Algorithm
Android powered 56 percent of smartphones sold in the last three months. Sad thing is it can?t lower the rating of ios!
How Windows 8 is faster, lighter and more efficient: WP7 Droid Bionic Android 2.3.4 HP TouchPad white ipods 72
U.S. News: AT&T begins sending throttling warnings to top data hogs: AT&T did away with its unlimited da... #iPhone
Can?t wait for the iphone 4s #letstalkiphone
Everybody put an iPhone up in the air one time #ripstevejobs
Table 2: Sample novel documents detected by our online algorithm.
Table 2 below shows a representative set of novel tweets identified by our online algorithm. Using
a completely automated process (refer to the full version [11]), we are able to detect breaking news
and trending relevant to the smartphone market, such as AT&T throttling data bandwidth, launch of
IPhone 4S, and the death of Steve Jobs.
8
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9
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3,946 | 4,572 | Random function priors for exchangeable arrays with
applications to graphs and relational data
James Robert Lloyd
Department of Engineering
University of Cambridge
Peter Orbanz
Department of Statistics
Columbia University
Zoubin Ghahramani
Department of Engineering
University of Cambridge
Daniel M. Roy
Department of Engineering
University of Cambridge
Abstract
A fundamental problem in the analysis of structured relational data like graphs,
networks, databases, and matrices is to extract a summary of the common structure underlying relations between individual entities. Relational data are typically
encoded in the form of arrays; invariance to the ordering of rows and columns
corresponds to exchangeable arrays. Results in probability theory due to Aldous,
Hoover and Kallenberg show that exchangeable arrays can be represented in terms
of a random measurable function which constitutes the natural model parameter in
a Bayesian model. We obtain a flexible yet simple Bayesian nonparametric model
by placing a Gaussian process prior on the parameter function. Efficient inference
utilises elliptical slice sampling combined with a random sparse approximation
to the Gaussian process. We demonstrate applications of the model to network
data and clarify its relation to models in the literature, several of which emerge as
special cases.
1
Introduction
Structured relational data arises in a variety of contexts, including graph-valued data [e.g. 1, 5],
micro-array data, tensor data [e.g. 27] and collaborative filtering [e.g. 21]. This data is typified by
expressing relations between 2 or more objects (e.g. friendship between a pair of users in a social
network). Pairwise relations can be represented by a 2-dimensional array (a matrix); more generally,
relations between d-tuples are recorded as d-dimensional arrays (d-arrays). We consider Bayesian
models of infinite 2-arrays (Xij )i,j?N , where entries Xij take values in a space X . Each entry
Xij describes the relation between objects i and j. Finite samples?relational measurements for n
objects?are n ? n-arrays. As the sample size increases, the data aggregates into a larger and larger
array. Graph-valued data, for example, corresponds to the case X = {0, 1}. In collaborative filtering
problems, the set of objects is subdivided into two disjoint sets, e.g., users and items.
Latent variable models for such data explain observations by means of an underlying structure or
summary, such as a low-rank approximation to an observed array or an embedding into a Euclidean
space. This structure is formalized as a latent (unobserved) variable. Examples include matrix
factorization [e.g. 4, 21], non-linear generalisations [e.g. 12, 27, 28], block modelling [e.g. 1, 10],
latent distance modelling [e.g. 5] and many others [e.g. 14, 17, 20].
Hoff [4] first noted that a number of parametric latent variable models for relational data are
exchangeable?an applicable assumption whenever the objects in the data have no natural ordering e.g., users in a social network or products in ratings data?and can be cast into the common
functional form guaranteed to exist by results in probability theory. Building on this connection,
1
0
U1
U2
0
0
U1
Pr{Xij = 1}
?
U2
1
1
1
Figure 1: Left: The distribution of any exchangeable random graph with vertex set N and edges
E = (Xij )i,j?N can be characterised by a random function ? : [0, 1]2 ? [0, 1]. Given ?, a graph
can be sampled by generating a uniform random variable Ui for each vertex i, and sampling edges as
Xij ? Bernoulli(?(Ui , Uj )). Middle: A heat map of an example function ?. Right: A 100 ? 100
symmetric adjacency matrix sampled from ?. Only unordered index pairs Xij are sampled in the
symmetric case. Rows and columns have been ordered by increasing value of Ui , rather than i.
we consider nonparametric models for graphs and arrays. Results of Aldous [2], Hoover [6] and
Kallenberg [7] show that random arrays that satisfy an exchangeability property can be represented
in terms of a random function. These representations have been further developed in discrete analysis for the special case of graphs [13]; this case is illustrated in Fig. 1. The results can be regarded as
a generalization of de Finetti?s theorem to array-valued data. Their implication for Bayesian modeling is that we can specify a prior for an exchangeable random array model by specifying a prior on
(measurable) functions. The prior is a distribution on the space of all functions that can arise in the
representation result, and the dimension of this space is infinite. A prior must therefore be nonparametric to have reasonably large support since a parametric prior concentrates on a finite-dimensional
subset. In the following, we model the representing function explicitly using a nonparametric prior.
2
Background: Exchangeable graphs and arrays
A fundamental component of every Bayesian model is a random variable ?, the parameter of the
model, which decouples the data. De Finetti?s theorem [9] characterizes this parameter for random
sequences: Let X1 , X2 , . . . be an infinite sequence of random variables, each taking values in a
common space X . A sequence is called exchangeable if its joint distribution is invariant under
arbitrary permutation of the indices, i.e., if
d
(X1 , X2 , . . .) = (X?(1) , X?(2) , . . .)
for all ? ? S? .
(2.1)
d
Here, = denotes equality in distribution, and S? is the set of all permutations of N that permute
a finite number of elements. De Finetti?s theorem states that, (Xi )i?N is exchangeable if and only
if there exists a random probability measure ? on X such that X1 , X2 , . . . | ? ?iid ?, i.e., conditioned on ?, the observations are independent and ?-distributed. From a statistical perspective, ?
represents common structure in the observed data?and thus a natural target of statistical inference?
whereas P [Xi |?] captures remaining, independent randomness in each observation.
2.1
De Finetti-type representations for random arrays
To specify Bayesian models for graph- or array-valued data, we need a suitable counterpart to de
Finetti?s theorem that is applicable when the random sequences in (2.1) are substituted by random
arrays X = (Xij )i,j?N . For such data, the invariance assumption (2.1) applied to all elements of X
is typically too restrictive: In the graph case Xij ? {0, 1}, for example, the probability of X would
then depend only on the proportion of edges present in the graph, but not on the graph structure.
Instead, we define exchangeability of random 2-arrays in terms of the simultaneous application of a
permutation to rows and columns. More precisely:
Definition 2.1. An array X = (Xij )i,j?N is called an exchangeable array if
d
for every ? ? S? .
(Xij ) = (X?(i)?(j) )
2
(2.2)
Since this weakens the hypothesis (2.1) by demanding invariance only under a subset of all permutations of N2 ?those of the form (i, j) 7? (?(i), ?(j))?we can no longer expect de Finetti?s theorem
to hold. The relevant generalization of the de Finetti theorem to this case is the following:
Theorem 2.2 (Aldous, Hoover). A random 2-array (Xij ) is exchangeable if and only if there is a
random (measurable) function F : [0, 1]3 ? X such that
d
(Xij ) = (F (Ui , Uj , Uij )).
(2.3)
for every collection (Ui )i?N and (Uij )i?j?N of i.i.d. Uniform[0, 1] random variables, where Uji =
Uij for j < i ? N.
2.2
Random graphs
The graph-valued data case X = {0, 1} is of particular interest. Here, the array X, interpreted
as an adjacency matrix, specifies a random graph with vertex set N. For undirected graphs, X is
symmetric. We call a random graph exchangeable if X satisfies (2.2).
For undirected graphs, the representation (2.3) simplifies further: there is a random function
? : [0, 1]2 ? [0, 1], symmetric in its arguments, such that
1 if Uij < ?(Ui , Uj )
F (Ui , Uj , Uij ) :=
(2.4)
0 otherwise
satisfies (2.3). Each variable Ui is associated with a vertex, each variable Uij with an edge. The
representation (2.4) is equivalent to the sampling scheme
U1 , U2 , . . . ?iid Uniform[0, 1]
and
Xij = Xji ? Bernoulli(?(Ui , Uj )) ,
(2.5)
which is illustrated in Fig. 1.
Recent work in discrete analysis shows that any symmetric measurable function [0, 1]2 ? [0, 1]
can be regarded as a (suitably defined) limit of adjacency matrices of graphs of increasing size
[13]?intuitively speaking, as the number of rows and columns increases, the array in Fig. 1 (right)
converges to the heat map in Fig. 1 (middle) (up to a reordering of rows and columns).
2.3
The general case: d-arrays
Theorem 2.2 can in fact be stated in a more general setting than 2-arrays, namely for random darrays, which are collections of random variables of the form (Xi1 ...id )i1 ,...,id ?N . Thus, a sequence
is a 1-array, a matrix a 2-array. A d-array can be interpreted as an encoding of a relation between
d-tuples. In this general case, an analogous theorem holds, but the random function F in (2.3)
is in general more complex: In addition to the collections U{i} and U{ij} of uniform variables, the
representation requires an additional collection U{ij }j?I for every non-empty subset I ? {1, . . . , d};
e.g., U{i1 i3 i4 } for d ? 4 and I = {1, 3, 4}. The representation (2.3) is then substituted by
F : [0, 1]2
d
?1
?? X
and
d
(Xi1 ,...,id ) =
(F (UI1 , . . . , UI(2d ?1) )) .
(2.6)
For d = 1, we recover a version of de Finetti?s theorem. For a discussion of convergence properties
of general arrays similar to those sketched above for random graphs, see [3].
Because we do not explicitly consider the case d > 2 in our experiments, we restrict our presentation of the model to the 2-array-valued case for simplicity. We note, however, that the model and
inference algorithms described in the following extend immediately to general d-array-valued data.
3
Model
To define a Bayesian model for exchangeable graphs or arrays, we start with Theorem 2.2: A distribution on exchangeable arrays can be specified by a distribution on measurable functions [0, 1]3 ?
X . We decompose the function F into two functions ? : [0, 1]2 ? W and H : [0, 1] ? W ? X for
a suitable space W, such that
d
(Xij ) =
(F (Ui , Uj , Uij )) = (H(Uij , ?(Ui , Uj ))) .
3
(3.1)
Such a decomposition always exists?trivially, choose W = [0, 1]2 . The decomposition introduces
a natural hierarchical structure. We initially sample a random function ??the model parameter in
terms of Bayesian statistics?which captures the structure of the underlying graph or array. The
(Ui ) then represent attributes of nodes or objects and H and the array (Uij ) model the remaining
noise in the observed relations.
Model definition. For the purpose of defining a Bayesian model, we will model ? as a continuous
function with a Gaussian process prior. More precisely, we take W = R and consider a zero-mean
Gaussian process prior on CW := C([0, 1]2 , W), the space of continuous functions from[0, 1]2 to
W, with kernel function ? : [0, 1]2 ? [0, 1]2 ? W. The full generative model is then:
? ? GP(0, ?)
U1 , U2 , . . . ?iid Uniform[0, 1]
Xij |Wij ? P [ . |Wij ]
where Wij = ?(Ui , Uj ) .
(3.2)
The parameter space of our the model is the infinite-dimensional space CW . Hence, the model is
nonparametric.
Graphs and real-valued arrays require different choices of P . In either case, the model first generates
the latent array W = (Wij ). Observations are then generated as follows:
Observed data
Sample space
P [Xij ? . |Wij ]
Graph
Real array
X = {0, 1}
X =R
Bernoulli(?(Wij ))
2
)
Normal(Wij , ?X
2
is a noise variance parameter.
where ? is the logistic function, and ?X
The Gaussian process prior favors smooth functions, which will in general result in more interpretable latent space embeddings. Inference in Gaussian processes is a well-understood problem,
and the choice of a Gaussian prior allows us to leverage the full range of inference methods available
for these models.
Discussion of modeling assumptions. In addition to exchangeability, our model assumes (i) that
the function ? is continuous?which implies measurability as in Theorem 2.2 but is a stronger
requirement?and (ii) that its law is Gaussian. Exchangeable, undirected graphs are always representable using a Bernoulli distribution for P [Xij ? . |Wij ]. Hence, in this case, (i) and (ii) are
indeed the only assumptions imposed by the model. In the case of real-valued matrices, the model
additionally assumes that the function H in (3.1) is of the form
d
H(Uij , ?(Ui , Uj )) = ?(Ui , Uj ) + ?ij
where
?ij ?iid Normal(0, ?) .
(3.3)
Another rather subtle assumption arises implicitly when the array X is not symmetric, i.e., not
guaranteed to satisfy Xij = Xji , for example, if X is a directed graph: In Theorem 2.2, the array
(Uij ) is symmetric even if X is not. The randomness in Uij accounts for both Xij and Xji which
means the conditional variables Xij |Wij and Xji |Wji are dependent, and a precise representation
would have to sample (Xij , Xji )|Wij , Wji jointly, a fact our model neglects in (3.2). However, it
can be shown that any exchangeable array can be arbitrarily well approximated by arrays which treat
Xij |Wij and Xji |Wji as independent [8, Thm. 2].
Remark 3.1 (Dense vs. sparse data). The methods described here address random arrays that are
dense, i.e., as the size of an n ? n array increases the number of non-zero entries grows as O(n2 ).
Network data is typically sparse, with O(n) non-zero entries. Density is an immediate
R consequence
of Theorem 2.2: For graph data the asymptotic proportion of present edges is p := ?(x, y)dxdy,
and the graph is hence either empty (for p = 0) or dense (since O(pn2 ) = O(n2 )). Analogous
representation theorems for sparse random graphs are to date an open problem in probability.
4
Related work
Our model has some noteworthy relations to the Gaussian process latent variable model (GPLVM);
a dimensionality-reduction technique [e.g. 11]. GPLVMs can be applied to 2-arrays, but doing so
makes the assumption that either the rows or the columns of the random array are independent [12].
In terms of our model, this corresponds to choosing kernels of the form ?U ? ?, where ? represents
4
a tensor product1 and ? represents an ?identity? kernel (i.e., the corresponding kernel matrix is the
identity matrix). From this perspective, the application of our model to exchangeable real-valued
arrays can be interpreted as a form of co-dimensionality reduction.
For graph data, a related parametric model is the eigenmodel of Hoff [4]. This model, also justified
by exchangeability arguments, approximates an array with a bilinear form, followed by some link
function and conditional probability distribution.
Available nonparametric models include the infinite relational model (IRM) [10], latent feature relational model (LFRM) [14], infinite latent attribute model (ILA) [17] and many others. A recent
development is the sparse matrix-variate Gaussian process blockmodel (SMGB) of Yan et al. [28].
Although not motivated in terms of exchangeability, this model does not impose an independence
assumptions on either rows or columns, in contrast to the GPLVM. The model uses kernels of the
form ?1 ? ?2 ; our work suggests that it may not be necessary to impose tensor product structure, which allows for inference with improved scaling. Roy and Teh [20] present a nonparametric
Bayesian model of relational data that approximates ? by a piece-wise constant function with a
specific hierarchical structure, which is called a Mondrian process in [20].
Some examples of the various available models can be succinctly summarized as follows:
Graph data
Random function model
Latent class [26]
IRM [10]
Latent distance [5]
Eigenmodel [4]
LFRM [14]
ILA [17]
SMGB [28]
?
? GP (0, ?)
Wij
= mUi Uj where Ui ? {1, . . . , K}
Wij
= mUi Uj where Ui ? {1, . . . , ?}
Wij
= ?|Ui ? Uj |
Wij
= Ui0 ?Uj
Wij
= Ui0 ?Uj where Ui ? {0, 1}?
P
(d)
?
Wij
=
d IUid IUjd ?Uid Ujd where Ui ? {0, . . . , ?}
?
? GP (0, ?1 ? ?2 )
Real-valued array data
Random function model
Mondrian process based [20]
PMF [21]
GPLVM [12]
?
?
Wij
?
5
?
=
=
?
GP (0, ?)
piece-wise constant random function
Ui0 Vj
GP (0, ? ? ?)
Posterior computation
We describe Markov Chain Monte Carlo (MCMC) algorithms for generating approximate samples
from the posterior distribution of the model parameters given a partially observed array. Most importantly, we describe a random subset-of-regressors approximation that scales to graphs with hundreds
of nodes. Given the relatively straightforward nature of the proposed algorithms and approximations,
we refer the reader to other papers whenever appropriate.
5.1
Latent space and kernel
Theorem 2.2 is not restricted to the use of uniform distributions for the variables Ui and Uij . The
proof remains unchanged if one replaces the uniform distributions with any non-atomic probability
measure on a Borel space. For the purposes of inference, normal distributions are more convenient,
and we henceforth use U1 , U2 , . . . ?iid N (0, Ir ) for some integer r.
Since we focus on undirected graphical data, we require the symmetry condition Wij = Wji . This
can be achieved by constructing the kernel function in the following way
?(?1 , ?2 )
=
?
? (?1 , ?2 )
=
1
?
? (?1 , ?2 ) + ?
? (?1 , ??2 ) + ? 2 I
2
s2 exp(?|?1 ? ?2 |2 /(2`2 ))
(Symmetry + noise)
(RBF kernel)
(5.1)
(5.2)
1
We define the tensor product of kernel functions as follows: (?U ? ?V )((u1 , v1 ), (u2 , v2 )) =
?U (u1 , u2 ) ? ?V (v1 , v2 ).
5
where ?k = (Uik , Ujk ), ??k = (Ujk , Uik ) and s, `, ? represent a scale factor, length scale and noise
respectively (see [e.g. 19] for a discussion of kernel functions). We collectively denote the kernel
parameters by ?.
5.2
Sampling without approximating the model
In the simpler case of a real-valued array X, we construct an MCMC algorithm over the variables
(U, ?, ?X ) by repeatedly slice sampling [16] from the conditional distributions
?i | ??i , ?X , U, X
?X | ?, U, X
and
Uj | U?j , ?, ?X , X
(5.3)
where ?X is the noise variance parameter used when modelling real valued data introduced in section
3. Let N = |U{i} | denote the number of rows in the observed array, let ? be the set of all pairs
(Ui , Uj ) for all observed relations Xij , let O = |?| denote the number of observed relations, and let
K represent the O ? O kernel matrix between all points in ?. Changes to ? affect every entry in
the kernel matrix K and so, naively, the computation of the Gaussian likelihood of X takes O(O3 )
time. The cubic dependence on O seems unavoidable, and thus this naive algorithm is unusable for
all but small data sets.
5.3
A random subset-of-regressor approximation
To scale the method to larger graphs, we apply a variation of a method known as Subsets-ofRegressors (SoR) [22, 23, 25]. (See [18] for an excellent survey of this and other sparse approximations.) The SoR approximation replaces the infinite dimensional GP with a finite dimensional
approximation. Our approach is to treat both the inputs and outputs of the GP as latent variables.
In particular, we introduce k Gaussian distributed pseudoinputs ? = (?1 , . . . , ?k ) and define target
values Tj = ?(?j ). Writing K?? for the kernel matrix formed from the pseudoinputs ?, we have
(?i ) ?iid N (0, I2r )
and
T | ? ? N (0, K?? ).
(5.4)
The idea of the SoR approximation is to replace Wij with the posterior mean conditioned on (?, T ),
?1
W = K?? K??
T,
(5.5)
where K?? is the kernel matrix between the latent embeddings ? and the pseudoinputs ?. By considering random pseudoinputs, we construct an MCMC analogue of the techniques proposed in [24].
The conditional distribution T | U, ?, ?, (?X ), X is amenable to elliptical slice sampling [15]. All
other random parameters, including the (Ui ), can again be sampled from their full conditional distributions using slice sampling. The sampling algorithms require that one computes expressions
involving (5.5). As a result they cost at most O(k 3 O) time.
6
Experiments
We evaluate the model on three different network data sets. Two of these data sets?the high school
and NIPS co-authorship data?have been extensively analyzed in the literature. The third data set,
a protein interactome, was previously noted by Hoff [4] to be of interest since it exhibits both block
structure and transitivity.
Data set
Recorded data
Vertices
High school
NIPS
Protein
high school social network
densely connected subset of coauthorship network
protein interactome
90
234
230
Reference
e.g. [4]
e.g. [14]
e.g. [4]
We compare performance of our model on these data sets to three other models, probabilistic matrix
factorization (PMF) [21], Hoff?s eigenmodel, and the GPLVM (see also Sec. 4). The models are
chosen for comparability, since they all embed nodes into a Euclidean latent space. Experiments for
all three models were performed using reference implementations by the respective authors.2
2
Implementations are available for PMF at http://www.mit.edu/~rsalakhu/software.html; for the
eignmodel at http://cran.r-project.org/src/contrib/Descriptions/eigenmodel.html; and for the GPLVM at
http://www.cs.man.ac.uk/~neill/collab/ .
6
Figure 2: Protein interactome data. Left: Interactome network. Middle: Sorted adjacency matrix.
The network exhibits stochastic equivalence (visible as block structure in the matrix) and homophily
(concentration of points around the diagonal). Right: Maximum a posteriori estimate of the function
?, corresponding to the function in Fig. 1 (middle).
Model
Method
Iterations [burn-in]
PMF [21]
Eigenmodel [4]
GPLVM [12]
Random function model
stochastic gradient
MCMC
stochastic gradient
MCMC
1000
10000 [250]
20 sweeps
1000 [200]
We use standard normal priors on the latent variables
U and pseudo points ?, and log normal priors for kernel parameters. Parameters are chosen to favor slice
sampling acceptance after a reasonable number of iterations, as evaluated over a range of data sets, summarized in the table on the right. Balancing computational demands, we sampled T 50 times per iteration
whilst all other variables were sampled once per iteration.
Algorithm parameters
author defaults
author defaults
author defaults
(see below)
length scale
scale factor
target noise
U
?
log mean
std
width
1
2
0.1
-
0.5
0.5
0.5
-
0.5
0.5
0.1
4
2
We performed 5-fold cross validation, predicting links in a held out partition given 4 others. Where
the models did not restrict their outputs to values between 0 and 1, we truncated any predictions
lying outside this range. The following table reports average AUC (area under receiver operating
characteristic) for the various models, with numbers for the top performing model set in bold. Significance of results is evaluated by means of a t-test with a p-value of 0.05; results for models not
distinguishable from the top performing model in terms of this t-test are also set in bold.
AUC results
Data set
Latent dimensions
PMF
Eigenmodel
GPLVM
RFM
High school
1
2
0.747
0.742
0.744
0.815
0.792
0.806
0.775
0.827
3
1
NIPS
2
3
1
0.792
0.806
0.782
0.820
0.729
0.789
0.888
0.907
0.789
0.818
0.876
0.914
0.820
0.845
0.883
0.919
0.787
0.805
0.877
0.903
Protein
2
0.810
0.866
0.883
0.910
3
0.841
0.882
0.873
0.912
The random function model outperforms the other models in all tests. We also note that in all
experiments, a single latent dimension suffices to achieve better performance, even when the other
models use additional latent dimensions.
The posterior distribution of ? favors functions defining random array distributions that explain the
data well. In this sense, our model fits a probability distribution. The standard inference methods
for GPLVM and PMF applied to relational data, in contrast, are designed to fit mean squared error,
and should therefore be expected to show stronger performance under a mean squared error metric.
As the following table shows, this is indeed the case.
7
RMSE results
Data set
Latent dimensions
PMF
Eigenmodel
GPLVM
RFM
High school
1
2
0.245
0.244
0.244
0.239
0.242
0.238
0.241
0.234
3
1
NIPS
2
3
1
0.240
0.236
0.239
0.235
0.141
0.141
0.112
0.114
0.135
0.132
0.109
0.111
0.130
0.124
0.106
0.110
0.151
0.149
0.139
0.138
Protein
2
0.142
0.142
0.137
0.136
3
0.139
0.138
0.138
0.136
An arguably more suitable metric is comparison in terms of conditional edge probability i.e.,
P (X{ij} | W{ij} ) for all i, j in the held out data. These cannot, however, be computed in a meaningful manner for models such as PMF and GPLVM, which assign a Gaussian likelihood to data. The
next table hence reports only comparisons to the eigenmodel.
Negative log conditional edge probability3
Data set
High school
NIPS
Latent dimensions
1
2
3
1
2
3
Eigenmodel
RFM
220
205
210
199
200
201
88
65
81
57
75
56
1
96
78
Protein
2
3
92
75
86
75
Remark 6.1 (Model complexity and lengthscales). Figure 2 provides a visualisation of ? when
modeling the protein interactome data using 1 latent dimension. The likelihood of the smooth peak
is sensitive to the lengthscale of the Gaussian process representation of ?. A Gaussian process prior
introduces the assumption that ? is continuous. Continuous functions are dense in the space of measurable functions, i.e., any measurable function can be arbitrarily well approximated by a continuous
one. The assumption of continuity is therefore not restrictive, but rather the lengthscale of the Gaussian process determines the complexity of the model a priori. The nonparametric prior placed on
? allows the posterior to approximate any function if supported by the data, but by sampling the
lengthscale we allow the model to quickly select an appropriate level of complexity.
7
Discussion and conclusions
There has been a tremendous amount of research into modelling matrices, arrays, graphs and relational data, but nonparametric Bayesian modeling of such data is essentially uncharted territory. In
most modelling circumstances, the assumption of exchangeability amongst data objects is natural
and fundamental to the model. In this case, the representation results [2, 6, 7] precisely map out the
scope of possible Bayesian models for exchangeable arrays: Any such model can be interpreted as
a prior on random measurable functions on a suitable space.
Nonparametric Bayesian statistics provides a number of possible priors on random functions, but the
Gaussian process and its modifications are the only well-studied model for almost surely continuous
functions. For this choice of prior, our work provides a general and simple modeling approach
that can be motivated directly by the relevant representation results. The model results in both
interpretable representations for networks, such as a visualisation of a protein interactome, and has
competitive predictive performance on benchmark data.
Acknowledgments
The authors would like to thank David Duvenaud, David Knowles and Konstantina Palla for helpful
discussions. PO was supported by an EPSRC Mathematical Sciences Postdoctoral Research Fellowship (EP/I026827/1). ZG is supported by EPSRC grant EP/I036575/1. DMR is supported by a
Newton International Fellowship and Emmanuel College.
3
The precise calculation implemented is ? log(P (X{ij} | W{ij} )) ? 1000 / (Number of held out edges).
8
References
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9
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3,947 | 4,573 | Scalable Inference of Overlapping Communities
Prem Gopalan David Mimno Sean M. Gerrish Michael J. Freedman David M. Blei
{pgopalan,mimno,sgerrish,mfreed,blei}@cs.princeton.edu
Department of Computer Science
Princeton University
Princeton, NJ 08540
Abstract
We develop a scalable algorithm for posterior inference of overlapping communities in large networks. Our algorithm is based on stochastic variational inference
in the mixed-membership stochastic blockmodel (MMSB). It naturally interleaves
subsampling the network with estimating its community structure. We apply our
algorithm on ten large, real-world networks with up to 60,000 nodes. It converges
several orders of magnitude faster than the state-of-the-art algorithm for MMSB,
finds hundreds of communities in large real-world networks, and detects the true
communities in 280 benchmark networks with equal or better accuracy compared
to other scalable algorithms.
1
Introduction
A central problem in network analysis is to identify communities, groups of related nodes with
dense internal connections and few external connections [1, 2, 3]. Classical methods for community
detection assume that each node participates in a single community [4, 5, 6]. This assumption is
limiting, especially in large real-world networks. For example, a member of a large social network
might belong to overlapping communities of co-workers, neighbors, and school friends.
To address this problem, researchers have developed several methods for detecting overlapping communities in observed networks. These methods include algorithmic approaches [7, 8] and probabilistic models [2, 3, 9, 10]. In this paper, we focus on the mixed-membership stochastic blockmodel
(MMSB) [2], a probabilistic model that allows each node of a network to exhibit a mixture of
communities. The MMSB casts community detection as posterior inference: Given an observed
network, we estimate the posterior community memberships of its nodes.
The MMSB can capture complex community structure and has been adapted in several ways [11,
12]; however, its applications have been limited because its corresponding inference algorithms
have not scaled to large networks [2]. In this work, we develop algorithms for the MMSB that
scale, allowing us to study networks that were previously out of reach for this model. For example,
we analyzed social networks with as many as 60,000 nodes. With our method, we can use the
MMSB to analyze large networks, finding approximate posteriors in minutes with networks for
which the original algorithm takes hours. When compared to other scalable methods for overlapping
community detection, we found that the MMSB gives better predictions of new connections and
more closely recovers ground-truth communities. Further, we can now use the MMSB to compute
descriptive statistics at scale, such as which nodes bridge communities.
The original MMSB algorithm optimizes the variational objective by coordinate ascent, processing
every pair of nodes in each iteration [2]. This algorithm is inefficient, and it quickly becomes
intractable for large networks. In this paper, we develop stochastic optimization algorithms [13, 14]
to fit the variational distribution, where we obtain noisy estimates of the gradient by subsampling
the network.
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Figure 1: Figure 1(a) shows communities (see ?2) discovered in a co-authorship network of 1,600 researchers [16] by an a-MMSB model with 50 communities. The color of author nodes indicates their most
likely posterior community membership. The size of nodes indicates bridgeness [17], a measure of participation in multiple communities. Figure 1(b) shows a graphical model of the a-MMSB. The prior over multinomial
? is a symmetric Dirichlet distribution. Priors over Bernoulli ? are Beta distributions.
Our algorithm alternates between subsampling from the network and adjusting its estimate of the
underlying communities. While this strategy has been used in topic modeling [15], the MMSB
introduces new challenges because the Markov blanket of each node is much larger than that of
a document. Our simple sampler usually selects unconnected nodes (due to sparse real-world networks). We develop better sampling methods that focus more on the informative data in the network,
e.g., the observed links, and thus make inference even faster.
2
Modeling overlapping communities
In this section, we introduce the assortative mixed-membership stochastic blockmodel (a-MMSB),
a statistical model of networks that models nodes participating in multiple communities. The aMMSB is a subclass of the mixed-membership stochastic blockmodel (MMSB) [2].1
Let y denote the observed links of an undirected network, where yab = 1 if nodes a and b are
linked and 0 otherwise. Let K denote the number of communities. Each node a is associated with
community memberships ?a , a distribution over communities; each community is associated with a
community strength ?k ? (0, 1), which captures how tightly its members are linked. The probability
that two nodes are linked is governed by the similarity of their community memberships and the
strength of their shared communities.
We capture these assumptions in the following generative process of a network.
1. For each community k, draw community strength ?k ? Beta(?).
2. For each node a, draw community memberships ?a ? Dirichlet(?).
3. For each pair of nodes a and b,
(a) Draw interaction indicator za?b ? ?a .
(b) Draw interaction indicator za?b ? ?b .
(c) Draw link yab ? Bernoulli(r), where
?k if za?b,k = za?b,k = 1,
r=
if za?b 6= za?b .
(1)
1
We use a subclass of the MMSB models that is appropriate for community detection in undirected networks. In particular, we assume assortativity, i.e., that links imply that nodes are similar. We call this special
case the assortative MMSB or a-MMSB. In ?2 we argue why the a-MMSB is more appropriate for community
detection than the MMSB. We note that our algorithms are immediately applicable to the MMSB as well.
2
Figure 1(b) represents the corresponding joint distribution of hidden and observed variables. The aMMSB defines a single parameter to govern inter-community links. This captures assortativity?if
two nodes are linked, it is likely that the latent community indicators were the same.
The full MMSB differs from the a-MMSB in that the former uses one parameter for each of the
K 2 ordered pairs of communities. When the full MMSB is applied to undirected networks, two
hypotheses compete to explain a link between each pair of nodes: either both nodes exhibit the same
community or they are in different communities that link to each other.
We analyze data with a-MMSB via the posterior distribution over latent variables
p(?1:N , z, ?1:K |y, ?, ?). The posterior lets us form a predictive distribution of unseen links
and measure latent network properties of the observed data. The posterior over ?1:N represents the
community memberships of the nodes, and the posterior over the interaction indicator variables
z identifies link communities in the network [8]. For example, in a social network one member?s
link to another might arise because they are from the same high school while another might arise
because they are co-workers. With an estimate of this latent structure, we can characterize the
network in interesting ways. In Figure 1(a), we sized author nodes according to their expected
posterior bridgeness [17], a measure of participation in multiple communities (see ?5).
3
Stochastic variational inference
Our goal is to compute the posterior distribution p(?1:N , z, ?1:K |y, ?, ?). Exact inference is intractable, so we use variational inference [18]. Traditional variational inference is a coordinate ascent algorithm. In the context of the MMSB (and the a-MMSB), coordinate ascent iterates between
analyzing all O(N 2 ) node pairs and updating the community memberships of the N nodes [2].
In this section, we will derive a stochastic variational inference algorithm. Our algorithm iterates
between sampling random pairs of nodes and updating node memberships. This avoids the periteration O(N 2 ) computation and allows us to scale to large networks.
3.1
Variational inference in a-MMSB
In variational inference, we define a family of distributions over the hidden variables q(?, ?, z) and
find the member of that family that is closest to the true posterior. (Closeness is measured with
KL divergence.) We use the mean-field family, under which each variable is endowed with its own
distribution and its own variational parameter. This allows us to tractably optimize the parameters
to find a local minimum of the KL divergence. For the a-MMSB, the variational distributions are
q(za?b = k) = ?a?b,k ;
q(?a ) = Dirichlet(?a ; ?p );
q(?k ) = Beta(?k ; ?k ).
(2)
The posterior over link community assignments z is parameterized by the per-interaction memberships ?, the node community distributions ? by the community memberships ?, and the link
probability ? by the community strengths ?. Notice that ? is of dimension K ? 2, and ? is of
dimension N ? K.
Minimizing the KL divergence between q and the true posterior is equivalent to optimizing an evidence lower bound (ELBO) L, a bound on the log likelihood of the observations. We obtain this
bound by applying Jensen?s inequality [18] to the data likelihood. The ELBO is
log p(y|?, ?) ? L(y, ?, ?, ?) , Eq [log p(y, ?, z, ?|?, ?)] ? Eq [log q(?, ?, z)].
(3)
The right side of Eq. 3 factorizes to
P
P
P
P
L = k Eq [log p(?k |?k )] ? k Eq [log q(?k |?k )] + n Eq [log p(?n |?)] ? n Eq [log q(?n |?n )]
P
+ a,b Eq [log p(za?b |?a )] + Eq [log p(za?b |?b )]
(4)
P
? a,b Eq [log q(za?b |?a?b )] ? Eq [log q(za?b |?a?b )]
P
+ a,b Eq [log p(yab |za?b , za?b , ?)]
Notice the first line in Eq. 4 contains summations over communities and nodes; we call these global
terms. They relate to the global variables, which are the community strengths ? and per-node
memberships ?. The remaining lines contain summations over all node pairs, which we call local
terms. They depend on both the global and local variables, the latter being the per-interaction
memberships ?. This distinction is important in the stochastic optimization algorithm.
3
3.2
Stochastic optimization
Our goal is to develop a variational inference algorithm that scales to large networks. We will use
stochastic variational inference [14], which optimizes the ELBO with respect to the global variational parameters using stochastic gradient ascent. Stochastic gradient algorithms follow noisy
estimates of the gradient with a decreasing step-size. If the expectation of the noisy gradient is equal
to the gradient and if the step-size decreases according to a certain schedule, then we are guaranteed
convergence to a local optimum [13]. Subsampling the data to form noisy gradients scales inference
as we avoid the expensive all-pairs sums in Eq. 4.
Stochastic variational inference is a coordinate ascent algorithm that iteratively updates local and
global parameters. For each iteration, we first subsample the network and compute optimal local
parameters for the sample, given the current settings of the global parameters. We then update the
global parameters using a stochastic natural gradient2 computed from the subsampled data and local
parameters. We call the first phase the local step and the second phase the global step [14].
The selection of subsamples in each iteration provides a way to plug in a variety of network subsampling algorithms. However, to maintain a correct stochastic optimization algorithm of the variational
objective, the subsampling method must be valid. That is, the natural gradients estimated from the
subsample must be unbiased estimates of the true gradients.
The global step. The global step updates the global community strengths ? and community memberships ? with a stochastic gradient of the ELBO in Eq. 4. Eq. 4 contains summations over all
O(N 2 ) node pairs. Now consider drawing a node pair (a, b) at random from a population distribution g(a, b) over the M = N (N ? 1)/2 node pairs. We can rewrite the ELBO as a random function
of the variational parameters that includes the global terms and the local terms associated only with
(a, b). The expectation of this random function is equal in objective to Eq. 4. For example, the
fourth term in Eq. 4 is rewritten as
P
1
(5)
a,b Eq [log p(yab |za?b , za?b , ?)] = Eg [ g(a,b) Eq [log p(yab |za?b , za?b , ?)]]
Evaluating the rewritten Eq. 4 for a node pair sampled from g gives a noisy but unbiased estimate of
the ELBO. Following [15], the stochastic natural gradients computed from a sample pair (a, b) are
t
??a,k
=?k +
1
t
g(a,b) ?a?b,k
??tk,i =?k,i +
t?1
? ?a,k
1
g(a,b) ?a?b,k
? ?a?b,k ? yab,i ? ?t?1
k,i ,
(6)
(7)
where yab,0 = yab , and yab,1 = 1?yab . In practice, we sample a ?mini-batch? S of pairs per update,
to reduce noise.
The intuition behind the above update is analogous to Online LDA [15]. When a single pair (a, b) is
sampled, we are computing the setting of ? that would be optimal (given ?t ) if our entire network
were a multigraph consisting of the interaction between a and b repeated 1/g(a, b) times.
Our algorithm has assumed that the subset of node pairs S are sampled independently. We can relax
this assumption by defining a distribution over predefined sets of links. These sets can be defined
using prior information about the pairs, such as network topology, which lets us take advantage of
more sophisticated sampling strategies. For example, we can define a set for each node, with each
set consisting of the node?s adjacent links or non-links. Each iteration we set S to one of these sets
sampled at random from the N sets.
In order to ensure that set-based sampling results in unbiased gradients, we specify two constraints
on sets. First, we assume that the union of these sets s is the total set of all node pairs, U : U = ?i si .
Second, we assume that every pair (a, b) occurs in some constant number of sets c and that c ? 1.
With these conditions satisfied, we can again rewrite Eq. 4 as the sum over its global terms and an
expectation over the local terms. Let h(t) be a distribution over predefined sets of node pairs. For
example, the fourth term in Eq. 4 can be rewritten using
P
P
1 1
(8)
(a,b)?st Eq [log p(yab |za?b , za?b , ?)]]
a,b Eq [log p(yab |za?b , za?b , ?)] = Eh [ c h(t)
2
Stochastic variational inference uses natural gradients [19] of the ELBO. Computing natural gradients
(along with subsampling) leads to scalable variational inference algorithms [14].
4
Algorithm 1 Stochastic a-MMSB
K
1: Initialize ? = (?n )N
n=1 , ? = (?k )k=1 randomly.
2: while convergence criteria is not met do
3:
Sample a subset S of node pairs.
4:
L-step: Optimize (?a?b , ?a?b ) ?(a, b) ? S
5:
Compute the natural gradients ?? tn ?n, ??tk ?k
6:
G-step: Update (?, ?) using Eq. 9.
7:
Set ?t = (?0 + t)?? ; t ? t + 1.
8: end while
The natural gradient of the random functions in Eq. 5 and Eq. 8 with respect to the global variational
parameters (?, ?) is a noisy but unbiased estimate of the natural gradient of the ELBO in Eq. 4.
However we subsample, the global step follows the noisy gradient with an appropriate step-size,
? ? ? + ?t ?? t ; ? ? ? + ?t ??t .
(9)
P 2
P
We require that t ?t < ? and t ?t = ? for convergence to a local optimum [13]. We set
?t , (?0 + t)?? , where ? ? (0.5, 1] is the learning rate and ?0 ? 0 downweights early iterations.
The local step. The local step optimizes the interaction parameters ? with respect to a subsample
of the network. Recall that there is a per-interaction membership parameter for each node pair?
?a?b and ?a?b ?representing the posterior approximation of which communities are active in determining whether there is a link. We optimize these parameters in parallel. The update for ?a?b
given ya,b is
t?1
Eq [log(1 ? ?k )]
?ta?b,k |y = 0 ? exp{Eq [log ?a,k ] + ?a?b,k
t?1
?ta?b,k |y = 1 ? exp{Eq [log ?a,k ] + ?a?b,k
Eq [log ?k ] + (1 ? ?t?1
a?b,k ) log .
(10)
The updates for ?a?b are symmetric. This is natural gradient ascent with a step-size of one.
We present the full Stochastic a-MMSB algorithm in Algorithm 1. Each iteration subsamples the
network and computes the local and global updates. We have derived this algorithm with node pairs
sampled from arbitrary population distributions g(a, b) or h(t). One advantage of this approach
is that we can explore various subsampling techniques without compromising the correctness of
Algorithm 1. We will discuss and study sampling methods in ?3.3 and ?5. First, however, we
discuss convergence and complexity.
Held-out sets and convergence criteria. We stop training on a network (the training set) when the
average change in expected log likelihood on held-out data (the validation set) is less than 0.001%.
The test and validation sets used in ?5 have equal parts links and non-links, selected randomly
from the network. A 50% links validation set poorly represents the severe class imbalance between
links and non-links in real-world networks. However, a validation set matching the network sparsity
would have too few links. Therefore, we compute the validation log likelihood at network sparsity by
reweighting the average link and non-link log likelihood (estimated from the 50% links validation
set) by their respective proportions in the network. We use a separate validation set to choose
learning parameters and study sensitivity to K.
Per-iteration complexity. Our L-step can be computed in O(nK), where n is the number of node
pairs sampled in each iteration. This is unlike MMSB, where the ? updates incur a cost quadratic
in K. Step 6 requires that all nodes must be updated in each iteration. The time for a G-step in
Algorithm 1 is O(N K) and the total memory required is O(N K).
3.3
Sampling strategies
Our algorithm allows us flexibility around how the subset of pairs is sampled, as long as the expectation of the stochastic gradient is equal to the true gradient. There are several ways we can take
advantage of this. We can sample based on informative pairs of nodes, i.e., ones that help us better
assess the community structure. We can also subsample to make data processing easier, for example, to accomodate a stream of links. Finally, large, real-world networks are often sparse, with links
5
accounting for less than 0.1% of all node pairs (see Figure 2). While we should not ignore non-links,
it may help to give preferential attention to links. These intuitions are captured in the following four
subsampling methods.
Random pair sampling. The simplest method is to sample node pairs uniformly at random. This
1
method is an instance of independent pair sampling, with g(a, b) (used in Eq. 5) equal to N (N ?1)/2
.
Random node sampling. This method focuses on local neighborhoods of the network. A set
consists of all the pairs that involve one of the N nodes. At each iteration, we sample a set uniformly
at random from the N sets, so h(t) (used in Eq. 8) is N1 . Since each pair involves two nodes, each
link appears in two sets, so c (also used in Eq. 8) is 2. By reweighting the terms corresponding to
pairs in the sampled set, we maintain a correct stochastic optimization.
Stratified random pair sampling. This method samples links independently, but focuses on observed links. We divide the M node pairs into two strata: links and non-links. Each iteration
either samples a mini-batch of links or samples a mini-batch of non-links. If the non-link stratum is
sampled, and N0 is the estimated total number of non-links, then
(
1
if yab = 0,
g(a, b) = N0
(11)
0
if yab = 1
The population distribution when the link strata is sampled is symmetric.
Stratified random node sampling. This method combines set-based sampling and stratified sampling to focus on observed links in local neighborhoods. For each node we define a ?link set?
consisting of all its links, and m ?non-link sets? that partition its non-links. Since the number of
non-links associated with each node is usually large, dividing them into many sets allows the computation in each iteration to be fast. At each iteration, we first select a random node and either select
its link set or sample one of its m non-link sets, uniformly at random. To compute Eq. 8 we define
the number of sets that contain each pair, c = 2, and the population distribution over sets
1
if t is a link set,
h(t) = 2N1
(12)
if t is a non-link set.
2N m
Stratified random node sampling gives the best gains in convergence speed (see ?5).
4
Related work
Newman et al. [3] described a model of overlapping communities in networks (?the Poisson model?)
where the number of links between two nodes is a Poisson random variable. Recently, other researchers have proposed latent feature network models [20, 21] that compute the probabilities of
links based on the interactions between binary features associated with each node. Efficient inference algorithms for these models exploit model-specific approximations that allow scaling in the
number of links. These ideas do not extend to the MMSB. Further, these algorithms do not explicitly
leverage network sampling. In contrast, the ideas in Algorithm 1 apply to a number of models [14].
It subsamples both links and non-links in an inner loop for scalability.
Other scalable algorithms include Clique Percolation (CP) [7] and Link Clustering (LC) [8], which
are based on heuristic clique-finding and hierarchical clustering, respectively. These methods are
fast in practice, although the underlying problem is NP-complete. Further, because they are not
statistical models, there is no clear mechanism for predicting new observations or model checking.
In the next section we will compare our method to these alternative scalable methods. Compared
to the Poisson model, we will show that the MMSB gives better predictions. Compared to CP and
LC, which do not provide predictions, we will show that the MMSB more reliably recovers the true
community structure.
5
Empirical study
In this section, we evaluate the efficiency and accuracy of Stochastic a-MMSB (AM). First, we
evaluate its efficiency on 10 real-world networks. Second, we demonstrate that stratified sampling
6
Figure 2: Network datasets. N is the number of nodes, K max is the maximum number of communities
analyzed and d is the percent of node pairs that are links. The time until convergence for the different methods
stoch
are Tcstoch and Tcbatch , while the time required for 90% of the perplexity at a-MMSB?s convergence is T90%
.
N
K max
d(%)
stoch
T90%
Tcstoch
Tcbatch
TYPE
1.1K
1.6K
5.2K
9.9K
12K
18.7K
27.8K
37K
40.4K
58.2K
19
100
300
32
32
32
512
158
300
64
1.2
0.3
0.1
0.05
0.16
0.11
0.09
0.03
0.02
0.01
1.7m
7.2m
2.3h
7.3h
36m
13.8h
8d
1.5d
4.6d
8d
3.4m
11.7m
4h
8.7h
2.8h
22.1h
10.3d
2.5d
5.2d
9.5d
40.5m
2.2h
> 29h
> 67h
> 67h
> 67h
-
TRANSP.
COLLAB .
COLLAB .
COLLAB .
COLLAB .
COLLAB .
CITE
EMAIL
COLLAB .
SOCIAL
time (hours)
10 20 30 40 50 60
Perplexity
?
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time (hours)
40
20
10
0
10 20 30 40 50 60
3
4
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0
5
stratified node
node
stratified pair
pair
10
15
time (hours)
10 20 30 40 50 60
hep?th2 27K
20
Perplexity
15 20 25 30 35
25
15
20
Perplexity
10
stratified node
node
stratified pair
pair
?
0
0
astro?ph 19K
?
?
astro?ph 19K
30
40
30
20
0
hep?ph 12K
SOURCE
[22]
[16]
[23]
[23]
[23]
[23]
[23],[24]
[25]
[26]
[27]
0
10 20 30 40 50 60
hep?ph 12K
15 20 25 30 35 40
0
hep?th 10K
10
10
20
30
online
batch
0
Perplexity
40
gravity 5K
0
US - AIR
NETSCIENCE
RELATIVITY
HEP - TH
HEP - PH
ASTRO - PH
HEP - TH 2
ENRON
COND - MAT
BRIGHTKITE
0 10 20 30 40 50 60
DATA SET
25
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stratified node
node
stratified pair
pair
20
30
40
time (hours)
Figure 3: Stochastic a-MMSB (with random pair sampling) scales better and finds communities as good as
batch a-MMSB on real networks (Top). Stratified random node sampling is an order of magnitude faster than
other sampling methods on the hep-ph, astro-ph and hep-th2 networks (Bottom).
significantly improves convergence speed on real networks. Third, we compare our algorithm with
leading algorithms in terms of accuracy on benchmark graphs and ability to predict links.
We measure convergence by computing the link prediction accuracy on a validation set. We set
aside a validation and a test set, each having 10% of the network links and an equal number of
non-links (see ?3.2). We approximate the probability that a link exists between two nodes using
posterior expectations of ? and ?. We then calculate perplexity, which is the exponential of the
average predictive log likelihood of held-out node pairs.
For random pair and stratified random pair sampling, we use a mini-batch size S = N/2 for graphs
with N nodes. For the stratified random node sampling, we set the number of non-link sets m = 10.
Based on experiments, we set the parameters ? = 0.5 and ?0 = 1024. We set hyperparameters ? =
1/K and {?1 , ?0 } proportional to the expected number of links and non-links in each community.
We implemented all algorithms in C++.
Comparing scalability to batch algorithms. AM is an order of magnitude faster than standard
batch inference for a-MMSB [2]. Figure 2 shows the time to convergence for four networks3 of
varying types, node sizes N and sparsity d. Figure 3 shows test perplexity for batch vs. stochastic
inference. For many networks, AM learns rapidly during the early iterations, achieving 90% of the
converged perplexity in less than 70% of the full convergence time. For all but the two smallest
networks, batch inference did not converge within the allotted time. AM lets us efficiently fit a
mixed-membership model to large networks.
Comparing sampling methods. Figure 3 shows that stratified random node sampling converges an
order of magnitude faster than random node sampling. It is statistically more efficient because the
observations in each iteration include all the links of a node and a random sample of its non-links.
3
Following [1], we treat the directed citation network hep-th2 as an undirected network.
7
sparse
training
dense
0.6
0.5
0.4
0.3
0.2
0.1
0.0
NMI
NMI
0.5
0.4
0.3
0.2
0.1
0.0
AM PM LC CP
AM PM LC CP
AM PM LC CP
K=5
training
test
K=40
?2
?4
?6
?8
?10
PM
AM PM LC CP
K=20
AM
PM
AM
PM
AM
held?out log likelihood
10% noisy
held?out log likelihood
0 noise
K=5
K=20
test
K=40
?1
?2
?3
?4
?5
PM
AM
PM
AM
PM
AM
(a)
(b)
(c)
(d)
Figure 4: Figures (a) and (b) show that Stochastic a-MMSB (AM) outperforms the Poisson model (PM),
Clique Percolation (CP), and Link Clustering (LC) in accurately recovering overlapping communities in 280
benchmark networks [28]. Each figure shows results on a binary partition of the 280 networks. Accuracy
is measured using normalized mutual information (NMI) [28]; error bars denote the 95% confidence interval
around the mean NMI. Figures (c) and (d) show that a-MMSB generalizes to new data better than PM on the
netscience and us-air network, respectively. Each algorithm was run with 10 random initializations per K.
Figure 3 also shows that stratified random pair sampling converges ?1x?2x faster than random pair
sampling.
Comparing accuracy to scalable algorithms. AM can recover communities with equal or better
accuracy than the best scalable algorithms: the Poisson model (PM) [3], Clique percolation (CP) [7]
and Link clustering (LC) [8]. We measure the ability of algorithms to recover overlapping communities in synthetic networks generated by the benchmark tool [28].4 Our synthetic networks reflect
real-world networks by modeling noisy links and by varying community densities from sparse to
dense. We evaluate using normalized mutual information (NMI) between discovered communities
and the ground truth communities [28]. We ran PM and a-MMSB until validation log likelihood
changed by less than 0.001%. CP and LC are deterministic, but results vary between parameter
settings. We report the best solution for each model.5
Figure 4 shows results for the 280 synthetic networks split in two ways. AM outperforms PM,
LC, and CP on noisy networks and networks with sparse communities, and it matches the best
performance in the noiseless case and the dense case. CP performs best on networks with dense
communities?they tend to have more k-cliques?but with a larger margin of error than AM.
Comparing predictive accuracy to PM. Stochastic a-MMSB also beats PM [3], the best scalable probabilistic model of overlapping communities, in predictive accuracy. On two networks, we
evaluated both algorithms? ability to predict held out links and non-links. We ran both PM and aMMSB until their validation log likelihood changed less than 0.001%. Figures 4(c) and 4(d) show
training and testing likelihood. PM overfits, while the a-MMSB generalizes well.
Using the a-MMSB as an exploratory tool. AM opens the door to large-scale exploratory analysis
of real-world networks. In addition to the co-authorship network in Figure 1(a), we analyzed the
?cond-mat? collaboration network [26] with the number of communities set to 300. This network
contains 40,421 scientists and 175,693 links. In the supplement, we visualized the top authors in
the network by a measure of their participation in different communities (bridgeness [17]). Finding
such bridging nodes in a network is an important task in disease prevention and marketing.
Acknowledgments
D.M. Blei is supported by ONR N00014-11-1-0651, NSF CAREER 0745520, AFOSR FA9550-091-0668, the Alfred P. Sloan foundation, and a grant from Google.
4
We
generated
280
networks
for
combinations
of
these
parameters:
#nodes?
{400}; #communities?{5, 10}; #nodes with at least 3 overlapping communities?{100}; community
N
sizes?{equal, unequal}, when unequal, the community sizes are in the range [ 2K
, 2N
]; average node
K
N
N
N
N
degree? {0.1 K , 0.15 K , .., 0.35 K , 0.4 K }, the maximum node degree=2?average node degree; % links of a
node that are noisy? {0, 0.1}; random runs?{1,..,5}.
5
CP finds a solution per clique size; LC finds a solution per threshold at which the dendrogram is cut [8] in
steps of 0.1 from 0 to 1; PM and a-MMSB find a solution ?K ? {k0 , k0 + 10} where k0 is the true number
of communities?increasing by 10 allows for potentially a larger number of communities to be detected; aMMSB also finds a solution for each of random pair or stratified random pair sampling methods with the
hyperparameters ? set to the default or set to fit dense clusters.
8
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9
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3,948 | 4,574 | Iterative Thresholding Algorithm for Sparse Inverse
Covariance Estimation
Dominique Guillot
Dept. of Statistics
Stanford University
Stanford, CA 94305
Bala Rajaratnam
Dept. of Statistics
Stanford University
Stanford, CA 94305
Benjamin T. Rolfs
ICME
Stanford University
Stanford, CA 94305
[email protected]
[email protected]
[email protected]
Arian Maleki
Dept. of ECE
Rice University
Houston, TX 77005
Ian Wong
Dept. of EE and Statistics
Stanford University
Stanford, CA 94305
[email protected]
[email protected]
Abstract
The `1 -regularized maximum likelihood estimation problem has recently become
a topic of great interest within the machine learning, statistics, and optimization
communities as a method for producing sparse inverse covariance estimators. In
this paper, a proximal gradient method (G-ISTA) for performing `1 -regularized
covariance matrix estimation is presented. Although numerous algorithms have
been proposed for solving this problem, this simple proximal gradient method is
found to have attractive theoretical and numerical properties. G-ISTA has a linear
rate of convergence, resulting in an O(log ?) iteration complexity to reach a tolerance of ?. This paper gives eigenvalue bounds for the G-ISTA iterates, providing
a closed-form linear convergence rate. The rate is shown to be closely related to
the condition number of the optimal point. Numerical convergence results and
timing comparisons for the proposed method are presented. G-ISTA is shown to
perform very well, especially when the optimal point is well-conditioned.
1
Introduction
Datasets from a wide range of modern research areas are increasingly high dimensional, which
presents a number of theoretical and practical challenges. A fundamental example is the problem
of estimating the covariance matrix from a dataset of n samples {X (i) }ni=1 , drawn i.i.d from a pdimensional, zero-mean, Gaussian distribution with covariance matrix ? ? Sp++ , X (i) ? Np (0, ?),
where Sp++ denotes the space of p ? p symmetric, positive definite matrices. When n ? p the maxi? is the sample covariance matrix S = 1 Pn X (i) X (i)T . A
mum likelihood covariance estimator ?
i=1
n
problem however arises when n < p, due to the rank-deficiency in S. In this sample deficient case,
common throughout several modern applications such as genomics, finance, and earth sciences, the
matrix S is not invertible, and thus cannot be directly used to obtain a well-defined estimator for the
inverse covariance matrix ? := ??1 .
A related problem is the inference of a Gaussian graphical model ([27, 14]), that is, a sparsity
pattern in the inverse covariance matrix, ?. Gaussian graphical models provide a powerful means
of dimensionality reduction in high-dimensional data. Moreover, such models allow for discovery
of conditional independence relations between random variables since, for multivariate Gaussian
data, sparsity in the inverse covariance matrix encodes conditional independences. Specifically, if
1
X = (Xi )pi=1 ? Rp is distributed as X ? Np (0, ?), then (??1 )ij = ?ij = 0 ?? Xi ?
? Xj |{Xk }k6=i,j , where the notation A ?? B|C denotes the conditional independence of A and
B given the set of variables C (see [27, 14]). If a dataset, even one with n p is drawn from a
normal distribution with sparse inverse covariance matrix ?, the inverse sample covariance matrix
S ?1 will almost surely be a dense matrix, although the estimates for those ?ij which are equal to 0
may be very small in magnitude. As sparse estimates of ? are more robust than S ?1 , and since such
sparsity may yield easily interpretable models, there exists significant impetus to perform sparse
inverse covariance estimation in very high dimensional low sample size settings.
Banerjee et al. [1] proposed performing such sparse inverse covariance estimation by solving the
`1 -penalized maximum likelihood estimation problem,
??? = arg min
? log det ? + hS, ?i + ? k?k1 ,
p
(1)
??S++
P
where ? > 0 is a penalty parameter, hS, ?i = Tr (S?), and k?k1 =
i,j |?ij |. For ? > 0,
Problem (1) is strongly convex and hence has a unique solution, which lies in the positive definite
cone Sp++ due to the log det term, and is hence invertible. Moreover, the `1 P
penalty induces sparsity
in ??? , as it is the closest convex relaxation of the 0 ? 1 penalty, k?k0 = i,j I(?ij 6= 0), where
I(?) is the indicator function [5]. The unique optimal point of problem (1), ??? , is both invertible
(for ? > 0) and sparse (for sufficiently large ?), and can be used as an inverse covariance matrix
estimator.
In this paper, a proximal gradient method for solving Problem (1) is proposed. The resulting ?graphical iterative shrinkage thresholding algorithm?, or G-ISTA, is shown to converge at a linear rate to
??? , that is, its iterates ?t are proven to satisfy
?t+1 ? ???
? s
?t ? ???
,
(2)
F
F
for a fixed worst-case contraction constant s ? (0, 1), where k?kF denotes the Frobenius norm.
The convergence rate s is provided explicitly in terms of S and ?, and importantly, is related to the
condition number of ??? .
The paper is organized as follows. Section 2 describes prior work related to solution of Problem (1).
The G-ISTA algorithm is formulated in Section 3. Section 4 contains the convergence proofs of
this algorithm, which constitutes the primary mathematical result of this paper. Numerical results
are presented in Section 5, and concluding remarks are made in Section 6.
2
Prior Work
While several excellent general convex solvers exist (for example, [11] and [4]), these are not always adept at handling high dimensional problems (i.e., p > 1000). As many modern datasets
have several thousands of variables, numerous authors have proposed efficient algorithms designed
specifically to solve the `1 -penalized sparse maximum likelihood covariance estimation problem (1).
These can be broadly categorized as either primal or dual methods. Following the literature, we refer
to primal methods as those which directly solve Problem (1), yielding a concentration estimate. Dual
methods [1] yield a covariance matrix by solving the constrained problem,
minimize
U ?Rp?p
? log det(S + U ) ? p
subject to kU k? ? ?,
(3)
where the primal and dual variables are related by ? = (S + U )?1 . Both the primal and dual problems can be solved using block methods (also known as ?row by row? methods), which sequentially
optimize one row/column of the argument at each step until convergence. The primal and dual block
problems both reduce to `1 -penalized regressions, which can be solved very efficiently.
2
2.1
Dual Methods
A number of dual methods for solving Problem (1) have been proposed in the literature. Banerjee
et al. [1] consider a block coordinate descent algorithm to solve the block dual problem, which
reduces each optimization step to solving a box-constrained quadratic program. Each of these
quadratic programs is equivalent to performing a ?lasso? (`1 -regularized) regression. Friedman et al.
[10] iteratively solve the lasso regression as described in [1], but do so using coordinate-wise descent. Their widely used solver, known as the graphical lasso (glasso) is implemented on CRAN.
Global convergence rates of these block coordinate methods are unknown. D?Aspremont et al. [9]
use Nesterov?s smooth approximation scheme, which produces an ?-optimal
? solution in O(1/?) iterations. A variant of Nesterov?s smooth method is shown to have a O(1/ ?) iteration complexity
in [15, 16].
2.2
Primal Methods
Interest in primal methods for solving Problem (1) has been growing for many reasons. One important reason stems from the fact that convergence within a certain tolerance for the dual problem does
not necessarily imply convergence within the same tolerance for the primal.
Yuan and Lin [30] use interior point methods based on the max-det problem studied in [26]. Yuan
[31] use an alternating-direction method, while Scheinberg et al. [24] proposes a similar method and
show a sublinear convergence rate. Mazumder and Hastie [18] consider block-coordinate descent
approaches for the primal problem, similar to the dual approach taken in [10]. Mazumder and
Agarwal [17] also solve the primal problem with block-coordinate descent, but at each iteration
perform a partial as opposed to complete block optimization, resulting in a decreased computational
complexity per iteration. Convergence rates of these primal methods have not been considered in the
literature and hence theoretical guarantees are not available. Hsieh et al. [13] propose a second-order
proximal point algorithm, called QUIC, which converges superlinearly locally around the optimum.
3
Methodology
In this section, the graphical iterative shrinkage thresholding algorithm (G-ISTA) for solving the
primal problem (1) is presented. A rich body of mathematical and numerical work exists for general
iterative shrinkage thresholding and related methods; see, in particular, [3, 8, 19, 20, 21, 25]. A brief
description is provided here.
3.1
General Iterative Shrinkage Thresholding (ISTA)
Iterative shrinkage thresholding algorithms (ISTA) are general first-order techniques for solving
problems of the form
minimize F (x) := f (x) + g(x),
x?X
(4)
where X is a Hilbert space with inner product h?, ?i and associated norm k?k, f : X ? R is a
continuously differentiable, convex function, and g : X ? R is a lower semi-continuous, convex
function, not necessarily smooth. The function f is also often assumed to have Lipschitz-continuous
gradient ?f , that is, there exists some constant L > 0 such that
k?f (x1 ) ? ?f (x2 )k ? L kx1 ? x2 k
(5)
for any x1 , x2 ? X .
For a given lower semi-continuous convex function g, the proximity operator of g, denoted by
proxg : X ? X , is given by
1
2
proxg (x) = arg min g(y) + kx ? yk ,
(6)
y?X
2
It is well known (for example, [8]) that x? ? X is an optimal solution of problem (4) if and only if
x? = prox?g (x? ? ??f (x? ))
3
(7)
for any ? > 0. The above characterization suggests a method for optimizing problem (4) based on
the iteration
xt+1 = prox?t g (xt ? ?t ?f (xt ))
(8)
for some choice of step size, ?t . This simple method is referred to as an iterative shrinkage thresholding algorithm (ISTA). For a step size ?t ? L1 , the ISTA iterates xt are known to satisfy
1
?
F (xt ) ? F (x ) ' O
, ?t,
(9)
t
where x? is some optimal point, which is to say, they converge to the space of optimal points at a
sublinear rate. If no Lipschitz constant L for ?f is known, the same convergence result still holds
for ?t chosen such that
f (xt+1 ) ? Q?t (xt+1 , xt ),
(10)
where Q? (?, ?) : X ? X ? R is a quadratic approximation to f , defined by
Q? (x, y) = f (y) + hx ? y, ?f (y)i +
1
2
kx ? yk .
2?
(11)
See [3] for more details.
3.2
Graphical Iterative Shrinkage Thresholding (G-ISTA)
The general method described in Section 3.1 can be adapted to the sparse inverse covariance estimation Problem (1). Using the notation introduced in Problem (4), define f, g : Sp++ ? R by
f (X) = ? log det(X) + hS, Xi and g(X) = ? kXk1 . Both are continuous convex functions defined on Sp++ . Although the function ?f (X) = S ? X ?1 is not Lipschitz continuous over Sp++ ,
it is Lipschitz continuous within any compact subset of Sp++ (See Lemma 2 of the Supplemental
section).
Lemma 1 ([1, 15]). The solution of Problem (1), ??? , satisfies ?I ??? ?I, for
p ? ? Tr(S)
1
?=
, ?= min
,? ,
kSk2 + p?
?
(12)
and
(
?=
?
min{1T S ?1 1, (p ? ? p?)
S ?1
2 ? (p ? 1)?}
21T (S + ?2 I)?1 1 ? Tr((S + ?2 I)?1 )
if S ? Sp++
otherwise,
(13)
where I denotes the p ? p dimensional identity matrix and 1 denotes the p-dimensional vector of
ones.
Note that f + g as defined is a continuous, strongly convex function on Sp++ . Moreover, by Lemma
2 of the supplemental section, f has a Lipschitz continuous gradient when restricted to the compact
domain aI ? bI. Hence, f and g as defined meet the conditions described in Section 3.1.
The proximity operator of ? kXk1 for ? > 0 is the soft-thresholding operator, ?? : Rp?p ? Rp?p ,
defined entrywise by
[?? (X)]i,j = sgn(Xi,j ) (|Xi,j | ? ?)+ ,
(14)
where for some x ? R, (x)+ := max(x, 0) (see [8]). Finally, the quadratic approximation Q?t of f ,
as in equation (11), is given by
Q?t (?t+1 , ?t ) = ? log det(?t ) + hS, ?t i + h?t+1 ? ?t , S ? ??1
t i+
1
2
k?t+1 ? ?t kF .
2?t
(15)
The G-ISTA algorithm for solving Problem (1) is given in Algorithm 1. As in [3], the algorithm
uses a backtracking line search for the choice of step size. The procedure terminates when a prespecified duality gap is attained. The authors found that an initial estimate of ?0 satisfying [?0 ]ii =
4
(Sii + ?)?1 works well in practice. Note also that the positive definite check of ?t+1 during Step
(1) of Algorithm 1 is accomplished using a Cholesky decomposition, and the inverse of ?t+1 is
computed using that Cholesky factor.
Algorithm 1: G-ISTA for Problem (1)
input : Sample covariance matrix S, penalty parameter ?, tolerance ?, backtracking constant
c ? (0, 1), initial step size ?1,0 , initial iterate ?0 . Set ? := 2?.
while ? > ? do
j
(1) Line search: Let ?t be the largest
element of {c ?t,0 }j=0,1,... so that for
?1
?t+1 = ??t ? ?t ? ?t (S ? ?t ) , the following are satisfied:
?t+1 0
and
f (?t+1 ) ? Q?t (?t+1 , ?t ),
for Q?t as defined in (15).
(2) Update iterate: ?t+1 = ??t ? ?t ? ?t (S ? ??1
t )
(3) Set next initial step, ?t+1,0 . See Section 3.2.1.
(4) Compute duality gap:
? = ? log det(S + Ut+1 ) ? p ? log det ?t+1 + hS, ?i + ? k?t+1 k1 ,
where (Ut+1 )i,j = min{max{([??1
t+1 ]i,j ? Si,j ), ??}, ?}.
end
output: ?-optimal solution to problem (1), ??? = ?t+1 .
3.2.1
Choice of initial step size, ?0
Each iteration of Algorithm 1 requires an initial step size, ?0 . The results of Section 4 guarantee
that any ?0 ? ?min (?t )2 will be accepted by the line search criteria of Step 1 in the next iteration.
However, in practice this choice of step is overly cautious; a much larger step can often be taken.
Our implementation of Algorithm 1 chooses the Barzilai-Borwein step [2]. This step, given by
?t+1,0 =
Tr ((?t+1 ? ?t )(?t+1 ? ?t ))
,
?1
Tr ((?t+1 ? ?t )(??1
t ? ?t+1 ))
(16)
is also used in the SpaRSA algorithm [29], and approximates the Hessian around ?t+1 . If a certain number of maximum backtracks do not result in an accepted step, G-ISTA takes the safe step,
?min (?t )2 . Such a safe step can be obtained from ?max (??1
t ), which in turn can be quickly approximated using power iteration.
4
Convergence Analysis
In this section, linear convergence of Algorithm 1 is discussed. Throughout the section, ?t (t =
1, 2, . . . ) denote the iterates of Algorithm 1, and ??? the optimal solution to Problem (1) for ? > 0.
The minimum and maximum eigenvalues of a symmetric matrix A are denoted by ?min (A) and
?max (A), respectively.
Theorem 1. Assume that the iterates ?t of Algorithm 1 satisfy aI ?t bI, ?t for some fixed
constants 0 < a < b. If ?t ? a2 , ?t, then
?t+1 ? ???
? max 1 ? ?t , 1 ? ?t
?t ? ???
.
(17)
2
2
F
F
b
a
Furthermore,
1. The step size ?t which yields an optimal worst-case contraction bound s(?t ) is ? =
2
a?2 +b?2 .
2. The optimal worst-case contraction bound corresponding to ? =
s(?) : = 1 ?
5
2
2
1 + ab 2
2
a?2 +b?2
is given by
Proof. A direct proof is given in the appendix. Note that linear convergence of proximal gradient
methods for strongly convex objective functions in general has already been proven (see Supplemental section).
It remains to show that there exist constants a and b which bound the eigenvalues of ?t , ?t. The
existence of such constants follows directly from Theorem 1, as ?t lie in the bounded domain
{? ? Sp++ : f (?) + g(?) < f (?0 ) + g(?0 )}, for all t. However, it is possible to specify the
constants a and b to yield an explicit rate; this is done in Theorem 2.
2
Theorem 2. Let ? > 0, define ? and ? as in Lemma 1,
and
assume
?t ? ??
, ?t. Then?the iterates
0
0
?
?t of Algorithm 1 satisfy ?I ?t b I, ?t, with b = ?? 2 + ?0 ? ??
F ? ? + p(? + ?).
Proof. See the Supplementary section.
Importantly, note that the bounds of Theorem 2 depend explicitly on the bound of ??? , as given by
Lemma 1. These eigenvalue bounds on ?t+1 , along with Theorem 1, provide a closed form linear
convergence rate for Algorithm 1. This rate depends only on properties of the solution.
Theorem 3. Let ? and ? be as in Lemma 1. Then for a constant step size ?t := ? < ?2 , the iterates
of Algorithm 1 converge linearly with a rate of
s(?) = 1 ?
2?2
<1
?
?2 + (? + p(? ? ?))2
(18)
Proof. By Theorem 2, for ? < ?2 , the iterates ?t satisfy
?I ?t
???
2 +
?0 ? ???
F I
for all t. Moreover, since ?I ?? ?I, if ?I ?0 ?I (for instance, by taking ?0 =
(S + ?I)?1 or some multiple of the identity) then this can be bounded as:
?
??
+
?0 ? ???
? ? + ?p
?0 ? ???
(19)
F
2
2
?
? ? + p(? ? ?).
(20)
Therefore,
?I ?t (? +
?
p(? ? ?)) I,
(21)
and the result follows from Theorem 1.
Remark 1. Note that the contraction constant (equation 18) of Theorem 3 is closely related to the
condition number of ??? ,
?max (??? )
?
?(??? ) =
?
?min (??? )
?
as
1?
?2
2?2
2?2
?1? 2
? 1 ? 2?(??? )?2 .
?
2
+ (? + p(? ? ?))
? + ?2
(22)
Therefore, the worst case bound becomes close to 1 as the conditioning number of ??? increases.
5
Numerical Results
In this section, we provide numerical results for the G-ISTA algorithm. In Section 5.2, the theoretical results of Section 4 are demonstrated. Section 5.3 compares running times of the G-ISTA,
glasso [10], and QUIC [13] algorithms. All algorithms were implemented in C++, and run on an
Intel i7 ? 2600k 3.40GHz ? 8 core with 16 GB of RAM.
6
5.1
Synthetic Datasets
Synthetic data for this section was generated following the method used by [16, 17]. For a fixed p, a
p dimensional inverse covariance matrix ? was generated with off-diagonal entries drawn i.i.d from
a uniform(?1, 1) distribution. These entries were set to zero with some fixed probability (in this
case, either 0.97 or 0.85 to simulate a very sparse and a somewhat sparse model). Finally, a multiple
of the identity was added to the resulting matrix so that the smallest eigenvalue was equal to 1. In
this way, ? was insured to be sparse, positive definite, and well-conditioned. Datsets of n samples
were then generated by drawing i.i.d. samples from a Np (0, ??1 ) distribution. For each value of p
and sparsity level of ?, n = 1.2p and n = 0.2p were tested, to represent both the n < p and n > p
cases.
problem
p = 2000
n = 400
nnz(?) = 3%
p = 2000
n = 2400
nnz(?) = 3%
p = 2000
n = 400
nnz(?) = 15%
p = 2000
n = 2400
nnz(?) = 15%
?
algorithm
nnz(??? )/?(??? )
glasso
QUIC
G-ISTA
nnz(??? )/?(??? )
glasso
QUIC
G-ISTA
nnz(??? )/?(??? )
glasso
QUIC
G-ISTA
nnz(??? )/?(??? )
glasso
QUIC
G-ISTA
0.03
time/iter
27.65%/48.14
1977.92/11
1481.80/21
145.60/437
14.56%/10.25
667.29/7
211.29/10
14.09/47
27.35%/64.22
2163.33/11
1496.98/21
251.51/714
19.98%/17.72
708.15/6
301.35/10
28.23/88
0.06
time/iter
15.08%/20.14
831.69/8
257.97/11
27.05/9
3.11%/2.82
490.90/6
24.98/7
3.51/13
15.20%/28.50
862.39/8
318.57/12
47.35/148
5.49%/4.03
507.04/6
491.54/17
4.08/16
0.09
time/iter
7.24%/7.25
604.42/7
68.49/8
8.05/27
0.91%/1.51
318.24/4
5.16/5
2.72/10
7.87%/11.88
616.81/7
96.25/9
7.96/28
65.47%/1.36
313.88/4
4.12/5
1.95/7
0.12
time/iter
2.39%/2.32
401.59/5
15.25/6
3.19/12
0.11%/1.18
233.94/3
1.56/4
2.20/8
2.94%/2.87
48.47/7
23.62/7
3.18/12
0.03%/1.09
233.16/3
1.34/4
1.13/4
Table 1: Timing comparisons for p = 2000 dimensional datasets, generated as in Section 5.1.
Above, nnz(A) is the percentage of nonzero elements of matrix A.
5.2
Demonstration of Convergence Rates
The linear convergence rate derived for G-ISTA in Section 4 was shown to be heavily dependent on
the conditioning of the final estimator. To demonstrate these results, G-ISTA was run on a synthetic
dataset, as described in Section 5.1, with p = 500 and n = 300. Regularization parameters of
? = 0.75, 0.1, 0.125, 0.15, and 0.175 were used. Note as ? increases, ??? generally becomes
better conditioned. For each value of ?, the numerical optimum was computed to a duality gap of
10?10 using G-ISTA. These values of ? resulted in sparsity levels of 81.80%, 89.67%, 94.97%,
97.82%, and 99.11%, respectively. G-ISTA was then run again, and the Frobenius norm argument
errors at each iteration were stored. These errors were plotted on a log scale for each value of ?
to demonstrate the dependence of the convergence rate on condition number. See Figure 1, which
clearly demonstrates the effects of conditioning.
5.3
Timing Comparisons
The G-ISTA, glasso, and QUIC algorithms were run on synthetic datasets (real datasets are
presented in the Supplemental section) of varying p, n and with different levels of regularization, ?.
All algorithms were run to ensure a fixed duality gap, here taken to be 10?5 . This comparison used
efficient C++ implementations of each of the three algorithms investigated. The implementation of
G-ISTA was adapted from the publicly available C++ implementation of QUIC Hsieh et al. [13].
Running times were recorded and are presented in Table 1. Further comparisons are presented in the
Supplementary section.
Remark 2. The three algorithms variable ability to take advantage of multiple processors is an
important detail. The times presented in Table 1 are wall times, not CPU times. The comparisons
were run on a multicore processor, and it is important to note that the Cholesky decompositions and
7
? = 0.075, ?(???) = 7.263
2
? = 0.1, ?(?? ) = 3.9637
10
?
? = 0.125, ?(???) = 2.3581
? = 0.15, ?(???) = 1.6996
? = 0.175, ?(?? ) = 1.3968
0
||?t??*?||F
10
?
?2
10
?4
10
?6
10
50
100
150
200 250
iteration
300
350
400
450
Figure 1: Semilog plot of
?t ? ???
F vs. iteration number t, demonstrating linear convergence
rates of G-ISTA, and dependence of those rates on ?(??? ).
inversions required by both G-ISTA and QUIC take advantage of multiple cores. On the other hand,
the p2 dimensional lasso solve of QUIC and p-dimensional lasso solve of glasso do not. For this
reason, and because Cholesky factorizations and inversions make up the bulk of the computation
required by G-ISTA, the CPU time of G-ISTA was typically greater than its wall time by a factor
of roughly 4. The CPU and wall times of QUIC were more similar; the same applies to glasso.
6
Conclusion
In this paper, a proximal gradient method was applied to the sparse inverse covariance problem.
Linear convergence was discussed, with a fixed closed-form rate. Numerical results have also been
presented, comparing G-ISTA to the widely-used glasso algorithm and the newer, but very fast,
QUIC algorithm. These results indicate that G-ISTA is competitive, in particular for values of
? which yield sparse, well-conditioned estimators. The G-ISTA algorithm was very fast on the
synthetic examples of Section 5.3, which were generated from well-conditioned models. For poorly
conditioned models, QUIC is very competitive. The Supplemental section gives two real datasets
which demonstrate this. For many practical applications however, obtaining an estimator that is
well-conditioned is important ([23, 28]). To conclude, although second-order methods for the sparse
inverse covariance method have recently been shown to perform well, simple first-order methods
cannot be ruled out, as they can also be very competitive in many cases.
8
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9
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3,949 | 4,575 | Selective Labeling via Error Bound Minimization
Quanquan Gu? , Tong Zhang? , Chris Ding? , Jiawei Han?
Department of Computer Science, University of Illinois at Urbana-Champaign
?
Department. of Statistics, Rutgers University
?
Department. of Computer Science & Engineering, University of Texas at Arlington
[email protected], [email protected], [email protected], [email protected]
?
Abstract
In many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. This motivates us to study a problem as follows: given a label budget, how to select data points to label such that the learning
performance is optimized. We propose a selective labeling method by analyzing
the out-of-sample error of Laplacian regularized Least Squares (LapRLS). In particular, we derive a deterministic out-of-sample error bound for LapRLS trained
on subsampled data, and propose to select a subset of data points to label by minimizing this upper bound. Since the minimization is a combinational problem, we
relax it into continuous domain and solve it by projected gradient descent. Experiments on benchmark datasets show that the proposed method outperforms the
state-of-the-art methods.
1 Introduction
The performance of (semi-)supervised learning methods typically depends on the amount of labeled
data. Roughly speaking, the more the labeled data, the better the learning performance will be.
However, in many practical machine learning problems, the acquisition of labeled data is often
expensive and/or time consuming. To overcome this problem, active learning [9, 10] was proposed,
which iteratively queries the oracle (labeler) to obtain the labels at new data points. Representative
methods include support vector machine (SVM) active learning [19, 18], agnostic active learning
[2, 5, 14], etc. Due to the close interaction between the learner and the oracle, active learning can be
advantageous to achieve better learning performance. Nevertheless, in many real-world applications,
such an interaction may not be feasible. For example, when one turns to Amazon Mechanical Turk1
to label data, the interaction between the learner and the labeling workers is very limited. Therefore,
standard active learning is not very practical in this case.
Another potential solution to the label deficiency problem is semi-supervised learning [7, 22, 21, 4],
which aims at combining a small number of labeled data and a large amount of unlabeled data to
improve the learning performance. In a typical setting of semi-supervised learning, a small set of
labeled data is assumed to be given at hand or randomly generated in practice. However, randomly
selecting (uniformly sampling) data points to label is unwise because not all the data points are
equally informative. It is desirable to obtain a labeled subset which is most beneficial for semisupervised learning.
In this paper, based on the above motivation, we investigate a problem as follows: given a fixed label
budget, how to select a subset of data points to label such that the learning performance is optimized.
We refer to this problem as selective labeling, in contrast to conventional random labeling. To
achieve the goal of selective labeling, it is crucial to consider the out-of-sample error of a specific
learner. We choose Laplacian Regularized Least Squares (LapRLS) as the learner [4] because it is a
1
https://www.mturk.com/
1
state-the-art semi-supervised learning method, and takes many linear regression methods as special
cases (e.g., ridge regression [15]). We derive a deterministic out-of-sample error bound for LapRLS
trained on subsampled data, which suggests to select the data points to label by minimizing this
upper bound. The resulting selective labeling method is a combinatorial optimization problem. In
order to optimize it effectively and efficiently, we relax it into a continuous optimization problem,
and solve it by projected gradient descent algorithm followed by discretization. Experiments on
benchmark datasets show that the proposed method outperforms the state-of-the-art methods.
The remainder of this paper is organized as follows. In Section 2, we briefly review manifold
regularization and LapRLS. In Section 3, we derive an out-of-sample error bound for LapRLS on
subsampled data, and present a selective labeling criterion by minimizing the this bound, followed
by its optimization algorithm. We discuss the connections between the proposed method and several
existing experimental design approaches in Section 4. The experiments are demonstrated in Section
5. We conclude this paper in Section 6.
2
Review of Laplacian Regularized Least Squares
Given a data set {(x1 , y1 ), . . . , (xn , yn )} where xi ? Rd and yi ? {?1}, Laplacian Regularized
Least Squares (LapRLS) [4] aims to learn a linear function f (x) = wT x. In order to estimate and
preserve the geometrical and topological properties of the data, LapRLS [4] assumes that if two data
points xi and xj are close in the intrinsic geometry of the data distribution, the labels of this two
points are also close to each other. Let f (x) be a? function that maps the original data point x in a
compact submanifold M to R, we use ||f ||2M = x?M || ?M f ||2 dx to measure the smoothness of
f along the geodesics in the intrinsic manifold of the data, where ?M f is the gradient of f along
the manifold M. Recent study on spectral graph theory [8] has demonstrated that ||f ||2M can be
discretely approximated through a nearest neighbor graph on a set of data points. Given an affinity
matrix W ? Rn?n of the graph, ||f ||2M is approximated as:
1?
||f ||2M ?
||fi ? fj ||22 Wij = f T Lf ,
(1)
2 ij
where fi is a?shorthand for f (xi ), f = [f1 , . . . , fn ]T , D is a diagonal matrix, called degree matrix,
n
with Dii = j=1 Wij , and L = D ? W is the combinatorial graph Laplacian [8]. Eq. (1) is called
Manifold Regularization. Intuitively, the regularization incurs a heavy penalty if neighboring points
xi and xj are mapped far apart.
Based on manifold regularization, LapRLS solves the following optimization problem,
arg min ||XT w ? y||22 +
w
?A
?I T
||w||22 +
w XLXT w,
2
2
(2)
where ?A , ?I > 0 are positive regularization parameters, X = [x1 , . . . , xn ] is the design matrix, y = [y1 , . . . , yn ]T is the response vector, ||w||2 is ?2 regularization of linear function, and
wT XLXT w is manifold regularization of f (x) = wT x. When ?I = 0, LapRLS reduces to ridge
regression [15]. A bias term b can be incorporated into the form by expanding the weight vector and
input feature vector as w ? [w; b] and x ? [x; 1]. Note that Eq. (2) is a supervised version of
LapRLS, because only labeled data are used in manifold regularization. Although our derivations
are based on this version in the rest of the paper, the results can be extended to semi-supervised
version of LapRLS straightforwardly.
3
3.1
The Proposed Method
Problem Formulation
The generic problem of selective labeling is as follows. Given a set of data points X =
{x1 , . . . , xn }, namely the pool of candidate data points, our goal is to find a subsample L ?
{1, . . . , n}, which contains the most informative |L| = l points.
To derive a selective labeling approach for LapRLS, we first derive an out-of-sample error bound of
LapRLS.
2
3.2
Out-of-Sample Error Bound of LapRLS
We define the function class of LapRLS as follows.
Definition 1. The function class of LapRLS is FB = {x ? wT x | ?A ||w||22 + ?I wT XLXT w ?
B}, where X = [x1 , . . . , xn ], and B > 0 is a constant.
Consider the following linear regression model,
y = XT w? + ?,
(3)
T
?
where X = [x1 , . . . , xn ] is the design matrix, y = [y1 , . . . , yn ] is the response vector, w is the
true weight vector which is unknown, and ? = [?1 , . . . , ?n ]T is the noise vector with ?i an unknown
noise with zero mean. We assume that different observations have noises that are independent, but
with equal variance ? 2 .
Moreover, we assume that the true weight vector w? satisfies
?A ||w? ||22 + ?I (w? )T XLXT w? ? B,
(4)
which implies that the true hypothesis belongs to the function class of LapRLS in Definition 1. In
this case, the approximation error vanishes and the excess error equals to the estimation error. Note
that this assumption can be relaxed with more effort, under which we can derive a similar error
bound as below. For simplicity, the following derivations are built upon the assumption in Eq. (4).
In selective labeling, we are interested in estimating w? using LapRLS in Eq. (2) from a subsample
L ? {1, . . . , n}. Denote the subsample of X by XL , the subsample of y by yL , and the subsample
of ? by ?L . The solution of LapRLS is given by
? L = (XL XTL + ?A I + ?I XL LL XTL )?1 XL yL ,
w
(5)
where I is an identity matrix, LL is the graph Laplacian computed based on XL , which is a principal
submatrix of L.
In the following, we will present a deterministic out-of-sample error bound for LapRLS trained on
the subsampled data, which is among the main contributions of this paper.
Theorem 2. For any fixed V = [v1 , . . . , vm ] and X = [x1 , . . . , xn ], and a subsample L of X, the
expected error of LapRLS trained on L in predicting the true response VT w? is upper bounded as
(
)
? L ? VT w? ||22 ? (B + ? 2 )tr VT (XL XTL + ?A I + ?I XL LL XTL )?1 V .
E||VT w
(6)
Proof. Let ML = ?A I + ?I XL LL XTL . Given L, the expected error (where the expectation is w.r.t.
?L ) is given by
? L ? VT w? ||22
E||VT w
= E||VT (XL XTL + ML )?1 XL yL ? VT w? ||22
= ||VT (XL XTL + ML )?1 XL XTL w? ? VT w? ||22 + E||VT (XL XTL + ML )?1 XL ?L ||22 ,(7)
|
{z
} |
{z
}
A1
A2
?
where the second equality follows from yL = XL w + ?L . Now we bound the two terms in the
right hand side respectively.
The first term is bounded by
1
1
= ||VT (XL XTL + ML )?1 ML w? ||22 ? ||VT (XL XTL + ML )?1 ML2 ||2F ||ML2 w? ||22
(
)
= Btr VT (XL XTL + ML )?1 ML (XL XTL + ML )?1 V
(
)
? Btr VT (XL XTL + ML )?1 V
(8)
where the first inequality is due to Cauchy Schwarz?s inequality, and the second inequality follows
from dropping the negative term.
A1
Similarly, the second term can be bounded by
(
)
A2 ? ? 2 tr VT (XL XTL + ML )?1 XL XTL (XL XTL + ML )?1 V
(
)
? ? 2 tr VT (XL XTL + ML )?1 V ,
E[?L ?TL ]
(9)
where the first equality uses
? ? I, and it becomes equality if ?i are independent and
identically distributed (i.i.d.). Combing Eqs. (8) and (9) completes the proof.
2
3
Note that in the above theorem, the sample V could be either the same as or different from the
sample X. Sometimes, we are also interested in the expected estimation error of w? as follows.
Theorem 3. For any fixed X, and a subsample L of X, the expected error of LapRLS trained on L
in estimating the true weight vector w? is upper bounded as
)
(
? L ? w? ||22 ? (B + ? 2 )tr (XL XTL + ?A I + ?I XL LL XTL )?1
E||w
(10)
The proof of this theorem follows similar derivations of Theorem 2.
3.3
The Criterion of Selective Labeling
From Theorem 2, we can see that given a subsample L of X, the expected prediction error of
LapRLS on V is upper bounded by Eq. (6). In addition, the right hand side of Eq. (6) does not
depend on the labels, i.e., y. More importantly, the error bound derived in this paper is deterministic,
which is unlike those probabilistic error bounds derived based on Rademacher complexity [3] or
algorithmic stability [6]. Since those probabilistic error bounds only hold for i.i.d. sample rather
than a particular sample, they cannot provide a criterion to choose a subsample set for labeling due
to the correlation between the pool of candidate points and the i.i.d. sample. On the contrary, the
deterministic error bound does not suffer from such a kind of problem. Therefore, it provides a
natural criterion for selective labeling.
In detail, given a pool of candidate data points, i.e., X, we propose to find a subsample L of
{1, . . . , n}, by minimizing the follow objective function
)
(
arg min tr XT (XL XTL + ?I XL LL XTL + ?A I)?1 X ,
(11)
L?{1,...,n}
where we simply assume V = X. The above problem is a combinatorial optimization problem.
Finding the global optimal solution is NP-hard. One potential way to solve it is greedy forward
(or backward) selection. However, it is inefficient. Here we propose an efficient algorithm, which
solves its continuous relaxation.
3.4
Reformulation
We introduce a selection matrix S ? Rn?l , which is defined as
{
1, if xi is selected as the j-point in L
Sij =
0,
otherwise.
(12)
It is easy to check that each column of S has one and only one 1, and each row has at most one 1.
The constraint set for S can be defined as
S1 = {S|S ? {0, 1}n?l , ST 1 = 1, S1 ? 1},
(13)
where 1 is a vector of all ones, or equivalently,
S2 = {S|S ? {0, 1}n?l , ST S = I},
(14)
where I is an identity matrix.
Based on S, we have XL = XS and LL = ST LS. Thus, Eq. (11) can be equivalently reformulated
as
(
)
arg min tr XT (XSST XT + ?I XSST LSST XT + ?A I)?1 X
S?S2
(
)
= arg min tr XT (XSST L? SST XT + ?A I)?1 X ,
(15)
S?S2
?
where L = I + ?I L. The above optimization problem is still a discrete optimization. Let
S3 = {S|S ? 0, ST S = I},
(16)
where we relax the binary constraint on S into nonnegative constraint. Note that S3 is a matching
polytope [17]. Then we solve the following continuous optimization,
(
)
arg min tr XT (XSST L? SST XT + ?A I)?1 X .
(17)
S?S3
4
We derive a projected gradient descent algorithm to find a local optimum of Eq. (17). We first ignore
the nonnegative constraint on S. Since ST S = I, we introduce a Lagrange multiplier ? ? Rl?l ,
thus the Lagrangian function is
(
)
(
)
L(S) = tr XT (XSST L? SST XT + ?A I)?1 X + tr ?(ST S ? I) .
(18)
The derivative of L(S) with respect to S is2
?L
= ?2(XT BXSST L? S + L? SST XT BXS) + 2S?,
(19)
?S
where B = A?1 (XXT )A?1 and A = XSST L? SST XT + ?I. Note that the computational burden
of the derivative is A?1 , which is the inverse of a d ? d matrix. To overcome this problem, we use
the Woodbury matrix identity [12]. Then A?1 can be computed as
(
)?1
1
1
1 T T
?1
T ?
?1
(20)
A = I ? 2 XS (S L S) + S X XS
ST XT ,
?
?
?
where ST L? S is a l ? l matrix, whose inverse can be solved efficiently when l ? d.
To determine the Lagrange multiplier ?, left multiplying Eq. (19) by ST , and using the fact that
ST S = I, we obtain
? = ST XT BXSST L? S + ST L? SST XT BXS.
(21)
Substituting the Lagrange multiplier ? back into Eq. (19), we can obtain the derivative depending
only on S. Thus we can use projected gradient descent to find a local optimal solution for Eq. (17).
In each iteration, it takes a step proportional to the negative of the gradient of the function at the
current point, followed by a projection back into the nonnegative set.
3.5
Discretization
Till now, we have obtained a local optimal solution S? by projected gradient descent. However,
this S? contains continuous values. In other words, S? ? S3 . In order to determine which l data
points to select, we need to project S? into S1 . We use a simple greedy procedure to conduct the
discretization: we first find the largest element in S (if there exist multiple largest elements, we
choose any one of them), and mark its row and column; then from the unmarked columns and rows
we find the largest element and also mark it; this procedure is repeated until we find l elements.
4
Related Work
We notice that our proposed method shares similar spirit with optimal experimental design3 in statistics [1, 20, 16], whose intent is to select the most informative data points to learn a function which
has minimum variance of estimation, or minimum variance of prediction.
For example, A-Optimal Design (AOD) minimizes the expected variance of the model parameter. In
particular, for ridge regression, it optimizes the following criterion,
(
)
arg min tr (XL XTL + ?A I)?1 ,
(22)
L?{1,...,n}
where I is an identity matrix. We can recover this criterion by setting ?I = 0 in Theorem 3.
However, the pitfall of AOD is that it does not characterize the quality of predictions on the data,
which is essential for classification or regression.
To overcome the shortcoming of A-optimal design, Yu et al. [20] proposed a Transdutive Experimental Design (TED) approach. TED selects the samples which minimize the expected predictive
variance of ridge regression on the data,
(
)
(23)
arg min tr XT (XL XTL + ?A I)?1 X .
L?{1,...,n}
2
The calculation of the derivative is non-trivial, please refer to the supplementary material for detail.
Some literature also call it active learning, while our understand is there is no adaptive interaction between
the learner and the oracle within optimal experimental design. Therefore, it is better to call it nonadaptive
active learning.
3
5
Although TED is motivated by minimizing the variance of the prediction, it is very interesting to
demonstrate that the above criterion is coinciding with minimizing the out-of-sample error bound
in Theorem 2 with ?I = 0. The reason is that
( for ridge regression, the) upper bounds of the bias
and variance terms share a common factor tr XT (XL XTL + ?A I)?1 X . This is a very important
observation because it explains why TED performs very well even though its criterion is minimizing
the variance of the prediction. Furthermore, TED can be seen as a special case of our proposed
method.
He et al. [16] proposed Laplacian Optimal Design (LOD), which selects data points that minimize
the expected predictive variance of Laplacian regularized least squares [4] on the data,
(
)
arg min tr XT (?I XLXT + XL XTL + ?A I)?1 X ,
(24)
L?{1,...,n}
where the graph Laplacian L is computed on all the data points in the pool, i.e., X. LOD selects
the points by XL XTL while leaving the graph Laplacian term XLXT fixed. However, our method
selects the points by XL XTL as well as the graph Laplacian term i.e., XL LL XTL . This difference
is essential, because our criterion has a strong theoretical foundation, i.e., minimizing the out-ofsample error bound of LapRLS. This explains the non-significant improvement of LOD over TED.
Admittedly, the term XL LL XTL in our method raised a challenge for optimization. Yet it has been
well-solved by the projected gradient descent algorithm derived in previous section.
We also notice that similar problem was studied for graphs [13]. However, their method cannot be
applied to our setting, because their input is restricted to the adjacency matrix of a graph.
5
Experiments
In this section, we evaluate the proposed method on both synthetic and real-world datasets, and
compare it with the state-of-the-art methods. All the experiments are conducted in Matlab.
5.1
Compared Methods
To demonstrate the effectiveness of our proposed method, we compare it with the following baseline
approaches: Random Sampling (Random) uniformly selects data points from the pool as training
data. It is the simplest baseline for label selection. A-Optimal Design (AOD) is a classic experimental design method proposed in the community of statistics. There is a parameter ?A to be
tuned. Transductive Experiment Design (TED) is proposed in [20], which is the state-of-the-art
(non-adaptive) active learning method. There is a parameter ?A to be tuned. Laplacian Optimal
Design (LOD) [16] is an extension of TED, which incorporates the manifold structure of the data.
Selective Labeling via Error Bound Minimization (Bound) is the proposed method. There are two
tunable parameters ?A and ?I in both LOD and Bound.
Both LOD and Bound use graph Laplacian. To compute it, we first normalize each data point into a
vector with unit ?2 -norm. Then we construct a 5-NN graph and use the cosine distance to measure
the similarity between data points throughout of our experiments.
Note that the problem setting of our study is to select a batch of data points to label without training
a classifier. Therefore, we do not compare our method with typical active learning methods such as
SVM active learning [19, 18] and agnostic active learning [2].
After selecting the data points by the above methods, we train a LapRLS [4] as the learner to do
classification. There are two parameters in LapRLS, i.e., ?A and ?I .
5.2
Synthetic Dataset
To get an intuitive picture of how the above methods (except random sampling, which is trivial)
work differently, we show their experimental results on a synthetic dataset in Figure 1. This dataset
contains two circles, each of which constitutes a class. It has strong manifold structure. We let
the compared methods select 8 data points. As can be seen, the data points selected by AOD are
concentrated on the inner circle (belonging to one class), which are not able to train a classifier.
The data points selected by TED, LapIOD and Bound are distributed on both inner and outer circles
6
2.5
2.5
2.5
2
2
2
1.5
1.5
1.5
54
8 1
1
0.5
2 3
6
1
861
3
2
0
-0.5
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2.5
-2.5
-1
-1.5
-2
-2.5
-2.5
(a) AOD
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(b) TED
-2.5
-2.5
1
7
-1.5
-2
-2
4
-0.5
4
5
-1
2
8
6
0
2
-0.5
4
-1
-1.5
1
0.5
0
-0.5
7
-1
13
68
0.5
75
5
2
1.5
1
0.5
0
2.5
7
-2
-2
-1.5
-1
-0.5
0
0.5
(c) LOD
1
1.5
2
2.5
-2.5
-2.5
3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(d) Bound
Figure 1: Selected points (the red marks) on the two circles dataset by (a) AOD; (b) TED; (c) LOD;
and (d) Bound.
(belonging to different classes), which are good at training a learner. Furthermore, the 8 data points
selected by Bound are uniformly distributed on the two circles, four from the inner circle, and the
other four from the outer circle, which can better represent the original data.
5.3
Real Datasets & Parameter Settings
In the following, we use three real-world benchmark datasets to evaluate the compared methods.
wdbc is the Wisconsin Diagnostic Breast Cancer data set, which is from UCI machine learning
repository4 . It aims at predicting the breast cancer as benign or malignant based on the digitalized
images. There are 357 positive samples and 212 negative samples. Each sample has 32 attributes.
ORL face database5 contains 10 images for each of the 40 human subjects, which were taken at
different times, varying the lighting, facial expressions and facial details. The original images (with
256 gray levels) have size 92 ? 112, which are resized to 32 ? 32 for efficiency.
Isolet was first used in [11]. It contains 150 people who spoke each letter of the alphabet twice. The
speakers are grouped into sets of 30 speakers each, and we use the first group, referred to Isolet1.
Each sample is represented by a 617-dimensional feature vector.
For each data set, we randomly select 20% data as held-out set for model selection, and the rest 80%
data as work set. In order to randomize the experiments, in each run of experiments, we restrict the
training data (pool of candidate data points) to be selected from a random sampling of 50% work
set (which accounts for 40% of the total data). The remaining half data (40% of the total data) is
used as test set. Once the labeled data are selected, we train a semi-supervised version of LapRLS,
which uses both labeled and unlabeled data (all the training data) for manifold regularization. We
report the classification result on the test set. This random split was repeated 10 times, thus we can
compute the mean and standard deviation of the classification accuracy.
The parameters of compared methods (See Section 5.1) are tuned by 2-fold cross validation on the
held-out set. For the parameters of LapRLS, we use the same parameters of LOD (or Bound) for
LapRLS. For the wdbc dataset, the chosen parameters are ?A = 0.001, ?I = 0.01. For ORL,
?A = 0.0001, ?I = 0.001. For Isolet1, ?A = 0.01, ?I = 0.001.
For wdbc, we let the compared methods incrementally choose {2, 4, . . . , 20} points to label, for
ORL, we incrementally choose {80, 90, . . . , 150} points for labeling, and for Isolet1, we choose
{30, 40, . . . , 120} points to query.
5.4
Results on Real Datasets
The experimental results are shown in Figure 2. In all subfigures, the x-axis represents the number of
labeled points, while the y-axis is the averaged classification accuracy on the test data over 10 runs.
In order to show some concrete results, we also list the accuracy and running time (in second) of
all the compared methods on the three datasets with 2, 80 and 30 labeled data points respectively in
4
5
http://archive.ics.uci.edu/ml/
http://www.cl.cam.ac.uk/Research/DTG/attarchive:pub/data
7
80
95
90
75
90
85
70
65
75
Random
AOD
TED
LOD
Bound
70
65
2
4
6
8
10
12
#Labeled data
14
(a) wdbc
16
18
Accuracy
80
Accuracy
Accuracy
85
80
75
70
65
20
80
90
100
110
120
130
#Labeled data
(b) ORL
140
150
60
55
Random
AOD
TED
LOD
Bound
Random
AOD
TED
LOD
Bound
50
45
160
40
30
40
50
60
70
80
#Labeled data
90
100
110
120
(c) Isolet1
Figure 2: Comparison of different methods on (a) wdbc; (b) ORL; and (c) Isolet1 using LapRLS.
Table 1: Classification accuracy (%) and running time (in second) of compared methods on the three
datasets.
Dataset
wdbc (2 labeled)
ORL (80 labeled)
Isolet1 (30 labeled)
Acc
time
Acc
time
Acc
time
Random 69.47?14.56
?
72.00?4.05
?
44.36?3.09
?
AOD
68.59?12.46 0.0 65.17?3.14 32.2 40.27?2.24 7.4
TED
68.33?10.68 0.0 80.33?2.94 39.6 55.98?2.54 41.1
LOD
63.48?8.38
0.1 80.25?2.64 41.7 57.79?1.87 41.5
Bound
88.68?2.82
0.3 83.25?3.17 23.4 61.99?2.14 17.4
Table 1. For each dataset, we did paired t-tests between the proposed method and the other methods
in the 95% confidence interval. If it is significant over all the other methods, the corresponding entry
of Bound is bolded.
We observe that the proposed selective labeling method greatly outperforms the other methods at
most cases. AOD is usually worse than random sampling. The reason is that minimizing the variance
of model parameter does not guarantee the quality of predictions on the data. TED performs very
well. As we mentioned before, the criterion of TED coincides with minimizing the out-of-sample
error bound of ridge regression. This explains its good empirical performance. The performance
of LOD is slightly better than TED. This is because LOD incorporates the geometric structure into
TED. The superior performance of our method is attributed to its theoretical foundation, which
guarantees that the learner (LapRLS) can achieve small error on the test data. In addition, the running
time of our method is comparable to or even less than the running time of the other methods.
One may argue that the above comparison is not fair because we use LapRLS as the learner, which
tends to fit the proposed method. Therefore, we also compare different methods using ridge regression (RR) as the learner. We find that our proposed method is also much better than the other
methods using RR. For the space limit, we omit the results here and put them in the supplemental
material.
6
Conclusions
The main contributions of this paper are: (1) We present a deterministic out-of-sample error bound
for LapRLS; (2) we present a selective labeling method by minimizing this upper bound; and (3) we
present a simple yet effective algorithm to optimize the criterion for selective labeling.
Acknowledgement
The work was supported in part by U.S. National Science Foundation grants IIS-0905215, CNS0931975, the U.S. Army Research Laboratory under Cooperative Agreement No. W911NF-09-20053 (NS-CTA), the U.S. Air Force Office of Scientific Research MURI award FA9550-08-1-0265,
and MIAS, a DHS-IDS Center for Multimodal Information Access and Synthesis at UIUC. We
would like to thank the anonymous reviewers for their helpful comments.
8
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3,950 | 4,576 | A Unifying Perspective of Parametric Policy Search
Methods for Markov Decision Processes
David Barber
Department of Computer Science
University College London
[email protected]
Thomas Furmston
Department of Computer Science
University College London
[email protected]
Abstract
Parametric policy search algorithms are one of the methods of choice for the optimisation of Markov Decision Processes, with Expectation Maximisation and natural gradient ascent being popular methods in this field. In this article we provide a unifying perspective of these two algorithms by showing that their searchdirections in the parameter space are closely related to the search-direction of
an approximate Newton method. This analysis leads naturally to the consideration of this approximate Newton method as an alternative optimisation method for
Markov Decision Processes. We are able to show that the algorithm has numerous desirable properties, absent in the naive application of Newton?s method, that
make it a viable alternative to either Expectation Maximisation or natural gradient ascent. Empirical results suggest that the algorithm has excellent convergence
and robustness properties, performing strongly in comparison to both Expectation
Maximisation and natural gradient ascent.
1
Markov Decision Processes
Markov Decision Processes (MDPs) are the most commonly used model for the description of sequential decision making processes in a fully observable environment, see e.g. [5]. A MDP is
described by the tuple {S, A, H, p1 , p, ?, R}, where S and A are sets known respectively as the
state and action space, H ? N is the planning horizon, which can be either finite or infinite, and
{p1 , p, ?, R} are functions that are referred as the initial state distribution, transition dynamics, policy (or controller) and the reward function. In general the state and action spaces can be arbitrary
sets, but we restrict our attention to either discrete sets or subsets of Rn , where n ? N. We use
boldface notation to represent a vector and also use the notation z = (s, a) to denote a state-action
pair. Given a MDP the trajectory of the agent is determined by the following recursive procedure:
Given the agent?s state, st , at a given time-point, t ? NH , an action is selected according to the policy, at ? ?(?|st ); The agent will then transition to a new state according to the transition dynamics,
st+1 ? p(?|at , st ); this process is iterated sequentially through all of the time-points in the planning horizon, where the state of the initial time-point is determined by the initial state distribution
s1 ? p1 (?). At each time-point the agent receives a (scalar) reward that is determined by the reward
function, where this function depends on the current action and state of the environment. Typically
the reward function is assumed to be bounded, but as the objective is linear in the reward function
we assume w.l.o.g that it is non-negative.
The most widely used objective in the MDP framework is to maximise the total expected reward of
the agent over the course of the planning horizon. This objective can take various forms, including
an infinite planning horizon, with either discounted or average rewards, or a finite planning horizon.
The theoretical contributions of this paper are applicable to all three frameworks, but for notational
ease and for reasons of space we concern ourselves with the infinite horizon framework with discounted rewards. In this framework the boundedness of the objective function is ensured by the
1
introduction of a discount factor, ? ? [0, 1), which scales the rewards of the various time-points in a
geometric manner. Writing the objective function and trajectory distribution directly in terms of the
parameter vector then, for any w ? W, the objective function takes the form
?
X
t?1
U (w) =
Ept (a,s;w) ? R(a, s) ,
(1)
t=1
where we have denoted the parameter space by W and have used the notation pt (a, s; w) to represent the marginal p(st = s, at = a; w) of the joint state-action trajectory distribution
H?1
Y
p(a1:H , s1:H ; w) = ?(aH |sH ; w)
p(st+1 |at , st )?(at |st ; w) p1 (s1 ), H ? N.
(2)
t=1
Note that the policy is now written in terms of its parametric representation, ?(a|s; w).
It is well known that the global optimum of (1) can be obtained through dynamic programming, see
e.g. [5]. However, due to various issues, such as prohibitively large state-action spaces or highly
non-linear transition dynamics, it is not possible to find the global optimum of (1) in most real-world
problems of interest. Instead most research in this area focuses on obtaining approximate solutions,
for which there exist numerous techniques, such as approximate dynamic programming methods [6],
Monte-Carlo tree search methods [19] and policy search methods, both parametric [27, 21, 16, 18]
and non-parametric [2, 25].
This work is focused solely on parametric policy search methods, by which we mean gradient-based
methods, such as steepest and natural gradient ascent [23, 1], along with Expectation Maximisation
[11], which is a bound optimisation technique from the statistics literature. Since their introduction
[14, 31, 10, 16] these methods have been the centre of a large amount of research, with much of it
focusing on gradient estimation [21, 4], variance reduction techniques [30, 15], function approximation techniques [27, 8, 20] and real-world applications [18, 26]. While steepest gradient ascent has
enjoyed some success it is known to suffer from some substantial issues that often make it unattractive in practice, such as slow convergence and susceptibility to poor scaling of the objective function
[23]. Various optimisation methods have been introduced as an alternative, most notably natural
gradient ascent [16, 24, 3] and Expectation Maximisation [18, 28], which are currently the methods
of choice among parametric policy search algorithms. In this paper our primary focus is on the
search-direction (in the parameter space) of these two methods.
2
Search Direction Analysis
In this section we will perform a novel analysis of the search-direction of both natural gradient
ascent and Expectation Maximisation. In gradient-based algorithms of Markov Decision Processes
the update of the policy parameters take the form
wnew = w + ?M(w)?w U (w),
(3)
+
where ? ? R is the step-size parameter and M(w) is some positive-definite matrix that possibly
depends on w. It is well-known that such an update will increase the total expected reward, provided
that ? is sufficiently small, and this process will converge to a local optimum of (1) provided the
step-size sequence is appropriately selected. While EM doesn?t have an update of the form given
in (3) we shall see that the algorithm is closely related to such an update. It is convenient for later
reference to note that the gradient ?w U (w) can be written in the following form
?w U (w) = Ep? (z;w)Q(z;w) ?w log ?(a|s; w) ,
(4)
where we use the expectation notation E[?] to denote the integral/summation w.r.t. a non-negative
function. The term p? (z; w) is a geometric weighted average of state-action occupancy marginals
given by
?
X
p? (z; w) =
? t?1 pt (z; w),
t=1
while the term Q(z; w) is referred to as the state-action value function and is equal to the total
expected future reward from the current time-point onwards, given the current state-action pair, z,
2
and parameter vector, w, i.e.
Q(z; w) =
?
X
t=1
t?1
0
Ept (z0 ;w) ? R(z ) z1 = z .
This is a standard result and due to reasons of space we have omitted the details, but see e.g. [27] or
section(6.1) of the supplementary material for more details.
An immediate issue concerning updates of the form (3) is in the selection of the ?optimal? choice
of the matrix M(w), which clearly depends on the sense in which optimality is defined. There
are numerous reasonable properties that are desirable of such an update, including the numerical
stability and computational complexity of the parameter update, as well as the rate of convergence
of the overall algorithm resulting from these updates. While all reasonable criteria the rate of convergence is of such importance in an optimisation algorithm that it is a logical starting point in our
analysis. For this reason we concern ourselves with relating these two parametric policy search algorithms to the Newton method, which has the highly desirable property of having a quadratic rate
of convergence in the vicinity of a local optimum. The Newton method is well-known to suffer from
problems that make it either infeasible or unattractive in practice, but in terms of forming a basis for
theoretical comparisons it is a logical starting point. We shall discuss some of the issues with the
Newton method in more detail in section(3). In the Newton method the matrix M(w) is set to the
negative inverse Hessian, i.e.
M(w) = ?H?1 (w),
where H(w) = ?w ?Tw U (w).
where we have denoted the Hessian by H(w). Using methods similar to those used to calculate the
gradient, it can be shown that the Hessian takes the form
H(w) = H1 (w) + H2 (w),
(5)
where
H1 (w) =
H2 (w) =
?
X
t=1
?
X
Ep(z1:t ;w) ? t?1 R(zt )?w log p(z1:t ; w)?Tw log p(z1:t ; w) ,
(6)
t?1
T
Ep(z1:t ;w) ? R(zt )?w ?w log p(z1:t ; w) .
(7)
t=1
We have omitted the details of the derivation, but these can be found in section(6.2) of the supplementary material, with a similar derivation of a sample-based estimate of the Hessian given in
[4].
2.1
Natural Gradient Ascent
To overcome some of the issues that can hinder steepest gradient ascent an alternative, natural
gradient, was introduced in [16]. Natural gradient ascent techniques originated in the neural network
and blind source separation literature, see e.g. [1], and take the perspective that the parameter space
has a Riemannian manifold structure, as opposed to a Euclidean structure. Deriving the steepest
ascent direction of U (w) w.r.t. a local norm defined on this parameter manifold (as opposed to w.r.t.
the Euclidean norm, which is the case in steepest gradient ascent) results in natural gradient ascent.
We denote the quadratic form that induces this local norm on the parameter manifold by G(w), i.e.
d(w)2 = wT G(w)w. The derivation for natural gradient ascent is well-known, see e.g. [1], and its
application to the objective (1) results in a parameter update of the form
wk+1 = wk + ?k G?1 (wk )?w U (wk ).
In terms of (3) this corresponds to M(w) = G?1 (w). In the case of MDPs the most commonly
used local norm is given by the Fisher information matrix of the trajectory distribution, see e.g.
[3, 24], and due to the Markovian structure of the dynamics it is given by
T
G(w) = ?Ep? (z;w) ?w ?w log ?(a|s; w) .
(8)
We note that there is an alternate, but equivalent, form of writing the Fisher information matrix, see
e.g. [24], but we do not use it in this work.
3
In order to relate natural gradient ascent to the Newton method we first rewrite the matrix (7) into
the following form
T
H2 (w) = Ep? (z;w)Q(z;w) ?w ?w log ?(a|s; w) .
(9)
For reasons of space the details of this reformulation of (7) are left to section(6.2) of the supplementary material. Comparing the Fisher information matrix (8) with the form of H2 (w) given in (9) it is
clear that natural gradient ascent has a relationship with the approximate Newton method that uses
H2 (w) in place of H(w). In terms of (3) this approximate Newton method corresponds to setting
M(w) = ?H2?1 (w). In particular it can be seen that the difference between the two methods lies
in the non-negative function w.r.t. which the expectation is taken in (8) and (9). (It also appears
that there is a difference in sign, but observing the form of M(w) for each algorithm shows that
this is not the case.) In the Fisher information matrix the expectation is taken w.r.t. to the geometrically weighted summation of state-action occupancy marginals of the trajectory distribution, while
in H2 (w) there is an additional weighting from the state-action value function. Hence, H2 (w)
incorporates information about the reward structure of the objective function, whereas the Fisher
information matrix does not, and so it will generally contain more information about the curvature
of the objective function.
2.2
Expectation Maximisation
The Expectation Maximisation algorithm, or EM-algorithm, is a powerful optimisation technique
from the statistics literature, see e.g. [11], that has recently been the centre of much research in
the planning and reinforcement learning communities, see e.g. [10, 28, 18]. A quantity of central
importance in the EM-algorithm for MDPs is the following lower-bound on the log-objective
log U (w) ? Hentropy (q(z1:t , t)) + Eq(z1:t ,t) log ? t?1 R(zt )p(z1:t ; w) ,
(10)
where Hentropy is the entropy function and q(z1:t , t) is known as the ?variational distribution?. Further
details of the EM-algorithm for MDPs and a derivation of (10) are given in section(6.3) of the
supplementary material or can be found in e.g. [18, 28]. The parameter update of the EM-algorithm
is given by the maximum (w.r.t. w) of the ?energy? term,
Q(w, wk ) = Ep? (z;wk )Q(z;wk ) log ?(a|s; w) .
(11)
Note that Q is a two-parameter function, where the first parameter occurs inside the expectation
and the second parameter defines the non-negative function w.r.t. the expectation is taken. This
decoupling allows the maximisation over w to be performed explicitly in many cases of interest.
For example, when the log-policy is quadratic in w the maximisation problems is equivalent to a
least-squares problem and the optimum can be found through solving a linear system of equations.
It has previously been noted, again see e.g. [18], that the parameter update of steepest gradient
ascent and the EM-algorithm can be related through this ?energy? term. In particular the gradient
(4) evaluated at wk can also be written as follows ?w|w=wk U (w) = ?10
w|w=wk Q(w, wk ), where
10
we use the notation ?w to denote the first derivative w.r.t. the first parameter, while the update of
the EM-algorithm is given by wk+1 = argmaxw?W Q(w, wk ). In other words, steepest gradient
ascent moves in the direction that most rapidly increases Q(w, wk ) w.r.t. the first variable, while the
EM-algorithm maximises Q(w, wk ) w.r.t. the first variable. While this relationship is true, it is also
quite a negative result. It states that in situations where it is not possible to explicitly perform the
maximisation over w in (11) then the alternative, in terms of the EM-algorithm, is this generalised
EM-algorithm, which is equivalent to steepest gradient ascent. Considering that algorithms such as
EM are typically considered because of the negative aspects related to steepest gradient ascent this
is an undesirable alternative. It is possible to find the optimum of (11) numerically, but this is also
undesirable as it results in a double-loop algorithm that could be computationally expensive. Finally,
this result provides no insight into the behaviour of the EM-algorithm, in terms of the direction of
its parameter update, when the maximisation over w in (11) can be performed explicitly.
Instead we provide the following result, which shows that the step-direction of the EM-algorithm
has an underlying relationship with the Newton method. In particular we show that, under suitable
4
regularity conditions, the direction of the EM-update, i.e. wk+1 ? wk , is the same, up to first order,
as the direction of an approximate Newton method that uses H2 (w) in place of H(w).
Theorem 1. Suppose we are given a Markov Decision Process with objective (1) and Markovian
trajectory distribution (2). Consider the update of the parameter through Expectation Maximisation
at the k th iteration of the algorithm, i.e.
wk+1 = argmaxw?W Q(w, wk ).
Provided that Q(w, wk ) is twice continuously differentiable in the first parameter we have that
wk+1 ? wk = ?H2?1 (wk )?w|w=wk U (w) + O(kwk+1 ? wk k2 ).
(12)
Additionally, in the case where the log-policy is quadratic the relation to the approximate Newton
method is exact, i.e. the second term on the r.h.s. (12) is zero.
Proof. The idea of the proof is simple and only involves performing a Taylor expansion of
?10
w Q(w, wk ). As Q is assumed to be twice continuously differentiable in the first component
this Taylor expansion is possible and gives
10
20
2
?10
w Q(wk+1 , wk ) = ?w Q(wk , wk ) + ?w Q(wk , wk )(wk+1 ? wk ) + O(kwk+1 ? wk k ). (13)
As wk+1 = argmaxw?W Q(w, wk ) it follows that ?10
w Q(wk+1 , wk ) = 0. This means that, upon
ignoring higher order terms in wk+1 ? wk , the Taylor expansion (13) can be rewritten into the form
?1 10
wk+1 ? wk = ??20
?w Q(wk , wk ).
w Q(wk , wk )
(14)
= ?w|w=wk U (w) and
The proof is completed by observing that ?10
w Q(wk , wk )
Q(w
,
w
)
=
H
(w
).
The
second
statement
follows
because
in
the
case where the log-policy
?20
k
k
2
k
w
is quadratic the higher order terms in the Taylor expansion vanish.
2.3
Summary
In this section we have provided a novel analysis of both natural gradient ascent and Expectation
Maximisation when applied to the MDP framework. Previously, while both of these algorithms have
proved popular methods for MDP optimisation, there has been little understanding of them in terms
of their search-direction in the parameter space or their relation to the Newton method. Firstly, our
analysis shows that the Fisher information matrix, which is used in natural gradient ascent, is similar
to H2 (w) in (5) with the exception that the information about the reward structure of the problem
is not contained in the Fisher information matrix, while such information is contained in H2 (w).
Additionally we have shown that the step-direction of the EM-algorithm is, up to first order, an
approximate Newton method that uses H2 (w) in place of H(w) and employs a constant step-size
of one.
3
An Approximate Newton Method
A natural follow on from the analysis in section(2) is the consideration of using M(w) = ?H2?1 (w)
in (3), a method we call the full approximate Newton method from this point onwards. In this section
we show that this method has many desirable properties that make it an attractive alternative to other
parametric policy search methods. Additionally, denoting the diagonal matrix formed from the
diagonal elements of H2 (w) by D2 (w), we shall also consider the method that uses M(w) =
?D2?1 (w) in (3). We call this second method the diagonal approximate Newton method.
Recall that in (3) it is necessary that M(w) is positive-definite (in the Newton method this corresponds to requiring the Hessian to be negative-definite) to ensure an increase of the objective. In
general the objective (1) is not concave, which means that the Hessian will not be negative-definite
over the entire parameter space. In such cases the Newton method can actually lower the objective
and this is an undesirable aspect of the Newton method. An attractive property of the approximate
Hessian, H2 (w), is that it is always negative-definite when the policy is log?concave in the policy
parameters. This fact follows from the observation that in such cases H2 (w) is a non-negative mixture of negative-definite matrices, which again is negative-definite [9]. Additionally, the diagonal
5
terms of a negative-definite matrix are negative and so D2 (w) is also negative-definite when the
controller is log-concave.
To motivate this result we now briefly consider some widely used policies that are either log-concave
or blockwise log-concave. Firstly, consider the Gibb?s policy, ?(a|s; w) ? exp wT ?(a, s), where
?(a, s) ? Rnw is a feature vector. This policy is widely used in discrete systems and is logconcave in w, which can be seen from the fact that log ?(a|s; w) is the sum of a linear term and
a negative log-sum-exp term, both of which are concave [9]. In systems with a continuous stateaction space a common choice of controller is ?(a|s; wmean , ?) = N (a|K?(s) + m, ?(s)), where
wmean = {K, m} and ?(s) ? Rnw is a feature vector. The notation ?(s) is used because there
are cases where is it beneficial to have state dependent noise in the controller. This controller is not
jointly log-concave in wmean and ?, but it is blockwise log-concave in wmean and ??1 . In terms of
wmean the log-policy is quadratic and the coefficient matrix of the quadratic term is negative-definite.
In terms of ??1 the log-policy consists of a linear term and a log-determinant term, both of which
are concave.
In terms of evaluating the search direction it is clear from the forms of D2 (w) and H2 (w) that
many of the pre-existing gradient evaluation techniques can be extended to the approximate Newton
framework with little difficulty. In particular, gradient evaluation requires calculating the expectation
of the derivative of the log-policy w.r.t. p? (z; w)Q(z; w). In terms of inference the only additional
calculation necessary to implement either the full or diagonal approximate Newton methods is the
calculation of the expectation (w.r.t. to the same function) of the Hessian of the log-policy, or its
diagonal terms. As an example in section(6.5) of the supplementary material we detail the extension
of the recurrent state formulation of gradient evaluation in the average reward framework, see e.g.
[31], to the approximate Newton method. We use this extension in the Tetris experiment that we
consider in section(4). Given ns samples and nw parameters the complexity of these extensions
scale as O(ns nw ) for the diagonal approximate Newton method, while it scales as O(ns n2w ) for the
full approximate Newton method.
An issue with the Newton method is the inversion of the Hessian matrix, which scales with O(n3w ) in
the worst case. In the standard application of the Newton method this inversion has to be performed
at each iteration and in large parameter systems this becomes prohibitively costly. In general H(w)
will be dense and no computational savings will be possible when performing this matrix inversion.
The same is not true, however, of the matrices D2 (w) and H2 (w). Firstly, as D2 (w) is a diagonal
matrix it is trivial to invert. Secondly, there is an immediate source of sparsity that comes from
taking the second derivative of the log-trajectory distribution in (7). This property ensures that any
(product) sparsity over the control parameters in the log-trajectory distribution will correspond to
sparsity in H2 (w). For example, in a partially observable Markov Decision Processes where the
policy is modeled through a finite state controller, see e.g. [22], there are three functions to be
optimised, namely the initial belief distribution, the belief transition dynamics and the policy. When
the parameters of these three functions are independent H2 (w) will be block-diagonal (across the
parameters of the three functions) and the matrix inversion can be performed more efficiently by
inverting each of the block matrices individually. The reason that H(w) does not exhibit any such
sparsity properties is due to the term H1 (w) in (5), which consists of the non-negative mixture of
outer-product matrices. The vector in these outer-products is the derivative of the log-trajectory
distribution and this typically produces a dense matrix.
A undesirable aspect of steepest gradient ascent is that its performance is affected by the choice of
basis used to represent the parameter space. An important and desirable property of the Newton
method is that it is invariant to non-singular linear (affine) transformations of the parameter space,
see e.g. [9]. This means that given a non-singular linear (affine) mapping T ? Rnw ?nw , the Newton
? (w) = U (T w) is related to the Newton update of the original objective
update of the objective U
through the same linear (affine) mapping, i.e. v + ?vnt = T w + ?wnt , where v = T w and ?vnt
and ?wnt denote the respective Newton steps. In other words running the Newton method on U (w)
? (T ?1 w) will give identical results. An important point to note is that this desirable property
and U
is maintained when using H2 (w) in an approximate Newton method, while using D2 (w) results
in a method that is invariant to rescaling of the parameters, i.e. where T is a diagonal matrix with
non-zero elements along the diagonal. This can be seen by using the linearity of the expectation
operator and a proof of this statement is provided in section(6.4) of the supplementary material.
6
20
Completed Lines
400
?2
15
10
5
0
?10
?8
?6
?4
?1
?2
0
300
200
100
0
0
2
(a) Policy Trace
20
40
60
80
Training Iterations
100
(b) Tetris Problem
Figure 1: (a) An empirical illustration of the affine invariance of the approximate Newton method,
performed on the two state MDP of [16]. The plot shows the trace of the policy during training
for the two different parameter spaces, where the results of the latter have been mapped back into
the original parameter space for comparison. The plot shows the two steepest gradient ascent traces
(blue cross and blue circle) and the two traces of the full approximate Newton method (red cross
and red circle). (b) Results of the tetris problem for steepest gradient ascent (black), natural gradient
ascent (green), the diagonal approximate Newton method (blue) and the full approximate Newton
method (red).
4
Experiments
The first experiment we performed was an empirical illustration that the full approximate Newton
method is invariant to linear transformations of the parameter space. We considered the simple two
state example of [16] as it allows us to plot the trace of the policy during training, since the policy
has only two parameters. The policy was trained using both steepest gradient ascent and the full
approximate Newton method and in both the original and linearly transformed parameter space. The
policy traces of the two algorithms are plotted in figure(1.a). As expected steepest gradient ascent is
affected by such mappings, whilst the full approximate Newton method is invariant to them.
The second experiment was aimed at demonstrating the scalability of the full and diagonal approximate Newton methods to problems with a large state space. We considered the tetris domain, which
is a popular computer game designed by Alexey Pajitnov in 1985. See [12] for more details. Firstly,
we compared the performance of the full and diagonal approximate Newton methods to other parametric policy search methods. Tetris is typically played on a 20 ? 10 grid, but due to computational
costs we considered a 10 ? 10 grid in the experiment. This results in a state space with roughly
7 ? 2100 states. We modelled the policy through a Gibb?s distribution, where we considered a feature vector with the following features: the heights of each column, the difference in heights between
adjacent columns, the maximum height and the number of ?holes?. Under this policy it is not possible to obtain the explicit maximum over w in (11) and so a straightforward application of EM is not
possible in this problem. We therefore compared the diagonal and full approximate Newton methods
with steepest and natural gradient ascent. Due to reasons of space the exact implementation of the
experiment is detailed in section(6.6) of the supplementary material. We ran 100 repetitions of the
experiment, each consisting of 100 training iterations, and the mean and standard error of the results
are given in figure(1.b). It can be seen that the full approximate Newton method outperforms all of
the other methods, while the performance of the diagonal approximate Newton method is comparable to natural gradient ascent. We also ran several training runs of the full approximate Newton
method on the full-sized 20 ? 10 board and were able to obtain a score in the region of 14, 000
completed lines, which was obtained after roughly 40 training iterations. An approximate dynamic
programming based method has previously been applied to the Tetris domain in [7]. The same set
of features were used and a score of roughly 4, 500 completed lines was obtained after around 6
training iterations, after which the solution then deteriorated.
In the third experiment we considered a finite horizon (controlled) linear dynamical system. This
allowed the search-directions of the various algorithms to be computed exactly using [13] and removed any issues of approximate inference from the comparison. In particular we considered a
3-link rigid manipulator, linearized through feedback linearisation, see e.g. [17]. This system has a
7
Normalised Total Expected Reward
Normalised Total Expected Reward
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
200
400
Training Time
600
(a) Model-Based Linear System
1
0.9
0.8
0.7
0.6
0
200
400
600
Training Iterations
800
(b) Model-Free Non-Linear System
Figure 2: (a) The normalised total expected reward plotted against training time, in seconds, for the
3-link rigid manipulator. The plot shows the results for steepest gradient ascent (black), EM (blue),
natural gradient ascent (green) and the approximate Newton method (red), where the plot shows the
mean and standard error of the results. (b) The normalised total expected reward plotted against
training iterations for the synthetic non-linear system of [29]. The plot shows the results for EM
(blue), steepest gradient ascent (black), natural gradient ascent (green) and the approximate Newton
method (red), where the plot shows the mean and standard error of the results.
6-dimensional state space, 3-dimensional action space and a 22-dimensional parameter space. Further details of the system can be found in section(6.7) of the supplementary material. We ran the
experiment 100 times and the mean and standard error of the results plotted in figure(2.a). In this
experiment the approximate Newton method found substantially better solutions than either steepest gradient ascent, natural gradient ascent or Expectation Maximisation. The superiority of the
results in comparison to either steepest or natural gradient ascent can be explained by the fact that
H2 (w) gives a better estimate of the curvature of the objective function. Expectation Maximisation
performed poorly in this experiment, exhibiting sub-linear convergence. Steepest gradient ascent
performed 3684 ? 314 training iterations in this experiment which, in comparison to the 203 ? 34
and 310 ? 40 iterations of natural gradient ascent and the approximate Newton method respectively,
illustrates the susceptibility of this method to poor scaling. In the final experiment we considered the
synthetic non-linear system considered in [29]. Full details of the system and the experiment can be
found in section(6.8) of the supplementary material. We ran the experiment 100 times and the mean
and standard error of the results are plotted in figure(2.b). Again the approximate Newton method
outperforms both steepest and natural gradient ascent. In this example only the mean parameters of
the Gaussian controller are optimised, while the parameters of the noise are held fixed, which means
that the log-policy is quadratic in the policy parameters. Hence, in this example the EM-algorithm
is a particular (less general) version of the approximate Newton method, where a fixed step-size
of one is used throughout. The marked difference in performance between the EM-algorithm and
the approximate Newton method shows the benefit of being able to tune the step-size sequence.
In this experiment we considered five different step-size sequences for the approximate Newton
method and all of them obtained superior results than the EM-algorithm. In contrast only one of
the seven step-size sequences considered for steepest and natural gradient ascent outperformed the
EM-algorithm.
5
Conclusion
The contributions of this paper are twofold: Firstly we have given a novel analysis of Expectation
Maximisation and natural gradient ascent when applied to the MDP framework, showing that both
have close connections to an approximate Newton method; Secondly, prompted by this analysis
we have considered the direct application of this approximate Newton method to the optimisation of
MDPs, showing that it has numerous desirable properties that are not present in the naive application
of the Newton method. In terms of empirical performance we have found the approximate Newton
method to perform consistently well in comparison to EM and natural gradient ascent, highlighting
its viability as an alternative to either of these methods. At present we have only considered actor
type implementations of the approximate Newton method and the extension to actor-critic methods
is a point of future research.
8
References
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9
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3,951 | 4,577 | Near-Optimal MAP Inference
for Determinantal Point Processes
Jennifer Gillenwater Alex Kulesza Ben Taskar
Computer and Information Science
University of Pennsylvania
{jengi,kulesza,taskar}@cis.upenn.edu
Abstract
Determinantal point processes (DPPs) have recently been proposed as computationally efficient probabilistic models of diverse sets for a variety of applications,
including document summarization, image search, and pose estimation. Many
DPP inference operations, including normalization and sampling, are tractable;
however, finding the most likely configuration (MAP), which is often required in
practice for decoding, is NP-hard, so we must resort to approximate inference.
This optimization problem, which also arises in experimental design and sensor
placement, involves finding the largest principal minor of a positive semidefinite
matrix. Because the objective is log-submodular, greedy algorithms have been
used in the past with some empirical success; however, these methods only give
approximation guarantees in the special case of monotone objectives, which correspond to a restricted class of DPPs. In this paper we propose a new algorithm
for approximating the MAP problem based on continuous techniques for submodular function maximization. Our method involves a novel continuous relaxation of
the log-probability function, which, in contrast to the multilinear extension used
for general submodular functions, can be evaluated and differentiated exactly and
efficiently. We obtain a practical algorithm with a 1/4-approximation guarantee
for a more general class of non-monotone DPPs; our algorithm also extends to
MAP inference under complex polytope constraints, making it possible to combine DPPs with Markov random fields, weighted matchings, and other models.
We demonstrate that our approach outperforms standard and recent methods on
both synthetic and real-world data.
1
Introduction
Informative subset selection problems arise in many applications where a small number of items
must be chosen to represent or cover a much larger set; for instance, text summarization [1, 2],
document and image search [3, 4, 5], sensor placement [6], viral marketing [7], and many others.
Recently, probabilistic models extending determinantal point processes (DPPs) [8, 9] were proposed
for several such problems [10, 5, 11]. DPPs offer computationally attractive properties, including
exact and efficient computation of marginals [8], sampling [12, 5], and (partial) parameter estimation [13]. They are characterized by a notion of diversity, as shown in Figure 1; points in the plane
sampled from a DPP (center) are more spread out than those sampled independently (left).
However, in many cases we would like to make use of the most likely configuration (MAP inference,
right), which involves finding the largest principal minor of a positive semidefinite matrix. This is
an NP-hard problem [14], and so we must resort to approximate inference methods. The DPP
probability is a log-submodular function, and hence greedy algorithms are natural; however, the
standard greedy algorithm of Nemhauser and Wolsey [15] offers an approximation guarantee of
1 1/e only for non-decreasing (monotone) submodular functions, and does not apply for general
1
Independent
DPP sample
DPP MAP
Figure 1: From left to right, a set of points in the plane sampled independently at random, a sample
drawn from a DPP, and an approximation of the DPP MAP set estimated by our algorithm.
DPPs. In addition, we are are often interested in conditioning MAP inference on knapsack-type
budget constraints, matroid constraints, or general polytope constraints. For example, we might
consider a DPP model over edges of a bipartite graph and ask for the most likely set under the
one-to-one matching constraint. In this paper we propose a new algorithm for approximating MAP
inference that handles these types of constraints for non-monotone DPPs.
Recent work on non-monotone submodular function optimization can be broadly split into combinatorial versus continuous approaches. Among combinatorial methods, modified greedy, local search
and simulated annealing algorithms provide certain constant factor guarantees [16, 17, 18] and have
been recently extended to optimization under knapsack and matroid constraints [19, 20]. Continuous methods [21, 22] use a multilinear extension of the submodular set function to the convex hull
of the feasible sets and then round fractional solutions obtained by maximizing in the interior of the
polytope. Our algorithm falls into the continuous category, using a novel and efficient non-linear
continuous extension specifically tailored to DPPs. In comparison to the constant-factor algorithms
for general submodular functions, our approach is more efficient because we have explicit access to
the objective function and its gradient. In contrast, general submodular functions assume a simple
function oracle and need to employ sampling to estimate function and gradient values in the polytope interior. We show that our non-linear extension enjoys some of the critical properties of the
standard multilinear extension and propose an efficient algorithm that can handle solvable polytope
constraints. Our algorithm compares favorably to greedy and recent ?symmetric? greedy [18] methods on unconstrained simulated problems, simulated problems under matching constraints, and a
real-world matching task using quotes from political candidates.
2
Background
Determinantal point processes (DPPs) are distributions over subsets that prefer diversity. Originally,
DPPs were introduced to model fermions in quantum physics [8], but since then they have arisen
in a variety of other settings including non-intersecting random paths, random spanning trees, and
eigenvalues of random matrices [9, 23, 12]. More recently, they have been applied as probabilistic
models for machine learning problems [10, 13, 5, 11].
Formally, a DPP P on a set of items Y = {1, 2, . . . , N } is a probability measure on 2Y , the set of
all subsets of Y. For every Y ? Y we have:
P(Y ) / det(LY )
(1)
where L is a positive semidefinite matrix. LY ? [Lij ]i,j2Y denotes the restriction of L to the entries
indexed by elements of Y , and det(L; ) = 1. If L is written as a Gram matrix, L = B > B, then the
quantity det(LY ) can be interpreted as the squared volume spanned by the column vectors Bi for
i 2 Y . If Lij = Bi> Bj is viewed as a measure of similarity between items i and j, then when i and
j are similar their vectors are relatively non-orthogonal, and therefore sets including both i and j
will span less volume and be less probable. This is illustrated in Figure 2. As a result, DPPs assign
higher probability to sets that are diverse under L.
2
Figure 2: (a) The DPP probability of a set Y depends on the volume spanned by vectors Bi for
i 2 Y . (b) As length increases, so does volume. (c) As similarity increases, volume decreases.
The normalization constant in Equation (1) can be computed explicitly thanks to the identity
X
det(LY ) = det(L + I) ,
(2)
Y
where I is the N ? N identity matrix. In fact, a variety of probabilistic inference operations can
be performed efficiently, including sampling, marginalization, and conditioning [12, 24]. However,
the maximum a posteriori (MAP) problem arg maxY det(LY ) is NP-hard [14]. In many practical
situations it would be useful to approximate the MAP set; for instance, during decoding, online
training, etc.
2.1
Submodularity
A function f : 2Y ! R is called submodular if it satisfies
f (X [ {i})
f (Y [ {i})
f (X)
f (Y )
(3)
whenever X ? Y and i 62 Y . Intuitively, the contribution made by a single item i only decreases as
the set grows. Common submodular functions include the mutual information of a set of variables
and the number of cut edges leaving a set of vertices of a graph. A submodular function f is called
nondecreasing (or monotone) when X ? Y implies f (X) ? f (Y ).
It is possible to show that log det(LY ) is a submodular function: entropy is submodular, and the
entropy of a Gaussian is proportional to log det(?Y ) (plus a linear term in |Y |), where ? is the
covariance matrix. Submodular functions are easy to minimize, and a variety of algorithms exist
for approximately maximizing them; however, to our knowledge none of these existing algorithms
simultaneously allows for general polytope constraints on the set Y , offers an approximation guarantee, and can be implemented in practice without expensive sampling to approximate the objective.
We provide a technique that addresses all three criteria for the DPP MAP problem, although approximation guarantees for the general polytope case depend on the choice of rounding algorithm
and remain an open problem. We use the submodular maximization algorithm of [21] as a starting
point.
3
MAP Inference
We seek an approximate solution to the generalized DPP MAP problem arg maxY 2S log det(LY ),
where S ? [0, 1]N and Y 2 S means that the characteristic vector I(Y ) is in S. We will assume that
S is a down-monotone, solvable polytope; down-monotone means that for x, y 2 [0, 1]N , x 2 S
implies y 2 S whenever x
y (that is, whenever xi
yi 8i), and solvable means that for any
linear objective function g(x) = a> x, we can efficiently find x 2 S maximizing g(x).
One common approach for approximating discrete optimization problems is to replace the discrete
variables with continuous analogs and extend the objective function to the continuous domain. When
the resulting continuous optimization is solved, the result may include fractional variables. Typically, a rounding scheme is then used to produce a valid integral solution. As we will detail below,
3
we use a novel non-linear continuous relaxation that has a nice property: when the polytope is unconstrained, S = [0, 1]N , our method will (essentially) always produce integral solutions. For more
complex polytopes, a rounding procedure is required.
When the objective f (Y ) is a submodular set function, as in our setting, the multilinear extension
can be used to obtain certain theoretical guarantees for the relaxed optimization scheme described
above [21, 25]. The multilinear extension is defined on a vector x 2 [0, 1]N :
XY Y
F (x) =
xi
(1 xi )f (Y ) .
(4)
Y
i2Y
i62Y
That is, F (x) is the expected value of f (Y ) when Y is the random set obtained by including element i with probability xi . Unfortunately, this expectation generally cannot be computed efficiently,
since it involves summing over exponentially many sets Y . Thus, to use the multilinear extension
in practice requires estimating its value and derivative via Monte Carlo techniques. This makes the
optimization quite computationally expensive, as well as introducing a variety of technical convergence issues.
Instead, for the special case of DPP probabilities we propose a new continuous extension that is efficiently computable and differentiable. We refer to the following function as the softmax extension:
XY Y
F? (x) = log
xi
(1 xi ) exp(f (Y )) .
(5)
Y
i2Y
i62Y
See the supplementary material for a visual comparison of Equations (4) and (5). While the softmax
extension also involves a sum over exponentially many sets Y , we have the following theorem.
Theorem 1. For a positive semidefinite matrix L and x 2 [0, 1]N ,
XY Y
xi
(1 xi ) det(LY ) = det(diag(x)(L I) + I) .
(6)
Y
i2Y
i62Y
All proofs are included in the supplementary material.
Corollary 2. For f (Y ) = log det(LY ), we have F? (x) = log det(diag(x)(L I) + I) and
@ ?
F (x) = tr((diag(x)(L I) + I) 1 (L I)i ) ,
@xi
where (L I)i denotes the matrix obtained by zeroing all except the ith row of L I.
(7)
Corollary 2 says that softmax extension for the DPP MAP problem is computable and differentiable
in O(N 3 ) time. Using a variant of gradient ascent (Section 3.1), this will be sufficient to efficiently
find a local maximum of the softmax extension over an arbitrary solvable polytope. It then remains
to show that this local maximum comes with approximation guarantees.
3.1
Conditional gradient
When the optimization polytope S is simple?for instance, the unit cube [0, 1]N ?we can apply
generic gradient-based optimization methods like L-BFGS to rapidly find a local maximum of the
softmax extension. In situations where we are able to efficiently project onto the polytope S, we can
apply projected gradient methods. In the general case, however, we assume only that the polytope is
solvable. In such settings, we can use the conditional gradient algorithm (also known as the FrankWolfe algorithm) [26, 27]. Algorithm 1 describes the procedure; intuitively, at each step we move to
a convex combination of the current point and the point maximizing the linear approximation of the
function given by the current gradient. This ensures that we move in an increasing direction while
remaining in S. Note that finding y requires optimizing a linear function over S; this step is efficient
whenever the polytope is solvable.
3.2
Approximation bound
In order to obtain an approximation bound for the DPP MAP problem, we consider the two-phase
optimization in Algorithm 2, originally proposed in [21]. The second call to LOCAL - OPT is necessary in theory; however, in practice it can usually be omitted with minimal loss (if any). We will
show that Algorithm 2 produces a 1/4-approximation.
4
Algorithm 1 LOCAL - OPT
Input: function F? , polytope S
x
0
while not converged do
y
arg maxy0 2S rF? (x)> y 0
?
arg max?0 2[0,1] F? (?0 x + (1
x
?x + (1 ?)y
end while
Output: x
Algorithm 2 Approximating the DPP MAP
Input: kernel L, polytope S
Let F? (x) = log det(diag(x)(L I) + I)
? , S)
x
LOCAL - OPT (F
? , S \{y 0 | y 0 ? 1 x})
y
LOCAL - OPT (F
?
x : F? (x) > F? (y)
Output:
y : otherwise
?0 )y)
We begin by proving that the continuous extension F? is concave in positive directions, although it
is not concave in general.
Lemma 3. When u, v
0, we have
@2 ?
F (x + su + tv) ? 0
@s@t
wherever 0 < x + su + tv < 1.
Corollary 4. F? (x + tv) is concave along any direction v
(8)
0 (equivalently, v ? 0).
Corollary 4 tells us that a local optimum x of F? has certain global properties?namely, that F? (x)
F? (y) whenever y ? x or y x. This leads to the following result from [21].
Lemma 5. If x is a local optimum of F? (?), then for any y 2 [0, 1]N ,
2F? (x)
F? (x _ y) + F? (x ^ y) ,
(9)
where (x _ y)i = max(xi , yi ) and (x ^ y)i = min(xi , yi ).
Following [21], we now define a surrogate function F? ? . Let Xi ? [0, 1] be a subset of the unit
interval representing xi = |Xi |, where |Xi | denotes the measure of Xi . (Note that this representation
is overcomplete, since there are in general many subsets of [0, 1] with measure xi .) F? ? is defined on
X = (X1 , X2 , . . . , XN ) by
F? ? (X ) = F? (x),
x = (|X1 |, |X2 |, . . . , |XN |) .
(10)
Lemma 6. F? ? is submodular.
Lemmas 5 and 6 suffice to prove the following theorem, which appears for the multilinear extension
in [21], bounding the approximation ratio of Algorithm 2.
Theorem 7. Let F? (x) be the softmax extension of a nonnegative submodular function f (Y ) =
log det(LY ), let OPT = maxx0 2S F? (x0 ), and let x and y be local optima of F? in S and S \
{y 0 | y 0 ? 1 x}, respectively. Then
max(F? (x), F? (y))
1
OPT
4
1
max log det(LY ) .
4 Y 2S
(11)
Note that the softmax extension is an upper bound on the multilinear extension, thus Equation (11)
is at least as tight as the corresponding result in [21].
Corollary 8. Algorithm 2 yields a 1/4-approximation to the DPP MAP problem whenever
log det(LY )
0 for all Y . In general, the objective value obtained by Algorithm 2 is bounded
below by 14 (OPT p0 ) + p0 , where p0 = minY log det(LY ).
In practice, filtering of near-duplicates can be used to keep p0 from getting too small; however, in
our empirical tests p0 did not seem to have a significant effect on approximation quality.
5
3.3
Rounding
When the polytope S is unconstrained, it is easy to show that the results of Algorithm 1?and, in
turn, Algorithm 2?are integral (or can be rounded without loss).
Theorem 9. If S = [0, 1]N , then for any local optimum x of F? , either x is integral or at least one
fractional coordinate xi can be set to 0 or 1 without lowering the objective.
More generally, however, the polytope S can be complex, and the output of Algorithm 2 needs to
be rounded. We speculate that the contention resolution rounding schemes proposed in [21] for the
multilinear extension F may be extensible to F? , but do not attempt to prove so here. Instead, in our
experiments we apply pipage rounding [28] and threshold rounding (rounding all coordinates up or
down using a single threshold), which are simple and seem to work well in practice.
3.4
Model combination
In addition to theoretical guarantees and the empirical advantages we demonstrate in Section 4, the
proposed approach to the DPP MAP problem offers a great deal of flexibility. Since the general
framework of continuous optimization is widely used in machine learning, this technique allows
DPPs to be easily combined with other models. For instance, if S is the local polytope for a Markov
random field, then, augmenting the objective with the (linear) log-likelihood of the MRF?additive
linear objective terms do not affect the lemmas proved above?we can approximately compute the
MAP configuration of the DPP-MRF product model. We might in this way model diverse objects
placed in a sequence, or fit to an underlying signal like an image. Empirical studies of these possibilities are left to future work.
4
Experiments
To illustrate the proposed method, we compare it to the widely used greedy algorithm of Nemhauser
and Wolsey [15] (Algorithm 3) and the recently proposed deterministic ?symmetric? greedy algorithm [18], which has a 1/3 approximation guarantee for unconstrained non-monotone problems. Note that, while a naive implementation of the arg max in Algorithm 3 requires evaluating the objective for each item in U , here we can exploit the fact that DPPs are closed under conditioning to compute all necessary values with only two matrix inversions [5]. We report baseline runtimes using this optimized greedy algorithm, which is about 10 times faster than
the naive version at N = 200. The code and data for all experiments can be downloaded from
http://www.seas.upenn.edu/?jengi/dpp-map.html.
4.1
Synthetic data
As a first test, we approximate the MAP configuration for DPPs with random kernels drawn from a
Wishart distribution. Specifically, we choose L = B > B, where B 2 RN ?N has entries drawn independently from the standard normal distribution, bij ? N (0, 1). This results in L ? WN (N, I), a
Wishart distribution with N degrees of freedom and an identity covariance matrix. This distribution
has several desirable properties: (1) in terms of eigenvectors, it spreads its mass uniformly over all
unitary matrices [29], and (2) the probability density of eigenvalues 1 , . . . , N is
! N QN
N
2
X
Y j=i+1 ( i
j)
exp
,
(12)
i
((N i)!)2
i=1
i=1
the first term of which deters the eigenvalues from being too large, and the second term of which
encourages the eigenvalues to be well-separated [30]. Property (1) implies that we will see a variety
of eigenvectors, which play an important role in the structure of a DPP [5]. Property (2) implies that
interactions between these eigenvectors will be important, as no one eigenvalue is likely to dominate.
Combined, these properties suggest that samples should encompass a wide range of DPPs.
Figure 3a shows performance results on these random kernels in the unconstrained setting. Our
proposed algorithm outperforms greedy in general, and the performance gap tends to grow with
the size of the ground set, N . (We let N vary in the range [50, 200] since prior work with DPPs
6
0.5
0
?0.5
50
100
150
200
log prob. ratio (vs. greedy)
log prob. ratio (vs. sym gr.)
log prob. ratio (vs. greedy)
1
6
4
2
0
50
100
4
2
0
50
100
2
1
150
200
3
2
1
0
50
100
150
N
N
(a)
(b)
150
200
150
200
N
time ratio (vs. greedy)
3
time ratio (vs. sym greedy)
time ratio (vs. greedy)
4
100
200
6
N
N
0
50
150
8
200
15
10
5
0
50
100
N
(c)
Figure 3: Median and quartile log probability ratios (top) and running time ratios (bottom) for 100
random trials. (a) The proposed algorithm versus greedy on unconstrained problems. (b) The proposed algorithm versus symmetric greedy on unconstrained problems. (c) The proposed algorithm
versus greedy on constrained problems. Dotted black lines indicate equal performance.
in real-world scenarios [5, 13] has typically operated in this range.) Moreover, Figure 3a (bottom)
illustrates that our method is of comparable efficiency at medium N , and becomes more efficient as
N grows. Despite the fact that the symmetric greedy algorithm [18] has an improved approximation
guarantee of 1/3, essentially the same analysis applies to Figure 3b.
Figure 3c summarizes the performance of our algorithm in a constrained setting. To create plausible
constraints, in this setting we generate two separate random matrices B (1) and B (2) , and then select
(1)
(2)
(1)
(2)
random pairs of rows (Bi , Bj ). Averaging (Bi + Bj )/2 creates one row of the matrix B;
we then set L = B > B. The constraints require that if xk corresponding to the (i, j) pair is 1, no
other xk0 can have first element i or second element j; i.e., the pairs cannot overlap. Since exact
duplicate pairs produce identical rows in L, they are never both selected and can be pruned ahead
of time. This means our constraints are of a form that allows us to apply pipage rounding to the
possibly fractional result. Figure 3c shows even greater gains over greedy in this setting; however,
enforcing the constraints precludes using fast methods like L-BFGS, so our optimization procedure
is in this case somewhat slower than greedy.
4.2
Matched summarization
Finally, we demonstrate our approach using real-world data. Consider the following task: given a
set of documents, select a set of document pairs such that the two elements within a pair are similar,
but the overall set of pairs is diverse. For instance, we might want to compare the opinions of various
authors on a range of topics?or even to compare the statements made at different points in time by
the same author, e.g., a politician believed to have changed positions on various issues.
In this vein, we extract all the statements made by the eight main contenders in the 2012 US Republican primary debates: Bachmann, Cain, Gingrich, Huntsman, Paul, Perry, Romney, and Santorum.
See the supplementary material for an example of some of these statements. Each pair of candidates
(a, b) constitutes one instance of our task. The task output is a set of statement pairs where the
first statement in each pair comes from candidate a and the second from candidate b. The goal of
optimization is to find a set that is diverse (contains many topics, such as healthcare, foreign policy,
immigration, etc.) but where both statements in each pair are topically similar.
Before formulating a DPP objective for this task, we perform some pre-processing. We filter short
statements, leaving us with an average of 179 quotes per candidate (min = 93, max = 332 quotes).
7
log probability ratio (SoftMax / Greedy)
Algorithm 3 Greedy MAP for DPPs
Input: kernel L, polytope S
Y
;, U
Y
while U is not empty do
i?
arg maxi2U log det(LY [{i} )
if log det(LY [{i? } ) < log det(LY )
then
break
end if
Y
Y [ {i? }
U
{i | i 62 Y, I(Y [ {i}) 2 S}
end while
Output: Y
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
?
Figure 4: Log ratio of the objective value
achieved by our method to that achieved by
greedy for ten settings of match weight .
We parse the quotes, keeping only nouns. We further filter nouns by document frequency, keeping
only those that occur in at least 10% of the quotes. Then we generate a feature matrix W where Wqt
is the number of times term t appears in quote q. This matrix is then normalized so that kWq k2 = 1,
where Wq is the qth row of W . For a given pair of candidates (a, b) we compute the quality of each
(a) (b)
possible quote pair (qi , qj ) as the dot product of their rows in W . While the model will naturally
ignore low-quality pairs, for efficiency we throw away such pairs in pre-processing. For each of
(a)
(a) (b)
candidate a?s quotes qi we keep a pair with quote j = arg maxj 0 quality(qi , qj 0 ) from candidate
b, and vice-versa. The scores of the unpruned quotes, which we denote r, are re-normalized to span
the [0, 1] range. To create a feature vector describing each pair, we simply add the corresponding
pair of quote feature vectors and re-normalize, forming a new W matrix.
Our task is to select some high-quality representative subset of the unpruned quote pairs. We formulate this as a DPP objective with kernel L = M SM , where Sij is a measurement of similarity
between quote pairs i and j, and M is a diagonal
matrix with Mii representing the match quality of
p
pair i. We set S = W W T and diag(M ) = exp( r), where is a hyperparameter. Large places
more emphasis on picking high-quality pairs than on making the overall set diverse.
To help limit the number of pairs selected when optimizing the objective, we add some constraints.
For each candidate we cluster their quotes using k-means on the word feature vectors and impose
the constraint that no more than one quote per cluster can be selected. We round the final solution
using the threshold rounding scheme described in Section 3.3.
Figure 4 shows the result of optimizing this constrained objective, averaged over all 56 candidate
pairs. For all settings of we outperform greedy. In general, we observe that our algorithm is most
improved compared to greedy when the constraints are in play. In this case, when is small the
constraints are less relevant, since the model has an intrinsic preference for smaller sets. On the
other hand, when is very large the algorithms must choose as many pairs as possible in order to
maximize their score; in this case the constraints play an important role.
5
Conclusion
We presented a new approach to solving the MAP problem for DPPs based on continuous algorithms
for submodular maximization. Unlike the multilinear extension used in the general case, the softmax
extension we propose is efficiently computable and differentiable. Furthermore, it allows for general
solvable polytope constraints, and yields a guaranteed 1/4-approximation in a subclass of DPPs. Our
method makes it easy to combine DPPs with other models like MRFs or matching models, and is
faster and more reliable than standard greedy methods on synthetic and real-world problems.
Acknowledgments
This material is based upon work supported under a National Science Foundation Graduate Research
Fellowship, Sloan Research Fellowship, and NSF Grant 0803256.
8
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9
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3,952 | 4,578 | Approximating Concavely Parameterized
Optimization Problems
S?oren Laue
Friedrich-Schiller-Universit?at Jena
Germany
[email protected]
Joachim Giesen
Friedrich-Schiller-Universit?at Jena
Germany
[email protected]
Jens K. Mueller
Friedrich-Schiller-Universit?at Jena
Germany
[email protected]
Sascha Swiercy
Friedrich-Schiller-Universit?at Jena
Germany
[email protected]
Abstract
We consider an abstract class of optimization problems that are parameterized
concavely in a single parameter, and show that the solution path along the ?
parameter can always be approximated
with
accuracy
?
>
0
by
a
set
of
size
O(1/
?). A
?
lower bound of size ?(1/ ?) shows that the upper bound is tight up to a constant
factor. We also devise an algorithm
that calls a step-size oracle and computes an
?
approximate path of size O(1/ ?). Finally, we provide an implementation of the
oracle for soft-margin support vector machines, and a parameterized semi-definite
program for matrix completion.
1
Introduction
Problem description. Let D be a set, I ? R an interval, and f : I ? D ? R such that
(1) f (t, ?) is bounded from below for every t ? I, and
(2) f (?, x) is concave for every x ? D.
We study the parameterized optimization problem h(t) = minx?D f (t, x).
A solution x?t ? D is called optimal at parameter value t if f (t, x?t ) = h(t), and x ? D is called
an ?-approximation at t if ?(t, x) := f (t, x) ? h(t) ? ?. Of course it holds ?(t, x?t ) = 0. A subset
P ? D is called an ?-path if P contains an ?-approximation for every t ? I. The size of a smallest
?-approximation path is called the ?-path complexity of the parameterized optimization problem.
The aim of this paper is to derive upper and lower bounds on the path complexity, and to provide
efficient algorithms to compute ?-paths.
Motivation. The rather abstract problem from above is motivated by regularized optimization
problems that are abundant in machine learning, i.e., by problems of the form
min f (t, x) := r(x) + t ? l(x),
x?D
where r(x) is a regularization- and l(x) a loss term. The parameter t controls the trade-off between
regularization and loss. Note that here f (?, x) is always linear and hence concave in the parameter t.
1
Previous work. Due to the widespread use of regularized optimization methods in machine learning regularization path following algorithms have become an active area of research. Initially, exact
path tracking methods have been developed for many machine learning problems [16, 18, 3, 9]
starting with the algorithm for SVMs by Hastie et al. [10]. Exact tracking algorithms tend to be
slow and numerically unstable as they need to invert large matrices. Also, the exact regularization
path can be exponentially large in the input size [5, 14]. Approximation algorithms can overcome
these problems [4]. Approximation path algorithms with approximation guarantees have been developed for SVMs with square loss [6], the LASSO [14], and matrix completion and factorization
problems [8, 7].
?
Contributions. We provide a structural upper bound in O(1/ ?) for the ?-path complexity for
the abstract problem class described above. We?show that this bound is tight up to a multiplicative
constant by constructing a lower bound in ?(1/ ?). Finally, we devise a generic algorithm to compute ?-paths that calls a problem specific oracle providing a step-size
? certificate. If such a certificate
exists, then the algorithm computes a path of complexity in O(1/ ?). Finally, we demonstrate the
implementation of the oracle for standard SVMs and a matrix completion problem.
? Resulting in the
first algorithms for both problems that compute ?-paths of complexity in O 1/ ? . Previously, no
approximation path algorithms have been known for standard SVMs but only a heuristic [12] and an
approximation algorithm for square loss SVMs [6] with complexity in O(1/?). The best approximation path algorithm for matrix completion also has complexity in ?
O(1/?).
To our knowledge, the
only known approximation path algorithm with complexity in O 1/ ? is [14] for the LASSO.
2
Upper Bound
Here we show that any problem that fits the problem definition
from the introduction for a compact
?
interval I = [a, b] has an ?-path with complexity in O(1/ ?).
Let (a, b) be the interior of [a, b] and let g : (a, b) ? R be concave, then g is continuous and has a
0
0
(t), respectively, at every point t ? I (see for example [15]).
(t) and g+
left- and right derivative g?
Note that f (?, x) is concave by assumption and h is concave as the minimum over a family of
concave functions.
0
0
Lemma 1. For all t ? (a, b), h0? (t) ? f?
(t, x?t ) ? f+
(t, x?t ) ? h0+ (t).
Proof. For all t0 < t it holds h(t0 ) ? f (t0 , x?t ) and hence h(t) ? h(t0 ) ? f (t, x?t ) ? f (t0 , x?t ) which
implies
h(t) ? h(t0 )
f (t, x?t ) ? f (t0 , x?t )
0
h0? (t) := lim
?
lim
=: f?
(t, x?t ).
t0 ?t
t0 ?t
t ? t0
t ? t0
0
0
0
The inequality f+
(t, x?t ) ? h0+ (t) follows analogously, and f?
(t, x?t ) ? f+
(t, x?t ) follows after
?
some algebra from the concavity of f (?, xt ) and the definition of the derivatives (see [15]).
Definition 2. Let I = [a, b] be a compact interval, ? > 0, and t0 = a. Let
Tk = t | t ? (tk?1 , b] such that ?(t, x?tk?1 ) := f (t, x?tk?1 ) ? h(t) = ? ,
and tk = min Tk for all integral k > 0 such that Tk 6= ?. Finally, let
P ? = {x?tk | k ? N such that Tk 6= ?}.
?1
2
Lemma 3. Let s1 , . . . , sn ? R>0 , then (s1 + . . . + sn )(s?1
1 + . . . + sn ) ? n .
Proof. The claim holds for n = 1 as s1 s?1
= 1 = 12 . Assume the claim holds for n ? 1 and
1
?1
?1
let a = s1 + . . . + sn?1 and b = s1 + . . . + s?1
n?1 . The rectangle with side lengths asn and
?1
?1
bsn has circumference 2(asn + bsn ) and area asn bsn ?
= ab. Since the square minimizes the
circumference for a given area we have 2(as?1
+
bs
)
?
4
ab. The claim for n now follows from
n
n
?
?
?1
2
2
2
(a + sn )(b + s?1
n ) = ab + asn + bsn + 1 ? ab + 2 ab + 1 = ( ab + 1) ? ((n ? 1) + 1) = n .
2
Lemma 4. The size of P ? is at most
q
?
(b ? a)(h0? (a) ? h0? (b)) /? ? O 1/ ? .
Proof. Let a = t0 ? t1 ? . . . be the sequence from Definition 2. Define ?k = tk+1 ? tk and
?k = h0? (tk ) ? h0? (tk+1 ). We have
0
?k ?k ? (f?
(tk , x?tk ) ? h0? (tk+1 ))(tk+1 ? tk )
f (tk+1 , x?tk ) ? f (tk , x?tk ) h(tk+1 ) ? h(tk )
?
?
(tk+1 ? tk )
tk+1 ? tk
tk+1 ? tk
= f (tk+1 , x?tk ) ? h(tk+1 ) = ?(tk+1 , x?tk ),
where the first inequality follows from Lemma 1 and the second inequality follows from concavity
and the definition of derivatives (see [15]).
Thus, there exists sk > 0 such that ?k ? ?sk and ?k ? s?1
k . It follows from Lemma 3 that
?n2 ?
?
?1
?(s1 + . . . + sn )(s?1
1 + . . . + sn )
(b ? a)(?1 + . . . + ?n )
?
(b ?
?
(?1 + . . . + ?n )(?1
0
a)(h? (a) ? h0? (b)),
+ . . . + ?n )
? h0? (t) for t ? b (which can be proved from conwhere the last inequality follows from
cavity, see again [15]). Hence,
the size of P ? must be finite, or more
q the sequence (tk ) and thus
specifically n is bounded by
(b ? a)(h0? (a) ? h0? (b)) /?.
h0? (b)
Theorem 5. P ? is an ?-path for I = [a, b].
Proof. For any x ? D, ?(?, x) is a continuous function. Hence, x?tk is an ?-approximation for all
t ? [tk , tk+1 ], because if there would be t ? (tk , tk+1 ] with ?(t, x?tk ) > ?, then by continuity, there
would be also t0 ? (tk , tk+1 ) with ?(t, x?tk ) = ? which contradicts the minimality of tk+1 . The
claim of the theorem follows since the proof of Lemma 4 shows that the sequence (tk ) is finite and
hence the intervals [tk , tk+1 ] cover the whole [a, b].
3
Lower Bound
Here we show that there exists a problem
that fits the problem description from the introduction
?
whose ?-path complexity is in ?(1/ ?). This shows that the upper bound from the previous section
is tight up to a constant.
Let I = [a, b], D = R, f (t, x) = 12 x2 ? tx and thus
2
1 2
1
1
h(t) = min
x ? tx = x?t ? tx?t = ? t2 ,
x?R
2
2
2
where the last equality follows from the convexity and differentiability of f (t, x) in x which together
?
?
imply ?f
?x (t, xt ) = xt ? t = 0.
For ? > 0 and x ? R let Ix = t ? [a, b] ?(t, x) := 21 x2 ? tx + 12 t2 ? ? , which is an interval
?
since 12 x2 ? tx + 12 t2 is a quadratic function in t. The length of this interval is 2 2? independent
?
of x. Hence, the ?-path complexity for the problem is at least (b ? a)/2 2?.
Let us compare this lowerqbound with the upper from the previous
section which gives for the
q
(b?a)2
0
0
? . Hence the upper
specific problem at hand,
(b ? a)(h? (a) ? h? (b)) /? =
= b?a
?
?
?
bound is tight up to constant of at most 2 2.
4
Generic Algorithm
So far we have only discussed structural complexity ?
bounds for ?-paths. Now we give a generic
algorithm to compute an ?-path of complexity in O(1/ ?). When applying the generic algorithm to
3
a specific problem a plugin-subroutine PATH P OLYNOMIAL needs to be implemented for the specific
problem. The generic algorithm builds on the simple idea that has been introduced in [6] to compute
an (?/?)-approximation (for ? > 1) and only update this approximation along the parameter interval I = [a, b] when it fails to be an ?-approximation. The plugin-subroutine PATH P OLYNOMIAL
provides a bound on the step-size for the algorithm, i.e., a certificate for how long the approximation
is valid along the interval I. Hence we describe the idea behind the construction of this certificate
first.
4.1
Step-size certificate and algorithm
We always consider a problem that fits the problem description from the introduction.
Definition 6. Let P be the set of all concave polynomials p : I ? R of degree at most 2. For
t ? I, x ? D and ? > 0 let
Pt (x, ?) := {p ? P | p ? h, f (t, x) ? p(t) ? ?},
0
where p ? h means p(t ) ? h(t0 ) for all t0 ? I.
Note that P contains constant and linear polynomials with second derivative p00 = 0 and quadratic
polynomials with constant second derivative p00 < 0. If Pt (x, ?) 6= ?, then x is an ?-approximation
at parameter value t, because there exists p ? P such that ?(t, x) ? f (t, x) ? p(t) ? ?.
Definition 7. [Step-size] For t ? I = [a, b], p ? P, ? > 0, and ? > 1, let ?t := t ? a and
?t (p, ?) =
?
, if p00 < 0 and ?t > 0.
? ?t2 |p00 |
The step-size is given as
?
(1)
?
: p00 = 0
? ?t (p)
(2)
?t (p, ?) =
?t (p, ?) : p00 < 0, ?t (p, ?) ?
?
? (3)
?t (p, ?) : p00 < 0, ?t (p, ?) ?
where
(1)
?t (p)
1
2
1
2
?t (? ? 1)
s
2
2?
1
1
2
=
+ ?t ?t (p, ?) ?
? ?t ?t (p, ?) +
|p00 |
2
2
s
2?
1
=
1? ?
00
|p |
?
=
(2)
?t (p, ?)
(3)
?t (p, ?)
To simplify the notation we will skip the argument ? of the step-size ?t whenever the value of ? is
obvious from the context.
(2)
(3)
Observation
8. If ?t (p, ?) = 1/2, then ?t (p) = ?t (p), because ?t (p, ?) = 1/2 implies ?t =
q
2?
? |p00 | .
Lemma 9. For t ? (a, b), x ? D, ? > 0 and ? > 1. If there exists p ? Pt (x, ?/?), then x is an
?-approximation for all t0 ? [t, b] with t0 ? t + ?t (p).
Proof. Let g : [a, b] ? R be the following linear function,
g(t0 ) = (t0 ? t)
p(t) + ?/? ? p(a)
?
+ p(t) + .
t?a
?
Then, for all t0 ? [t, b],
f (t0 , x) ? (t0 ? t)
f (t, x) ? f (a, x)
p(t) + ?/? ? p(a)
?
+ f (t, x) ? (t0 ? t)
+ p(t) + = g(t0 )
t?a
t?a
?
4
where the first inequality follows from the concavity of f (?, x), and the second inequality follows
from f (t, x) ? p(t) ? ?/? and from p(a) ? h(a) ? f (a, x). Thus, x is an ?-approximation for all
t0 ? [t, b] that satisfy g(t0 ) ? p(t0 ) ? ? because
?(t0 , x) = f (t0 , x) ? h(t0 ) ? f (t0 , x) ? p(t0 ) ? g(t0 ) ? p(t0 ) ? ?.
We finish the proof by considering three cases.
(i) If p00 = 0, then g(t0 ) ? p(t0 ) is a linear function in t0 , and g(t0 ) ? p(t0 ) ? ? solves to t0 ? t ?
(1)
?t (? ? 1) = ?t (p) = ?t (p).
(ii) If p00 < 0, then g(t0 ) ? p(t0 ) is a quadratic polynomial in t0 with second derivative ?p00 > 0,
(2)
and the equation g(t0 )?p(t0 ) ? ? solves to t0 ?t ? ?t (p). Note that we do not need the condition
?t (p) ? 1/2 here.
(iii) The caseq
p00 < 0 and ?t (p) ? 1/2 can
q be reduced to Case (ii). From ?t (p) ? 1/2 we obtain
2?
and
thus
a
?
t
?
?. Let p? the restriction of p onto the interval
t ? a = ?t ? |p2?
00 |?
|p00 |? =: a
[?
a, b] and ??t = t ? a
?, then p?00 = p00 , and thus ?t (?
p) = ?/ ? ??2 |?
p00 | = 1 . Hence by Observation 8,
t
(3)
(3)
2
(2)
?t (p) = ?t (?
p) = ?t (?
p). The claim follows from Case (ii).
Assume now that we have an oracle PATH P OLYNOMIAL available that on input t ? (a, b) and
?/? > 0 returns x ? D and p ? Pt (x, ?/?), then the following algorithm G ENERIC PATH returns an
?-path if it terminates.
Algorithm 1 G ENERIC PATH
Input: f : [a, b] ? D ? R that fits the problem description, and ? > 0
Output: ?-path for the interval [a, b]
choose t? ? (a, b)
P := C OMPUTE PATH (f, t?, ?)
define f? : [a, b] ? D ? R, (t, x) 7? f (a + b ? t, x) [then f? also fits the problem description]
P := P ? C OMPUTE PATH (f?, a + b ? t?, ?)
return P
Algorithm 2 C OMPUTE PATH
Input: f : [a, b] ? D ? R that fits the problem description, t? ? (a, b) and ? > 0
Output: ?-path for the interval [t?, b]
t := t? and P := ?
while t ? b do
(x, p) := PATH P OLYNOMIAL t, ?/?
P := P ? {x}
t := min b, t + ?t (p)
end while
return P
4.2
Analysis of the generic algorithm
The running time of the algorithm G ENERIC PATH is essentially determined by the complexity of
the computed path times the cost of the oracle PATH ?
P OLYNOMIAL. In the following we show that
the complexity of the computed path is at most O(1/ ?).
?
Observation 10. For c ? R let ?c : R ? ? R, x 7? x2 + c ? x. Then we have
>
|c|
1. limx?? ?c (x) = 0
2. ?0c (x) = ?xx2 +c ? 1 for the derivative of ?c . Thus, ?0c (x) > 0 for c < 0 and ?c is
monotonously increasing.
5
(2)
Furthermore, ?t (p) =
q
r
2?
|p00 |
+ ?t2 ?t (p) ?
1
2
2
1
2
2
?t2 ?t (p) + ? ?
=
r
?t2 ?t (p) + ? ?
=
1 2
2
1
2
+ ?t2 ?(1 ? ?) ? ?t ?t (p) +
1
2
+ ?t2 ?(1 ? ?) ? ?t ?t (p) + ? ?
= ??2 ?(1??) ?t ?t (p) + ? ?
t
? ?t ?t (p) +
1
2
1
2
+ ?t (? ? 1)
+ ?t (? ? 1).
Lemma 11. Given t ? I and p ? P, then ?t (p) is continuous in |p00 |.
(2)
(3)
Proof. The continuity for |p00 | > 0 follows from the definitions of ?t (p) and ?t (p), and from
Observation 8. Since ?t (p) > 1/2 for small |p00 | the continuity at |p00 | = 0 follows from Observation 10, because
(2)
(1)
lim ?t (p) = lim
??t2 ?(1??) (?t ? (?t (p) + ? ? 1/2)) + ?t (? ? 1) = ?t (? ? 1) = ?t (p),
00
|p00 |?0
|p |?0
where we have used ?t (p) ? ? as |p00 | ? 0.
Lemma 12. Given t ? I and p1 , p2 ? P, then ?t (p1 ) ? ?t (p2 ) if |p001 | ? |p002 |.
Proof. The claim is that ?t (p) is monotonously decreasing in |p00 |. Since ?t is continuous in |p00 |
(1)
(2)
(3)
by Lemma 11 it is enough to check the monotonicity of ?t (p), ?t (p) and ?t (p). The mono(1)
(3)
tonicity of ?t (p) and ?t (p) follows directly from the definitions of the latter. The monotonicity
(2)
of ?t (p) follows from Observation 10 since we have
1
(2)
?t (p) = ??t2 ?(1??) ?t ?t (p) + ? ?
+ ?t (? ? 1),
2
(2)
and thus ?t (p) is monotonously decreasing in |p00 | because ?t2 ?(1 ? ?) < 0 and ?t (p) is
monotonously decreasing in |p00 |.
Lemma 13. Given t ? I and p ? P, then ?t (p) is monotonously increasing in ?t and hence in t.
Proof. Since ?t (p) is continuous in ?t by Observation 8 it is enough to check the monotonicity of
(1)
(2)
(3)
(1)
(3)
?t (p), ?t (p) and ?t (p). The monotonicity of ?t (p) and ?t (p) follows directly from the
(2)
definitions of the latter. It remains
toshow the monotonicity of ?t (p) for ?t (p) ? 21 . For c ? 0
let ??1 : R>0 ? R, y 7?
1
2
?c (??1
c (y)) = y. Apparently,
c
y ? y . The notation is justified because for
??1
c is monotonously decreasing, and we have
??1
c (y) > 0 we have
(2)
?1
?1
?t (p) = ?c1 (??1
c2 (?t )) ? ?t = ?c1 (?c2 (?t )) ? ?c2 (?c2 (?t )),
1
with c1 = |p2?00 | and c2 = c?1 . Note that ??1
c2 (?t ) > 0 since ?t (p) ? 2 , and c2 < c1 since ? > 1.
0
0
?1
Because ?c1 ? ?c2 < 0 for c1 > c2 , both ?c2 and ?c1 ? ?c2 are monotonously decreasing in their
(2)
respective arguments. Hence, ?t (p) is monotonously increasing in ?t .
Theorem 14. If there exists p ? P and ?? > 0 such that |q 00 | ? |p00 | for all q that are returned by the
oracle?PATH
1 terminates after at most
P OLYNOMIAL on input t ? [a, b] and ? ? ??. Then Algorithm
?
O 1/ ? steps, and thus returns an ?-path of complexity in O(1/ ?).
Proof. For all t ? [t?, b], where t? ? (a, b) is chosen in algorithm G ENERIC PATH, we have ?t (q) ?
?t (p) ? ?t?(p). Here the first inequality is due to Lemma 12 and the second inequality is due
to Lemma 13. Hence, the number of steps in the first call of C OMPUTE PATH is upper bounded by
(b? t?)/(min{?t?(p), b? t?})+1. Similarly, the number of steps in the second call of C OMPUTE PATH
is upper bounded by (t? ? a)/(min{?a+b?t?(p), t? ? a}) + 1.
6
(1)
For the asymptotic behavior, observe that ?t?(p) = ?t? (p) does not depend on ? for p00 = 0. For
|p00 | > 0 observe that lim??0 ?t?(p, ?) = 0. Hence, there exists ?? > 0 such that ?t?(p, ?) < 1/2 and
(3)
?t? (p, ?) ? b ? t? for all ? < ??, and thus
r
?
?
|p00 |
b ? t?
1
b ? t?
?
?
+ 1 = (3)
(b
?
t
)
+
1
?
O
.
+1 =
?
2? ? ? 1
?
min{?t?(p), b ? t?}
?t? (p)
?
Analogously, (t? ? a)/(min{?a+b?t?(p), t? ? a}) + 1 ? O 1/ ? , which completes the proof.
5
Applications
Here we demonstrate on two examples that Lagrange duality can be a tool for implementing the oracle PATH P OLYNOMIAL in the generic path algorithm. This approach obtains the step-size certificate
from an approximate solution that has to be computed anyway.
5.1
Support vector machines
Given data points xi ? Rd together with labels yi ? {?1} for i = 1, . . . , n. A support vector
machine (SVM) is the following parameterized optimization problem
!
n
X
1
2
T
min
kwk + t
max{0, 1 ? yi (w xi + b)} =: f (t, w)
2
w?Rd ,b?R
i=1
parameterized in the regularization parameter t ? [0, ?). The Lagrangian dual of the SVM is given
as
1
s.t. 0 ? ?i ? t, y T ? = 0,
maxn ? ?T K? + 1T ? =: d(?)
??R
2
where K = AT A, A = (y1 x1 , . . . , yn xn ) ? Rd?n and y = (y1 , . . . , yn ) ? Rn .
Algorithm 3 PATH P OLYNOMIAL SVM
Input: t ? (0, ?) and ? > 0
Output: w ? Rd and p ? Pt (w, ?)
compute a primal solution w ? Rd and a dual solution ? ? Rn such that f (t, w) ? d(?) < ?
define p : I ? R, t0 7? d ?t0 /t
return (w, p)
Lemma 15. Let (w, p) be the output of PATH P OLYNOMIAL SVM on input t > 0 and ? > 0, then
p ? Pt (w, ?) and |p00 | ? max0???1
?
?T K ?
? . [Hence, Theorem 14 applies here.]
?
Proof. Let ? be the dual solution computed by PATH P OLYNOMIAL SVM and p be the polynomial
defined in PATH P OLYNOMIAL SVM. Then,
2
t0 1 T
t0
1
? K? + 1T ? and thus p00 (t0 ) = ? 2 ?T K? ? 0
2
t 2
t
t
since K is positive semidefinite. Hence, p ? P. For p ? Pt (w, ?), it remains to show that p ? h =
minw?Rd f (?, w) and f (t, w) ? p(t) ? ?. The latter follows immediately from p(t) = d(?). For
t0 > 0 let ?0 = ?t0 /t, then ?0 is feasible for the dual SVM at parameter value t0 since ? is feasible
for the dual SVM at t. It follows, p(t0 ) = d(?0 ) ? h(t0 ) = minw?Rd f (?, w). Finally, observe that
?
?T K ?
?.
?i ? t implies |p00 | = t12 ?T K? ? max0???1
?
p(t0 ) = ?
The same results hold when using any positive kernel K. In the kernel case one has the following
primal SVM (see [2]),
(
!)
!
n
n
X
X
1 T
min
??Rm ,b
2
? K? + t ?
max
0, 1 ? yi
i=1
?j yj Kij + b
j=1
7
=: f (t, ?)
.
We have implemented the algorithm G ENERIC PATH for SVMs in Matlab using LIBSVM [1] as the
SVM solver. To assess the practicability of the proposed algorithm we ran it on several datasets
taken from the LIBSVM website. For each dataset we have measured the size of the computed
?-path (number of nodes) for t ? [0.1, 10] and ? ? {2?i | i = 2, . . . , 10}. Figure 5.1 shows the
size of paths as a function of ? using double logarithmic plots. A straight line plot with slope ? 21
?
corresponds to an empirical path complexity that follows the function 1/ ?.
1/sqrt(epsilon)
a1a
duke
fourclass scale
mushrooms
w1a
# nodes
# nodes
1/sqrt(epsilon)
a1a
duke
fourclass scale
mushrooms
w1a
1
10
?3
10
?2
?1
10
?3
10
10
epsilon
?2
?1
10
10
epsilon
(a) Path complexity for a linear SVM
5.2
1
10
(b) Path complexity for a SVM with Gaussian
kernel exp(??ku ? vk22 ) for ? = 0.5
Matrix completion
Matrix completion asks for a completion X of an (n ? m)-matrix Y that has been observed only at
the indices in ? ? {1, . . . , m} ? {1, . . . , n}. The problem can be solved by the following convex
semidefinite optimization approach, see [17, 11, 13],
X
2
1
A X
0.
min
Xij ? Yij + t ? tr(A) + tr(B) s.t.
XT B
2
X?Rn?m , A?Rn?n , B?Rm?m
(i,j)??
The Lagrangian dual of this convex semidefinite program is given as
X 1
tI ?
max ?
0, and ?ij = 0 if (i, j) ?
/ ?.
?2ij + ?ij Yij s.t.
?T tI
2
??Rn?m
(i,j)??
? for X
? = (X, A, B) be the primal objective function at parameter value t, and d(?) be
Let f (t, X)
the dual objective function. Analogously to the SVM case we have the following:
Algorithm 4 PATH P OLYNOMIAL M ATRIX C OMPLETION
Input: t ? (0, ?) and ? > 0
? and p ? Pt (X,
? ?)
Output: X
? and a dual solution ? ? Rn?m such that f (t, X)
? ? d(?) < ?
compute a primal solution X
0
0
define p : I ? R, t 7? d t /t ?
? p)
return (X,
? p) be the output of PATH P OLYNOMIAL M ATRIXCOMPLETION on inLemma 16. Let (X,
? ?) and |p00 | ? max ?
? 2
put t > 0 and ? > 0, then p ? Pt (X,
??F1 k?kF , where
tI
?
Ft = ? ? Rn?m
0, ?ij = 0, ?(i, j) ?
/? .
?T tI
The proof for Lemma 16 is similar to the proof of Lemma 15, and Lemma 16 shows that Theorem 14
can be applied here.
Acknowledgments
schaft (GI-711/3-2).
This work has been supported by a grant of the Deutsche Forschungsgemein-
8
References
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9
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3,953 | 4,579 | A nonparametric variable clustering model
Konstantina Palla?
University of Cambridge
[email protected]
David A. Knowles?
Stanford University
[email protected]
Zoubin Ghahramani
University of Cambridge
[email protected]
Abstract
Factor analysis models effectively summarise the covariance structure of high dimensional data, but the solutions are typically hard to interpret. This motivates attempting to find a disjoint partition, i.e. a simple clustering, of observed variables
into highly correlated subsets. We introduce a Bayesian non-parametric approach
to this problem, and demonstrate advantages over heuristic methods proposed to
date. Our Dirichlet process variable clustering (DPVC) model can discover blockdiagonal covariance structures in data. We evaluate our method on both synthetic
and gene expression analysis problems.
1
Introduction
Latent variables models such as principal components analysis (Pearson, 1901; Hotelling, 1933; Tipping and Bishop, 1999; Roweis, 1998) and factor analysis (Young, 1941) are popular for summarising high dimensional data, and can be seen as modelling the covariance of the observed dimensions.
Such models may be used for tasks such as collaborative filtering, dimensionality reduction, or data
exploration. For all these applications sparse factor analysis models can have advantages in terms
of both predictive performance and interpretability (Fokoue, 2004; Fevotte and Godsill, 2006; Carvalho et al., 2008). For example, data exploration might involve investigating which variables have
significant loadings on a shared factor, which is aided if the model itself is sparse. However, even
using sparse models interpreting the results of a factor analysis can be non-trivial since a variable
will typically have significant loadings on multiple factors.
As a result of these problems researchers will often simply cluster variables using a traditional
agglomerative hierarchical clustering algorithm (Vigneau and Qannari, 2003; Duda et al., 2001).
Interest in variable clustering exists in many applied fields, e.g. chemistry (Basak et al., 2000a,b)
and acturial science (Sanche and Lonergan, 2006). However, it is most commonly applied to gene
expression analysis (Eisen et al., 1998; Alon et al., 1999; D?haeseleer et al., 2005), which will also
be the focus of our investigation. Note that variable clustering represents the opposite regime to the
usual clustering setting where we partition samples rather than dimensions (but of course a clustering
algorithm can be made to work like this simply by transposing the data matrix). Typical clustering
algorithms, and their probabilistic mixture model analogues, consider how similar entities are (e.g.
in terms of Euclidean distance) rather how correlated they are, which would be closer in spirit to the
ability of factor analysis to model covariance structure. While using correlation distance (one minus
the Pearson correlation coefficient) between variables has been proposed for clustering genes with
heuristic methods, the corresponding probabilistic model appears not to have been explored to the
best of our knowledge.
?
These authors contributed equally to this work
1
To address the general problem of variable clustering we develop a simple Bayesian nonparametric
model which partitions observed variables into sets of highly correlated variables. We denote our
method DPVC for ?Dirichlet Process Variable Clustering?. DPVC exhibits the usual advantages
over heuristic methods of being both probabilistic and non-parametric: we can naturally handle
missing data, learn the appropriate number of clusters from data, and avoid overfitting.
The paper is organised as follows. Section 2 describes the generative process. In Section 3 we
note relationships to existing nonparametric sparse factor analysis models, Dirichlet process mixture
models, structure learning with hidden variables, and the closely related ?CrossCat? model (Shafto
et al., 2006). In Section 4 we describe efficient MCMC and variational Bayes algorithms for performing posterior inference in DPVC, and point out computational savings resulting from the simple
nature of the model. In Section 5 we present results on synthetic data where we test the method?s
ability to recover a ?true? partitioning, and then focus on clustering genes based on gene expression
data, where we assess predictive performance on held out data. Concluding remarks are given in
Section 6.
2
The Dirichlet Process Variable Clustering Model
Consider observed data {yn ? RD : n = 1, .., N } where we have D observed dimensions and N
samples. The D observed dimensions correspond to measured variables for each sample, and our
goal is to cluster these variables. We partition the observed dimensions d = {1, ..., D} according
to the Chinese restaurant process (Pitman, 2002, CRP). The CRP defines a distribution over partitionings (clustering) where the maximum possible number of clusters does not need to be specified
a priori. The CRP can be described using a sequential generative process: D customers enter a
Chinese restaurant one at a time. The first customer sits at some table and each subsequent customer
sits at table k with mk current customers with probability proportional to mk , or at a new table with
probability proportional to ?, where ? is a parameter of the CRP. The seating arrangement of the
customers at tables corresponds to a partitioning of the D customers. We write
(c1 , ..., cD ) ? CRP(?),
cd ? N
(1)
where cd = k denotes that variable d belongs to cluster k. The CRP partitioning allows each
dimension to belong only to one cluster. For each cluster k we have a single latent factor
xkn ? N (0, ?x2 )
(2)
which models correlations between the variables in cluster k. Given these latent factors, real valued
observed data can be modeled as
ydn = gd xcd n + dn
(3)
where gd is a factor loading for dimension d, and dn ? N (0, ?d2 ) is Gaussian noise. We place a
Gaussian prior N (0, ?g2 ) on every element gd independently. It is straightforward to generalise the
model by substituting other noise models for Equation 3, for example using a logistic link for binary
data ydn ? {0, 1}. However, in the following we will focus on the Gaussian case.
To improve the flexibility of the model, we put Inverse Gamma priors on ?g2 and ?d2 and a Gamma
prior on the CRP concentration parameter ? as follows:
? ? G(1, 1)
?g2 ? IG(1, 1)
?d2 ? IG(1, 0.1)
Note that we fix ?x = 1 due to the scale ambiguity in the model.
3
Related work
Since DPVC is a hybrid mixture/factor analysis model there is of course a wealth of related work,
but we aim to highlight a few interesting connections here.
DPVC can be seen as a simplification of the infinite factor analysis models proposed by Knowles
and Ghahramani (2007) and Rai and Daum?e III (2008), which we will refer to as Non-parametric
2
x1
y1
y2
x2
y3
y4
x3
y5
y6
Figure 1: Graphical model structure that could be learnt using the model, corresponding to cluster
assignments c = {1, 1, 1, 2, 2, 3}. Gray nodes represent the D = 6 observed variables yd and white
nodes represent the K = 3 latent variables xk .
Sparse Factor Analysis (NSFA). Where they used the Indian buffet process to allow dimensions
to have non-zero loadings on multiple factors, we use the Chinese restaurant process to explicitly
enforce that a dimension can be explained by only one factor. Obviously this will not be appropriate
in all circumstances, but where it is appropriate we feel it allows easier interpretation of the results.
To see the relationship more clearly, introduce the indicator variable zdk = I[cd = k]. We can then
write our model as
yn = (G ? Z)xn + n
(4)
where G is a D ? K Gaussian matrix, and ? denotes elementwise multiplication. Replacing our
Chinese restaurant process prior on Z with an Indian buffet prior recovers an infinite factor analysis
model. Equation 4 has the form of a factor analysis model. It is straightforward to show that the
conditional covariance of y given the factor loading matrix W := G ? Z is ?x2 WWT + ?2 I.
Analogously for DPVC we find
2
?x gd gd0 + ?d2 ?dd0 , cd = cd0
cov(ydn , yd0 n |G, c) =
(5)
0,
otherwise
Thus we see the covariance structure implied by DPVC is block diagonal: only dimensions belonging to the same cluster have non-zero covariance.
The obvious probabilistic approach to clustering genes would be to simply apply a Dirichlet process
mixture (DPM) of Gaussians, but considering the genes (our dimensions) as samples, and our samples as ?features? so that the partitioning would be over the genes. However, this approach would
not achieve the desired result of clustering correlated variables, and would rather cluster together
variables close in terms of Euclidean distance. For example two variables which have the relationship yd = ayd0 for a = ?1 (or a = 2) are perfectly correlated but not close in Euclidean space;
a DPM approach would likely fail to cluster these together. Also, practitioners typically choose either to use restrictive diagonal Gaussians, or full covariance Gaussians which result in considerably
greater computational cost than our method (see Section 4.3).
DPVC can also be seen as performing a simple form of structure learning, where the observed
variables are partitioned into groups explained by a single latent variable. This is subset of the
structures considered in Silva et al. (2006), but we maintain uncertainty over the structure using a
fully Bayesian analysis. Figure 1 illustrates this idea.
DPVC is also closely related to CrossCat (Shafto et al., 2006). CrossCat also uses a CRP to partition
variables into clusters, but then uses a second level of independent CRPs to model the dependence
of variables within a cluster. In other words whereas the latent variables x in Figure 1 are discrete
variables (indicating cluster assignment) in CrossCat, they are continuous variables in DPVC corresponding to the latent factors. For certain data the CrossCat model may be more appropriate but
our simple factor analysis model is more computationally tractable and often has good predictive
performance as well. The model of Niu et al. (2012) is related to CrossCat in the same way that
NSFA is related to DPVC, by allowing an observed dimension to belong to multiple features using
the IBP rather than only one cluster using the CRP.
4
Inference
We demonstrate both MCMC and variational inference for the model.
3
Algorithm 1 Marginal conditional
1: for m = 1 to M do
2:
?(m) ? P (?)
3:
Y (m) ? P (Y |?(m) )
4: end for
4.1
Algorithm 2 Successive conditional
1:
2:
3:
4:
5:
6:
?(1) ? P (?)
Y (1) ? P (Y |?(1) )
for m = 2 to M do
?(m) ? Q(?|?(m?1) , Y (m?1) )
Y (m) ? P (Y |?(m) )
end for
MCMC
We use a partially collapsed Gibbs sampler to explore the posterior distribution over all latent variables g, c, X as well as hyperparameters ?d2 , ?g2 and ?. The Gibbs update equations for the factor
loadings g, factors X, noise variance ?d2 and ?g2 are standard, and therefore only sketched out below with the details deferred to supplementary material. The Dirichlet concentration parameter ? is
sampled using slice sampling (Neal, 2003). We sample the cluster assignments c using Algorithm 8
of Neal (2000), with g integrated out but instantiating X. Updating the factor loading matrix G is
done elementwise, sampling from
gdk |Y, G?dk , C, X, ?g , ?x , ?d , ? ? N (??g , ??1
(6)
g )
The factors X can be jointly sampled as
X:n |Y, G, C, ?g , ?x , ?d , ? ? N (?X:n , ??1
(7)
X:n )
When sampling the cluster assignments, c we found it beneficial to integrate out g, while instantiating X. We require
Z
P (cd = k|yd: , xk: , ?g , c?d ) = P (cd |c?d ) P (yd: |xk: , gd )p(gd |?g )dgd
the calculation of which is given in the supplementary material, along with expressions for
??g , ?g , ?X:n and ?X:n .
We confirm the correctness of our algorithm using the joint distribution testing methodology of
Geweke (2004). There are two ways to sample from the joint distribution, P (Y, ?) over parameters,
? = {g, c, X} and data, Y defined by a probabilistic model such as DPVC. The first we will refer
to as ?marginal-conditional? sampling, shown in Algorithm 1. Both steps here are straightforward:
sampling from the prior followed by sampling from the likelihood model. The second way, referred
to as ?successive-conditional? sampling is shown in Algorithm 2, where Q represents a single (or
multiple) iteration(s) of our MCMC sampler. To validate our sampler we can then check, either
informally or using hypothesis tests, whether the samples drawn from the joint P (Y, ?) in these two
different ways appear to have come from the same distribution.
We apply this method to our DPVC sampler with just N = D = 2, and all hyperparameters fixed
as follows: ? = 1, ?d = 0.1, ?g = 1, ?x = 1. We draw 104 samples using both the marginalconditional and successive-conditional procedures. We look at various characteristics of the samples, including the number of clusters and the mean of X. The distribution of the number of features
under the successive-conditional sampler matches that under the marginal-conditional sampler almost perfectly. Under the correct successive-conditional sampler the average number of clusters is
1.51 (it should be 1.5): a hypothesis test did not reject the null hypothesis that the means of the two
distributions are equal. While this cannot completely guarantee correctness of the algorithm and
code, 104 samples is a large number for such a small model and thus gives strong evidence that our
algorithm is correct.
4.2
Variational inference
We use Variational Message Passing (Winn and Bishop, 2006) under the Infer.NET framework (Minka et al., 2010) to fit an approximate posterior q to the true posterior p, by minimising the
Kullback-Leibler divergence
Z
KL(q||p) = ?H[q(v)] ? q(v) log p(v)dv
(8)
4
R
where H[q(v)] = ? q(v) log q(v)dv is the entropy and v = {w, g, c, X, ?d2 , ?g2 }, where w is
introduced so that the Dirichlet process can be approximated as
w ? Dirichlet(?/T, ..., ?/T )
cd ? Discrete(w)
(9)
(10)
where we have truncated to allow a maximum of T clusters. Where not otherwise specified we
choose T = D so that every dimension could use its own cluster if this is supported by the data.
Note that the Dirichlet process is recovered in the limit T ? ?.
We use a variational posterior of the form
q(v) = qw (w)q?g2 (?g2 )
D
Y
qcd (cd )q?d2 (?d2 )qgd |cd (gd |cd )
N
Y
qxnd (xnd )
(11)
n=1
d=1
where qw is a Dirichlet distribution, each qcd is a discrete distribution on {1, .., T }, q?g2 and q?d2 are
Inverse Gamma distributions and qnd and qgd |cd are univariate Gaussian distributions. We found that
using the structured approximation qgd |cd (gd |cd ) where the variational distribution on gd is conditional on the cluster assignment cd gave considerably improved performance. Using the representation of the Dirichlet process in Equation 10 this model is conditionally conjugate (i.e. all variables
have exponential family distributions conditioned on their Markov blanket) so the VB updates are
standard and therefore omitted here.
Due to the symmetry of the model under permutation of the clusters, we are require to somehow
break symmetry initially. We experimented with initialising either the variational distribution over
the factors qxnd (xnd ) with mean N (0, 0.1) and variance 1 or each cluster assignments distribution
qcd (cd ) to a sample from a uniform Dirichlet. We found initialising the cluster assignments gave
considerably better solutions on average. We also typically ran the algorithm L = 10 times and took
the solution with the best lower bound on the marginal likelihood.
We also experimented with using Expectation Propagation (Minka, 2001) for this model but found
that the algorithm often diverged, presumably because of the multimodality in the posterior. It might
be possible to alleviate this using damping, but we leave this to future work.
4.3
Computational complexity
DPVC enjoys some computational savings compared to NSFA. For both models sampling the factor loadings matrix is O(DKN ), where K is the number of active features/clusters. However, for
DPVC sampling the factors X is considerably cheaper. Calculating the diagonal precision matrix is
O(KD) (compared to O(K 2 D) for the precision in NSFA), and finding the square root of the diagonal elements is negligible at O(K) (compared to a O(K 3 ) Cholesky decomposition for NSFA).
Finally both models require an O(DKN ) operation to calculate the conditional mean of X. Thus
where NSFA is O(DKN + DK 2 + K 3 ), DPVC is only O(DKN ), which is the same complexity
as k-means or Expectation Maximisation (EM) for mixture models with diagonal Gaussian clusters. Note that mixture models with full covariance clusters would typically cost O(DKN 3 ) in this
setting due to the need to perform Cholesky decompositions on N ? N matrices.
5
Results
We present results on synthetic data and two gene expression data sets. We show comparisons
to k-means and hierarchical clustering, for which we use the algorithms provided in the Matlab
statistics toolbox. We also compare to our implementation of Bayesian factor analysis (see for
example Kaufman and Press (1973) or Rowe and Press (1998)) and the non-parametric sparse factor
analysis (NSFA) model of (Knowles and Ghahramani, 2011). We experimented with three publicly
available implementations of DPM of Gaussian using full covariance matrices, but found that none
of them were sufficiently numerically robust to cope with the high dimensional and sometimes
ill conditioned gene expression data analysed in Section 5. To provide a similar comparison we
implemented a DPM of diagonal covariance Gaussians using a collapsed Gibbs sampler.
5
1.1
1
0.9
RAND index
0.8
0.7
0.6
0.5
DPVC MCMC
0.4
K?means (distance)
0.3
K?means (correlation)
DPVC VB
0.2
0.1
1
10
2
10
sample size, N
3
10
Figure 2: Performance of DPVC compared to k-means at recoverying the true partitioning used to
simulate the data.
Dataset
DPVC
NSFA
DPM
FA (K = 5)
FA (K = 10)
FA (K = 20)
Breast cancer ?0.876 ? 0.024 ?0.634 ? 0.038 ?1.348 ? 0.108 ?1.129 ? 0.043 ?1.275 ? 0.056 ?1.605 ? 0.072
Yeast
?0.849 ? 0.012 ?0.653 ? 0.061 ?1.397 ? 0.419 ?1.974 ? 1.925 ?1.344 ? 0.165 ?1.115 ? 0.052
Table 1: Predictive performance (mean log predictive loglikelihood over the test elements) results
on two gene expression datasets.
5.1
Synthetic data
In order to test the ability of the models to recover a true underlying partitioning of the variables
into correlated groups we use synthetic data. We generate synthetic data with D = 20 dimensions
partitioned into K = 5 equally sized clusters (of four variables). Within each cluster we sample
analoguously to our model: sample xkn ? N (0, 1) for all k, n, then gd ? N (0, 1) for all d and
finally sample ydn ? N (gd xcd n , 0.1) for all d, n. We vary the sample size N and perform 10
repeats for each sample size. We compare k-means (with the true number of clusters 5) using
Euclidean distance and correlation distance, and DPVC with inference using MCMC or variational
Bayes. To compare the inferred and true partitions we calculate the well known Rand index, which
varies between 0 and 1, with 1 denoting perfect recovering of the true clustering. The results are
shown in Figure 2. We see that the MCMC implementation of DPVC consistently outperforms
the k-means methods. As expected given the nature of the data simulation, k-means using the
correlation distance performs better than using Euclidean distance. DPVC VB?s performance is
somewhat disappointing, suggesting that even the structured variational posterior we use is a poor
approximation of the true posterior. We emphasise that k-means is given a significant advantage:
it is provided with the true number of clusters. In this light, the performance of DPVC MCMC is
impressive, and the seemingly poor performance of DPVC VB is more forgivable (DPVC VB used
a truncation level T = D = 20).
5.2
Breast cancer dataset
We assess these algorithms in terms of predictive performance on the breast cancer dataset of West
et al. (2007), including 226 genes across 251 individuals. The samplers were found to have converged after around 500 samples according to standard multiple chain convergence measures, so
1000 MCMC iterations were used for all models. The predictive log likelihood was calculated using
every 10th sample form the final 500 samples. We ran 10 repeats holding out a different random 10%
of the the elements of the matrix as test data each time. The results are shown in Table 1. We see
that NSFA performs the best, followed by DPVC. This is not surprising and is the price DPVC pays
for a more interpretable solution. However, DPVC does outperform both the DPM and the finite
(non-sparse) factor analysis models. We also ran DPVC VB on this dataset but its performance was
6
20
20
20
40
40
40
60
60
60
80
80
80
100
100
100
120
120
120
140
140
140
160
160
160
180
180
180
200
200
200
220
220
50
100
150
200
220
50
100
150
200
50
100
150
200
Figure 3: Clustering of the covariance structure. Left: k-means using correlation distance. Middle:
Agglomerative heirarchical clustering using average linkage and correlation distance. Right: DPVC
MCMC.
significantly below that of the MCMC method, with a predictive log likelihood of ?1.154 ? 0.010.
Performing a Gene Ontology enrichment analysis we find clusters enriched for genes involved in
both cell cycle regulation and cell division, which is biologically reasonable in a cancer orientated
dataset
On this relatively small dataset it is possible to visualise the D ? D empirical correlation matrix
of the data, and investigate what structure our clustering has uncovered, as shown in Figure 3. The
genes have been reordered in each plot according three different clusterings coming from k-means,
hierarchical clustering and DPVC (MCMC, note we show the clustering corresponding to the posterior sample with the highest joint probability). For both k-means and hierarchical clustering it
was necessary to ?tweak? the number of clusters to give a sensible result. Hierarchical clustering in
particular appeared to have a strong bias towards putting the majority of the genes in one large cluster/clade. Note that such a visualisation is straightforward only because we have used a clustering
based method rather than a factor analysis model, emphasising how partitionings can be more useful
summaries of data for certain tasks than low dimensional embeddings.
5.3
Yeast in varying environmental conditions
We use the data set of (Gasch et al., 2000), a collection of N = 175 non-cell-cycle experiments
on S. cerevisiae (yeast), including conditions such as heat shock, nitrogen depletion and amino acid
starvation. Measurements are available for D = 6152 genes. Again we ran 10 repeats holding
out a different random 10% of the the elements of the matrix as test data each time. The results
shown in Table 1 are broadly consistent with our findings for the breat cancer dataset: DPVC sits
between NSFA and the less performant DPM and FA models. Running 1000 iterations of DPVC
MCMC on this dataset takes around 1.2 hours on a standard dual core desktop running at 2.5GHz
with 4Gb RAM. Unfortunately we were unable to run the VB algorithm on a dataset of this size due
to memory constraints.
6
Discussion
We have introduced DPVC, a model for clustering variables into highly correlated subsets. While,
as expected, we found the predictive performance of DPVC is somewhat worse than that of state of
the art nonparametric sparse factor analysis models (e.g. NSFA), DPVC outperforms both nonparametric mixture models and Bayesian factor analysis models when applied to high dimensional data
such as gene expression microarrays. For a practitioner we see interpretability as the key advantage
of DPVC relative to a model such as NSFA: one can immediately see which groups of variables are
correlated, and use this knowledge to guide further analysis. An example use one could envisage
would be using DPVC in an analoguous fashion to principal components regression: regressing a
dependent variable against the inferred factors X. Regression coefficients would then correspond to
the predictive ability of the clusters of variables.
7
7
Acknowledgements
This work was supported by the Engineering and Physical Sciences Research Council (EPSRC)
under Grant Number EP/I036575/1 and EP/H019472/1.
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9
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3,954 | 458 | A Connectionist Learning Approach to Analyzing
Linguistic Stress
Prahlad Gupta
Department of Psychology
Carnegie Mellon University
Pittsburgh, PA 15213
David S. Touretzky
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
Abstract
We use connectionist modeling to develop an analysis of stress systems in terms
of ease of learnability. In traditional linguistic analyses, learnability arguments
determine default parameter settings based on the feasibilty of logicall y deducing
correct settings from an initial state. Our approach provides an empirical alternative to such arguments. Based on perceptron learning experiments using data
from nineteen human languages, we develop a novel characterization of stress
patterns in terms of six parameters. These provide both a partial description of the
stress pattern itself and a prediction of its learnability, without invoking abstract
theoretical constructs such as metrical feet. This work demonstrates that machine learning methods can provide a fresh approach to understanding linguistic
phenomena.
1 LINGUISTIC STRESS
The domain of stress systems in language is considered to have a relatively good linguistic
theory, called metrical phonologyl. In this theory, the stress patterns of many languages
can be described concisely, and characterized in terms of a set of linguistic "parameters,"
such as bounded vs. unbounded metrical feet, left vs. right dominant feet, etc. 2 In many
languages, stress tends to be placed on certain kinds of syllables rather than on others; the
former are termed heavy syllables, and the latter light syllables. Languages that distinguish
lFor an overview of the theory, see [Goldsmith 90, chapter 4].
2See [Dresher 90] for one such parameter scheme.
225
226
Gupta and Touretzky
OUTPUT UNIT
(PERCEPTRON)
Input
INPUT LAYER (2 x 13 units)
Figure 1: Perceptron model used in simulations.
between heavy and light syllables are termed quantity-sensitive (QS), while languages that
do not make this distinction are termed quantity-insensitive (QI). In some QS languages,
what counts as a heavy syllable is a closed syllable (a syllable that ends in a consonant),
while in others it is a syllable with a long vowel. We examined the stress patterns of
nineteen QI and QS systems, summarized and exemplified in Table 1. The data were drawn
primarily from descriptions in [Hayes 80].
2 PERCEPTRON SIMULATIONS
In separate experiments, we trained a perceptron to produce the stress pattern of each of
these languages. 1\\10 input representations were used. In the syllabic representation, used
for QI patterns only, a syllable was represented as a [11] vector, and [00] represented no
syllable. In the weight-string representation, which was necessary for QS languages, the
input patterns used were [1 0] for a heavy syllable, [0 1] for a light syllable, and [00] for
no syllable. For stress systems with up to two levels of stress, the output targets used in
training were 1.0 for primary stress, 0.5 for secondary stress, and 0 for no stress. For stress
systems with three levels of stress, the output targets were 0.6 for secondary stress, 0.35 for
tertiary stress, and 1.0 and 0 respectively for primary stress and no stress. The input data set
for all stress systems consisted of all word-forms of up to seven syllables. With the syllabic
input representation there are 7 of these, and with the weight-string representation, there are
255. The perceptron's input array was a buffer of 13 syllables; each word was processed
one syllable at a time by sliding it through the buffer (see Figure 1). The desired output at
each step was the stress level of the middle syllable of the buffer. Connection weights were
adjusted at each step using the back-propagation learning algorithm [Rumelhart 86]. One
epoch consisted of one presentation of the entire training set. The network was trained for
as many epochs as necessary to ensure that the stress value produced by the perceptron was
within 0.1 of the target value, for each syllable of the word, for all words in the training set.
A learning rate of 0.05 and momentum of 0.90 was used in all simulations. Initial weights
were uniformly distributed random values in the range ?0.5. Each simulation was run at
least three times, and the learning times averaged.
Connectionist Learning and Linguistic Stress
REF LANGUAGE
DESCRIPTION OF S1RESS PATIERN
Quantity-Insensitive Languages:
Ll
Latvian
Fixed word-initial stress.
L2
French
Fixed word-final stress.
L3
Maranungku Primary stress on first syllable, secondary stress on alternate
succeeding syllables.
Weri
L4
Primary stress on last syllable, secondary stress on alternate
preceding syllables.
L5
Garawa
Primary stress on first syllable, secondary stress on penultimate syllable, tertiary stress on alternate syllables preceding
the penUlt, no stress on second syllable.
Lakota
L6
Primary stress On second syllable.
L7
Swahili
Primary stress on penultimate syllable.
L8
Paiute
Primary stress on second syllable, secondary stress on alternate succeeding syllables.
Warao
L9
Primary stress on penultimate syllable. secondary stress on
alternate preceding syllables.
Quantity-Sensitive Languages:
LlO Koya
Primary stress on first syllable, secondary stress on heavy
syllables.
(Heavy = closed syllable or syllable with long vowel.)
L11
Eskimo
(Primary) stress on final and heavy syllables.
(Heavy = closed syllable.)
L12 Gurkhali
Primary stress on first syllable except when first syllable light
and second syllable heavy.
(Heavy = long vowel.)
Primary stress on last syllable except when last is light and
L13 Yapese
penultimate heavy.
(Heavy = long vowel.)
Primary stress on first syllable if heavy. else on second sylL14 Ossetic
lable.
(Heavy = long vowel.)
L15 Rotuman
Primary stress on last syllable if heavy. else on penultimate
syllable.
JHeavy =long vowel.)
L16 Komi
Primary stress on first heavy syllable. or on last syllable if
none heavy.
(Heavy = long vowel.)
L17 Cheremis
Primary stress on last heavy syllable. or on first syllable if
none heavy.
(Heavy = long vowel.)
Primary stress on first heavy syllable. or on first syllable if
L18 Mongolian
none heavy.
(Heavy =long vowel.)
L19 Mayan
Primary stress on last heavy syllable. or on last syllable if
none heavy.
(Heavy = long vowel.)
EXAMPLES
SlSOSOSOSOSoSo
SOSoSoSOSOSOSI
SlSOS2S0S2S0S2
S2S0S2S0S2S0S1
SlSOSOS3S0S2S0
SOSI SOSOSoSOSo
SOSOSOSOSOSlSO
SOSlSOS2SOS2S0
SOS2S0S2SOS1S0
LILoLoH2LoLoLo
LILoLoLoLoLoLo
LOLoLoHILoLoLl
LOLoLoLoLoLoLI
LILoL?J-fJLoLoLo
L?HI L0J-fJL?LoLo
LOLoL?J-fJLoLoLl
L?J-fJL?J-fJL?HIL?
?
HI L?L?J-fJL?L L?
LOLILoLoLoLoLo
LOLoL?J-fJLoLoHl
LOLoLoLoLoLlLo
L?L?H1L?L?J-fJL?
LOLoLoLoLoLoLl
LOL?J-fJLoLoH1L
LILoLoLoLoLoLo
?
L?L ?H1LoL?J-fJL?
LILoLoLoLoLoLo
LOL?J-fJLoLoHlLo
LOLoLoLoLoLoLI
Table 1: Stress patterns: description and example stress assignment. Examples are of
stress assignment in seven-syllable words. Primary stress is denoted by the superscript 1
(e.g., Sl), secondary stress by the superscript 2, tertiary stress by the superscript 3, and no
stress by the superscript O. "S" indicates an arbitrary syllable, and is used for the QI stress
patterns. For QS stress patterns, "H" and "L" are used to denote Heavy and Light syllables,
respectivel y.
227
228
Gupta and Touretzky
3 PRELIMINARY ANALYSIS OF LEARNABILITY OF STRESS
The learning times differ considerably for {Latvian, French}, {Maranungku, Weri} ,
{Lakota, Polish} and Garawa, as shown in the last column of Table 2. Moreover, Paiute
and Warao were unlearnable with this mode1. 3 Differences in learning times for the various stress patterns suggested that the factors ("parameters") listed below are relevant in
determining learnability.
1. Inconsistent Primary Stress (IPS): it is computationally expensive to learn the pattern
if neither edge receives primary stress except in mono- and di-syllables; this can be
regarded as an index of computational complexity that takes the values {O, I}: 1 if an
edge receives primary stress inconsistently, and 0, otherwise.
2. Stress clash avoidance (SeA): if the components of a stress pattern can potentially
lead to stress clash4, then the language may either actually permit such stress clash, or
it may avoid it. This index takes the values {O, I}: 0 if stress clash is permitted, and
1 if stress clash is avoided.
3. Alternation (AIt): an index of learnability with value 0 if there is no alternation, and
value 1 if there is. Alternation refers to a stress pattern that repeats on alternate
syllables.
4. Multiple Primary Stresses (MPS): has value 0 if there is exactly one primary stress,
and value 1 if there is more then one primary stress. It has been assumed that a repeating
pattern of primary stresses will be on alternate, rather than adjacent syllables. Thus,
[Alternation=O] implies [MPS=O]. Some of the hypothetical stress patterns examined
below include ones with more than one primary stress; however, as far as is known,
no actually occurring QI stress pattern has more than one primary stress.
5. Multiple Stress Levels (MSL): has value 0 if there is a single level of stress (primary
stress only), and value 1 otherwise.
Note that it is possible to order these factors with respect to each other to form a fivedigit binary string characterizing the ease/difficulty of learning. That is, the computational
complexity of learning a stress pattern can be characterized as a 5-bit binary number whose
bits represent the five factors above, in decreasing order of significance. Table 2 shows
that this characterization captures the learning times of the QI patterns quite accurately. As
an example of how to read Thble 2, note that Garawa takes longer to learn than Latvian
(165 vs. 17 epochs). This is reflected in the parameter setting for Garawa, "01101", being
lexicographically greater than that for Latvian, "00000". A further noteworthy point is
that this framework provides an account of the non-learnability of Paiute and Warao, viz,.
that stress patterns whose parameter string is lexicographically greater than "10000" are
unlearnable by the perceptron.
4 TESTING THE QI LEARNABILITY PREDICTIONS
We devised a series of thirty artificial QI stress patterns (each a variation on some language
in Table 1) to examine our parameter scheme in more detail. The details of the patterns
3They were learnable in a three-layer model, which exhibited a similar ordering of learning times
[Gupta 92].
4Placement of stress on adjacent syllables.
229
Connectionist Learning and Linguistic Stress
IPS
SCA
Alt
MPS
MSL
0
0
0
0
0
0
0
1
0
1
0
1
1
0
1
0
0
0
1
0
1
0
1
0
1
QI LANGUAGES
Latvian
French
Maranungku
Weri
Garawa
Lakota
Swahili
Paiute
Warao
REF
Ll
L2
L3
L4
L5
L6
L7
L8
L9
EPOCHS
(syllabic)
17
16
37
34
165
255
254
**
**
Table 2: Preliminary analysis of learning times for QI stress systems. using the syllabic
input representation. IPS=Inconsistent Primary Stress; SCA=Stress Clash Avoidance;
Alt=Altemation; MPS=Multiple Primary Stresses; MSL=Multiple Stress Levels. References LI-L9 refer to Table 1.
II
Agg
0
I IPS
0
SCA
0
0
0
0
0
0
0
0
0
I Alt I MPS I MSL
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0.25
1
0
1
0
0
0
1
0
0
0.50
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
1
1
0
0
0
0
0
2
0
0
0
0
0
" QI LANGS
Latvian
French
Maranungku
Weri
Garawa
Lakota
Swahili
Paiute
Warao
I REF I TIME
Ll
L2
L3
L4
L5
L6
L7
L8
L9
" QS LANGS
I REF I TIME II
2
2
Koya
Eskimo
LI0
Lll
2
3
Gurkhali
Yapese
Ossetic
Rotuman
L12
L13
L14
L15
19
19
30
29
Komi
Cheremis
Mongolian
Mayan
L16
L17
U8
U9
216
212
2306
2298
3
3
7
10
10
**
**
Table 3: Summary of results and analysis of QI and QS learning (using weightstring input representations). Agg=Aggregative Information; IPS=Inconsistent Primary
Stress; SCA=Stress Clash Avoidance; Alt=Altemation; MPS=Multiple Primary Stresses;
MSL=Multiple Stress Levels. References index into Table 1. Time is learning time in
epochs.
230
Gupta and Touretzky
are not crucial for present purposes (see [Gupta 92] for details). What is important to
note is that the learnability predictions generated by the analytical scheme described in
the previous section show good agreement with actual perceptron learning experiments on
these patterns.
The learning results are summarized in Table 4. It can be seen that the 5-bit characterization
fits the learning times of various actual and hypothetical patterns reasonably well (although
there are exceptions - for example, the hypothetical stress patterns with reference numbers
h21 through h25 have a higher 5-bit characterization than other stress patterns, but lower
learning times.) Thus, the "complexity measure" suggested here appears to identify a
number of factors relevant to the learnability of QI stress patterns within a minimal twolayer connectionist architecture. It also assesses their relative impacts. The analysis is
undoubtedly a Simplification, but it provides a completely novel framework within which
to relate the various learning results. The important point to note is that this analytical
framework arises from a consideration of (a) the nature of the stress systems, and (b) the
learning results from simulations. That is, this framework is empirically based, and makes
no reference to abstract constructs of the kind that linguistic theory employs. Nevertheless,
it provides a descriptive framework, much as the linguistic theory does.
5
INCORPORATING QS SYSTEMS INTO THE ANALYSIS
Consideration of the QS stress patterns led to refinement of the IPS parameter without
changing its setting for the QI patterns. This parameter is modified so that its value indicates
the proportion of cases in which primary stress is not assigned at the edge of a word.
Additionally, through analysis of connection weights for QS patterns, a sixth parameter,
Aggregative Information, is added as a further index of computational complexity.
6. Aggregative Information (Agg) : has value 0 if no aggregative information is required
(single-positional information suffices); 1 if one kind of aggregative information is
required; and 2 if two kinds of aggregative information are required.
Detailed discussion of the analysis leading to these refinements is beyond the scope of
this paper; the interested reader is referred to [Gupta 92]. The point we wish to make
here is that, with these modifications, the same parameter scheme can be used for both
the QI and QS language classes, with good learnability predictions within each class, as
shown in Table 3. Note that in this table, learning times for all languages are reported in
terms of the weight-string representation (255 input patterns) rather than the unweighted
syllabic representation (7 input patterns) used for the initial QI studies. Both the QI and QS
results fall into a single analysis within this generalized parameter scheme and weight-string
representation, but with a less perfect fit than the within-class results.
6
DISCUSSION
Traditional linguistic analysis has devised abstract theoretical constructs such as "metrical
foot" to describe linguistic stress systems. Learnability arguments were then used to
determine default parameter settings (e.g., whether feet should by default be assumed
to be bounded or unbounded, left or right dominant, etc.) based on the feasibility of
logically deducing correct settings from an initial state. As an example, in one analysis
Connectionist Learning and Linguistic Stress
IPS
SCA
Alt
MPS
MSL
LANGUAGE
REF
0
0
0
Ll
L2
0
0
1
Latvian
French
Latvian2stress
Latvian3stress
French2stress
French 3stress
Latvian2edge
Latvian2edge2stress
0
0
1
1
1
1
0
0
0
1
0
1
1
1
0
1
1
1
0
0
1
0
0
1
0
1
1
1
1
0
0
1
1
1
0
0
1
1
1
1
1
0
1
1
1
0
1
0
1
0
1
1
1
0
0
0
0
1
1
1
0
1
0
1
1
1
1
1
1
1
0
1
1
1
hI
h2
h3
h4
h5
h6
EPOCHS
(syllabic)
17
16
21
11
23
14
30
37
impossible
Maranungku
Weri
Maranungku3stress
Weri3stress
Latvian2edge2stress-alt
Garawa-SC
Garawa2stress-SC
Maranungku 1stress
Weri 1stress
Latvian2edge-alt
Garawal stress-SC
Latvian2edge2stress-lalt
L3
L4
h7
h8
h9
hl0
hll
h12
h13
h14
h15
h16
37
34
43
41
58
38
50
61
65
78
88
85
impossible
Garawa-non-alt
Latvian3stress2edge-SCA
Latvian2edge-SCA
Latvian2edge2stress-SCA
Garawa
Garawa2stress
Latvian2edge2stress-alt-SCA
Garawalstress
Latvian2edge-alt-SCA
Latvian2edge2stress-lalt-SCA
Lakota
Swahili
Lakota2stress
Lakota2edge
Lakota2edge2stress
Paiute
Warao
Lakota-alt
Lakota2stress-alt
h17
h18
h19
h20
L5
h21
h22
h23
h24
h25
L6
L7
h26
h27
h28
L8
L9
h29
h30
164
163
194
206
165
71
91
121
126
129
255
254
**
**
**
**
**
**
**
Table 4: Analysis of Quantity-Insensitive learning using the syllabic input representation. IPS=Inconsistent Primary Stress; SCA=Stress Clash Avoidance; Alt=Altemation;
MPS=Multiple Primary Stresses; MSL=Multiple Stress Levels. References LI-L9 index
into Table 1.
231
232
Gupta and Touretzky
[Dresher 90, p. 191], "metrical feet" are taken to be "iterative" by default, since there is
evidence that can cause revision of this default if it turns out to be the incorrect setting,
but there might not be such disconfirming evidence if the feet were by default taken to be
"non-iterative". We provide an alternative to logical deduction arguments for determining
"markedness" of parameter values, by measuring learnability (and hence markedness)
empirically. The parameters of our novel analysis generate both a partial description of
each stress pattern and a prediction of its learnability. Furthermore, our parameters encode
linguistically salient concepts (e.g., stress clash avoidance) as well as concepts that have
computational significance (single-positional vs. aggregative information.) Although our
analyses do not explicitly invoke theoretical linguistic constructs such as metrical feet, there
are suggestive similarities between such constructs and the weight patterns the perceptron
develops [Gupta 91].
In conclusion, this work offers a fresh perspective on a well-studied linguistic domain, and
suggests that machine learning techniques in conjunction with more traditional tools might
provide the basis for a new approach to the investigation of language.
Acknowledgements
We would like to acknowledge the feedback prOvided by Deirdre Wheeler throughout
the course of this work. The first author would like to thank David Evans for access
to exceptional computing facilities at Carnegie Mellon's Laboratory for Computational
Linguistics, and Dan Everett, Brian MacWhinney, Jay McClelland, Eric Nyberg, Brad
Pritchett and Steve Small for helpful discussion of earlier versions of this paper. Of course,
none of them is responsible for any errors.
The second author was supported by a grant from Hughes Aircraft Corporation, and by the
Office of Naval Research under contract number NOOOI4-86-K-0678.
References
[Dresher 90] Dresher, B., & Kaye, J., A Computational Learning Model for Metrical
Phonology, Cognition 34, 137-195.
[Goldsmith 90] Goldsmith, J., Autosegmental and Metrical Phonology, Basil Blackwell,
Oxford, England, 1990.
[Gupta 91] Gupta, P. & Touretzky, D., What a perceptron reveals about metrical phonology.
Proceedings of the Thirteenth Annual Conference of the Cognitive Science Society, 334339. Lawrence Erlbaum, Hillsdale, NJ, 1991.
[Gupta 92] Gupta, P. & Touretzky, D., Connectionist Models and Linguistic Theory: Investigations of Stress Systems in Language. Manuscript.
[Hayes 80] Hayes, B., A Metrical Theory of Stress Rules, doctoral dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1980. Circulated by the Indiana
University Linguistics Club, 1981.
[Rumelhart 86] Rumelhart, D., Hinton, G., & Williams, R, Learning Internal Representations by Error Propagation, in D. Rumelhart, J. McClelland & the PDP Research Group.
Parallel Distributed Processing. Volume 1: Foundations, MIT Press, Cambridge, MA,
1986.
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3,955 | 4,580 | Hierarchical Optimistic Region Selection driven by
Curiosity
Odalric-Ambrym Maillard
Lehrstuhl f?ur Informationstechnologie
Montanuniversit?at Leoben
Leoben, A-8700, Austria
[email protected]
Abstract
This paper aims to take a step forwards making the term ?intrinsic motivation?
from reinforcement learning theoretically well founded, focusing on curiositydriven learning. To that end, we consider the setting where, a ?xed partition P of
a continuous space X being given, and a process ? de?ned on X being unknown,
we are asked to sequentially decide which cell of the partition to select as well as
where to sample ? in that cell, in order to minimize a loss function that is inspired
from previous work on curiosity-driven learning. The loss on each cell consists of
one term measuring a simple worst case quadratic sampling error, and a penalty
term proportional to the range of the variance in that cell. The corresponding
problem formulation extends the setting known as active learning for multi-armed
bandits to the case when each arm is a continuous region, and we show how an
adaptation of recent algorithms for that problem and of hierarchical optimistic
sampling algorithms for optimization can be used in order to solve this problem.
The resulting procedure, called Hierarchical Optimistic Region SElection driven
by Curiosity (HORSE.C) is provided together with a ?nite-time regret analysis.
1
Introduction
In this paper, we focus on the setting of intrinsically motivated reinforcement learning (see Oudeyer
and Kaplan [2007], Baranes and Oudeyer [2009], Schmidhuber [2010], Graziano et al. [2011]),
which is an important emergent topic that proposes new dif?cult and interesting challenges for the
theorist. Indeed, if some formal objective criteria have been proposed to implement speci?c notions
of intrinsic rewards (see Jung et al. [2011], Martius et al. [2007]), so far, many - and only - experimental work has been carried out for this problem, often with interesting output (see Graziano et al.
[2011], Mugan [2010], Konidaris [2011]) but unfortunately no performance guarantee validating a
proposed approach. Thus proposing such an analysis may have great immediate consequences for
validating some experimental studies.
Motivation. A typical example is the work of Baranes and Oudeyer [2009] about curiosity-driven
learning (and later on Graziano et al. [2011], Mugan [2010], Konidaris [2011]), where a precise
algorithm is de?ned together with an experimental study, yet no formal goal is de?ned, and no
def
analysis is performed as well. They consider a so-called sensory-motor space X = S ?M ? [0, 1]d
where S is a (continuous) state space and M is a (continuous) action space. There is no reward, yet
one can consider that the goal is to actively select and sample subregions of X for which a notion of
?learning progress? - this intuitively measures the decay of some notion of error when successively
sampling into one subregion - is maximal. Two key components are advocated in Baranes and
Oudeyer [2009], in order to achieve successful results (despite that success is a fuzzy notion):
? The use of a hierarchy of regions, where each region is progressively split into sub-regions.
1
? Splitting leaf-regions in two based on the optimization of the dissimilarity, amongst the
regions, of the learning progress. The idea is to identify regions with a learning complexity that is a globally constant in that region, which also provides better justi?cation for
allocating samples between identi?ed regions.
We believe it is possible to go one step towards a full performance analysis of such algorithms, by
relating the corresponding active sampling problem to existing frameworks.
Contribution. This paper aims to take a step forwards making the term ?intrinsic motivation? from
reinforcement learning theoretically well founded, focusing on curiosity-driven learning. We introduce a mathematical framework in which a metric space (which intuitively plays the role of the
state-action space) is divided into regions and a learner has to sample from an unknown random function in a way that reduces a notion of error measure the most. This error consists of two terms, the
?rst one is a robust measure of the quadratic error between the observed samples and their unknown
mean, the second one penalizes regions with non constant learning complexity, thus enforcing the
notion of curiosity. The paper focuses on how to choose the region to sample from, when a partition
of the space is provided.
The resulting problem formulation can be seen as a non trivial extension of the setting of active
learning in multi-armed bandits (see Carpentier et al. [2011] or Antos et al. [2010]), where the main
idea is to estimate the variance of each arm and sample proportionally to it, to the case when each
arm is a region as opposed to a point. In order to deal with this dif?culty, the maximal and minimal
variance inside each region is tracked by means of a hierarchical optimization procedure, in the spirit
of the HOO algorithm from Bubeck et al. [2011]. This leads to a new procedure called Hierarchical
Optimistic Region SElection driven by Curiosity (HORSE.C) for which we provide a theoretical
performance analysis.
Outline. The outline of the paper is the following. In Section 2 we introduce the precise setting and
de?ne the objective function. Section 3 de?nes our assumptions. Then in Section 4 we present the
HORSE.C algorithm. Finally in Section 5, we provide the main Theorem 1 that gives performance
guarantees for the proposed algorithm.
2
Setting: Robust region allocation with curiosity-inducing penalty.
Let X assumed to be a metric space and let Y ? Rd be a normed space, equipped with the Euclidean
norm || ? ||. We consider an unknown Y-valued process de?ned on X , written ? : X ? M+
1 (Y),
(Y)
refers
to
the
set
of
all
probability
measures
on
Y,
such
that
for
all
x
?
X
,
the
random
where M+
1
variable Y ? ?(x) has mean ?(x) ? Rd and covariance matrix ?(x) ? Md,d (R) assumed to be
def
diagonal. We thus introduce for convenience the notation ?(x) = trace(?(x)), where trace is
the trace operator (this corresponds to the variance in dimension 1). We call X the input space or
sampling space, and Y the output space or value space.
def
Intuition Intuitively when applied to the setting of Baranes and Oudeyer [2009], then X = S ? A
is the space of state-action pairs, where S is a continuous state space and A a continuous action
def
space, ? is the transition kernel of an unknown MDP, and ?nally Y = S. This is the reason why
d
we consider Y ? R and not only Y ? R as would seem more natural. One difference is that
we assume (see Section 3) that we can sample anywhere in X , which is a restrictive yet common
assumption in the reinforcement learning literature. How to get rid of this assumption is an open
and challenging question that is left for future work.
Sampling error and robustness Let us consider a sequential sampling process on X , i.e. a process
that samples at time t a value Yt ? ?(Xt ) at point Xt , where Xt ? F<t is a measurable function of
the past inputs and outputs {(Xs , Ys )}s<t . It is natural to look at the following quantity, that we call
t
average noise vector ?t :
?
def 1
?t =
Ys ? ?(Xs ) ? Rd .
t s=1
One interesting property is that this is a martingale difference sequence, which means that the norm
of this vector enjoys a concentration property. More precisely (see [Maillard, 2012, Lemma 1] in
the extended version of the paper), we have for all?deterministic
t
?t > 0
1 1?
2
?(Xs ) .
E[ ||?t || ] = E
t
t s=1
2
A similar property holds for a region R ? X that has been sampled nt (R) times, and in order to be
robust against a bad sampling strategy inside a region, it is natural to look at the worst case error,
that we de?ne as
def supx?R ?(x)
.
eR (nt ) =
nt (R)
One reason for looking at robustness is that for instance, in the case we work with an MDP, we are
generally not completely free to choose the sample Xs ? S ? A: we can only choose the action and
the next state is generally given by Nature. Thus, it is important to be able to estimate this worst
case error so as to prevent from bad situations.
Goal Now let P be a ?xed, known partition of the space X and consider the following game. The
goal of an algorithm is, at each time step t, to propose one point xt where to sample the space
X , so that its allocation of samples {nt (R)}R?P (that is, the number of points sampled in each
region) minimizes some objective function. Thus, the algorithm is free to sample everywhere in
each region, with the goal that the total number of points chosen in each region is optimal in some
sense. A simple candidate for this objective function would be the following
?
?
def
LP (nt ) = max eR (nt ) ; R ? P ,
however, in order to incorporate a notion of curiosity, we would also like to penalize regions that
have a variance term ? that is non homogeneous (i.e. the less homogeneous, the more samples we
allocate). Indeed, if a region has constant variance, then we do not really need to understand more
its internal structure, and thus its better to focus on an other region that has very heterogeneous
variance. For instance, one would like to split such a region in several homogeneous parts, which
is essentially the idea behind section C.3 of Baranes and Oudeyer [2009]. We thus add a curiositypenalization term to the previous objective function, which leads us to de?ne the pseudo-loss of an
def
allocation nt = {nt (R)}R?P in the following way:
?
?
def
(1)
LP (nt ) = max eR (nt ) + ?|R|(max ?(x) ? min ?(x)) ; R ? P .
x?R
x?R
Indeed, this means that we do not want to focus just on regions with high variance, but also trade-off
with highly heterogeneous regions, which is coherent with the notion of curiosity (see Oudeyer and
Kaplan [2007]). For convenience, we also de?ne the pseudo-loss of a region R by
def
LR (nt ) = eR (nt ) + ?|R|(max ?(x) ? min ?(x)) .
x?R
x?R
Regret The regret (or loss) of an allocation algorithm at time T is de?ned as the difference between
the cumulated pseudo-loss of the allocations nt = {nR,t }R?P proposed by the algorithm and that
of the best allocation strategy n?t = {n?R,t }R?P at each time steps; we de?ne
def
RT =
T
?
t=|P|
LP (nt ) ? LP (n?t ) ,
where an optimal allocation at time t is de?ned by
?
?
?
nt (R) = t .
n?t ? argmin LP (nt ) ; {nt (R)}R?P is such that
R?P
Note that the sum starts at t = |P| for a technical reason, since for t < |P|, whatever the allocation,
there is always at least one region with no sample, and thus LP (nt ) = ?.
Example 1 In the special case when X = {1, . . . , K} is ?nite with K ? T , and when P is the
complete partition (each cell corresponds to exactly one point), the penalization term is canceled.
Thus the problem reduces to the choice of the quantities nt (i) for each arm i, and the loss of an
?
?
allocation simply becomes
?(i)
def
L(nt ) = max
;1?i?K .
nt (i)
This almost corresponds to the already challenging setting analyzed for instance in Carpentier et al.
[2011] or Antos et al. [2010]. The difference is that we are interested in the cumulative regret of
our allocation instead of only the regret suffered for the last round as considered in Carpentier et al.
whereas they consider the mean sampling
[2011] or Antos et al. [2010]. Also we directly target n?(i)
t (i)
error (but both terms are actually of the same order). Thus the setting we consider can be seen as
a generalization of these works to the case when each arm corresponds to a continuous sampling
domain.
3
3
Assumptions
In this section, we introduce some mild assumptions. We essentially assume that the unknown
distribution is such that it has a sub-Gaussian noise, and a smooth mean and variance functions.
These are actually very mild assumptions. Concerning the algorithm, we consider it can use a
partition tree of the space, and that this one is essentially not degenerated (a typical binary tree that
satis?es all the following assumptions is such that each cell is split in two children of equal volume).
Such assumptions on trees have been extensively discussed for instance in Bubeck et al. [2011].
Sampling At any time, we assume that we are able to sample at any point in X , i.e. we assume we
have a generative model1 of the unknown distribution ?.
Unknown distribution We assume that ? is sub-Gaussian, meaning that for all ?xed x ? X
?? ? Rd ln E exp[??, Y ? ?(X)?] ?
and has diagonal covariance matrix in each point2 .
?T ?(x)?
,
2
The function ? is assumed to be Lipschitz w.r.t a metric ?1 , i.e. it satis?es
?x, x? ? X ||?(x) ? ?(x? )|| ? ?1 (x, x? ) .
Similarly, the function ? is assumed to be Lipschitz w.r.t a metric ?2 , i.e. it satis?es
?x, x? ? X |?(x) ? ?(x? )| ? ?2 (x, x? ) .
Hierarchy We assume that Y is a convex and compact subset of [0, 1]d . We consider an in?nite
binary tree T whose nodes correspond to regions of X . A node is indexed by a pair (h, i), where
h ? 0 is the depth of the nodes in T and 0 ? i < 2h is the position of the node at depth h. We write
R(h, i) ? X the region associated with node (h, i). The regions are ?xed in advance, are all assumed
to be measurable with positive measure, and must satisfy that for each h ? 1, {R(h, i)}0?i<2h is a
def
partition of X that is compatible with depth h ? 1, where R(0, 0) = X ; in particular for all h ? 0,
for all 0 ? i < 2h , then
R(h, i) = R(h + 1, 2i) ? R(h + 1, 2i + 1) .
In dimension d, a standard way to de?ne such a tree is to split each parent node in half along the
largest side of the corresponding hyper-rectangle, see Bubeck et al. [2011] for details.
For a ?nite sub-tree Tt of T , we write Leaf (Tt ) for the set of all leaves of Tt . For a region (h, i) ?
Tt , we denote by Ct (h, i) the set of its children in Tt , and by Tt (h, i) the subtree of Tt starting with
root node (h, i).
Algorithm and partition The partition P is assumed to be such that each of its regions R corresponds to one region R(h, i) ? T ; equivalently, there exists a ?nite sub-tree T0 ? T such that
Leaf (T0 ) = P. An algorithm is only allowed to expand one node of Tt at each time step t. In the
sequel, we write indifferently P ? T and (h, i) ? T or P and R(h, i) ? X to refer to the partition
or one of its cell.
Exponential decays Finally, we assume that the ?1 and ?2 diameters of the region R(h, i) as well as
its volume |R(h, i)| decay at exponential rate in the sense that there exists positive constants ?, ?1 ,
?2 and c, c1 , c2 such that for all h ? 0, then |R(h, i)| ? c? h ,
max ?1 (x, x? ) ? c1 ?1h and ? max ?2 (x, x? ) ? c2 ?2h .
?
x ,x?R(h,i)
x ,x?R(h,i)
Similarly, we assume that there exists positive constants c? ? c, c?1 ? c1 and c?2 ? c2 such that for
all h ? 0, then |R(h, i)| ? c? ? h ,
max ?1 (x, x? ) ? c?1 ?1h and ? max ?2 (x, x? ) ? c?2 ?2h .
?
x ,x?R(h,i)
x ,x?R(h,i)
This assumption is made to avoid degenerate trees and for general purpose only. It actually holds
for any reasonable binary tree.
1
using the standard terminology in Reinforcement Learning.
this assumption is only here to make calculations easier and avoid nasty technical considerations that
anyway do not affect the order of the ?nal regret bound but only concern second order terms.
2
4
4
Allocation algorithm
In this section, we now introduce the main algorithm of this paper in order to solve the problem
considered in Section 2. It is called Hierarchical Optimistic Region SElection driven by Curiosity.
Before proceeding, we need to de?ne some quantities.
4.1
High-probability upper-bound and lower-bound estimations
Let us consider the following (biased) estimator
t
t
1 ?
1 ?
def
? 2t (R) =
?
||Ys ||2 I{Xs ? R} ? ||
Ys I{Xs ? R}||2 .
Nt (R) s=1
Nt (R) s=1
t (R)?1
, it has more importantly a positive bias
Apart from a small multiplicative biased by a factor NN
t (R)
due to the fact that the random variables do not share the same mean; this phenomenon is the same
as the estimation of the average variance for independent
but?
non i.i.d variables with different means
?n
n
{?i }i?n , where the bias would be given by n1 i=1 [?i ? n1 j=1 ?j ]2 (see Lemma 5). In our case,
it is thus always non negative, and under the assumption that ? is Lipschitz w.r.t the metric ?1 , it is
fortunately bounded by d1 (R)2 , the diameter of R w.r.t the metric ?1 .
We then introduce the two following key quantities, de?ned for all x ? R and ? ? [0, 1] by
?
t
?
1 ?
d ln(2d/?) d ln(2d/?)
def
2
? t (R) + (1 + 2 d)
+
+
?2 (Xs , x)I{Xs ? R},
Ut (R, x, ?) = ?
2Nt (R)
2Nt (R)
Nt (R) s=1
?
t
?
d ln(2d/?)
1 ?
def
? 2t (R) ? (1 + 2 d)
? d1 (R)2 ?
Lt (R, x, ?) = ?
?2 (Xs , x)I{Xs ? R} .
2Nt (R)
Nt (R) s=1
Note that we would have preferred to replace the terms involving ln(2d/?) with a term depending
on the empirical variance, in the spirit of Carpentier et al. [2011] or Antos et al. [2010]. However,
contrary to the estimation of the mean, extending the standard results valid for i.i.d data to the case
of a martingale difference sequence is non trivial for the estimation of the variance, especially due
to the additive bias resulting from the fact that the variables may not share the same mean, but also
to the absence of such results for U-statistics (up to the author?s knowledge). For that reason such
an extension is left for future work.
The following results (we provide the proof in [Maillard, 2012, Appendix A.3]) show that
Ut (R, x, ?) is a high probability upper bound on ?(x) while Lt (R, x, ?) is a high probability lower
bound on ?(x).
Proposition 1 Under the assumptions that Y is a convex subset of [0, 1]d , ? is sub-Gaussian, ? is
Lipschitz w.r.t. ?2 and R ? X is compact and convex, then
?
?
P ?x ? X ; Ut (R, x, ?) ? ?(x) ? t? .
Similarly, under the same assumptions, then
?
?
P ?x ? X ; Lt (R, x, ?) ? ?(x) ? b(x, R, Nt (R), ?) ? t? ,
where we introduced for convenience the quantity
b(x, R, n, ?)
def
=
?
2 max
?2 (x, x ) + d1 (R) + 2(1 + 2 d)
?
x ?R
?
2
?
d ln(2d/?) d ln(2d/?)
+
.
2n
2n
Now on the other other hand, we have that (see the proof in [Maillard, 2012, Appendix A.3])
Proposition 2 Under the assumptions that Y is a convex subset of [0, 1]d , ? is sub-Gaussian, ? is
Lipschitz w.r.t. ?1 , ? is Lipschitz w.r.t. ?2 and R ? X is compact and convex, then
?
?
P ?x ? X ; Ut (R, x, ?) ? ?(x) + b(x, R, Nt (R), ?) ? t? .
Similarly, under the same assumptions, then
?
?
P ?x ? X ; Lt (R, x, ?) ? ?(x) ? t? .
5
4.2
Hierarchical Optimistic Region SElection driven by Curiosity (HORSE.C).
The pseudo-code of the HORSE.C algorithm is presented in Figure 1 below. This algorithm relies
on the estimation of the quantities maxx?R ?(x) and minx?R ?(x) in order to de?ne which point
Xt+1 to sample at time t + 1. It is chosen by expanding a leaf of a hierarchical tree Tt ? T , in an
optimistic way, starting with a tree T0 with leaves corresponding to the partition P.
The intuition is the following: let us consider a node (h, i) of the tree Tt expanded by the algorithm
at time t. The maximum value of ? in R(h, i) is thus achieved for one of its children node (h? , i? ) ?
Ct (h, i). Thus if we have computed an upper bound on the maximal value of ? in each child, then
we have an upper bound on the maximum value of ? in R(h, i). Proceeding in a similar way for the
lower bound, this motivates the following two recursive de?nitions:
?
?
? + ? ?
?
def
+
? ?
? t (h , i ; ?) ; (h , i ) ? Ct (h, i)
? t (h, i; ?) = min
max Ut (R(h, i), x, ?) , max ?
,
?
x?R(h,i)
??
?
t (h, i; ?)
def
=
max
?
? ? ? ?
?
? t (h , i ; ?) ; (h? , i? ) ? Ct (h, i)
min Lt (R(h, i), x, ?) , min ?
x?R(h,i)
?
.
These values are used in order to build an optimistic estimate of the quantity LR(h,i) (Nt ) in region
(h, i) (step 4), and then to select in which cell of the partition we should sample (step 5). Then the
?+
algorithm chooses where to sample in the selected region so as to improve the estimations of ?
t and
?
? t . This is done by alternating (step 6.) between expanding a leaf following a path that is optimistic
?
??
?+
according to ?
t (step 7,8,9), or according to ?
t (step 11.)
Thus, at a high level, the algorithm performs on each cell (h, i) ? P of the given partition two
hierarchical searches, one for the maximum value of ? in region R(h, i) and one for its minimal
value. This can be seen as an adaptation of the algorithm HOO from Bubeck et al. [2011] with the
main difference that we target the variance and not just the mean (this is more dif?cult). On the other
hand, there is a strong link between step 5, where we decide to allocate samples between regions
{R(h, i)}(h,i)?P , and the CH-AS algorithm from Carpentier et al. [2011].
5
Performance analysis of the HORSE.C algorithm
In this section, we are now ready to provide the main theorem of this paper, i.e. a regret bound on
the performance of the HORSE.C algorithm, which is the main contribution of this work. To this
end, we make use of the notion of near-optimality dimension, introduced in Bubeck et al. [2011],
and that measures a notion of intrinsic dimension of the maximization problem.
De?nition (Near optimality dimension) For c > 0, the c-optimality dimension of ? restricted to
the region R with respect to the pseudo-metric ?2 is de?ned as
?
?
?
?
ln(N (Rc? , ?2 , ?))
def
,
0
where
R
=
x
?
R
;
?(x)
?
max
?(x)
?
?
,
max lim sup
c?
x?R
ln(??1 )
??0
and where N (Rc? , ?2 , ?) is the ?-packing number of the region Rc? .
Let d+ (h0 , i0 ) be the c-optimality dimension of ? restricted to the region R(h0 , i0 ) (see e.g. Bubeck
def
et al. [2011]), with the constant c = 4(2c2 + c21 )/c?2 . Similarly, let d? (h0 , i0 ) be the c-optimality
dimension of ?? restricted to the region R(h0 , i0 ). Let us ?nally de?ne the biggest near-optimality
dimension of ? over each cell of the partition P to be
?
?
?
?
def
d? = max max d+ (h0 , i0 ), d? (h0 , i0 ) ; (h0 , i0 ) ? P .
Theorem 1 (Regret bound for HORSE.C) Under the assumptions of Section 3 and if moreover
?12 ? ?2 , then for all ? ? [0, 1], the regret of the Hierarchical Optimistic Region SElection driven by
Curiosity procedure parameterized with ? is bounded with probability higher than 1 ? 2? as follows.
?
?
T
?
?
?
1
h0
+
2?c?
RT ?
B h0 , n?t (h0 , i0 ), ?t ,
max
?
nt (h0 , i0 )
(h0 ,i0 )?P
t=|P|
6
Algorithm 1 The HORSE.C algorithm.
Require: An in?nite binary tree T , a partition P ? T , ? ? [0, 1], ? ? 0
6?
1: Let T0 be such that Leaf (T0 ) = P, and ?i,t = ?2 i2 (2t+1)|P|t
3 , t := 0.
2: while true do
3:
de?ne for each region (h, i) ? Tt the estimated loss
? +
?
? + (h, i; ?)
def ?
? t (h, i; ?) ? ?
??
+ ?|R(h, i)| ?
L?t (h, i) = t
t (h, i; ?) ,
Nt (R(h, i))
where ? = ?Nt (R(h,i)),t , where by convention L?t (h, i) if it is unde?ned.
4:
choose the next region of the current partition P ? T to sample
?
?
def
(Ht+1 , It+1 ) = argmax L?t (h, i) ; (h, i) ? P .
5:
6:
if Nt (R(h, i)) = n is odd then
sequentially select a path of children of (Ht+1 , It+1 ) in Tt de?ned by the initial node
def
0
0
(Ht+1
, It+1
) = (Ht+1 , It+1 ), and then
?
?
j+1 j+1 def
j
j
?+
, It+1 ) = argmax ?
(Ht+1
t (h, i; ?n,t ) ; (h, i) ? Ct (Ht+1 , It+1 ) ,
j
7:
8:
9:
10:
11:
12:
13:
j
t+1
t+1
until j = jt+1 is such that (Ht+1
, It+1
) ? Leaf (Tt ).
jt+1
jt+1
expand the node (Ht+1 , It+1 ) in order to de?ne Tt+1 and then de?ne the candidate child
?
?
def
jt+1
jt+1
?+
(ht+1 , it+1 ) = argmax ?
t (h, i; ?n,t ) ; (h, i) ? Ct+1 (Ht+1 , It+1 ) .
sample at point Xt+1 and receive the value Yt+1 ? ?(Xt+1 ), where
?
?
def
Xt+1 = argmax Ut (R(ht+1 , it+1 ), x, ?n,t ) ; x ? R(ht+1 , it+1 ) ,
else
?+
??
proceed similarly than steps 6,7,8 with ?
t replaced with ?
t .
end if
t := t + 1.
end while
where ?t is a shorthand notation for the quantity ?n?t (h0 ,i0 ),t?1 , where n?t (h0 , i0 ) is the optimal
allocation at round t for the region (h0 , i0 ) ? P and where
?
?
?
?
d ln(2d/?k,t ) d ln(2d/?k,t )
def
h
2 2h
+
B(h0 , k, ?k,t ) = min 2c2 ?2 + c1 ?1 + 2(1 + 2 d)
,
h0 ?h
2Nh0 (h, k)
2Nh0 (h, k)
in which we have used the following quantity
?
?
?
?
d ln(2d/?k,t )
1
def
h?h0
2
k?2
.
[2 + 4 d + d ln(2d/?k,t )/2]
Nh0 (h, k) =
C(c?2 ?2h )?d?
2(2c2 ?2h + c21 ?12h )2
Note that the assumption ?12 ? ?2 is only here so that d? can be de?ned w.r.t the metric ?2 only.
We can remove it at the price of using instead a metric mixing ?1 and ?2 together and of much
more technical considerations. Similarly, we could have expressed the result using the local values
d+ (h0 , i0 ) instead of the less precise d? (neither those, nor d? need to be known by the algorithm).
The full proof of this theorem is reported in the appendix. The main steps of the proof are as follows.
First we provide upper and lower con?dence bounds for the estimation of the quantities Ut (R, x, ?)
and Lt (R, x, ?). Then, we lower-bound the depth of the subtree of each region (h0 , i0 ) ? P that
contains a maximal point argmaxx?R(h0 ,i0 ) ?(x), and proceed similarly for a minimal point. This
uses the near-optimality dimension of ? and ?? in the region R(h0 , i0 ), and enables to provide an
?+
??
upper bound on ?
t (h, i; ?) as well as a lower bound on ?
t (h, i; ?). This then enables us to deduce
bounds relating the estimated loss L?t (h, i) to the true loss LR(h,i) (Nt ). Finally, we relate the true
loss of the current allocation to the one using the optimal one n?t+1 (h0 , i0 ) by discussing whether a
region has been over or under sampled. This ?nal part is closed in spirit to the proof of the regret
bound for CH-AS in Carpentier et al. [2011].
In order to better understand the gain in Theorem 1, we provide the following corollary that gives
more insights about the order of magnitude of the regret.
7
?
def
?d ?
Corollary 1 Let ? = 1+ln max{2, ?2 ? } . Under the assumptions of Theorem 1, assuming that
the partition P of the
X is well behaved, i.e. that for all (h0 , i0 ) ? P, then n?t+1 (h0 , i0 ) grows
? space
? 1 ?2h0 ? ?
, then for all ? ? [0, 1], with probability higher than 1 ? 2? we
at least at speed O ln(t) ?2
have
? ?
T
1 ?
?
?? ln(t) ? 2?
1
h0
+ 2?c?
RT = O
.
max
?
n?t (h0 , i0 )
(h0 ,i0 )?P nt (h0 , i0 )
t=|P|
This regret term has to be compared with the typical range of the cumulative loss of the optimal
allocation strategy, that is given by
? +
?
T
T
?
?
?(h0 ,i0 )
?
h0 +
+
2?c?
LP (n?t ) =
max
(?
?
?
(h0 ,i0 )
(h0 ,i0 )) ,
n?t (h0 , i0 )
(h0 ,i0 )?P
t=|P|
t=|P|
def
def
where
= maxx?R(h0 ,i0 ) ?(x), and similarly ??
(h0 ,i0 ) = minx?R(h0 ,i0 ) ?(x). Thus,
this shows that, after normalization, the relative regret on each cell (h0 , i0 ) is roughly of order
1
? ln(t) ? 2?
? 1
1
, i.e. decays at speed n?t (h0 , i0 ) 2? . This shows that we are not only able
?+ (h0 ,i0 ) n?
t (h0 ,i0 )
to compete with the performance of the best allocation strategy, but we actually achieve the exact
same performance with multiplicative factor 1, up to a second order term. Note also that, when
speci?ed to the case of Example 1, the order of this regret is competitive with the standard results
from Carpentier et al. [2011].
?+
(h0 ,i0 )
The lost of the variance term ?+ (h0 , i0 )?1 (that is actually a constant) here comes from the fact
that we are only able to use Hoeffding?s like bounds for the estimation of the variance. In order
to remove it, one would need empirical Bernstein?s bounds for variance estimation in the case of
martingale difference sequences. This is postponed to future work.
6
Discussion
In this paper, we have provided an algorithm together with a regret analysis for a problem of online
allocation of samples in a ?xed partition, where the objective is to minimize a loss that contains a
penalty term that is driven by a notion of curiosity. A very speci?c case (?nite state space) already
corresponds to a dif?cult question known as active learning in the multi-armed bandit setting and
has been previously addressed in the literature (e.g. Antos et al. [2010], Carpentier et al. [2011]). We
have considered an extension of this problem to a continuous domain where a ?xed partition of the
space as well as a generative model of the unknown dynamic are given, using our curiosity-driven
loss function as a measure of performance. Our main result is a regret bound for that problem,
that shows that our procedure is ?rst order optimal, i.e. achieves the same performance as the best
possible allocation (thus with multiplicative constant 1).
We believe this result contributes to ?lling the important gap that exists between existing algorithms
for the challenging setting of intrinsic reinforcement learning and a theoretical analysis of such, the
HORSE.C algorithm being related in spirit to, yet simpler and less ambitious the RIAC algorithm
from Baranes and Oudeyer [2009]. Indeed, in order to achieve the objective that tries to address
RIAC, one should ?rst remove the assumption that the partition is given: One trivial solution is to
run the HORSE.C algorithm in episodes of doubling length, starting with the trivial partition, and to
select at the end of each a possibly better partition based on computed con?dence intervals, however
making ef?cient use of previous samples and avoiding a blow-up of candidate partitions happen to
be a challenging question; then one should relax the generative model assumption (i.e. that we can
sample wherever we want), a question that shares links with a problem called autonomous exploration. Thus, even if the regret analysis of the HORSE.C algorithm is already a strong, new result
that is interesting independently of such dif?cult speci?c goals and of the reinforcement learning
framework (no MDP structure is required), those questions are naturally left for future work.
Acknowledgements The research leading to these results has received funding from the European
Community?s Seventh Framework Programme (FP7/2007-2013) under grant agreement no 270327
(CompLACS) and no 216886 (PASCAL2).
8
References
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9
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3,956 | 4,581 | Nonparametric Max-Margin Matrix Factorization for
Collaborative Prediction
Minjie Xu, Jun Zhu and Bo Zhang
State Key Laboratory of Intelligent Technology and Systems (LITS)
Tsinghua National Laboratory for Information Science and Technology (TNList)
Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
[email protected],{dcszj,dcszb}@mail.tsinghua.edu.cn
Abstract
We present a probabilistic formulation of max-margin matrix factorization and
build accordingly a nonparametric Bayesian model which automatically resolves
the unknown number of latent factors. Our work demonstrates a successful example that integrates Bayesian nonparametrics and max-margin learning, which are
conventionally two separate paradigms and enjoy complementary advantages. We
develop an efficient variational algorithm for posterior inference, and our extensive empirical studies on large-scale MovieLens and EachMovie data sets appear
to justify the aforementioned dual advantages.
1
Introduction
Collaborative prediction is a task of predicting users? potential preferences on currently unrated
items (e.g., movies) based on their currently observed preferences and their relations with others?.
One typical setting formalizes it as a matrix completion problem, i.e., to fill in missing entries (or,
preferences) into a partially observed user-by-item matrix. Often there is extra information available
(e.g., users? age, gender; movies? genre, year, etc.) [10] to help with the task.
Among other popular approaches, factor-based models have been used extensively in collaborative
prediction. The underlying idea behind such models is that there is only a small number of latent
factors influencing the preferences. In a linear factor model, a user?s rating of an item is modeled as
a linear combination of these factors, with user-specific coefficients and item-specific factor values.
Thus, given a N ? M preference matrix for N users and M items, a K-factor model fits it with
a N ? K coefficient matrix U and a M ? K factor matrix V as U V > . Various computational
methods have been successfully developed to implement such an idea, including probabilistic matrix
factorization (PMF) [13, 12] and deterministic reconstruction/approximation error minimization,
e.g., max-margin matrix factorization (M3 F) with hinge loss [14, 11, 16].
One common problem in latent factor models is how to determine the number of factors, which
is unknown a priori. A typical solution relies on some general model selection procedure, e.g.,
cross-validation, which explicitly enumerates and compares many candidate models and thus can
be computationally expensive. On the other hand, probabilistic matrix factorization models have
lend themselves naturally to leverage recent advances in Bayesian nonparametrics to bypass explicit
model selection [17, 1]. However, it remains largely unexplored how to borrow such advantages into
deterministic max-margin matrix factorization models, particularly the very successful M3 F.
To address the above problem, this paper presents infinite probabilistic max-margin matrix factorization (iPM3 F), a nonparametric Bayesian-style M3 F model that utilizes nonparametric Bayesian
techniques to automatically resolve the unknown number of latent factors in M3 F models. The first
key step towards iPM3 F is a general probabilistic formulation of the standard M3 F, which is based
on the maximum entropy discrimination principle [4]. We can then principally extend it to a non1
parametric model, which in theory has an unbounded number of latent factors. To avoid overfitting
we impose a sparsity-inducing Indian buffet process prior on the latent coefficient matrix, selecting
only an appropriate number of active factors. We develop an efficient variational method to infer
posterior distributions and learn parameters (if ever exist) and our extensive empirical results on
MovieLens and EachMovie demonstrate appealing performances.
The rest of the paper is structured as follows. In Section 2, we briefly review the formalization
of max-margin matrix factorization; In Section 3, we present a general probabilistic formulation
of M3 F, and then its nonparametric extension and a fully Bayesian formulation; In Section 4, we
discuss how to perform learning and inference; In Section 5, we give empirical results on 2 prevalent
collaborative filtering data sets; And finally, we conclude in Section 6.
2
Max-margin matrix factorization
Given a preference matrix Y ? RN ?M , which is partially observed and usually sparse, we denote
the observed entry indices by I. The task of traditional matrix factorization is to find a low-rank
matrix X ? RN ?M to approximate Y under some loss measure, e.g., the commonly used squared
error, and use Xij as the reconstruction of the missing entries Yij wherever ij ?
/ I. Max-margin
matrix factorization (M3 F) [14] extends the model by using a sparsity-inducing norm regularizer
for a low-norm factorization and adopting hinge loss for the error measure, which is applicable to
binary, discrete ordinal, or categorical data. For the binary case where Yij ? {?1} and one predicts
3
by Ybij = sign(Xij ), the optimization problem of MX
F is defined as
min
X
kXk? + C
h (Yij Xij ) ,
(1)
ij?I
where h(x) = max(0, 1 ? x) is the hinge loss and kXk? is the nuclear norm of X. M3 F can be
equivalently reformulated as a semi-definite programming (SDP) and thus learned using standard
SDP solvers, but it is unfortunately very slow and can only scale up to thousands of users and items.
As shown in [14], the nuclear norm can be written in a variational form, namely
kXk? =
min
X=U V
>
1
kU k2F + kV k2F .
2
(2)
Based on the equivalence, a fast M3 F model is proposed in [11], which uses gradient descent to
solve an equivalent problem, only on U and V instead
X
1
min
U,V
2
kU k2F + kV k2F + C
h Yij Ui Vj> ,
(3)
ij?I
where U ? RN ?K is the user coefficient matrix, V ? RM ?K the item factor matrix, and K the
number of latent factors. We use Ui to denote the ith row of U , and Vj likewise.
The fast M3 F model can scale up to millions of users and items. But one unaddressed resulting
problem is that it needs to specify the unknown number of latent factors, K, a priori. Below we
present a nonparametric Bayesian approach, which effectively bypasses the model selection problem
and produces very robust prediction. We also design a blockwise coordinate descent algorithm that
directly solves problem (3) rather than working on a smoothing relaxation [11], and it turns out to
be as efficient and accurate. To save space, we defer this part to Appendix B.
3
Nonparametric Bayesian max-margin matrix factorization
Now we present the nonparametric Bayesian max-margin matrix factorization models. We start with
a brief introduction to maximum entropy discrimination, which lays the basis for our methods.
3.1
Maximum entropy discrimination
We consider the binary classification setting since it suffices for our model. Given a set of training
data {(xd , yd )}D
d=1 (yd ? {?1}) and a discriminant function F (x; ?) parameterized by ?, maximum entropy discrimination (MED) [4] seeks to learn a distribution p(?) rather than perform a
point estimation of ? as is the case with standard SVMs that typically lack a direct probabilistic
interpretation. Accordingly, MED takes expectation over the original discriminant function with
respect to p(?) and has the new prediction rule
y? = sign (Ep [F (x; ?)]) .
2
(4)
To find p(?), MED solves the following relative-entropic regularized risk minimization problem
X
min KL (p(?)kp0 (?)) + C
h` (yd Ep [F (xd ; ?)]) ,
(5)
p(?)
d
where p0 (?) is the pre-specified prior distribution of ?, KL(pkp0 ) the Kullback-Leibler divergence,
or relative entropy, between two distributions, C the regularization constant and h` (x) = max(0, `?
x) (` > 0) the generalized hinge loss.
By defining F as the log-likelihood ratio of a Bayesian generative model1 , MED provides an elegant
way to integrate discriminative max-margin learning and Bayesian generative modeling. In fact,
MED subsumes SVM as a special case and has been extended to incorporate latent variables [5, 18]
and perform structured output prediction [21]. Recent work has further extended MED to unite
Bayesian nonparametrics and max-margin learning [20, 19], which have been largely treated as
isolated topics, for learning better classification models. The present work contributes by introducing
a novel generalization of MED to handle the challenging matrix factorization problems.
3.2
Probabilistic max-margin matrix factorization
Like PMF [12], we treat U and V as random variables, whose joint prior distribution is denoted by
p0 (U, V ). Then, our goal is to infer their posterior distribution p(U, V )2 after a set of observations
have been provided. We first consider the binary case where Yij takes value from {?1}. If the
factorization, U and V , is given, we can naturally define the discriminant function F as
F ((i, j); U, V ) = Ui Vj> .
(6)
Furthermore, since both U and V are random variables, we need to resolve the uncertainty in order to
derive a prediction rule. Here, we choose the canonical MED approach, namely the expectation operator, which is linear and has shown promise in [18, 19], rather than the log-marginalized-likelihood
ratio approach [5], which requires an extra likelihood model. Hence, substituting the discriminant
function (6) into (4), we have the prediction rule
Ybij = sign Ep [Ui Vj> ] .
(7)
Then following the principle of MED learning, we define probabilistic max-margin matrix factorization (PM3 F) as solving the following optimization problem
min
p(U,V )
KL(p(U, V )kp0 (U, V )) + C
X
ij?I
h` Yij Ep [Ui Vj> ] .
(8)
Note that our probabilistic formulation is strictly more general than the original M3 F model, which
is in fact a special case of PM3 F under a standard Gaussian
prior Q
and a mean-field assumption
Q
on p(U, V ). Specifically, if we assume p0 (U, V ) = i N (Ui |0, I) j N (Vj |0, I) and p(U, V ) =
Q
Q
p(U )p(V ), then one can prove p(U ) = i N (Ui |?i , I), p(V ) = j N (Vj |?j , I) and PM3 F reduces
accordingly to a M3 F problem (3), namely
min
?,?
X
1
(k?k2F + k?k2F ) + C
h` Yij ?i ?>
.
j
2
ij?I
(9)
Ratings: For ordinal ratings Yij ? {1, 2, . . . , L}, we use the same strategy as in [14] to define the
loss function. Specifically, we introduce thresholds ?0 ? ?1 ? ? ? ? ? ?L , where ?0 = ?? and
?L = +?, to discretize R into L intervals. The prediction rule is changed accordingly to
Ybij = max r|Ep [Ui Vj> ] ? ?r + 1.
(10)
In a hard-margin setting, we would require that
?Yij ?1 + ` ? Ep [Ui Vj> ] ? ?Yij ? `.
(11)
While in a soft-margin setting, we define the loss as
ij ?1
X YX
ij?I
1
2
r=1
h` (Ep [Ui Vj> ] ? ?r ) +
L?1
X
r=Yij
X L?1
X r
h` (?r ? Ep [Ui Vj> ]) =
h` Tij (?r ? Ep [Ui Vj> ]) (12)
ij?I r=1
F can also be directly specified without any reference to probabilistic models [4], as is our case.
We abbreviated the posterior p(U, V |Y ) since we don?t specify the likelihood p(Y |U, V ) anyway.
3
where
Tijr
=
(
+1 for r ? Yij
. The loss thus defined is an upper bound to the sum of absolute
?1 for r < Yij
differences between the predicted ratings and the true ratings, a loss measure closely related to
Normalized Mean Absolute Error (NMAE) [7, 14].
Furthermore, we can learn a more flexible model to capture users? diverse rating criteria by replacing
user-common thresholds ?r in the prediction rule (10) and the loss (12) with user-specific ones ?ir .
Finally, we may as well treat the additionally introduced thresholds ?ir as random variables and infer
their posterior distribution, hereby giving the full PM3 F model as solving
min
p(U,V,?)
3.3
KL(p(U, V, ?)kp0 (U, V, ?)) + C
X L?1
X
ij?I r=1
h` Tijr (Ep [?ir ] ? Ep [Ui Vj> ]) .
(13)
Infinite PM3 F (iPM3 F)
As we have stated, one common problem with finite factor-based models, including PM3 F, is that we
need to explicitly select the number of latent factors, i.e., K. In this section, we present an infinite
PM3 F model which, through Bayesian nonparametric techniques, automatically adapts and selects
the number of latent factors during learning.
Without loss of generality, we consider learning a binary3 coefficient matrix Z ? {0, 1}N ?? . For
finite-sized binary matrices, we may define their prior as given by a Beta-Bernoulli process [8].
While in the infinite case, we allow Z to have an infinite number of columns. Similar to the
nonparametric matrix factorization model [17], we adopt IBP prior over unbounded binary matrices
as previously established in [3] and furthermore, we focus on its stick-breaking construction [15],
which facilitates the development of efficient inference algorithms. Specifically, let ?k ? (0, 1) be
a parameter associated with each column of Z (with respect to its left-ordered equivalent class).
Then the IBP prior can be described as given by the following generative process
i.i.d. for i = 1, . . . , N
Zik ? Bernoulli(?k )
?1 = ?1 , ?k = ?k ?k?1 =
k
Y
i=1
?i , where ?i ? Beta(?, 1)
(?k),
i.i.d. for i = 1, . . . , +?.
(14)
(15)
This process results in a descending sequence of ?k . Specifically, given a finite data set (N < +?),
the probability of seeing the kth factor decreases exponentially with k and the number of active
factors K+ follows a Poisson(?HN ), where HN is the N th harmonic number. Alternatively, we
can use a Beta process prior over Z as in [9].
As for the counterpart, we place an isotropic Gaussian prior over the item factor matrix V . Prior
specified, we may follow the above probabilistic framework to perform max-margin training, with
U replaced by Z. In summary, the stick-breaking construction for the IBP prior results in an
augmented iPM3 F problem for binary data as
min
p(?,Z,V )
KL(p(?, Z, V )kp0 (?, Z, V )) + C
ij?I
where p0 (?, Z, V ) = p0 (?)p0 (Z|?)p0 (V ) with
?k ? Beta(?, 1)
Zik |? ? Bernoulli(?k )
Vjk ? N (0, ? 2 )
X
h` Yij Ep [Zi Vj> ] ,
(16)
i.i.d. for k = 1, . . . , +?,
i.i.d. for i = 1, . . . , N (?k),
i.i.d. for j = 1, . . . , M, k = 1, . . . , +?.
For ordinal ratings, we augment the iPM3 F problem from (13) likewise and, apart from adopting the
same prior assumptions for ?, Z and V , assume p0 (?) = p0 (?|?, Z, V ) with
?ir ? N (?r , ? 2 )
i.i.d. for i = 1, . . . , N, r = 1, . . . , L ? 1,
where ?1 < ? ? ? < ?L?1 are specified as a prior guidance towards an ascending sequence of largemargin thresholds.
3
Learning real-valued coefficients can be easily done as in [3] by defining U = Z ? W , where W is a
real-valued matrix and ? denotes the Hadamard product or element-wise product.
4
3.4
The fully Bayesian model (iBPM3 F)
To take iPM3 F one step further towards a Bayesian-style model, we introduce priors for hyperparameters and perform fully-Bayesian inference [12], where model parameters and hyperparameters are integrated out when making prediction. This approach naturally fits in our MEDbased model thanks to the adoption of the expectation operator when defining prediction rule (7)
and (10). Another observation is that the hyper-parameter ? in a way serves the same role as the
regularization constant C, and thus we also try simplifying the model by omitting C in iBPM3 F.
We admit though, however many level of hyper-parameters are stacked and treated as stochastic and
integrated out, there always exists a gap between our model and a canonical Bayesian one since
we reject a likelihood. We believe the connection is better justified under the general regularized
Bayesian inference framework [19] with a trivial non-informative likelihood.
Here we use the same Gaussian-Wishart prior over the latent factor matrix V as well as its
hyper-parameters ? and ?, thus yielding a doubly augmented problem
for binary dataas
X
min
p(?,Z,?,?,V )
h` Yij Ep [Zi Vj> ] ,
KL(p(?, Z, ?, ?, V )kp0 (?, Z, ?, ?, V )) +
(17)
ij?I
where we?ve omitted the regularization constant C and set p0 (?, Z, ?, ?, V ) to be factorized as
p0 (?)p0 (Z|?)p0 (?, ?)p0 (V |?, ?), with ? and Z enjoying the same priors as in iPM3 F and
(?, ?) ? GW(?0 , ?0 , W0 , ?0 ) = N (?|?0 , (?0 ?)?1 )W(?|W0 , ?0 ),
Vj |?, ? ? N (Vj |?, ??1 )
i.i.d. for j = 1, . . . , M .
And note that exactly the same process applies as well to the full model for ordinal ratings.
4
Learning and inference under truncated mean-field assumptions
Now, we briefly discuss how to perform learning and inference in iPM3 F. For iBPM3 F, similar
procedures are applicable. We defer all the details to Appendix D for saving space. Specifically, we introduce a simple variational inference method to approximate the optimal posterior,
which turns out to perform well in practice. We make the following truncated mean-field assumption
p(?, Z, V ) = p(?)p(Z)p(V ) =
K
Y
k=1
where K is the truncation level and
p(?k ) ?
N Y
K
Y
i=1 k=1
p(Zik ) ? p(V ),
i.i.d. for k = 1, . . . , K,
i.i.d. for i = 1, . . . , N, k = 1, . . . , K.
?k ? Beta(?k1 , ?k2 )
Zik ? Bernoulli(?ik )
(18)
(19)
(20)
Note that we make no further assumption on the functional form of p(V ) and that we factorize p(Z)
into element-wise i.i.d. p(Zik ) and parameterize it with Bernoulli(?ik ) merely out of the pursuit of
a simpler denotation for subsequent deduction. Actually it can be shown that p(Z) indeed enjoys all
these properties given the mildest truncated mean-field assumption p(?, Z, V ) = p(?)p(Z)p(V ).
For ordinal ratings, we make an additional mean-field assumption
p(?, Z, V, ?) = p(?, Z, V )p(?),
(21)
where p(?, Z, V ) is treated exactly the same as for binary data and p(?) is left in free forms.
One noteworthy point is that given p(Z), we may calculate the expectation of the posterior effective
dimensionality of the latent factor space as
!
Ep [K+ ] =
K
X
k=1
1?
N
Y
(1 ? ?ik ) .
(22)
i=1
Then the problem can be solved using an iterative procedure that alternates between optimizing each
component at a time, as outlined below (We defer the details to Appendix D.):
Infer p(V ): The linear discriminant function and the isotropic Gaussian prior on V leads to an
QM
isotropic Gaussian posterior p(V ) = j=1 N (Vj |?j , ? 2 I) while the M mean vectors ?j can be
obtained via solving M independent binary SVMs
min
?j
X
1
k?j k2 + C
h` Yij ?j ?i> .
2? 2
i|ij?I
5
(23)
Infer p(?) and p(Z): Since ? is marginalized before exerting any influence in the loss term, its
update is independent of the loss and hence we adopt the same update rules as in [2]; The subproblem on p(Z) decomposes into N independent convex optimization problems, one for each ?i as
min
?i
K
X
k=1
X
EZ [log p(Zik )] ? E?,Z [log p0 (Zik |?)] + C
h` Yij ?i ?>
,
j
(24)
j|ij?I
P
where EZ [log p(Zik )] = ?ik log ?ik +(1??ik ) log(1??ik ), E?,Z [log p0 (Zik |?)] = ?ik kj=1 E? [log ?j ]+
Q
Q
(1 ? ?ik )E? [log(1 ? kj=1 ?j )] and E? [log ?j ] = ?(?k1 ) ? ?(?k1 + ?k2 ), E? [log(1 ? kj=1 ?j )] ? L?k ,
where L?k in turn is the multivariate lower bound as in [2]. We may use the similar subgradient
technique as in [19] to approximately solve for ?i . Here we introduce an alternative solution, which
is as efficient and guarantees convergence as iteration goes on. We update ?i via coordinate descent,
with each conditional optimal ?ik sought by binary search. (See Appendix D.1.3 for details.)
QN QL?1
Infer p(?): p(?) remains an isotropic Gaussian as p(?) = i=1 r=1 N (?ir |%ir , ? 2 ) and the mean
%ir of each component is solution to the corresponding subproblem
min
%ir
X
1
2
r
>
(%
?
?
)
+
C
h
T
(%
?
?
?
)
,
ir
r
ir
i
`
ij
j
2? 2
(25)
j|ij?I
to which the binary search solver for each ?ik also applies. Note that as ? ? +?, the Gaussian
distribution regresses to a uniform distribution and problem (25) reduces accordingly to the corresponding conditional subproblem for ? in the original M3 F (Appendix B.3).
5
Experiments and discussions
We conduct experiments on the MovieLens 1M and EachMovie data sets, and compare our results
with fast M3 F [11] and two probabilistic matrix factorization methods, PMF [13] and BPMF [12].
Data sets: The MovieLens data set contains 1,000,209 anonymous ratings (ranging from 1 to 5) of
3,952 movies made by 6,040 users, among which 3,706 movies are actually rated and every user
has at least 20 ratings. The EachMovie data set contains 2,811,983 ratings of 1,628 movies made by
72,916 users, among which 1,623 movies are actually rated and 36,656 users has at least 20 ratings.
As in [7, 11], we discarded users with fewer than 20 ratings, leaving us with 2,579,985 ratings.
There are 6 possible rating values, {0, 0.2, . . . , 1} and we mapped them to {1, 2, . . . , 6}.
Protocol: As in [7, 11], we test our method in a pure collaborative prediction setting, neglecting any
external information other than the user-item-rating triplets in the data sets. We adopt as well the
all-but-one protocol to partition the data set into training set and test set, that is to randomly withhold
one of the observed ratings from each user into test set and use the rest as training set. Validation set,
when needed, is constructed likewise from the constructed training set. Also as described in [7], we
consider both weak and strong generalization. For weak, the training ratings for all users are always
available, so a single-stage training process will suffice; while for strong, training is first carried out
on a subset of users, and then keeping the learned latent factor matrix V fixed, we train the model a
second time on the other users for their user profiles (coefficients Z and thresholds ?) and perform
prediction on these users only. We partition the users accordingly as in [7, 11], namely 5,000 and
1,040 users for weak and strong respectively in MovieLens, and 30,000 and 6,565 in EachMovie.
We repeat the random partition thrice. We compute Normalized Mean Absolute Error (NMAE) as
the error measure and report the averaged performance.4
Implementation details: We perform cross-validation to choose the best regularization constant C
for iPM3 F as well as to guide early-stopping during the learning process. The candidate C values
are the same 11 values which are log-evenly distributed between 0.13/4 and 0.12 as in [11]. We
set the truncation level K = 100 (same for M3 F and PMF models), ? = 3, ? = 1, ? = 1.5`;
?1 , . . . , ?L?1 are set to be symmetric with respect to 0, with a step-size of 2`; We set the margin
parameter ` = 9. Although M3 F is invariant to ` (Appendix B.4), we find that setting ` = 9 achieved
a good balance between performance and training time (Figure 1). The difference is largely believed
to attribute to the uniform convergence standard we used when solving SVM subproblems. Finally,
for iBPM3 F, we find that although removing C can achieve competitive results with iPM3 F, keeping
C will produce even better performance. Hence we learn iBPM3 F using the selected C for iPM3 F.
6
Table 1: NMAE performance of different models on MovieLens and EachMovie.
Algorithm
M3 F [11]
PMF [13]
BPMF [12]
M3 F?
iPM3 F
iBPM3 F
5.1
MovieLens
weak
strong
.4156 ? .0037 .4203 ? .0138
.4332 ? .0033 .4413 ? .0074
.4235 ? .0023 .4450 ? .0085
.4176 ? .0016 .4227 ? .0072
.4031 ? .0030 .4135 ? .0109
.4050 ? .0029 .4089 ? .0146
EachMovie
weak
strong
.4397 ? .0006 .4341 ? .0025
.4466 ? .0016 .4579 ? .0016
.4352 ? .0014 .4445 ? .0005
.4348 ? .0023 .4301 ? .0034
.4211 ? .0019 .4224 ? .0051
.4268 ? .0029 .4403 ? .0040
Experimental results
Table 1 presents the NMAE performance of different models, where the performance of M3 F is
cited from the corresponding paper [11] and represents the state-of-the-art. We observe that iPM3 F
significantly outperforms M3 F, PMF and BPMF in terms of the NMAE error measure on both data
sets for both settings. Moreover, we find that the fully Bayesian formulation of iPM3 F achieves
comparable performances in most cases as iPM3 F and that our coordinate descent algorithm for
M3 F (M3 F? ) performs quite similar to the original gradient descent algorithm for M3 F.
In summary, the effect of endowing M3 F models with a probabilistic formulation is intriguing in
that not only the performance of the model is largely improved but with the help of Bayesian nonparametric techniques, the effort of selecting the number of latent factors is saved as well.
Another observation from Table 1 is that in gener- Table 2: NMAE on the purged EachMovie.
al almost all models perform worse on EachMovie
Algorithm
weak
strong
than on MovieLens. A closer investigation finds that
3
M
F
[11]
.4009
?
.0012
.4028
? .0064
the EachMovie data set has a special rating. When
PMF [13] .4153 ? .0016 .4329 ? .0059
a user has rated an item as zero star, he might either
BPMF [12] .4021 ? .0011 .4119 ? .0062
express a genuine dislike or, when the weight of the
M3 F?
.4059 ? .0012 .4095 ? .0052
rating is less than 1, indicate that he never plans to
.3954 ? .0026 .3977 ? .0034
iPM3 F
see that movie since it just ?sounds awful?. Ideally
iBPM3 F
.3982 ? .0021 .4026 ? .0067
we should treat such a declaration as less authoritative than a regular rating of zero star and hence omit it from the data set. We have tried this setting
by removing these special ratings.5 Table 2 presents the NMAE results of different models. Again,
the coordinate descent M3 F performs comparably with fast M3 F; iPM3 F performs better than all the
other methods; And iBPM3 F performs comparably with iPM3 F.
5
0.46
9.5
2000
x 10
0.5
partition #1
partition #2
partition #3
NMAE
time
0.44
1000
0.43
500
0.46
1
9
25
49
margin parameter: `
100
400
0
900
0.44
8
0.42
7.5
0.42
0.5
partition #1
partition #2
partition #3
8.5
NMAE
1500
Objective value
NMAE
0.45
Average time per iteration (s)
9
0.48
7
0.4
0
10
20
30
# of iterations
40
50
0.38
0
10
20
30
# of iterations
40
50
Figure 1: Influence of ` on M3 F. Figure 2: Objective values dur- Figure 3: NMAE during the
We fixed ` = 9 across the exper- ing the training of iPM3 F on training of iPM3 F on MovieLeniments.
MovieLens 1M.
s 1M.
Closer analysis of iPM3 F
5.2
The posterior dimensionality: As indicated in Eq. (22), we may calculate the expectation of the
effective dimensionality K+ of the latent factor space to roughly have a sense of how the iPM3 F
model automatically chooses the latent dimensionality. Since we take ? = 3 in the IBP prior (15)
and N ? 104 , the expected prior dimensionality ?HN is about 30. We find that when the truncation
level K is set small, e.g., 60 or 80, the expected posterior dimensionality very quickly saturates,
4
5
Note that M3 F models output discretized ordinal ratings while PMF models output real-valued ratings.
After discarding users with less than 20 normal ratings, we are left with 35,281 users and 2,315,060 ratings.
7
Table 3: Performance of iPM3 F with and without probabilistic treatment of ?
Algorithm
w/ prob.
w/o prob.
margin
MovieLens
.4031 ? .0030
.4056 ? .0043
.0024 ? .0013
EachMovie
.4211 ? .0019
.4256 ? .0011
.0045 ? .0016
pEachMovie
.3954 ? .0026
.4026 ? .0023
.0072 ? .0045
often within the first few iterations; While for sufficiently large Ks, e.g., 150 or 200, iPM3 F tends to
output a sparse Z of expected dimensionality around 135 or 110 respectively. (For each truncation
level, we rerun our model and perform cross-validation to select the best regularization constant C.)
This interesting observation verifies our model?s capability of automatic model complexity control.
Stability: As Figure 2 and 3 shows, iPM3 F performs quite stably against 3 different randomly partitioned subsets. iBPM3 F expresses a similar trait, but the test performance does not keep dropping
with the decreasing of the objective value. Therefore we use a validation set to guide the earlystopping during the learning process, terminating when validation error starts to rebound.
Treating thresholds ?: When predicting ordinal ratings, the introduced thresholds ? are very important since they underpin the large-margin principle of max-margin matrix factorization models.
Nevertheless without a proper probabilistic treatment, the subproblems on thresholds (25) are not
strictly convex, very often giving rise to a section of candidate thresholds that are ?equally optimal?
for the solution. Under our probabilistic model however, we can easily get rid of this non-strict
convexity by introducing for them a Gaussian prior as stated above in section 3.3. We compare performances of iPM3 F both with and without the probabilistic treatment of ? and as shown in Table 3,
the improvement is outstanding.
Finally, Table 4 presents the running time of vari- Table 4: Running time of different models.
ous models on both EachMovie and MovieLens data
Algorithm MovieLens EachMovie Iters
sets. For M3 F, the original paper [11] reported about
M3 F [11]
?5h
?15h
100
5h on MovieLens with a standard 3.06Ghz Pentium
PMF
[13]
8.7m
25m
50
4 CPU and about 15h on EachMovie, which are fairBPMF
[12]
19m
1h
50
ly acceptable for factorizing a matrix with millions
3 ?
M
F
4h
10h
50
3
of entries. Our current implementations of M F and
U, V
3.8h
9.5h
iPM3 F consume about 4.5h and 10h on MovieLens
?
125s
750s
and EachMovie respectively with a 3.00Ghz Core i5
3
iPM
F
4.6h
5.5h
50
CPU. A closer investigation discovers that most of
V
4.3h
4.3h
the running time is spent on learning U (or Z) and
?
18m
1h
V in PM3 F models, which breaks down into a set of
struct
SVM optimization problems that are learned by SVM
. More efficient SVM solvers can be immediately applied to further improve the efficiency. Furthermore, the blockwise coordinate descent
algorithm can naturally be parallelized, since the sub-problems of learning different Ui (or Vj ) are
not coupled. We leave this improvement in future work.
6
Conclusions
We?ve presented an infinite probabilistic max-margin matrix factorization method, which utilizes the
advantages of nonparametric Bayesian techniques to bypass the model selection problem of maxmargin matrix factorization methods. We?ve also developed efficient blockwise coordinate descent
algorithms for variational inference and performed extensive evaluation on two large benchmark
data sets. Empirical results demonstrate appealing performance.
Acknowledgments
This work is supported by the National Basic Research Program (973 Program) of China (Nos.
2013CB329403, 2012CB316301), National Natural Science Foundation of China (Nos. 91120011,
61273023), and Tsinghua University Initiative Scientific Research Program (No. 20121088071).
8
References
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9
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experimental:1 m3:31 select:2 indian:4 outstanding:1 incorporate:1 |
3,957 | 4,582 | Learning Networks of Heterogeneous Influence
Nan Du? Le Song? Alex Smola? Ming Yuan?
Georgia Institute of Technology? , Google Research?
[email protected] [email protected]
[email protected] [email protected]
Abstract
Information, disease, and influence diffuse over networks of entities in both natural systems and human society. Analyzing these transmission networks plays
an important role in understanding the diffusion processes and predicting future
events. However, the underlying transmission networks are often hidden and incomplete, and we observe only the time stamps when cascades of events happen.
In this paper, we address the challenging problem of uncovering the hidden network only from the cascades. The structure discovery problem is complicated by
the fact that the influence between networked entities is heterogeneous, which can
not be described by a simple parametric model. Therefore, we propose a kernelbased method which can capture a diverse range of different types of influence
without any prior assumption. In both synthetic and real cascade data, we show
that our model can better recover the underlying diffusion network and drastically
improve the estimation of the transmission functions among networked entities.
1
Introduction
Networks have been powerful abstractions for modeling a variety of natural and artificial systems
that consist of a large collection of interacting entities. Due to the recent increasing availability
of large-scale networks, network modeling and analysis have been extensively applied to study
the spreading and diffusion of information, ideas, and even virus in social and information networks (see e.g., [17, 5, 18, 1, 2]). However, the process of influence and diffusion often occurs in
a hidden network that might not be easily observed and identified directly. For instance, when a
disease spreads among people, epidemiologists can know only when a person gets sick, but they
can hardly ever know where and from whom he (she) gets infected. Similarly, when consumers
rush to buy some particular products, marketers can know when purchases occurred, but they cannot
track in further where the recommendations originally came from [12]. In all such cases, we could
observe only the time stamp when a piece of information has been received by a particular entity,
but the exact path of diffusion is missing. Therefore, it is an interesting and challenging question
whether we can uncover the diffusion paths based just on the time stamps of the events.
There are many recent studies on estimating correlation or causal structures from multivariate timeseries data (see e.g., [2, 6, 13]). However, in these models, time is treated as discrete index and not
modeled as a random variable. In the diffusion network discovery problem, time is treated explicitly as a continuous variable, and one is interested in capturing how the occurrence of event at one
node affects the time for its occurence at other nodes. This problem recently has been explored by
a number of studies in the literature. Specifically, Meyers and Leskovec inferred the diffusion network by learning the infection probability between two nodes using a convex programming, called
CONNIE [14]. Gomez-Rodriguez et al. inferred the network connectivity using a submodular optimization, called NETINF [4]. However, both CONNIE and NETINF assume that the transmission
model for each pair of nodes is fixed with predefined transmission rate. Recently, Gomez-Rodriguez
et al. proposed an elegant method, called NETRATE [3], using continuous temporal dynamics
model to allow variable diffusion rates across network edges. NETRATE makes fewer number of
assumptions and achieves better performance in various aspects than the previous two approaches.
However, the limitation of NETRATE is that it requires the influence model on each edge to have a
1
0.1
histogram
exp
rayleigh
KernelCascade
pdf
pdf
0.2
0.08
0.1
0.1
histogram
exp
rayleigh
KernelCascade
0.06
0.04
0.02
0
0
10
20
30
t(hours)
40
50
(a) Pair 1
0
0
histogram
exp
rayleigh
KernelCascade
0.08
pdf
0.3
0.06
0.04
0.02
20
t(hours)
40
0
0
50
t(hours)
(b) Pair 2
100
(c) Pair 3
Figure 1: The histograms of the interval between the time when a post appeared in one site and the time when
a new post in another site links to it. Dotted and dash lines are density fitted by NETRATE. The solid lines are
given by KernelCascade.
fixed parametric form, such as exponential, power-law, or Rayleigh distribution, although the model
parameters learned from cascades could be different.
In practice, the patterns of information diffusion (or a spreading disease) among entities can be quite
complicated and different from each other, going far beyond what a single family of parametric
models can capture. For example, in twitter, an active user can be online for more than 12 hours a
day, and he may instantly respond to any interesting message. However, an inactive user may just
log in and respond once a day. As a result, the spreading pattern of the messages between the active
user and his friends can be quite different from that of the inactive user.
Another example is from the information diffusion in a blogsphere: the hyperlinks between posts can
be viewed as some kind of information flow from one media site to another, and the time difference
between two linked posts reveal the pattern of diffusion. In Figure 1, we examined three pairs of
media sites from the MemeTracker dataset [3, 9], and plotted the histograms of the intervals between
the the moment when a post first appeared in one site and the moment when it was linked by a new
post in another site. We can observe that information can have very different transmission patterns
for these pairs. Parametric models fitted by NETRATE may capture the simple pattern in Figure 1(a),
but they might miss the multimodal patterns in Figure 1(b) and Figure 1(c). In contrast, our method,
called KernelCascade, is able to fit both data accurately and thus can handle the heterogeneity.
In the reminder of this paper, we present the details of our approach KernelCascade. Our key idea
is to model the continuous information diffusion process using survival analysis by kernelizing the
hazard function. We obtain a convex optimization problem with grouped lasso type of regularization
and develop a fast block-coordinate descent algorithm for solving the problem. The sparsity patterns
of the coefficients provide us the structure of the diffusion network. In both synthetic and real world
data, our method can better recover the underlying diffusion networks and drastically improve the
estimation of the transmission functions among networked entities.
2
Preliminary
In this section, we will present some basic concepts from survival analysis [7, 8], which are essential
for our later modeling. Given a nonnegative random variable T corresponding to the time
R twhen an
event happens, let f (t) be the probability density function of T and F (t) = P r(T ? t) = 0 f (x)dx
be its cumulative distribution function. The probability that an event does not happen up to time t
is thus given by the survival function S(t) = P r(T ? t) = 1 ? F (t). The survival function is a
continuous and monotonically decreasing function with S(0) = 1 and S(?) = limt?? S(t) = 0.
Given f (t) and S(t), we can define the instantaneous risk (or rate) that an event has not happened
yet up to time t but happens at time t by the hazard function
P r(t ? T ? t + ?t|T ? t)
f (t)
h(t) = lim
=
.
(1)
?t?0
?t
S(t)
With this definition, h(t)?t will be the approximate probability that an event happens in [t, t + ?t)
given that the event has not happened yet up to t. Furthermore, the hazard function h(t) is also
d
related to the survival function S(t) via the differential equation h(t) = ? dt
log S(t), where we
have used f (t) = ?S 0 (t). Solving the differential equation with boundary condition S(0) = 1, we
can recover the survival function S(t) and the density function f (t) based on the hazard function
h(t), i.e.,
Z t
Z t
S(t) = exp ?
h(x) dx
and f (t) = h(t) exp ?
h(x) dx .
(2)
0
0
2
a
b
c
t0
t1
a
b
t2
t3
c
t4
t0
t1
a
b
t3
c
t2
d
e
d
e
d
e
(a) Hidden network
(b) Node e gets infected at time t4
(c) Node e survives
Figure 2: Cascades over a hidden network. Solid lines in panel(a) represent connections in a hidden network.
In panel (b) and (c), filled circles indicate infected nodes while empty circles represent uninfected ones. Node
a, b, c and d are the parents of node e which got infected at t0 < t1 < t2 < t3 respectively and tended to infect
node e. In panel (b), node e survives given node a, b and c shown in green dash lines. However, it was infected
by node d. In panel (c), node e survives even though all its parents got infected.
3
Modeling Cascades using Survival Analysis
We use survival analysis to model information diffusion for networked entities. We will largely
follow the presentation of Gomez-Rodriguez et al. [3], but add clarification when necessary. We
assume that there is a fixed population of N nodes connected in a directed network G = (V, E).
Neighboring nodes are allowed to directly influence each other. Nodes along a directed path may
influence each other only through a diffusion process. Because the true underlying network is unknown, our observations are only the time stamps when events occur to each node in the network.
The time stamps are then organized as cascades, each of which corresponds to a particular event.
For instance, a piece of news posted on CNN website about ?Facebook went public? can be treated
as an event. It can spread across the blogsphere and trigger a sequence of posts from other sites
referring to it. Each site will have a time stamp when this particular piece of news is being discussed
and cited. The goal of the model is to capture the interplay between the hidden diffusion network
and the cascades of observed event time stamps.
More formally, a directed edge, j ? i, is associated with an transmission function fji (ti |tj ), which
is the conditional likelihood of an event happening to node i at time ti given that the same event has
already happened to node j at time tj . The transmission function attempts to capture the temporal
dependency between the two successive events for node i and j. In addition, we focus on shiftinvariant transmission functions whose value only depends on the time difference, i.e., fji (ti |tj ) =
fji (ti ? tj ) = fji (?ji ) where ?ji := ti ? tj . Given the likelihood function, we can compute the
corresponding survival function Sji (?ji ) and hazard function hji (?ji ). When there is no directed
edge j ? i, the transmission function and hazard function are both identically zeros, i.e., fji (?ji ) =
0 and hji (?ji ) = 0, but the survival function is identically one, i.e., Sji (?ji ) = 1. Therefore, the
structure of the diffusion network is reflected in the non-zero patterns of a collection of transmission
functions (or hazard functions).
A cascade is an N -dimensional vector tc := (tc1 , . . . , tcN )> with i-th dimension recording the time
stamp when event c occurs to node i. Furthermore, tci ? [0, T c ] ? {?}, and the symbol ? labels
nodes that have not been influenced during observation window [0, T c ] ? it does not imply that
nodes are never influenced. The ?clock?
is set to 0 at the start of each cascade. A dataset can
contain a collection, C, of cascades t1 , . . . , t|C| . The time stamps assigned to nodes by a cascade
induce a directed acyclic graph (DAG) by defining node j as the parent of i if tj < ti . Thus, it
is meaningful to refer to parents and children within a cascade [3], which is different from the
parent-child structural relation on the true underlying diffusion network. Since the true network is
inferred from many cascades (each of which imposes its own DAG structure), the inferred network
is typically not a DAG.
The likelihood `(tc ) of a cascade induced by event c is then simply a product of all individual
likelihood `i (tc ) that event c occurs to each node i. Depending on whether event c actually occurs
to node i in the data, we can compute this individual likelihood as:
Event c did occur at node i. We assume that once an event occurs at node i under the influence of
a particular parent j in a cascade, the same event will not happen again. In Figure 2(b), node e is
susceptible given its parent a, b, c and d. However, only node d is the first parent who infects node e.
Because each parent could be equally likely to first influence node i, the likelihood is just a simple
sum over the likelihoods of the mutually disjoint events that node i has survived from the influence
of all the other parents except the first parent j, i.e.,
X
Y
X
Y
c
`+
fji (?cji )
Ski (?cki ) =
hji (?cji )
Ski (?cki ).
(3)
i (t ) =
j:tcj <tci
j:tcj <tci
k:k6=j,tck <tci
3
k:tck <tci
Event c did not occur at node i. In other words, node i survives from the influence of all parents (see Figure 2(c) for illustration). The likelihood is a product of survival functions, i.e.,
Y
c
`?
Sji (T c ? tj ).
(4)
i (t ) =
tj ?T c
Combining the above two scenarios together, we can obtain the overall likelihood of a cascade tc by
multiplying together all individual likelihoods, i.e.,
Y
Y
c
c
`?
`+
`(tc ) =
(5)
i (t ) ?
i (t ) .
tci >T c
|
tci ?T c
{z
}
uninfected nodes
|
{z
infected nodes
}
Therefore, the likelihood of all
Q cascades is a product of the these individual cascade likelihoods, i.e. `({t1 , . . . , t|C| }) = c=1,...,|C| `(tc ). In the end, we take the negative log of this likelihood function and regroup all terms associated with edges pointing to node i together to derive
?
?
X ?X X
X
X
?
log S(?cji ) +
h(?cji )?
(6)
L({t1 , . . . , t|C| }) = ?
log
?
c
c
c
i
j {c|tc <tc }
c
{c|ti 6T }
{tj <ti }
i
j
There are two interesting implications from this negative log likelihood function. First, the function
can be expressed using only the hazard and the survival function. Second, the function is decomposed into additive contribution from each node i. We can therefore estimate the hazard and survival
function for each node separately. Previously, Gomez-Rodriguez et al. [3] used parametric hazard
and survival functions, and they estimated the model parameters using the convex programming. In
contrast, we will instead formulate an algorithm using kernels and grouped parameter regularization,
which allows us to estimate complicated hazard and survival functions without overfitting.
4
KernelCascade for Learning Diffusion Networks
This section presents our kernel method for uncovering diffusion networks from cascades. Our key
idea is to kernelize the hazard function used in the negative log-likelihood in (6), and then estimate
the parameters using grouped lasso type of optimization.
4.1
Kernelizing survival analysis
Kernel methods are powerful tools for generalizing classical linear learning approaches to analyze
nonlinear relations. A kernel function, k : X ? X ? R, is a real-valued positive definite symmetric
function iff. for any set of points {?1 , ?2 , . . . , ?m } ? X the kernel matrix K with entris Kls :=
k(?l , ?s ) is positive definite. We want to model heterogeneous transmission functions, fji (?ji ),
from j to i. Rather than directly kernelizing the transmission function, we kernelize the hazard
function instead, by assuming that it is a linear combination of m kernel functions, i.e.,
m
X
l
hji (?ji ) =
?ji
k(?l , ?ji ),
(7)
l=1
where we fix one argument of each kernel function, k(?l , ?), to a point ?l in a uniform grid of m
locations in the range of (0, maxc T c ]. To achieve fully nonparametric modeling of the hazard
function, we can let m grow as we see more cascades. Alternatively, we can also place a nonlinear basis function on each time point in the observed cascades. For efficiency consideration,
we will use a fixed uniform grid in our later experiments. Since the hazard function is always
positive, we use positive kernel functions and require the weights to be positive, i.e., k(?, ?) ? 0
l
and ?ji
? 0 to capture such constraint. For simplicity of notation, we will define vectors ?ji :=
1
m >
(?ji , . . . , ?ji
) , and k(?ji ) := (k(?1 , ?ji ), . . . , k(?m , ?ji ))> . Hence, the hazard function can be
written as hji (?ji ) = ?>
ji k(?ji ).
In addition, the survival function and likelihood function can also be kernelizedRbased on their
?
respective relation with the hazard function in (2). More specifically, let gl (?ji ) := 0 ji k(?l , x)dx
and the corresponding vector g(?ji ) := (g1 (?ji ), . . . , gm (?ji ))> . We then can derive
>
Sji (?ji ) = exp ??>
and
fji (?ji ) = ?>
(8)
ji g(?ji )
ji k(?ji ) exp ??ji g(?ji ) .
In the formulation, we need to perform integration over the kernel function to compute gl (?ji ). This
can be done efficiently for many kernels, such as the Gaussian RBF kernel, the Laplacian kernel,
the Quartic kernel, and the Triweight kernel. In later experiments, we mainly focus on the Gaussian
4
RBF kernel, k(?l , ?s ) = exp(?k?l ? ?s k2 /(2? 2 )), and derive a closed form solution for gl (?ji ) as
?
Z ?ji
2??
?l ? ?ji
?l
?
gl (?ji ) =
k(?l , x) dx =
erfc
? erfc ?
,
(9)
2
2?
2?
0
R
2
?
where erfc(t) := ?2? t e?x dx is the error function. Yet, our method is not limited to the particular RBF kernel. If there is no closed form solution for the one-dimensional integration, we can use a
large number of available numerical integration methods for this purpose [15]. We note that given a
dataset, both the vector k(?ji ) and g(?ji ) need to be computed only once as a preprocessing, and
then can be reused in the algorithm.
4.2
Estimating sparse diffusion networks
Next we plug in the kernelized hazard function and survival function into the likelihood of cascades
in (6). Since the negative log likelihood is separable for each node i, we can optimize the set of
variables {?ji }N
j=1 separately. As a result, the negative log likelihood for the data associated with
node i can be estimated as X X
X
X
c
c
?>
(10)
log
?>
Li {?ji }N
ji k(?ji ).
ji g(?ji ) ?
j=1 =
c
c
c
j {c|tc <tc }
c
{tj <ti }
{c|ti 6T }
i
j
A desirable feature of this function is that it is convex in its arguments, {?ji }N
j=1 , which allows us
to bring various convex optimization tools to solve the problem efficiently.
In addition, we want to induce a sparse network structure from the data and avoid overfitting.
Basically, if the coefficients ?ji = 0, then there is no edge (or direct influence) from node j
to i. For P
this purpose, we will impose grouped lasso type of regularization on the coefficients
?ji , i.e., ( j k?ji k)2 [16, 19]. Grouped lasso type of regularization has the tendency to select a
small number of groups of non-zero coefficients but push other groups of coefficients to be zero.
Overall, the optimization problem trades off between the data likelihood term and the group sparsity
of the coefficients
2
X
min Li {?ji }N
k?ji k ,
s.t. ?ji ? 0, ?j,
(11)
j=1 + ?
{?ji }N
j=1
j
where ? is the regularization parameter. After we obtain a sparse solution from the above optimization, we obtain partial network structures, each of which centers around a particular node i.
We can then join all the partial structures together and obtain the overall diffusion network. The
corresponding hazard function along each edge can also be obtained from (8).
4.3
Optimization
We note that (11) is a nonsmooth optimization problem because of the regularizer. There are
many ways to solve the optimization problem, and we will illustrate this using a simple algorithm originating from multiple kernel learning [16, 19]. In this approach, an additional set of
variables are introduced to turn the
P optimization problem into a smooth optimization problem.
More specifically, let ?i ? 0 and j ?j = 1. Then using Cauchy-Schwartz inequality, we have
P
P
P
P
P
1/2 1/2
2
2
( j k?ji k)2 = ( j (k?ji k /?j )?j )2 ? ( j k?ji k /?j )( j ?j ) = j k?ji k /?j , where
the equality holds when
X
?j = k?ji k /
k?ji k .
(12)
j
With these additional variables, ?j , we can solve an alternative smooth optimization problem, which
is jointly convex in both ?ji and ?j
X
X k?ji k2
min
Li {?ji }N
+?
, s.t. ?ji ? 0, ?j ? 0,
?j = 1, ?j.
(13)
j=1
?j
{?ji ,?j }N
j=1
j
j
There are many ways to solve the convex optimization problem in (13). In this paper, we used
a block coordinate descent approach alternating between the optimization of ?ji and ?j . More
specifically, when we fixed ?ji , we can obtain the best ?j using the closed form formula in (12);
when we fixed ?j , we can optimize over ?ji using, e.g., a projected gradient method. The overall
algorithm pseudocodes are given in Algorithm 1. Moreover, we can speed up the optimization in
three ways. First, because the optimization is independent for each node i, the overall process can
be easily parallelized into N separate sub-problems. Second, we can prune the possible nodes that
were never infected before node i in any cascade where i was infected. Third, if we further assume
that all the edges from the same node belong to the same type of models, especially when the sample
5
size is small, the N edges could share a common set of m parameters, and thus we can only estimate
N ? m parameters in total.
Algorithm 1: KernelCascade
Initialize the diffusion network G to be empty;
for i = 1 to N do
N
Intialize {?ji }N
j=1 and {?j }j=1 ;
repeat
N
Update {?ji }N
j=1 using projected gradient method with {?j }j=1 from last update;
N
N
Update {?j }j=1 using formula (12) with {?ji }j=1 from last update;
until convergence;
Extract the sparse neighborhood N (i) of node i from nonzero ?ji ;
Join N (i) to the diffusion network G
5
Experimental Results
We will evaluate KernelCascade on both realistic synthetic networks and real world networks. We
compare it to NETINF [4] and NETRATE [3], and we show that KernelCascade can perform significantly better in terms of both recovering the network structures and the transmission functions.
5.1
Synthetic Networks
Network generation. We first generate synthetic networks that mimic the structural properties of
real networks. These synthetic networks can then be used for simulation of information diffusion.
Since the latent networks for generating cascades are known in advance, we can perform detailed
comparisons between various methods. We use Kronecker generator [10] to examine two types of
networks with directed edges: (i) the core-periphery structure [11], which mimics the information
diffusion process in real world networks, and (ii) the Erd?os-R?enyi random networks.
Influence function. For each edge j ? i in a network G, we will assign it a mixture of two Rayleigh
distributions: fji (t|?,
a1 , b1 , a2 ,b2 ) = ?R1 (t|a1 , b1 ) + (1 ? ?)R2 (t|a2 , b2 ) where Ri (t|ai , bi ) =
2
2
t?ai
2
i
exp ? t?a
, t > ai , and ? ? (0, 1) is a mixing proportion. We examine
t?ai
bi
bi
three different parameter settings for the transmission function: (1) all edges in network G have
the same transmission function p(t) = f (t|0.5, 10, 1, 20, 1); (2) all edges in network G have the
same transmission function q(t) = f (t|0.5, 0, 1, 20, 1); and (3) all edges in network G are uniformly
randomly assigned to either p(t) or q(t).
Cascade generation. Given a network G and the collection of transmission functions fji for each
edge, we generate a cascade from G by randomly choosing a node of G as the root of the cascade.
The root node j is then assigned to time stamp tj = 0. For each neighbor node i pointed by j,
its event time ti is sampled from fji (t). The diffusion process will continue by further infecting
the neighbors pointed by node i in a breadth-first fashion until either the overall time exceed the
predefined observation time window T c or there is no new node being infected. If a node is infected
more than once by multiple parents, only the first infection time stamp will be recorded.
Experiment setting and evaluation metric. We consider a combination of two network topologies (i)-(ii) with three different transmission function settings (1)-(3), which results in six different
experimental settings. For each setting, we randomly instantiate the network topologies and transmission functions for 10 times and then vary the number of cascades from 50, 100, 200, 400, 800 to
1000. For KernelCascade, we use a Gaussian RBF kernel. The kernel bandwidth ? is chosen using
median pairwise distance between grid time points. The regularization parameter is chosen using
two fold cross-validation. NETINF requires the desired number of edges as input, and we give it an
advantage and supply the true number of edges to it. For NETRATE, we experimented with both
exponential and Rayleigh transmission function.
We compare different methods in terms of (1) F 1 score for the network recovery. F 1 :=
2?precision?recall
precision+recall , where precision is the fraction of edges in the inferred network that also present in
the true network and recall is the fraction of edges in the true network that also present in the inferred
network; (2) KL divergence between the estimated transmission function and the true transmission
function, averaged over all edges in a network; (3) the shape of the fitted transmission function
compared to the true transmission function.
6
0.6
netinf
netrate(rayleigh)
netrate(exp)
KernelCascade
0.4
0.2
500
num of cascades
0.6
0.4
0.2
0
0
1000
(a) Core-Periphery, p(t)
netinf
netrate(rayleigh)
netrate(exp)
KernelCascade
0.4
0.2
500
num of cascades
0.4
0.8
0.6
0.4
0.2
500
num of cascades
0
0
1000
netinf
netrate(rayleigh)
netrate(exp)
KernelCascade
500
num of cascades
1000
log(KL Distance)
6
4
netrate(rayleigh)
netrate(exp)
KernelCascade
2
0
0
1000
8
500
num of cascades
(a) Core-Periphery, p(t)
(b) Core-Periphery, q(t)
8
8
6
4
netrate(rayleigh)
netrate(exp)
KernelCascade
1000
4
0
0
4
netrate(rayleigh)
netrate(exp)
KernelCascade
2
500
num of cascades
1000
(c) Core-Periphery, mix p(t), q(t)
8
6
2
6
0
0
1000
log(KL Distance)
log(KL Distance)
netrate(rayleigh)
netrate(exp)
KernelCascade
log(KL Distance)
log(KL Distance)
4
500
num of cascades
1000
(c) Core-Periphery, mix p(t), q(t)
8
500
num of cascades
500
num of cascades
1
0.6
0
0
netinf
netrate(rayleigh)
netrate(exp)
KernelCascade
(e) Random, q(t)
(f) Random, mix p(t), q(t)
Figure 3: F1 Scores for network recovery.
6
0
0
0
0
0.2
8
2
0.4
1000
netinf
netrate(rayleigh)
netrate(exp)
KernelCascade
0.8
1000
(d) Random, p(t)
0
0
500
num of cascades
average F1
0.6
2
0.6
1
average F1
average F1
1
0.8
0.2
(b) Core-Periphery, q(t)
0.8
0
0
netinf
netrate(rayleigh)
netrate(exp)
KernelCascade
log(KL Distance)
0
0
1
0.8
average F1
1
average F1
average F1
1
0.8
netrate(rayleigh)
netrate(exp)
KernelCascade
500
num of cascades
1000
6
4
netrate(rayleigh)
netrate(exp)
KernelCascade
2
0
0
500
num of cascades
1000
(d)Random, p(t)
(e) Random, q(t)
(f) Random, mix p(t), q(t)
Figure 4: KL Divergence between the estimated and the true transmission function.
F1 score for network recovery. From Figure 3, we can see that in all cases, KernelCascade performs consistently and significantly better than NETINF and NETRATE. Furthermore, its performance also steadily increases as we increase the number of cascades, and finally KernelCascade recovers the entire network with around 1000 cascades. In contrast, the competitor methods seldom
fully recover the entire network given the same number of cascades. We also note that the performance of NETRATE is very sensitive to the choice of the transmission function (exponential vs.
Rayleigh). For instance, depending on the actual data generating process, the performance of NETRATE with Rayleigh model can vary from the second best to the worst.
KL divergence for transmission function. Besides better network recovery, KernelCascade also
estimates the transmission function better. In all cases we experimented, KernelCascade leads to
drastic improvement in recovering the transmission function (Figure 4). We also observe that as we
increase the number of cascades, KernelCascade adapts better to the actual transmission function. In
contrast, the performance of NETRATE with exponential model does not improve with increasing
number of cascades, since the parametric model assumption is incorrect. We note that NETINF does
not recover the transmission function, and hence there is no corresponding curve in the plot.
Visualization of the transmission function. We also visualize the estimated transmission function
for an edge from different methods in Figure 5. We can see that KernelCascade captures the essential
7
0.5
0.5
KernelCascade
exp
rayleigh
data
0.4
0.3
pdf
pdf
0.4
0.2
0.1
KernelCascade
exp
rayleigh
data
0.3
0.2
0.1
0
0
10
t
20
0
0
30
10
t
20
30
(a) An edge with transmission function p(t) (b) An edge with transmission function q(t)
Figure 5: Estimated transmission function of a single edge based on 1000 cascades against the true
transmission function (blue curve).
(a) KernelCascade
(b) NETINF
(c) NETRATE
Figure 6: Estimated network of top 32 sites. Edges in grey are correctly uncovered, while edges
highlighted in red are either missed or estimated falsely.
features of the true transmission function, i.e., bi-modal behavior, while the competitor methods miss
out the important statistical feature completely.
5.2
Real world dataset
Finally, we use the MemeTracker dataset [3] to compare NETINF, NETRATE and KernelCascade.
In this dataset, the hyperlinks between articles and posts can be used to represent the flow of information from one site to another site. When a site publishes a new post, it will put hyperlinks to
related posts in some other sites published earlier as its sources. Later as it also becomes ?older?, it
will be cited by other newer posts as well. As a consequence, all the time-stamped hyperlinks form
a cascade for particular piece of information (or event) flowing among different sites. The networks
formed by these hyperlinks are used to be the ground truth. We have extracted a network consisting
of top 500 sites with 6,466 edges and 11,530 cascades from 7,181,406 posts in a month, and we
want to recover the the underlying networks. From Table 1, we can see that KernelCascade achieves
a much better F 1 score for network recovery compared to other methods. Finally, we visualize the
estimated sub-network structure for the top 32 sites in Figure 6. By comparison, KernelCascade has
a relatively better performance with fewer misses and false predictions.
Table 1: Network recovery results from MemeTracker dataset.
methods
precision recall
F1
predicted edges
NETINF
0.62
0.62 0.62
6466
NETRATE(exp)
0.93
0.23 0.37
1600
KernelCascade
0.79
0.66 0.72
5368
6
Conclusion
In this paper, we developed a flexible kernel method, called KernelCascade, to model the latent diffusion processes and to infer the hidden network with heterogeneous influence between each pair
of nodes. In contrast to previous state-of-the-art, such as NETRATE, NETINF and CONNIE, KernelCascade makes no restricted assumption on the specific form of the transmission function over
network edges. Instead, it can infer it automatically from the data, which allows each pair of nodes to
have a different type of transmission model and better captures the heterogeneous influence among
entities. We obtain an efficient algorithm and demonstrate experimentally that KernelCascade can
significantly outperforms previous state-of-the-art in both synthetic and real data. In future, we will
explore the combination of kernel methods, sparsity inducing norms and other point processes to
address a diverse range of social network problems.
Acknowledgement: L.S. is supported by NSF IIS-1218749 and startup funds from Gatech.
8
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9
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3,958 | 4,583 | Symmetric Correspondence Topic Models for
Multilingual Text Analysis
Kosuke Fukumasu?
Koji Eguchi?
Eric P. Xing?
Graduate School of System Informatics, Kobe University, Kobe 657-8501, Japan
?
School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA
?
[email protected], [email protected], [email protected]
Abstract
Topic modeling is a widely used approach to analyzing large text collections. A
small number of multilingual topic models have recently been explored to discover latent topics among parallel or comparable documents, such as in Wikipedia.
Other topic models that were originally proposed for structured data are also applicable to multilingual documents. Correspondence Latent Dirichlet Allocation
(CorrLDA) is one such model; however, it requires a pivot language to be specified in advance. We propose a new topic model, Symmetric Correspondence LDA
(SymCorrLDA), that incorporates a hidden variable to control a pivot language,
in an extension of CorrLDA. We experimented with two multilingual comparable datasets extracted from Wikipedia and demonstrate that SymCorrLDA is more
e?ective than some other existing multilingual topic models.
1
Introduction
Topic models (also known as mixed-membership models) are a useful method for analyzing large
text collections [1, 2]. In topic modeling, each document is represented as a mixture of topics, where
each topic is represented as a word distribution. Latent Dirichlet Allocation (LDA) [2] is one of the
well-known topic models. Most topic models assume that texts are monolingual; however, some
can capture statistical dependencies between multiple classes of representations and can be used for
multilingual parallel or comparable documents. Here, a parallel document is a merged document
consisting of multiple language parts that are translations from one language to another, sometimes
including sentence-to-sentence or word-to-word alignments. A comparable document is a merged
document consisting of multiple language parts that are not translations of each other but instead
describe similar concepts and events. Recently published multilingual topic models [3, 4], which
are the equivalent of Conditionally Independent LDA (CI-LDA) [5, 6], can discover latent topics
among parallel or comparable documents. SwitchLDA [6] was modeled by extending CI-LDA. It
can control the proportions of languages in each multilingual topic. However, both CI-LDA and
SwitchLDA preserve dependencies between languages only by sharing per-document multinomial
distributions over latent topics, and accordingly the resulting dependencies are relatively weak.
Correspondence LDA (CorrLDA) [7] is another type of topic model for structured data represented
in multiple classes. It was originally proposed for annotated image data to simultaneously model
words and visual features, and it can also be applied to parallel or comparable documents. In the
modeling, it first generates topics for visual features in an annotated image. Then only the topics
associated with the visual features in the image are used to generate words. In this sense, visual
features can be said to be the pivot in modeling annotated image data. However, when CorrLDA
is applied to multilingual documents, a language that plays the role of the pivot (a pivot language1 )
1
Note that the term ?pivot language? does not have exactly the same meaning as that commonly used in the
machine translation community, where it means an intermediary language for translation between more than
three languages.
1
must be specified in advance. The pivot language selected is sensitive to the quality of the multilingual topics estimated with CorrLDA. For example, a translation of a Japanese book into English
would presumably have a pivot to the Japanese book, but a set of international news stories would
have pivots that di?er based on the country an article is about. It is often di?cult to appropriately
select the pivot language. To address this problem, which we call the pivot problem, we propose
a new topic model, Symmetric Correspondence LDA (SymCorrLDA), that incorporates a hidden
variable to control the pivot language, in an extension of CorrLDA. Our SymCorrLDA addresses the
problem of CorrLDA and can select an appropriate pivot language by inference from the data.
We evaluate various multilingual topic models, i.e., CI-LDA, SwitchLDA, CorrLDA, and our SymCorrLDA, as well as LDA, using comparable articles in di?erent languages (English, Japanese, and
Spanish) extracted from Wikipedia. We first demonstrate through experiments that CorrLDA outperforms the other existing multilingual topic models mentioned, and then show that our SymCorrLDA
works more e?ectively than CorrLDA in any case of selecting a pivot language.
2 Multilingual Topic Models with Multilingual Comparable Documents
Bilingual topic models for bilingual parallel documents that have word-to-word alignments have
been developed, such as those by [8]. Their models are directed towards machine translation, where
word-to-word alignments are involved in the generative process. In contrast, we focus on analyzing
dependencies among languages by modeling multilingual comparable documents, each of which
consists of multiple language parts that are not translations of each other but instead describe similar
concepts and events. The target documents can be parallel documents, but word-to-word alignments
are not taken into account in the topic modeling. Some other researchers explored di?erent types
of multilingual topic models that are based on the premise of using multilingual dictionaries or
WordNet [9, 10, 11]. In contrast, CI-LDA and SwitchLDA only require multilingual comparable
documents that can be easily obtained, such as from Wikipedia, when we use those models for
multilingual text analysis. This is more similar to the motivation of this paper. Below, we introduce
LDA-style topic models that handle multiple classes and can be applied to multilingual comparable
documents for the above-mentioned purposes.
2.1
Conditionally Independent LDA (CI-LDA)
CI-LDA [5, 6] is an extension of the LDA model to handle multiple classes, such as words and
citations in scientific articles. The CI-LDA framework was used to model multilingual parallel or
comparable documents by [3] and [4]. Figure 1(b) shows a graphical model representation of CILDA for documents in L languages, and Figure 1(a) shows that of LDA for reference. D, T , and
Nd respectively indicate the number of documents, number of topics, and number of word tokens
that appear in a specific language part in a document d. The superscript ?(?)? indicates the variables
corresponding to a specific language part in a document d. For better understanding, we show below
the process of generating a document according to the graphical model of the CI-LDA model.
1.
2.
3.
For all D documents, sample ?d ? Dirichlet(?)
(?)
For all T topics and for all L languages, sample ?(?)
t ? Dirichlet(? )
For each of the Nd(?) words w(?)
in
language
?
(?
?
{1,
?
?
?
,
L})
of
document
d:
i
a.
Sample a topic z(?)
?
Multinomial(?
)
d
i
(?)
b.
Sample a word w(?)
i ? Multinomial(? (?) )
zi
For example, when we deal with Japanese and English bilingual data, w(1) and w(2) are a Japanese
and an English word, respectively. CI-LDA preserves dependencies between languages only by
sharing the multinomial distributions with parameters ?d . Accordingly, there are substantial chances
that some topics are assigned only to a specific language part in each document, and the resulting
dependencies are relatively weak.
2.2
SwitchLDA
Similarly to CI-LDA, SwitchLDA [6] can be applied to multilingual comparable documents. However, di?erent from CI-LDA, SwitchLDA can adjust the proportions of multiple di?erent languages
for each topic, according to a binomial distribution for bilingual data or a multinomial distribution for data of more than three languages. Figure 1(c) depicts a graphical model representation of
SwitchLDA for documents in L languages. The generative process is described below.
2
(a) LDA
(b) CI-LDA
(c) SwitchLDA
Figure 1: Graphical model representations of (a) LDA, (b) CI-LDA, and (c) SwitchLDA
1.
2.
3.
For all D documents, sample ?d ? Dirichlet(?)
For all T topics:
(?)
a.
For all L languages, sample ?(?)
t ? Dirichlet(? )
b.
Sample ?t ? Dirichlet(?)
For each of the Nd words wi in document d:
a.
Sample a topic zi ? Multinomial(?d )
b.
Sample a language label si ? Multinomial(?zi )
i)
c.
Sample a word wi ? Multinomial(?(s
zi )
Here, ?t indicates a multinomial parameter to adjust the proportions of L di?erent languages for
topic t. If all components of hyperparameter vector ? are large enough, SwitchLDA becomes equivalent to CI-LDA. SwitchLDA is an extension of CI-LDA to give emphasis or de-emphasis to specific
languages for each topic. Therefore, SwitchLDA may represent multilingual topics more flexibly;
however, it still has the drawback that the dependencies between languages are relatively weak.
2.3
Correspondence LDA (CorrLDA)
CorrLDA [7] can also be applied to multilingual comparable documents. In the multilingual setting,
this model first generates topics for one language part of a document. We refer to this language as a
pivot language. For the other languages, the model then uses the topics that were already generated
in the pivot language. Figure 2(a) shows a graphical model representation of CorrLDA assuming L
languages, when p is the pivot language that is specified in advance. Here, Nd(?) (? ? {p, 2, ? ? ? , L})
denotes the number of words in language ? in document d. The generative process is shown below:
1.
2.
3.
For all D documents? pivot language parts, sample ?d(p) ? Dirichlet(?(p) )
(?)
For all T topics and for all L languages (including the pivot language), sample ?(?)
t ? Dirichlet(? )
For each of the Nd(p) words w(p)
in
the
pivot
language
p
of
document
d:
i
(p)
a.
Sample a topic z(p)
i ? Multinomial(?d )
(p)
b.
Sample a word wi ? Multinomial(?(p)
(p) )
4.
For each of the Nd(?) words w(?)
? (? ?) {2, ? ? ? , L}) of document d:
i in language
( (p)
a.
Sample a topic y(?)
?
Uni
f
orm
z
,
?
?
?
,
z(p)(p)
i
1
zi
b.
(?)
Sample a word w(?)
i ? Multinomial(? (?) )
Nd
yi
This model can capture more direct dependencies between languages, due to the constraints that topics have to be selected from the topics selected in the pivot language parts. However, when CorrLDA
is applied to multilingual documents, a pivot language must be specified in advance. Moreover, the
pivot language selected is sensitive to the quality of the multilingual topics estimated with CorrLDA.
3 Symmetric Correspondence Topic Models
When CorrLDA is applied to parallel or comparable documents, this model first generates topics
for one language part of a document, which we refer to this language as a pivot language. For the
other languages, the model then uses the topics that were already generated in the pivot language.
CorrLDA has the great advantage that it can capture more direct dependency between languages;
3
(a) CorrLDA
(b) SymCorrLDA
(c) alternative SymCorrLDA
Figure 2: Graphical model representations of (a) CorrLDA, (b) SymCorrLDA, and (c) its variant
however, it has a disadvantage that it requires a pivot language to be specified in advance. Since
the pivot language may di?er based on the subject, such as the country a document is about, it is
often di?cult to appropriately select the pivot language. To address this problem, we propose
Symmetric Correspondence LDA (SymCorrLDA). This model generates a flag that specifies a pivot
language for each word, adjusting the probability of being pivot languages in each language part
of a document according to a binomial distribution for bilingual data or a multinomial distribution
for data of more than three languages. In other words, SymCorrLDA estimates from the data the
best pivot language at the word level in each document. The pivot language flags may be assigned
to the words in the originally written portions in each language, since the original portions may be
described confidently and with rich vocabulary. Figure 2(b) shows a graphical model representation
of SymCorrLDA. SymCorrLDA?s generative process is shown as follows, assuming L languages:
1.
For all D documents:
For all L languages, sample ?d(?) ? Dirichlet(?(?) )
Sample ?d ? Dirichlet(?)
(?)
For all T topics and for all L languages, sample ?(?)
t ? Dirichlet(? )
(?)
(?)
For each of the Nd words wi in language ? (? ? {1, ? ? ? , L}) of document d:
a.
Sample a pivot language flag xi(?) ? Multinomial(?d )
(?)
b.
If (xi(?)=?), sample a topic z(?)
i ? Multinomial(?
d )
)
( (m)
(?)
(?)
c.
If (xi =m,?), sample a topic yi ? Uni f orm z1 , ? ? ? , z(m)(m)
Md
(
)
(?)
(?)
d. Sample a word w(?)
i ? Multinomial ? x(?) =? ? (?) + (1 ? ? x(?) =? )? (?)
a.
b.
2.
3.
zi
i
yi
i
The pivot language flag xi(?) = ? for an arbitrary language ? indicates that the pivot language for the
word wi(?) is its own language ?, and xi(?) = m indicates that the pivot language for w(?)
i is another
language m di?erent from its own language ?. The indicator function ? takes the value 1 when the
designated event occurs and 0 if otherwise. Unlike CorrLDA, the uniform distribution at Step 3-c is
not based on the topics that are generated for all Nd(m) words with the pivot language flags, but based
only on the topics that are already generated for Md(m) (Md(m) ? Nd(m) ) words with the pivot language
flags at each step while in the generative process.2 The full conditional probability for collapsed
Gibbs sampling of this model is given by the following equations, assuming symmetric Dirichlet
priors parameterized by ?(?) , ?(?) (? ? {1, ? ? ? , L}), and ?:
(?)
(?) (?)
(?) (?)
P(zi(?) = t, xi(?) = ?|w(?)
i = w , z?i , w?i , x?i , ? , ? , ?) ?
W (?) T
TD
C (?) ? + ?
Ctd,?i
+ ?(?)
nd?,?i + ?
w t,?i
?
??
?
?
(?)
(?)
(?)
nd?,?i + j,? nd j + L?
? C T? D
? C W ?T
+
T
?
+ W (?) ?(?)
t
w(?)
(?)
t d,?i
(?)
w
(?)
(1)
t,?i
(?)
(?) (?) (m)
(?)
P(yi(?) = t, xi(?) = m|w(?)
i = w , y?i , z , w?i , x?i , ? , ?) ?
Md(m) words may indeed di?er in size at the step of generating each word in the generative process. However, this is not problematic for inference, such as by collapsed Gibbs sampling, where any topic is first randomly assigned to every word, and a more appropriate topic is then re-assigned to each word, based on the
topics previously assigned to all Nd(m) words, not Md(m) words, with the pivot language flags.
2
4
Table 1: Summary of bilingual data
No. of documents
No. of word types (vocab)
No. of word tokens
Table 2: Summary of trilingual data
Japanese
English
229,855
124,046
173,157
61,187,469
80,096,333
Japanese
English
Spanish
No. of documents
90,602
No. of word types (vocab) 70,902
98,474
96,191
No. of word tokens
25,952,978 33,999,988 25,701,830
(?)
C W(?) ?T + ?(?)
T D(m)
Ctd
ndm,?i + ?
w t,?i
?
?
??
(?)
ndm,?i + j,m nd j + L? Nd(m)
? C W ?T
+ W (?) ?(?)
(?)
w
(?)
w
(2)
t,?i
(?)
(?)
(?)
(?)
(?)
(?)
where w(?) = {w(?)
i }, z = {zi }, and x = {xi }. W and Nd respectively indicate the total number of
vocabulary words (word types) in the specified language, and the number of word tokens that appear
in the specified language part of document d. nd? and ndm are the number of times, for an arbitrary
word i ? {1, ? ? ? , Nd(?) } in an arbitrary language j ? {1, ? ? ? , L} of document d, the flags xi( j) = ? and
T D(?)
xi( j) = m respectively are allocated to document d. Ctd
indicates the (t, d) element of a T ? D
topic-document count matrix, meaning the number of times topic t is allocated to the document d?s
W (?) T
language part specified in parentheses. Cwt
indicates the (w, t) element of a W (?) ? T word-topic
count matrix, meaning the number of times topic t is allocated to word w in the language specified
in parentheses. The subscript ??i? indicates when wi is removed from the data.
Now we slightly modify SymCorrLDA by replacing Step 3-c in its generative process by:
(m)
3-c. If (xi(?)=m,?), sample a topic y(?)
i ? Multinomial(?d )
Figure 2(c) shows a graphical model representation of this alternative SymCorrLDA. In this model,
non-pivot topics are dependent on the distribution behind the pivot topics, not dependent directly on
the pivot topics as in the original SymCorrLDA. By this modification, the generative process is more
naturally described. Accordingly, Eq. (2) of the full conditional probability is replaced by:
(?)
(?) (?) (m)
(?)
P(yi(?) = t, xi(?) = m|w(?)
i = w , y?i , z , w?i , x?i , ? , ?) ?
(?)
C W(?) ?T + ?(?)
T D(m)
+ ?(m)
Ctd
ndm,?i + ?
w t,?i
?
??
??
(?)
T D(m)
ndm,?i + j,m nd j + L?
?
? C W ?T
+ T ?(m)
+ W (?) ?(?)
(?)
t C t? d
w
(?)
w
(3)
t,?i
As you can see in the second term of the right-hand side above, the constraints are relaxed by this
modification so that topics do not always have to be selected from the topics selected for the words
with the pivot language flags, di?erently from that of Eq. (2). We will show through experiments
how the modification a?ects the quality of the estimated multilingual topics, in the following section.
4
Experiments
In this section, we demonstrate some examples with SymCorrLDA, and then we compare multilingual topic models using various evaluation methods. For the evaluation, we use held-out loglikelihood using two datasets, the task of finding an English article that is on the same topic as that
of a Japanese article, and a task with the languages reversed.
4.1
Settings
The datasets used in this work are two collections of Wikipedia articles: one is in English and
Japanese, the other is in English, Japanese, and Spanish, and articles in each collection are connected
across languages via inter-language links, as of November 2, 2009. We extracted text content from
the original Wikipedia articles, removing link information and revision history information. We used
WP2TXT3 for this purpose. For English articles, we removed 418 types of standard stop words [12].
For Spanish articles, we removed 351 types of standard stop words [13]. As for Japanese articles,
we removed function words, such as symbols, conjunctions and particles, using part-of-speech tags
annotated by MeCab4 . The statistics of the datasets after preprocessing are shown in Tables 1 and
2. We assumed each set of Wikipedia articles connected via inter-language links between two (or
3
4
http://wp2txt.rubyforge.org/
http://mecab.sourceforge.net/
5
0th iteration
5th iteration
20th iteration
50th iteration
80000
70000
? d = (1,0,0)
Europe
Austria
Physics
frequency
60000
50000
Shogi
(Japanese chess)
Japanese
Language
Mount Fuji
Mobile phone
? d ,1
40000
0.0
30000
0.5
Western art
history
20000
Personal
computer
Sony
1.0
Bull?gh"ng
Horyu?-ji
(Horyu Temple)
10000
? d = (0,1,0)
0
0
0.2
0.4
0.6
0.8
(a) Examples with bilingual data
1
?d,1
Figure 3: Change of frequency distribution of ?d,1 according to number of iterations
Proporon of
Japanese pivot
? d = (0,0,1)
Figure 4: Document titles and corresponding ?d
nen (year)
ingurando (England)
daihyo? (representave)
rigu (league)
sizun (season)
Topic 269
species
insects
eggs
body
larvae
0.0
rui (species)
shu (species)
karada (body)
konchu? (insect)
dobutsu (animal)
Topic 13
japan
osaka
kyoto
hughes
japanese
osaka
kyoto
shi (city)
nen (year)
kobe
0.5
Topic 201
ireland
irish
scotland
sco!sh
dublin
Europe
(b) Examples with trilingual data
Topic 251
united
cup
manchester
manager
league
NFL
airurando (Ireland)
suko"orando (Scotland)
nen (year)
daburin (Dublin)
kitaairurando (Northern Ireland)
1.0
Topic 426
car
vehicle
vehicles
cars
truck
kuruma (car)
jidosha(automobile)
sharyo? (vehicle)
unten (driving)
torakku (truck)
Topic 59
castle
ba"le
oda
hideyoshi
nobunaga
nobunaga
shiro (castle)
hideyoshi
shi(surname)
oda
Figure 5: Topic examples and corresponding proportion of pivots assigned to Japanese. An English
translation for each Japanese word follows in parentheses, except for Japanese proper nouns.
three) languages as a comparable document that consists of two (or three) language parts. To carry
out the evaluation in the task of finding counterpart articles that we will describe later, we randomly
divided the Wikipedia document collection at the document level into 80% training documents and
20% test documents. Furthermore, to compute held-out log-likelihood, we randomly divided each
of the training documents at the word level into 90% training set and 10% held-out set.
We first estimated CI-LDA, SwitchLDA, CorrLDA, and SymCorrLDA and its alternative version
(?SymCorrLDA-alt?) as well as LDA for a baseline, using collapsed Gibbs sampling with the training
set. In addition, we estimated a special implementation of SymCorrLDA, setting ?d in a simple way
for comparison, where the pivot language flag for each word is randomly selected according to the
proportion of the length of each language part (?SymCorrLDA-rand?).
For all the models, we assumed symmetric Dirichlet hyperparameters ? = 50/T and ? = 0.01, which
have often been used in prior work [14]. We imposed the convergence condition of collapsed Gibbs
sampling, such that the percentage change of held-out log-likelihood is less than 0.1%. For SymCorrLDA, we assumed symmetric Dirichlet hyperparameters ? = 1. For SwitchLDA, we assumed
symmetric Dirichlet hyperparameters ? = 1. We investigated the e?ect of ? in SymCorrLDA and
? in SwitchLDA; however, the held-out log-likelihood was almost constant when varying these hyperparameters. LDA does not distinguish languages, so for a baseline we assumed all the language
parts connected via inter-language links to be mixed together as a single document.
4.2 Pivot assignments
Figure 3 demonstrates how the frequency distribution of the pivot language-flag (binomial) parameter ?d,1 for the Japanese language with the bilingual dataset5 in SymCorrLDA changes while in
iterations of collapsed Gibbs sampling. This figure shows that the pivot language flag is randomly
assigned at the initial state, and then it converges to an appropriate bias for each document as the iterations proceed. We next demonstrate how the pivot language flags are assigned to each document.
Figure 4(a) shows the titles of eight documents and the corresponding ?d when using the bilingual
data (T = 500). If ?d,1 is close to 1, the article can be considered to be more related to a subject on
Japanese or Japan. In contrast, if ?d,1 is close to 0 and therefore ?d,2 = 1 ? ?d,1 is close to 1, the
article can be considered to be more related to a subject on English or English-speaking countries.
Therefore, a pivot is assigned considering the language biases of the articles. Figure 4(b) shows
the titles of six documents and the corresponding ?d = (?d,1 , ?d,2 , ?d,3 ) when using the trilingual
5
The parameter for English was ?d,2 = 1 ? ?d,1 in this case.
6
Table 3: Per-word held-out log-likelihood with
bilingual data. Boldface indicates the best result
in each column.
T=500
Japanese
English
LDA
CI-LDA
SwitchLDA
CorrLDA1
CorrLDA2
SymCorrLDA
SymCorrLDA-alt
SymCorrLDA-rand
-8.127
-8.136
-8.139
-7.463
-7.777
-7.433
-7.476
-7.483
-8.633
-8.644
-8.641
-8.403
-8.197
-8.175
-8.206
-8.222
T=1000
Japanese
English
-7.992
-8.008
-8.012
-7.345
-7.663
-7.317
-7.358
-7.373
Table 4: Per-word held-out log-likelihood with
trilingual data. Boldface indicates the best result
in each column.
T=500
T=1000
Japanese English Spanish Japanese English Spanish
-8.530
-8.549
-8.549
-8.346
-8.109
-8.084
-8.116
-8.137
CorrLDA1
CorrLDA2
CorrLDA3
SymCorrLDA
SymCorrLDA-alt
-7.408
-7.655
-7.794
-7.394
-7.440
-8.512
-8.198
-8.460
-8.178
-8.209
-8.667
-8.467
-8.338
-8.289
-8.330
-7.305
-7.572
-7.700
-7.287
-7.330
-8.393
-8.122
-8.383
-8.093
-8.120
-8.545
-8.401
-8.274
-8.215
-8.254
data (T = 500). Here, ?d,1 , ?d,2 , and ?d,3 respectively indicate the pivot language-flag (multinomial)
parameters corresponding to Japanese, English, and Spanish parts in each document. We further
demonstrate the proportions of pivot assignments at the topic level. Figure 5 shows the content of
6 topics through 10 words with the highest probability for each language and for each topic when
using the bilingual data (T = 500), some of which are biased to Japanese (Topics 13 and 59) or
English (Topics 201 and 251), while the others have almost no bias. It can be seen that the pivot bias
to specific languages can be interpreted.
4.3
Held-out log-likelihood
By measuring the held-out log-likelihood, we can evaluate the quality of each topic model. The
higher the held-out log-likelihood, the greater the predictive ability of the model. In this work,
we estimated multilingual topic models with the training set and computed the log-likelihood of
generating the held-out set that was mentioned in Section 4.1.
Table 3 shows the held-out log-likelihood of each multilingual topic model estimated with the bilingual dataset when T = 500 and 1000. Note that the held-out log-likelihood (i.e., the micro-average
per-word log-likelihood of the 10% held-out set) is shown for each language in this table, while
the model estimation was performed over the 90% training set in all the languages. Hereafter, CorrLDA1 refers to the CorrLDA model that was estimated when Japanese was the pivot language. As
described in Section 2.3, the CorrLDA model first generates topics for the pivot language part of a
document, and for the other language parts of the document, the model then uses the topics that were
already generated in the pivot language. CorrLDA2 refers to the CorrLDA model when English was
the pivot language. As the results in Table 3 show, the held-out log-likelihoods of CorrLDA1 and
CorrLDA2 are much higher than those of the other prior models: CI-LDA, SwitchLDA, and LDA,
in both cases. This is because CorrLDA can capture direct dependencies between languages, due to
the constraints that topics have to be selected from the topics selected in the pivot language parts.
On the other hand, CI-LDA and SwitchLDA are too poorly constrained to e?ectively capture the
dependencies between languages, as mentioned in Sections 2.1 and 2.2. In particular, CorrLDA1
has the highest held-out log-likelihood among all the prior models for Japanese, while CorrLDA2
is the best among all the prior models for English. This is probably due to the fact that CorrLDA
can estimate topics from the pivot language parts (Japanese in the case of CorrLDA1) without any
specific constraints; however, great constraints (topics having to be selected from the topics selected
in the pivot language parts) are imposed for the other language parts. In SymCorrLDA, the held-out
log-likelihood for Japanese is larger than that of CorrLDA1 (and the other models), and the held-out
log-likelihood for English is larger than that of CorrLDA2. This is probably because SymCorrLDA
estimates the pivot language appropriately adjusted for each word in each document. Next, we compare SymCorrLDA and its alternative version (SymCorrLDA-alt). We observed in Table 3 that the
held-out log-likelihood of SymCorrLDA-alt is smaller than that of the original SymCorrLDA, and
comparable to CorrLDA?s best. This is because the constraints in SymCorrLDA-alt are relaxed so
that topics do not always have to be selected from the topics selected for the words with the pivot
language flags.
For further consideration, let us examine the results of the simplified implementation:
SymCorrLDA-rand, which we defined in Section 4.1. SymCorrLDA-rand?s held-out log-likelihood
lies even below CorrLDA?s best. These results reflect the fact that the performance of SymCorrLDA in its full form is inherently a?ected by the nature of the language biases in the multilingual
comparable documents, rather than merely being a?ected by the language part length.
7
Table 4 shows the held-out log-likelihood with the trilingual data when T = 500 and 1000. Here,
CorrLDA3 refers to the CorrLDA model that was estimated when Spanish was the pivot language.
As you can see in this table, SymCorrLDA?s held-out log-likelihood is larger than CorrLDA?s best.
SymCorrLDA can estimate the pivot language appropriately adjusted for each word in each document in the trilingual data, as with the bilingual data. SymCorrLDA-alt behaves similarly as with
the bilingual data.
For both the bilingual and trilingual data, the improvements with SymCorrLDA were statistically
significant, compared to each of the other models, according to the Wilcoxon signed-rank test at the
5% level in terms of the word-by-word held-out log-likelihood. As for the scalability, SymCorrLDA
is as scalable as CorrLDA because the time complexity of SymCorrLDA is the same order as that of
CorrLDA: the number of topics times the sum of vocabulary size in each language. On clock time,
SymCorrLDA does pay some extra, such as around 40% of the time for CorrLDA in the case of the
bilingual data, for allocating the pivot language flags.
4.4
Finding counterpart articles
Given an article, we can find its unseen counterpart articles in other languages using a multilingual topic model. To evaluate this task, we experimented with the bilingual dataset. We estimated
document-topic distributions of test documents for each language, using the topic-word distributions
that were estimated by each multilingual topic model with training documents. We then evaluated
the performance of finding English counterpart articles using Japanese articles as queries, and vice
versa. For estimating the document-topic distributions of test documents, we used re-sampling of
LDA using the topic-word distribution estimated beforehand by each multilingual topic model [15].
We then computed the Jensen-Shannon (JS) divergence [16] between a document-topic distribution
of Japanese and that of English for each test document. Each held-out English-Japanese article pair
connected via an inter-language link is considered to be on the same topic; therefore, JS divergence
of such an article pair is expected to be small if the latent topic estimation is accurate. We first
assumed each held-out Japanese article to be a query and the corresponding English article to be
relevant, and evaluated the ranking of all the test articles of English in ascending order of the JS
divergence; then we conducted the task with the languages reversed.
Table 5 shows the results of mean reciprocal Table 5: MRR in counterpart article finding task.
rank (MRR), when T = 500 and 1000. The re- Boldface indicates the best result in each column.
ciprocal rank is defined as the multiplicative inJapanese to English
English to Japanese
verse of the rank of the counterpart article corT=500
T=1000
T=500
T=1000
responding to each query article, and the mean
LDA
0.0743
0.1027
0.0870
0.1262
CI-LDA
0.1426
0.1464
0.1697
0.1818
reciprocal rank is the average of it over all the
SwitchLDA
0.1357
0.1347
0.1668
0.1653
query articles. CorrLDA works much more efCorrLDA1
0.2987
0.3281
0.2863
0.3111
CorrLDA2
0.2829
0.3063
0.3161
0.3464
fectively than the other prior models: CI-LDA,
SymCorrLDA 0.3256
0.3592
0.3348
0.3685
SwitchLDA, and LDA, and overall, SymCorrLDA works the most e?ectively. We observed that the improvements with SymCorrLDA were
statistically significant according to the Wilcoxon signed-rank test at the 5% level, compared with
each of the other models. Therefore, it is clear that SymCorrLDA estimates multilingual topics the
most successfully in this experiment.
5
Conclusions
In this paper, we compared the performance of various topic models that can be applied to multilingual documents, not using multilingual dictionaries, in terms of held-out log-likelihood and in the
task of cross-lingual link detection. We demonstrated through experiments that CorrLDA works significantly more e?ectively than CI-LDA, which was used in prior work on multilingual topic models.
Furthermore, we proposed a new topic model, SymCorrLDA, that incorporates a hidden variable to
control a pivot language, in an extension of CorrLDA. SymCorrLDA has an advantage in that it does
not require a pivot language to be specified in advance, while CorrLDA does. We demonstrated that
SymCorrLDA is more e?ective than CorrLDA and the other topic models, through experiments
with Wikipedia datasets using held-out log-likelihood and in the task of finding counterpart articles
in other languages. SymCorrLDA can be applied to other kinds of data that have multiple classes of
representations, such as annotated image data. We plan to investigate this in future work.
8
Acknowledgments We thank Sinead Williamson, Manami Matsuura, and the anonymous reviewers for valuable discussions and comments. This work was supported in part by the Grant-in-Aid for
Scientific Research (#23300039) from JSPS, Japan.
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9
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3,959 | 4,584 | Structure estimation for discrete graphical models:
Generalized covariance matrices and their inverses
Martin J. Wainwright
Departments of Statistics and EECS
University of California, Berkeley
Berkeley, CA 94720
[email protected]
Po-Ling Loh
Department of Statistics
University of California, Berkeley
Berkeley, CA 94720
[email protected]
Abstract
We investigate a curious relationship between the structure of a discrete graphical
model and the support of the inverse of a generalized covariance matrix. We show
that for certain graph structures, the support of the inverse covariance matrix of
indicator variables on the vertices of a graph reflects the conditional independence
structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby
addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. Based on our population-level results, we
show how the graphical Lasso may be used to recover the edge structure of certain classes of discrete graphical models, and present simulations to verify our
theoretical results.
1
Introduction
Graphical model inference is now prevalent in many fields, running the gamut from computer vision
and civil engineering to political science and epidemiology. In many applications, learning the
edge structure of an underlying graphical model is of great importance?for instance, a graphical
model may be used to represent friendships between people in a social network, or links between
organisms with the propensity to spread an infectious disease [1]. It is well known that zeros in the
inverse covariance matrix of a multivariate Gaussian distribution indicate the absence of an edge
in the corresponding graphical model. This fact, combined with techniques in high-dimensional
statistical inference, has been leveraged by many authors to recover the structure of a Gaussian
graphical model when the edge set is sparse (e.g., see the papers [2, 3, 4, 5] and references therein).
Recently, Liu et al. [6, 7] introduced the notion of a nonparanormal distribution, which generalizes
the Gaussian distribution by allowing for univariate monotonic transformations, and argued that the
same structural properties of the inverse covariance matrix carry over to the nonparanormal.
However, the question of whether a relationship exists between conditional independence and the
structure of the inverse covariance matrix in a general graph remains unresolved. In this paper, we
focus on discrete graphical models and establish a number of interesting links between covariance
matrices and the edge structure of an underlying graph. Instead of only analyzing the standard covariance matrix, we show that it is often fruitful to augment the usual covariance matrix with higherorder interaction terms. Our main result has a striking corollary in the context of tree-structured
graphs: for any discrete graphical model, the inverse of a generalized covariance matrix is always
(block) graph-structured. In particular, for binary variables, the inverse of the usual covariance matrix corresponds exactly to the edge structure of the tree. We also establish several corollaries that
apply to more general discrete graphs. Our methods are capable of handling noisy or missing data
in a seamless manner.
1
Other related work on graphical model selection for discrete graphs includes the classic ChowLiu algorithm for trees [8]; nodewise logistic regression for discrete models with pairwise interactions [9, 10]; and techniques based on conditional entropy or mutual information [11, 12]. Our
main contribution is to present a clean and surprising result on a simple link between the inverse
covariance matrix and edge structure of a discrete model, which may be used to derive inference
algorithms applicable even to data with systematic corruptions.
The remainder of the paper is organized as follows: In Section 2, we provide brief background and
notation on graphical models, and describe the classes of augmented covariance matrices we will
consider. In Section 3, we state our main results on the relationship between the support of generalized inverse covariance matrices and the edge structure of a discrete graphical model. We relate our
population-level results to concrete algorithms that are guaranteed to recover the edge structure of a
discrete graph with high probability. In Section 4, we report the results of simulations used to verify
our theoretical claims. For detailed proofs, we refer the reader to the technical report [13].
2
Background and problem setup
In this section, we provide background on graphical models and exponential families. We then work
through a simple example that illustrates the phenomena and methodology studied in this paper.
2.1
Graphical models
An undirected graph G = (V, E) consists of a collection of vertices V = {1, 2, . . . , p} and a
collection of unordered vertex pairs E ? V ? V , meaning no distinction is made between edges
(s, t) and (t, s). We associate to each vertex s ? V a random variable Xs taking values in some
space X . The random vector X := (X1 , . . . , Xp ) is a Markov random field with respect to G if
XA ?
? XB | XS whenever S is a cutset of A and B, meaning every path from A to B in G must pass
through S. We have used the shorthand XA := {Xs : s ? A}. In particular, Xs ?? Xt | X\{s,t}
whenever (s, t) ?
/ E.
By the Hammersley-Clifford theorem for strictly positive distributions [14], the Markov properties
imply a factorization of the distribution of X:
Y
P(x1 , . . . , xp ) ?
?C (xC ),
(1)
C?C
where C is the set of all cliques (fully-connected subsets of V ) and ?C (xC ) are the corresponding
clique potentials. The factorization (1) may alternatively be represented in terms of an exponential
family associated with the clique structure of G. For each clique C ? C, we define a family of
sufficient statistics {?C;? : X |C| ? R, ? ? IC } associated with variables in C, where IC indexes
the sufficient statistics corresponding to C. We also introduce a canonical parameter ?C;? ? R
associated with each sufficient statistic ?C;? . For a given assignment of canonical parameters ?, we
may express the clique potentials as
X
?C (xC ) =
?C;? ?C;? (xC ) := h?C , ?C i,
??IC
so equation (1) may be rewritten as
P? (x1 , . . . , xp ) = exp
X
h?C , ?C i ? A(?) ,
(2)
C?C
where A(?) := log
P
x?X p
exp
P
C?C h?C ,
?C i is the (log) partition function.
Note that for a graph with only pairwise interactions, we have C = V ? E. If we associate the
function ?s (xs ) = xs with clique {s} and the function ?st (xs , xt ) = xs xt with edge (s, t), the
factorization (2) becomes
X
X
P? (x1 , . . . , xp ) = exp
? s xs +
?st xs xt ? A(?) .
(3)
s?V
(s,t)?E
2
When X = {0, 1}, this family of distributions corresponds to the inhomogeneous Ising model.
When X = R (and with certain additional restrictions on the weights), the family (3) corresponds
to a Gauss-Markov random field. Both of these models are minimal exponential families, meaning
the sufficient statistics are linearly independent [15].
For a discrete graphical model with X = {0, 1, . . . , m?1}, it is convenient to make use of sufficient
statistics involving indicator functions. For clique C, define the subset of configurations
|C|
X0
= {J = (j1 , . . . , j|C| ) | j` 6= 0 for all ` = 1, . . . , |C|},
|C|
|C|
for which no variables take the value 0. Then |X0 | = (m ? 1)|C| . For any configuration J ? X0 ,
we define the indicator function
1 if xC = J,
?C;J (xC ) =
0 otherwise,
and consider the family of models
X
h?C , ?C i ? A(?) ,
P? (x1 , . . . , xp ) = exp
where xj ? X = {0, 1, . . . , m ? 1},
(4)
C?C
P
|C|
with h?C , ?C i = J?X |C| ?C;J ?C;J (xC ). Note in particular that when m = 2, X0 is a singleton
0
state containing the vector of all ones, and the sufficient statistics are given by
Y
?C;J (xC ) =
xs ,
for C ? C and J = {1}|C| ;
s?C
i.e., the indicator functions may simply be expressed as products of variables appearing in the clique.
When the graphical model has only pairwise interactions, elements of C have cardinality at most
two, and the model (4) clearly reduces to the Ising model (3). Finally, as with the equation (3), the
family (4) is a minimal exponential family.
2.2
Covariance matrices and beyond
Consider the usual covariance matrix ? = cov(X1 , . . . , Xp ). When X is Gaussian, it is a wellknown consequence of the Hammersley-Clifford theorem that the entries of the precision matrix
? = ??1 correspond to rescaled conditional correlations [14]. The magnitude of ?st is a scalar
multiple of the correlation of Xs and Xt conditioned on X\{s,t} , and encodes the strength of the
edge (s, t). In particular, the sparsity pattern of ?st reflects the edge structure of the graph: ?st = 0
if and only if Xs ?
? Xt | X\{s,t} . For more general distributions, however, Corr(Xs , Xt | X\{s,t} )
is a function of X\{s,t} , and it is not known whether the entries of ? have any relationship with the
strengths of edges in the graph.
Nonetheless, it is tempting to conjecture that inverse covariance matrices, and more generally, inverses of higher-order moment matrices, might be related to graph structure. Let us explore this
possibility by considering a simple example, namely the binary Ising model (3) with X = {0, 1}.
Example 1. Consider a simple chain graph on four nodes, as illustrated in Figure 1(a). In terms
of the factorization (3), let the node potentials be ?s = 0.1 for all s ? V and the edge potentials
be ?st = 2 for all (s, t) ? E. For a multivariate Gaussian graphical model defined on G, standard
theory predicts that the inverse covariance matrix ? = ??1 of the distribution is graph-structured:
?st = 0 if and only if (s, t) ?
/ E. Surprisingly, this is also the case for the chain graph with binary
variables: a little computation show that ? takes the form shown in panel (f). However, this statement
is not true for the single-cycle graph shown in panel (b), with added edge (1, 4). Indeed, as shown
in panel (g), the inverse covariance matrix has no nonzero entries at all. But for a more complicated
graph, say the one in (e), we again observe a graph-structured inverse covariance matrix.
Still focusing on the single-cycle graph in panel (b), suppose that instead of considering the ordinary covariance matrix, we compute the covariance matrix of the augmented random vector
(X1 , X2 , X3 , X4 , X1 X3 ), where the extra term X1 X3 is represented by the dotted edge shown
3
(a) Chain
(b) Single cycle
?
9.80
??3.59
?chain = ?
? 0
0
?3.59
34.30
?4.77
0
0
?4.77
34.30
?3.59
(c) Edge augmented
?
0
0 ?
?.
?3.59?
9.80
(f)
(d) With 3-cliques
?
51.37
??5.37
?loop = ?
??0.17
?5.37
?5.37
51.37
?5.37
?0.17
?0.17
?5.37
51.37
?5.37
(e) Dino
?
?5.37
?0.17?
?,
?5.37?
51.37
(g)
Figure 1. (a)?(e) Different examples of graphical models. (f) Inverse covariance for chain-structured
graph in (a). (g) Inverse covariance for single-cycle graph in (b).
in panel (c). The 5 ? 5 inverse of this generalized covariance matrix takes the form
?
?
1.15 ?0.02 1.09 ?0.02 ?1.14
??0.02 0.05 ?0.02
0
0.01 ?
?
?
?aug = 103 ? ? 1.09 ?0.02 1.14 ?0.02 ?1.14? .
??0.02
0
?0.02 0.05
0.01 ?
?1.14 0.01 ?1.14 0.01
1.19
This matrix safely separates nodes 1 and 4, but the entry corresponding to the phantom edge (1, 3) is
not equal to zero. Indeed, we would observe a similar phenomenon if we chose to augment the graph
by including the edge (2, 4) rather than (1, 3). Note that the relationship between entries of ?aug and
the edge strength is not direct; although the factorization (3) has no potential corresponding to the
augmented ?edge? (1, 3), the (1, 3) entry of ?aug is noticeably larger in magnitude than the entries
corresponding to actual edges with nonzero potentials. This example shows that the usual inverse
covariance matrix is not always graph-structured, but computing generalized covariance matrices
involving higher-order interaction terms may indicate graph structure.
Now let us consider a more general graphical model that adds the 3-clique interaction terms shown
in panel (d) to the usual Ising terms. We compute the covariance matrix of the augmented vector
?(X) = X1 , X2 , X3 , X4 , X1 X2 , X2 X3 , X3 X4 , X1 X4 , X1 X3 , X1 X2 X3 , X1 X3 X4 ? {0, 1}11 .
Empirically, we find that the 11?11 inverse of the matrix cov(?(X)) continues to respect aspects of
the graph structure:
Q in particular, thereQare zeros in position (?, ?), corresponding to the associated
functions X? = s?? Xs and X? = s?? X? , whenever ? and ? do not lie within the same maximal clique. (For instance, this applies to the pairs (?, ?) = ({2}, {4}) and (?, ?) = ({2}, {1, 4}).)
The goal of this paper is to understand when certain inverse covariances do (and do not) capture
the structure of a graphical model. The underlying principles behind the behavior demonstrated in
Example 1 will be made concrete in Theorem 1 and its corollaries in the next section.
3
Main results and consequences
We now state our main results on the structure of generalized inverse covariance matrices and graph
structure. We present our results in two parts: one concerning statements at the population level,
and the other concerning statements at the level of statistical consistency based on random samples.
3.1
Population-level results
Our main result concerns a connection between the inverses of generalized inverse covariance matrices associated with the model (4) and the structure of the graph. We begin with some notation.
e = (V, E)
e with no
Recall that a triangulation of a graph G = (V, E) is an augmented graph G
chordless 4-cycles. (For instance, the single cycle in panel (b) is a chordless 4-cycle, whereas panel
4
e
(c) shows a triangulated graph. The dinosaur graph in panel (e) is also triangulated.) The edge set E
corresponds to the original edge set E plus the additional edges added to form the triangulation. In
general, G admits many different triangulations; the results we prove below will hold for any fixed
triangulation of G.
We also require some notation for defining generalized covariance matrices. Let S be a collection
of subsets of vertices, and define the random vector
|C|
?(X; S) = ?S;J , J ? X0 , S ? S ? C ,
(5)
consisting of all sufficient statistics over cliques in S. We will often be interested in situations where
S contains all subsets of a given set. For a subset A ? V , we let pow(A) denote the set of all
non-empty subsets of A. (For instance, pow({1, 2}) = {1, 2, (1, 2)}.) Furthermore, for a collection
of subsets S, we let pow(S) be the set of all subsets {pow(S), S ? S}, discarding any duplicates
that arise. We are now ready to state our main theorem regarding the support of a certain type of
generalized inverse covariance matrix.
Theorem 1. [Triangulation and block graph-structure.] Consider an arbitrary discrete graphical
model of the form (4), and let T be the set of maximal cliques in any triangulation of G. Then the
inverse ? of the augmented covariance matrix cov(?(X; pow(T ))) is block graph-structured in the
following sense:
(a) For any two subsets A and B which are not subsets of the same maximal clique, the block
?(pow(A), pow(B)) is zero.
(b) For almost all parameters ?, the entire block ?(pow(A), pow(B)) is nonzero whenever A
and B belong to a common maximal clique.
The proof of this result relies on convex analysis and the geometry of exponential families [15, 16].
In particular, in any minimal exponential family, there is a one-to-one correspondence between
exponential parameters (?? in our notation) and mean parameters (?? = E[?? (X)]). This correspondence is induced by the Fenchel-Legendre duality between the log partition function A and its
dual A? , and allows us to relate ? to the graph structure.
Note that when the original graph G is a tree, the graph is already triangulated and the set T in
Theorem 1 is equal to the edge set E. Hence, Theorem 1 implies that the inverse ? of the augmented
covariance matrix with sufficient statistics for all vertices and edges is graph-structured, and blocks
of nonzeros in ? correspond to edges in the graph. In particular, the (m ? 1)p ? (m ? 1)p submatrix
?V,V corresponding to sufficient statistics of vertices is block graph-structured; in the case when
m = 2, the submatrix ?V,V is simply the p ? p block corresponding to the vector (X1 , . . . , Xp ).
When G is not triangulated, however, we may need to invert a larger augmented covariance matrix
and include sufficient statistics over pairs (s, t) ?
/ E, as well.
In fact, it is not necessary to take the set of sufficient statistics over all maximal cliques, and we
may consider a slightly smaller augmented covariance matrix. Recall that any triangulation T gives
rise to a junction tree representation of G, where nodes of the junction tree are subsets of V corresponding to maximal cliques in T , and the edges are intersections of adjacent cliques known as
separator sets [15]. The following corollary involves the generalized covariance matrix containing
only sufficient statistics for nodes and separator sets of T :
Corollary 1. Let S be the set of separator sets in any triangulation of G, and let ? be the inverse of
e
cov(?(X; V ? pow(S))). Then ?V,V is block graph-structured: ?s,t = 0 whenever (s, t) ?
/ E.
The proof of this corollary is based on applying the block matrix inversion formula [17] to express
?V,V in terms of the matrix ? from Theorem 1. Panel (c) of Example 1 and the associated matrix
?aug provides a concrete instance of this corollary in action. In panel (c), the single separator set in
the triangulation is {1, 3}, so augmenting the usual covariance matrix with the additional sufficient
statistic X1 X3 and taking the inverse should yield a graph-structured matrix. Indeed, edge (2, 4)
e and as predicted by Corollary 1, we observe that ?aug (2, 4) = 0.
does not belong to E,
Note that V ? pow(S) ? pow(T ), and the set of sufficient statistics considered in Corollary 1 is
generally much smaller than the set of sufficient statistics considered in Theorem 1. Hence, the generalized covariance matrix of Corollary 1 has a smaller dimension than the generalized covariance
matrix of Theorem 1, and is much more tractable for estimation.
5
Although Theorem 1 and Corollary 1 are clean results at the population level, however, forming the
proper augmented covariance matrix requires some prior knowledge of the graph?namely, which
edges are involved in a suitable triangulation. In the case of a graph with only singleton separator
sets, Corollary 1 specializes to the following useful corollary, which only involves the covariance
matrix over indicators of vertices of G:
Corollary 2. For any graph with singleton separator sets, the inverse matrix ? of the ordinary
covariance matrix cov(?(X; V )) is graph-structured. (This class includes trees as a special case.)
Again, we may relate this corollary to Example 1?the inverse covariance matrices for the tree graph
in panel (a) and the dinosaur graph in panel (e) are exactly graph-structured. Indeed, although the
dinosaur graph is not a tree, it possesses the nice property that the only separator sets in its junction
tree are singletons.
Corollary 1 also guarantees that inverse covariances may be partially graph-structured, in the sense
that (?V,V )st = 0 for any pair of vertices (s, t) separable by a singleton separator set. This is
because for any such pair (s, t), we form a junction tree with two nodes, one containing s and one
containing t, and apply Corollary 1 to conclude that (?V,V )st = 0. Indeed, the matrix ?V,V over
singleton vertices is agnostic to which triangulation we choose for the graph.
In settings where there exists a junction tree representation of the graph with only singleton separator
sets, Corollary 2 has a number of useful implications for the consistency of methods that have
traditionally only been applied for edge recovery in Gaussian graphical models. In such settings,
Corollary 2 implies that it suffices to estimate the support of ?V,V from the data.
3.2
Consequences for graphical Lasso for trees
Moving beyond the population level, we now establish results concerning the statistical consistency
of methods for graph selection in discrete graphical models, based on i.i.d. draws from a discrete
graph. We describe how a combination of our population-level results and some concentration
inequalities may be leveraged to analyze the statistical behavior of log-determinant methods for discrete tree-structured graphical models, and suggest extensions of these methods when observations
are systematically corrupted by noise or missing data.
Given p-dimensional random variables (X1 , . . . , Xp ) with covariance ?? , consider the estimator
X
b ? arg min{trace(??)
b ? log det(?) + ?n
?
|?st |},
(6)
?0
s6=t
?
b is an estimator for ? . For multivariate Gaussian data, this program is an `1 -regularized
where ?
maximum likelihood estimate known as the graphical Lasso, and is a well-studied method for recovering the edge structure in a Gaussian graphical model [18, 19, 20]. Although the program (6)
has no relation to the MLE in the case of a discrete graphical model, it is still useful for estimating
?? := (?? )?1 , and our analysis shows the surprising result that the program is consistent for reb of the
covering the structure of any tree-structured Ising model. We consider a general estimate ?
covariance matrix ? such that
r
log p
?
?
b
P k? ? ? kmax ? ?(? )
? c exp(??(n, p))
(7)
n
for functions ? and ?, where k ? kmax denotes the elementwise `? -norm. In the case of fullyb = 1 Pn xi xT ? xxT is the usual
observed i.i.d. data with sub-Gaussian parameter ? 2 , where ?
i
i=1
n
sample covariance, this bound holds with ?(?? ) = ? 2 and ?(n, p) = c0 log p.
In addition, we require a certain mutual incoherence condition on the true covariance matrix ?? to
control the correlation of non-edge variables with edge variables in the graph. Let ?? = ?? ? ?? ,
where ? denotes the Kronecker product. Then ?? is a p2 ? p2 matrix indexed by vertex pairs. The
incoherence condition is given by
maxc k??eS (??SS )?1 k1 ? 1 ? ?,
? ? (0, 1],
(8)
e?S
where S := {(s, t) : ??st 6= 0} is the set of vertex pairs corresponding to nonzero elements of
the precision matrix ?? ?equivalently, the edge set of the graph, by our theory on tree-structured
discrete graphs. For more intuition on the mutual incoherence condition, see Ravikumar et al. [4].
6
Our global edge recovery algorithm proceeds as follows:
Algorithm 1 (Graphical Lasso).
b of the true covariance matrix ?.
1. Form a suitable estimate ?
b
2. Optimize the graphical Lasso program (6) with parameter ?n , denoting the solution by ?.
b at level ?n to obtain an estimate of ?? .
3. Threshold the entries of ?
We then have the following consistency result, a straightforward consequence of the graph structure
b
of ?? and concentration properties of ?:
Corollary 3. Suppose we have a tree-structured Ising model with degree at most d, satisfying the
2
b the sample covariance
mutual incoherence condition (8).
Algorithm 1 with ?
q If n % d log p, then
q
matrix and parameters ?n ? c?1 logn p and ?n = c2 c?1 logn p + ?n recovers all edges (s, t) with
|??st | > ?n /2, with probability at least 1 ? c exp(?c0 log p).
Hence, if |??st | > ?n /2 for all edges (s, t) ? E, Corollary 3 guarantees that the log-determinant
method plus thresholding recovers the full graph exactly. In the case of the standard sample covariance matrix, this method has been implemented by Banerjee et al. [18]; our analysis establishes consistency of their method for discrete trees. The scaling n % d2 log p is unavoidable, as
shown by information-theoretic analysis [21], and also appears in other past work on Ising models [10, 9, 11]. Our analysis also has a cautionary message: the proof of Corollary 3 relies heavily
on the population-level result in Corollary 2, which ensures that ?? is tree-structured. For a general
graph, we have no guarantees that ?? will be graph-structured (e.g., see panel (b) in Figure 1), so
the graphical Lasso (6) is inconsistent in general.
On the positive side, if we restrict ourselves to tree-structured graphs, the estimator (6) is attractive,
b of the population covariance ?? that satisfies the deviation
since it relies only on an estimate ?
condition (7). In particular, when the samples {xi }ni=1 are contaminated by noise or missing data,
b of ?? . Furthermore, the program (6) is always convex
all we require is a sufficiently good estimate ?
b
even when the estimator ? is not positive semidefinite (as will often be the case for missing/corrupted
data).
As a concrete example of how we may correct the program (6) to handle corrupted data, consider
the case when each entry of xi is missing independently with probability ?, and the corresponding
observations zi are zero-filled for missing entries. A natural estimator is
!
n
X
1
1
T
b=
zi zi ? M ?
zz T ,
(9)
?
n i=1
(1 ? ?)2
where ? denotes elementwise division by the matrix M with diagonal entries (1 ? ?) and offdiagonal entries (1 ? ?)2 , correcting for the bias in both the mean and second moment terms. The
deviation condition (7) may be shown to hold w.h.p., where ?(?? ) scales with (1 ? ?) (cf. Loh and
b and a subsequent version of
Wainwright [22]). Similarly, we may derive an appropriate estimator ?
Algorithm 1 in situations when the data are systematically contaminated by other forms of additive
or multiplicative corruption.
Generalizing to the case of m-ary discrete graphical models with m > 2, we may easily modify
the program (6) by replacing the elementwise `1 -penalty by the corresponding group `1 -penalty,
where the groups are the indicator variables for a given vertex. Precise theoretical guarantees may
be derived from results on the group graphical Lasso [23].
4
Simulations
Figure 2 depicts the results of simulations we performed to test our theoretical predictions. In all
cases, we generated binary Ising models with node weights 0.1 and edge weights 0.3 (using spin
{?1, 1} variables). The five curves show the results of our graphical Lasso method applied to
the dinosaur graph in Figure 1. Each curve plots the probability of success in recovering the 15
7
edges of the graph, as a function of the rescaled sample size logn p , where p = 13. The leftmost
(red) curve corresponds to the case of fully-observed covariates (? = 0), whereas the remaining
four curves correspond to increasing missing data fractions ? ? {0.05, 0.1, 0.15, 0.2}, using the
corrected estimator (9). We observe that all five runs display a transition from success probability 0
to success probability 1 in roughly the same range of the rescaled sample size, as predicted by our
theory. Indeed, since the dinosaur graph has only singleton separators, Corollary 2 ensures that the
inverse covariance matrix is exactly graph-structured. Note that the curves shift right as the fraction
? of missing data increases, since the problem becomes harder.
success prob, avg over 1000 trials
success prob vs. sample size for dino graph with missing data
1
0.8
0.6
0.4
rho = 0
rho = 0.05
rho = 0.1
rho = 0.15
rho = 0.2
0.2
0
0
100
200
n/log p
300
400
500
Figure 2. Simulation results for our graphical Lasso method on binary Ising models, allowing for
missing data in the observations. The figure shows simulation results for the dinosaur graph. Each
point represents an average over 1000 trials. The horizontal axis gives the rescaled sample size logn p .
5
Discussion
The correspondence between the inverse covariance matrix and graph structure of a Gauss-Markov
random field is a classical fact, with many useful consequences for efficient estimation of Gaussian
graphical models. It has long been an open question as to whether or not similar properties extend
to a broader class of graphical models. In this paper, we have provided a partial affirmative answer
to this question and developed theoretical results extending such relationships to discrete undirected
graphical models.
As shown by our results, the inverse of the ordinary covariance matrix is graph-structured for special
subclasses of graphs with singleton separator sets. More generally, we have shown that it is worthwhile to consider the inverses of generalized covariance matrices, formed by introducing indicator
functions for larger subsets of variables. When these subsets are chosen to reflect the structure
of an underlying junction tree, the edge structure is reflected in the inverse covariance matrix. Our
population-level results have a number of statistical consequences for graphical model selection. We
have shown how our results may be used to establish consistency (or inconsistency) of the standard
graphical Lasso applied to discrete graphs, even when observations are systematically corrupted
by mechanisms such as additive noise and missing data. As noted by an anonymous reviewer, the
Chow-Liu algorithm might also potentially be modified to allow for missing or corrupted observations. However, our proposed method and further offshoots of our population-level result may be
applied even in cases of non-tree graphs, which is beyond the scope of the Chow-Liu algorithm.
Acknowledgments
PL acknowledges support from a Hertz Foundation Fellowship and an NDSEG Fellowship. MJW
and PL were also partially supported by grants NSF-DMS-0907632 and AFOSR-09NL184. The
authors thank the anonymous reviewers for helpful feedback.
8
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9
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3,960 | 4,585 | Multi-scale Hyper-time Hardware Emulation of
Human Motor Nervous System Based on Spiking
Neurons using FPGA
C. Minos Niu
Department of Biomedical Engineering
University of Southern California
Los Angeles, CA 90089
[email protected]
Sirish K. Nandyala
Department of Biomedical Engineering
University of Southern California
Los Angeles, CA 90089
[email protected]
Won Joon Sohn
Department of Biomedical Engineering
University of Southern California
Los Angeles, CA 90089
[email protected]
Terence D. Sanger
Department of Biomedical Engineering
Department of Neurology
Department of Biokinesiology
University of Southern California
Los Angeles, CA 90089
[email protected]
Abstract
Our central goal is to quantify the long-term progression of pediatric neurological diseases, such as a typical 10-15 years progression of child dystonia. To this
purpose, quantitative models are convincing only if they can provide multi-scale
details ranging from neuron spikes to limb biomechanics. The models also need
to be evaluated in hyper-time, i.e. significantly faster than real-time, for producing
useful predictions. We designed a platform with digital VLSI hardware for multiscale hyper-time emulations of human motor nervous systems. The platform is
constructed on a scalable, distributed array of Field Programmable Gate Array
(FPGA) devices. All devices operate asynchronously with 1 millisecond time
granularity, and the overall system is accelerated to 365x real-time. Each physiological component is implemented using models from well documented studies
and can be flexibly modified. Thus the validity of emulation can be easily advised
by neurophysiologists and clinicians. For maximizing the speed of emulation,
all calculations are implemented in combinational logic instead of clocked iterative circuits. This paper presents the methodology of building FPGA modules
emulating a monosynaptic spinal loop. Emulated activities are qualitatively similar to real human data. Also discussed is the rationale of approximating neural
circuitry by organizing neurons with sparse interconnections. In conclusion, our
platform allows emulating pathological abnormalities such that motor symptoms
will emerge and can be analyzed. It compels us to test the origins of childhood
motor disorders and predict their long-term progressions.
1
Challenges of studying developmental motor disorders
There is currently no quantitative model of how a neuropathological condition, which mainly affects
the function of neurons, ends up causing the functional abnormalities identified in clinical examinations. The gap in knowledge is particularly evident for disorders in developing human nervous
systems, i.e. childhood neurological diseases. In these cases, the ultimate clinical effect of cellu1
lar injury is compounded by a complex interplay among the child?s injury, development, behavior,
experience, plasticity, etc. Qualitative insight has been provided by clinical experiences into the
association between particular types of injury and particular types of outcome. Their quantitative
linkages, nevertheless, have yet to be created ? neither in clinic nor in cellular physiological tests.
This discrepancy is significantly more prominent for individual child patients, which makes it very
difficult to estimate the efficacy of treatment plans. In order to understand the consequence of injury and discover new treatments, it is necessary to create a modeling toolset with certain design
guidelines, such that child neurological diseases can be quantitatively analyzed.
Perhaps more than any other organ, the brain necessarily operates on multiple spatial and temporal
scales. On the one hand, it is the neurons that perform fundamental computations, but neurons have
to interact with large-scale organs (ears, eyes, skeletal muscles, etc.) to achieve global functions.
This multi-scale nature worths more attention in injuries, where the overall deficits depend on both
the cellular effects of injuries and the propagated consequences. On the other hand, neural processes
in developmental diseases usually operate on drastically different time scales, e.g. spinal reflex in
milliseconds versus learning in years. Thus when studying motor nervous systems, mathematical
modeling is convincing only if it can provide multi-scale details, ranging from neuron spikes to limb
biomechanics; also the models should be evaluated with time granularity as small as 1 millisecond,
meanwhile the evaluation needs to continue trillions of cycles in order to cover years of life.
It is particularly challenging to describe the multi-scale nature of human nervous system when modeling childhood movement disorders. Note that for a child who suffered brain injury at birth, the full
development of all motor symptoms may easily take more than 10 years. Therefore the millisecondbased model needs to be evaluated significantly faster than real-time, otherwise the model will fail
to produce any useful predictions in time. We have implemented realistic models for spiking motoneurons, sensory neurons, neural circuitry, muscle fibers and proprioceptors using VLSI and programmable logic technologies. All models are computed in Field Programmable Gate Array (FPGA)
hardware in 365 times real-time. Therefore one year of disease progression can be assessed after
one day of emulation. This paper presents the methodology of building the emulation platform. The
results demonstrate that our platform is capable of producing physiologically realistic multi-scale
signals, which are usually scarce in experiments. Successful emulations enabled by this platform
will be used to verify theories of neuropathology. New treatment mechanisms and drug effects can
also be emulated before animal experiments or clinical trials.
2
Methodology of multi-scale neural emulation
A. Human arm
B. Monosynaptic spinal loop
C. Inner structure of muscle spindle
Gamma
Secondary
dynamic
Gamma output
input
static
Primary
input
output
Bag 1
?MN
Bag 2
Chain
Figure 1: Illustration of the multi-scale nature of motor nervous system.
The motor part of human nervous system is responsible for maintaining body postures and generating voluntary movements. The multi-scale nature of motor nervous system is demonstrated in Fig.1.
When the elbow (Fig.1A) is maintaining a posture or performing a movement, a force is established by the involved muscle based on how much spiking excitation the muscle receives from its ?motoneurons (Fig.1B). The ?-motoneurons are regulated by a variety of sensory input, part of which
comes directly from the proprioceptors in the muscle. As the primary proprioceptor found in skeletal
muscles, a muscle spindle is another complex system that has its own microscopic Multiple-InputMultiple-Output structure (Fig.1C). Spindles continuously provide information about the length and
lengthening speed of the muscle fiber. A muscle with its regulating motoneurons, sensory neurons and proprioceptors constitutes a monosynaptic spinal loop. This minimalist neurophysiological
2
structure is used as an example for explaining the multi-scale hyper-time emulation in hardware.
Additional structures can be added to the backbone set-up using similar methodologies.
2.1
Modularized architecture for multi-scale models
Decades of studies on neurophysiology provided an abundance of models characterizing different
components of the human motor nervous system. The informational characteristics of physiological
components allowed us to model them as functional structures, i.e. each of which converting input
signals to certain outputs. In particular, within a monosynaptic spinal loop illustrated in Fig.1B,
stretching the muscle will elicit a chain of physiological activities in: muscle stretch ? spindle ?
sensory neuron ? synapse ? motoneuron ? muscle contraction. The adjacent components must
have compatible interfaces, and the interfacing variables must also be physiologically realistic. In
our design, each component is mathematically described in Table 1:
Table 1: Functional definition of neural models
COMPONENT
Neuron
Synapse
Muscle
Spindle
MATHEMATICAL DEFINITION
S(t) = fneuron (I, t)
I(t) = fsynapse (S, t)
? t)
T (t) = fmuscle (S, L, L,
?
A(t) = fspindle (L, L, ?dynamic , ?static , t)
all components are modeled as black-box functions that map the inputs to the outputs. The meanings
of these mathematical definitions are explained below. This design allows existing physiological
models to be easily inserted and switched. In all models the input signals are time-varying, e.g.
I = I(t), L = L(t) , etc. The argument of t in input signals are omitted throughout this paper.
2.2
Selection of models for emulation
Models were selected in consideration of their computational cost, physiological verisimilitude, and
whether it can be adapted to the mathematical form defined in Table 1.
Model of Neuron
The informational process for a neuron is to take post-synaptic current I as the input, and produce
a binary spike train S in the output. The neuron model adopted in the emulation was developed by
Izhikevich [1]:
v0
u0
= 0.04v 2 + 5v + 140 ? u + I
= a(bv ? u)
(1)
(2)
if v = 30 mV, then v ? c, u ? u + d
where a, b, c, d are free parameters tuned to achieve certain firing patterns. Membrane potential v
directly determines a binary spike train S(t) that S(t) = 1 if v ? 30, otherwise S(t) = 0. Note that
v in Izhikevich model is in millivolts and time t is in milliseconds. Therefore the coefficients in eq.1
need to be adjusted in correspondence to SI units.
Model of Synapse
When a pre-synaptic neuron spikes, i.e. S(0) = 1, an excitatory synapse subsequently issues an
Excitatory Post-Synaptic Current (EPSC) that drives the post-synaptic neuron. Neural recording of
hair cells in rats [2] provided evidence that the time profile of EPSC can be well characterized using
the equations below:
(
t
? t
Vm ? e ?d Vm ? e? ?r Vm
if t ? 0
I(t) =
(3)
0
otherwise
The key parameters in a synapse model is the time constants for rising (?r ) and decaying (?d ). In
our emulation ?r = 0.001 s and ?r = 0.003 s.
3
Model of Muscle force and electromyograph (EMG)
The primary effect of skeletal muscle is to convert ?-motoneuron spikes S into force T , depending
? We used Hill?s muscle model
on the muscle?s instantaneous length L and lengthening speed L.
in the emulation with parameter tuning described in [3]. Another measurable output of muscle is
electromyograph (EMG). EMG is the small skin current polarized by motor unit action potential
(MUAP) when it travels along muscle fibers. Models exist to describe the typical waveform picked
by surface EMG electrodes. In this project we chose to implement the one described in [4].
Model of Proprioceptor
Spindle is a sensory organ that provides the main source of proprioceptive information. As can
be seen in Fig.1C, a spindle typically produces two afferent outputs (primary Ia and secondary II)
? There
according to its gamma fusimotor drives (?dynamic and ?static ) and muscle states (L and L).
is currently no closed-form models describing spindle functions due to spindle?s significant nonlinearity. On representative model that numerically approximates the spindle dynamics was developed by Mileusnic et al. [5]. The model used differential equations to characterize a typical cat
soleus spindle. Eqs.4-10 present a subset of this model for one type of spindle fiber (bag1):
!
?dynamic
x?0 =
? x0 /?
(4)
?dynamic + ?2bag1
x?1
x?2
= x2
1
=
[TSR ? TB ? TP R ? ?1 x0 ]
M
(5)
(6)
where
TSR
TB
TP R
CSS
=
KSR (L ? x1 ? LSR0 )
(7)
0.3
= (B0 + B1 x0 ) ? (x1 ? R) ? CSS ? |x2 |
= KP R (x1 ? LP R0 )
2
=
?1
1 + e?1000x2
(8)
(9)
(10)
Eq.8 and 10 suggest that evaluating the spindle model requires multiplication, division as well as
more complex arithmetics like polynomials and exponentials. The implementation details are described in Section 3.
2.3
Neuron connectivity with sparse interconnections
Although the number of spinal neurons (~1 billion) is significantly less compared to that of cortical
neurons (~100 billion), a fully connected spinal network still means approximately 2 trillion synaptic
endings [6]. Implementing such a huge number of synapses imposes a major challenge, if not
impossible, given limited hardware resource.
In this platform we approximated the neural connectivity by sparsely connecting sensory neurons
to motoneurons as parallel pathways. We do not attempt to introduce the full connectivity. The
rationale is that in a neural control system, the effect of a single neuron can be considered as mapping
current state x to change in state x? through a band-limited channel. Therefore when a collection of
neurons are firing stochastically, the probability of x? depends on both x and the firing behavior s
(s = 1 when spiking, otherwise s = 0) of each neuron, as such:
p(x|x,
? s) = p(x|s
? = 1)p(s = 1|x) + p(x|s
? = 0)p(s = 0|x)
(11)
Eq.11 is a master equation that determines a probability flow on the state. From the Kramers-Moyal
expansion we can associate this probability flow with a partial differential equation:
i
?
X
?
?
p(x, t) =
?
D(i) (x)p(x, t)
(12)
?t
?x
i=1
where D(i) (x) is a time-invariant term that modifies the change of probability density based on its
i-th gradient.
4
Under certain conditions [7, 8], D(i) (x) for i > 2 all vanish and therefore the probability flow can
be described deterministically using a linear operator L:
?
?
? 2 (2)
D
(x)
p(x, t) = Lp(x, t)
(13)
p(x, t) = ? D(1) (x) +
?t
?x
?x2
This means that various Ls can be superimposed to achieve complex system dynamics (illustrated
in Fig.2A).
B. Equivalent network with
sparse interconnections
A. Neuron function as superimposed
linear operators
SN
Sensory
Input
+
SN
SN
SN
?MN
?MN
?MN
Motor
Output
?MN
Figure 2: Functions of neuron population can be described as the combination of linear operators
(A). Therefore the original neural function can be equivalently produced by sparsely connected
neurons formalizing parallel pathways (B).
As a consequence, the statistical effect of two fully connected neuron populations is equivalent to
ones that are only sparsely connected, as long as the probability flow can be described by the same L.
For a movement task, in particular, it is the statistical effect from the neuron ensemble onto skeletal
muscles that determines the global behavior. Therefore we argue that it is feasible to approximate
the spinal cord connectivity by sparsely interconnecting sensory and motor neurons (Fig.2B). Here
a pool of homogenous sensory neurons projects to another pool of homogeneous ?-motoneurons.
Pseudorandom noise is added to the input of all homogeneous neurons within a population. It is
worth noting that this approximation significantly reduces the number of synapses that need to be
implemented in hardware.
3
Hardware implementation on FPGA
We select FPGA as the implementation device due to its inherent parallelism that resembles the nervous system. FPGA is favored over GPU or clustered CPUs because it is relatively easy to network
hundreds of nodes under flexible protocols. The platform is distributed on multiple nodes of Xilinx
Spartan-6 devices. The interfacing among FPGAs and computers is created using OpalKelly development board XEM6010. The dynamic range of variables is tight in models of Izhikevich neuron,
synapse and EMG. This helps maintaining the accuracy of models even when they are evaluated in
32-bit fixed-point arithmetics. The spindle model, in contrast, requires floating-point arithmetics due
to its wide dynamic range and complex calculations (see eq.4-10). Hyper-time computations with
floating-point numbers are resource consuming and therefore need to be implemented with special
attentions.
3.1
Floating-point arithmetics in combinational logic
Our arithmetic implementations are compatible with IEEE-754 standard. Typical floating-point
arithmetic IP cores are either pipe-lined or based on iterative algorithms such as CORDIC, all of
which require clocks to schedule the calculation. In our platform, no clock is provided for model
evaluations thus all arithmetics need to be executed in pure combinational logic. Taking advantage
of combinational logic allows all model evaluations to be 1) fast, the evaluation time depends entirely on the propagating and settling time of signals, which is on the order of microseconds, and 2)
parallel, each model is evaluated on its own circuit without waiting for any other results.
Our implementations of adder and multiplier are inspired by the open source project ?Free FloatingPoint Madness?, available at http://www.hmc.edu/chips/. Please contact the authors of this paper if
the modified code is needed.
5
Fast combinational floating-point division
Floating-point division is even more resource demanding than multiplications. We avoided directly
implementing the dividing algorithm by approximating it with additions and multiplications. Our
approach is inspired by an algorithm described in [9], which provides a good approximation of the
inverse square root for any positive number x within one Newton-Raphson iteration:
1
x
Q(x) = ? ? x(1.5 ? ? x2 )
2
x
(x > 0)
(14)
Q(x) can be implemented only using floating-point adders and multipliers. Thereby any division
with a positive divisor can be achieved if two blocks of Q(x) are concatenated:
a
a
(15)
= ? ? = a ? Q(b) ? Q(b) (b > 0)
b
b? b
This algorithm has been adjusted to also work with negative divisors (b < 0).
Numerical integrators for differential equations
Evaluating the instantaneous states of differential equation models require a fixed-step numerical
integrator. Backward Euler?s Method was chosen to balance the numerical error and FPGA usage:
x?
xn+1
= f (x, t)
= xn + T f (xn+1 , tn+1 )
(16)
(17)
where T is the sampling interval. f (x, t) is the derivative function for state variable x.
3.2
Asynchronous spike-based communication between FPGA chips
Clock
Spike
clean
count
Counter
1
1
2
1
2
3
Figure 3: Timing diagram of asynchronous spike-based communication
FPGA nodes are networked by transferring 1-bit binary spikes to each other. Our design allowed
the sender and the receiver to operate on independent clocks without having to synchronize. The
timing diagram of the spike-based communication is shown in Fig.3. The sender issues Spike with
a pulse width of 1/(365 ? Femu ) second. Each Spike then triggers a counting event on the receiver,
meanwhile each Clock first reads the accumulated spike count and subsequently cleans the counter.
Note that the phase difference between Spike and Clock is not predictable due to asynchronicity.
3.3
Serialize neuron evaluations within a homogeneous population
Different neuron populations are instantiated as standalone circuits. Within in each population,
however, homogeneous neurons mentioned in Section 2.3 are evaluated in series in order to optimize
FPGA usage.
Within each FPGA node all modules operate with a central clock, which is the only source allowed
to trigger any updating event. Therefore the maximal number of neurons that can be serialized
(Nserial ) is restrained by the following relationship:
Ffpga = C ? Nserial ? 365 ? Femu
(18)
Here Ffpga is the fastest clock rate that a FPGA can operate on; C = 4 is the minimal clock cycles
needed for updating each state variable in the on-chip memory; Femu = 1 kHz is the time granularity of emulation (1 millisecond), and 365 ? Femu represents 365x real-time. Consider that Xilinx
6
Spartan-6 FPGA devices peaks at 200MHz central clock frequency, the theoretical maximum of
neurons that can be serialized is
Nserial 6 200 MHz/(4 ? 365 ? 1 kHz) ? 137
(19)
In the current design we choose Nserial = 128.
4
Results: emulated activities of motor nervous system
Figure 4 shows the implemented monosynaptic spinal loop in schematics and in operation. Each
FPGA node is able to emulate monosynaptic spinal loops consisting of 1,024 sensory and 1,024 motor neurons, i.e. 2,048 neurons in total. The spike-based asynchronous communication is successful
between two FPGA nodes. Note that the emulation has to be significantly slowed down for on-line
plotting. When the emulation is at full speed (365x real-time) the software front-end is not able to
visualize the signals due to limited data throughput.
128 SNs 128 ?MNs
SN
?MN
128 SNs 128 ?MNs
SN
?MN
...
8 parallel pathways
2,048 neurons
Figure 4: The neural emulation platform in operation. Left: Neural circuits implemented for each
FPGA node including 2,048 neurons. SN = Sensory Neuron; ?MN = ?-motoneuron. Center: One
working FPGA node. Right: Two FPGA nodes networked using asynchronous spiking protocol.
The emulation platform successfully created multi-scale information when the muscle is externally
stretched (Fig.5A). We also tested if our emulated motor system is able to produce the recruitment
order and size principles observed in real physiological data. It has been well known that when
a voluntary motor command is sent to the ?-motoneuron pool, the motor units are recruited in an
order that small ones get recruited first, followed by the big ones [10]. The comparison between
our results and real data are shown in Fig.5B, where the top panel shows 20 motor unit activities
emulated using our platform, and the bottom panel shows decoded motor unit activities from real
human EMG [11]. No qualitative difference was found.
5
Discussion and future work
We designed a hardware platform for emulating the multi-scale motor nervous activities in hypertime. We managed to use one node of single Xilinx Spartan-6 FPGA to emulate monosynaptic
spinal loops consisting of 2,048 neurons, associated muscles and proprioceptors. The neurons are
organized as parallel pathways with sparse interconnections. The emulation is successfully accelerated to 365x real-time. The platform can be scaled by networking multiple FPGA nodes, which
is enabled by an asynchronous spike-based communication protocol. The emulated monosynaptic
spinal loops are capable of producing reflex-like activities in response to muscle stretch. Our results
of motor unit recruitment order are compatible with the physiological data collected in real human
subjects.
There is a question of whether this stochastic system turns out chaotic, especially with accumulated
errors from Backward Euler?s integrator. Note that the firing property of a neuron population is
usually stable even with explicit noise [8], and spindle inputs are measured from real robots so the
integrator errors are corrected at every iteration. To our knowledge, the system is not critically
sensitive to the initial conditions or integrator errors. This question, however, is both interesting and
important for in-depth investigations in the future.
7
It has been shown [12] that replicating classic types of spinal interneurons (propriospinal, Iaexcitatory, Ia-inhibitory, Renshaw, etc.) is sufficient to produce stabilizing responses and rapid
reaching movement in a wrist. Our platform will introduce those interneurons to describe the known
spinal circuitry in further details. Physiological models will also be refined as needed. For the
purpose of modeling movement behavior or diseases, Izhikevich model is a good balance between
verisimilitude and computational cost. Nevertheless when testing drug effects along disease progression, neuron models are expected to cover sufficient molecular details including how neurotransmitters affect various ion channels. With the advancing of programmable semiconductor technology, it
is expected to upgrade our neuron model to Hodgkin-Huxley?s. For the muscle models, Hill?s type
of model does not fit the muscle properties accurately enough when the muscle is being shortened.
Alternative models will be tested.
Other studies showed that the functional dexterity of human limbs ? especially in the hands ? is
critically enabled by the tendon configurations and joint geometry [13]. As a result, if our platform
is used to understand whether known neurophysiology and biomechanics are sufficient to produce
able and pathological movements, it will be necessary to use this platform to control human-like
limbs. Since the emulation speed can be flexibly adjusted from arbitrarily slow to 365x real-time,
when speeded to exactly 1x real-time the platform will function as a digital controller with 1kHz
refresh rate.
The main purpose of the emulation is to learn how certain motor disorders progress during childhood
development. This first requires the platform to reproduce motor symptoms that are compatible with
clinical observations. For example it has been suggested that muscle spasticity in rats is associated
with decreased soma size of ?-motoneurons [14], which presumably reduced the firing threshold of
neurons. Thus when lower firing threshold is introduced to the emulated motoneuron pool, similar
EMG patterns as in [15] should be observed. It is also necessary for the symptoms to evolve with
neural plasticity. In the current version we presume that the structure of each component remains
time invariant. In the future work Spike Timing Dependent Plasticity (STDP) will be introduced
such that all components are subject to temporal modifications.
B. Verify motor unit recruitment pattern
A. Multi-scale activities from emulation
Emulation
1s
Stretch
Spindle Ia
Sensory post-synaptic current
Real Data
Motoneurons
Muscle Force
EMG
Figure 5: A) Physiological activity emulated by each model when the muscle is sinusoidally
stretched. B) Comparing the emulated motor unit recruitment order with real experimental data.
Acknowledgments
The authors thank Dr. Gerald Loeb for helping set up the emulation of spindle models. This project
is supported by NIH NINDS grant R01NS069214-02.
8
References
[1] Izhikevich, E. M. Simple model of spiking neurons. IEEE transactions on neural networks / a publication
of the IEEE Neural Networks Council 14, 1569?1572 (2003).
[2] Glowatzki, E. & Fuchs, P. A. Transmitter release at the hair cell ribbon synapse. Nature neuroscience 5,
147?154 (2002).
[3] Shadmehr, R. & Wise, S. P. A Mathematical Muscle Model. In Supplementary documents for ?Computational Neurobiology of Reaching and Pointing?, 1?18 (MIT Press, Cambridge, MA, 2005).
[4] Fuglevand, A. J., Winter, D. A. & Patla, A. E. Models of recruitment and rate coding organization in
motor-unit pools. Journal of neurophysiology 70, 2470?2488 (1993).
[5] Mileusnic, M. P., Brown, I. E., Lan, N. & Loeb, G. E. Mathematical models of proprioceptors. I. Control
and transduction in the muscle spindle. Journal of neurophysiology 96, 1772?1788 (2006).
[6] Gelfan, S., Kao, G. & Ruchkin, D. S. The dendritic tree of spinal neurons. The Journal of comparative
neurology 139, 385?411 (1970).
[7] Sanger, T. D. Neuro-mechanical control using differential stochastic operators. In Engineering in
Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, 4494?4497
(2010).
[8] Sanger, T. D. Distributed control of uncertain systems using superpositions of linear operators. Neural
computation 23, 1911?1934 (2011).
[9] Lomont, C. Fast inverse square root (2003). URL http://www.lomont.org/Math/Papers/
2003/InvSqrt.pdf.
[10] Henneman, E. Relation between size of neurons and their susceptibility to discharge. Science (New York,
N.Y.) 126, 1345?1347 (1957).
[11] De Luca, C. J. & Hostage, E. C. Relationship between firing rate and recruitment threshold of motoneurons in voluntary isometric contractions. Journal of neurophysiology 104, 1034?1046 (2010).
[12] Raphael, G., Tsianos, G. A. & Loeb, G. E. Spinal-like regulator facilitates control of a two-degree-offreedom wrist. The Journal of neuroscience : the official journal of the Society for Neuroscience 30,
9431?9444 (2010).
[13] Valero-Cuevas, F. J. et al. The tendon network of the fingers performs anatomical computation at a
macroscopic scale. IEEE transactions on bio-medical engineering 54, 1161?1166 (2007).
[14] Brashear, A. & Elovic, E. Spasticity: Diagnosis and Management (Demos Medical, 2010), 1 edn.
[15] Levin, M. F. & Feldman, A. G. The role of stretch reflex threshold regulation in normal and impaired
motor control. Brain research 657, 23?30 (1994).
9
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tested:2 |
3,961 | 4,586 | Online Sum-Product Computation over Trees
Fabio Vitale
Department of Computer Science
University of Milan
20135 Milan, Italy
[email protected]
Mark Herbster
Stephen Pasteris
Department of Computer Science
University College London
London WC1E 6BT, England, UK
{m.herbster, s.pasteris}@cs.ucl.ac.uk
Abstract
We consider the problem of performing efficient sum-product computations in an
online setting over a tree. A natural application of our methods is to compute
the marginal distribution at a vertex in a tree-structured Markov random field.
Belief propagation can be used to solve this problem, but requires time linear
in the size of the tree, and is therefore too slow in an online setting where we
are continuously receiving new data and computing individual marginals. With
our method we aim to update the data and compute marginals in time that is no
more than logarithmic in the size of the tree, and is often significantly less. We
accomplish this via a hierarchical covering structure that caches previous local
sum-product computations. Our contribution is three-fold: we i) give a linear time
algorithm to find an optimal hierarchical cover of a tree; ii) give a sum-productlike algorithm to efficiently compute marginals with respect to this cover; and iii)
apply ?i? and ?ii? to find an efficient algorithm with a regret bound for the online
allocation problem in a multi-task setting.
1
Introduction
The use of graphical models [1, 2] is ubiquitous in machine learning. The application of the
batch sum-product algorithm to tree-structured graphical models, including hidden Markov models, Kalman filtering and turbo decoding, is surveyed in [3]. Our aim is to adapt these techniques to
an online setting.
In our online model we are given a tree and a fixed set of parameters. We then receive a potentially unbounded online sequence of ?prediction requests? and ?data updates.? A prediction request
indicates a vertex for which we then return the posterior marginal at that vertex. Each data update
associates a new ?factor? to that vertex. Classical belief propagation requires time linear in the size
of the tree for this task. Our algorithm requires time linear in the height of an optimal hierarchical cover of this tree. The height of the cover is in the worst case logarithmic in the size the tree.
Thus our per trial prediction/update time is at least an exponential improvement over classical belief
propagation.
The paper is structured as follows. In Section 2 we introduce our notation leading to our definition
of an optimal hierarchical cover. In Section 3 we give our optimal hierarchical covering algorithm.
In Section 4 we show how we may use this cover as a structure to cache computations in our sumproduct-like algorithm. Finally, in Section 5 we give a regret bound and a sketch of an application
of our techniques to an online multi-task allocation [4] problem.
Previous work. Pearl [5] introduced belief propagation for Bayes nets which computes marginals
in time linear in the size of the tree. In [6] an algorithm for the online setting was given for a Bayes
net on a tree T which required O(log |V (T )|) time per marginalization step, where |V (T )| is the
number of vertices in the tree. In this work we consider a Markov random field on a tree. We give
an algorithm whose performance is bounded by O(?? (T )). The term ?? (T ) is the height of our
1
optimal hierarchical cover which is upper bounded by O(min(log |V (T )|, diameter(T ))), but may
in fact be exponentially smaller.
2 Hierarchical cover of a tree
In this section we introduce our notion of a hierarchical cover of a tree and its dual the decomposition tree.
Graph-theoretical preliminaries. A graph G is a pair of sets (V, E) such that E is a set of
unordered pairs of distinct elements from V . The elements of V are called vertices and those of
E are called edges. In order to avoid ambiguities deriving from dealing with different graphs, in
some cases we will highlight the membership to graph G denoting these sets as V (G) and E(G)
respectively. With slight abuse of notation, by writing v ? G, we mean v ? V (G). S is a subgraph
G (we write S ? G) iff V (S) ? V (G) and E(S) = {(i, j) : i, j ? V (S), (i, j) ? E(G)}. Given
any subgraph S ? G, we define its boundary (or inner border) ?G (S) and its neighbourhood (or
outer border) NG (S) as: ?G (S) := {i : i ? S, j ?
/ S, (i, j) ? E(G)}, and NG (S) := {j : i ?
S, j ?
/ S, (i, j) ? E(G)}. With slight abuse of notation, NG (v) := NG ({v}), and thus the degree
of a vertex v is |NG (v)|. Given any graph G, we define the set of its leaves as leaves(G) := {i ?
G : |NG (i)| = 1}, and its interior G? := {i ? G : |NG (i)| =
# 1}. A path P in a graph G is
defined by a sequence of distinct vertices $v1 , v2 , ..., vm % of G, such that for all i < m we have that
(vi , vi+1 ) ? E(G). In this case we say that v1 and vm are connected by the subgraph P . A tree T is a
graph in which for all v, w ? T there exists a unique path connecting v with w. In this paper we will
only consider trees with a non-empty edge set and thus the vertex set will always have a cardinality
of at least 2. The distance dT (v, w) between v, w ? T is the path length |E(P )|. The pair (T, r)
denotes a rooted tree T with root vertex r. Given a rooted tree (T, r) and any vertex i ? V (T ), the
(proper) descendants of i are all vertices that can be connected with r via paths P ? T containing
i (excluding i). Analogously, the (proper) ancestors of i are all vertices that lie on the path P ? T
connecting i with r (excluding i). We denote the set of all descendants (resp. all ancestors) of i by
?rT (i) (resp. ?rT (i)). We shall omit the root r when it is clear from the context. Vertex i is the parent
(resp. child) of j, which is denoted by ?rT (j) (resp. i ? ?rT (j)) if (i, j) ? E(T ) and i ? ?rT (j) (resp.
i ? ?rT (j)). Given a tree T we use the notation S ? T only if S is a tree and subgraph of T . The
height of a rooted tree (T, r) is the maximum length of a path P ? T connecting the root to any
vertex: hr (T ) := maxv?T dT (v, r). The diameter ?(T ) of a tree T is defined as the length of the
longest path between any two vertices in T .
2.1 The hierarchical cover of a tree
In this section we describe a splitting process that recursively decomposes a given tree T . A (decomposition) tree (D, r) identifies this splitting process, generating a tree-structured collection S of
subtrees that hierarchically cover the given tree T .
This process recursively splits at each step a subtree of T (that we call a ?component?) resulting from
some previous splits. More precisely, a subtree S ? T is split into two or more subcomponents and
the decomposition of S depends only on the choice of a vertex v ? S ? , which we call splitting
vertex, in the following way. The splitting vertex v ? S ? of S induces the split set ?(S, v) =
{S1 , . . . , S|NS (v)| } which is the unique set of S?s subtrees which overlap at a vertex v, uniquely,
that represent a cover for S, i.e., it satisfies (i) ?S ! ??(S,v) S # = S and (ii) {v} = Si ? Sj for all
1 ? i < j ? |NS (v)|. Thus the split may be visualized by considering the forest F resulting from
removing a vertex from S, but afterwards each component S1 , . . . , S|NS (v)| of F has the ?removed
vertex? v added back to it. A component having only two vertices is called atomic, since it cannot
be split further. We indicate with S v ? T the component subtree whose splitting vertex is v, and
we denote atomic components by S (i,j) , where E(S (i,j) ) = {(i, j)}. We finally denote by S the
set of all component subtrees obtained by this splitting process. Since the method is recursive, we
can associate a rooted tree (D, r), with T ?s decomposition into a hierarchical cover, whose internal
vertices are the splitting vertices of the splitting process. Its leaves correspond to the single edges
(of E(T )) of each atomic component, and a vertex ?parent-child? relation c ? ?rD (p) corresponds
to the ?splits-into? relation S c ? ?(S p , p) (see Figure 1).
We will now formalize the splitting process by defining the hierarchical cover S of a tree T , which
is a key concept used by our algorithm.
2
Definition 1. A hierarchical cover S of a tree T is a tree-structured collection of subtrees that
hierarchically cover the tree T satisfying the following three properties:
1. T ? S ,
2. for all S ? S with S ? #= ? there exists an x ? S ? such that ?(S, x) ? S ,
3. for all S, R ? S such that S #? R and R #? S, we have |V (R) ? V (S)| ? 1.
The above definition recursively generates a cover. The splitting process that generates a hierarchical
cover S of T is formalized as rooted tree (D, r) in the following definition.
Definition 2. If S is a hierarchical cover of T we define the associated decomposition tree (D, r)
as a rooted tree, whose vertex set V (D) := T ? ? E(T ) where D? = T ? and leaves(D) = E(T ),
such that the following three properties hold:
1. S r = T ,
2. for all c, p ? D? , c ? ?rD (p) iff S c ? ?(S p , p) ,
3. for all (c, p) ? E(T ) 1 , we have (c, p) ? ?rD (p) iff S (c,p) ? ?(S p , p) .
The following lemma shows that with any given hierarchical cover S it is possible to associate a
unique decomposition tree (D, r).
Lemma 3. A hierarchical cover S of T defines a unique decomposition tree (D, r) such that if
S ? S there exists a v ? V (D) such that S = S v and if v, w ? V (D) and v #= w, then S v #= S w .
For a given hierarchical cover S in the following we define the height and the exposure: two
properties which measure different senses of the ?size? of a cover. The height of a hierarchical cover
S is the height of the associated decomposition tree D. Note that the height of a decomposition tree
D may be exponentially smaller than the height of T , since, for example, it is not difficult to show
that there exists a decomposition tree isomorphic to a binary tree when the input tree T is a path
graph. If R ? T and SR is a hierarchical cover of R, we define the exposure of SR (with respect to
tree T ) as maxQ?SR |?T (Q)|. Thus the exposure is a measure relative to a ?containing? tree (which
can be the input tree T itself) and the height is independent of any containing tree.
In Section 4 the covering subtrees correspond to cached ?joint distributions,? which are defined on
the boundary vertices of the subtrees, and require memory exponential in the boundary size. Thus
we are interested in covers with small exposure.
We now define a measure of the optimal height with respect to a given exposure value.
Definition 4. A hierarchical cover with exposure at most k is called a k-hierarchical cover. Given
any subtree R ? T , the k-decomposition potential ?k (R) of R is the minimum height of all hierarchical covers of SR with exposure (with respect to T ) not larger than k. The ?-decomposition potential ?? (R) is the minimum height of all hierarchical covers of R. If |?T (R)| > k then ?k (R) := ?.
Let?s consider some examples. Given a star graph, i.e., a graph with a single central vertex and any
number of adjacent vertices, there is in fact only one possible hierarchical cover obtained by splitting
the central vertex so that ?? (star) = 1. For path graphs, ?? (path) = ?(log |path|), as mentioned
above. An interesting example is a star with path graphs rather than single edges. Specifically, a
|star-path|
star-path may be formed by a set of log |star-path| path graphs P1 , P2 , . . . each with log |star-path|
edges. These path graphs are then joined at a central vertex. In this case we have ?? (star-path) =
O(log log(|star-path|)); as each path has a hierarchical cover of height O(log log(|star-path|)), each
of these path covers may then be joined to create a cover of the star-path. In Theorem 6 we will
see the generic bound ?? (T ) ? O(min(?(T ), log |V (T )|)). The star-path thus illustrates that the
bound may be exponentially loose.
In Theorem 6 we will see that ?2 (T ) ? 2?? (T ). Thus we may restrict our algorithm to hierarchical
covers with an exposure of 2 at very little cost in efficiency. Hence, we will now focus our attention
on 2-hierarchical covers.
2-Hierarchical covers. Given any element Q #= T in a 2-hierarchical cover of T then |?T (Q)| ?
{1, 2}. Consider the case in which ?T (Q) = {v, w}, i.e. |?T (Q)| = 2. Then Q can be specified by
1
Observe that (c, p) ? E(T ) implies c, p ? V (T ) and (c, p) ? leaves(D).
3
! "
the two vertices v, w and defined as follows: Q := wv := argmaxS?T (|V (S)| : v, w ? leaves(S)),
that is the maximal subtree of T , having v and w among its leaves.
Consider now the case in which ?T (Q) = {w}, i.e. |?T (Q)| = 1. Q is now defined as the T ?s
subtree containing vertex w together with all the descendents ?w
T (z) where z ? NT (w). Hence, a
subtree such as Q can be uniquely determined by the w?s neighbor
z ? NT (w). In order to denote
! "
subtree Q in this case we use the following notation: Q := w!z . Observe
! that
" one can also represent
a ?boundary one? subtree with the previous notation by writing Q := w" , where # is any 2 chosen
leaf of T belonging to ?w
T (z) (see Figure 1).
(2, s)-Hierarchical covers. We now introduce the notion of (2, s)-hierarchical covers (which, for
simplicity, we shall also call (2, s)-covers) with respect to a rooted tree (T, s). This notion explicitly
depends on a given vertex s ? V (T ), which, for the sake of simplicity, will be assumed to be a leaf
of T . (2, s)-Hierarchical covers are guaranteed to not be much larger than a 2-hierarchical cover
(see Theorem 6). They are also amenable to a bottom-up construction.
Definition 5. Given any subtree R ? T , a 2-hierarchical cover SR is a (2, s)-hierarchical cover of
R if, !for"all S ? SR \ {T }, there exists v,!w"? S where v ? ?sT (w) such that (case 1: |?T (Q)| = 1)
w
s
2
S= w
!
v , or (case 2: |?T (Q)| = 2) S = v . In the former case v ? ?T (w). We define ?s (R) to be
the minimal height of any possible (2, s)-hierarchical cover of R ? T .
Thus every subtree of a (2, s)-hierarchical cover is necessarily ?oriented? with respect to a root s.
3
Computing an optimal hierarchical cover
From a ?big picture? perspective, a (2, s)-hierarchical cover G is recursively constructed in a bottomup fashion: in the initialization phase G contains only the atomic components convering T , i.e. the
ones formed only by a pair of adjacent vertices of V (T ). We have then at this stage |G| = |E(T )|.
Then G grows step by step through the addition of new covering subtrees of T . At each time step
t, at least one subtree of T is added to G. All the subtrees added at each step t must strictly contain
only subtrees added before step t.
We now introduce the formal description of our method for constructing a (2, s)-hierarchical cover
G. As we said, the construction of G proceeds in incremental steps. At each step t the method
operates on a tree Tt , whose vertices are part of V (T ). The construction of Tt is accomplished
starting by Tt?1 (if t > 0) in such a way that V (Tt ) ? V (Tt?1 ), where T0 is set to be the subtree of
(T, s) containing the root and all the internal vertices.
During each step t all the while-loop instructions of Figure 1 are executed: (1) some vertices (the
black ones in Figure 1) are selected through a depth-first visit (during the backtracking steps) of
Tt starting from s 3 , (2) for each selected vertex v, subtree S v is obtained from merging subtrees
added to G in previous steps and overlapping at vertex v, (3) in order to create tree Tt+1 from
Tt the previously selected vertices of Tt are removed, (4) the edge set E(Tt+1 ) is created from
E(Tt ) in such a way to preserve the Tt ?s structure, but all the edges incident to the vertices removed
from V (Tt ) (the black vertices Figure 1) in the while-loop step 3 need to be deleted. The possible
disconnection that would arise by the removal of these parts of Tt is avoided by completing the
construction of Et+1 through the addition of some new edges. These additional edges are not part
of E(T ) and link each vertex v with its grand-parent in Tt if vertex v?s parent was deleted (see the
dashed line edges in Figure 1) during the construction of Tt+1 from Tt . In the final while-loop step
the variable t gets incremented by 1.
Basically, the key for obtaining optimality with this construction method can be explained with the
following observation. At each time step t, when we add a covering subtree S v for some vertex
v ? V (Tt ) selected by the algorithm (black vertices in Figure 1), the whole (2, s)-cover of S v
becomes completely contained in G and its height is t + 1, which can be proven to be the minimum
possible height of a (2, s)-cover of S v . Hence, at each time step t we construct the t + 1-th level
(in the hierarchical nested sense) of G in such a way to achieve local optimality for all elements
contained in all levels smaller or equal to t + 1. As the next theorem states, the running of the
algorithm is linear in |V (T )|.
2
This representation is not necessarily unique, as if !1 , !2 ? leaves(T ) ? Q, we have
3
Observe that s is the unique vertex belonging to V (Tt ) for all time steps t ? 0.
4
!w"
!1
=
! w "# !w"$
!2
=
"
z
.
Theorem 6. Given a rooted tree (T, s), the algorithm in Figure 1 outputs G, an optimal (2, s)hierarchical cover in time linear in |V (T )| of height ?2s (T ) which is bounded as ?? (T ) ? ?2 (T ) ?
?2s (T ) ? 2?? (T ) ? O(min(log |V (T )|, ?(T ))) .
Before we provide the detailed description of the algorithm for constructing an optimal (2, s)hierarchical cover we need some ancillary definitions. We call a vertex v ? V (Tt ) \ {s} mergeable
(at time t) if and only if either (i) v ? leaves(Tt ) or (ii) v has a single child in Tt and that child is not
mergeable. If v ? V (Tt ) \ {s} is mergeable we write v ? Mt . We also use the following shorthands
for making more intuitive our notation: We set ctv := ?sTt (v) when |?sTt (v)| = 1, ptv := ?sTt (v) when
v #= s and gvt := ?sTt (ptv ) when v, ptv #= s. Finally, given u, u# ? V (T ) such that u# ? ?sT (u), we
indicate with with ?sT (u 1? u# ) the child of u which is ancestor of u# in T .
?????????????????????
Input: Rooted tree (T, s).
?????????????????????
Initialisation: T0 ??T ? ? {s};
t ? 0;
?
!
s
"
Else G ? G ?
v
"
z
#
v
*
#
)
!!
*
&
'
!$
!!
($
#
3. V (Tt+1 ) ? V (Tt ) \ Mt .
4. E(Tt+1 ) ? {(v, ptv ) : v, ptv ? V (Tt+1 )}?
{(v, gvt ) : v, gvt ? V (Tt+1 ),
ptv &? V (Tt+1 )}.
'
!"
5. t ? t + 1.
%
"
#
)
*
&
!+
!!
(%
!
%
*
&
!%
$
"
.
!"
!"
$
.
)
!+
!#
!"
!
%
"
!$
'
!pt "
ctv
%
&
2. For all v ? Mt , merge as follows:
If v ? leaves(Tt ) then
z ? ?sT (ptv (? v).
G?G?
$
"
1. Construct Mt via depth-first search
of Tt from s.
(!
!
$
G ? ?Tv(v) : v ? V (T ) \ {s} .
?????????????????????
#
$
While V (Tt ) &= {s}
!pt "
(+
!
!$
!%
,-./01-23-45-26786(/
'
)
!+
!!
!"
!$
!%
,-./01-23-45-26786(696(/
?????????????????????
:45-260;/.74<1-460;6(/
=-.5->?@-6A-./01-2
Output: Optimal (2, s)-hierarchical cover G of T .
B<?/.--26>44-46/76/C-6D$E2FGH0-.>.1C01>@617A-.
?????????????????????
Figure 1: Left: Pseudocode for the linear time construction algorithm for an optimal (2, s)-hierarchical cover.
Right: Pictorial explanation of the pseudocode and the details of the hierarchical cover.
In order to clarify the method, we describe some of the details of the cover and some merge operations that
are performed in the diagram. Vertex 1 is the root vertex s. In each component, depicted as enclosed in a
line, the black node is the splitting vertex, i.e., a mergeable vertex of the tree Tt . The boundary definition may
be clarified by highlighting, for instance, that ?T (S 2 ) = {4} and ?T (S 10 ) = {8, 12}. Subtree S 2 contains
vertices 1, 2, 3 and 4. Vertex 2 is the splitting vertex of S 2 . ?(S 2 , 2) = {S (1,2) , S (2,3) , S (2,4) }, i.e., at
time t = 0, S 2 is formed by merging the three atomic subtrees S (1,2) , S (2,3) and S (2,4) , which were added
in the initialization step. These three subtrees overlap at only vertex 2, which is depicted in black because it
is mergeable in T0 . For what concerns the decomposition tree (D, r), we have ?rD (5) = {(4, 5), 6}, which
implies that S 5 is therefore formed by the atomic component S (4,5) and the non-atomic component S 6 . At time
t = 1, S 12 is obtained by merging S 10 together with S 13 , which have been both created at time t = 0. Observe
that in T1 vertex 12 is a leaf and the z variable in the while-loop step 2 is assigned to vertex 10 (v and and
ptv is respectively vertex 12 and
with the square bracket notation we
! "8). Regarding! the
" subtree
! 8 " representation
!8"
8
can write, for instance, S 2 = 14 and S 12 = 10
" (? 11 ? 14 ). Observe that, according to the definition
of a (2, s)-hierarchical cover, we have 4 ? ?1T (1) and 10 ? ?1T (8). Finally, notice that the height of the
(2, s)-hierarchical cover of S v is equal to t + 1 iff v is depicted in black in Tt .
4
Online marginalization
In this section we introduce our algorithm for efficiently computing marginals by summing over
products of variables in a tree topology. Formally our model is specified by a triple (T, ?, D) where
5
T is a tree, ? = (?e,l,m : e ? E(T ), l ? INk , m ? INk ) so that ?e is a positive symmetric k ? k
matrix and D = (dv,c : v ? V (T ), c ? INk ) is a |V (T )| ? k matrix. In a probabilistic setting it
is natural to view each normalized ?e as a stochastic symmetric ?transition? matrix and the ?data?
D as a right stochastic matrix corresponding to ?beliefs? about k different labels at each vertex in
T . In our online setting ? is a fixed parameter and D is changing over time and thus an element in
a sequence (D1 , . . . , Dt , . . .) where successive elements only differ in a single row. Thus at each
point at time we receive information at a single vertex.
In our intended application (see Section 5) of the model there is no necessary ?randomness? in the
generation of the data. However the language of probability provides a natural metaphor we use for
V (T )
our computed quantities. Thus a (k-ary) labeling of T is a vector ? ? L with L := INk
and its
?probability? with respect to (?, D) is
%
%
1
p(?|?, D) :=
?(i,j),?(i),?(j)
dv,?(v) ,
(1)
Z
(i,j)?E(T )
&
with the normalising constant Z := ??L
the marginal probability at a vertex v as
'
v?V (T )
(i,j)?E(T ) ?(i,j),?(i),?(j)
p(v ? a|?, D) :=
(
??L : ?(v)=a
'
v?V (T )
dv,?(v) . We denote
(2)
p(?|?, D) .
Using the hierarchical cover for efficient online marginalization. In the previous section we
discussed a method to compute a hierarchical cover of a tree T with optimal height ?2s (T ) in time
linear in T . In this subsection we will show how these covering components form a covering set of
cached ?marginals??. So that we may either compute p(v ? a|?, D) or update a single row of the
data matrix D and recompute the changed cached marginals all in time linear in ?2s (T ).
Definition 7. Given a tree S ? T , the potential function, ?TS : L(?T (S)) ? R with respect to
(?, D) is defined by:
?
??
?
(
%
%
?) :=
?(v,w),?(v),?(w)
dv,?(v)
(3)
?TS (?
??L(S) : ?(?T (S))=?
?
(v,w)?E(S)
v?S\?T (S)
Where L(X) := INX
k with X ? V (T ) is thus the restriction of L to X and likewise if ? ? L then
?(X) ? L(X) is the restriction of ? to X. For each tree in our hierarchical cover S ? S we will
have an associated potential function. Intuitively each of these potential functions summarize the
information in their interior by the marginal function defined on their boundary. Thus trees S ? S
with a boundary size of 1 require k values to be cached, the ??? weights; while boundary size 2
trees requires k 2 values, the ??? weights. This clarifies our motivation to find a cover with both
small height and exposure. We also cache ? weights that represent the product of ? weights; these
weights allow efficient computation on high degree vertices. The set of cached values necessary for
fast online computation correspond to these three types of weights of which there is a linear quantity
and on any given update or marginalization step only O(?2s (T )) of them are accessed.
Definitions of weights and potentials. Given a tree T and a hierarchical cover S it is isomorphic to
a decomposition tree (D, r). The decomposition tree will serve a dual purpose. First, each vertex z ?
D will serve as a ?name? for a tree S z ? S. Second, in the same way that the ?messages passing?
in belief propagation the follows the topology of the input tree, the structure of our computations
follows the decomposition tree D. We now introduce our notations for computing and traversing
the decomposition tree. As the cover has trees with one or two boundary vertices (excepting T
which has none) we define the corresponding vertices of the decomposition tree, Ci := {z ? D :
|?T (S z )| = i} for i ? {1, 2}. In this section since we are concerned with the traversal of (D, r)
we abbreviate ?D , ?D as both ? , ? respectively as convenient. As ?D (v) is a set of children, we
define the following functions to select specific children, )(v) := w if w ? ?(v), ?(v) ? ?T (S (w) )
for v ? D? ? (C1 ? C2 ) and *(v) := w if w ? ?(v), w #= )(v) for w ? C2 and v ? D? ? C2 .
When clear from the context we will use )v for )(v) as well as *v for *(v). We also need notation
for the potentially two boundary vertices of a tree S v ? S if v ? D \ {r}. Observe that for
v ? C1 ? C2 one boundary vertex of S v is necessarily v? :=? v and if v ? C2 there exists an ancestor
v? of v in D of so that {v,
? v?} = ?T (S v ). We also extend the split notation to pick out the specific
6
v
?a (v) := ?TS (v? ? a),
v
?ab (v) := ?TS (v? ? a, v? ? b),
?(T,v,v)
?
?a# (v) := dva
? ?T
)#a (v)
:=
?(T,v,v)
?
(v
?T
(v? ? a),
? a),
(v ? C1 )
?a (v) := dva
(v ? C2 )
?a (v) := dva
'
?a (w),
(v ? V (T ))
?TR (v ? a),
(v ? V (T ))
w??(v)?C1
'
R??(T,v)
?(T,?
v ,v)
(v ?V (T )\{r})
?a$ (v) := dv?a ?T
(v ?V (T )\{r})
)$a (v)
:=
?(T,v,?
v)
(v
?T
(?
v ? a),
(v ? C2 )
? a),
(v ? C2 )
Table 1: Weight definitions
complementary subtrees of T resulting from a split thus ?(T, p, q) := Q ? ?(T, p) if q ? Q and
define ?(T, p, q) := ?{R ? ?(T, p) : q #? R}. Observe that T = ?(T, p, q) ? ?(T, p, q) and
{p} = ?(T, p, q) ? ?(T, p, q). We shall use the notation (v1 ? a1 , v2 ? a2 , . . . , vm ? am ) to
represent a labeling of {v1 , v2 , . . . , vm } that maps vi to ai . In Table 1 we now give the weights
used in our online marginalization algorithm. The ?a , ?ab , ?a weights are cached values maintained
by the algorithm and the weights ?a , ?a$ , ?a% , -$a , and -%a are temporary values4 computed ?on-thefly.? The indices a, b ? INk and thus the memory requirements of our algorithm are linear in the
cardinality of the tree and quadratic in the number of labels.
Identities for weights and potentials. For the following lemma we introduce the notion of the
extension of a labelling. We extend by a vertex v ? V (T ) and a label a ? INk , the labelling
? ? L(X) to the labelling ?av ? L(X ? {v}) which satisfies ?av (v) = a and ?av (X) = ?.
Lemma 8. Given a tree, S ? T , and a vertex v ? S then if v ? S \ ?T (S)
%
(
%
?TR (?(?T (R)))
?TS (?)=
dva
?TR (?av (?T (R))) else if v ? ?T (S) then ?TS (?) =
a?INk
R??(S,v)
R??(S,v)
Thus a direct consequence of Lemma 8 is that we can compute the marginal probability at v as
? (v)
p(v ? a|?, D) = & a ? (v) . The recursive application of such factorizations is the basis of our
b?INk
b
algorithm (these factorizations are summarized in Table 2 in the technical appendices).
Algorithm initialization and complexity. In Figure 2 we give our algorithm for computing the
marginals at vertices with respect to (?, D). A number of our identities assumed for a given vertex
that it is in the interior of the tree and hence in the interior of decomposition tree. Thus before we
find the hierarchical cover of our input tree we extend the tree by adding a ?dummy? edge from
each leaf of the tree to a new dummy vertex. These dummy edges play no role except to simplify
notation. The hierarchical cover is then found on this enlarged tree; the cover height may at most
only increase by one. By setting the values in dummy edges and vertices in ? and D to one, this
ensures that all marginal computations are unchanged.
The running time of the algorithm is as follows. The computation of the hierarchical cover5 is linear
in |V (T )| as is the initialization step. The update and marginalization are linear in cover height
?? (T ). The algorithm also scales quadratically in k on the marginalization step and cubically in k
on update as the merge of two C2 trees require the multiplication of two k ? k matrices. Thus for
example if the set of possible labels is linear in the size of the tree classical belief propagation may
be faster.
Finally we observe that we may reduce the cubic dependence to a quadratic dependence on k via a
cover with the height bounded by the diameter of T as opposed to ?? (T ). This follows as the only
cubic step is in the update of a non-atomic (non-edge) ?-potential. Thus if we can build a cover,
with only atomic ?-potentials the running time will scale with k quadratically. We accomplish this
by modifying the cover algorithm (Figure 1) to only merge leaf vertices. Observe that the height of
this cover is now O(diameter(T )); and we have a hierarchical factorization into ?-potentials and
only atomic ?-potentials.
5
Multi-task learning in the allocation model with T REE -H EDGE
We conclude by sketching a simple online learning application to multi-task learning that is
amenable to our methods. The inspiration is that we have multiple tasks and a given tree structure that describes our prior expectation of ?relatedness? between tasks (see e.g., [7, Sec. 3.1.3]).
4
5
Note: if for ?a (v) if the product is empty then the product evaluates to 1; and if v ? C1 then )$a (v) := 1.
The construction of the decomposition tree may be simultaneously accomplished with the same complexity.
7
Marginalization (vertex v ? D? ) :
1. w ? r
2. ?a (w) ? ?a (r)
3. while(w &= v)
4.
w ? ?v (w)
5.
if(w ? C1 )
6.
?a# (w) ? &
?a (?(w))/?a (w)
7.
)#a (w) ? b ?ab (*(w))?b# (w)
8.
?a (w) = ?a (w))#a (w)
9.
else
10.
if(w = *(?(w)))
11.
?a# (w) ? )$a (?(w))?a (?(w))
12.
?a$ (w) ? ?a# (?(w))
13.
else
14.
?a# (w) ? )#a (?(w))?a (?(w))
$
15.
?a$ (w) ?
&?a (?(w))
#
16.
)a (w) ? &b ?b# (w)?ab (*(w))
17.
)$a (w) ? b ?b$ (w)?ab (+(w))
18.
?a (w) ? )#a (w))$a (w)?a (w)
19.
&
20. Output: ?a (v)/( b ?b (v))
Initialization: The ?, ? and ? weights are initialised in a
bottom-up fashion on the decomposition tree - we initialise
the weights of a vertex after we have initialised the weights
of all its children. Specifically, we first do a depth-first search
of D starting from r: When we reach an edge (v, w) ?
E(T ), if neither v or w is a leaf then we set ?ab ((v, w)) ?
?(v,w),a,b otherwise assuming w is a leaf we set ?a (v) ? 1
(dummy edge). When we reach a vertex, v ? V (T ), for
the last time (i.e. 'just before we backtrack from v) then
set: ?a (v) ? dva w??(v)?C1 ?a (w), and if v ? C2 then
&
?ab (v) ?& c ?ca (*(v))?cb (+(v))?c (v), or if v ? C1 then
?a (v) ? c ?ca (*(v))?c (v).
Update (vertex v ? D? ; data d ? [0, ?)k ):
a
; dv ? d; w ? v
1. ?a (v) ? ?a (v) ddva
2. while(w &= r)
3.
if(w ? C1 )
4.
?aold ? ?a&
(w)
5.
?a (w) ? c ?ca (*(w))?c (w)
6.
?a (?(w)) ? ?a (?(w))?a (w)/?aold
7.
else
&
8.
?ab (w) ? c ?ca (*(w))?cb (+(w))?c (w)
9.
w ? ?(w)
Figure 2: Algorithm: Initialization, Marginalization and Update
1. Parameters: A triple (T, ?, D1 ) and ? ? (0, ?).
2. For t = 1 to ! do
3.
Receive: v t ? V (T )
4.
Predict: p?t = (p(v t ? a|?, Dt ))a?INk
5.
Receive: y t ? [0, 1]k
6.
Incur loss: Lmix (y t , p?t )
t
7.
Update: Dt+1 = Dt ; Dt+1 (v t ) = (?
pt (a)e??y (a) )a?INk
Figure 3: T REE -H EDGE
Thus each vertex represents a task and if we have an edge between vertices then a priori we expect
those tasks to be related. Thus the hope is that information received for one task (vertex) will allow
us to improve our predictions on another task. For us each of these tasks is an allocation task as
addressed often with the H EDGE algorithm [4]. A similar application of the H EDGE algorithm in
multi-task learning was given in [8]. Their the authors considered a more challenging set-up where
the task structure is unknown and the hope is to do well if there is a posteriori a small clique of
closely related tasks. Our strong assumption of prior ?tree-structured? knowledge allows us to obtain a very efficient algorithm and sharp bounds which are not directly comparable to their results.
Finally, this set-up is also closely related to online graph labeling problem as in e.g., [9, 10, 11].
Thus the set-up is as follows. We incorporate our prior knowledge of task-relatedness with the triple
(T, ?, D1 ). Then on a trial t, the algorithm is given a v t ? V (T ), representing the task. The
&
algorithm then gives a non-negative prediction vector p?t ? {p : ka=1 p(a) = 1} for task v t and
t
k
receives an outcome y ? [0, 1] . It then suffers a mixture loss Lmix (y t , p?t ) := y t ? p?t . The aim is
to predict to minimize this loss. We give the algorithm in Figure 3. The notation follows Section 4
and the method therein implies that on each trial we can predict and update in O(?? (T )) time. We
obtain the following theorem (a proof sketch is contained in appendix C of the long version).
Theorem 9. Given a tree T , a vertex sequence $v 1 , . . . , v " % and an outcome sequence $y 1 , . . . , y " %
the loss of the T REE -H EDGE algorithm with the parameters (?, D1 ) and ? > 0 is, for all labelings
V (T )
? ? INk , bounded by
*
) !
!
(
( t
1
?
ln 2
t
t
t
t=1
Lmix (y , p? ) ? c?
y (?(v )) +
t=1
? log2 p(?|?, D1 )
with c? :=
1 ? e??
.
(4)
Acknowledgements. We would like to thank David Barber, Guy Lever and Massimiliano Pontil for valuable
discussions. We, also, acknowledge the financial support of the PASCAL 2 European Network of Excellence.
8
References
[1] David Barber. Bayesian Reasoning and Machine Learning. Cambridge University Press, 2012.
[2] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.
[3] Frank R. Kschischang, Brenden J. Frey, and Hans Andrea Loeliger. Factor graphs and the
sum-product algorithm. IEEE Transactions on Information Theory, 47(2):498?519, 2001.
[4] Yoav Freund and Robert E Schapire. A decision-theoretic generalization of on-line learning
and an application to boosting. Journal of Computer and System Sciences, 55(1):119?139,
1997.
[5] Judea Pearl. Reverend Bayes on inference engines: A distributed hierarchical approach. In
Proc. Natl. Conf. on AI, pages 133?136, 1982.
[6] Arthur L. Delcher, Adam J. Grove, Simon Kasif, and Judea Pearl. Logarithmic-time updates
and queries in probabilistic networks. J. Artif. Int. Res., 4:37?59, February 1996.
[7] Theodoros Evgeniou, Charles A. Micchelli, and Massimiliano Pontil. Learning multiple tasks
with kernel methods. Journal of Machine Learning Research, 6:615?637, 2005.
[8] Jacob Abernethy, Peter L. Bartlett, and Alexander Rakhlin. Multitask learning with expert
advice. In COLT, pages 484?498, 2007.
[9] Mark Herbster, Massimiliano Pontil, and Lisa Wainer. Online learning over graphs. In ICML,
pages 305?312. ACM, 2005.
[10] Mark Herbster, Guy Lever, and Massimiliano Pontil. Online prediction on large diameter
graphs. In NIPS, pages 649?656. MIT Press, 2008.
[11] Nicol`o Cesa-Bianchi, Claudio Gentile, and Fabio Vitale. Fast and optimal prediction on a
labeled tree. In COLT, 2009.
9
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3,962 | 4,587 | Phoneme Classification using Constrained Variational
Gaussian Process Dynamical System
Sungrack Yun
Qualcomm Korea
Seoul, South Korea
[email protected]
Hyunsin Park
Department of EE, KAIST
Daejeon, South Korea
[email protected]
Sanghyuk Park
Department of EE, KAIST
Daejeon, South Korea
[email protected]
Jongmin Kim
Department of EE, KAIST
Daejeon, South Korea
[email protected]
Chang D. Yoo
Department of EE, KAIST
Daejeon, South Korea
[email protected]
Abstract
For phoneme classification, this paper describes an acoustic model based on the
variational Gaussian process dynamical system (VGPDS). The nonlinear and nonparametric acoustic model is adopted to overcome the limitations of classical hidden Markov models (HMMs) in modeling speech. The Gaussian process prior
on the dynamics and emission functions respectively enable the complex dynamic
structure and long-range dependency of speech to be better represented than that
by an HMM. In addition, a variance constraint in the VGPDS is introduced to
eliminate the sparse approximation error in the kernel matrix. The effectiveness
of the proposed model is demonstrated with three experimental results, including
parameter estimation and classification performance, on the synthetic and benchmark datasets.
1
Introduction
Automatic speech recognition (ASR), the process of automatically translating spoken words into
text, has been an important research topic for several decades owing to its wide array of potential
applications in the area of human-computer interaction (HCI). The state-of-the-art ASR systems
typically use hidden Markov models (HMMs) [1] to model the sequential articulator structure of
speech signals. There are various issues to consider in designing a successful ASR and certainly
the following two limitations of an HMM need to be overcome. 1) An HMM with a first-order
Markovian structure is suitable for capturing short-range dependency in observations and speech
requires a more flexible model that can capture long-range dependency in speech. 2) Discrete latent
state variables and sudden state transitions in an HMM have limited capacity when used to represent
the continuous and complex dynamic structure of speech. These limitations must be considered
when seeking to improve the performance of an ASR.
To overcome these limitations, various models have been considered to model the complex structure
of speech. For example, the stochastic segment model [2] is a well-known generalization of the
HMM that represents long-range dependency over observations using a time-dependent emission
function. And the hidden dynamical model [3] is used for modeling the complex nonlinear dynamics
of a physiological articulator.
Another promising research direction is to consider a nonparametric Bayesian model for nonlinear
probabilistic modeling of speech. Owing to the fact that nonparametric models do not assume any
1
fixed model structure, they are generally more flexible than parametric models and can allow dependency among observations naturally. The Gaussian process (GP) [4], a stochastic process over
a real-valued function, has been a key ingredient in solving such problems as nonlinear regression
and classification. As a standard supervised learning task using the GP, Gaussian process regression
(GPR) offers a nonparametric Bayesian framework to infer the nonlinear latent function relating the
input and the output data. Recently, researchers have begun focusing on applying the GP to unsupervised learning tasks with high-dimensional data, such as the Gaussian process latent variable
model (GP-LVM) for reduction of dimensionality [5-6]. In [7], a variational inference framework
was proposed for training the GP-LVM. The variational approach is one of the sparse approximation approaches [8]. The framework was extended to the variational Gaussian process dynamical
system (VGPDS) in [9] by augmenting latent dynamics for modeling high-dimensional time series
data. High-dimensional time series have been incorporated in many applications of machine learning such as robotics (sensor data), computational biology (gene expression data), computer vision
(video sequences), and graphics (motion capture data). However, no previous work has considered
the GP-based approach for speech recognition tasks that involve high-dimensional time series data.
In this paper, we propose a GP-based acoustic model for phoneme classification. The proposed
model is based on the assumption that the continuous dynamics and nonlinearity of the VGPDS can
be better represent the statistical characteristic of real speech than an HMM. The GP prior over the
emission function allows the model to represent long-range dependency over the observations of
speech, while the HMM does not. Furthermore, the GP prior over the dynamics function enables
the model to capture the nonlinear dynamics of a physiological articulator.
Our contributions are as follows: 1) we introduce a GP-based model for phoneme classification tasks
for the first time, showing that the model has the potential of describing the underlying characteristics of speech in a nonparametric way; 2) we propose a prior for hyperparameters and a variance
constraint that are specially designed for ASR; and 3) we provide extensive experimental results and
analyses to reveal clearly the strength of our proposed model.
The remainder of the paper is structured as follows: Section 2 introduces the proposed model after a
brief description of the VGPDS. Section 3 provides extensive experimental evaluations to prove the
effectiveness of our model, and Section 4 concludes the paper with a discussion and plans for future
work.
2
2.1
Acoustic modeling using Gaussian Processes
Variational Gaussian Process Dynamical System
The VGPDS [9] models time series data by assuming that there exist latent states that govern the
data. Let Y = [[y11 , ? ? ? yN 1 ]T , ? ? ? , [y1D , ? ? ? yN D ]T ] ? RN ?D , t = [t1 , ? ? ? , tN ]T ? RN
+ , and
T
T
N ?Q
X = [[x11 , ? ? ? xN 1 ] , ? ? ? , [x1Q , ? ? ? xN Q ] ] ? R
be observed data, time, and corresponding
latent state, where N , D, and Q(< D) are the number of samples, the dimension of the observation
space, and the dimension of the latent space, respectively. In the VGPDS, these variables are related
as follows:
xnj
= gj (tn ) + ?nj ,
?nj ? N (0, 1/?jx ),
yni
= fi (xn ) + ni ,
ni ? N (0, 1/?iy ),
(1)
where fi (x) ? GP(?fi (x), kif (x, x0 )) and gj (t) ? GP(?gj (t), kjg (t, t0 )) are the emission function
from the latent space to the i-th dimension of the observation space and the dynamics function
from the time space to the j-th dimension of the latent space, respectively. Here, n ? {1, ? ? ? , N },
i ? {1, ? ? ? , D}, and j ? {1, ? ? ? , Q}. In this paper, a zero-mean function is used for all GPs. Fig.
1 shows graphical representations of HMM and VGPDS. Although the Gaussian process dynamical
model (GPDM) [10], which involves an auto-regressive dynamics function, is also a GP-based model
for time-series, it is not considered in this paper.
The marginal likelihood of the VGPDS is given as
Z
p(Y|t) = p(Y|X)p(X|t)dX.
2
(2)
Figure 1: Graphical representations of (left) the left-to-right HMM and (right) the VGPDS: In the
left figure, yn ? RD and xn ? {1, ? ? ? , C} are observations and discrete latent states. In the right
figure, yni , fni , xnj , gnj , and tn are observations, emission function points, latent states, dynamics
function points, and times, respectively. All function points in the same plate are fully connected.
Since the integral in Eq. (2) is not tractable, a variational method is used by introducing a variational
distribution q(X). A variational lower bound on the logarithm of the marginal likelihood is
Z
p(Y|X)p(X|t)
log p(Y|t) ?
q(X) log
dX
q(X)
Z
Z
q(X)
=
q(X) log p(Y|X)dX ? q(X) log
dX
p(X|t)
= L ? KL(q(X)||p(X|t)).
(3)
By the assumption of independence over the observation dimension, the first term in Eq. (3) is given
as
D Z
D
X
X
L=
q(X) log p(yi |X)dX =
Li .
(4)
i=1
i=1
In [9], a variational approach which involves sparse approximation of the covariance matrix obtained
from GP is proposed. The variational lower bound on Li is given as
"
#
? i |1/2
1 T
?iy
(?iy )N/2 |K
(? 2 yi Wi yi )
? ?1 ?2i )),
(?0i ? Tr(K
(5)
Li ? log
e
?
i
? i |1/2
2
(2?)N/2 |?iy ?2i + K
? i )?1 ?T . Here, K
? i ? RM ?M is a kernel matrix calcuwhere Wi = ?iy IN ? (?iy )2 ?1i (?iy ?2i + K
1i
? ? RM ?Q that are used for sparse
lated using the i-th kernel function and inducing input variables X
approximation of the full kernel matrix Ki . The closed-form of the statistics {?0i , ?1i , ?2i }D
i=1 ,
which are functions of variational parameters and inducing points, can be found in [9]. In the secQQ
QN QQ
ond term of Eq. (3), p(X|t) = j=1 p(xj ) and q(X) = n j=1 N (?nj , snj ) are the prior for
the latent state and the variational distribution that is used for approximating the posterior of the
latent state, respectively.
The parameter set ?, which consists of the hyperparameters {? f , ? g } of the kernel functions,
the noise variances {? y , ? x }, the variational parameters {[?n1 , ? ? ? , ?nQ ], [sn1 , ? ? ? , snQ ]}N
n=1 of
? is estimated by maximizing the lower bound on log p(Y|t)
q(X), and the inducing input points X,
in Eq. (3) using a scaled conjugate gradient (SCG) algorithm.
2.2
Acoustic modeling using VGPDS
For several decades, HMM has been the predominant model for acoustic speech modeling. However,
as we mentioned in Section 1, the model suffers from two major limitations: discrete state variables
and first-order Markovian structure which can model short-range dependency over the observations.
3
To overcome such limitations of the HMM, we propose an acoustic speech model based on the
VGPDS, which is a nonlinear and nonparametric model that can be used to represent the complex
dynamic structure of speech and long-range dependency over observations of speech. In addition,
to fit the model to large-scale speech data, we describe various implementation issues.
2.2.1
Time scale modification
The time length of each phoneme segment in an utterance varies with various conditions such as
position of the phoneme segment in the utterance, emotion, gender, and other speaker and environment conditions. To incorporate this fact into the proposed acoustic model, the time points tn are
modified as follows:
n?1
tn =
,
(6)
N ?1
where n and N are the observation index and the number of observations in a phoneme segment,
respectively. This time scale modification makes all phoneme signals have unit time length.
2.2.2
Hyperparameters
To compute the kernel matrices in Eq. (5), the kernel function must be defined. We use the radial
basis function (RBF) kernel for the emission function f as follows:
?
?
Q
X
f
k f (x, x0 ) = ?f exp ??
?j (xj ? x0j )2 ? ,
(7)
j=1
where ?f and ?jf are the RBF kernel variance and the j-th inverse length scale, respectively. The
RBF kernel function is adopted for representing smoothness of speech. For the dynamics function
g, the following kernel function is used:
k g (t, t0 ) = ?g exp ?? g (t ? t0 )2 + ?tt0 + b,
(8)
where ? and b are linear kernel variance and bias, respectively. The above dynamics kernel, which
consists of both linear and nonlinear components, is used for representing the complex dynamics of
the articulator. All hyperparameters are assumed to be independent in this paper.
In [11], same kernel function parameters are shared over all dimensions of human-motion capture
data and high-dimensional raw video data. However, this extensive sharing of the hyperparameters is unsuitable for speech modeling. Even though each dimension of observations is normalized in advance to have unit variance, the signal-to-noise ratio (SNR) is not consistent over all
dimensions. To handle this problem, this paper considers each dimension to be modeled independently using different kernel function parameters. Therefore, the hyperparameter sets are defined as
g
g
Q
f
f
g
}}D
? f = {?if , {?1i
, ? ? ? , ?Qi
i=1 and ? = {?j , ?j , ?j , bj }j=1 .
2.2.3
Priors on the hyperparameters
In the parameter estimation of the VGPDS, the SCG algorithm does not guarantee the optimal solution. To overcome this problem, we place the following prior on the hyperparameters of the kernel
functions as given below
p(?) ? exp(?? 2 /?
? ),
f
(9)
g
where ? ? {? , ? } and ?? are the hyper-parameter and the model parameter of the prior, respectively. In this paper, ?? is set to the sample variance for the hyperparameters of the emission kernel
functions, and ?? is set to 1 for the hyperparameters of the dynamics kernel functions. Uniform
priors are adopted for other hyperparameters, then the parameters of the VGPDS are estimated by
maximizing the joint distribution p(Y, ?|t) = p(Y|t, ?)p(?).
2.2.4
Variance constraint
In the lower bound of Eq. (5), the second term on the right-hand side is the regularization term that
represents the sparse approximation error of the full kernel matrix Ki . Note that with more inducing
4
input points, approximation error becomes smaller. However, only a small number of inducing
input points can be used owing to the limited availability of computational power, which increases
the effect of the regularization term.
To mitigate this problem, we introduce the following constraint on the diagonal terms of the covariance matrix as given below:
Tr(hKi iq(X) )
+ 1/?iy = ?i2 ,
N
(10)
where hKi iq(X) and ?i2 are the expectation of the full kernel matrix Ki and the sample variance of
the i-th dimension of the observation, respectively. This constraint is designed so that the variance
of each observation calculated from the estimated model is equal to the sample variance. By using
?0i = Tr(hKi iq(X) ), the inverse noise variance parameter is obtained directly by ?iy = (?i2 ?
? log ? y
?0i /N )?1 without separate gradient-based optimization. Then, the partial derivative ??0i i =
1
N ? 2 ??0i is used for SCG-based optimization. In Section 3.1, the effectiveness of the variance
constraint is demonstrated empirically.
2.3
Classification
For classification with trained VGPDSs, maximum-likelihood (ML) decoding is used. Let D(l) =
{Y(l) , t(l) } and ?(l) be the observation and parameter sets of the l-th VGPDS, respectively. Given
the test data D? = {Y? , t? }, the classification result ?l ? {1, ? ? ? , L} can be obtained by
?l =
=
3
arg max log p(Y? |t? , Y(l) , t(l) , ?(l) )
l
arg max log
l
p(Y(l) , Y? |t(l) , t? , ?(l) )
.
p(Y(l) |t(l) , ?(l) )
(11)
Experiments
To evaluate the effectiveness of the proposed model, three different kinds of experiments have been
designed:
1. Parameter estimation: validating the effectiveness of the proposed variance constraint (Section 2.2.4) on model parameter estimation
2. Two-class classification using synthetic data: demonstrating explicitly the advantages of
the proposed model over the HMM with respect to the degree of dependency over the
observations
3. Phoneme classification: evaluating the performance of the proposed model on real speech
data
Each experiment is described in detail in the following subsections. In this paper, the proposed
model is referred to as the constrained-VGPDS (CVGPDS).
3.1
Parameter estimation
In this subsection, the experiments of parameter estimation on synthetic data are described. Synthetic data are generated by using a phoneme model that is selected from the trained models in
Section 3.3 and then modified. The RBF kernel variances of the emission functions and the emission noise variances are modified from the selected model. In this experiment, the emission noise
variances and inducing input points are estimated, while all other parameters are fixed to the true
values used in generating the data.
Fig. 2 shows the parameter estimation results. The estimates of the 39-dimensional noise variance
of the emission functions are shown with the true noise variances, the true RBF kernel variances, and
the sample variances of the synthetic data. The top row denotes the estimation results without the
variance constraint, and the bottom row with the variance constraint. By comparing the two figures
5
Figure 2: Results of parameter estimation: (top-left) VGPDS with M = 5, (top-right) VGPDS with
M = 30, and (bottom) CVGPDS with M = 5
on the top row, we can confirm that the estimation result of the noise variance with M = 30 inducing
input points is better than that with M = 5 inducing input points. This result is obvious in the sense
that smaller values of M produce more errors in the sparse approximation of the covariance matrix.
However, both noise variance estimates are still different from the true values. By comparing the
top and bottom rows, we can see that the proposed CVGPDS outperforms the VGPDS in terms of
parameter estimation. Remarkably, the estimation result of the CVGPDS with M = 5 inducing input
points is much better than the result of the VGPDS with M = 30. Based on these observations, we
can conclude that the proposed CVGPDS is considerably more robust to the sparse approximation
error compared to the VGPDS, as we claimed in Section 2.2.4.
3.2
Two-class classification using synthetic data
This section aims to show that when there is strong dependency over the observations, the proposed
CVGPDS is a more appropriate model than the HMM for the classification task. To this end, we
first generated several sets of two-class classification datasets with different degrees of dependency
over the observations. The considered classification task is to map each input segment to one of two
class labels. Using s ? {1, ..., S} as the segment index, the synthetic dataset D = {Ys , ts , ls }Ss=1
consists of S segments, where the s-th segment has Ns samples. Here, Ys ? RNs ?D , ts ? RNs ,
and ls are the observation data, time, and class label of the s-th segment, respectively. The synthetic
dataset is generated as follows:
? Mean and kernel functions of two GPs gj (t) and fi (x) are defined as
gj (t) : ?gj (t) = aj t + bj ,
kjg (t, t0 ) = 1t=t0
PZi
f
fi (x) : ?i (x) = z=1 wz N (x; mzi ?zi ), kif (x, x0 ) = ?i exp(??i ||x ? x0 ||)
(12)
where {aj , bj }, {wz , mzi , ?zi }, and {?i , ?i } are respectively the parameters of the linear,
Gaussian mixture, and RBF kernel functions. The superscript z denotes the component
index of the Gaussian mixture, and Zi is the number of components in fi (x).
6
? For the s-th segment, {Ys , ts , ls },
1. ls is selected as either class 1 or 2.
2. Ns is randomly selected from interval [20, 30], and ts is obtained by using Eq. (6).
3. From ts , the mean vector ?gj (ts ) and covariance matrix Kgj are obtained for j =
1, ..., Q. Let Xs ? RNs ?Q be the latent state of the s-th segment. Then, the j-th column of Xs is generated by the Ns -dimensional Gaussian distribution N (?gj (ts ), Kgj ).
4. From Xs , the mean vector ?fi (Xs ) and covariance matrix Kfi are obtained for i =
1, ..., D. Then, the i-th column of Ys is generated by the Ns -dimensional Gaussian
distribution, N (?fi (Xs ), Kfi ).
Note that parameter ?i controls the degree of dependency over the observations. For instance, if ?i
decreases, the off-diagonal terms of the emission kernel matrix Kfi increase, which means stronger
correlations over the observations.
The experimental setups are as follows. The synthesized dataset consists of 200 segments in total
(100 segments per class). The dimensions of the latent space and observation space are set to Q = 2
and D = 5, respectively. We use 6(= Zi ) components for the mean function of the emission kernel
function. In this experiment, three datasets are synthesized and used to compare the CVGPDS and
the HMM. When generating each dataset, we use two different ?i values, one for each class, while
all other parameters in Eq. (12) are shared between the two classes. As a result, the degree of
correlation between the observations is the only factor that distinguishes the two classes. The three
generated datasets have different degrees of correlation over the observations, as a result of setting
different ?i values for generating each dataset. In particular, the third dataset is constructed with two
limitations of HMM such that it is well represented by an HMM. This could be achieved simply by
changing the form of the mean function ?gj (t) from a linear to a step function, and setting ?i = ? so
that each data sample is generated independently of the others. In the third dataset, the two classes
are set to have different ?i values. The classification experiments are conducted using an HMM and
CVGPDS.
Table 1: Classification accuracy for the two-class synthetic datasets (10-fold CV average [%]):
All parameters except ?i are set to be equal for classes 1 and 2.
In the case of ?i = ?, ?i are set to be different.
?i (class 1 : class 2)
0.1 : 0.5
1.0 : 2.0
?:?
HMM
61.0
68.5
88.5
CVGPDS
78.0
79.0
92.0
Table 1 summarizes the classification performance of the HMM and CVGPDS for the three synthetic
datasets. Remarkably, in all cases, the proposed CVGPDS outperforms the HMM, even in the case
of ?i = ? (the fourth column), where we assumed the dataset follows HMM-like characteristics.
Comparing the second and the third columns of Table 1, we can see that the performance of the
HMM degrades by 6.5% as ?i becomes smaller, while the proposed CVGPDS almost maintains
its performance with only 1.0% reduction. This result demonstrates the superiority of the proposed
CVGPDS in modeling data with strong correlations over the observations. Apparently, the HMM
failed to distinguish the two classes with different degree of dependency over the observations. In
contrast, the proposed CVGPDS distinguishes the two classes more effectively by capturing the
different degrees of inter-dependencies over the observations incorporated in each class.
3.3
Phoneme classification
In this section, phoneme classification experiments is described on real speech data from the TIMIT
database. The TIMIT database contains a total of 6300 phonetically rich utterances, each of which
is manually segmented based on 61 phoneme transcriptions. Following the standard regrouping of
phoneme labels [11], 61 phonemes are reduced to 48 phonemes selected for modeling. As observations, 39-dimensional Mel-frequency cepstral coefficients (MFCCs) (13 static coefficients, ?, and
7
??) extracted from the speech signals with standard 25 ms frame size, and 10 ms frame shifts are
used. The dimension of the latent space is set to Q = 2.
For the first phoneme classification experiment, 100 segments per phoneme are randomly selected
using the phoneme boundary provided information in the TIMIT database. The number of inducing
input points is set to M = 10. A 10-fold cross-validation test was conducted to evaluate the proposed
model in comparison with an HMM that has three states and a single Gaussian distribution with a
full covariance matrix per state. Parameters of the HMMs are estimated by using the conventional
expectation-maximization (EM) algorithm with a maximum likelihood criterion.
Table 2: Classification accuracy on the 48-phoneme dataset (10-fold CV average [%]):
100 segments are used for training and testing each phoneme model
HMM
VGPDS
CVGPDS
49.19
48.17
49.36
Table 2 shows the experimental results of a 48-phoneme classification. Compared to the HMM and
VGPDS, the proposed CVGPDS performs more effectively.
For the second phoneme classification experiment, the TIMIT core test set consisting of 192 sentences is used for evaluation. We use the same 100 segments for training the phoneme models as in
the first phoneme classification experiment. The size of the training dataset is smaller than that of
conventional approaches due to our limited computational ability. When evaluating the models, we
merge the labels of 48 phonemes into the commonly used 39 phonemes [11]. Given speech observations with boundary information, a sequence of log-likelihoods is obtained, and then a bigram is
constructed to incorporate linguistic information into the classification score. In this experiment, the
number of inducing input points is set to M = 5.
Table 3: Classification accuracy on the TIMIT core test set [%]:
100 segments are used for training each phoneme model
HMM
VGPDS
CVGPDS
57.83
61.44
61.54
Table 3 shows the experimental results of phoneme classification for the TIMIT core test set. As
shown by the results in Table 2, the proposed CVGPDS performed better than the HMM and VGPDS. However, the classification accuracies in Table 3 are lower than the state-of-the-art phoneme
classification results [12-13]. The reasons for low accuracy are as follows: 1) insufficient amount
of data is used for training the model owing to limited availability of computational power; 2) a
mixture model for the emission is not considered. These remaining issues need to be addressed for
improved performance.
4
Conclusion
In this paper, a VGPDS-based acoustic model for phoneme classification was considered. The proposed acoustic model can represent the nonlinear latent dynamics and dependency among observations by GP priors. In addition, we introduced a variance constraint on the VGPDS. Although the
proposed model could not achieve the state-of-the-art performance of phoneme classification, the
experimental results showed that the proposed acoustic model has potential for speech modeling.
For future works, extension to phonetic recognition and mixture of the VGPDS will be considered.
Acknowledgments
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the
Korea government(MEST) (No.2012-0005378 and No.2012-0000985)
8
References
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[5] N. D. Lawrence, ?Probabilistic non-linear principal component analysis with Gaussian process latent variable models,? Journal of Machine Learning Research (JMLR), Vol.6, pp.1783-1816, 2005.
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in Neural Information Processing Systems (NIPS), 2007.
9
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3,963 | 4,588 | Nystr?om Method vs Random Fourier Features:
A Theoretical and Empirical Comparison
Tianbao Yang? , Yu-Feng Li? , Mehrdad Mahdavi\ , Rong Jin\ , Zhi-Hua Zhou?
?
Machine Learning Lab, GE Global Research, San Ramon, CA 94583
\
Michigan State University, East Lansing, MI 48824
?
National Key Laboratory for Novel Software Technology, Nanjing University, 210023, China
[email protected],mahdavim,[email protected],liyf,[email protected]
Abstract
Both random Fourier features and the Nystr?om method have been successfully
applied to efficient kernel learning. In this work, we investigate the fundamental
difference between these two approaches, and how the difference could affect
their generalization performances. Unlike approaches based on random Fourier
features where the basis functions (i.e., cosine and sine functions) are sampled
from a distribution independent from the training data, basis functions used by
the Nystr?om method are randomly sampled from the training examples and are
therefore data dependent. By exploring this difference, we show that when there
is a large gap in the eigen-spectrum of the kernel matrix, approaches based on
the Nystr?om method can yield impressively better generalization error bound than
random Fourier features based approach. We empirically verify our theoretical
findings on a wide range of large data sets.
1
Introduction
Kernel methods [16], such as support vector machines, are among the most effective learning methods. These methods project data points into a high-dimensional or even infinite-dimensional feature
space and find the optimal hyperplane in that feature space with strong generalization performance.
One limitation of kernel methods is their high computational cost, which is at least quadratic in the
number of training examples, due to the calculation of kernel matrix. Although low rank decomposition approaches (e.g., incomplete Cholesky decomposition [3]) have been used to alleviate the
computational challenge of kernel methods, they still require computing the kernel matrix. Other approaches such as online learning [9] and budget learning [7] have also been developed for large-scale
kernel learning, but they tend to yield performance worse performance than batch learning.
To avoid computing kernel matrix, one common approach is to approximate a kernel learning problem with a linear prediction problem. It is often achieved by generating a vector representation of
data that approximates the kernel similarity between any two data points. The most well known
approaches in this category are random Fourier features [13, 14] and the Nystr?om method [20, 8].
Although both approaches have been found effective, it is not clear what are their essential difference, and which method is preferable under which situations. The objective of this work is to
understand the difference between these two approaches, both theoretically and empirically
The theoretical foundation for random Fourier transform is that a shift-invariant kernel is the Fourier
transform of a non-negative measure [15]. Using this property, in [13], the authors proposed to
represent each data point by random Fourier features. Analysis in [14] shows that, the generalization
error bound for kernel learning based on random Fourier features is given by O(N ?1/2 + m?1/2 ),
where N is the number of training examples and m is the number of sampled Fourier components.
1
An alternative approach for large-scale kernel classification is the Nystr?om method [20, 8] that
approximates the kernel matrix by a low rank matrix. It randomly samples a subset of training
b for the random samples. It then represents each data
examples and computes a kernel matrix K
point by a vector based on its kernel similarity to the random samples and the sampled kernel matrix
b Most analysis of the Nystr?om method follows [8] and bounds the error in approximating the
K.
kernel matrix. According to [8], the approximation error of the Nystr?om method, measured in
spectral norm 1 , is O(m?1/2 ), where m is the number of sampled training examples. Using the
arguments in [6], we expected an additional error of O(m?1/2 ) in the generalization performance
caused by the approximation of the Nystr?om method, similar to random Fourier features.
Contributions In this work, we first establish a unified framework for both methods from the
viewpoint of functional approximation. This is important because random Fourier features and the
Nystr?om method address large-scale kernel learning very differently: random Fourier features aim
to approximate the kernel function directly while the Nystr?om method is designed to approximate
the kernel matrix. The unified framework allows us to see a fundamental difference between the
two methods: the basis functions used by random Fourier features are randomly sampled from a
distribution independent from the training data, leading to a data independent vector representation;
in contrast, the Nystr?om method randomly selects a subset of training examples to form its basis
functions, leading to a data dependent vector representation. By exploring this difference, we show
that the additional error caused by the Nystr?om method in the generalization performance can be
improved to O(1/m) when there is a large gap in the eigen-spectrum of the kernel matrix. Empirical
studies on a synthetic data set and a broad range of real data sets verify our analysis.
2
A Unified Framework for Approximate Large-Scale Kernel Learning
Let D = {(x1 , y1 ), . . . , (xN , yN )} be a collection of N training examples, where xi ? X ? Rd ,
yi ? Y. Let ?(?, ?) be a kernel function, H? denote the endowed Reproducing Kernel Hilbert Space,
and K = [?(xi , xj )]N ?N be the kernel matrix for the samples in D. Without loss of generality,
we assume ?(x, x) ? 1, ?x ? X . Let (?i , vi ), i = 1, . . . , N be the eigenvalues and eigenvectors
of K ranked in the descending order of eigenvalues. Let V = [Vij ]N ?N = (v1 , . . . , vN ) denote
b = {b
bm } denote the randomly
the eigenvector matrix. For the Nystr?om method, let D
x1 , . . . , x
b = [?(b
bj )]m?m denote the corresponding kernel matrix. Similarly, let
sampled examples, K
xi , x
bi , v
b ranked in the descending order of eigenvalues, and
bi ), i ? [m]} denote the eigenpairs of K
{(?
b
b
bm ). We introduce two linear operators induced by examples in D and
V = [Vij ]m?m = (b
v1 , . . . , v
b i.e.,
D,
N
m
1 X
1 X
LN [f ] =
?(xi , ?)f (xi ), Lm [f ] =
?(b
xi , ?)f (b
xi ).
(1)
N i=1
m i=1
It can be shown that both LN and Lm are self-adjoint operators. According to [18], the eigenvalbi /m, i ? [m], respectively, and their corresponding
ues of LN and Lm are ?i /N, i ? [N ] and ?
normalized eigenfunctions ?j , j ? [N ] and ?
bj , j ? [m] are given by
N
1 X
?j (?) = p
Vi,j ?(xi , ?), j ? [N ],
?j i=1
m
1 Xb
?
bj (?) = q
Vi,j ?(b
xi , ?), j ? [m].
b
i=1
?j
(2)
? ) = exp(?kx ?
To make our discussion concrete, we focus on the RBF kernel 2 , i.e., ?(x, x
? k22 /[2? 2 ]), whose inverse Fourier transform is given by a Gaussian distribution p(u) =
x
N (0, ? ?2 I) [15]. Our goal is to efficiently learn a kernel prediction function by solving the following optimization problem:
min
f ?HD
N
?
1 X
kf k2H? +
`(f (xi ), yi ),
2
N i=1
1
(3)
We choose the bound based on spectral norm according to the discussion in [6].
The improved bound obtained in the paper for the Nystrom method is valid for any kernel matrix that
satisfies the eigengap condition.
2
2
where HD = span(?(x1 , ?), . . . , ?(xN , ?)) is a span over all the training examples 3 , and `(z, y) is
a convex loss function with respect to z. To facilitate our analysis, we assume maxy?Y `(0, y) ? 1
and `(z, y) has a bounded gradient |?z `(z, y)| ? C. The high computational cost of kernel learning
arises from the fact that we have to search for an optimal classifier f (?) in a large space HD .
Given this observation, to alleviate the computational cost of kernel classification, we can reduce
space HD to a smaller space Ha , and only search for the solution f (?) ? Ha . The main challenge is
how to construct such a space Ha . On the one hand, Ha should be small enough to make it possible
to perform efficient computation; on the other hand, Ha should be rich enough to provide good approximation for most bounded functions in HD . Below we show that the difference between random
Fourier features and the Nystr?om method lies in the construction of the approximate space Ha . For
each method, we begin with a description of a vector representation of data, and then connect the
vector representation to the approximate large kernel machine by functional approximation.
Random Fourier Features The random Fourier features are constructed by first sampling Fourier components u1 , . . . , um from p(u), projecting each example x to u1 , . . . , um
separately, and then passing them through sine and cosine functions, i.e., zf (x) =
>
>
>
(sin(u>
1 x), cos(u1 x), . . . , sin(um x), cos(um x)). Given the random Fourier features, we then
learn a linear machine f (x) = w> zf (x) by solving the following optimization problem:
min2m
w?R
N
?
1 X
`(w> zf (xi ), yi ).
kwk22 +
2
N i=1
(4)
To connect the linear machine (4) to the kernel machine in (3) by a functional approximation, we can
construct a functional space Haf = span(s1 (?), c1 (?), . . . , sm (?), cm (?)), where sk (x) = sin(u>
k x)
f
,
we
have
and ck (x) = cos(u>
x).
If
we
approximate
H
in
(3)
by
H
D
a
k
min
f ?Hfa
N
?
1 X
kf k2H? +
`(f (xi ), yi ).
2
N i=1
(5)
The following proposition connects the approximate kernel machine in (5) to the linear machine
in (4). Proofs can be found in supplementary file.
Proposition 1 The approximate kernel machine in (5) is equivalent to the following linear machine
min2m
w?R
N
? >
1 X
w (w ? ?) +
`(w> zf (xi ), yi ),
2
N i=1
s/c
c >
s
where ? = (?1s , ?1c , ? ? ? , ?m
, ?m
) and ?i
(6)
= exp(? 2 kui k22 /2).
Comparing (6) to the linear machine based on random Fourier features in (4), we can see that other
s/c
than the weights {?i }m
i=1 , random Fourier features can be viewed as to approximate (3) by restricting the solution f (?) to Haf .
The Nystr?om Method The Nystr?om method approximates the full kernel matrix K by first sambr =
b1 , ? ? ? , x
bm , and then constructing a low rank matrix by K
pling m examples, denoted by x
? >
?
b
b
b
b
bj )]N ?m , K = [?(b
bj )]m?m , K is the pseudo inverse of K,
Kb K Kb , where Kb = [?(xi , x
xi , x
b
and r denotes the rank of K. In order to train a linear machine, we can derive a vector representab1 , . . . , ?
br ) and
b r?1/2 Vbr> (?(x, x
b r = diag(?
b1 ), . . . , ?(x, x
bm ))> , where D
tion of data by zn (x) = D
>
b
b
br ). It is straightforward to verify that zn (xi ) zn (xj ) = [Kr ]ij . Given the vector
Vr = (b
v1 , . . . , v
representation zn (x), we then learn a linear machine f (x) = w> zn (x) by solving the following
optimization problem:
N
?
1 X
2
min kwk2 +
`(w> zn (xi ), yi ).
w?Rr 2
N i=1
3
We use HD , instead of H? in (3), owing to the representer theorem [16].
3
(7)
In order to see how the Nystr?om method can be cast into the unified framework of approximating the
large scale kernel machine by functional approximation, we construct the following functional space
Han = span(?
b1 , . . . , ?
br ), where ?
b1 , . . . , ?
br are the first r normalized eigenfunctions of the operator
Lm . The following proposition shows that the linear machine in (7) using the vector representation
of the Nystr?om method is equivalent to the approximate kernel machine in (3) by restricting the
solution f (?) to an approximate functional space Han .
Proposition 2 The linear machine in (7) is equivalent to the following approximate kernel machine
minn
f ?Ha
N
?
1 X
kf k2H? +
`(f (xi ), yi ),
2
N i=1
(8)
Although both random Fourier features and the Nystr?om method can be viewed as variants of the
unified framework, they differ significantly in the construction of the approximate functional space
Ha . In particular, the basis functions used by random Fourier features are sampled from a Gaussian
distribution that is independent from the training examples. In contrast, the basis functions used by
the Nystr?om method are sampled from the training examples and are therefore data dependent.
This difference, although subtle, can have significant impact on the classification performance. In
the case of large eigengap, i.e., the first few eigenvalues of the full kernel matrix are much larger than
the remaining eigenvalues, the classification performance is mostly determined by the top eigenvectors. Since the Nystr?om method uses a data dependent sampling method, it is able to discover the
subspace spanned by the top eigenvectors using a small number of samples. In contrast, since random Fourier features are drawn from a distribution independent from training data, it may require a
large number of samples before it can discover this subspace. As a result, we expect a significantly
lower generalization error for the Nystr?om method.
To illustrate this point, we generate a synthetic data set consisted of two balanced classes with a
total of N = 10, 000 data points generated from uniform distributions in two balls of radius 0.5
centered at (?0.5, 0.5) and (0.5, 0.5), respectively. The ? value in the RBF kernel is chosen by
cross-validation and is set to 6 for the synthetic data. To avoid a trivial task, 100 redundant features,
each drawn from a uniform distribution on the unit interval, are added to each example. The data
points in the first two dimensions are plotted in Figure 1(a) 4 , and the eigenvalue distribution is
shown in Figure 1(b). According to the results shown in Figure 1(c), it is clear that the Nystr?om
method performs significantly better than random Fourier features. By using only 100 samples, the
Nystr?om method is able to make perfect prediction, while the decision made by random Fourier features based method is close to random guess. To evaluate the approximation error of the functional
space, we plot in Figure 1(e) and 1(f), respectively, the first two eigenvectors of the approximate
kernel matrix computed by the Nystr?om method and random Fourier features using 100 samples.
Compared to the eigenvectors computed from the full kernel matrix (Figure 1(d)), we can see that
the Nystr?om method achieves a significantly better approximation of the first two eigenvectors than
random Fourier features.
Finally, we note that although the concept of eigengap has been exploited in many studies of kernel
learning [2, 12, 1, 17], to the best of our knowledge, this is the first time it has been incorporated in
the analysis for approximate large-scale kernel learning.
3
Main Theoretical Result
?
?
Let fm
be the optimal solution to the approximate kernel learning problem in (8), and let fN
be the
?
solution to the full version of kernel learning in (3). Let f be the optimal solution to
?
min
F (f ) = kf k2H? + E [`(f (x), y)] ,
f ?H?
2
where E[?] takes expectation over the joint distribution P (x, y). Following [10], we define the excess
risk of any classifier f ? H? as
?(f ) = F (f ) ? F (f ? ).
4
Note that the scales of the two axes in Figure 1(a) are different.
4
(9)
Synthetic data
Synthetic data
0
1
100
10
0.9
90
?1
0.7
0.6
0.5
0.4
0.3
0.2
0.5
10
2000
4000
6000
8000
10000
0.01
0
?0.01
2000
4000
6000
8000
0.4N
0.6N
0.8N
0.01
0.0095
0
10000
2000
4000
6000
8000
0.02
0
?0.02
2000
4000
6000
8000
5
10
20
50
# random samples
100
(c) Classification accuracy vs
the number of samples
10000
0.04
?0.04
0
Nystrom Method
Random Fourier Features
60
N
0.0105
Eigenvector 2
0.02
0.2N
(b) Eigenvalues (in logarithmic scale) vs. rank. N is the
total number of data points.
Eigenvector 1
0.01
Eigenvector 2
Eigenvector 1
0.0105
70
40
rank
?5
1
Eigenvector 1
0
80
50
Eigenvector 2
?0.5
(a) Synthetic data: the first two
dimensions
?0.02
0
?3
10
?4
1st dimension
0.0095
0
?2
10
10
0.1
0
?1
accuaracy
10
Eigenvalues/N
2nd dimension
0.8
10000
0.04
0.02
0
?0.02
?0.04
0
2000
4000
6000
8000
10000
2000
4000
6000
8000
10000
0.05
0
?0.05
0
(d) the first two eigenvectors of the (e) the first two eigenvectors com- (f) the first two eigenvectors comfull kernel matrix
puted by Nystr?om method
puted by random Fourier features
Figure 1: An Illustration Example
?
by the generalization
Unlike [6], in this work, we aim to bound the generalization performance of fm
?
performance of fN , which better reflects the impact of approximating HD by Han .
In order to obtain a tight bound, we exploit the local Rademacher complexity [10]. Define ?(?) =
P
1/2
N
2
2
. Let ?e as the solution to ?e2 = ?(e
?) where the existence and uniqueness
i=1 min(? , ?i )
N
q
6 ln N
. According
of ?e are determined by the sub-root property of ?(?) [4], and = max ?e,
N
to [10], we have 2 = O(N ?1/2 ), and when the eigenvalues of kernel function follow a p-power law,
?
?
). Section 4
) by ?(fN
it is improved to 2 = O(N ?p/(p+1) ). The following theorem bounds ?(fm
will be devoted to the proof of this theorem.
Theorem 1 For 162 e?2N ? ? ? 1, ?r+1 = O(N/m) and
2 ln(2N 3 )
+
m
(?r ? ?r+1 )/N = ?(1) ? 3
r
2 ln(2N 3 )
m
!
,
with a probability 1 ? 3N ?3 , we have
?
?(fm
)
?
?
3?(fN
)
1e 2
1
+ O +
,
?
m
e suppresses the polynomial term of ln N .
where O(?)
Theorem 1 shows that the additional error caused by the approximation of the Nystr?om method is
improved to?O(1/m) when there is a large gap between ?r and ?r+1 . Note that the improvement
from O(1/ m) to O(1/m) is very significant from the theoretical viewpoint, because it is well
known that the generalization error for kernel learning is O(N ?1/2 ) [4]5 . As a result, to achieve
a similar performance as the standard kernel learning, the number of required samples has to be
5
It is possible to achieve a better generalization error bound of O(N ?p/(p+1) ) by assuming the eigenvalues
of kernel matrix follow a p-power law [10]. However, large eigengap doest not immediately indicate power law
distribution for eigenvalues and and consequently a better generalization error.
5
?
O(N ) if the additional error caused by the kernel approximation is bounded by O(1/ m), leading
to a high computational cost. On the other hand, with O(1/m) bound for the additional
error caused
?
by the kernel approximation, the number of required samples is reduced to N , making it more
practical for large-scale kernel learning.
We also note that the improvement made for the Nystr?om method relies on the property that Han ?
HD and therefore requires data dependent basis functions. As a result, it does not carry over to
random Fourier features.
4
Analysis
In this section, we present the analysis that leads to Theorem 1. Most of the proofs can be found in
?
the supplementary materials. We first present a theorem to show that the excessive risk bound of fm
b
is related to the matrix approximation error kK ? Kr k2 .
Theorem 2 For 162 e?2N ? ? ? 1, with a probability 1 ? 2N ?3 , we have
!
b r k2
2
kK ? K
?
?
?N
+
+e
,
?(fm ) ? 3?(fN ) + C2
?
N?
where C2 is a numerical constant.
In the sequel, we let Kr be the best rank-r approximation matrix for K. By the triangle inequality,
b r k2 ? kK ? Kr k2 + kKr ? K
b r k2 ? ?r+1 + kKr ? K
b r k2 , we thus proceed to bound
kK ? K
b r k2 . Using the eigenfunctions of Lm and LN , we define two linear operators Hr and H
br
kKr ? K
as
r
r
X
X
b r [f ](?) =
Hr [f ](?) =
?i (?)h?i , f iH? , H
?
bi (?)h?
bi , f iH? ,
(10)
i=1
i=1
b r k2 is related to the linear operator
where f ? H? . The following theorem shows that kKr ? K
br .
?H = Hr ? H
br > 0 and ?r > 0, we have
Theorem 3 For ?
1/2
1/2
b r ? Kr k2 ? N kL ?HL k2 ,
kK
N
N
where kLk2 stands for the spectral norm of a linear operator L.
1/2
1/2
Given the result in Theorem 3, we move to bound the spectral norm of LN ?HLN . To this
end, we assume a sufficiently large eigengap ? = (?r ? ?r+1 )/N . The theorem below bounds
1/2
1/2
kLN ?HLN k2 using matrix perturbation theory [19].
Theorem 4 For ? = (?r ? ?r+1 )/N > 3kLN ? Lm kHS , we have
1/2
1/2
kLN ?HLN k2 ? ?
r
where ? = max
4kLN ? Lm kHS
,
? ? kLN ? Lm kHS
!
2kLN ? Lm kHS
?r+1
,
.
N ? ? kLN ? Lm kHS
Remark To utilize the result in Theorem 4, we consider the case when ?r+1 = O(N/m) and
? = ?(1). We have
1
1/2
1/2
2
?
kLN ?HLN k2 ? O max
kLN ? Lm kHS , kLN ? Lm kHS
.
m
?
1/2
1/2
Obviously, in order to achieve O(1/m) bound for kLN ?HLN k2 , we need an O(1/ m) bound
for kLN ? Lm kHS , which is given by the following theorem.
6
Theorem 5 For ?(x, x) ? 1, ?x ? X , with a probability 1 ? N ?3 , we have
r
2 ln(2N 3 )
2 ln(2N 3 )
kLN ? Lm kHS ?
+
.
m
m
Theorem 5 directly follows from Lemma 2 of [18]. Therefore, by assuming the conditions in Theb r k2 ?
orem 1 and combining results from Theorems 3, 4, and 5, we immediately have kK ? K
O (N/m). Combining this bound with the result in Theorem 2 and using
the
union
bound,
we
have,
1
?
?
with a probability 1 ? 3N ?3 , ?(fm
) ? 3?(fN
) + C? 2 + m
+ e?N . We complete the proof of
Theorem 1 by using the fact e?N < 1/N ? 1/m.
5
Empirical Studies
To verify our theoretical findings, we evaluate the empirical performance of the Nystr?om method
and random Fourier features for large-scale kernel learning. Table 1 summarizes the statistics of the
six data sets used in our study, including two for regression and four for classification. Note that
datasets C PU, C ENSUS, A DULT and F OREST were originally used in [13] to verify the effectiveness of random Fourier features. We evaluate the classification performance by accuracy, and the
performance of regression by mean square error of the testing data.
We use uniform sampling in the Nystr?om method owing to its simplicity. We note that the empirical
performance of the Nystr?om method may be improved by using a different implementation [21,
11]. We download the codes from the website http://berkeley.intel-research.net/
arahimi/c/random-features for the implementation of random Fourier features. A RBF
kernel is used for both methods and for all the datasets. A ridge regression package from [13] is used
for the two regression tasks, and LIBSVM [5] is used for the classification tasks. All parameters
are selected by a 5-fold cross validation. All experiments are repeated ten times, and prediction
performance averaged over ten trials is reported.
Figure 2 shows the performance of both methods with varied number of random samples. Note
that for large datasets (i.e., C OVTYPE and F OREST), we restrict the maximum number of random
samples to 200 because of the high computational cost. We observed that for all the data sets, the
Nystr?om method outperforms random Fourier features 6 . Moreover, except for C OVTYPE with 10
random samples, the Nystr?om method performs significantly better than random Fourier features,
according to t-tests at 95% significance level. We finally evaluate that whether the large eigengap
condition, the key assumption for our main theoretical result, holds for the data sets. Due to the
large size, except for C PU, we compute the eigenvalues of kernel matrix based on 10, 000 randomly
selected examples from each dataset. As shown in Figure 3 (eigenvalues are in logarithm scale),
we observe that the eigenvalues drop very quickly as the rank increases, leading to a significant gap
between the top eigenvalues and the remaining eigenvalues.
6
Conclusion and Discussion
We study two methods for large-scale kernel learning, i.e., the Nystr?om method and random Fourier
features. One key difference between these two approaches is that the Nystr?om method uses data
6
We note that the classification performance of A DULT data set reported in Figure 2 does not match with
the performance reported in [13]. Given the fact that we use the code provided by [13] and follow the same
cross validation procedure, we believe our result is correct. We did not use the KDDCup dataset because of the
problem of oversampling, as pointed out in [13].
Table 1: Statistics of data Sets
TASK
Reg.
Reg.
Class.
DATA # TRAIN
CPU
6,554
CENSUS 18,186
ADULT 32,561
# TEST
819
2,273
16,281
#Attr.
21
119
123
TASK
Class.
Class.
Class.
7
DATA # TRAIN
COD-RNA 59,535
COVTYPE 464,810
FOREST 522,910
# TEST
271,617
116,202
58,102
#Attr.
8
54
54
CPU
1.5
1
0.5
10
20
1
0.5
70
50
40
10
20
30
50 100 200 500 1000
# random samples
80
80
75
75
60
accuracy(%)
70
70
65
60
50
10
20
50
100
Nystrom Method
Random Fourier Features
60
Nystrom Method
Random Fourier Features
80
40
1.5
10
20
200
# random samples
55
500
20
50
100
# random samples
100
200
500
1000
Nystrom Method
Random Fourier Features
70
65
60
Nystrom Method
Random Fourier Features
10
50
# random samples
FOREST
COVTYPE
accuracy(%)
accuracy(%)
90
80
2
# random samples
COD_RNA
100
2.5
0
50 100 200 500 1000
90
Nystrom Method
Random Fourier Features
accuracy(%)
mean square error
mean square error
3
Nystrom Method
Random Fourier Features
2
0
ADULT
CENSUS
2.5
55
200
10
20
50
100
# random samples
200
Figure 2: Comparison of the Nymstr?om method and random Fourier features. For regression tasks,
the mean square error (with std.) is reported, and for classification tasks, accuracy (with std.) is
reported.
CPU
0
CENSUS
0
10
ADULT
0
10
10
?4
10
?6
10
rank
?8
0.2N
0.4N
0.6N
0.8N
rank
0.2N
0.4N
0.6N
10
N
?2
Eigenvalues/N
?4
10
?6
10
rank
?8
0.2N
0.4N
0.6N
N
0.6N
0.8N
N
0.8N
N
FOREST
0
?2
10
?4
10
?6
10
10
0.4N
10
rank
?8
0.8N
0.2N
COVTYPE
0
rank
?10
0.8N
10
10
?6
10
?8
10
N
?4
10
10
COD?RNA
0
Eigenvalues/N
?6
10
?8
10
10
?4
10
Eigenvalues/N
10
10
?2
10
Eigenvalues/N
Eigenvalues/N
Eigenvalues/N
?2
?2
10
0.2N
0.4N
0.6N
?2
10
?4
10
?6
10
rank
?8
0.8N
N
10
0.2N
0.4N
0.6N
Figure 3: The eigenvalue distributions of kernel matrices. N is the number of examples used to
compute eigenvalues.
dependent basis functions while random Fourier features introduce data independent basis functions.
This difference leads to an improved analysis for kernel learning approaches based on the Nystr?om
method. We show that when there is a large eigengap of kernel matrix, the approximation error
of Nystr?om method can be improved to O(1/m), leading to a significantly better generalization
performance than random Fourier features. We verify the claim by an empirical study.
As implied from our study, it is important to develop data dependent basis functions for large-scale
kernel learning. One direction we plan to explore is to improve random Fourier features by making
the sampling data dependent. This can be achieved by introducing a rejection procedure that rejects
the sample Fourier components when they do not align well with the top eigenfunctions estimated
from the sampled data.
Acknowledgments
This work was partially supported by ONR Award N00014-09-1-0663, NSF IIS-0643494, NSFC
(61073097) and 973 Program (2010CB327903).
8
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9
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3,964 | 4,589 | Repulsive Mixtures
Vinayak Rao
Gatsby Computational Neuroscience Unit
University College London
[email protected]
Francesca Petralia
Department of Statistical Science
Duke University
[email protected]
David B. Dunson
Department of Statistical Science
Duke University
[email protected]
Abstract
Discrete mixtures are used routinely in broad sweeping applications ranging from
unsupervised settings to fully supervised multi-task learning. Indeed, finite mixtures and infinite mixtures, relying on Dirichlet processes and modifications, have
become a standard tool. One important issue that arises in using discrete mixtures is low separation in the components; in particular, different components can
be introduced that are very similar and hence redundant. Such redundancy leads
to too many clusters that are too similar, degrading performance in unsupervised
learning and leading to computational problems and an unnecessarily complex
model in supervised settings. Redundancy can arise in the absence of a penalty on
components placed close together even when a Bayesian approach is used to learn
the number of components. To solve this problem, we propose a novel prior that
generates components from a repulsive process, automatically penalizing redundant components. We characterize this repulsive prior theoretically and propose
a Markov chain Monte Carlo sampling algorithm for posterior computation. The
methods are illustrated using synthetic examples and an iris data set.
Key Words: Bayesian nonparametrics; Dirichlet process; Gaussian mixture model; Model-based
clustering; Repulsive point process; Well separated mixture.
1
Introduction
Discrete mixture models characterize the density of y ? Y ? <m as
f (y) =
k
X
ph ?(y; ?h )
(1)
h=1
where p = (p1 , . . . , pk )T is a vector of probabilities summing to one, and ?(?; ?) is a kernel depending on parameters ? ? ?, which may consist of location and scale parameters. In analyses of
finite mixture models, a common concern is over-fitting in which redundant mixture components
located close together are introduced. Over-fitting can have an adverse impact on predictions and
degrade unsupervised learning. In particular, introducing components located close together can
lead to splitting of well separated clusters into a larger number of closely overlapping clusters. Ideally, the criteria for selecting k in a frequentist analysis and the prior on k and {?h } in a Bayesian
analysis should guard against such over-fitting. However, the impact of the criteria used and prior
chosen can be subtle.
1
Recently, [1] studied the asymptotic behavior of the posterior distribution in over-fitted Bayesian
mixture models having more components than needed. They showed that a carefully chosen prior
will lead to asymptotic emptying of the redundant components. However, several challenging practical issues arise. For their prior and in standard Bayesian practice, one assumes that ?h ? P0
independently a priori. For example, if we consider a finite location-scale mixture of multivariate
Gaussians, one may choose P0 to be multivariate Gaussian-inverse Wishart. However, the behavior
of the posterior can be sensitive to P0 for finite samples, with higher variance P0 favoring allocation
to fewer clusters. In addition, drawing the component-specific parameters from a common prior
tends to favor components located close together unless the variance is high.
Sensitivity to P0 is just one of the issues. For finite samples, the weight assigned to redundant
components is often substantial. This can be attributed to non- or weak identifiability. Each mixture
component can potentially be split into multiple components having the same parameters. Even
if exact equivalence is ruled out, it can be difficult to distinguish between models having different
degrees of splitting of well-separated components into components located close together. This
issue can lead to an unnecessarily complex model, and creates difficulties in estimating the number
of components and component-specific parameters. Existing strategies, such as the incorporation
of order constraints, do not adequately address this issue, since it is difficult to choose reasonable
constraints in multivariate problems and even with constraints, the components can be close together.
The problem of separating components has been studied for Gaussian mixture models ([2]; [3]).
Two Gaussians can be separated by placing an arbitrarily chosen lower bound on the distance between their means. Separated Gaussians have been mainly utilized to speed up convergence of the
Expectation-Maximization (EM) algorithm. In choosing a minimal separation level, it is not clear
how to obtain a good compromise between values that are too low to solve the problem and ones
that are so large that one obtains a poor fit. To avoid such arbitrary hard separation thresholds, we
instead propose a repulsive prior that smoothly pushes components apart.
In contrast to the vast majority of the recent Bayesian literature on discrete mixture models, instead
of drawing the component-specific parameters {?h } independently from a common prior P0 , we
propose a joint prior for {?1 , . . . , ?k } that is chosen to assign low density to ?h s located close
together. The deviation from independence is specified a priori by a pair of repulsion parameters.
The proposed class of repulsive mixture models will only place components close together if it
results in a substantial gain in model fit. As we illustrate, the prior will favor a more parsimonious
representation of densities, while improving practical performance in unsupervised learning. We
provide strong theoretical results on rates of posterior convergence and develop Markov chain Monte
Carlo algorithms for posterior computation.
2
Bayesian repulsive mixture models
2.1
Background on Bayesian mixture modeling
Considering the finite mixture model in expression (1), a Bayesian specification is completed by
choosing priors for the number of components k, the probability weights p, and the componentspecific parameters ? = (?1 , . . . , ?k )T . Typically, k is assigned a Poisson or multinomial prior, p a
Dirichlet(?) prior with ? = (?1 , . . . , ?k )T , and ?h ? P0 independently, with P0 often chosen to
be conjugate to the kernel ?. Posterior computation can proceed via a reversible jump Markov chain
Monte Carlo algorithm involving moves for adding or deleting mixture components. Unfortunately,
in making a k ? k + 1 change in model dimension, efficient moves critically depend on the choice
of proposal density. [4] proposed an alternate Markov chain Monte Carlo method, which treats the
parameters as a marked point process, but does not have clear computational advantages relative to
reversible jump.
It has become popular to use over-fitted mixture models in which k is chosen as a conservative
upper bound on the number of components under the expectation that only relatively few of the
components will be occupied by subjects in the sample. From a practical perspective, the success of
over-fitted mixture models has been largely due to ease in computation.
As motivated in [5], simply letting ?h = c/k for h = 1, . . . , k and a constant c > 0 leads to an
approximation to a Dirichlet process mixture model for the density of y, which is obtained in the
2
limit as k approaches infinity. An alternative finite approximation to a Dirichlet process mixture is
obtained by truncating the stick-breaking representation of [6], leading to a similarly simple Gibbs
sampling algorithm [7]. These approaches are now used routinely in practice.
2.2
Repulsive densities
We seek a prior on the component parameters in (1) that automatically favors spread out components near the support of the data. Instead of generating the atoms ?h independently from P0 , one
could generate them from a repulsive process that automatically pushes the atoms apart. This idea
is conceptually related to the literature on repulsive point processes [8]. In the spatial statistics literature, a variety of repulsive processes have been proposed. One such model assumes that points are
clustered spatially, with the cluster centers having a Strauss density [9], that is p(k, ?) ? ? k ?r(?)
where k is the number of clusters, ? > 0, 0 < ? ? 1 and r(?) is the number of pairwise centers that
lie within a pre-specified distance r of each other. A possibly unappealing feature is that repulsion
is not directly dependent on the pairwise distances between the clusters. We propose an alternative
class of priors, which smoothly push apart components based on pairwise distances.
Definition 1. A density h(?) is repulsive if for any ? > 0 there is a corresponding > 0 such that
h(?) < ? for all ? ? ? \ G , where G = {? : d(?s , ?i ) > ; s = 1, . . . , k; i < s} and d is a metric.
Depending on the specification of the metric d(?s , ?j ), a prior satisfying definition 1 may limit overfitting or favor well separated clusters. When d(?s , ?j ) is the distance between sub-vectors of ?s and
?j corresponding to only locations the proposed prior favors well separated clusters. Instead, when
d(?s , ?j ) is the distance between the sth and jth kernel, a prior satisfying definition 1 limits overfitting in density estimation. Though both cases can be implemented, in this paper we will focus
exclusively on the clustering problem. As a convenient class of repulsive priors which smoothly
push components apart, we propose
?(?) = c1
k
Y
!
g0 (?h ) h(?),
(2)
h=1
with c1 being the normalizing constant that depends on the number of components k. The proposed
prior is related to a class of point processes from the statistical physics and spatial statistics
literature referred to as Gibbs processes [10]. We assume g0 : ? ? <+ and h : ?k ? [0, ?) are
continuous with respect to Lesbesgue measure, and h is bounded above by a positive constant c2
and is repulsive according to definition 1. It follows that density ? defined in (2) is also repulsive.
A special hardcore repulsion is produced if the repulsion function is zero when at least one pairwise
distance is smaller than a pre-specified threshold. Such a density implies choosing a minimal
separation level between the atoms. As mentioned in the introduction, we avoid such arbitrary
hard separation thresholds by considering repulsive priors that smoothly push components apart. In
particular, we propose
Ytwo repulsion functions defined as
h(?) =
g{d(?s , ?j )}
(3)
h(?) = min g{d(?s , ?j )}
(4)
{(s,j)?A}
{(s,j)?A}
with A = {(s, j) : s = 1, . . . , k; j < s} and g : <+ ? [0, M ] a strictly monotone differentiable
function with g(0) = 0, g(x) > 0 for all x > 0 and M < ?. It is straightforward to show that h
in (3) and (4) is integrable and satisfies definition 1. The two alternative repulsion functions differ
in their dependence on the relative distances between components, with all the pairwise distances
playing a role in (3), while (4) only depends on the minimal separation. A flexible choice of g
corresponds to
g{d(?s , ?j )} = exp ? ? {d(?s , ?j )}?? ,
(5)
where ? > 0 is a scale parameter and ? is a positive integer controlling the rate at which g approaches
zero as d(?s , ?j ) decreases. Figure 1 shows contour plots of the prior ?(?1 , ?2 ) defined as (2) with
g0 being the standard normal density, the repulsive function defined as (3) or (4) and g defined as
(5) for different values of (?, ?). As ? and ? increase, the prior increasingly favors well separated
components.
3
(I)
(II)
5
5
0
0
?5
?5
?5
0
5
?5
(III)
0
5
(IV)
5
5
0
0
?5
?5
?5
0
5
?5
0
5
Figure 1: Contour plots of the repulsive prior ?(?1 , ?2 ) under (3), either (4) or (5) and (6) with
hyperparameters (?, ?) equal to (I)(1, 2), (II)(1, 4), (III)(5, 2) and (IV )(5, 4)
2.3
Theoretical properties
Pk0
Let the true density f0 : <m ? <+ be defined as f0 = h=1
p0h ?(?0h ) with ?0h ? ? and ?0j s
such that there exists an 1 > 0 such that min{(s,j):s<j} d(?0s , ?0j ) ? 1 with d being the Euclidean
Pk
distance. Let f = h=1 ph ?(?h ) with ?h ? ?. Let ? ? ? with ? = (?1 , . . . , ?k )T and ? satisfying
definition 1. Let p ? ? with ? = Dirichlet(?) and k ? ? with ?(k = k0 ) > 0. Let ? = (p, ?).
These assumptions on f0 and f will be referred to as condition B0. Let ? be the prior induced on
??
j=1 Fk , where Fk is the space of all distributions defined as (1).
We will focus on ? being a location parameter, though the results
can be extended to location-scale
R
kernels. Let | ? |1 denote the L1 norm and KL(f0 , f ) = f0 log(f0 /f ) refer to the KullbackLeibler (K-L) divergence between f0 and f . Density f0 belongs to the K-L support of the prior ?
if ?{f : KL(f0 , f ) < } > 0 for all > 0. The next lemma provides sufficient conditions under
which the true density is in the K-L support of the prior.
Lemma 1. Assume condition B0 is satisfied with m = 1. Let D0 be a compact set containing
parameters (?01 , . . . , ?0k0 ). Suppose ? ? ? with ? satisfying definition 1. Let ? and ? satisfy the
following conditions:
A1. for any y ? Y, the map ? ? ?(y; ?) is uniformly continuous
A2. for any y ? Y, ?(y; ?) is bounded above by a constant
R
A3. f0 log sup??D0 ?(?) ? log {inf ??D0 ?(?)} < ?
A4. ? is continuous with respect to Lebesgue measure and for any vector x ? ?k with
min{(s,j):s<j} d(xs , xj ) ? ? for some ? > 0 there is a ? > 0 such that ?(?) > 0 for all ?
satisfying ||? ? x||1 < ?
Then f0 is in the K-L support of the prior ?.
Lemma 2. The repulsive density in (2) with h defined as either (3) or (4) satisfies condition A4 in
lemma 1.
The next lemma formalizes the posterior rate of concentration for univariate location mixtures of
Gaussians.
Lemma 3. Let condition B0 be satisfied, let m = 1 and ? be the normal kernel depending on a
location parameter ? and a scale parameter ?. Assume that condition (i), (ii) and (iii) of theorem
3.1 in [11] and assumption A4 in lemma 1 are satisfied. Furthermore, assume that
C1) the joint density ? leads to exchangeable random variables and
for all k the marginal density
of the location parameter ?1 satisfies ?m (|?1 | ? t) . exp ?q1 t2 for a given q1 > 0
4
C2) there are constants u1 , u2 , u3 > 0, possibly depending on f0 , such that for any ? u3
?(||? ? ?0 ||1 ? ) ? u1 exp(?u2 k0 log(1/))
Then the posterior rate of convergence relative to the L1 metric is n = n?1/2 log n.
Lemma 3 is essentially a modification of theorem 3.1 in [11] to the proposed repulsive mixture
model. Lemma 4 gives sufficient conditions for ? to satisfy condition C1 and C2 in lemma 3.
Lemma 4. Let ? be defined as (2) and h be defined as either (3) or (4), then ? satisfies condition
C2 in lemma 3. Furthermore, if for a positive constant n1 the function g0 satisfies g0 (|x| ? t) .
exp(?n1 t2 ), ? satisfies condition C1 in lemma 3.
As motivated above, when the number of mixture components is chosen to be unnecessarily large, it
is appealing for the posterior distribution of the weights of the extra components to be concentrated
near zero. Theorem 1 formalizes the rate of concentration with increasing sample size n. One
of the main assumptions required in theorem 1 is that the posterior rate of convergence relative to
the L1 metric is ?n = n?1/2 (log n)q with q ? 0. We provided the contraction rate, under the
proposed prior specification and univariate Gaussian kernel, in lemma 3. However, theorem 1 is a
more general statement and it applies to multivariate mixture density of any kernel.
Theorem 1. Let assumptions B0 ? B5 be satisfied. Let ? be defined as (2) and h be defined as
either (3) or (4). If ?
? = max(?1 , . . . , ?k ) < m/2 and for positive constants r1 , r2 , r3 the function
g satisfies g(x) ? r1 xr2 for 0 ? x < r3 then
" (
!
)#
k
X
0
?1/2
q(1+s(k0 ,?)/sr2 )
lim lim sup En P
min
p?(i) > M n
(log n)
=0
M ?? n??
{??Sk }
i=k0 +1
with s(k0 , ?) = k0 ? 1 + mk0 + ?
? (k ? k0 ), sr2 = r2 + m/2 ? ?
? and Sk the set of all possible
permutations of {1, . . . , k}.
Assumptions (B1 ? B5) can be found in the supplementary material. Theorem 1 is a modification
of theorem 1 in [1] to the proposed repulsive mixture model. Theorem 1 implies that the posterior
expectation of weights of the extra components is of order O n?1/2 (log n)q(1+s(k0 ,?)/sr2 ) . When
g is defined as (5), parameters r1 and r2 can be chosen such that r1 = ? and r2 = ?.
When the number of components is unknown, with only an upper bound known, the posterior rate
of convergence is equivalent to the parametric rate n?1/2 [12]. In this case, the rate in theorem 1
is n?1/2 under usual priors or the repulsive prior. However, in our experience using usual priors,
the sum of the extra components can be substantial in small to moderate sample sizes, and often
has high variability. As we show in Section 3, for repulsive priors the sum of the extra component
weights is close to zero and has small variance for small as well as large sample sizes. On the
other hand, when an upper bound on the number of components is unknown, the posterior rate of
concentration is n?1/2 (log n)q with q > 0. In this case, according to theorem 1, using the proposed
prior specification the logarithmic factor in theorem 1 of [1] can be improved.
2.4
Parameter calibration and posterior computation
The parameters involved in the repulsion function h are chosen such that a priori, with high probability, the clusters will be adequately separated. Consider the case where ? is a location-scale kernel
with location and scale parameters (?, ?) and is symmetric about ?. Here, it is natural to relate
the separation of two densities to the distance between their location parameters. The following
definition introduces the concept of separation level between two densities.
Definition 2. Let f1 and f2 be two densities having location-scale parameters (?1 , ?1 ) and (?2 , ?2 )
respectively, with ?1 , ?2 ? ? and ?1 , ?2 ? ?. Given a metric t(?, ?), a positive constant c and a
function ? : ? ? ? ? <+ , f1 and f2 are c-separated if
t(?1 , ?2 ) ? c?(?1 , ?2 )1/2
Definition 2 is in the spirit of [2] but generalized to any symmetric location-scale kernel. A mixture
of k densities is c-separated if all pairs of densities are c-separated. The parameters of the repulsion
5
(II)
(I)
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
?10
?5
0
5
0
10
?2
0
1
3
0.8
2
0.6
1
0.4
0
0.2
?1
0
?3
?2
?1
0
2
(IV)
(III)
1
2
?2
?2
3
?1
0
1
2
3
Figure 2: (I) Student?s t density, (II) two-components mixture of poorly (solid) and well separated
(dot-dash) Gaussian densities, referred as (IIa, IIb), (III) mixture of poorly (dot-dash) and well
separated (solid) Gaussian and Pearson densities, referred as (IIIa, IIIb), (IV ) two-components
mixture of two-dimensional non-spherical Gaussians
function, (?, ?), will be chosen such that, for an a priori chosen separation level c, definition 2
is satisfied with high probability. In practice, for a given pair (?, ?), we estimate the probability
of pairwise c-separation empirically by simulating N replicates of (?h , ?h ) for each component
h = 1, . . . , k from the prior. The appropriate values (?, ?) are obtained by starting with small values,
and increasing until the pre-specified pairwise c-separated probability is reached. In practice, only ?
will be calibrated to reach a particular probability value. This is because ? controls the rate at which
the density tends to zero as two components approach but not the separation level across them. In
practice we have found that ? = 2 provides a good default value and we fix ? at this value in all our
applications below.
A possible issue with the proposed repulsive mixture prior is that the full conditionals are nonstandard, complicating posterior computation. To address this, we propose a data augmentation scheme,
introducing auxiliary slice variables to facilitate sampling [13]. This algorithm is straightforward
to implement and is efficient by MCMC standards. Further details can be found in the supplementary material. It will be interesting in future work to develop fast approximations to MCMC for
implementation of repulsive mixture models, such as variational methods for approximating the full
posterior and optimization methods for obtaining a maximum a posteriori estimate. The latter approach would provide an alternative to usual maximum likelihood estimation via the EM algorithm,
which provides a penalty on components located close together.
3
Synthetic examples
Synthetic toy examples were considered to assess the performance of the repulsive prior in density
estimation, classification and emptying the extra components. Figure 2 plots the true densities in the
various synthetic cases that we considered. For each synthetic dataset, repulsive and non-repulsive
mixture models were compared considering a fixed upper bound on the number of components; extra
components should be assigned small probabilities and hence effectively excluded. The auxiliary
variable sampler was run for 10, 000 iterations with a burn-in of 5, 000. The chain was thinned by
keeping every 10th simulated draw. To overcome the label switching problem, the samples were
post-processed following the algorithm of [14]. Details on parameters involved in the true densities
and choice of prior distributions can be found in the supplementary material.
Table 1 shows summary statistics of the K-L divergence, the misclassification error and the sum of
extra weights under repulsive and non-repulsive mixtures with six mixture components as the upper
bound. Table 1 shows also the misclassification error resulting from hierarchical clustering [15]. In
practice, observations drawn from the same mixture component were considered as belonging to the
same category and for each dataset a similarity matrix was constructed. The misclassification error
was established in terms of divergence between the true similarity matrix and the posterior similar6
ity matrix. As shown in table 1, the K-L divergences under repulsive and non-repulsive mixtures
become more similar as the sample size increases. For smaller sample sizes, the results are more
similar when components are very well separated. Since a repulsive prior tends to discourage overlapping mixture components, a repulsive model might not estimate the density quite as accurately
when a mixture of closely overlapping components is needed. However, as the sample size increases,
the fitted density approaches the true density regardless of the degree of closeness among clusters.
Again, though repulsive and non-repulsive mixtures perform similarly in estimating the true density,
repulsive mixtures place considerably less probability on extra components leading to more interpretable clusters. In terms of misclassification error, the repulsive model outperforms the other two
approaches while, in most cases, the worst performance was obtained by the non-repulsive model.
Potentially, one may favor fewer clusters, and hence possibly better separated clusters, by penalizing
the introduction of new clusters more through modifying the precision in the Dirichlet prior for the
weights; in the supplemental materials, we demonstrate that this cannot solve the problem.
Table 1: Mean and standard deviation of K-L divergence, misclassification error and sum of extra
weights resulting from non-repulsive (N-R) and repulsive (R) mixtures with a maximum number of
clusters equal to six under different synthetic data scenarios
n=100
I
IIb
IIIa
IIIb
IV
I
IIa
IIb
IIIa
IIIb
IV
K-L divergence
N-R 0?05 0?03
0?07
0?05
0?08
0?22
0?00
0?01
0?01
0?00
0?01
0?02
0?03
0?01
0?02
0?02
0?03
0?04
0?00
0?00
0?00
0?00
0?00
0?00
0?03
0?08
0?09
0?07
0?09
0?24
0?01
0?01
0?01
0?01
0?01
0?03
0?02
0?02
0?03
0?03
0?03
0?04
0?00
0?00
0?00
0?00
0?00
0?00
Misclassification
HCT 0?12 0?11
N-R 0?68 0?26
0?41
0?06
0?12
0?17
0?78
0?05
0?21
0?13
0?45
0?65
0?42
0?24
0?14
0?03
0?42
0?14
0?09
0?02
0?20
0?19
R
0?09
0?10
0?05
0?09
0?06
0?05
0?11
0?08
0?04
0?08
0?03
0?02
0?06
0?09
0?00
0?05
0?00
0?09
0?05
0?08
0?00
0?03
0?00
0?18
0?05
0?04
0?02
0?03
0?01
0?03
0?05
0?02
0?02
0?03
0?01
0?01
Sum of extra weights
N-R 0?30 0?21 0?09
0?29
R
R
4
n=1000
IIa
0?16
0?07
0?13
0?30
0?21
0?03
0?16
0?03
0?10
0?11
0?07
0?09
0?07
0?07
0?11
0?11
0?04
0?10
0?03
0?03
0?01
0?01
0?01
0?01
0?01
0?08
0?01
0?00
0?00
0?00
0?00
0?26
0?01
0?01
0?01
0?01
0?01
0?05
0?01
0?00
0?00
0?00
0?00
0?03
Real data
We assessed the clustering performance of the proposed method on a real dataset. This dataset
consists of 150 observations from three different species of iris each with four measurements. This
dataset was previously analyzed by [16] and [17] proposing new methods to estimate the number of
clusters based on minimizing loss functions. They concluded the optimal number of clusters was
two. This result did not agree with the number of species due to low separation in the data between
two of the species. Such point estimates of the number of clusters do not provide a characterization
of uncertainty in clustering in contrast to Bayesian approaches.
Repulsive and non-repulsive mixtures were fitted under different choices of upper bound on the
number of components. Since the data contains three true biological clusters, with two of these
having similar distributions of the available features, we would expect the posterior to concentrate on two or three components. Posterior means and standard deviations of the three highest
weights were (0?30, 0?23, 0?13) and (0?05, 0?04, 0?04) for non-repulsive and (0?60, 0?30, 0?04) and
(0?04, 0?03, 0?02) for repulsive under six components. Clearly, repulsive priors lead to a posterior
more concentrated on two components, and assign low probability to more than three components.
7
25
20
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 3: Posterior density of the total probability weight assigned to more than three components
in the Iris data under a max of 6 or 10 components for non-repulsive (6:solid, 10:dash-dot) and
repulsive (6:dash, 10:dot) mixtures.
Figure 3 shows the density of the total probability assigned to the extra components. This quantity
was computed considering the number of species as the true number of clusters. According to
figure 3, our repulsive prior specification leads to extra component weights very close to zero
regardless of the upper bound on the number of components. The posterior uncertainty is also
small. Non-repulsive mixtures assign large weight to extra components, with posterior uncertainty
increasing considerably as the number of components increases.
Discussions
We have proposed a new repulsive mixture modeling framework, which should lead to substantially
improved unsupervised learning (clustering) performance in general applications. A key aspect is
soft penalization of components located close together to favor, without sharply enforcing, well separated clusters that should be more likely to correspond to the true missing labels. We have focused
on Bayesian MCMC-based methods, but there are numerous interesting directions for ongoing research, including fast optimization-based approaches for learning mixture models with repulsive
penalties.
Acknowledgments
This research was partially supported by grant 5R01-ES-017436-04 from the National Institute of
Environmental Health Sciences (NIEHS) of the National Institutes of Health (NIH) and DARPA
MSEE.
8
References
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[2] S. Dasgupta. Learning Mixtures of Gaussians. Proceedings of the 40th Annual Symposium on Foundations
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[3] S. Dasgupta and L. Schulman. A Probabilistic Analysis of EM for Mixtures of Separated, Spherical
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[4] M. Stephens. Bayesian Analysis of Mixture Models with an Unknown Number of Components - An
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[5] H. Ishwaran and M. Zarepour. Dirichlet Prior Sieves in Finite Normal Mixtures. Statistica Sinica, 12:941?
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[8] M. L. Huber and R. L. Wolpert. Likelihood-Based Inference for Matern Type-III Repulsive Point Processes. Advances in Applied Probability, 41:958?977, 2009.
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[10] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer, 2008.
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[12] H. Ishwaran, L. F. James, and J. Sun. Bayesian Model Selection in Finite Mixtures by Marginal Density
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[13] Paul Damien, Jon Wakefield, and Stephen Walker. Gibbs Sampling for Bayesian Non-Conjugate and
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[14] M. Stephens. Dealing with label switching in mixture models. Journal of the Roya; statistical society B,
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[15] H. Locarek-Junge and C. Weihs. Classification as a Tool for Research. Springer, 2009.
[16] C. Sugar and G. James. Finding the number of clusters in a data set: an information theoretic approach.
Journal of the American Statistical Association, 98:750?763, 2003.
[17] J. Wang. Consistent selection of the number of clusters via crossvalidation. Biometrika, 97:893?904,
2010.
9
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3,965 | 459 | Kernel Regression and
Backpropagation Training with Noise
Petri Koistinen and Lasse Holmstrom
Rolf Nevanlinna Institute, University of Helsinki
Teollisuuskatu 23, SF-0051O Helsinki, Finland
Abstract
One method proposed for improving the generalization capability of a feedforward network trained with the backpropagation algorithm is to use
artificial training vectors which are obtained by adding noise to the original training vectors. We discuss the connection of such backpropagation
training with noise to kernel density and kernel regression estimation. We
compare by simulated examples (1) backpropagation, (2) backpropagation
with noise, and (3) kernel regression in mapping estimation and pattern
classification contexts.
1
INTRODUCTION
Let X and Y be random vectors taking values in R d and RP, respectively. Suppose
that we want to estimate Y in terms of X using a feedforward network whose
input-output mapping we denote by y g(x, w). Here the vector w includes all the
weights and biases of the network. Backpropagation training using the quadratic
loss (or error) function can be interpreted as an attempt to minimize the expected
loss
'\(w) ElIg(X, w) _ Y1I2.
(1)
Suppose that EIIYW < 00. Then the regression function
=
=
(2)
m(x) = E[YIX = x].
minimizes the loss Ellb(X) - YI1 2 over all Borel measurable mappings b. Therefore,
backpropagation training can also be viewed as an attempt to estimate m with the
network g.
1033
1034
Koistinen and Holmstrom
In practice, one cannot minimize -' directly because one does not know enough
about the distribution of (X, Y). Instead one minimizes a sample estimate
(3)
in the hope that weight vectors w that are near optimal for ~n are also near optimal
for -'. In fact, under rather mild conditions the minimizer of ~n actually converges
towards the minimizing set of weights for -' as n -+ 00, with probability one (White,
1989). However, if n is small compared to the dimension of w, minimization of ~n
can easily lead to overfitting and poor generalization, i.e., weights that render ~n
small may produce a large expected error -'.
Many cures for overfitting have been suggested. One can divide the available samples into a training set and a validation set, perform iterative minimization using
the training set and stop minimization when network performance over the validation set begins to deteriorate (Holmstrom et al., 1990, Weigend et al., 1990). In
another approach, the minimization objective function is modified to include a term
which tries to discourage the network from becoming too complex (Weigend et al.,
1990). Network pruning (see, e.g., Sietsma and Dow, 1991) has similar motivation.
Here we consider the approach of generating artificial training vectors by adding
noise to the original samples. We have recently analyzed such an approach and
proved its asymptotic consistency under certain technical conditions (Holmstrom
and Koistinen, 1990).
2
ADDITIVE NOISE AND KERNEL REGRESSION
Suppose that we have n original training vectors (Xi, Yi) and want to generate
artificial training vectors using additive noise. If the distributions of both X and Y
are continuous it is natural to add noise to both X and Y components of the sample.
However, if the distribution of X is continuous and that of Y is discrete (e.g., in
pattern classification), it feels more natural to add noiRe to the X components only.
In Figure 1 we present sampling procedures for both ca~es. In the x-only case the
additive noise is generated from a random vector Sx with density Kx whereas in the
x-and-y case the noise is generated from a random vector SXy with density Kxy.
Notice that we control the magnitude of noise with a scalar smoothing parameter
h > O.
In both cases the sampling procedures can be thought of as generating random
samples from new random vectors Xk n ) and y~n) . Using the same argument as
in the Introduction we see that a network trained with the artificial samples tends
to approximate the regression function E[y~n) IXkn)]. Generate I uniformly on
{1, ... , n} and denote by I and I( .11 = i) the density and conditional density of
Xk n ). Then in the x-only case we get
n
m~n)(Xkn)) := E[y~n)IXkn)] = LYiP(I
i=l
= iIXkn))
Kernel Regression and Backpropagation Training with Noise
Procedure 2.
(Add noise to both x and y)
Procedure 1.
(Add noise to x only)
1. Select i E {I, ... , n} with equal
probability for each index.
2. Draw a sample (sx, Sy) from
density /{Xy on Rd+p.
1. Select i E {I, ... , n} with equal
probability for each index.
2. Draw a sample Sx from density
I<x on Rd.
x~n)
3. Set
(n)
Yh
Xi
+ hsx
3. Set
x~n)
+ hsx
Yi + h Sy.
Xi
Yh(n)
Yi?
Figure 1: Two Procedures for Generating Artificial Training Vectors.
tt
f(X~n)II = i)P(I
n
=
Denoting
Yi
/{x
= i)
=
f(Xh n ))
tt
n
h-d/{x?Xkn ) - xi)/h). n- 1
Yi 2:7=1 n- 1 h- d/{x?Xkn ) - xi)/h)?
by k we obtain
(n)( ) _
mh
X
-
2:~=1 k?x - xi)/h)Yi
",0
.
L..,j=1 k?x - xi)/h)
(4)
We result in the same expression also in the x-and-y case provided that
fY/{Xy(x,y)dy
0 and that we take k(x)
fI<Xy(x,y)dy (Watson, 1964).
The expression (4) is known as the (N adaraya-Watson) kernel regression estimator
(Nadaraya, 1964, Watson, 1964, Devroye and Wagner, 1980).
=
=
A common way to train a p-class neural network classifier is to train the network
to associate a vector x from class j with the j'th unit vector (0, ... ,0,1,0, ... ,0).
It is easy to see that then the kernel regression estimator components estimate the
class a posteriori probabilities using (Parzen-Rosenblatt) kernel density estimators
for the class conditional densities. Specht (1990) argues that such a classifier can
be considered a neural network. Analogously, a kernel regression estimator can be
considered a neural network though such a network would need units proportional
to the number of training samples. Recently Specht (1991) has advocated using
kernel regression and has also presented a clustering variant requiring only a fixed
amount of units. Notice also the resemblance of kernel regression to certain radial
basis function schemes (Moody and Darken, 1989, Stokbro et al., 1990).
An often used method for choosing h is to minimize the cross-validated error (HardIe
and Marron, 1985, Friedman and Silverman, 1989)
(5)
Another possibility is to use a method suggested by kernel density estimation theory
(Duin, 1976, Habbema et al., 1974) whereby one chooses that h maximizing a crossvalidated (pseudo) likelihood function
o
Lxy (h)
= II f:,L(Xi, yd,
i=1
o
Lx(h)
= IT f:'h,i(Xi),
i=1
(6)
1035
1036
Koistinen and Holmstrom
I;r
where
i (I; h i) is a kernel density estimate with kernel Kxy (Kx) and smoothing para~~ter h hut with the i'th sample point left out.
3
EXPERIMENTS
In the first experiment we try to estimate a mapping go from noisy data (x, y),
go(X)+Ny =asinX+b+Ny ,
UNI( -71",71"),
Ny '" N(O, (72),
Y
a=0.4,b=0.5
=
X '"
(7
O.l.
Here UNI and N denote the uniform and the normal distribution. We experimented with backpropagation, backpropagation with noise and kernel regression.
Backpropagation loss function was minimized using Marquardt's method. The network architecture was FN-1-13-1 with 40 adaptable weights (a feedforward network
with one input, 13 hidden nodes, one output, and logistic activation functions in
the hidden and output layers). We started the local optimizations from 3 different
random initial weights and kept the weights giving the least value for ~n. Backpropagation training with noise was similar except that instead of the original n vectors
we used IOn artificial vectors generated with Procedure 2 using SXy '" N(O, 12 ).
Magnitude of noise was chosen with the criterion Lxy (which, for backpropagation,
gave better results than M). In the kernel regression experiments SXy was kept the
same. Table 1 characterizes the distribution of J, the expected squared distance of
the estimator 9 (g(., w) or m~n) from go,
J = E[g(X) - gO(X)]2.
Table 2 characterizes the distribution of h chosen according to the criteria Lxy and
M and Figure 2 shows the estimators in one instance. Notice that, on the average,
kernel regression is better than backpropagation with noise which is better than
plain backpropagation. The success of backpropagation with noise is partly due to
the fact that (7 and n have here been picked favorably. Notice too that in kernel
regression the results with the two cross-validation methods are similar although
the h values they suggest are clearly different .
In the second experiment we trained classifiers for a four-dimensional two-class
problem with equal a priori probabilities and class-conditional densities N(J.ll, C 1 )
and N(J.l2' C2),
J.ll
= 2.32[1 0 0 O]T, C = 14;
1
J.l2
= 0, C2 = 414.
An FN-4-6-2 with 44 adaptable weights was trained to associate vectors from class 1
with [0.9 O.l]T and vectors from class 2 with [0.1 0.9jT. We generated n/2 original
vectors from each class and a total of IOn artificial vectors using Procedure 1 with
Sx '" N(O, 14). We chose the smoothing parameters, hI and h2' separately for
the two classes using the criterion Lx: hi was chosen by evaluating Lx on class
i samples only. We formed separate kernel regression estimators for each class;
the i'th estimator was trained to output 1 for class i vectors and 0 for the other
sample vectors. The M criterion then produces equal values for hI and h2. The
classification rule was to classify x to class i if the output corresponding to the
i'th class was the maximum output. The error rates are given in Table 3. (The
error rate of the Bayesian classifier is 0.116 in this task.) Table 4 summarizes the
distribution of hI and h2 as selected by Lx and M .
Kernel Regression and Backpropagation Training with Noise
Table 1: Results for Mapping Estimation. Mean value (left) and standard deviation
(right) of J based on 100 repetitions are given for each method.
BP
BP+noise,
Lxy
n
40
80
.0218
.00764
.016
.0048
.0104
.00526
.0079
.0018
Kernel regression
M
Lxy
.00446 .0022
.00365 .0019
.00250 .00078 .00191 .00077
Table 2: Values of h Suggested by the Two Cross-validation Methods in the Mapping Estimation Experiment. Mean value and standard deviation based on 100
repetitions are given.
Lxy
n
40
80
0.149
0.114
M
0.020
0.011
0.276
0.241
0.086
0.062
Table 3: Error Rates for the Different Classifiers. Mean value and standard deviation based on 25 repetitions are given for each method.
BP+noise,
BP
Lx
n
44
88
176
Kernel regression
.281
.264
.210
.054
.028
.023
.189
.163
.145
Lx
.018
.011
.0lD
.201
.182
.164
M
.022
.010
.0089
.207
.184
.164
.027
.013
.011
Table 4: Values of hl and h2 Suggested by the Two Cross-validation Methods in
the Classification Experiment. Mean value and standard deviation based on 25
repetitions are given.
Lx
n
44
88
176
M
hl
.818
.738
.668
h2
.078
.056
.048
1.61
1.48
1.35
.14
.11
.090
hl = h2
1.14 .27
1.01 .19
.868 .lD
1037
1038
Koistinen and Holmstrom
4
CONCLUSIONS
Additive noise can improve the generalization capability of a feedforward network
trained with the backpropagation approach. The magnitude of the noise cannot
be selected blindly, though. Cross-validation-type procedures seem to suit well for
the selection of noise magnitude. Kernel regression, however, seems to perform well
whenever backpropagation with noise performs well. If the kernel is fixed in kernel
regression, we only have to choose the smoothing parameter h, and the method is
not overly sensitive to its selection.
References
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Bruckmann, G., editor, COMPSTAT 1974, pages 101-110, Wien. Physica Verlag.
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selection in nonparametric regression function estimation. The Annals of Statistics, 13(4):1465-148l.
[Holmstrom and Koistinen, 1990] Holmstrom, 1. and Koistinen, P. (1990). Using
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[Holmstrom et al., 1990] Holmstrom, L., Koistinen, P., and Ilmoniemi, R. J. (1990).
Classification of un averaged evoked cortical magnetic fields. In Proc. IJCNN-90WASH DC, pages II: 359-362. Lawrence Erlbaum Associates.
[Moody and Darken, 1989] Moody, J. and Darken, C. (1989). Fast learning in networks of locally-tuned processing units. Neural Computation, 1:281-294.
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neural networks that generalize. Neural Networks, 4:67-79.
[Specht, 1991] Specht, D. (1991). A general regression neural network. IEEE Transactions on Neural Networks, 2(6):568-576.
[Specht, 1990] Specht, D. F. (1990). Probabilistic neural networks. Neural Networks, 3(1):109-118.
[Stokbro et al., 1990] Stokbro, K., Umberger, D., and Hertz, J. (1990). Exploiting
neurons with localized receptive fields to learn chaos. NORDITA preprint.
[Watson, 1964] Watson, G. (1964). Smooth regression analysis. Sankhyii Ser. A,
26:359-372.
[Weigend et al., 1990] Weigend, A., Huberman, B., and Rumelhart, D. (1990). Predicting the future: A connectionist approach. International Journal of Neural
Systems, 1(3):193-209.
Kernel Regression and Backpropagation Training with Noise
[White, 1989] White, H. (1989). Learning in artificial neural networks: A statistical
perspective. Neural Computation, 1:425-464.
1.5.....--.....----.---.....--....----.---.....--....----,
o
1
0
0.5
0
0
-0.54
--- true
-3
kernel
0
-2
3
2
4
1.5
?
-
..
':
..
? 0
1
0.5
0
..
.
..
-0.54
-3
-2
-
..
'
-I
0
1
----
BP
-
BP+noise
2
3
4
Figure 2: Results From a Mapping Estimation Experiment. Shown are the n = 40
original vectors (o's), the artificial vectors (dots), the true function asinx + band
the fitting results using kernel regression, backpropagation and backpropagation
with noise. Here h = 0.16 was chosen with Lxy. Values of J are 0.0075 (kernel
regression), 0.014 (backpropagation with noise) and 0.038 (backpropagation) .
1039
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3,966 | 4,590 | Kernel Latent SVM for Visual Recognition
Yang Wang
Department of Computer Science
University of Manitoba
[email protected]
Weilong Yang
School of Computing Science
Simon Fraser University
[email protected]
Greg Mori
School of Computing Science
Simon Fraser University
[email protected]
Arash Vahdat
School of Computing Science
Simon Fraser University
[email protected]
Abstract
Latent SVMs (LSVMs) are a class of powerful tools that have been successfully
applied to many applications in computer vision. However, a limitation of LSVMs
is that they rely on linear models. For many computer vision tasks, linear models are suboptimal and nonlinear models learned with kernels typically perform
much better. Therefore it is desirable to develop the kernel version of LSVM. In
this paper, we propose kernel latent SVM (KLSVM) ? a new learning framework
that combines latent SVMs and kernel methods. We develop an iterative training algorithm to learn the model parameters. We demonstrate the effectiveness of
KLSVM using three different applications in visual recognition. Our KLSVM formulation is very general and can be applied to solve a wide range of applications
in computer vision and machine learning.
1
Introduction
We consider the problem of learning discriminative classification models for visual recognition. In
particular, we are interested in models that have the following two characteristics: 1) can be used on
weakly labeled data; 2) have nonlinear decision boundaries.
Linear classifiers are a class of popular learning methods in computer vision. In the case of binary
classification, they are prediction models in the form of f (x) = w> x, where x is the feature vector,
and w is a vector of model parameters1 . The classification decision is based on the value of f (x).
Linear classifiers are amenable to efficient and scalable learning/inference ? an important factor in
many computer vision applications that involve high dimension features and large datasets. The
person detection algorithm in [2] is an example of the success of linear classifiers in computer
vision. The detector is trained by learning a linear support vector machine based on HOG descriptors
of positive and negative examples. The model parameter w in this detector can be thought as a
statistical template for HOG descriptors of persons.
The reliance on a rigid template w is a major limitation of linear classifiers. As a result, the learned
models usually cannot effectively capture all the variations (shape, appearance, pose, etc.) in natural
images. For example, the detector in [2] usually only works well when a person is in an upright
posture.
In the literature, there are two main approaches for addressing this limitation. The first one is to
introduce latent variables into the linear model. In computer vision, this is best exemplified by the
success of deformable part models (DPM) [5] for object detection. DPM captures shape and pose
variations of an object class with a root template covering the whole object and several part templates. By allowing these parts to deform from their ideal locations with respect to the root template,
DPM provides more flexibility than a rigid template. Learning a DPM involves solving a latent
1
Without loss of generality, we assume linear models without the bias term.
1
SVM (LSVM) [5, 17] ? an extension of regular linear SVM for handling latent variables. LSVM
provides a general framework for handling ?weakly labeled data? arising in many applications. For
example, in object detection, the training data are weakly labeled because we are only given the
bounding boxes of the objects without the detailed annotation for each part. In addition to modeling
part deformation, another popular application of LSVM is to use it as a mixture model where the
mixture component is represented as a latent variable [5, 6, 16].
The other main approach is to directly learn a nonlinear classifier. The kernel method [1] is a
representative example along this line of work. A limitation of kernel methods is that the learning is
more expensive than linear classifiers on large datasets, although efficient algorithms exist for certain
types of kernels (e.g. histogram intersection kernel (HIK) [10]). One possible way to address the
computational issue is to use nonlinear mapping to convert the original feature into some higher
dimensional space, then apply linear classifiers in the high dimensional space [14].
Latent SVM and kernel methods represent two different, yet complementary approaches for learning classification models that are more expressive than linear classifiers. They both have their own
advantages and limitations. The advantage of LSVM is that it provides a general and elegant formulation for dealing with many weakly supervised problems in computer vision. The latent variables
in LSVM can often have some intuitive and semantic meanings. As a result, it is usually easy to
adapt LSVM to capture various prior knowledge about the unobserved variables in various applications. Examples of latent variables in the literature include part locations in object detection [5],
subcategories in video annotation [16], object localization in image classification [8], etc. However,
LSVM is essentially a parametric model. So the capacity of these types of models is limited by the
parametric form. In contrast, kernel methods are non-parametric models. The model complexity is
implicitly determined by the number of support vectors. Since the number of support vectors can
vary depending on the training data, kernel methods can adapt their model complexity to fit the data.
In this paper, we propose kernel latent SVM (KLSVM) ? a new learning framework that combines
latent SVMs and kernel methods. As a result, KLSVM has the benefits of both approaches. On
one hand, the latent variables in KLSVM can be something intuitive and semantically meaningful.
On the other hand, KLSVM is nonparametric in nature, since the decision boundary is defined
implicitly by support vectors. We demonstrate KLSVM on three applications in visual recognition:
1) object classification with latent localization; 2) object classification with latent subcategories; 3)
recognition of object interactions.
2
Preliminaries
In this section, we introduce some background on latent SVM and on the dual form of SVMs used
for deriving kernel SVMs. Our proposed model in Sec. 3 will build upon these two ideas.
Latent SVM: We assume a data instance is in the form of (x, h, y), where x is the observed variable
and y is the class label. Each instance is also associated with a latent variable h that captures some
unobserved information about the data. For example, say we want to learn a ?car? model from a
set of positive images containing cars and a set of negative images without cars. We know there is
a car somewhere in a positive image, but we do not know its exact location. In this case, h can be
used to represent the unobserved location of the car in the image. In this paper, we consider binary
classification for simplicity, i.e. y ? {+1, ?1}. Multi-class classification can be easily converted
to binary classification, e.g. using one-vs-all or one-vs-one strategy. To simplify the notation, we
also assume the latent variable h takes its value from a discrete set of labels h ? H. However, our
formulation is general. We will show how to deal with more complex h in Sec. 3.2 and in one of the
experiments (Sec. 4.3).
In latent SVM, the scoring function of sample x is defined as fw (x) = maxh w> ?(x, h), where
?(x, h) is the feature vector defined for the pair of (x, h). For example, in the ?car model? example,
?(x, h) can be a feature vector extracted from the image patch P
at location h of the image x. The
objective function of LSVM is defined as L(w) = 12 ||w||2 + C i max(0, 1 ? yi fw (xi )). LSVM
is essentially a non-convex optimization problem. However, the learning problem becomes convex
once the latent variable h is fixed for positive examples. Therefore, we can train the LSVM by
an iterative algorithm that alternates between inferring h on positive examples and optimizing the
model parameter w.
Dual form with fixed h on positive examples : Due to its nature of non-convexity, it is not straightforward to derive the dual form for the general LSVM. Therefore, as a starting point, we first consider a simpler scenario assuming h is fixed (or observed) on the positive training examples. As
previously mentioned, the LSVM is then relaxed to a convex problem with this assumption. Note
that we will relax this assumption in Sec. 3. In the above ?car model? example, this means that
we have the ground-truth bounding boxes of the cars in each image. More formally, we are given
2
M +N
M positive samples {xi , hi }M
i=1 , and N negative samples {xj }j=M +1 . Inspired by linear SVMs,
>
our goal is to find a linear discriminant fw (x, h) = w ?(x, h) by solving the following quadratic
program:
X
X
1
P(w? ) = min ||w||2 + C1
?i + C 2
?j,h
(1a)
w,? 2
i
j,h
s.t. w> ?(xi , hi ) ? 1 ? ?i , ?i ? {1, 2, ..., M },
(1b)
>
?w ?(xj , h) ? 1 ? ?j,h ?j ? {M + 1, M + 2, ..., M + N }, ?h ? H
?i ? 0, ?j,h ? 0 ?i, ?j, ?h ? H
(1c)
(1d)
Similar to standard SVMs, {?i } and {?j,h } are the slack variables for handling soft margins.
It is interesting to note that the optimization problem in Eq. 1 is almost identical to that of standard
linear SVMs. The only difference lies in the constraint on the negative training examples (Eq. 1c).
Since we assume h?s are not observed on negative images, we need to enumerate all possible values
for h?s in Eq. 1c. Intuitively, this means every image patch from a negative image (i.e. non-car
image) is not a car.
It is easy to show that Eq. 1 is convex. Similar to the dual form of standard SVMs, we can derive
the dual form of Eq. 1 as follows:
X
XX
XX
1 X
?i +
?j,h ? ||
D(?? , ? ? ) = max
?i ?(xi , hi ) ?
?j,h ?(xj , h)||2 (2a)
?,?
2
i
j
i
j
h
s.t.
h
0 ? ?i ? C1 , ?i; 0 ? ?j,h ? C2 , ?j, ?h ? H
?
(2b)
?
?
The optimal primal parameters w for Eq. 1 and the optimal dual parameters (? , ? ) for Eq. 2 are
related as follows:
X
XX
?
w? =
?i? ?(xi , hi ) ?
?j,h
?(xj , h)
(3)
i
j
h
Let us define ? to be the concatenations of {?i : ?i} and {?j,h : ?j, ?h ? H}, so |?| = M +N ?|H|.
Let ? be a |?| ? D matrix where D is the dimension of ?(x, h). ? is obtained by stacking together
{?(xi , hi ) : ?i} and {??(xj , h) : ?j, ?h ? H}. We also define Q = ??> and 1 to be a vector of
all 1?s. Then Eq. 2a can be rewritten as (we omit the linear constraints on ? for simplicity):
1
max ?> ? 1 ? ?> Q?
?
2
(4)
The advantage of working with the dual form in Eq. 4 is that it only involves a socalled kernel matrix Q.
Each entry of Q is a dot-product of two vectors in the
form of ?(x, h)> ?(x0 , h0 ). We can replace the dot-product with any other kernel functions in the form of k(?(x, h), ?(x0 , h0 )) to get nonlinear classifiers [1].
The scornew
ing function
can be kernelized as follows: f (xnew
P for the testing images x
) =
P P ?
?
new
new
new
new
maxhnew
,h
)) ? j h ?j,h k(?(xj , h), ?(x
,h
)) .
i ?i k(?(xi , hi ), ?(x
Another important, yet often overlooked fact is that the optimal values of the two quadratic programs
in Eqs. 1 and 2 have some specific meanings. They correspond to the inverse of the (soft) margin of
the resultant SVM classifier [9, 15]: P(w? ) = D(?? , ? ? ) = SVM 1margin . In the next section, we will
exploit this fact to develop the kernel latent support vector machines.
3
Kernel Latent SVM
Now we assume the variables {hi }M
i=1 on the positive training examples are unobserved. If the scoring function used for classification is in the form of f (x) = maxh w> ?(x, h), we can use the LSVM
formulation [5, 17] to learn the model parameters w. As mentioned earlier, the limitation of LSVM
is the linearity assumption of w> ?(x, h). In this section, we propose kernel latent SVM (KLSVM)
? a new latent variable learning method that only requires a kernel function K(x, h, x0 , h0 ) between
a pair of (x, h) and (x0 , h0 ).
Note that when {hi }M
i=1 are observed on the positive training examples, we can plug them in Eq. 2
to learn a nonlinear kernelized decision function that separates the positive and negative examples.
3
M
When {hi }M
i=1 are latent, an intuitive thing to do is to find the labeling of {hi }i=1 so that when
we plug them in and solve for Eq. 2, the resultant nonlinear decision function separates the two
classes as widely as possible. In other words, we look for a set of {h?i } which can maximize the
SVM margin (equivalent to minimizing D(?? , ? ? , {hi })). The same intuition was previously used
to develop the max-margin clustering method in [15]. Using this intuition, we write the optimal
function value of the dual form as D(?? , ? ? , {hi }) since now it implicitly depends on the labelings
{hi }. We can jointly find the labelings {hi } and solve for (?? , ? ? ) by the following optimization
problem:
min D(?? , ? ? , {hi })
(5a)
{hi }
= min max
{hi } ?,?
X
?i +
i
XX
j
s.t. 0 ? ?i ? C1 , ?i;
h
XX
1 X
?j,h ? ||
?i ?(xi , hi ) ?
?j,h ?(xj , h)||2 (5b)
2 i
j
h
0 ? ?j,h ? C2 , ?j, ?h ? H
(5c)
The most straightforward way of solving Eq. 5 is to optimize D(?? , ? ? , {hi }) for every possible
combination of values for {hi }, and then take the minimum. When hi takes its value from a discrete set of K possible choices (i.e. |H| = K), this naive approach needs to solve M K quadratic
programs. This is obviously too expensive. Instead, we use the following iterative algorithm:
? Fix ? and ?, compute the optimal {hi }? by
XX
1 X
?i ?(xi , hi ) ?
?j,h ?(xj , h)||2
(6)
{hi }? = arg max ||
2 i
{hi }
j
h
? Fix {hi }, compute the optimal (?? , ? ? ) by
?
?
?X
?
XX
X
X
X
1
?j,h ? ||
(?? , ? ? ) = arg max
?i +
?j,h ?(xj , h)||2
(7)
?i ?(xi , hi ) ?
?
?
2 i
?,?
i
j
j
h
h
The optimization problem in Eq. 7 is a quadratic program similar to that of a standard dual SVM.
As a result, Eq. 7 can be kernelized as Eq. 4 and solved using standard dual solver in regular SVMs.
In Sec. 3.1, we describe how to kernelize and solve the optimization problem in Eq. 6.
3.1
Optimization over {hi }
The complexity of a simple enumeration approach for solving Eq. 6 is again O(M K ), which is
clearly too expensive for practical purposes. Instead, we solve it iteratively using an algorithm
similar to co-ordinate ascent. Within an iteration, we choose one positive training example t. We
update ht while fixing hi for all i 6= t. The optimal h?t can be computed as follows:
h?t = arg max ||?t ?(xt , ht ) +
ht
X
?i ?(xi , hi ) ?
XX
j
i:i6=t
?j,h ?(xj , h)||2
?>
?
? arg max ||?t ?(xt , ht )||2 + 2 ?
ht
(8a)
h
X
?i ?(xi , hi ) ?
XX
j
i:i6=t
?j,h ?(xj , h)? ?t ?(xt , ht ) (8b)
h
By replacing the dot-product ?(x, h)> ?(x0 , h0 ) with a kernel function k(?(x, h), ?(x0 , h0 )), we obtain the kernerlized version of Eq. 8(b) as follows
X
h?t = arg max ?t ?t k(?(xt , ht ), ?(xt , ht )) + 2
?i ?t k(?(xi , hi ), ?(xt , ht ))
ht
i:i6=t
?2
XX
j
?j,h ?t k(?(xj , h), ?(xt , ht ))
(9)
h
It is interesting to notice that if the t-th example is not a support vector (i.e. ?t = 0), the function
value of Eq. 9 will be zero regardless of the value of ht . This means in KLSVM we can improve the
training efficiency by only performing Eq. 9 on positive examples corresponding to support vectors.
For other positive examples (non-support vectors), we can simply set their latent variables the same
4
as the previous iteration. Note that in LSVM, the inference during training needs to be performed
on every positive example.
Connection to LSVM: When a linear kernel is used, the inference problem (Eq. 8) has a very
interesting connection to LSVM in [5]. Recall that for linear kernels, the model parameters w and
dual variables (?, ?) are related by Eq. 3. Then Eq. 8 becomes:
>
?t ?(xt , ht )
(10a)
h?t = arg max ||?t ?(xt , ht )||2 + 2 w ? ?t ?(xt , hold
t )
ht
1
>
? arg max ?t w> ?(xt , ht ) + ?t2 ||?(xt , ht )||2 ? ?t2 ?(xt , hold
t ) ?(xt , ht )
2
ht
(10b)
where hold
t is the value of latent variable of the t-th example in the previous iteration. Let us consider the situation when ?t 6= 0 and the feature vector ?(x, h) is l2 normalized, which is commonly used in computer vision. In this case, ?t2 ?(xt , ht )> ?(xt , ht ) is a constant, and we have
>
old
old >
old
?(xt , hold
t ) ?(xt , ht ) > ?(xt , ht ) ?(xt , ht ) if ht 6= ht . Then Eq. 10 is equivalent to:
>
h?t = arg max w> ?(xt , ht ) ? ?t ?(xt , hold
t ) ?(xt , ht )
(11)
ht
Eq. 11 is very similar to the inference problem in LSVM, i.e., h?t = arg maxht w> ?(xt , ht ), but
>
with an extra term ?t ?(xt , hold
t ) ?(xt , ht ) which penalizes the choice of ht for being the same
old
value as previous iteration ht . This has a very appealing intuitive interpretation. If the t-th positive
example is a support vector, the latent variable hold from previous iteration causes this example to
lie very close to (or even on the wrong side) the decision boundary, i.e. the example is not wellseparated. During the current iteration, the second term in Eq. 11 penalizes hold to be chosen again
since we already know the example will not be well-separated if we choose hold again. The amount
>
of penalty depends on the magnitudes of ?t and ?(xt , hold
t ) ?(xt , ht ). We can interpret ?t as how
?
old >
old
?bad? ht is, and ?(xt , ht ) ?(xt , ht ) as how close ht is to hold
t . Eq. 11 penalizes the new ht to
old
be ?close? to ?bad? ht .
3.2
Composite Kernels
So far we have assumed that the latent variable h takes its value from a discrete set of labels. Given
a pair of (x, h) and (x0 , h0 ), the types of kernel function k(x, h; x0 , h0 ) we can choose from are still
limited to a handful of standard kernels (e.g. Gaussian, RBF, HIK, etc). In this section, we consider
more interesting cases where h involves some complex structures. This will give us two important
benefits. First of all, it allows us to exploit structural information in the latent variables. This is in
analog to structured output learning (e.g. [12, 13]). More importantly, it gives us more flexibility to
construct new kernel functions by composing from simple kernels.
Before we proceed, let us first motivate the composite kernel with an example application. Suppose
we want to detect some complex person-object interaction (e.g. ?person riding a bike?) in an image.
One possible solution is to detect persons and bikes in an image, then combine the results by taking
into account of their relationship (i.e. ?riding?). Imagine we already have kernel functions corresponding to some components (e.g. person, bike) of the interaction. In the following, we will show
how to compose a new kernel for the ?person riding a bike? classifier from those components.
We denote the latent variable using ~h to emphasize that now it is a vector instead of a single discrete
value. We denote it as ~h = (z1 , z2 , ...), where zu is the u-th component of ~h and takes its value
from a discrete set of possible labels. For the structured latent variable, it is assumed that there are
certain dependencies between some pairs of (zu , zv ). We can use an undirected graph G = (V, E) to
capture the structure of the latent variable, where a vertex u ? V corresponds to the label zu , and an
edge (u, v) ? E corresponds to the dependency between zu and zv . As a concrete example, consider
the ?person riding a bike? recognition problem. The latent variable in this case has two components
~h = (zperson , zbike ) corresponding to the location of person and bike, respectively. On the training
data, we have access to the ground-truth bounding box of ?person riding a bike? as a whole, but not
the exact location of ?person? or ?bike? within the bounding box. So ~h is latent in this application.
The edge connecting zperson and zbike captures the relationship (e.g. ?riding on?, ?next to?, etc.)
between these two objects.
Suppose we already have kernel functions corresponding to the vertices and edges in the graph, we
can then define the composite kernel as the summation of the kernels over all the vertices and edges.
5
Figure 1: Visualization of how the latent variable (i.e. object location) changes during the learning. The red
bounding box corresponds to the initial object location. The blue bounding box corresponds to the object
location after the learning.
Method
BOF + linear SVM BOF + kernel SVM linear LSVM
KLSVM
Acc (%)
45.57 ? 4.23
50.53 ? 6.53
75.07 ? 4.18 84.49 ? 3.63
Table 1: Results on the mammal dataset. We show the mean/std of classification accuracies over five rounds of
experiments.
K(?(x, ~h), ?(x0 , ~h0 )) =
X
ku (?(x, zu ), ?(x0 , zu0 )) +
u?V
X
kuv (?(x, zu , zv ), ?(x0 , zu0 , zv0 )) (12)
(u,v)?E
When the latent variable ~h forms a tree structure, there exist efficient inference algorithms for
solving Eq. 9, such as dynamic programming. It is also possible for Eq. 12 to include kernels
defined on higher-order cliques in the graph, as long as we have some pre-defined kernel functions
for them.
4
Experiments
We evaluate KLSVM in three different applications of visual recognition. Each application has a
different type of latent variables. For these applications, we will show that KLSVM outperforms
both the linear LSVM [5] and the regular kernel SVM. Note that we implement the learning of
linear LSVM by ourselves using the same iterative algorithm as the one in [5].
4.1
Object Classification with Latent Localization
Problem and Dataset: We consider object classification with image-level supervision. Our training
data only have image-level labels indicating the presence/absence of each object category in an
image. The exact object location in the image is not provided and is considered as the latent variable
h in our formulation. We define the feature vector ?(x, h) as the HOG feature extracted from the
image at location h. During testing, the inference of h is performed by enumerating all possible
locations of the image.
We evaluate our algorithm on the mammal dataset [8] which consists of 6 mammal categories. There
are about 45 images per category. For each category, we use half of the images for training and the
remaining half for testing. We assume the object size is the same for the images of the same category,
which is a reasonable assumption for this dataset. This dataset was used to evaluate the linear LSVM
in [8].
Results: We compare our algorithm with linear LSVM. To demonstrate the benefit of using latent
variables, we also compare with two simple baselines using linear and kernel SVMs based on bag-offeatures (BOF) extracted from the whole image (i.e. without latent variables). For both baselines, we
aggregate the quantized HOG features densely sampled from the whole image. Then, the features are
fed into the standard linear SVM and kernel SVM respectively. We use the histogram intersection
kernel (HIK) [10] since it has been proved to be successful for vision applications, and efficient
learning/inference algorithms exist for this kernel.
We run the experiments for five rounds. In each round, we randomly split the images from each
category into training and testing sets. For both linear LSVM and KLSVM, we initialize the latent
variable at the center location of each image and we set C1 = C2 = 1. For both algorithms, we use
one-versus-one classification scheme. We use the HIK kernel in the KLSVM. Table 1 summarizes
the mean and standard deviations of the classification accuracies over five rounds of experiments.
Across all experiments, both linear LSVM and KLSVM achieve significantly better results than
approaches using BOF features from the whole image. This is intuitively reasonable since most of
images on this dataset share very similar scenes. So BOF feature without latent variables cannot
capture the subtle differences between each category. Table 1 also shows KLSVM significantly
outperforms linear LSVM.
Fig. 1 shows examples of how the latent variables change on some training images during the learning of the KLSVM. For each training image, the location of the object (latent variable h) is initialized
to the center of the image. After the learning algorithm terminates, the latent variables accurately
locate the objects.
6
Figure 2: Visualization of some testing examples from the ?bird? (left) and ?boat? (right) categories. Each row
corresponds to a subcategory. We can see that visually similar images are grouped into the same subcategory.
Method
non-latent linear SVM linear LSVM non-latent kernel SVM
KLSVM
Acc (%)
50.69 ? 0.38
53.13 ? 0.63
52.98 ? 0.22
55.17 ? 0.27
Table 2: Results on CIFAR10 Dataset. We show the mean/std of classification accuracies over five folds of
experiments. Each fold uses a different batch of the training data.
4.2
Object Classification with Latent Subcategory
Problem and Dataset: Our second application is also on object classification. But here we consider a different type of latent variable. Objects within a category usually have a lot of intra-class
variations. For example, consider the images for the ?bird? category shown in the left column of
Fig. 2. Even though they are examples of the same category, they still exhibit very large appearance
variations. It is usually very difficult to learn a single ?bird? model that captures all those variations.
One way to handle the intra-class variation is to split the ?bird? category into several subcategories.
Examples within a subcategory will be more visually similar than across all subcategories. Here we
use the latent variable h to indicate the subcategory an image belongs to. If a training image belongs
to the class c, its subcategory label h takes value from a set Hc of subcategory labels corresponding
to the c-th class. Note that subcategories are latent on the training data, so they may or may not have
semantic meanings.
The feature vector ?(x, h) is defined as a sparse vector whose feature dimension is |Hc | times of
the dimension of ?(x), where ?(x) is the HOG descriptor extracted from the image x. In the
experiments, we set |Hc | = 3 for all c?s. Then we can define ?(x, h = 1) = (?(x); 0; 0), ?(x, h =
2) = (0; ?(x); 0), and so on. Similar models have been proposed to address the viewpoint changing
in object detection [6] and semantic variations in YouTube video tagging [16].
We use the CIFAR10 [7] dataset in our experiment. It consists of images from ten classes including
airplane, automobile, bird, cat, etc. The training set has been divided into five batches and each
batch contains 10000 images. There are in total 10000 test images.
Results: Again we compare with three baselines: linear LSVM, non-latent linear SVM, non-latent
kernel SVM. Similarly, we use HIK kernel for the kernel-based methods. For non-latent approaches,
we simply feed feature vector ?(x) to SVMs without using any latent variable.
We run the experiments in five folds. Each fold use a different training batch but the same testing
batch. We set C1 = C2 = 0.01 for all the experiments and initialize the subcategory labels of
training images by k-means clustering. Table 2 summarizes the results. Again, KLSVM outperforms
other baseline approaches. It is interesting to note that both linear LSVM and KLSVM outperform
their non-latent counterparts, which demonstrates the effectiveness of using latent subcategories in
object classification. We visualize examples of the correctly classified testing images from the ?bird?
and ?boat? categories in Fig. 2. Images on the same row are assigned the same subcategory labels.
We can see that visually similar images are automatically grouped into the same subcategory.
4.3
Recognition of Object Interaction
Problem and Dataset: Finally, we consider an application where the latent variable is more complex and requires the composite kernel introduced in Sec. 3.2. We would like to recognize complex
interactions between two objects (also called ?visual phrases? [11]) in static images. We build a
dataset consisting of four object interaction classes, i.e. ?person riding a bicycle?, ?person next to
a bicycle?, ?person next to a car? and ?bicycle next to a car? based on the visual phrase dataset in
[11]. Each class contains 86?116 images. Each image is only associated with one of the four object
interaction label. There is no ground-truth bounding box information for each object. We use 40
images from each class for training and the rest for testing.
Our approach: We treat the locations of objects as latent variables. For example, when learning
the model for ?person riding a bicycle?, we treat the locations of ?person? and ?bicycle? as latent
variables. In this example, each image is associated with latent variables ~h = (z1 , z2 ), where
z1 denotes the location of the ?person? and z2 denotes the location of the ?bicycle?. To reduce
the search space of inference, we first apply off-the-shelf ?person? and ?bicycle? detectors [5] on
7
Method
BOF + linear SVM BOF + kernel SVM linear LSVM
KLSVM
Acc(%)
42.92
58.46
46.33 ? 1.4
66.42 ? 0.99
Table 3: Results on object interaction dataset. For the approaches using latent variables, we show the mean/std
of classification accuracies over five folds of experiments.
Figure 3: Visualization of how latent variables (i.e. object locations) change during the learning. The left image
is from the ?person riding a bicycle? category, and the right image is from the ?person next to a car? category.
Yellow bounding boxes corresponds to the initial object locations. The blue bounding boxes correspond to the
object locations after the learning.
each image. For each object, we generate five candidate bounding boxes which form a set Zi ,
i.e. |Z1 | = |Z2 | = 5 and zi ? Zi . Then, the inference of ~h is performed by enumerating 25
combinations of z1 and z2 . We also assume there are certain dependencies between the pair of
(z1 , z2 ). Then the kernel between two images can be defined as follows:
X
K(?(x, ~h), ?(x0 , ~h0 )) =
ku (?(x, zu ), ?(x0 , zu0 )) + kp (?(z1 , z2 ), ?(z10 , z20 ))
(13)
u={1,2}
We define ?(x, zu ) as the bag-of-features (BOF) extracted from the bounding box zu in the image x.
For each bounding box, we split the region uniformly into four equal quadrants. Then we compute
the bag-of-features for each quadrant by aggregating quantized HOG features. The final feature
vector is the concatenation of these four bag-of-features histograms. This feature representation
is similar to the spatial pyramid feature representation. In our experiment, we choose HIK for
ku (?). The kernel kp (?) captures the spatial relationship between z1 and z2 such as above, below,
overlapping, next-to, near, and far. Here ?(z1 , z2 ) is a sparse binary vector and its k-th element is
set to 1 if the corresponding k-th relation is satisfied between bounding boxes z1 and z2 . Note that
kp (?) does not depend on the images. Similar representation has been used in [4]. We define kp (?)
as a simple linear kernel.
Results: We compare with the simple BOF + linear SVM, and BOF + kernel SVM approaches.
These two baselines use the same BOF feature representation as our approach except that the features
are extracted from the whole image. We choose the HIK in the kernel SVM. Note that this is a
strong baseline since [3] has shown that a similar pyramid feature representation with kernel SVM
achieves top performances on the task of person-object interaction recognition. The other baseline
is the standard linear LSVM, in which we build the feature vector ?(x, h) by simply concatenating
both unary features and pairwise features, i.e. ?(x, h) = [?(x, z1 ); ?(x, z2 ); ?(z1 , z2 )]. Again, we
set C1 = C2 = 1 for all experiments. We run the experiments for five rounds for approaches using
latent variables. In each round, we randomly initialize the choices of z1 and z2 . Table 3 summarizes
the results. The kernel latent SVM that uses HIK for ku (?) achieves the best performance.
Fig. 3 shows examples of how the latent variables change on some training images during the learning of the KLSVM. For each training image, both latent variables z1 and z2 are randomly initialized
to one of five candidate bounding boxes. As we can see, the initial bounding boxes can accurately
locate the target objects but their spatial relations are different to ground-truth labels. After learning
algorithm terminates, the latent variables not only locate the target objects, but more importantly
they also capture the correct spatial relationship between objects.
5
Conclusion
We have proposed kernel latent SVM ? a new learning framework that combines the benefits of
LSVM and kernel methods. Our learning framework is very general. The latent variables can not
only be a single discrete value, but also be more complex values with interdependent structures. Our
experimental results on three different applications in visual recognition demonstrate that KLSVM
outperforms using LSVM or using kernel methods alone. We believe our work will open the
possibility of constructing more powerful and expressive prediction models for visual recognition.
Acknowledgement: This work was supported by a Google Research Award and NSERC.
Yang Wang was partially supported by a NSERC postdoc fellowship.
8
References
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9
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3,967 | 4,591 | Multilabel Classification using Bayesian Compressed
Sensing
Ashish Kapoor? , Prateek Jain? and Raajay Viswanathan?
?
Microsoft Research, Redmond, USA
?
Microsoft Research, Bangalore, INDIA
{akapoor, prajain, t-rviswa}@microsoft.com
Abstract
In this paper, we present a Bayesian framework for multilabel classification using
compressed sensing. The key idea in compressed sensing for multilabel classification is to first project the label vector to a lower dimensional space using a
random transformation and then learn regression functions over these projections.
Our approach considers both of these components in a single probabilistic model,
thereby jointly optimizing over compression as well as learning tasks. We then
derive an efficient variational inference scheme that provides joint posterior distribution over all the unobserved labels. The two key benefits of the model are that a)
it can naturally handle datasets that have missing labels and b) it can also measure
uncertainty in prediction. The uncertainty estimate provided by the model allows
for active learning paradigms where an oracle provides information about labels
that promise to be maximally informative for the prediction task. Our experiments
show significant boost over prior methods in terms of prediction performance over
benchmark datasets, both in the fully labeled and the missing labels case. Finally,
we also highlight various useful active learning scenarios that are enabled by the
probabilistic model.
1
Introduction
Large scale multilabel classification problems arise in several practical applications and has recently
generated a lot of interest with several efficient algorithms being proposed for different settings
[1, 2]. A primary reason for thrust in this area is due to explosion of web-based applications, such
as Picasa, Facebook and other online sharing sites, that can obtain multiple tags per data point.
For example, users on the web can annotate videos and images with several possible labels. Such
applications have provided a new dimension to the problem as these applications typically have
millions of tags. Most of the existing multilabel methods learn a decision function or weight vector
per label and then combine the decision functions in a certain manner to predict labels for an unseen
point [3, 4, 2, 5, 6]. However, such approaches quickly become infeasible in real-world as the
number of labels in such applications is typically very large. For instance, traditional multi-label
classification techniques based on 1-vs-all SVM [7] is prohibitive because of both large train and
test times.
To alleviate this problem, [1] proposed a compressed sensing (CS) based method that exploits the
fact that usually the label vectors are very sparse, i.e., the number of positive labels/tags present
in a point is significantly less than the total number of labels. Their algorithm uses the following result from the CS literature: an s-sparse vector in RL can be recovered efficiently using
K = O(s log L/s) measurements. Their method projects label vectors into a s log L/s dimensional
space and learns a regression function in the projected space (independently for each dimension).
For test points, the learnt regression function is applied in the reduced space and then standard recovery algorithms from CS literature are used to obtain sparse predicted labels [8, 9]. However, in
1
this method, learning of the decision functions is independent of the sparse recovery and hence in
practice, it requires several measurements to match accuracy of the standard baseline methods such
as 1-vs-all SVM. Another limitation of this method is that the scheme does not directly apply when
labels are missing, a common aspect in real-world web applications. Finally, the method does not
lend itself naturally to uncertainty analysis that can be used for active learning of labels.
In this paper, we address some of the issues mentioned above using a novel Bayesian framework
for multilabel classification. In particular, we propose a joint probabilistic model that combines
compressed sensing [10, 11] with a Bayesian learning model on the projected space. Our model
can be seen as a Bayesian co-training model, where the lower dimensional projected space can be
thought of as latent variables. And these latent variables are generated by two different views: a)
using a random projection of the label vector, b) using a (linear) predictor over the input data space.
Hence, unlike the method of [1], our model can jointly infer predictions in the projected space and
projections of the label vector. This joint inference leads to more efficient utilization of the latent
variable space and leads to significantly better accuracies than the method of [1] while using same
number of latent variables K.
Besides better prediction performance, there are several other advantages offered by our probabilistic
model. First, the model naturally handles missing labels as the missing labels are modeled as random variables that can be marginalized out. Second, the model enables derivation of a variational
inference method that can efficiently compute joint posterior distribution over all the unobserved
random variables. Thus, we can infer labels not only for the test point but also for all the missing
labels in the training set. Finally, the inferred posterior over labels provide an estimate of uncertainty
making the proposed method amenable to active learning.
Active learning is an important learning paradigm that has received a lot of attention due to the
availability of large unlabeled data but paucity of labels over these data sets. In the traditional active
learning setting (for binary/multiclass classification), at each round the learner actively seeks labels
for a selected unlabeled point and updates its models using the provided label. Several criteria, such
as uncertainty [12], expected informativeness [13, 14], reduction in version space [15], disagreement among a committee of classifiers [16], etc. have been proposed. While heuristics have been
proposed [17] in the case of 1-vs-all SVMs, it is still unclear how these methods can be extended to
multilabel classification setting in a principled manner. Our proposed model naturally handles the
active learning task as the variational inference procedure provides the required posteriors which can
guide information acquisition. Further, besides the traditional active learning scenario, where all the
labels are revealed for a selected data, the model leads to extension of information foraging to more
practical and novel scenarios. For example, we introduce active diagnosis, where the algorithm only
asks about labels for the test case that potentially can help with prediction over the rest of the unobserved tags. Similarly, we can extend to a generalized active learning setting, where the method
seeks answer to questions of the type: ?does label ?A? exists in data point x?. Such extensions are
made feasible due to the Bayesian interpretation of the multilabel classification task.
We demonstrate the above mentioned advantages of our model using empirical validation on benchmark datasets. In particular, experiments show that the method significantly outperforms ML-CS
based method by [1] and also obtains accuracies matching 1-vs-all SVM while projecting onto Kdimensional space that is typically less than half the total number of labels. We expect these gains
to become even more significant for datasets with larger number of labels. We also show that the
proposed framework is robust to missing labels and actually outperforms 1-vs-all SVM with about
85-95% missing labels while using K = .5L only. Finally, we demonstrate that our active learning
strategies select significantly more informative labels/points than the random selection strategy.
2
Approach
Assume that we are given a set of training data points X = {xi } with labels Y = {yi }, where each
yi = [yi1 , .., yiL ] ? [0, 1]L is a multilabel binary vector of size L. Further, let us assume that there
are data points in the training set for which we have partially observed labeled vectors that leads to
the following partitioning: X = XL ? XP . Here the subscripts L and P indicate fully and partially
labeled data respectively. Our goal then is to correctly predict all the labels for data in the test set
XU . Further, we also seek an active learning procedure that would request as few labels as possible
from an oracle to maximize classification rate over the test set.
If we treat each label independently then standard machine learning procedures could be used to train
individual classifiers and this can even be extended to do active learning. However, such procedures
2
can be fairly expensive when the number of labels is huge. Further, these methods would simply
ignore the missing data, thus may not utilize statistical relationship amongst the labels. Recent
techniques in multilabel classification alleviate the problem of large output space [1, 18], but cannot
handle the missing data cases. Finally, there are no clear methods of extending these approaches for
active learning.
We present a probabilistic graphical model that builds upon ideas of compressed sensing and utilizes
statistical relations across the output space for prediction and active information acquisition. The
key idea in compressed sensing is to consider a linear transformation of the L dimensional label
vector y to a K dimensional space z, where K L, via a random matrix ?. The efficiency in the
classification system is improved by considering regression functions to the compressed vectors z
instead of the true label space. The proposed framework considers Gaussian process priors over the
compressed label space and has the capability to propagate uncertainties to the output label space by
considering the constraints imposed by the random projection matrix. There are several benefits of
the proposed method: 1) first it naturally handles missing data by marginalizing over the unobserved
labels, 2) the Bayesian perspective leads to valid probabilities that reflect the true uncertainties
in the system, which in turn helps guide active learning procedures, 3) finally, the experiments
show that the model significantly outperforms state-of-the-art compressed sensing based multilabel
classification methods.
2.1 A Model for Multilabel Classification with Bayesian Compressed Sensing
We propose a model that simultaneously handles two key aspects: first is the task of compressing
and recovering the label vector yi to and from the lower dimensional representation zi . Second,
given an input data xi the problem is estimating low dimensional representation in the compressed
space. Instead of separately solving each of the tasks, the proposed approach aims at achieving better
performance by considering both of these tasks jointly, thereby modeling statistical relationships
amongst different variables of interest.
Figure 1 illustrates the factor graph corresponding to the proposed model. For every data point xi ,
the output labels yi influence the compressed latent vector zi via the random projection matrix ?.
These compressed signals in turn also get influenced by the d-dimensional feature vector xi via the
K different linear regression functions represented as a d ? K matrix W. Consequently, the role of
zi is not only to compress the output space but also to consider the compatibility with the input data
point. The latent variable W corresponding to the linear model has a spherical Gaussian prior and
is motivated by Gaussian Process regression [19]. Note that when zi is observed, the model reduces
to simple Gaussian Process regression.
One of the critical assumptions in compressed sensing is that the output labels yi is sparse. The
proposed model induces this constraint via a zero-mean Gaussian prior on each of the labels (i.e.
yij ? N (0, 1/?ij )), where the precision ?ij of the normal distribution follows a Gamma prior
?ij ? ?(a0 , b0 ) with hyper-parameters a0 and b0 . The Gamma prior has been earlier proposed
in the context of Relevance Vector Machine (RVM) [20] as it not only induces sparsity but also is a
conjugate prior to the precision ?ij of the zero mean Gaussian distributions. Intuitively, marginalizing the precision in the product of Gamma priors and the Gaussian likelihoods leads to a potential
function on the labels that is a student-t distribution and has a significant probability mass around
zero. Thus, the labels yij naturally tend to zero unless they need to explain observed data. Finally, the
conjugate-exponential form between the precisions ?i and the output labels yi leads to an efficient
inference procedure that we describe later in the paper.
Note that, for labeled training data xi ? XL all the labels yi are observed, while only some or
none of the labels are observed for the partially labeled and test cases respectively. The proposed
model ties the input feature vectors X to the output space Y via the compressed representations Z
according to the following distribution:
N
Y
1
p(W)
fxi (w, zi )g? (yi , zi )h?i (yi )p(?i )
Z
i=1
QK
where Z is the partition function (normalization term), p(W) = i=1 N (wi , 0, I) is the spherical
QL
Gaussian prior on the linear regression functions and p(?i ) = j=1 ?(?ij ; a0 , b0 ) is the product of
Gamma priors on each individual label. Finally, the potentials fxi (?, ?), g? (?, ?) and h?i (?) take the
p(Y, Z, W, [?i ]N
i=1 |X, ?) =
3
??
1
??
?? ?
2
?
?=1
?? ? ?(0,
??1
?
?(?? ; ?0 , ?0 )
???
?1?
0
0
???
??2
?1
)
???
?? (?? , ?? )
??1 ??2
???
??
?
?=1 ?(?? ; 0, ?)
?
??? ?, ??
??
? = 1 ?? ?
Figure 1: A Bayesian model for multilabel classification via compressed sensing. The input data is
xi with multiple labels yi , which are fully observed for the case of fully labeled training data set L,
partially observed for training data with missing labels P, or completely unobserved as in test data
U. The latent variables zi indicate the compressed label space, and ?i with independent Gamma
priors enforce the sparsity. The set of regression functions described by W is also a latent random
variable and is connected across all the data points.
following form:
fxi (W, zi ) = e?
||WT xi ?zi ||2
2? 2
, g? (yi , zi ) = e
?
||?yi ?zi ||2
2?2
, h?i (yi ) =
L
Y
j=1
N (yij ; 0,
1
?ij
).
Intuitively, the potential term fxi (W, zi ) favors configurations that are aligned with output of the
linear regression function when applied to the input feature vector. Similarly, the term g? (yi , zi )
favors configurations that are compatible with the output compressive projections determined by
?. Finally, as described earlier, h?i (yi ) enforces sparsity in the output space. The parameters ? 2
and ?2 denote noise parameters and determine how tight the relation is between the labels in the
output space, the compressed space and the regression coefficients. By changing the value of these
parameters we can emphasize or de-emphasize the relationship between the latent variables.
In summary, our model provides a powerful framework for modeling multilabel classification using
compressive sensing. The model promises statistical efficiency by jointly considering compressive
sensing and regression within a single model. Moreover, as we will see in the next section this model
allows efficient numerical procedures for inferring unobserved labels by resolving the constraints
imposed by the potential functions and the observed data. The model naturally handles the case of
missing data (incomplete labels) by automatically marginalizing the unobserved data as a part of
the inference mechanism. Finally, the probabilistic nature of the approach provides us with valid
probabilistic quantities that can be used to perform active selection of the unlabeled points.
2.2 Inference
First, consider the simpler scenario where the training data set only consists of fully labeled instances
XL with labels YL . Thus our aim is to infer p(YU |X, YL , ?), the posterior distribution over
unlabeled data. Performing exact inference is prohibitive in this model primarily due to the following
reason. First, notice that the joint distribution is a product of a Gaussian (Spherical prior on W and
compatibility terms with zi ) and non-Gaussian terms (the Gamma priors). Along with these sparsity
terms, the projection of the label space into the compressed space precludes usage of exact inference
via a junction tree algorithm. Thus, we resort to approximate inference techniques. In particular we
perform an approximate inference by maximizing the variational lower bound by assuming that the
posterior over the unobserved random variable W, YU , Z and [?i ]N
i=1 can be factorized:
Z
p(Y, Z, W, [?i ]N
i=1 |X, ?)
F =
q(YU )q(Z)q(W)q([?i ]N
)
log
i=1
q(YU )q(Z)q(W)q([?i ]N
YU ,Z,W,[?]N
i=1 )
i=1
Z
? log
p(Y, Z, W, [?i ]N
i=1 |X, ?)
YU ,Z,W,[?]N
i=1
4
Here, the posteriors on the precisions ?i are assumed to be Gamma distributed while the rest
of the distributions are constrained to be Gaussian. Further, each of these joint posterior
denQ
sities are assumed to have the following per data point factorization: q(YU ) =
q(y
i ),
i?U
Q
QN Ql
j
q(Z) = i?U ?L q(zi ) and q([?]N
)
=
q(?
).
Similarly
the
posterior
over
the
regresi=1
i
i=1
j=1
QK
sion functions has a per dimension factorization: q(W) = i=1 q(wi ). The approximate inference
algorithm aims to compute good approximations to the real posteriors by iteratively optimizing the
above described variational bound. Specifically, given the approximations q t (yi ) ? N (?tyi , ?tyi )
(similar forms for zi and wi ) and q t (?ij ) ? ?(atij , btij ) from the tth iteration the update rules are as
follows:
t
T ?2 ?1
Update for q t+1 (yi ): ?t+1
?] ,
yi = [diag(E(?i )) + ? ?
0
Update for q t+1 (?ij ): at+1
ij = aij + 0.5,
t+1 T ?2 t
?t+1
?zi ,
yi = ?yi ? ?
0
t+1
t+1
2
bt+1
ij = bij + 0.5[?yi (j, j) + [?yi (j)] ],
?2
Update for q t+1 (zi ): ?t+1
I + ??2 I]?1 ,
zi = [?
t+1 ?2 t+1 T
?t+1
[?W ] xi + ??2 ??t+1
zi = ?zi [?
yi ],
?2
Update for q t+1 (wi ): ?t+1
XXT + I]?1 ,
wi = [?
?2 t+1
T
?t+1
?wi X[?t+1
wi = ?
z (i)] .
Alternating between the above described updates can be considered as message passing between
the low-dimensional regression outputs and higher dimensional output labels, which in turn are
constrained to be sparse. By doing the update on q(yi ), the algorithms attempts to explain the
compressed signal zi using sparsity imposed by the precisions ?i . Similarly, by updating q(zi )
and q(W) the inference procedures reasons about a compressed representation that is most efficient
in terms of reconstruction. By iterating between these updates the model consolidates information
from the two key components, compressed sensing and regression, that constitute the system and is
more effective than doing these tasks in isolation.
Also note that the most expensive step is in the first update for computing ?t+1
yi , which if naively
implemented would require an inversion of an L ? L matrix. However, this inversion can be computed easily using Sherman-Morrison-Woodbury formula, which in turn reduces the complexity of
the update to O(K 3 + K 2 L). The only other significant update is the posterior computation q(w)
that is O(d3 ), where d is the dimensionality of the feature space. Consequently, this scheme is fairly
efficient and has time complexity similar to that of other non-probabilistic approaches. Finally, note
that straightforward extension to non-linear regression functions can be done via the kernel trick.
Handling Missing Labels in Training Data: The proposed model and the inference procedure
naturally handles the case of missing labels in the training set via the variational inference. Lets
consider a data point xp with set of partially observed labels ypo . If we denote ypu as the set of
unobserved labels, then all the above mentioned update steps stay the same except for the one that
updates q(zp ), which takes the following form:
t+1 ?2 T t+1
o
?t+1
xp ?W + ??2 ?uo [?t+1
u ; yp ]].
zp = ?zp [?
yp
Here ?uo denotes re-ordering of the columns on ? according to the indices of the observed and
unobserved labels. Intuitively, the compressed signal zp now considers compatibility with the unobserved labels, while taking into account the observed labels, and in doing so effectively facilitates
message passing between all the latent random variables.
Handling a Test Point: While it might seem that the above mentioned framework works in the
transductive setting, we here show such is not the case and that the framework can seamlessly handle test data in an inductive setting. Note that given a training set, we can recover the posterior
distribution q(W) that summarizes the regression parameter. This posterior distribution is sufficient
for doing inference on a test point x? . Intuitively, the key idea is that the information about the
training set is fully captured in the regression parameters, thus, the labels for the test point can be
simply recovered by only iteratively updating q(y? ), q(z? ) and q(?? ).
2.3 Active Learning
The main aim in active learning is to seek bits of information that would promise to enhance the
discriminatory power of the framework the most. When employed in a traditional classification
setting, the active learning procedure boils down to the task of seeking the label for one of the
unlabeled examples that promises to be most informative and then update the classification model
by incorporating it into the existing training set. However, multilabel classification enables richer
forms of active information acquisitions, which we describe below:
5
? Traditional Active Learning: This is similar to the active learning scenario in traditional
classification tasks. In particular, the goal is to select an unlabeled sample for which all the
labels will be revealed.
? Active Diagnosis: Given a test data point, at every iteration the active acquisition procedure
seeks a label for each test point that is maximally informative for the same and promises to
improve the prediction accuracy over the rest of the unknown labels.
Note that Active Diagnosis is highly relevant for real-world tasks. For example, consider the wikipedia page classification problem. Just knowing a few labels about the page can be immensely
useful in inferring the rest of the labels. Active diagnosis should be able to leverage the statistical
dependency amongst the output label space, in order to ask for labels that are maximally informative.
A direct generalization of the above two paradigms is a setting in which the active learning procedure
selects a label for one point in the training set. Specifically, the key difference between this scenario
and the traditional active learning is that only one label is chosen to be revealed for the selected data
point instead of the entire set of labels.
Non-probabilistic classification schemes, such as SVMs, can handle traditional active learning by
first establishing the confidence in the estimate of each label by using the distance from the classification boundary (margin) and then selecting the point that is closest to the margin. However, it is
fairly non-trivial to extend those approaches to tackle the active diagnosis and generalized information acquisition. On the other hand the proposed Bayesian model provides a posterior distribution
over the unknown class labels as well as other latent variables and can be used for active learning.
In particular, measures such as uncertainty or information gain can be used to guide the selective
sampling procedure for active learning. Formally, we can write these two selection criteria as:
Uncertainty: arg max H(yi )
yi ?YU
InfoGain: arg max H(YU /yi ) ? Eyi [H(YU /yi |yi )].
yi ?YU
Here, H(?) denotes Shannon entropy and is a measure of uncertainty. The uncertainty criterion seeks
to select the labels that have the highest entropy, whereas the information gain criterion seeks to
select a label that has the highest expected reduction in uncertainty over all the other unlabeled points
or unknown labels. Either of these criteria can be computed given the inferred posteriors; however
we note that the information gain criterion is far more expensive to compute as it requires repeated
inference by considering all possible labels for every unlabeled data point. The uncertainty criterion
on the other hand is very simple and often guides active learning with reasonable amount of gains.
In this work we will consider uncertainty as the primary active learning criterion. Finally, we?d like
to point that the different described forms of active learning can naturally be addressed with these
heuristics by appropriately choosing the set of possible candidates and the posterior distributions
over which the entropy is measured.
3
Experiments
In this section, we present experimental results using our methods on standard benchmark datasets.
The goals of our experiments are three-fold: a) demonstrate that the proposed jointly probabilistic
method is significantly better than the standard compressed sensing based method by [1] and gets
comparable accuracy to 1-vs-all SVM while projecting labels onto much smaller dimensionality
K compared to the total number of labels L, b) show robustness of our method to missing labels,
c) demonstrate various active learning scenarios and compare them against the standard baselines.
We use Matlab for all our implementations. We refer to our Compressed Sensing based Bayesian
Multilabel classification method as BML-CS . In BML-CS method, the hyper-parameters a0 and b0
are set to 10?6 , which in turn leads to a fairly uninformative prior. The noise parameters ? and ? are
found by maximizing the marginalized likelihood of the Gaussian Process Regression model [19].
We use liblinear for SVM implementation; error penalty C is selected using cross-validation. We
also implemented the multilabel classification method based on compressed sensing (ML-CS ) [1]
with CoSamp [8] being the underlying sparse vector recovery algorithm.
For our experiments, we use standard multilabel datasets. In particular, we choose datasets where
the number of labels is high. Such datasets generally tend to have only a few labels per data point
and the compressed sensing methods can exploit this sparsity to their advantage.
6
100
40
100
40
35
90
35
40
Precision (in %)
60
30
25
20
Precision (in %)
80
Precision (in %)
Precision (in %)
80
70
60
50
30
25
20
40
20
1 vs all SVM
BML?CS
ML?CS
0
0
20
40
60
80
(a) CAL500 dataset
100
1 vs all SVM
BML?CS
ML?CS
15
10
0
20
40
60
80
1 vs all SVM
BML?CS
ML?CS
30
100
(b) Bookmarks dataset
20
0
20
40
60
80
100
(c) RCV1 dataset
1 vs all SVM
BML?CS
ML?CS
15
10
0
20
40
60
80
100
(d) Corel5k dataset
Figure 2: Comparison of precision values (in top-1 label) for different methods with different values
of K, dimensionality of the compressed label space. The SVM baseline uses all the L labels. The
x-axis shows K as a percentage of the total number of labels L. Clearly, for each of the dataset
the proposed method obtains accuracy similar to 1-vs-all SVM method while projecting to only
K = L/2 dimensions. Also, our method consistently obtains significantly higher accuracies than
the CS method of [1] while using the same number of latent variables K.
K
10%
25%
50%
75%
100%
Top-3
SVM BML-CS ML-CS
0.04
0.36
0.38
0.48
0.74
0.61
0.44
0.75
0.53
0.70
0.61
K
10%
25%
50%
75%
100%
Top-3
SVM BML-CS ML-CS
0.33
0.19
0.65
0.59
0.75
0.75
0.69
0.75
0.71
0.75
0.72
Top-5
SVM BML-CS ML-CS
0.09
0.32
0.28
0.41
0.67
0.51
0.40
0.60
0.55
0.65
0.57
K
10%
25%
50%
75%
100%
Top-3
SVM BML-CS ML-CS
0.10
0.06
0.15
0.08
0.20
0.17
0.09
0.17
0.10
0.19
0.10
K
10%
25%
50%
75%
100%
Top-3
SVM BML-CS ML-CS
0.20
0.08
0.27
0.17
0.27
0.27
0.21
0.27
0.22
0.27
0.22
(a)
Top-5
SVM BML-CS ML-CS
0.07
0.04
0.10
0.05
0.14
0.12
0.06
0.13
0.07
0.13
0.07
(b)
Top-5
SVM BML-CS ML-CS
0.23
0.14
0.44
0.39
0.54
0.52
0.49
0.53
0.50
0.53
0.51
(c)
Top-5
SVM BML-CS ML-CS
0.15
0.06
0.22
0.14
0.22
0.23
0.17
0.23
0.18
0.23
0.17
(d)
Figure 3: Precision values obtained by various methods in retrieving 3 and 5 labels respectively.
First column in each table shows K as the fraction of number of labels L. 1-vs-all SVM requires
training L weight vectors, while both BML-CS and ML-CS trains K weight vectors. BML-CS is
consistently more accurate than ML-CS although its accuracy is not as close to SVM as it is for the
case of top-1 labels (see Figure 2).
For each of the algorithms we recover the top 1, 3, 5 most likely positive labels and set remaining
labels to be negative. For each value of t ? {1, 3, 5}, we report precision in prediction, i.e., fraction
of true positives to the total number of positives predicted.
3.1
Multilabel Classification Accuracies
We train both ML-CS and our method BML-CS on all datasets using different values of K, i.e.,
the dimensionality of the space of latent variables z for which weight vectors are learned. Figure 2
compares precision (in predicting 1 positive label) of our proposed method on four different datasets
for different values of K with the corresponding values obtained by ML-CS and SVM . Note that 1vs-all SVM learns all L > K weight vectors, hence it is just one point in the plot; we provide a line
for ease of comparison. It is clear from the figure that both BML-CS and ML-CS are significantly
worse than 1-vs-all SVM when K is very small compared to total number of labels L. However,
for around K = 0.5L, our method achieves close to the baseline (1-vs-all SVM) accuracy while
ML-CS still achieves significantly worse accuracies. In fact, even with K = L, ML-CS still obtains
significantly lower accuracy than SVM baseline.
In Figure 3 we tabulate precision for top-3 and top-5 retrieved positive labels. Here again, the
proposed method is consistently more accurate than ML-CS . However, it requires larger K to
obtain similar precision values as SVM. This is fairly intuitive as for higher recall rate the multilabel
problems become harder and hence our method requires more weight vectors to be learned per label.
3.2
Missing Labels
Next, we conduct experiments for multilabel classification with missing labels. Specifically, we
remove a fixed fraction of training labels randomly from each dataset considered. We then apply
7
Variation of precision with incomplete labels
86
79
Precision (in %)
Precision (in %)
80
BML?CS
SVM
88
84
82
80
77
76
75
74
76
80
73
0
95
70
78
78
85
90
Percentage of labels missing
80
BML?CS Active
BML?CS Rand
Precision (in %)
90
60
50
40
30
5
10
15
Active learning rounds (1 point per round)
20
BML?CS Active
BML?CS Rand
20
0
20
40
60
80
100
Active learning rounds (1 label per point per round)
(a)
(b)
(c)
Figure 4: (a) Precision (in retrieving the most likely positive label) obtained by BML-CS and SVM
methods on RCV1 dataset with varying fraction of missing labels. We observe that BML-CS obtains
higher precision values than baseline SVM.(k = 0.5L) (b) Precision obtained after each round
of active learning by BML-CS-Active method and by the baseline random selection strategy over
RCV1 dataset.(c) Precision after active learning, where one label per point is added to the training
set, in comparison with random baseline on RCV1 dataset. Parameters for (b) & (c): k = 0.1L.
Both (b) and (c), start with 100 points initially.
BML-CS as well as 1-vs-all SVM method to such training data. Since, SVM cannot directly handle
missing labels, we always set a missing label to be a negative label. In contrast, our method can
explicitly handle missing labels and can perform inference by marginalizing the unobserved tags.
As the number of positive labels is significantly smaller than the negative labels, when only a small
fraction of labels are removed, both SVM and BML-CS obtain similar accuracies to the case where
all the labels are present. However, as the number of missing labels increase there is a smooth dip
in the precision of the two methods. Figure 4 (a) compares precision obtained by BML-CS with the
precision obtained by 1-vs-all SVM. Clearly, our method performs better than SVM, while using
only K = .5L weight vectors.
3.3 Active Learning
In this section, we provide empirical results for some of the active learning tasks we discussed in
Section 2.3. For each of the tasks, we use our Bayesian multilabel method to compute the posterior
over the label vector. We then select the desired label/point appropriately according to each individual task. For each of the tasks, we compare our method against an appropriate baseline method.
Traditional Active Learning: The goal here is to select most informative points which if labeled
completely will increase the accuracy by the highest amount. We use uncertainty sampling where
we consider the entropy of the posterior over label vector as the selection criterion for BML-CSActive method. We compare the proposed method against the standard random selection baseline.
For these experiments, we initialize both the methods with an initial labeled dataset of 100 points
and then after each active learning round we seek all the labels for the selected training data point.
Figure 4 (b) compares precisions obtained by BML-CS-Active method with the precisions obtained
by the baseline method after every active learning round. After just 15 active learning rounds, our
method is able to gain about 6% of accuracy while random selection method do not provide any gain
in the accuracy.
Active Diagnosis: In this type of active learning, we query one label for each of the training points
in each round. For each training point, we choose a label with the most uncertainty and ask for
its label. Figure 4 (c) plots the improvement in precision values with number of rounds of active
learning, for estimating the top-1 label. From the plot, we can see that after just 20 rounds, choosing
points by uncertainty has an improvement of 20% over the random baseline.
4
Conclusion and Future Work
We presented a Bayesian framework for multilabel classification that uses compressive sensing.
The proposed framework jointly models the compressive sensing/reconstruction task with learning
regression over the compressed space. We present an efficient variational inference scheme that
jointly resolves compressed sensing and regression tasks. The resulting posterior distribution can
further be used to perform different flavors of active learning. Experimental evaluations highlight the
efficacy of the framework. Future directions include considering other structured prediction tasks
that are sparse and applying the framework to novel scenarios. Further, instead of myopic next best
information seeking we also seek to investigate non-myopic selective sampling where an optimal
subset of unlabeled data are selected.
8
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9
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3,968 | 4,592 | Kernel Hyperalignment
Alexander Lorbert & Peter J. Ramadge
Department of Electrical Engineering
Princeton University
Abstract
We offer a regularized, kernel extension of the multi-set, orthogonal Procrustes
problem, or hyperalignment. Our new method, called Kernel Hyperalignment,
expands the scope of hyperalignment to include nonlinear measures of similarity and enables the alignment of multiple datasets with a large number of base
features. With direct application to fMRI data analysis, kernel hyperalignment is
well-suited for multi-subject alignment of large ROIs, including the entire cortex.
We report experiments using real-world, multi-subject fMRI data.
1
Introduction
One of the goals of multi-set data analysis is forming qualitative comparisons between datasets. To
the extent that we can control and design experiments to facilitate these comparisons, we must first
ask whether the data are aligned. In its simplest form, the primary question of interest is whether
corresponding features among the datasets measure the same quantity. If yes, we say the data are
aligned; if not, we must first perform an alignment of the data.
The alignment problem is crucial to multi-subject fMRI data analysis, which is the motivation for
this work. An appreciable amount of effort is devoted to designing experiments that maintain the
focus of a subject. This is to ensure temporal alignment across subjects for a common stimulus.
However, with each subject exhibiting his/her own unique spatial response patterns, there is a need
for spatial alignment. Specifically, we want between subject correspondence of voxel j at TR i
(Time of Repetition). The typical approach taken is anatomical alignment [20] whereby anatomical landmarks are used to anchor spatial commonality across subjects. In linear algebra parlance,
anatomical alignment is an affine transformation with 9 degrees of freedom.
Recently, Haxby et al. [9] proposed Hyperalignment, a function-based alignment procedure. Instead
of a 9-parameter transformation, a higher-order, orthogonal transformation is derived from voxel
time-series data. The underlying assumption of hyperalignment is that, for a fixed stimulus, a subject?s time-series data will possess a common geometry. Accordingly, the role of alignment is to
find isometric transformations of the per-subject trajectories traced out in voxel space so that the
transformed time-series best match each other. Using their method, the authors were able to achieve
a between-subject classification accuracy on par with?and even greater than?within-subject accuracy.
Suppose that subject data are recorded in matrices X1:m ? Rt?n . This could be data from an
experiment involving m subjects, t TRs, and n voxels. We are interested in extending the regularized
hyperalignment problem
P
2
minimize
i<j kXi Ri ? Xj Rj kF
(1)
subject to RTk Ak Rk = I
k = 1, 2, . . . , m ,
where matrices A1:m ? Rn?n are symmetric and positive definite. In general, the above problem
manifests itself in many application areas. For example, when Ak = I we have hyperalignment or
1
a multi-set orthogonal Procrustes problem, commonly used in shape analysis [6, 7]. When Ak =
XTk Xk , (1) represents a form of multi-set Canonical Correlation Analysis (CCA) [12, 13, 8].
The success of hyperalignment engenders numerous questions and in this work we address two of
them. First, is hyperalignment scalable? In [9], the authors consider a subset of ventral temporal cortex (VT), using hundreds of voxels. The relatively-low voxel count alleviates a huge computational
cost and storage burden. However, the current method for solving (1) is infeasible when considering
many or all voxels, and therefore limits the scope of hyperalignment to a local alignment procedure.
For example, if n = 50,000 voxels, then storing the n ? n matrix for one subject requires over 18
gigabytes of memory. Moreover, computing a full SVD for a matrix this size is a tall order.
Coupled with scalability, we also ask whether we can include new features of our subjects? data.
For example, we may want to augment the input data with the associated second-order mixtures,
i.e., n voxels become ( n1 ) + ( n2 ) = n(n+1)/2 features. Again, for a reasonably-sized voxel count,
running hyperalignment is infeasible.
Addressing scalability and feature extension results in the main contribution of kernel hyperalignment. The inclusion of a large feature space motivates the use of kernel methods. Additionally,
numerous optimization problems that use the kernel trick possess global optimizers spanned by the
mapped examples. This is guaranteed by the Representer Theorem [14, 18]. Therefore, the two separate issues of scalability and feature extension are merged into a single problem through the use of
kernel methods. With kernel hyperalignment, the bottleneck shifts from voxel count to the number
of TRs times subjects (or the original inputs to the number of examples).
The problem we address in this paper is the alignment of multiple datasets in the same and extended
feature space. Multi-set data analysis by means of kernel methods has already been considered in
the framework of CCA [16, 1]. Our approach deviates from [1] and [15] because we focus on alignment and never leave feature space until training and testing. We use the kernel trick as a means
of navigating through a high-dimensional orthogonal group. Our CCA variant is more constrained,
and each dataset is assigned the same kernel, supplying us with a richer, single reproducing kernel
Hilbert space (RKHS) over a collection of m smaller and distinct ones. Allowing for subject-specific
kernels leads to the difficult problem of selecting them?a significantly harder problem than selecting a single kernel. In this respect, we assume a single kernel can provide the sought-after linearity
used for comparing multiple datasets.
The paper is organized as follows: in ?2 we review regularized hyperalignment, or the regularized
multi-set orthogonal Procrustes problem. Next, in ?3 we formulate its kernel variant, and in ?4 we
discuss classification with aligned data. We provided experimental results in ?5, and we conclude in
?6. All proofs are supplied in the Supplemental Material.
2
Hyperalignment
The hyperalignment problem of (1) is equivalent to [7]:
Pm
2
minimize
i=1 kXi Ri ? YkF
P
m
1
and RTk Ak Rk = I for k = 1, . . . , m .
subject to Y = m
j=1 Xj Rj
(2)
The matrix Y is the image centroid and serves as the catalyst for computing a solution: for dataset
i, fix a centroid and solve for Ri . This process cycles over all datasets for a specified number of
rounds, or until approximate convergence is reached (see Algorithm 1). The dynamic centroid Y
can be a sample mean or a leave-one-out (LOO) mean. Regardless of type, the last round should use
1/2
the fixed sample mean provided by the penultimate round. We can set Qk = Ak Rk , using the
1
symmetric, positive definite square root , yielding the key operation
?1
minimize kXk Ak 2 Qk ? Yk2F
(3)
subject to QTk Qk = I .
The above is the familiar orthogonal Procrustes problem [19] and is solved using the SVD of
?1
Ak 2 XTk Y.
1
In practice, we would use the Cholesky factorization of Ak . However, in deriving the kernel hyperalignment procedure it is necessary to familiarize the reader with this approach.
2
3
Kernel Hyperalignment
The previous section dealt with alignment based on the original data. In the context of optimization,
the alignment problem of (1) is indifferent to both data generation and data recording. There are,
however, implicit assumptions about these two processes. The data are generated according to a
common input signal, and each of the m datasets represents a specific view of this signal. In other
words, the matrices X1:m have row correspondence. The alignment problem of (1) seeks column
correspondence through a linear mapping of the original features.
In fMRI, the m views are manifested by m subjects experiencing a common, synchronous stimulus.
Each data matrix records fMRI time-series data: the rows are indexed by a TR and the columns are
indexed by a voxel. There are t TRs and n voxels per subject, i.e., Xk ? Rt?n . The synchrony of the
stimulus ensures row correspondence. Hyperalignment can be posed as the minimization problem
of (2) with Ak = I. Voxel (column) correspondence is then achieved via an orthogonal constraint
placed on each of the linear mappings. The orthogonal constraint present in hyperalignment follows
a subject-independent isometry assumption. We can view the time-series data of each subject as a
trajectory in Rn . For a fixed stimulus this trajectory is [approximately] identical?up to a rotationreflection?across subjects.
As stated above, we are assuming equivalence of the per-view information in its original form, but
we are not assuming that this information can be related through a linear mapping. Now suppose
there is a common set of N features?derived from each n-dimensional example?that does allow
for a linear relationship between views. Alternatively, there may be derivative features of interest
that lead to better alignment via a linear mapping. For example, it is conceivable that second-order
data, i.e., pairwise mixtures of the original data, obey a linear construct and may be a preferred
feature set for alignment. In general, we wish to formulate an alignment technique for this new
feature set. Rather than limit expression of the data to the n given coordinates, we consider an
N -coordinate representation, where N may be much greater than n.
Let Xi ? Rt?n have i0 -th row [xii0 ]T with xii0 ? Rn . We introduce the row-based mapping of Xi :
?
?
?1 (xi1 ) ?2 (xi1 ) ? ? ? ?N (xi1 )
?
..
.. ? ? Rt?N .
(4)
?(Xi ) = ? ...
.
. ?
?1 (xit )
?2 (xit )
???
?N (xit )
The N functions ?1:N : Rn ? R are used to derive N features from the original data. For matrix
Xi ? Rt?n let ?i = ?(Xi ). In general, for Xi ? Rt?n and Xj ? Rs?n , we define the Gram
matrix Kij , ?i ?Tj ? Rt?s . We also write Ki , Kii = ?i ?Ti . We assume that there is an
appropriate positive definite kernel, k? : Rn ? Rn ? R, so that we can leverage the kernel trick
[2, 10] and obtain the i0 j 0 -th element of Kij via
? x i0 , x j 0 ) .
(Kij )i0 j 0 = k(
i
j
(5)
Using the feature map ?(?), we form the regularized Kernel Hyperalignment problem:
P
2
minimize
i<j k?(Xi )Ri ? ?(Xj )Rj kF
subject to RTk Ak Rk = I for k = 1, . . . , m .
(6)
The latent variables are R1:m ? RN ?N and we are given symmetric, positive definite matrices
A1:m ? RN ?N . Although different than the original hyperalignment problem, obtaining a solution
to (6) is accomplished in the same way: fix a centroid and find the individual linear maps. To this
end, the key operation involves solving
arg min k?i R ? ?k2F
?1
arg min k?i Ai 2 Q ? ?k2F ,
or
RT Ai R=I
(7)
QT Q=I
where ?i = ?(Xi ), i ? 1, is the current, individual dataset under consideration and ? =
P
1
?
?
j?A ?j Rj is a centroid based on the current estimates of R1:m , denoted R1:m . The index set
|A|
A ? {1, . . . , m} determines how the estimated centroid is calculated (sample or LOO mean).
3
The difficulty of (7) lies in the size of N . Any of the well-known kernels correspond to an N so large
that direct computation is generally impractical. For example, if using second-order interactions as
the feature set, the number of unknowns in kernel hyperalignment is O(mn4 ) in contrast to O(mn2 )
unknowns for hyperalignment. Nevertheless, the minimization problem of (7) places us in familiar
territory of solving an orthogonal Procrustes problem.
Since we are now in feature space, the matrix Ai poses a problem unless we confine it to a specific
?1/2
form. For example, if Ai is random, finding Ai
would be infeasible for large N . Additionally,
T
the constraint Ri Ai Ri = I would lack any intuition. Therefore, we restrict Ai = ?I + ??Ti ?i
with ? > 0 and ? ? 0. As with regularized hyperalignment [22], when (?, ?) = (1, 0) we obtain
hyperalignment and when (?, ?) ? (0, 1) we obtain a form of CCA.
Let Ki have eigen-decomposition Vi ?i ViT , where ?i = diag{?i1 , . . . , ?it } or diagj {?ij } for
short. We introduce two symmetric, positive definite matrices: Bi = Vi diagj { ? 1
}ViT and
?+??ij
Ci =
Vi diagj { ?1ij ( ? 1
?+??ij
?
?1 )}VT .
i
?
? 12
Lemma 3.1. For Ai = ?I + ??Ti ?i we have Ai
=
?1 I
?
? 12
+ ?Ti Ci ?i and ?i Ai
= Bi ?i .
We can use Lemma 3.1 to transform (7) into
arg min kBi ?i Q ? ?k2F
or
h
i
P
1
?
arg max tr QT ?Ti Bi |A|
,
j?A Bj ?j Qj
(8)
QT Q=I
QT Q=I
? j is the current estimate of Qj . Solving for the matrix Q is still well beyond practical
where Q
computation. The following lemma is the gateway for managing this problem.
? ? St(N, d) and G
? ? O(d), then Q
? = IN ? U(I
? d ? G)
? U
? T ? O(N ).2
Lemma 3.2. If U
? = Id ) and Householder
Familiar applications of the above lemma include the identity matrix (G
? = ?Id ). If G
? is block diagonal with 2 ? 2 blocks of Givens rotations, then the
reflections (G
? taken two at a time, are the two-dimensional planes of rotation [7]. We therefore
columns of U,
?
refer to U as the plane support matrix.
?
Lemma 3.2 can be interpreted as a lifting mechanism for identity deviations. The difference Id ? G
? d ? G)
? U
? T = IN ? Q,
? ?lifts? this differrepresents a O(d) deviation from identity. Applying U(I
ence to a O(N ) deviation from identity. Reversing directions, we can also utilize Lemma 3.2 for
? = U(I
? d ? G)
? U
? T , the rank of the deviation, IN ? Q, is upper
compressing O(N ). From IN ? Q
bounded by d, producing a subset of O(N ).
Motivated by Lemma 3.2 we impose
Qi = IN ? U(I ? Gi )UT ,
(9)
where U ? St(N, r), Gi ? O(r), and 1 ? r ? N . Ideally, we want r small to benefit from a
reduced dimension. As is typically the case when using kernel methods, leveraging the Representer
Theorem shifts the dimensionality of the problem from the feature cardinality to the number of
examples, i.e., r = mt. We pool all of the data, forming the mt ? N matrix
T
,
(10)
?0 = ?T1 ?T2 ? ? ? ?Tm
? 21
and set U = ?T0 K0
? RN ?r with K0 = ?0 ?T0 assumed positive definite. As long as r ? N ,
?1
?1
?1
? 12
the orthogonality constraint is met because (?T0 K0 2 )T (?T0 K0 2 ) = K0 2 K0 K0
= Ir .
Theorem 3.3 (Hyperalignment Representer Theorem). Within the set of global minimizers of (6)
?1
?1
there exists a solution {R?1 , . . . , R?m } = {A1 2 Q?1 , . . . , Am 2 Q?m } that admits a representation
? 12
Q?i = IN ? U(I ? G?i )UT , where U = ?T0 K0
and G?i ? O(mt) (i = 1, . . . , m).
St(N, d) , {Z : Z ? RN ?d , ZT Z = Id } is the (N, d) Stiefel Manifold (N ? d), and
O(N ) , {Z : Z ? RN ?N , ZT Z = IN } is the orthogonal group of N ? N matrices.
2
4
? ?), ?, ?, X1:m ? Rt?n
Input: k(?,
Output: R1:m , linear maps in feature space
Initialize feature maps ?1 , . . . , ?m ? Rt?N
T
Initialize plane support ?0 = ?T1 ?T2 ? ? ? ?Tm
Initialize G1:m ? Rr?r as identity (r = mt)
foreach round do
foreach subject/view
i do
(
{1, 2, . . . , m}
sample mean
A?
{1, 2, . . . , m} \ {i} LOO mean
Input: X1:m ? Rt?n , A1:m ? Rn?n
Output: R1:m ? Rn?n
Initialize Q1:m as identity (n ? n)
?1/2
? i 1:m
Set X
? Xi Ai
foreach round do
foreach subject/view
i do
(
{1, 2, . . . , m}
sample mean
A?
{1, 2, . . . , m} \ {i} LOO mean
Y?
Y?
1 X ?
Xj Qj
|A| j?A
? ?
? V]
? ? SVD(B
? Ti Y)
[U
T
?V
?
Gi ? U
? ?
? V]
? ? SVD(X
? Ti Y)
[U
T
?
?
Qi ? UV
end
end
foreach subject/view i do
end
end
foreach subject/view i do
1
?2
Ri ? Ai
1 X ?
Bj Gj
|A| j?A
?1
?1
Qi ? I ? ?T0 K0 2 (Ir ? Gi )K0 2 ?0
?1
2
Ri ? Ai
Qi
Qi
end
end
Algorithm 2: Regularized Kernel Hyperalignment
Algorithm 1: Regularized Hyperalignment
When mt is large enough so that evaluating an SVD of numerous mt ? mt matrices is prohibitive,
we can first perform PCA-like reduction. Let K0 have eigen-decomposition V0 ?0 V0T , where the
nonnegative diagonal entries of ?0 are sorted in decreasing order. We set ?00 = V0T0 ?0 , where
?1/2
V00 is formed by the first r columns of V0 , and then use U = ?T00 K00 . In general, rather
2
than compute Q according to (7), involving N (N ?1)/2 = O(N ) degrees of freedom (when N is
finite), we end up with r(r?1)/2 = O(r2 ) degrees of freedom via the kernel trick.
? 21
? i = Bi Ki0 K
Let B
0
? Rt?r . We reduce (8) in terms of Gi and obtain (Supplementary Material)
?
?
??
X
1
?T ?
? jG
? j ?? ,
Gi = arg max tr ?GT B
B
(11)
i
|A|
G?O(r)
j?A
? j is the current estimate of Gj . Equation (11) is the classical orthogonal Procrustes probwhere G
i
h
? jG
? j , then a maximizer is given by U
?V
? T [7].
??
?V
? T is the SVD of GT B
?T 1 P
B
lem. If U
i
|A|
j?A
The kernel hyperalignment procedure is given in Algorithm 2. Using the approach taken in this
section also leads to an efficient solution of the standard orthogonal Procrustes problem for n ? 2t
(Supplementary Material). In turn, this leads to an efficient iterative solution for the hyperalignment
problem when n is large.
4
Alignment Assessment
An alignment procedure is not subject to the typical train-and-test paradigm. The lack of spatial
correspondence demands an align-train-test approach. We assume these three sets have withinsubject (or within-view) alignment. With all other parameters fixed, if the aligned test error is
smaller than the unaligned test error, there is strong evidence suggesting that alignment was the
underlying cause.
Kernel hyperalignment returns linear transformations R1:m that act on data living in feature space.
In general, we cannot directly train and test in the feature space due to its large size. We can,
however, learn from relational data. For example, we can compute distances between examples
and, subsequently, produce nearest neighbor classifiers. Assume (?, ?) = (1, 0), i.e., the R1:m
5
are orthogonal. If x1 ? Rn is a view-i example and x2 ? Rn is a view-j example, the respective
pre-aligned and post-aligned squared distances between the two examples are given by
? 1 , x1 ) + k(x
? 2 , x2 ) ? 2k(x
? 1 , x2 )
k?(xT1 ) ? ?(xT2 )k2F = k(x
? 1 , x1 ) + k(x
? 2 , x2 ) ? 2?(xT )Ri RT ?(xT )T .
k?(xT1 )Ri ? ?(xT2 )Rj k2F = k(x
1
j
2
(12)
(13)
The cross-term in (13) has not been expanded for a simple reason: it is too messy. We realized early
on that the alignment and training phase would be replete with lengthy expansions and, consequently,
sought to simplify matters with a computer science solution. Both binary and unary operations in
feature space can be accomplished with a simple class. Our Phi class stores expressions of the
following forms:
PK
PK
PK
T
bIN + k=1 ?(Xa(k) )T Mk ?(Xa(k) ) . (14)
k=1 Mk ?(Xa(k) )
k=1 ?(Xa(k) ) Mk
|
{z
} |
{z
} |
{z
}
Type 1
Type 2
Type 3
Each class instance stores matrices M1:K , scalar b, right address vector a, and left address vector a.
The address vectors are pointers to the input data. This allows for faster manipulation and smaller
memory allocation. Addition and subtraction require a common type. If types match, then the M
matrices must be checked for compatible sizes. Multiplication is performed for types 1 with 2, 1
with 3, 2 with 1, 3 with 2, and 3 with 3. The first of these cases, for example, produces a numeric
result via the kernel trick. We also define scalar multiplication and division for all types and matrix
multiplication for types 1 and 2. A transpose operator applies for all types and maps type 1 to 2,
2 to 1, and 3 to 3. More advanced operations, such as powers and inverses, are also possible. Our
implementation was done in Matlab.
The construction of the Phi class allows us to stay in feature space and avoid lengthy expansions. In
s?n
turn, this facilitates implementing the richer set of SVM classifiers. Let X?1 , . . . , Xm
be our
? ?R
s?N
training data with feature representation ??? = ?(X?? ) ? R
. Recall that kernel hyperalignment
seeks to align in feature space. Before alignment we might have considered K???? = ??? ?T?? ; we now
consider the Gram matrix (??? Ri )(???Rj )T = ??? Ri RTj ?T?? . If every row of X?? has a corresponding
label, we can train an SVM with
?
?
??1 R1 RT1 ?T?1 ??1 R1 RT2 ?T?2 ? ? ? ??1 R1 RTm ?Tm
?
??
?T
?
??1 R1
??1 R1
?
?
?
?
??
? ? ?? R2 RT ?T ??2 R2 RT2 ?T?2
KA? =? ... ??? ... ? =? 2 . 1 ?1
? , (15)
..
?
?
..
.
?m
?m
? Rm
? Rm
T
T T
T
?m
?m
? R m R 1 ??
? Rm Rm ?m
?
1
where KA? = KTA? ? Rms?ms denotes the aligned kernel matrix. The unaligned kernel matrix, KU? ,
is also an m ? m block matrix with ij-th block K????.
Using the dual formulation of an SVM, a classifier can be constructed from the relational data
exhibited among the examples [4]. Similar to a k-nearest neighbor classifier relying on pairwise
distances, an SVM relies on the kernel matrix. The kernel matrix is a matrix of inner products and
is therefore linear. This enables us to assess a partition-based alignment.
In fMRI, we perform two alignments?one for each hemisphere. Each alignment produces two
aligned kernel matrices, which we sum and then input into an SVM. Thus, linearity provides us the
means to handle finer partitions by simply summing the aligned kernel matrices.
6
Table 1: Seven label classification using movie-based alignment Below is the cross-validated,
between-subject classification accuracy (within-subject in brackets) with (?, ?) = (1, 0). Four
hundred TRs per subject were used for the alignment. Chance = 1/7 ? 14.29%.
Kernel
Linear
Quadratic
Gaussian
Sigmoid
5
Ventral Temporal
Entire Cortex
2,997 voxels/hemisphere
133,590 voxels/hemisphere
Anatomical
Kernel Hyp.
Anatomical
Kernel Hyp.
35.71% [42.68%]
35.00% [43.32%]
36.25% [43.39%]
35.89% [43.21%]
48.57% [42.68%]
50.36% [42.32%]
48.57% [43.39%]
48.21% [43.21%]
34.64% [26.79%]
36.07% [25.54%]
36.07% [26.07%]
35.00% [26.79%]
36.25% [26.79%]
36.43% [25.54%]
36.43% [26.07%]
36.25% [26.79%]
Experiments
The data used in this section consisted of fMRI time-series data from 10 subjects who viewed a
movie and also engaged in a block-design visualization experiment [17]. Each subject saw Raiders
of the Lost Ark (1981) lasting a total of 2213 TRs. In the visualization experiment, subjects were
shown images belonging to a specific class for 16 TRs followed by 10 TRs of rest. The 7 classes
were: (1) female face, (2) male face, (3) monkey, (4) house, (5) chair, (6) shoe and (7) dog. There
were 8 runs total, and each run had every image class represented once.
We assess alignment by classification accuracy. To provide the same number of voxels per ROI for
all subjects, we first performed anatomical alignment. We then selected a contiguous block of 400
TRs from the movie data to serve as the per-subject input of the kernel hyperalignment. Next, we
extracted labeled examples from the visualization experiment by taking an offset time average of
each 16 TR class exposure. An offset of 6 seconds factored in the hemodynamic response. This
produced 560 labeled examples: 10 subjects ? 8 runs/subject ? 7 examples/run.
Kernel hyperalignment allows us to (a) use nonlinear measures of similarity, and (b) consider more
voxels for the alignment. Consequently, we (a) experiment with a variety of kernels, and (b) do not
need to pre-select or screen voxels as was done in [9]?we include them all. Table 1 features results
from a 7-label classification experiment. Recall that a linear kernel reduces to hyperalignment. We
classified using a multi-label ?-SVM [3]. We used the first 400 TRs from each subject?s movie data,
and aligned each hemisphere separately. The kernel functions are supplied in the Supplementary
Material. As observed in [9] and repeated here, hyperalignment leads to increased between-subject
accuracy and outperforms within-subject accuracy. Thus, we are extracting more common structure
across subjects. Whereas employing Algorithm 1 for 2,997 voxels is feasible (and slow), 133,590
voxels is not feasible at all.
To complete the picture, we plot the effects of regularization. Figure 1 displays the cross-validated,
between-subject classification accuracy for varying (?, ?) where ? = 1??. This traces out a route
from CCA (? ? 0) to hyperalignment (? = 1). When compared to the alignments in [9], our voxel
counts are orders of magnitude larger. For our four chosen kernels, hyperalignment (? = 1) presents
itself as the option with near-greatest accuracy.
Our results support the robustness of hyperalignment and imply that voxel selection may be a crucial
pre-processing step when dealing with the whole volume. More voxels mean more noisy voxels,
and hyperalignment does not distinguish itself from anatomical alignment when the entire cortex is
considered. We can visualize this phenomenon with Multidimensional Scaling (MDS) [21].
MDS takes as input all of the pairwise distances between subjects (the previous section discussed
distance calculations). Figure 2 depicts the optimal Euclidean representation of our 10 subjects before and after kernel hyperalignment ((?, ?) = (1, 0)) with respect to the first 400 TRs of the movie
data. Focusing on VT, kernel hyperalignment manages to cluster 7 of the 10 subjects. However,
when we shift to the entire cortex, we see that anatomical alignment has already succeeded in a similar clustering. Kernel hyperalignment manages to group the subjects closer together, and manifests
itself as a re-centering.
7
Quadratic Kernel
Gaussian Kernel
0.55
0.5
0.5
0.45
0.4
0.35
0.45
0.4
0.35
Sigmoid Kernel
0.55
0.5
BSC Accuracy
0.55
0.5
BSC Accuracy
0.55
BSC Accuracy
BSC Accuracy
Linear Kernel
0.45
0.4
0.35
0.45
0.4
0.35
0.3
0.3
0.3
0.3
0.25
0.25
0.25
0.25
0.2
0
0.2
0.4
0.6
? ( = 1-?)
0.8
1
0.2
0
0.2
0.4
0.6
? ( = 1-?)
0.8
1
0.2
0
0.2
0.4
0.6
? ( = 1-?)
0.8
1
0.2
0
0.2
0.4
0.6
? ( = 1-?)
0.8
1
Figure 1: Cross-validated between-subject classification accuracy (7 labels) as a function of the
regularization parameter, ? = 1??, for various kernels after alignment. The solid curves are for
Ventral Temporal and the dashed curves are for the entire cortex. Chance = 1/7 ? 14.29%.
Ventral Temporal
Entire Cortex
4
7
Linear Kernel
5
10
7
52
6
4
1
10
83
5
7
3
91
82 10
34
62
1
9
4
3
56
2
10
1
8
6
8
9
9
7
Gaussian Kernel
5
9
9
43
10218
9
9
16
2
8
10
2
65
4
3
2
5
4
7
8
3
8 6
4
10
10
1
6
7
3
7
5
7
1
Figure 2: Visualizing alignment with MDS Each locus pair approximates the normalized relationship among the 10 subjects in 2D - before (left) and after (right) applying kernel hyperalignment.
Centroids are translated to the origin and numbers correspond to individual subjects.
6
Conclusion
We have extended hyperalignment in both scale and feature space. Kernel hyperalignment can
handle a large number of original features and incorporate nonlinear measures of similarity. We have
also shown how to use the linear maps?applied in feature space?for post-alignment classification.
In the setting of fMRI, we have demonstrated successful alignment with a variety of kernels. Kernel
hyperalignment achieved better between-subject classification over anatomical alignment for VT.
There was no noticeable difference when we considered the entire cortex. Nevertheless, kernel
hyperalignment proved robust and did not degrade with increasing voxel count.
We envision a fruitful path for kernel hyperalignment. Empirically, we have noticed a tradeoff
between feature cardinality and classification accuracy, motivating the need for intelligent feature
selection within our established framework. Although we have limited our focus to fMRI data analysis, kernel hyperalignment can be applied to other research areas which rely on multi-set Procrustes
problems.
8
References
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9
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3,969 | 4,593 | Homeostatic plasticity in Bayesian spiking networks as
Expectation Maximization with posterior constraints
Stefan Habenschuss? , Johannes Bill? , Bernhard Nessler
Institute for Theoretical Computer Science, Graz University of Technology
{habenschuss,bill,nessler}@igi.tugraz.at
Abstract
Recent spiking network models of Bayesian inference and unsupervised learning
frequently assume either inputs to arrive in a special format or employ complex
computations in neuronal activation functions and synaptic plasticity rules. Here
we show in a rigorous mathematical treatment how homeostatic processes, which
have previously received little attention in this context, can overcome common
theoretical limitations and facilitate the neural implementation and performance of
existing models. In particular, we show that homeostatic plasticity can be understood as the enforcement of a ?balancing? posterior constraint during probabilistic inference and learning with Expectation Maximization. We link homeostatic
dynamics to the theory of variational inference, and show that nontrivial terms,
which typically appear during probabilistic inference in a large class of models,
drop out. We demonstrate the feasibility of our approach in a spiking WinnerTake-All architecture of Bayesian inference and learning. Finally, we sketch how
the mathematical framework can be extended to richer recurrent network architectures. Altogether, our theory provides a novel perspective on the interplay of
homeostatic processes and synaptic plasticity in cortical microcircuits, and points
to an essential role of homeostasis during inference and learning in spiking networks.
1
Introduction
Experimental findings from neuro- and cognitive sciences have led to the hypothesis that humans
create and maintain an internal model of their environment in neuronal circuitry of the brain during
learning and development [1, 2, 3, 4], and employ this model for Bayesian inference in everyday
cognition [5, 6]. Yet, how these computations are carried out in the brain remains largely unknown.
A number of innovative models has been proposed recently which demonstrate that in principle,
spiking networks can carry out quite complex probabilistic inference tasks [7, 8, 9, 10], and even
learn to adapt to their inputs near optimally through various forms of plasticity [11, 12, 13, 14, 15].
Still, in network models for concurrent online inference and learning, most approaches introduce
distinct assumptions: Both [12] in a spiking Winner-take-all (WTA) network, and [15] in a rate based
WTA network, identified the limitation that inputs must be normalized before being presented to the
network, in order to circumvent an otherwise nontrivial (and arguably non-local) dependency of the
intrinsic excitability on all afferent synapses of a neuron. Nessler et al. [12] relied on population
coded input spike trains; Keck et al. [15] proposed feed-forward inhibition as a possible neural
mechanism to achieve this normalization. A theoretically related issue has been encountered by
Deneve [7, 11], in which inference and learning is realized in a two-state Hidden Markov Model by
a single spiking neuron. Although synaptic learning rules are found to be locally computable, the
learning update for intrinsic excitabilities remains intricate. In a different approach, Brea et al. [13]
have recently proposed a promising model for Bayes optimal sequence learning in spiking networks
?
These authors contributed equally to this work.
1
in which a global reward signal, which is computed from the network state and synaptic weights,
modulates otherwise purely local learning rules. Also the recent innovative model for variational
learning in recurrent spiking networks by Rezende et al. [14] relies on sophisticated updates of
variational parameters that complement otherwise local learning rules.
There exists great interest in developing Bayesian spiking models which require minimal nonstandard neural mechanisms or additional assumptions on the input distribution: such models are
expected to foster the analysis of biological circuits from a Bayesian perspective [16], and to provide a versatile computational framework for novel neuromorphic hardware [17]. With these goals
in mind, we introduce here a novel theoretical perspective on homeostatic plasticity in Bayesian
spiking networks that complements previous approaches by constraining statistical properties of the
network response rather than the input distribution. In particular we introduce ?balancing? posterior
constraints which can be implemented in a purely local manner by the spiking network through a
simple rule that is strongly reminiscent of homeostatic intrinsic plasticity in cortex [18, 19]. Importantly, it turns out that the emerging network dynamics eliminate a particular class of nontrivial
computations that frequently arise in Bayesian spiking networks.
First we develop the mathematical framework for Expectation Maximization (EM) with homeostatic
posterior constraints in an instructive Winner-Take-all network model of probabilistic inference and
unsupervised learning. Building upon the theoretical results of [20], we establish a rigorous link
between homeostatic intrinsic plasticity and variational inference. In a second step, we sketch how
the framework can be extended to recurrent spiking networks; by introducing posterior constraints
on the correlation structure, we recover local plasticity rules for recurrent synaptic weights.
2
Homeostatic plasticity in WTA circuits as EM with posterior constraints
We first introduce, as an illustrative and representative example, a generative mixture model
p(z, y|V ) with hidden causes z and binary observed variables y, and a spiking WTA network N
which receives inputs y(t) via synaptic weights V . As shown in [12], such a network N can
implement probabilistic inference p(z|y, V ) through its spiking dynamics, and maximum likelihood learning through local synaptic learning rules (see Figure 1A). The mixture model comprises
PK
K binary and mutually exclusive components zk ? {0, 1},
k=1 zk = 1, each specialized on a
different N -dimensional input pattern:
p(y, z|V ) =
K
Y
?
bk zk
e
N
Y
zk
(?ki )yi ? (1 ? ?ki )1?yi
(1)
i=1
k=1
!
? log p(y, z|V ) =
X
k
with
X
zk
X
Vki yi ? Ak + ?bk
,
(2)
i
?
ebk = 1 and ?ki = ?(Vki ) and Ak =
X
log(1 + eVki ) ,
(3)
i
k
where ?(x) = (1 + exp(?x))?1 denotes the logistic function, and ?ki the expected activation of
input i under the mixture component k. For simplicity and notational convenience, we will treat the
prior parameters ?bk as constants throughout the paper. Probabilistic inference of hidden causes zk
based on an observed input y can be implemented by a spiking WTA network N of K neurons
which fire with the instantaneous spiking probability (for ?t ? 0),
euk (t)
p(zk spikes in [t, t + ?t]) = ?t ? rnet ? P u (t) ? p(zk = 1|y, V ) ,
(4)
j
je
P
?
with the input potential uk (t) =
i Vki yi (t) ? Ak + bk . Each WTA neuron k receives spiking inputs yi via synaptic weights Vki and responds with an instantaneous spiking probability
which depends exponentially on its input potential uk in accordance with biological findings [21].
Stochastic winner-take-all (soft-max) competition between the neurons is modeled via divisive
normalization (4) [22]. The input is defined as yi (t) = 1 if input neuron i emitted a spike within the
last ? milliseconds, and 0 otherwise, corresponding to a rectangular post-synaptic potential (PSP) of
length ? . We define zk (t) = 1 at spike times t of neuron k and zk (t) = 0 otherwise.
2
Figure 1: A. Spiking WTA network model. B. Input templates from MNIST database (digits 0-5)
are presented in random order to the network as spike trains (the input template switches after every
250ms, black/white pixels are translated to high/low firing rates between 20 and 90 Hz). C. Sketch
of intrinsic homeostatic plasticity maintaining a certain target average activation. D. Homeostatic
plasticity induces average firing rates (blue) close to target values (red). E. After a learning period,
each WTA neuron has specialized on a particular input motif. F. WTA output spikes during a test
phase before and after learning. Learning leads to a sparse output code.
In addition to the spiking input, each neuron?s potential uk features an intrinsic excitability ?Ak +?bk .
Note that, besides the prior constant ?bk , this excitability depends on the normalizing term Ak , and
hence on all afferent synaptic weights through (3): WTA neurons which encode strong patterns
with high probabilities ?ki require lower intrinsic excitabilities, while neurons with weak patterns
require larger excitabilities. In the presence of synaptic plasticity, i.e., time-varying Vki , it is unclear
how biologically realistic neurons could communicate ongoing changes in synaptic weights from
distal synaptic sites to the soma. This critical issue was apparently identified in [12] and [15]; both
papers circumvent the problem (in similar probabilistic models) by constraining the input y (and
also the synaptic weights in [15]) in order to maintain constant and uniform values Ak across all
WTA neurons.
Here, we propose a different approach to cope with the nontrivial computations Ak during inference
and learning in the network. Instead of assuming that the inputs y meet a normalization constraint,
we constrain the network response during inference, by applying homeostatic dynamics to the intrinsic excitabilities. This approach turns out to be beneficial in the presence of time-varying synaptic
weights, i.e., during ongoing changes of Vki and Ak . The resulting interplay of intrinsic and synaptic
plasticity can be best understood from the standard EM lower bound [23],
F (V , q(z|y)) = L(V ) ? h KL (q(z|y) || p(z|y, V ) ip? (y)
= h log p(y, z|V ) ip? (y)q(z|y) + h H(q(z|y)) ip? (y)
? E-step ,
(5)
? M-step ,
(6)
where L(V ) = hlog p(y|V )ip? (y) denotes the log-likelihood of the input under the model, KL (? || ?)
the Kullback-Leibler divergence, and H(?) the entropy. The decomposition holds for arbitrary distributions q. In hitherto proposed neural implementations of EM [11, 12, 15, 24], the network
implements the current posterior distribution in the E-step, i.e., q = p and KL (q || p) = 0. In
contrast, by applying homeostatic plasticity, the network response will be constrained to implement
a variational posterior from a class of ?homeostatic? distributions Q: the long-term average activation of each WTA neuron zk is constrained to an a priori defined target value. Notably, we will
see that the resulting network response q ? describes an optimal variational E-Step in the sense that
q ? (z|y) = arg minq?Q KL (q(z|y) || p(z|y, V )). Importantly, homeostatic plasticity fully regulates the intrinsic excitabilities, and as a side effect eliminates the non-local terms Ak in the E-step,
3
while synaptic plasticity of the weights Vki optimizes the underlying probabilistic model p(y, z|V )
in the M-step.
In summary, the network response implements q ? as the variational E-step, the M-Step can be performed via gradient ascent on (6) with respect to Vki . As derived in section 2.1, this gives rise to
the following temporal dynamics and plasticity rules in the spiking network, which instantiate a
stochastic version of the variational EM scheme:
X
uk (t) =
Vki yi (t) + bk ,
b? k (t) = ?b ? (rnet ? mk ? ?(zk (t) ? 1)) ,
(7)
i
V? ki (t) = ?V ? ?(zk (t) ? 1) ? (yj (t) ? ?(Vki )) ,
(8)
where ?(?) denotes the Dirac delta function, and ?b , ?V are learning rates (which were kept timeinvariant in the simulations with ?b = 10 ? ?V ). Note that (8) is a spike-timing dependent plasticity
rule (cf. [12]) and is non-zero only at post-synaptic spike times t, for which zk (t) = 1. The effect of
the homeostatic intrinsic plasticity rule (7) is illustrated in Figure 1C: it aims to keep the long-term
average activation of each WTA neuron k close to a certain target value mk . More precisely, if rk is
a neuron?s long-term average firing rate, then homeostatic plasticity will ensure that rkP
/rnet ? mk .
The target activations mk ? (0, 1) can be chosen freely with the obvious constraint that k mk = 1.
Note that (7) is strongly reminiscent of homeostatic intrinsic plasticity in cortex [18, 19].
We have implemented these dynamics in a computer simulation of a WTA spiking network N .
Inputs y(t) were defined by translating handwritten digits 0-5 (Figure 1B) from the MNIST
dataset [25] into input spike trains. Figure 1D shows that, at the end of a 104 s learning period,
homeostatic plasticity has indeed achieved that rk ? rnet ? mk . Figure 1E illustrates the patterns
learned by each WTA neuron after this period (shown are the ?ki ). Apparently, the WTA neurons have specialized on patterns of different intensity which correspond to different values of Ak .
Figure 1F shows the output spiking behavior of the circuit before and after learning in response to a
set of test patterns. The specialization to different patterns has led to a distinct sparse output code,
in which any particular test pattern evokes output spikes from only one or two WTA neurons. Note
that homeostasis forces all WTA neurons to participate in the competition, and thus prevents neurons
from becoming underactive if their synaptic weights decrease, and from becoming overactive if their
synaptic weights increase, much like the original Ak terms (which are nontrivial to compute for the
network). Indeed, the learned synaptic parameters and the resulting output behavior corresponds to
what would be expected from an optimal learning algorithm for the mixture model (1)-(3).1
2.1
Theory for the WTA model
In the following, we develop the three theoretical key results for the WTA model (1)-(3):
? Homeostatic intrinsic plasticity finds the network response distribution q ? (z|y) ? Q closest to the posterior distribution p(z|y, V ), from a set of ?homeostatic? distributions Q.
? The interplay of homeostatic and synaptic plasticity can be understood from the perspective
of variational EM.
? The critical non-local terms Ak defined by (3) drop out of the network dynamics.
E-step: variational inference with homeostasis
The variational distribution q(z|y) we consider for the model (1)-(3) is a 2N ? K dimensional object.
Since q describes a conditional probability distribution, it is non-negative and normalized for all y.
In addition, we constrain q to be a ?homeostatic? distribution q ? Q such that the average activation
of each hidden variable (neuron) zk equals an a-priori specified mean activation mk under the input
statistics p? (y). This is sketched in Figure 2. Formally we define the constraint set,
X
Q = {q : hzk ip? (y)q(z|y) = mk , for all k = 1 . . . K} ,
with
mk = 1 .
(9)
k
1
Without adaptation of intrinsic excitabilities, the network would start performing erroneous inference,
learning would reinforce this erroneous behavior, and performance would quickly break down. We have verified
this in simulations for the present WTA model: Consistently across trials, a small subset of WTA neurons
became dominantly active while most neurons remained silent.
4
Figure 2: A. Homeostatic posterior constraints in the WTA model: Under the variational distribution q, the average activation of each variable zk must equal mk . B. For each set of synaptic
weights V there exists a unique assignment of intrinsic excitabilities b, such that the constraints are
fulfilled. C. Theoretical decomposition of the intrinsic excitability bk into ?Ak , ?bk and ?k . D. During variational EM the bk predominantly ?track? the dynamically changing non-local terms ?Ak
(relative comparison between two WTA neurons from Figure 1).
The constrained maximization problem q ? (z|y) = arg maxq?Q F (V , q(z|y)) can be solved with
the help of Lagrange multipliers (cf. [20]). We find that the q ? which maximizes the objective
function F during the E-step (and thus minimizes the
PKL-divergence to the posterior p(z|y, V ))
has the convenient form q ? (z|y) ? p(z|y, V ) ? exp( k ?k? zk ) with some ?k? . Hence, it suffices to
consider distributions of the form,
X
X
q? (z|y) ? exp(
zk (
Vki yi + ?bk ? Ak + ?k )) ,
(10)
|
{z
}
i
k
=:bk
for the maximization problem. We identify ?k as the variational parameters which remain to be
optimized. Note that any distribution of this form can be implemented by the spiking network N
if the intrinsic excitabilities are set to bk = ?Ak + ?bk + ?k . The optimal variational distribution
q ? (z|y) = q?? (z|y) then has ? ? = arg max? ?(?), i.e. the variational parameter vector which
maximizes the dual [20],
X
X
X
?k zk )ip? (y) .
(11)
p(z|y, V ) exp(
?(?) =
?k mk ? hlog
k
z
k
Due to concavity of the dual, a unique global maximizer ? ? exists, and thus also the corresponding
optimal intrinsic excitabilities b?k = ?Ak +?bk +?k? are unique. Hence, the posterior constraint q ? Q
can be illustrated as in Figure 2B: For each synaptic weight configuration V there exists, under
a particular input distribution p? (y), a unique configuration of intrinsic excitabilities b such that
the resulting network output fulfills the homeostatic constraints. The theoretical relation between
the intrinsic excitabilities bk , the original nontrivial term ?Ak and the variational parameters ?k
is sketched in Figure 2C. Importantly, while bk is implemented in the network, Ak , ?k and ?bk
are not explicitly represented in the implementation anymore. Finding the optimal b in the dual
perspective, i.e. those intrinsic excitabilities which fulfill the homeostatic constraints, amounts to
gradient ascent ?? ?(?) on the dual, which leads to the following homeostatic learning rule for the
intrinsic excitabilities,
?bk ? ??k ?(?) = mk ? hzk ip? (y)q(z|y) .
(12)
Note that the intrinsic homeostatic plasticity rule (7) in the network corresponds to a sample-based
stochastic version of this theoretically derived adaptation mechanism (12). Hence, given enough
time, homeostatic plasticity will automatically install near-optimal intrinsic excitabilities b ? b? and
implement the correct variational distribution q ? up to stochastic fluctuations in b due to the nonzero learning rate ?b . The non-local terms Ak have entirely dropped out of the network dynamics,
since the intrinsic excitabilities bk can be arbitrarily initialized, and are then fully regulated by the
local homeostatic rule, which does not require knowledge of Ak .
As a side remark, note that although the variational parameters ?k are not explicitly present
in the implementation, they can be theoretically recovered from the network at any point, via
5
Figure 3: A. Input templates from MNIST dataset (digits 0,3 at a ratio 2:1, and digits 0,3,4 at a ratio
1:1:1) used during the first and second learning period, respectively. B. Learned patterns at the end
of each learning period. C. Network performance converges in the course of learning. F is a tight
lower bound to L. D. Illustration of pattern learning and re-learning dynamics in a 2-D projection in
the input space. Each black dot corresponds to the pattern ?ki of one WTA neuron k. Colored dots
are input samples from the training set (blue/green/red ? digits 0/3/4).
?k = bk + Ak ? ?bk . Notably, in all our simulations we have consistently found small absolute values of ?k , corresponding to a small KL-divergence between q ? and p.2 Hence, a major effect of the
local homeostatic plasticity rule during learning is to dynamically track and effectively implement
the non-local terms ?Ak . This is shown in Figure 2D, in which the relative excitabilities of two
WTA neurons bk ? bj are plotted against the corresponding non-local Ak ? Aj over the course of
learning in the first simulation (Figure 1).
M-step: interplay of synaptic and homeostatic intrinsic plasticity
During the M-step, we aim to increase the EM lower bound F in (6) w.r.t. the synaptic parameters V .
Gradient ascent yields,
?Vki F (V , q(z|y)) = h?Vki log p(y, z|V )ip? (y)q(z|y)
(13)
= h zk ? (yj ? ?(Vki )) ip? (y)q(z|y) ,
(14)
?
where q is the variational distribution determined during the E-step, i.e., we can set q = q . Note the
formal correspondence of (14) with the network synaptic learning rule (8). Indeed, if the network
activity implements q ? , it can be shown easily that the expected update of synaptic weights due to
the synaptic plasticity (8) is proportional to (14), and hence implements a stochastic version of the
theoretical M-step (cf. [12]).
2.2
Dynamical properties of the Bayesian spiking network with homeostasis
To highlight a number of salient dynamical properties emerging from homeostatic plasticity in the
considered WTA model, Figure 3 shows a simulation of the same network N with homeostatic
dynamics as in Figure 1, only with different input statistics presented to the network, and uniform
1
mk = K
. During the first 5000s, different writings of 0?s and 3?s from the MNIST dataset were
presented, with 0?s occurring twice as often as 3?s. Then the input distribution p? (y) abruptly
switched to include also 4?s, with each digit occurring equally often. The following observations
can be made: Due to the homeostatic constraint, each neuron responds on average to mk ? T out of T
presented inputs. As a consequence, the number of neurons which specialize on a particular digit is
2
This is assuming for simplicity uniform prior parameters ?bk . Note that a small KL-divergence is in fact
often observed during variational EM since F , which contains the negative KL-divergence, is being maximized.
6
directly proportional to the frequency of occurrence of that digit, i.e. 8:4 and 4:4:4 after the first and
second learning period, respectively (Figure 3B). In general, if uniform target activations mk are
chosen, output resources are allocated precisely in proportion to input frequency. Figure 3C depicts
the time course of the EM lower bound F as well as the average likelihood L (assuming uniform ?bk )
under the model during a single simulation run, demonstrating both convergence and tightness of
the lower bound. As expected due to the stabilizing dynamics of homeostasis, we found variability
in performance among different trials to be small (not shown). Figure 3D illustrates the dynamics
of learning and re-learning of patterns ?ki in a 2D projection of input patterns onto the first two
principal components.
3
Homeostatic plasticity in recurrent spiking networks
The neural model so far was essentially a feed-forward network, in which every postsynaptic spike
can directly be interpreted as one sample of the instantaneous posterior distribution [12]. The lateral
inhibition served only to ensure the normalization of the posterior. We will now extend the concept
of homeostatic processes as posterior constraints to the broader class of recurrent networks and
sketch the utility of the developed framework beyond the regulation of intrinsic excitabilities.
Recently it was shown in [9, 10] that recurrent networks of stochastically spiking neurons can in
principle carry out probabilistic inference through a sampling process. At every point in time, the
joint network state z(t) represents one sample of a posterior. However, [9] and [10] did not consider
unsupervised learning on spiking input streams.
For the following considerations, we divide the definition of the probabilistic model in two parts.
First, we define a Boltzmann distribution,
X
X
?bk zk + 1
? kj zk zj )/norm. ,
p(z) = exp(
W
(15)
2
k
j6=k
? kj = W
? jk as ?prior? for the hidden variables z which will be represented by a recurrently
with W
? kj as
connected network of K spiking neurons. For the purpose of this section, we treat ?bk and W
constants. Secondly, we define a conditional distribution in the exponential-family form [23],
X
p(y|z, V ) = exp(f0 (y) +
Vki zk fi (y) ? A(z, V )) ,
(16)
k,i
that specifies the likelihood of observable inputs y, given a certain network state z. This defines the
generative model p(y, z|V ) = p(z) p(y|z, V ).
We map this probabilistic model to the spiking network and define that for every k and every point
in time t the variable zk (t) has the value 1, if the corresponding neuron has fired within the time
window (t ? ?, t]. In accordance with the neural sampling theory, in order for a spiking network to
sample from the correct posterior p(z|y, V ) ? p(z) p(y|z, V ) given the input y, each neuron must
compute in its membrane potential the log-odd [9],
X
p(zk = 1|z \k , V ) X
? kj )zj ? . . .
=
Vki fi (y) ?Ak (V ) + ?bk +
uk = log
(?Akj (V ) + W
|
{z
}
p(zk = 0|z \k , V )
{z
}
|
i
j6=k
recurrent weight
|
{z
} intr. excitability
feedforward drive
(17)
where z \k = (z1 , . . . , zk?1 , zk+1 , . . . zK )T . The Ak , Akj , . . . are given by the decomposition of
A(z, V ) along the binary combinations of z as,
X
1X
A(z, V ) = A0 (V ) +
zk Ak (V ) +
zk zj Akj (V ) + . . .
(18)
2
j6=k
k
? kj . InNote, that we do not aim at this point to give learning rules for the prior parameters ?bk and W
stead we proceed as in the last section and specify a-priori desired properties of the average network
response under the input distribution p? (y),
ckj = hzk zj ip? (y)q(z|y)
and
7
mk = hzk ip? (y)q(z|y) .
(19)
Let us explore some illustrative configurations for mk and ckj . One obvious choice is closely re1
lated to the goal of maximizing the entropy of the output code by fixing hzk i to K
and hzk zj i
1
to hzk ihzj i = K 2 , thus enforcing second order correlations to be zero. Another intuitive choice
would be to set all hzk zj i very close to zero, which excludes that two neurons can be active simultaneously and thus recovers the function of a WTA. It is further conceivable to assign positive
correlation targets to groups of neurons, thereby creating populations with redundant codes. Finally,
with a topographical organization of neurons in mind, all three basic ideas sketched above might
be combined: one could assign positive correlations to neighboring neurons in order to create local cooperative populations, mutual exclusion at intermediate distance, and zero correlation targets
between distant neurons.
With this in mind, we can formulate the goal of learning for the network in the context of EM
with posterior constraints: we constrain the E-step such that the average posterior fulfills the chosen
targets, and adapt the forward weights V in the M-step according to (6). Analogous to the first-order
case, the variational solution of the E-step under these constraints takes the form,
?
?
X
X
1
q?,? (z|y) ? p(z|y, V ) ? exp ?
?k zk +
?kj zk zj ? ,
(20)
2
k
j6=k
with symmetric ?kl = ?lk as variational parameters. A neural sampling network N with input
weights Vki will sample from q?,? if the intrinsic excitabilities are set to bk = ?Ak + ?bk + ?k , and
? kj + ?kj . The variational parameters
the symmetric recurrent synaptic weights to Wkj = ?Akj + W
?, ? (and hence also b, W ) which optimize the dual problem ?(b, ?) are uniquely defined and
can be found iteratively via gradient ascent. Analogous to the last section, this yields the intrinsic
plasticity rule (12) for bk . In addition, we obtain for the recurrent synapses Wkj ,
?Wkj ? ckj ? hzk zj ip? (y)q(z|y) ,
(21)
which translates to an anti-Hebbian spike-timing dependent plasticity rule in the network implementation.
For any concrete instantiation of f0 (y), fi (y) and A(z, V ) in (16) it is possible to derive learning
rules for Vki for the M-step via ?Vki F (V , q). Of course not all models entail local synaptic learning
rules. In particular it might be necessaryQ
to assume conditional independence of the inputs y given
the network state z, i.e., p(y|z, V ) = i p(yi |z, V ). Furthermore, in order to fulfill the neural
computability condition (17) for neural sampling [9] with a recurrent network of point neurons, it
might be necessary to choose A(z, V ) such that terms of order higher than 2 vanish in the decomposition. This can be shown to hold, for example, in a model with conditionally independent Gaussian
distributed inputs yi . It is ongoing work to find further biologically realistic network models in the
sense of this theory and to assess their computational capabilities through computer experiments.
4
Discussion
Complex and non-local computations, which appear during probabilistic inference and learning, arguably constitute one of the cardinal challenges in the development of biologically realistic Bayesian
spiking network models. In this paper we have introduced homeostatic plasticity, which to the best
of our knowledge had not been considered before in the context of EM in spiking networks, as a
theoretically grounded approach to stabilize and facilitate learning in a large class of network models. Our theory complements previously proposed neural mechanisms and provides, in particular,
a simple and biologically realistic alternative to the assumptions on the input distribution made in
[12] and [15]. Indeed, our results challenge the hypothesis of [15] that feedforward inhibition is
critical for correctly learning the structure of the data with biologically plausible plasticity rules.
More generally, it turns out that the enforcement of a balancing posterior constraint often simplifies
inference in recurrent spiking networks by eliminating nontrivial computations. Our results suggest
a crucial role of homeostatic plasticity in the Bayesian brain: to constrain activity patterns in cortex
to assist the autonomous optimization of an internal model of the environment.
Acknowledgments. Written under partial support by the European Union - projects #FP7-269921
(BrainScaleS), #FP7-216593 (SECO), #FP7-237955 (FACETS-ITN), #FP7-248311 (AMARSi),
#FP7-216886 (PASCAL2) - and the Austrian Science Fund FWF #I753-N23 (PNEUMA).
8
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9
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3,970 | 4,594 | Controlled Recognition Bounds for Visual Learning
and Exploration
Vasiliy Karasev1
1
Alessandro Chiuso2
University of California, Los Angeles
2
Stefano Soatto1
University of Padova
Abstract
We describe the tradeoff between the performance in a visual recognition problem
and the control authority that the agent can exercise on the sensing process. We
focus on the problem of ?visual search? of an object in an otherwise known and
static scene, propose a measure of control authority, and relate it to the expected
risk and its proxy (conditional entropy of the posterior density). We show this analytically, as well as empirically by simulation using the simplest known model that
captures the phenomenology of image formation, including scaling and occlusions.
We show that a ?passive? agent given a training set can provide no guarantees on
performance beyond what is afforded by the priors, and that an ?omnipotent? agent,
capable of infinite control authority, can achieve arbitrarily good performance
(asymptotically). In between these limiting cases, the tradeoff can be characterized
empirically.
1
Introduction
We are interested in visual learning for recognition of objects and scenes embedded in physical space.
Rather than using datasets consisting of collections of isolated snapshots, however, we wish to
actively control the sensing process during learning. This is because, in the presence of nuisance
factors involving occlusion and scale changes, learning requires mobility [1]. Visual learning is thus a
process of discovery, literally uncovering occluded portions of an object or scene, and viewing it from
close enough that all structural details are revealed.1 We call this phase of learning exploration or
mapping, accomplished by actively controlling the sensor motion within a scene, or by manipulating
an object so as to discover all aspects.2
Once exploration has been performed, one has a model (or ?map? or ?representation?) of the scene or
object of interest. One can then attempt to detect, localize or recognize a particular object or scene, or
a class of them, provided intra-class variability has been exposed during exploration. This phase can
yield localization ? where one wishes to recognize a portion of a mapped scene and, as a byproduct,
infer the pose relative to the map ? or search where a particular object mapped during the exploration
phase is detected and localized within an otherwise known scene. This can also be interpreted as a
change detection problem, where one wishes to revisit a known map to detect changes. In the case
1
It has been shown [1] that mobility is required in order to reduce the Actionable Information Gap, the
difference between the complexity of a maximal invariant of the data and the minimal sufficient statistic of a
complete representation of the underlying scene.
2
Note that we are not suggesting that one should construct a three-dimensional (3-D) model of an object or a
scene for recognition, as opposed to using collections of 2-D images. From an information perspective, there is
no gain in replacing a collection of 2-D images with a 3-D model computed from them. What matters is how
these images are collected. The multiple images must portray the same scene or object, lest one cannot attribute
the variability in the data to nuisance factors as opposed to intrinsic variability of the object of interest. The
multiple images must enable establishing correspondence between different images of the same scene. Temporal
continuity enables that.
1
where a known object is sought in an unknown map, exploration and search have to be conducted
simultaneously.
Within this scenario, exploration and search can be framed as optimal control and optimal stopping
time problems. These relate to active vision (next-best-view generation), active learning, robotic
motion planning, sequential decision in the setting of partially-observable Markov decision processes
(POMDP) and a number of related fields (including Information Bottleneck, Value of Information)
and a vast literature that we cannot extensively review here. As often in this class of problems,
inference algorithms are essentially intractable, so we wish to design surrogate tasks and prove
performance bounds to ensure desirable properties of the surrogate solution.
In this manuscript we consider the problem of detecting and estimating discrete parameters of an
unknown object in a known environment. To this purpose we:
1. Describe the simplest model that includes scaling and occlusion nuisances, a two dimensional
?cartoon flatland,? and a test suite to perform simulation experiments. We derive an explicit
probability model to compute the posterior density given photometric measurements.
2. Discuss the tradeoff between performance in a visual decision task and the control authority
that the explorer possesses. This tradeoff is akin the tradeoff between rate and distortion
in a communication system, but it pertains to decision and control tasks, as opposed to the
transmission of data. We characterize this tradeoff for the simple case of a static environment,
where control authority relates to reachability and energy.
3. Discuss and test algorithms for visual search based on the maximization of the conditional
entropy of future measurements and the proxies of this quantity. These algorithms can
be used to locate an unknown object in unknown position of a known environment, or to
perform change detection in an otherwise known map, for the purpose of updating it.
4. Provide experimental validation of the algorithms, including regret and expected exploration
length.
1.1
Related prior work
Active search and recognition of objects in the scene has been one of the mainstays of Active
Perception in the eighties [2, 3], and has recently resurged (see [4] and references therein). The
problem can be formulated as a POMDP [5], solving which requires developing approximate, nearoptimal policies. Active recognition using next-best-view generation and object appearance is
discussed in [6] where authors use PCA to embed object images in a linear, low dimensional space.
The scheme does not incorporate occlusions or scale changes. More recently, information driven
sensor control for object recognition was used in [7, 8, 9], who deal with visual and sonar sensors,
but take features (e.g. SIFT, SURF) to be the observed data. A utility function that accounts for
occlusions, viewing angle, and distance to the object is proposed in [10] who aim to actively learn
object classifiers during the training stage. Exploration and learning of 3D object surface models by
robotic manipulation is discussed in [11]. The case of object localization (and tracking if object is
moving) is discussed in [12]; information-theoretic approach for solving this problem using a sensor
network is described in [13]. Both authors used realistic, nonlinear sensor models, which however are
different from photometric sensors and are not affected by the same nuisances. Typically, informationtheoretic utility functions used in these problems are submodular and thus can be efficiently optimized
by greedy heuristics [14, 15]. With regards to models, our work is different in several aspects: instead
of choosing the next best view on a sphere centered at the object, we model a cluttered environment
where the object of interest occupies a negligible volume and is therefore fully occluded when viewed
from most locations. Second, we wish to operate in a continuous environment, rather than in a world
that is discretized at the outset. Third, given the significance of quantization-scale and occlusions in a
visual recognition task, we model the sensing process such that it accounts for both.
2
Preliminaries
Let y 2 Y denote data3 (measurements) and x 2 X a hidden class variable from a finite alphabet
that we are interested in inferring. If prior p(x) and conditional distributions p(y|x) are known, the
3
Random variables will be displayed in boldface (e.g. y), and realizations in regular fonts (e.g. y).
2
expected risk can be written as
Pe =
Z
max p(xi |y))dy
p(y)(1
i
(1)
and minimized by Bayes? decision rule, which chooses the class label with maximum a posteriori
probability. If the distributions above are estimated empirically, the expected risk depends on the data
set. We are interested in controlling the data acquisition process so as to make this risk as small as
possible. We use the problem of visual search (finding a not previously seen object in a scene) as
a motivation. It is related to active learning and experimental design. In order to enforce temporal
continuity, we model the search agent (?explorer?) as a dynamical system of the form:
(
?t+1 = ?t
gt+1 = f (gt , ut )
(2)
yt = h(gt , ?) + nt
where gt denotes the pose state at time t, ut denotes the control, and ? denotes the scene that
describes the search environment ? a collection of objects (simply-connected surfaces supporting a
radiance function) of which the target x is one instance. Constraints on the controller enter through f ;
photometric nuisances, quantization and occlusions enter through the measurement map h. Additive
and unmodeled phenomena that affect observed data are incorporated into nt , the ?noise? term.
2.1
Signal models
The simplest model that includes both scaling and occlusion nuisances is the ?cartoon flatland?,
where a bounded subset of R2 is populated by self-luminous line segments, corresponding to clutter
objects. We denote an instance of this model, the scene, by ? = ( 1 , . . . , C ), which is a collection
of C objects k . The number of objects in the scene C is the clutter density parameter that can
possibly grow to be infinite in the limit. Each object is described by its center (ck ), length (lk ),
binary orientation (ok ), and radiance function supported on the segment ?k . This is the ?texture? or
?appearance? of the object, which in the simplest case can be assumed to be a constant function:
k
= (ck , lk , ok , ?k ) 2 [0, 1]3 ? {0, 1} ? [R2 ! R+ ]
(3)
An agent can move continuously throughout the search domain. We take the state gt 2 R2 to be its
current position, ut 2 R2 the currently exerted move, and assume trivial dynamics: gt+1 = gt + ut .
More complex agents where gt 2 SE(3) can be incorporated without conceptual difficulties.
The measurement model is that of an omnidirectional m-pixel camera, with each entry of yt 2 Rm
in (2) given by:
Z (i+ 12 ) 2?
Z 1
m
yt (i) =
?`(?,gt ) (z)d?d? + nt (i), with z = (? cos(?), ? sin(?))
(4)
(i
1 2?
2) m
0
where
is the angle subtended by each pixel. The integrand is a collection of radiance functions
which are supported on objects (line segments). Because of occlusions, only the closest objects that
intersect the pre-image contribute to the image. The index of the object (clutter or object of interest)
that contributes to the image is denoted by `(?, gt ) and is defined as:
?
?
n
o
l k lk
ok
`(?, gt ) = arg min k 9(sk , k ) 2 [
, ] ? R+ s.t. ck +
sk = g + g?(?) k (5)
1 ok
k
2 2
2?
m
Above, g and g?(?) = (cos(?), sin(?)) are current position and direction, respectively. ck , lk , and ok
are k-th segment center, length, and orientation. Condition ck + sk = g + g?(?) k encodes intersection
of ray g + g?(?) with a point on a segment k. The segment closest to viewer, i.e. one that is visible, has
the smallest k . Integration over 2?
m in (4) accounts for quantization, and the layer model (5) describes
occlusions. While the measurement model is non-trivial (in particular, it is not differentiable), it is
the simplest that captures the nuisance phenomenology. All unmodeled phenomena are lumped in the
additive term nt , which we assume to be zero-mean Gaussian ?noise? with covariance 2 I.
In order to design control sequences to minimize risk, we need to evaluate the uncertainty of future
measurements, those we have not yet measured, which are a function of the control action to be taken.
To that end, we write the probability model for computing the posterior and the predictive density.
3
We first describe the general case of visual exploration where the environment is unknown. We begin
with noninformative prior for objects k = 1, . . . , C
p(
k)
= p(ck )p(lk )p(ok )p(?k ) = U [0, Nc ]2 ? Exp( ) ? Ber(1/2) ? U [0, N? ]
(6)
where U ,Exp and Ber denote uniform, exponential, and Bernoulli distributions parameterized by
Nc , , and N? . Then p(?) = p( 1 , ..., C ). The posterior is then computed by Bayes rule4 :
p(?|y t , g t ) /
t
Y
? =1
p(y? |g? , ?)p(?) =
t
Y
? =1
N (y?
h(g? , ?);
2
I)p(?)
(7)
Above, N (z, ?) denotes the value of a zero-mean Gaussian density with covariance ? at z. The
density can be decomposed as a product of likelihoods since knowledge of environment (?) and
location (gt ) is sufficient to predict measurement yt up to Gaussian noise. The predictive distribution
(distribution of the next measurement conditioned on the past) is computed by marginalization:
Z
p(yt+1 |y t , g t , gt+1 ) =
p(?|y t , g t , gt+1 )p(yt+1 |?, y t , gt+1 )d?
(8)
Z
=
p(?|y t , g t )N (yt+1 h(gt+1 , ?), 2 I)d?
(9)
The marginalization above is essentially intractable. In this paper we focus on visual search of a
particular object in an otherwise known environment, so marginalization is only performed with
respect to a single object in the environment, x, whose parameters are discrete, but otherwise
analogous to (6):
p(x) = U {0, ..., Nc
1}2 ? Exp( ) ? Ber(1/2) ? U {0, . . . , N?
1}
(10)
We denote by xi , i = 1, ..., |X | object with parameters (ci , li , oi , ?i ) and write ?i = (xi , 1 , . . . , C )
to denote the scene with known clutter objects 1 , ..., C augmented by an unknown object xi . In this
case, we have:
t
Y
p(x|y t , g t ) /
N (y? h(g? , ?); 2 I)p(x)
(11)
? =1
p(yt+1 |y t , g t , gt+1 ) =
3
|X |
X
i=1
p(xi |y t , g t )N (yt+1
h(gt+1 , ?i ),
2
I)
(12)
The role of control in active recognition
It is clear from equations (11) and (12) that the history of agent?s positions g t plays a key role in
the process of acquiring new information on the object of interest x for the purpose of recognition.
This is encoded by the conditional density (11). In the context of the identification of the model (2),
one would say that data y t (a function of the scene and the history of positions) must be sufficiently
informative [16] on x, meaning that y t contains enough information to estimate x; this can be
measured e.g. through the Fisher information matrix if x is deterministic but unknown, or by the
posterior p(x|y t ) in a probabilistic setting. This depends upon whether ut is sufficiently exciting, a
?richness? condition that has been extensively used in the identification and adaptive control literature
[17, 18], which guarantees that the state trajectory g t explores the space of interest. If this condition
is not satisfied, there are limitations on the performance that can be attained during the search process.
There are two extreme cases which set an upper and lower bounds on recognition error:
1. Passive recognition: there is no active control, and instead a collection of vantage points g t
is given a-priori. Under this scenario it is easy to prove that, averaging over the possible
scenes and initial agent locations, the probability of error approaches chance (i.e. that given
by the prior distribution) as clutter density and/or the environment volume increase.
2. Full control on g t : if the control action can take the ?omnipotent agent? anywhere, and
infinite time is available to collect measurements, then the conditional entropy H(x|y t )
decreases asymptotically to zero thus providing arbitrarily good recognition rate in the limit.
4
.
.
superscript in e.g. y t indicates history of y up to t, i.e. y t = (y1 , . . . , yt ) and ytt+T = (yt , . . . yt+T )
4
In general, there is a tradeoff between the ability to gather new information through suitable control
actions, which we name ?control authority?, and the recognition rate. In the sequel we shall propose
a measure for the ?control authority? over the sensing process; later in the paper we will consider
conditional entropy as a proxy (upper bound) on probability of error and evaluate empirically how
control authority affects the conditional entropy decrease.
3.1
Control authority
Unlike the passive case, in the controlled scenario time plays an important role. This happens in two
ways. One is related to the ability to visit previously unexplored regions and therefore is related to
the reachable space under input and time constraints, the other is the effect of noise which needs
to be averaged. If objects in the scene move, this can be done only at an expense in energy, and
achieving asymptotic performance may not be possible under control limitations. This considerably
more complex scenario is beyond our scope in this paper. We focus on the simplest case of static
environment.
Control authority depends on (i) the controller u, as measured for instance by a norm5 kuk :
U [0, T ] ! R and (ii) on the geometry of the state space, the input-to-state map and on the environment.
We propose to measure control authority in the following manner: associate to each pair of locations
in the state space (go , gf ) and a given time horizon T the cost kuk required to move from go at time
t = 0 to gf at time t = T along a minimum cost path i.e.
.
J? (go , gf , T ) =
inf
kuk
(13)
u : gu (0)=go ,gu (T )=gf ?
where gu (t) is the state vector at time t under control u. If gf is not reachable from go in time T we
set J? (go , gf , T ) = 1. This will depend on the dynamical properties of the agent g? = f (g, u) (or
gt+1 = f (gt , ut ) for discrete time) as well as on the scene ? where the agent has to navigate through
while avoiding obstacles.
The control authority (CA) can be measured via the volume of the reachable space for fixed control
cost, and will be a function of the initial configuration g0 and of the scene ?, i.e.
.
CA(k, go , ?) = V ol{gf : J? (g0 , gf , k) ? 1}
(14)
If instead one is interested in average performance (e.g. w.r.t. the possible scene distributions with
fixed clutter density), a reasonable measure is the average of smallest volume (as g0 varies) of the
reachable space with a unit cost input
?
?
.
CA(k) = E? inf CA(k, go , ?)
(15)
go
If planning on an indefinitely long time horizon is allowed, then one would minimize J(go , gf , T )
over time T :
.
J(go , gf ) = inf J(go , gf , T )
(16)
T
with
0
.
CA1 = inf (V ol{gf : J(go , gf ) ? 1})
go
(17)
The figures CA(k, go , ?) in (14), CA(k) and CA1 in (17) are proxies of the exploration ability which,
in turn, is related to the ability to gather new information on the task at hand. The data acquisition
process can be regarded as an experiment design problem [16] where the choice of the control signal
guides the experiment. Control authority, as defined above, measures how much freedom one has
on the sampling procedure; the larger the CA, the more freedom the designer has. Hence, having
fixed (say) the number of snapshots of the scene one may consider the time interval over which these
snapshots can be taken, the designer is trying to maximize the information the data contains on the
task (making a decision on class label); this information is of course a nondecreasing function of CA.
More control authority corresponds to more freedom in the choice of which samples one is taking
(from which location and at which scale).
Therefore the risk, considered against CA(k) in (15), CA(k, go , ?) in (14) or CA1 in (17) will follow
a surface that depends on the clutter: For any given clutter (or clutter density), the risk will be a
monotonically non-increasing function of control authority CA(k). This is illustrated in Fig. 4.
5
This could be, for instance, total energy, (average) power, maximum amplitude and so on. We can assume
that the control is such that kuk ? 1
5
4
Control policy
Given gt , ?, and a finite control authority CA(k, gt , ?), in order to minimize average risk (1) with
respect to a sequence of control actions we formulate a finite k-step horizon optimal control problem:
Z
?t+k 1
t+k t
1
t+k
1
t+k
ut
= arg min
p(yt+1
|y , ut+k
) 1 max p(xi |y t , yt+1
, ut+k
) dyt+1
(18)
t
t
ut+k
t
i
1
which is unfortunately intractable. As is standard, we can settle for the greedy k = 1 case:
Z
u?t = arg min p(yt+1 |y t , ut ) 1 max p(xi |y t , yt+1 , ut ) dyt+1
ut
i
(19)
but it is still often impractical. We relax the problem further by choosing to minimize the upper
bound on Bayesian risk, of which a convenient one is the conditional entropy (see [19], which shows
Pe ? 12 H(x|y)): This implies that control action can be chosen by entropy minimization:
u?t = arg min H(x|y t , yt+1 , ut )
(20)
ut
Using chain rules of entropy, we can rewrite minimization of H(x|y t , yt+1 , ut ) as maximization of
conditional entropy of next measurement:
u?t = arg min H(x|y t , yt+1 , ut )
ut
=
arg min H(x|y t )
=
arg max H(yt+1 |y t , ut )
=
arg max H(yt+1 |y t , ut )
ut
I(yt+1 ; x|y t , ut )
ut
ut
H(yt+1 |y t , ut , x)
(21)
(22)
(23)
because H(yt+1 |y t , ut , x) = H(nt ) is due to Gaussian noise, since yt+1 = h(gt+1 ; ?) + nt+1 and
both gt+1 and ? are known (the only unknown object in the scene is x, and it is conditioned on).
H(yt+1 |y t , ut ) is the entropy of a Gaussian mixture distribution which can be easily approximated
by Monte Carlo, and for which both lower [20] and upper bounds [21] are known.
Since the controller has energy limitations, i.e. is unable to traverse the environment in one step,
optimization is taken over a small ball in R2 centered at current location gt . In practice, the set
of controls needs to be discretized and entropy computed for each action. However, rather than
myopically choosing the next control, we instead choose the next target position, in a ?bandit?
approach [22, 23]: maximization in (23) is taken with respect to all locations in the world, rather
than the set of controls (locations reachable in one step), and the agent takes the minimum energy
path toward the most informative location. Since this location is typically not reachable in a single
step, one can adopt a ?stubborn? strategy that follows the planned path to the target location before
choosing next action, and an ?indecisive? ? that replans as soon as additional information becomes
available as a consequence of motion. We demonstrate the characteristics of conditional entropy as a
criterion for planning in Fig. 1.
5
Experiments
In addition to evaluating ?indecisive? and ?stubborn? strategies, we also consider several different
uncertainty measures. Section 4 provided arguments for H(yt+1 |y t , g) (a ?max-ent? approach)
which is a proxy for minimization of Bayesian risk. Another option is to maximize covariance of
p(yt+1 |y t , g) (?max-var?), for example due to reduced computational cost. Alternatively, if we do not
wish to hypothesize future measurements and compute p(yt+1 |y t , g), we may search by approaching
the mode of the posterior distribution p(x|y t ) (?max-posterior?). To test average performance of
these strategies, we consider search in 100 environment instances, each containing 40 known clutter
objects and one unknown object. Clutter objects are sampled from the continuous prior distribution
(6) and unknown object is chosen from the prior (10) discretized to |X | ? 9000. Agent?s sensor has
m = 30 pixels, with additive noise set to half of the difference between object colors. Conditional
entropy of the next measurement, H(yt+1 |y t , gt+1 ), is calculated over the entire map, on a 16x16
grid. Search is terminated once residual entropy falls below a threshold value: H(x|y t ) < 0.001. We
are interested in average search time (expressed in terms of number of steps) and average regret, which
6
entropy
lower bound
upper bound
entropy
lower bound
entropy
upper bound
lower bound
upper bound
Figure 1: ?Value of measurement? described by conditional entropy H(yt+1 |y t , g) as a function of
location g. We focus on three special cases, and for each show entropy, its lower bound (see [20]),
and upper bound (based on Gaussian approximation, see [24]). In all cases, the agent is at the bottom
of the environment, and a small unknown object is at the top. The agent has made one measurement
(y1 ) and must now determine the best location to visit. The left three panels demonstrate a case of
scaling: object is seen, but due to noise and quantization its parameters are uncertain. Agent gains
information if putative object location (top) is approached. Middle three panels demonstrate partial
occlusion: a part of the object has been seen, and there is now a region (bottom right corner) that is
uninformative ? measurements taken there are predictable. Full occlusion is shown in the right three
panels. The object has not been seen (due to occluder in the middle of the environment) and the best
action is to visit new area. Notice that lower and upper bounds are maximized at the same point as
actual entropy. This was a common occurrence in many experiments that we did. Because we are
interested in the maximizing point, rather than the maximizing value, even if the bounds are loose,
using them for navigation can lead to reasonable results.
15
finish
finish
start
10
5
start
0
15
5
10
15
20
10
5
0
10
20
30
Figure 2: A typical run of ?indecisive? (left) and ?stubborn? (right) strategies. Objects are colored
according to their radiance and the unknown object is shown as a thick line. Traveled path is shown
in black. The thinner lines are the planned paths that were not traversed to the end because of
replanning. Stubborn explorer traverses each planned segment to its end. Right: Residual entropy
H(x|y t ) shown over time for the two strategies (top: ?indecisive?, bottom: ?stubborn?). Lower and
upper bounds on H(x|y t , yt+1 ) can be computed prior to measuring yt+1 using upper and lower
bounds on H(yt+1 |y t ). Sharp decrease occurs when object becomes visible.
we define as the excess fraction of the minimum energy path to the center of the unknown object (c0 )
.
,c0 ) J(xo ,c0 )
that the explorer takes: regret = cu (xoJ(x
. Because it is not always necessary to reach the
o ,c0 )
object to recognize it (viewing it closely from multiple viewpoints may be sufficient), this quantity is
an approximation to minimum search effort. We show an example of a typical problem instance in
Fig. 2. Statistics of strategies? performance are shown in Fig. 3. Minimum energy path and random
walk strategy play roles of lower and upper bounds. For each of the three uncertainty measures,
?indecisive? outperformed ?stubborn? in terms of both average path length and average regret, as
also shown in Table 1. Notice however that for specific problem instances ?indecisive? can be much
worse than ?stubborn? ? the curves for the two strategy types cross. Generally, ?max-ent? strategy
seems to perform best, followed by ?max-var?, and ?max-posterior?. ?Random-walk? strategy was
unable to find the object unless it was visible initially or became visible by chance. We next
indecisive
stubborn
Average search duration
max-ent max-var max-p(x|y t )
28.42
32.70
41.00
34.26
36.17
41.49
max-ent
1.27
1.71
Average regret
max-var max-p(x|y t )
1.44
1.96
1.78
2.19
Table 1: Search time statistics for different strategies.
7
Figure 3: Search time statistics for a 100 world test suite. Left: cumulative distribution of distance
until detection traveled by the max-entropy, max-posterior, max-variance explorers, and random
walker. Right: cumulative distribution of regret for the explorers.
prior entropy
environment volume
reachable volume
without clutter
Figure 4: Left: Control authority. The red dashed curve corresponds to reachable volume in the
absence of clutter. The black dashed line is the normalized maximum reachable volume in the
environment. Right: Residual entropy H(x|y t ), as a function of control authority and clutter density.
Black dashed line indicates H(x), entropy prior to taking any measurements. Lines correspond to
residual entropy for a given control authority averaged over the test suite; markers ? to residual
entropy on a specific problem instance. For certain scenes, agent is unable to significantly reduce
entropy because the object never becomes unoccluded (once object is seen, there is a sharp drop in
residual entropy, as shown in Fig. 2).
empirically evaluated explorer?s exploration ability under finite control authority. Reachable volume
was computed by Monte Carlo sampling, following (14)-(15) for several clutter density values. For
each clutter density, we generated 40 scene instances and tested ?indecisive? max-entropy strategy
with respect to control authority. Here |X | ? 2000, and other parameters remained as in previous
experiment. Fig. 4 empirically verifies discussion in Section 3.
6
Discussion
We have described a simple model that captures the phenomenology of nuisances in a visual search
problem, that includes uncertainty due to occlusion, scaling, and other ?noise? processes, and used
it to compute the entropy of the prediction density to be used as a utility function in the control
policy. We have then related the amount of ?control authority? the agent can exercise during the
data acquisition process with the performance in the visual search task. The extreme cases show
that if one is given a passively gathered dataset of an arbitrary number of images, performance
cannot be guaranteed beyond what is afforded by the prior. In the limit of infinite control authority,
arbitrarily good decision performance can be attained. In between, we have empirically characterized
the tradeoff between decision performance and control authority. We believe this to be a natural
extension of rate-distortion tradeoffs where the underlying task is not transmission and storage of
data, but usage of (visual) data for decision and control.
Acknowledgments
Research supported on ARO W911NF-11-1-0391 and DARPA MSEE FA8650-11-1-7154.
8
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9
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3,971 | 4,595 | Waveform Driven Plasticity in BiFeO3 Memristive
Devices: Model and Implementation
Christian Mayr, Paul Staerke, Johannes Partzsch, Rene Schueffny
Institute of Circuits and Systems
TU Dresden, Dresden, Germany
{christian.mayr,johannes.partzsch,rene.schueffny}@tu-dresden.de
Love Cederstroem
Zentrum Mikroelektronik Dresden AG
Dresden, Germany
[email protected]
Yao Shuai
Inst. of Ion Beam Physics and Materials Res.
Helmholtz-Zentrum Dresden-Rossendorf e.V.
Dresden, Germany
[email protected]
Nan Du, Heidemarie Schmidt
Professur Materialsysteme der Nanoelektronik
TU Chemnitz, Chemnitz, Germany
[email protected],[email protected]
Abstract
Memristive devices have recently been proposed as efficient implementations of
plastic synapses in neuromorphic systems. The plasticity in these memristive devices, i.e. their resistance change, is defined by the applied waveforms. This behavior resembles biological synapses, whose plasticity is also triggered by mechanisms that are determined by local waveforms. However, learning in memristive
devices has so far been approached mostly on a pragmatic technological level. The
focus seems to be on finding any waveform that achieves spike-timing-dependent
plasticity (STDP), without regard to the biological veracity of said waveforms or
to further important forms of plasticity. Bridging this gap, we make use of a plasticity model driven by neuron waveforms that explains a large number of experimental observations and adapt it to the characteristics of the recently introduced
BiFeO3 memristive material. Based on this approach, we show STDP for the
first time for this material, with learning window replication superior to previous
memristor-based STDP implementations. We also demonstrate in measurements
that it is possible to overlay short and long term plasticity at a memristive device
in the form of the well-known triplet plasticity. To the best of our knowledge, this
is the first implementations of triplet plasticity on any physical memristive device.
1
Introduction
Neuromorphic systems try to replicate cognitive processing functions in integrated circuits. Their
complexity/size is largely determined by the synapse implementation, as synapses are significantly
more numerous than neurons [1]. With the recent push towards larger neuromorphic systems and
higher integration density of these systems, this has resulted in novel approaches especially for the
synapse realization. Proposed solutions on the one hand employ nanoscale devices in conjuction
with conventional circuits [1] and on the other hand try to integrate as much synaptic functionality
(short- and long term plasticity, pulse shaping, etc) in as small a number of devices as possible. In
1
this context, memristive devices 1 as introduced by L. Chua [2] have recently been proposed as efficient implementations of plastic synapses in neuromorphic systems. Memristive devices offer the
possibility of having the actual learning mechanism, synaptic weight storage and synaptic weight
effect (i.e. amplification of the presynaptic current) all in one device, compared to the distributed
mechanisms in conventional circuit implementations [3]. Moreover, a high-density passive array
on top of a conventional semiconductor chip is possible [1]. The plasticity in these memristors,
i.e. their resistance change, is defined by the applied waveforms [4], which are fed into the rows
and columns of the memristive array by CMOS pre- and postsynaptic neurons [1]. This resembles biological synapses, whose plasticity is also triggered by mechanisms that are determined by
local waveforms [5, 6]. However, learning in memristors has so far been approached mostly on
a pragmatic technological level. The goal seems to be to find any waveform that achieves spiketiming-dependent plasticity (STDP) [4], without regard to the biological veracity of said waveforms
or to further important forms of plasticity [7].
Bridging this gap, we make use of a plasticity rule introduced by Mayr and Partzsch [6] which is
driven in a biologically realistic way by neuron waveforms and which explains a large number of
experimental observations. We adapt it to a model of the recently introduced BiFeO3 memristive
material [8]. Measurement results of the modified plasticity rule implemented on a sample device
are given, exhbiting configurable STDP behaviour and pulse triplet [7] reproduction.
2 Materials and Methods
2.1 Local Correlation Plasticity (LCP)
The LCP rule as introduced by Mayr and Partzsch [6] combines two local waveforms, the synaptic
conductance g(t) and the membrane potential u(t). Presynaptic activity is encoded in g(t), which
determines the conductance change due to presynaptic spiking. Postsynaptic activity in turn is signaled to the synapse by u(t). The LCP rule combines both in a formulation for the change of the
synaptic weight w that is similar to the well-known Bienenstock-Cooper-Munroe rule [9]:
dw
= B ? g(t) ? (u(t) ? ?u )
(1)
dt
In this equation, ?u denotes the voltage threshold between weight potentiation and depression,
which is normally set to the resting potential. Please note that coincident pre- and postsynaptic
activities are detected in this rule by multiplication: A weight change only occurs if both presynaptic
conductance is elevated and postsynaptic membrane potential is away from rest.
The waveforms for g(t) and u(t) are determined by the employed neuron model. Mayr et al. [6] use
a spike response model [10], with waveforms triggered at times of pre- and postsynaptic spikes:
? ? e?
g(t) = G
t?tpre
n
?pre
u(t) = Up,n ? ?(t ? tpost
n ) + Urefr ? e
tpre
n
t?tpost
? ? n
post
pre
for tpre
n ? t < tn+1 ,
(2)
for tpost
? t < tpost
n
n+1 ,
(3)
tpost
n
where
and
denote the n-th pre- and postsynaptic spike, respectively. The presynaptic con? and decay time constant ?pre . The postsynaptic
ductance waveform is an exponential with height G
potential at a spike is defined by a Dirac pulse with integral Up,n , followed by an exponential decay
with height Urefr (< 0) and membrane time constant ?post .
Following [6], postsynaptic adaptation is realised in the value of Up,n . For this, Up,n is decreased
from a nominal value Up if the postsynaptic pulse occurs shortly after another postsynaptic pulse:
?
Up,n = Up ? (1 ? e
post
tpost
?tn?1
n
?post
)
(4)
The time constant for the exponential decay in this equation is the same as the membrane time
constant.
1
In 1971 Leon Chua postulated the existence of a device where the current or voltage is directly controlled
by voltage flux or charge respectively, this was called a memristor. Using a general state space description
Chua and Kang later extended the theory to cover the very broad class of memristive devices [2]. Even though
the two terms are used interchangeably in other studies, since the devices used in this study do not fit the strict
definition of memristor, we will refer to them as memristive devices in the following.
2
g in nS
u in mV
?w in %
1.0
0.8
0.6
0.4
0.2
0.0
15
10
5
0
?5
?10
2.0
1.5
1.0
0.5
0.0
0
20
40
60
t in ms
80
100
120
Figure 1: Progression of the conductance g, the membrane potential u and the synapse weight w for
a sample spike pattern.
Figure 1 shows the pre- and postsynaptic waveforms, as well as the synaptic weight for a sample
spike train. For the simple waveforms, two principal weight change mechanisms are present: If
the presynaptic side is active at a postsynaptic spike, the weight is instantaneously increased by
the large elevation of the membrane potential. In contrast, all presynaptic activity falling into the
refractoriness period of the neuron (exponential decay after spike) integrates as a weight decrease.
As shown in [6], this simple model can replicate a multitude of experimental evidence, on par
with the most advanced (and complex) phenomenological plasticity models currently available. In
addition, the LCP rule directly links synaptic plasticity to other pre- and postsynaptic adaptation
processes by their influence on the local waveforms. This can be used to explain further experimental
results [6]. In Sec. 3.1, we will adapt the above rule equations to the characteristics of our memristive
device, which is introduced in the next section.
2.2
Memristive Device
Non-volatile passive analog memory has often been discussed for applications in neuromorphic
systems because of the space limitations of analog circuitry. However, until recently only a few
groups had access to sufficient materials and devices. Developments in the field of nano material
science, especially in the last decade, opened new possibilities for creating compact circuit elements
with unique properties.
Most notably after HP released information about their so-called Memristor [11] much effort has
been put in the analysis of thin film semiconductor-metal-metaloxide compounds. One of the commonly used materials in this class is BiFeO3 (BFO). The complete conducting mechanisms in BFO
are not fully understood yet, with partly contradictory results reported in literature, but it has been
confirmed that different physical effects are overlayed and dominate in different states. Particularly
the resistive switching effect seems promising for neuromorphic devices and will be discussed in
more detail. It has been shown in [12, 8] that the effect can appear uni- or bipolar and is highly
dependent on the processing regarding the substrate, growth method, doping, etc. [13].
We use BFO grown by pulsed laser deposition on Pt/Ti/SiO2 /Si substrate with an Au top contact,
see in Fig. 2. Memristors were fabricated with circular top plates, which were contacted with needle
probes, whereas the continuous bottom plate was contacted at one edge of the die. The BFO films
have a thickness of some 100nm. The created devices show a unipolar resistive switching with a
rectifying behavior. For a positive bias the device goes into a low resistive state (LRS) and stays
there until a negative bias is applied which resets it back to a high resistive state (HRS). The state
can be measured without influencing it by applying a low voltage of under 2V.
Figure 3 shows a voltage-current-diagram which indicates some of the characteristics of the device.
The measurement consists of three parts: 1) A rising negative voltage is applied which resets the
device from an intermediate level to HRS. 2) A rising voltage lowers the resistance exponentially.
3
Figure 2: Photograph of the fabricated memristive material that was used for the measurements.
3) A falling positive voltage does not affect the resistance anymore and the relation is nearly ohmic.
Because of the rectifying characteristic the current in LRS and HRS for negative voltages does not
exhibit as large a dynamic range as for positive voltages.
800
10?3
700
10?4
600
10?5
abs(I) in A
Im in uA
500
400
300
200
10?6
10?7
100
10?8
0
?100
?6
?4
?2
0
Vm in V
2
4
6
10?9?6
?4
?2
0
Vm in V
2
4
6
Figure 3: Voltage-current diagram of the device as linear and log-scale plot
2.3
Phenomenological Device Model
To apply the LCP model to the BFO device and enable circuit design, a simplified device model
is required. We have based our model on the framework of Chua and Kang [2]; that is, using an
output function (i.e., for current Im ) dependent on time, state and input (i.e., voltage Vm ). Recently,
this has been widely used for the modeling of memristive devices [11, 14, 15]. In contrast to many
memristive device models which are based on a sinh function for the output relationship (following
Yang et al. [14]), we model the BFO device as two semiconductor junctions. The junctions can
abstractly be described by a diode equation: Id = I0 (exp(qV /kT ) ? 1) [16]. In an attempt to catch
the basic characteristics, our device could be modeled employing two diode equations letting a state
variable, x, influence the output and roughly represent the conductance:
(
)
Im = h(x, Vm , t) = I01 ? (ed1 ?Vm (t) ? 1) ? I02 ? (e?d2 ?Vm (t) ? 1) ? x(t)
(5)
where Vm is the voltage over the device2 and the diode like equations guarantee a zero crossing
hysteresis. The use of parameters I0i and di now allows individual control of current characteristics
for negative and positive voltages, and as shown in the previous section these are rather asymmetric
for our BFO devices. For the purpose of modeling plasticity, our focus has been on the dynamic
behavior of the conductance change; this was investigated in some detail by Querlioz et al. [15] and
has served as the basis for our model of the state variable:
dx
= f (x, Vm , t) = ?(x) ? ?(Vm )
dt
2
(6a)
With sinh(z) = 1/2 ? (ez ? e?z ), our approach is not fundamentally different from using a sinh function.
4
In the above the functions ?(x) and ?(Vm ) relate to how the current state affects the state development and the effect of the applied voltage, respectively. ?(x) is described by an exponential function.
?
x?Gmin
?
? e??1 Gmax ?Gmin , Vm (t) > 0,
Gmax ?x
??
?(x) =
(6b)
e 2 Gmax ?Gmin , Vm (t) ? 0, x > Gmin ,
?
?
0,
else
In ?(Vm ) we again favor using separate exponential over sinh functions for increased controllability
of the different voltage domains (positive and negative). Here the parameters ?1 and ?2 govern the
voltage dependence of the state modification, with ?1 and ?2 scaling the result. With ?1 and ?2 , the
speed of state saturation is set:
{
(
)
?1 ? (e?1 Vm ? 1 ,) Vm (t) ? 0,
(6c)
?(Vm ) =
?2 ? 1 ? e??2 Vm , Vm (t) < 0,
For implementation, we have used one of the most prominent commercially available simulators
R
R
for custom analog and mixed-signal integrated circuit design, the Cadence?
Spectre?
. Using behavioral current sources, the equations for h(x, Vm , t) and f (x, Vm , t) can be implemented and
simulated with feasibility for circuit design. Depicted in Fig. 4 are the conductance change over
time, at different voltages, for model (Fig. 4a) and measurements (Fig. 4b). It can be seen how the
exponential dependency on device voltage gives rise to different levels of operation (Equations (5)
and (6c)). Also the saturation of conductance change for a given voltage is visible (Equation (6b)).
The sharp changes of current seen in the model are a result of our simplistic approach, whereas the
real devices show slower transitions. In addition, it can be noted that above 5 V the real device
appears to experience a significantly steeper rise in current. However, the target is to have reasonable characteristics in the region of operation below 5 V which is relevant in our plasticity rule
experiments.
1.6
Im (t)
Vm (t)
6
5
1.0
4
0.8
3
0.6
0.4
20
30
40
50
60
5
1.0
4
0.8
3
0.6
2
0.2
1
10
6
0.4
2
0.2
Im (t)
Vm (t)
1.2
Im in mA
Im in mA
1.2
0.0
0
1.4
Vm in Volt
1.4
0.0
0
70
t in s
Vm in Volt
1.6
1
10
20
30
40
50
60
70
t in s
(a)
(b)
Figure 4: Device current for different applied voltages for model (a) and measurement (b).
3
3.1
Results
Modified LCP
A nonlinearity or learning threshold is required in order to carry out the correlation operation
between pre- and postsynaptic waveforms that characterizes various forms of long term learning
[9, 17]. In the original LCP rule, this is done by the multiplication of pre- and postsynaptic waveforms, i.e. only coincident activity results in learning. Memristive devices are usually operated in
an additive manner, i.e. the pre- and postsynaptic waveforms are applied to both terminals of the
device, thus adding/subtracting their voltage curves. In order for the state of the memristive device
to only be affected by an overlap of both waveforms, a positive and negative modification threshold
is required [4]. As can be seen from equation 6c, the internal voltage driven state change ?(Vm )
is affected by two different parameters ?1 and ?2 which govern the thresholds for negative and
positive voltages. For our devices, these work out to effective modification thresholds of -2V and
5
Vpre in V
Vpost in V
?2
?1
0
1
2
Vpre-Vpost in V
4
2
0
?2
?Im in %
2
1
0
?1
?2
50
40
30
20
10
0
0
20
40
60
t in ms
80
100
120
Figure 5: Modification of the original LCP rule for the BFO memristive device, from top to bottom:
pre- and postsynaptic voltages/waveforms, exponential decay with ?pre resp. ?post (postsynaptic
waveform plotted as inverse to illustrate waveform function); resultant voltage difference across
memristive device and corresponding memristance modification thresholds (horizontal grey lines);
and memristance change as computed from the model of sec. 2.3
+2.3V. Thus, we need waveforms where coincident activity causes a voltage rise above the positive
threshold resp. a voltage drop below the negative threshold. In addition, we need a dependence
between voltage level and weight change, as the simplest method to differentiate between weights is
the voltage saturation characteristic in Fig. 3. That is, a single stimulus (e.g. pulse pairing in STDP)
should result in a distinctive memristive programming voltage, driving the memristive device into
the corresponding voltage saturation level via the (for typical experiments) 60 stimulus repetitions.
Apart from quantitative adjustments to the original LCP rule, this requires one qualitative adjustment. The presynaptic conductance waveform is now taken as a voltage trace and a short rectangular
pulse is added immediately before the exponential downward trace, arriving at a waveform similar
to the spike response model for the postsynaptic trace, see uppermost curve in Fig. 5. We call this
the modified LCP rule. For overlapping pre- and postsynaptic waveforms, the rectangular pulses of
both waveforms ?ride up? on the exponential slopes of their counterparts when looking at the voltage
difference Vm = Vpre ? Vpost across the memristive device for pre- and postsynaptic waveforms
applied to both terminals of the device (see third curve from top in Fig. 5). Since the rectangular
pulses are short compared to the exponential waveforms, they represent a constant voltage whose
amplitude depends on the time difference between both waveforms (as expressed by the exponential
slopes) as required above. Thus, as in the original LCP rule, the exponential slopes of pre- and postsynaptic neuron govern the STDP time windows. Repeated application of such a pre-post pairing
drives the memristive device in its corresponding voltage-dependent saturation level.
Similar to the original LCP rule, short term plasticity of the postsynaptic action potentials can now
be added to make the model more biologically realistic (e.g. with respect to the triplet learning
protocol [6]). We employ the same attenuation function as in equation 4, adjusting the duration of
the postsynaptic action potential, see second curve from top in Fig. 5.
Please note: One further important advantage of using this modified LCP rule is that both preand postsynaptic waveform are causal, i.e. they start only at the pre- respectively postsynaptic
pulse. This is in contrast to most currently proposed waveforms for memristive learning, i.e. these
waveforms have to start well in advance of the actual pulse [4], which requires preknowledge of a
pulse occurrence. Especially in an unsupervised learning context with self-driven neuron spiking,
this preknowledge is simply not existent.
6
120
120
?pre=15ms, ?post=35ms
?pre=30ms, ?post=50ms
80
60
60
40
40
20
0
20
0
?20
?20
?40
?40
?60
?60
?80
?200
?150
?100
?50
?pre=15ms, ?post=35ms
?pre=30ms, ?post=50ms
100
80
?Im in %
?Im in %
100
0
?t in ms
50
100
150
?80
?200
200
?150
?100
?50
(a)
0
?t in ms
50
100
150
200
(b)
Figure 6: Results for STDP protocol: (a) model simulation, (b) measurement with BFO memristive
device.
3.2
Measurement results
The waveforms developed in the previous section can be tested in actual protocols for synaptic plasticity. As a first step, we investigate the behaviour of the BFO memristive device in a standard
pair-based STDP experiment. For this, we apply 60 spike pairings of different relative timings at
a low repetition frequency (4Hz), comparable to biological measurement protocols [17]. Measurements were performed with a BFO memristive device as shown in Fig. 2. As shown in the model
simulations in Fig. 6a, the developed waveforms are transformed by the memristive device into
approx. exponentially decaying conductance changes. This is in good agreement with biological
measurements [17] and common STDP models [7]. The model results are confirmed in measurements for the BFO memristive device, as shown in Fig. 6b. Notably, the measurements result in
smooth, continuous curves. This is an expression of the continuous resistance change in the BFO
material, which results in a large number of stable resistance levels. This is in contrast e.g. to
memristive materials that rely on ferroelectric switching, which exhibit a limited number of discrete
resistance levels [18, 1]. Moreover, the nonlinear behaviour of the BFO memristive device has only
limited effect on the resulting STDP learning window. The resistance change is directly linked to
the applied waveforms. For example, as shown in Fig. 6, an increase in time constants results in correspondingly longer STDP time windows. Following our modeling approach, these time constants
are directly linked to the time constants of the underlying neuron and synapse model.
60
30
45
20
?t1 in ms
15
0
0
?10
?20
?15
?30
?30
?30
(a)
?20
0
?10
10
?t2 t in ms
20
? Im in %
30
10
30
(b)
Figure 7: Measurement results for the triplet protocol of Froemke and Dan [7]. (a) biological measurement data, adapted from [7], (b) measurement with BFO memristive device.
7
Experiments have shown that weight changes of single spike pairings, as expressed by STDP, are
nonlinearly integrated when occuring shortly after one another. Commonly, triplets of spikes are
used to investigate this effect, as carried out by [7]. The main deviation of these experimental results
compared to a pure STDP rule occur for the post-pre-post triplet [6], which can be attributed to
postsynaptic adaptation [7]. With this adaptation included in our waveforms (equation 4, as seen in
the action potential duration in the second curve from the top of Fig. 5), the BFO memristive device
measurements well resemble the post-pre-post results of [7]. The measurement results in Fig. 7b
show more depression than the biological data for the pre-post-pre triplet (upper left quadrant).
This is because changes in resistance need some time to build up after a stimulating pulse. In the
pre-post-pre case, the weight increase has not fully developed when it is overwritten by the second
presynaptic pulse, which results in weight decrease. This effect is dependent on the measured device
and the parameters of the stimulation waveforms (cf. Supplementary Material).
For keeping the stimulation waveforms as simple as possible, only postsynaptic adaptation has been
included. However, it has been shown that presynaptic short-term plasticity also has a strong influence on long-term learning [19, 6]. With our modeling approach, a model of short-term plasticity
can be easily connected to the stimulation waveforms by modulating the length of the presynaptic
pulse. Along the same lines, the postsynaptic waveform can be shifted by a slowly changing voltage
analogous to the original LCP rule (cf. Eq. 1) to introduce a metaplastic regulation of weight potentiation and depression [6]. Together, these extensions open up an avenue for the seamless integration
of different forms of plasticity in learning memristive devices.
3.3
Conclusion
Starting from a waveform-based general plasticity rule and a model of the memristive device, we
have shown a direct way to go from these premises to biologically realistic learning in a BiFeO3
memristive device. Employing the LCP rule for memristive learning has several advantages. As
a memristor is a two-terminal device, the separation of the learning in two waveforms in the LCP
rule lends itself naturally to employing it in a passive array of memristors [1, 4]. In addition, this
waveform-defined plasticity behaviour enables easy control of the STDP time windows, which is
further aided by the excellent multi-level memristive programming capability of the BiFeO3 memristive devices. There is only a very small number of memristors where plasticity has been shown at
actual devices at all [18, 1]. Among those, our highly-configurable, finely grained learning curves
are unique, other implementations exhibit statistical variations [1], can only assume a few discrete
levels [18] or the learning windows are device-inherent, i.e. cannot be adjusted [20]. This comes at
the price that in contrast to e.g. phase-change materials, BiFeO3 is not easily integrated on top of
CMOS [8].
The waveform-defined plasticity of the LCP rule enables the explicit inclusion of short term plasticity in long term memristive learning, as shown for the triplet protocol. As the pre- and postsynaptic
waveforms are generated in the CMOS neuron circuits below the memristive array [1], short term
plasticity can thus be added at little extra overall circuit cost and without modification of the memristive array itself. In contrast to our easily controlled short term plasticity, the only previous work
targeting memristive short term plasticity employed intrinsic (i.e. non-controllable) device properties [20]. To the best of our knowledge, this is the first time triplets or other higher-order forms of
plasticity have been shown for a physical memristive device.
In a wider neuroscience context, waveform defined plasticity as shown here could be seen as a
general computational principle, i.e. synapses are not likely to measure time differences as in naive
forms of STDP rules, they are more likely to react to local static [21] and dynamic [5] state variables.
Some interesting predictions could be derived from that, e.g. STDP time constants that are linked to
synaptic conductance changes or to the membrane time constant [22, 6]. These predictions could be
easily verified experimentally.
Acknowledgments
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under grant agreement no. 269459 (Coronet).
8
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9
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tested:1 |
3,972 | 4,596 | Fused sparsity and robust estimation for linear
models with unknown variance
Arnak S. Dalalyan
ENSAE-CREST-GENES
92245 MALAKOFF Cedex, FRANCE
[email protected]
Yin Chen
University Paris Est, LIGM
77455 Marne-la-Valle, FRANCE
[email protected]
Abstract
In this paper, we develop a novel approach to the problem of learning sparse representations in the context of fused sparsity and unknown noise level. We propose
an algorithm, termed Scaled Fused Dantzig Selector (SFDS), that accomplishes
the aforementioned learning task by means of a second-order cone program. A
special emphasize is put on the particular instance of fused sparsity corresponding
to the learning in presence of outliers. We establish finite sample risk bounds and
carry out an experimental evaluation on both synthetic and real data.
1
Introduction
Consider the classical problem of Gaussian linear regression1 :
Y = X? ? + ? ? ?,
? ? Nn (0, In ),
(1)
where Y ? Rn and X ? Rn?p are observed, in the neoclassical setting of very large dimensional
unknown vector ? ? . Even if the ambient dimensionality p of ? ? is larger than n, it has proven
possible to consistently estimate this vector under the sparsity assumption. The letter states that the
number of nonzero elements of ? ? , denoted by s and called intrinsic dimension, is small compared
to the sample size n. Most famous methods of estimating sparse vectors, the Lasso and the Dantzig
Selector (DS), rely on convex relaxation of `0 -norm penalty leading to a convex program that in? > 0, the Lasso and the DS [26, 4, 5, 3] are
volves the `1 -norm of ?. More precisely, for a given ?
defined as
L
1
2
b
?
? = arg minp
kY ? X?k2 + ?k?k1
(Lasso)
??R
2
b DS = arg min k?k1 subject to kX> (Y ? X?)k? ? ?.
?
?
(DS)
?
The performance of these algorithms depends heavily on the choice of the tuning parameter ?.
?
Several empirical and theoretical studies emphasized that ? should be chosen proportionally to the
noise standard deviation ? ? . Unfortunately, in most applications, the latter is unavailable. It is
therefore vital to design statistical procedures that estimate ? and ? in a joint fashion. This topic
received special attention in last years, cf. [10] and the references therein, with the introduction of
computationally efficient and theoretically justified ?-adaptive procedures the square-root Lasso [2]
(a.k.a. scaled Lasso [24]) and the `1 penalized log-likelihood minimization [20].
In the present work, we are interested in the setting where ? ? is not necessarily sparse, but for a
known q ? p matrix M, the vector M? ? is sparse. We call this setting ?fused sparsity scenario?.
1
We denote by In the n ? n identity matrix. For a vector v, we use the standard notation kvk1 , kvk2 and
kvk? for the `1 , `2 and `? norms, corresponding respectively to the sum of absolute values, the square root
of the sum of squares and the maximum of the coefficients of v.
1
The term ?fused? sparsity, introduced by [27], originates from the case where M? is the discrete
derivative of a signal ? and the aim is to minimize the total variation, see [12, 19] for a recent
overview and some asymptotic results. For general matrices M, tight risk bounds were proved in
[14]. We adopt here this framework of general M and aim at designing a computationally efficient
procedure capable to handle the situation of unknown noise level and for which we are able to
provide theoretical guarantees along with empirical evidence for its good performance.
This goal is attained by introducing a new procedure, termed Scaled Fused Dantzig Selector (SFDS),
which is closely related to the penalized maximum likelihood estimator but has some advantages in
terms of computational complexity. We establish tight risk bounds for the SFDS, which are nearly
as strong as those proved for the Lasso and the Dantzig selector in the case of known ? ? . We also
show that the robust estimation in linear models can be seen as a particular example of the fused
sparsity scenario. Finally, we carry out a ?proof of concept? type experimental evaluation to show
the potential of our approach.
2
Estimation under fused sparsity with unknown level of noise
2.1
Scaled Fused Dantzig Selector
We will only consider the case rank(M) = q ? p, which is more relevant for the applications
we have in mind (image denoising and robust estimation). Under this condition, one can find a
(p ? q) ? p matrix N such that the augmented matrix M = [M> N> ]> is of full rank. Let us denote
by mj the jth column of the matrix M ?1 , so that M ?1 = [m1 , ..., mp ]. We also introduce:
M ?1 = [M? , N? ],
M? = [m1 , ..., mq ] ? Rp?q ,
N? = [mq+1 , ..., mp ] ? Rp?(p?q) .
Given two positive tuning parameters ? and ?, we define the Scaled Fused Dantzig Selector (SFDS)
b ?
(?,
b) as a solution to the following optimization problem:
?
>
|m>
?
j X (X? ? Y )| ? ??kXmj k2 , j ? q;
?
?
>
N>
(P1)
minimize
kXmj k2 |(M?)j | subject to
? X (X? ? Y ) = 0,
?
?
j=1
?
n?? 2 + Y > X? ? kY k22 .
q
X
This estimator has several attractive properties: (a) it can be efficiently computed even for very large
scale problems using a second-order cone program, (b) it is equivariant with respect to the scale
transformations both in the response Y and in the lines of M and, finally, (c) it is closely related to
the penalized maximum likelihood estimator. Let us give further details on these points.
2.2
Relation with the penalized maximum likelihood estimator
One natural way to approach the problem of estimating ? ? in our setup is to rely on the standard
procedure of penalized log-likelihood minimization. If the noise distribution is Gaussian, ? ?
Nn (0, In ), the negative log-likelihood (up to irrelevant additive terms) is given by
`(Y , X; ?, ?) = n log(?) +
kY ? X?k22
.
2? 2
In the context of large dimension we are concerned with, i.e., when p/n is not small, the maximum
likelihood estimator is subject to overfitting and is of very poor quality. If it is plausible to expect
that the data can be fitted sufficiently well by a vector ? ? such that for some matrix M, only a
small fraction of elements of M? ? are nonzero, then one can considerably improve the quality of
estimation by adding a penalty term to the log-likelihood. However, the most appealing penalty,
the number of nonzero elements of M?, leads to a nonconvex optimization problem which cannot
be efficiently solved even for moderately large values of p. Instead, convex penalties of the form
P
j ?j |(M?)j |, where wj > 0 are some weights, have proven to provide high accuracy estimates
b PL , ?
at a relatively low computational cost. This corresponds to defining the estimator (?
bPL ) as the
2
minimizer of the penalized log-likelihood
q
2
X
? , X; ?, ?) = n log(?) + kY ? X?k2 +
`(Y
?j |(M?)j |.
2
2?
j=1
To ensure the scale equivariance, the weights ?j should be chosen inversely proportionally to ?:
?j = ? ?1 ?
? j . This leads to the estimator
q
kY ? X?k22 X |(M?)j |
PL
PL
b
(? , ?
+
b ) = arg min n log(?) +
?
?j
.
?,?
2? 2
?
j=1
Although this problem can be cast [20] as a problem of convex minimization (by making the change
of parameters ? = ?/? and ? = 1/?), it does not belong to the standard categories of convex
problems that can be solved either by linear programming or by second-order cone programming or
by semidefinite programming. Furthermore, the smooth part of the objective function is not Lipschitz which makes it impossible to directly apply most first-order optimization methods developed in
recent years. Our goal is to propose a procedure that is close in spirit to the penalized maximum likelihood but has the additional property of being computable by standard algorithms of second-order
cone programming.
To achieve this goal, at the first step, we remark that it can be useful to introduce a penalty term that
depends exclusively on ? and that prevents the estimator of ? ? from being too large or too small. One
can show that the only function (up to a multiplicative constant) that can serve as penalty without
breaking the property of scale equivariance is the logarithmic function. Therefore, we introduce an
additional tuning parameter ? > 0 and look for minimizing the criterion
q
n? log(?) +
kY ? X?k22 X |(M?)j |
+
?
?j
.
2? 2
?
j=1
(2)
If we make the change of variables ?1 = M?/?, ?2 = N?/? and ? = 1/?, we get a convex
function for which the first-order conditions [20] take the form
>
m>
? j sign({M?}j ),
j X (Y ? X?) ? ?
(3)
>
N>
? X (Y
(4)
? X?) = 0,
1
kY k22 ? Y > X? = ? 2 .
n?
(5)
Thus, any minimizer of (2) should satisfy these conditions. Therefore, to simplify the problem of
optimization
we propose to replace minimization of (2) by the minimization of the weighted `1 P
norm j ?
? j |(M?)j | subject to some constraints that are as close as possible to (3-5). The only
problem here is that the constraints (3) and (5) are not convex. The ?convexification? of these
constraints leads to the procedure described in (P1). As we explain below, the particular choice of
?
? j s is dictated by the desire to enforce the scale equivariance of the procedure.
2.3
Basic properties
b ?
A key feature of the SFDS is its scale equivariance. Indeed, one easily checks that if (?,
b) is a
b
solution to (P1) for some inputs X, Y and M, then ?(?, ?
b) will be a solution to (P1) for the inputs
X, ?Y and M, whatever the value of ? ? R is. This is the equivariance with respect to the scale
change in the response Y . Our method is also equivariant with respect to the scale change in M.
b ?
b ?
More precisely, if (?,
b) is a solution to (P1) for some inputs X, Y and M, then (?,
b) will be a
solution to (P1) for the inputs X, Y and DM, whatever the q ? q diagonal matrix D is. The latter
property is important since if we believe that for a given matrix M the vector M? ? is sparse, then
this is also the case for the vector DM? ? , for any diagonal matrix D. Having a procedure the output
of which is independent of the choice of D is of significant practical importance, since it leads to a
solution that is robust with respect to small variations of the problem formulation.
The second attractive feature of the SFDS is that it can be computed by solving a convex optimization problem of second-order cone programming (SOCP). Recall that an SOCP is a constrained
3
optimization problem that can be cast as minimization with respect to w ? Rd of a linear function
a> w under second-order conic constraints of the form kAi w + bi k2 ? c>
i w + di , where Ai s are
some ri ? d matrices, bi ? Rri , ci ? Rd are some vectors and di s are some real numbers. The
problem (P1) belongs well to this category, since it can be written as min(u1 + . . . + uq ) subject to
>
|m>
?j = 1, . . . , q;
j X (X? ? Y )| ? ??kXmj k2 ,
q
>
N>
4n?kY k22 ? 2 + (Y > X?)2 ? 2kY k22 ? Y > X?.
? X (X? ? Y ) = 0,
kXmj k2 |(M?)j | ? uj ;
Note that all these constraints can be transformed into linear inequalities, except the last one which
is a second order cone constraint. The problems of this type can be efficiently solved by various
standard toolboxes such as SeDuMi [22] or TFOCS [1].
2.4
Finite sample risk bound
To provide theoretical guarantees for our estimator, we impose the by now usual assumption of
restricted eigenvalues on a suitably chosen matrix. This assumption, stated in Definition 2.1 below,
was introduced and thoroughly discussed by [3]; we also refer the interested reader to [28].
D?efinition 2.1. We say that a n ? q matrix A satisfies the restricted eigenvalue condition RE(s, 1),
if
kA?k2
?
?
?(s, 1) = min
min
> 0.
nk?J k2
|J|?s k?J c k1 ?k?J k1
We say that A satisfies the strong restricted eigenvalue condition RE(s, s, 1), if
?
?(s, s, 1) = min
|J|?s
min
k?J c k1 ?k?J k1
?
kA?k2
> 0,
nk?J?J0 k2
where J0 is the subset of {1, ..., q} corresponding to the s largest in absolute value coordinates of ?.
For notational convenience, we assume that M is normalized in such a way that the diagonal ele>
ments of n1 M>
? X XM? are all equal to 1. This can always be done by multiplying M from the
left by a suitably chosen positive definite diagonal matrix. Furthermore, we will repeatedly use the
>
?1 > >
projector2 ? = XN? (N>
N? X onto the subspace of Rn spanned by the columns of
? X XN? )
XN? . We denote by r = rank{?} the rank of this projector which is typically very small compared
to n ? p, and is always smaller than n ? (p ? q). In all theoretical results, the matrices X and M are
assumed deterministic.
p
Theorem 2.1. Let us fix a tolerance level ? ? (0, 1) and define ? = 2n? log(q/?). Assume that
the tuning parameters ?, ? > 0 satisfy
p
(n ? r) log(1/?) + log(1/?)
?
r
?1? ?2
.
(6)
?
n
n
If the vector M? ? is s-sparse and the matrix (In ? ?)XM? satisfies the condition RE(s, 1) with
some ? > 0 then, with probability at least 1 ? 6?, it holds:
r
r
2? log(q/?) ? ? 2s log(1/?)
4
?
?
b
? + ? )s
+
(7)
kM(? ? ? )k1 ? 2 (b
?
n
?
n
p
p
2?s log(q/?)
b ? ? ? )k2 ? 2(b
+ ??
8 log(1/?) + r .
(8)
kX(?
? + ?? )
?
If, in addition, (In ? ?)XM? satisfies the condition RE(s, s, 1) with some ? > 0 then, with a
probability at least 1 ? 6?, we have:
r
r
4(b
? + ? ? ) 2s log(q/?) ? ? 2 log(1/?)
?
b
kM? ? M? k2 ?
+
(9)
?2
n
?
n
Moreover, with a probability at least 1 ? 7?, we have:
r
??
?kM? ? k1
s1/2 ? ? log(q/?)
2 log(1/?)
?
?
?1/2
?
b ? 1/2 +
+
+ (? + kM? k1 )?
.
(10)
1/2
n?
n
?
n??
2
Here and in the sequel, the inverse of a singular matrix is understood as MoorePenrose pseudoinverse.
4
Before looking at the consequences of these risk bounds in the particular case of robust estimation,
let us present some comments highlighting the claims of Theorem 2.1. The first comment is about
the conditions on the tuning parameters ? and ?. It is interesting to observe that the roles of these
parameters are very clearly defined: ? controls the quality of estimating ? ? while ? determines the
quality of estimating ? ? . One can note that all the quantities entering in the right-hand side of (6)
are known, so that it is not hard to choose ? and ? in such a way that they satisfy the conditions of
Theorem 2.1. However, in practice, this theoretical choice may be too conservative in which case it
could be a better idea to rely on cross validation.
The second remark is about the rates of convergence. According to (8), the rate of estimation
b ? ? ? )k2 is of the order of s log(q)/n, which is known
measured in the mean prediction loss n1 kX(?
2
?
as fast or parametric rate. The vector M? is also estimated with the nearly parametric rate in both
`1 and `2 -norms. To the best of our knowledge, this is the first work where such kind of fast rates
are derived in the context of fused sparsity with unknown noise-level. With some extra work, one
can check that if, for instance, ? = 1 and |? ? 1| ? cn?1/2 for some constant c, then the estimator
?
b has also a risk of the order of sn?1/2 . However, the price to pay for being adaptive with respect
to the noise level is the presence of kM? ? k1 in the bound on ?
b, which deteriorates the quality of
estimation in the case of large signal-to-noise ratio.
Even if Theorem 2.1 requires the noise distribution to be Gaussian, the proposed algorithm remains
valid in a far broader context and tight risk bounds can be obtained under more general conditions
on the noise distribution. In fact, one can see from the proof that we only need to know confidence
sets for some linear and quadratic functionals of ?. For instance, such kind of confidence sets can be
readily obtained in the case of bounded errors ?i using the Bernstein inequality. It is also worthwhile
to mention that the proof of Theorem 2.1 is not a simple adaptation of the arguments used to prove
analogous results for ordinary sparsity, but contains some qualitatively novel ideas. More precisely,
the cornerstone of the proof of risk bounds for the Dantzig selector [4, 3, 9] is that the true parameter
? ? is a feasible solution. In our case, this argument cannot be used anymore. Our proposal is then
e that simultaneously satisfies the following three conditions: M?
e has the
to specify another vector ?
? e
?
same sparsity pattern as M? , ? is close to ? and lies in the feasible set.
A last remark is about the restricted eigenvalue conditions. They are somewhat cumbersome in this
abstract setting, but simplify a lot when the concrete example of robust estimation is considered,
cf. the next section. At a heuristical level, these conditions require from the columns of XM? to
be not very strongly correlated. Unfortunately, this condition fails for the matrices appearing in
the problem of multiple change-point detection, which is an important particular instance of fused
sparsity. There are some workarounds to circumvent this limitation in that particular setting, see
[17, 11]. The extension of these kind of arguments to the case of unknown ? ? is an open problem
we intend to tackle in the near future.
3
Application to robust estimation
This methodology can be applied in the context of robust estimation, i.e., when we observe Y ? Rn
and A ? Rn?k such that the relation
Yi = (A? ? )i + ? ? ?i ,
iid
?i ? N (0, 1)
holds only for some indexes i ? I ? {1, ..., n}, called inliers. The indexes does not belonging to
I will be referred to as outliers. The setting we are interested in is the one frequently encountered
in computer vision [13, 25]: the dimensionality k of ? ? is small as compared to n but the presence
of outliers causes the complete failure of the least squares estimator. In what follows, we use the
standard assumption that the matrix n1 A> A has diagonal entries equal to one.
Following the ideas developed in [6, 7, 8, 18, 15], we introduce a new vector ? ? Rn that serves to
characterize the outliers. If an entry ?i of ? is nonzero, then the corresponding observation Yi is an
outlier. This leads to the model:
?
?
Y = A? ? + n? ? + ? ? ? = X? ? + ? ? ?, where X = [ n In A], and ? = [? ? ; ? ? ]> .
Thus, we have rewritten the problem of robust estimation in linear models as a problem of
estimation in high dimension under the fused sparsity scenario. Indeed, we have X ? Rn?(n+k)
5
b of ? ? for which ?
b
b = [In 0n?k ]?
and ? ? ? Rn+k , and we are interested in finding an estimator ?
contains as many zeros as possible. This means that we expect that the number of outliers is
significantly smaller than the sample size. We are thus in the setting of fused sparsity with
M = [In 0n?k ]. Setting N = [0k?n Ik ], we define the Scaled Robust Dantzig Selector (SRDS) as
b ?
b, ?
a solution (?,
b) of the problem:
minimize k?k1
??
?
nkA? + n ? ? Y k? ? ??,
?
?
?
?
A> (A? + n ? ? Y ) = 0,
subject to
?
?
?
?
n?? 2 + Y > (A? + n?) ? kY k22 .
(P2)
Once again, this can be recast in a SOCP and solved with great efficiency by standard algorithms.
Furthermore, the results of the previous section provide us with strong theoretical guarantees for the
SRDS. To state the corresponding result, we will need a notation for the largest and the smallest
singular values of ?1n A denoted by ? ? and ?? respectively.
p
Theorem 3.1. Let us fix a tolerance level ? ? (0, 1) and define
? = 2n? log(n/?). Assume that
p
the tuning parameters ?, ? > 0 satisfy ?? ? 1 ? nk ? n2
(n ? k) log(1/?) + log(1/?) . Let ?
denote the orthogonal projector onto the k-dimensional
subspace of Rn spanned by the columns of
?
?
A. If the vector ? is s-sparse and the matrix n(In ? ?) satisfies the condition RE(s, 1) with
some ? > 0 then, with probability at least 1 ? 5?, it holds:
r
r
4
2? log(n/?) ? ? 2s log(1/?)
?
?
kb
? ? ? k1 ? 2 (b
? + ? )s
+
,
(11)
?
n
?
n
r
r
2(b
? + ? ? ) 2s log(n/?)
2 log(1/?)
k(In ? ?)(b
? ? ? ? )k2 ?
+ ??
.
(12)
?
n
n
?
If, in addition, n (In ? ?) satisfies the condition RE(s, s, 1) with some ? > 0 then, with a probability at least 1 ? 6?, we have:
r
r
4(b
? + ? ? ) 2s log(n/?) ? ? 2 log(1/?)
?
kb
? ? ? k2 ?
+
?2
n
?
n
r
r
p
?
?
?
?
?
?
(
k
+
2 log(1/?))
?
2s
log(n/?)
?
2
log(1/?)
4(b
?
+
?
)
?
b ? ? k2 ?
?
k?
+
+
??2
?2
n
?
n
n
Moreover, with a probability at least 1 ? 7?, the following inequality holds:
r
?k? ? k1
s1/2 ? ? log(n/?)
2 log(1/?)
??
?
?
?1/2
+
+ (? + k? k1 )?
.
?
b ? 1/2 +
1/2
n?
n
?
n??
(13)
All the comments made after Theorem 2.1, especially those concerning the tuning parameters and
the rates of convergence, hold true for the risk bounds in Theorem 3.1 as well. Furthermore, the
restricted eigenvalue condition in the latter theorem is much simpler
? and deserves a special attention.
In particular, one can remark that the failure of RE(s, 1) for n(In ? ?) implies that there is
a unit vector ? in Im(A) such that |?(1) | + . . . + |?(n?s) | ? |?(n?s+1) | + . . . + |?(n) |, where
?(k) stands for the kth smallest (in absolute value) entry of ?. To gain a better understanding of
how restrictive this assumption is, let us consider the case where the rows a1 , . . . , an of A are
i.i.d. zero mean Gaussian vectors. Since ? ? Im(A), its coordinates ?i are also i.i.d. Gaussian
random variables (they can be considered N (0, 1) due to the homogeneity of the inequality we are
interested in). The inequality |?(1) | + . . . + |?(n?s) | ? |?(n?s+1) | + . . . + |?(n) | can be written
P
as n1 i |?i | ? n2 (|?(n?s+1) | + . . . + |?(n) |). While the left-hand side of this inequality tends to
?
E[|?1 |] > 0, the right-hand side is upper-bounded by 2s
maxi |?i |, which is on the order of 2s nlog n .
n
?
Therefore, if 2s nlog n is small, the condition RE(s, 1) is satisfied. This informal discussion can be
made rigorous by studying large deviations of the quantity max??Im(A)\{0} k?k? /k?k1 . A simple
?
sufficient condition entailing RE(s, 1) for n(In ? ?) is presented in the following lemma.
Pn
?
2skAk
Lemma 3.2. Let us set ?s (A) = inf u?Sk?1 n1 i=1 |ai u|? ?n2,? . If ?s (A) > 0, then n (In ?
p
?) satisfies both RE(s, 1) and RE(s, s, 1) with ?(s, 1) ? ?(s, s, 1) ? ?s (A)/ (? ? )2 + ?s (A)2 .
6
SFDS
b ? ? ? |2
|?
Lasso
?
Square-Root Lasso
b ? ? ? |2
|?
|b
??? |
b ? ? ? |2
|?
|b
? ? ?? |
( T, p, s? , ? ? )
Ave
StD
Ave
StD
Ave
StD
Ave
StD
Ave
StD
(200, 400, 2, .5)
(200, 400, 2, 1)
(200, 400, 2, 2)
(200, 400, 5, .5)
(200, 400, 5, 1)
(200, 400, 5, 2)
(200, 400, 10, .5)
(200, 400, 10, 1)
(200, 400, 10, 2)
0.04
0.09
0.23
0.06
0.20
0.34
0.10
0.19
1.90
0.03
0.05
0.17
0.01
0.05
0.11
0.01
0.09
0.20
0.18
0.42
0.75
0.28
0.56
0.34
0.36
0.27
4.74
0.14
0.35
0.55
0.11
0.10
0.21
0.02
0.26
1.01
0.07
0.16
0.31
0.13
0.31
0.73
0.15
0.31
0.61
0.05
0.11
0.21
0.09
0.04
0.25
0.00
0.04
0.08
0.06
0.13
0.25
0.11
0.25
0.47
0.10
0.19
1.80
0.04
0.09
0.18
0.06
0.02
0.29
0.01
0.09
0.04
0.20
0.46
0.79
0.18
0.66
0.69
0.36
0.27
3.70
0.14
0.37
0.56
0.27
0.05
0.70
0.02
0.26
0.48
Table 1: Comparing our procedure SFDS with the (oracle) Lasso and the SqRL on a synthetic dataset. The
b ? ? ? |2 and |b
average values and the standard deviations of the quantities |?
? ? ? ? | over 500 trials are reported.
They represent respectively the accuracy in estimating the regression vector and the level of noise.
The proof of the lemma can be found in the supplementary material.
One can take note that the problem (P2) boils down to computing (b
?, ?
b) as a solution to
?
?
nk(In ? ?)( n? ? Y )k? ? ??,
?
minimize k?k1 subject to
n?? 2 + n[(In ? ?)Y ]> ? ? k(In ? ?)Y k22 .
b = (A> A)?1 A> (Y ? ?n ?
b ).
and then setting ?
4
Experiments
For the empirical evaluation we use a synthetic dataset with randomly drawn Gaussian design matrix
X and the real-world dataset fountain-P113 , on which we apply our methodology for computing the
fundamental matrices between consecutive images.
4.1
Comparative evaluation on synthetic data
We randomly generated a n ? p matrix X with independent entries distributed according to the
standard normal distribution. Then we chose a vector ? ? ? Rp that has exactly s nonzero elements
all equal to one. The indexes of these elements were chosen at random. Finally, the response
Y ? Rn was computed by adding a random noise ? ? Nn (0, In ) to the signal X? ? . Once Y and X
available, we computed three estimators of the parameters using the standard sparsity penalization
(in order to be able to compare our approach to the others): the SFDS, the Lasso and the squareroot
tuning parameters for all these
p Lasso (SqRL). We used the ?universal?
p
p methods: (?, ?) =
( 2n log(p), 1) for the SFDS, ? = 2 log(p) for the SqRL and ? = ? ? 2 log(p) for the Lasso.
Note that the latter is not really an estimator but rather an oracle since it exploits the knowledge of
the true ? ? . This is why the accuracy in estimating ? ? is not reported in Table 1. To reduce the
well known bias toward zero [4, 23], we performed a post-processing for all of three procedures. It
consisted in computing least squares estimators after removing all the covariates corresponding to
vanishing coefficients of the estimator of ? ? . The results summarized in Table 1 show that the SFDS
is competitive with the state-of-the-art methods and, a bit surprisingly, is sometimes more accurate
than the oracle Lasso using the true variance in the penalization. We stress however that the SFDS
is designed for being applied in?and has theoretical guarantees for?the broader setting of fused
sparsity.
4.2
Robust estimation of the fundamental matrix
To provide a qualitative evaluation of the proposed methodology on real data, we applied the SRDS
to the problem of fundamental matrix estimation in multiple-view geometry, which constitutes an
3
available at http://cvlab.epfl.ch/?strecha/multiview/denseMVS.html
7
?
b
kb
? k0
100
? k0
n kb
1
2
3
4
5
6
7
8
9
10
Average
0.13
218
1.3
0.13
80
0.46
0.13
236
1.37
0.17
90
0.52
0.16
198
1.13
0.17
309
1.84
0.20
17
0.12
0.18
31
0.19
0.17
207
1.49
0.11
8
1.02
0.15
139.4
0.94
Table 2: Quantitative results on fountain dataset.
Figure 1: Qualitative results on fountain dataset. Top left: the values of ?
bi for the first pair of images. There
is a clear separation between outliers and inliers. Top right: the first pair of images and the matches classified
as wrong by SRDS. Bottom: the eleven images of the dataset.
essential step in almost all pipelines of 3D reconstruction [13, 25]. In short, if we have two images I
and I 0 representing the same 3D scene, then there is a 3?3 matrix F, called fundamental matrix, such
that a point x = (x, y) in I1 matches with the point x0 = (x0 , y 0 ) in I 0 only if [x; y; 1] F [x0 ; y 0 ; 1]> =
0. Clearly, F is defined up to a scale factor: if F33 6= 0, one can assume that F33 = 1. Thus, each
pair x ? x0 of matching points in images I and I 0 yields a linear constraint on the eight remaining
coefficients of F. Because of the quantification and the presence of noise in images, these linear
relations are satisfied up to some error. Thus, estimation of F from a family of matching points
{xi ? x0i ; i = 1, . . . , n} is a problem of linear regression. Typically, matches are computed by
comparing local descriptors (such as SIFT [16]) and, for images of reasonable resolution, hundreds
of matching points are found. The computation of the fundamental matrix would not be a problem in
this context of large sample size / low dimension, if the matching algorithms were perfectly correct.
However, due to noise, repetitive structures and other factors, a non-negligible fraction of detected
matches are wrong (outliers). Elimination of these outliers and robust estimation of F are crucial
steps for performing 3D reconstruction.
Here, we apply the SRDS to the problem of estimation of F for 10 pairs of consecutive images
provided by the fountain dataset [21]: the 11 images are shown at the bottom of Fig. 1. Using SIFT
descriptors, we found more than 17.000 point matches in most pairs of images among the 10 pairs
we are considering. The CPU time for computing each matrix using the SeDuMi solver [22] was
about 7 seconds, despite such a large dimensionality. The number of outliers and the estimated
noise-level for each pair of images are reported in Table 2. We also showed in Fig. 1 the 218 outliers
for the first pair of images. They are all indeed wrong correspondncies, even those which correspond
to the windows (this is due to the repetitive structure of the window).
5
Conclusion and perspectives
We have presented a new procedure, SFDS, for the problem of learning linear models with unknown
noise level under the fused sparsity scenario. We showed that this procedure is inspired by the
penalized maximum likelihood but has the advantage of being computable by solving a secondorder cone program. We established tight, nonasymptotic, theoretical guarantees for the SFDS with
a special attention paid to robust estimation in linear models. The experiments we have carried out
are very promising and support our theoretical results.
In the future, we intend to generalize the theoretical study of the performance of the SFDS to the case
of non-Gaussian errors ?i , as well as to investigate its power in variable selection. The extension to
the case where the number of lines in M is larger than the number of columns is another interesting
topic for future research.
8
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9
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3,973 | 4,597 | A Conditional Multinomial Mixture Model for
Superset Label Learning
Thomas G. Dietterich
EECS, Oregon State University
Corvallis, OR 97331
[email protected]
Li-Ping Liu
EECS, Oregon State University
Corvallis, OR 97331
[email protected]
Abstract
In the superset label learning problem (SLL), each training instance provides a set
of candidate labels of which one is the true label of the instance. As in ordinary
regression, the candidate label set is a noisy version of the true label. In this
work, we solve the problem by maximizing the likelihood of the candidate label
sets of training instances. We propose a probabilistic model, the Logistic StickBreaking Conditional Multinomial Model (LSB-CMM), to do the job. The LSBCMM is derived from the logistic stick-breaking process. It first maps data points
to mixture components and then assigns to each mixture component a label drawn
from a component-specific multinomial distribution. The mixture components can
capture underlying structure in the data, which is very useful when the model is
weakly supervised. This advantage comes at little cost, since the model introduces
few additional parameters. Experimental tests on several real-world problems with
superset labels show results that are competitive or superior to the state of the art.
The discovered underlying structures also provide improved explanations of the
classification predictions.
1
Introduction
In supervised classification, the goal is to learn a classifier from a collection of training instances,
where each instance has a unique class label. However, in many settings, it is difficult to obtain such
precisely-labeled data. Fortunately, it is often possible to obtain a set of labels for each instance,
where the correct label is one of the elements of the set.
For example, captions on pictures (in newspapers, facebook, etc.) typically identify all of the people
the picture but do not necessarily indicate which face belongs to each person. Imprecisely-labeled
training examples can be created by detecting each face in the image and defining a label set containing all of the names mentioned in the caption. A similar case arises in bird song classification [2]. In
this task, a field recording of multiple birds singing is divided into 10-second segments, and experts
identify the species of all of the birds singing in each segment without localizing each species to a
specific part of the spectrogram. These examples show that superset-labeled data are typically much
cheaper to acquire than standard single-labeled data. If effective learning algorithms can be devised
for superset-labeled data, then they would have wide application.
The superset label learning problem has been studied under two main formulations. In the multiinstance multi-label (MIML) formulation [15], the training data consist of pairs (Bi , Yi ), where
Bi = {xi,1 , . . . , xi,ni } is a set of instances and Yi is a set of labels. The assumption is that for every
instance xi,j ? Bi , its true label yi,j ? Yi . The work of Jie et al. [9] and Briggs et al. [2] learn
classifiers from such set-labeled bags.
In the superset label formulation (which has sometimes been confusingly called the ?partial label?
problem) [7, 10, 8, 12, 4, 5], each instance xn has a candidate label set Yn that contains the unknown
1
true label yn . This formulation ignores any bag structure and views each instance independently. It
is more general than the MIML formulation, since any MIML problem can be converted to a superset label problem (with loss of the bag information). Furthermore, the superset label formulation is
natural in many applications that do not involve bags of instances. For example, in some applications, annotators may be unsure of the correct label, so permitting them to provide a superset of the
correct label avoids the risk of mislabeling. In this paper, we employ the superset label formulation.
Other relevant work includes Nguyen et al. [12] and Cour et al. [5] who extend SVMs to handle
superset labeled data.
In the superset label problem, the label set Yn can be viewed as a corruption of the true label.
The standard approach to learning with corrupted labels is to assume a generic noise process and
incorporate it into the likelihood function. In standard supervised learning, it is common to assume
that the observed label is sampled from a Bernoulli random variable whose most likely outcome is
equal to the true label. In ordinary least-squares regression, the assumption is that the observed value
is drawn from a Gaussian distribution whose mean is equal to the true value and whose variance is
a constant ? 2 . In the superset label problem, we will assume that the observed label set Yn is drawn
from a set-valued distribution p(Yn |yn ) that depends only on the true label. When computing the
likelihood, this will allow us to treat the true label as a latent variable that can be marginalized away.
When the label information is imprecise, the learning algorithm has to depend more on underlying
structure in the data. Indeed, many semi-supervised learning methods [16] model cluster structure
of the training data explicitly or implicitly. This suggests that the underlying structure of the data
should also play important role in the superset label problem.
In this paper, we propose the Logistic Stick-Breaking Conditional Multinomial Model (LSB-CMM)
for the superset label learning problem. The model has two components: the mapping component
and the coding component. Given an input xn , the mapping component maps xn to a region k. Then
the coding component generates the label according to a multinomial distribution associated with k.
The mapping component is implemented by the Logistic Stick Breaking Process(LSBP) [13] whose
Bernoulli probabilities are from discriminative functions. The mapping and coding components are
optimized simultaneously with the variational EM algorithm.
LSB-CMM addresses the superset label problem in several aspects. First, the mapping component
models the cluster structure with a set of regions. The fact that instances in the same region often
have the same label is important for inferring the true label from noisy candidate label sets. Second,
the regions do not directly correspond to classes. Instead, the number of regions is automatically
determined by data, and it can be much larger than the number of classes. Third, the results of the
LSB-CMM model can be more easily interpreted than the approaches based on SVMs [5, 2]. The
regions provide information about how data are organized in the classification problem.
2
The Logistic Stick Breaking Conditional Multinomial Model
The superset label learning problem seeks to train a classifier f : Rd 7? {1, ? ? ? , L} on a given
d
dataset (x, Y ) = {(xn , Yn )}N
n=1 , where each instance xn ? R has a candidate label set Yn ?
{1, ? ? ? , L}. The true labels y = {yn }N
are
not
directly
observed.
The only information is that
n=1
the true label yn of instance xn is in the candidate set Yn . The extra labels {l|l 6= yn , l ? Yn }
causing ambiguity will be called the distractor labels. For any test instance (xt , yt ) drawn from the
same distribution as {(xn , yn )}N
n=1 , the trained classifier f should be able to map xt to yt with high
probability. When |Yn | = 1 for all n, the problem is a supervised classification problem. We require
|Yn | < L for all n; that is, every candidate label set must provide at least some information about
the true label of the instance.
2.1
The Model
As stated in the introduction, the candidate label set is a noisy version of the true label. To train
a classifier, we first need a likelihood function p(Yn |xn ). The key to our approach is to write this
PL
as p(Yn |xn ) = yn =1 p(Yn |yn )p(yn |xn ), where each term is the product of the underlying true
classifier, p(yn |xn ), and the noise model p(Yn |yn ). We then make the following assumption about
the noise distribution:
2
Figure 1: The LSB-CMM. Square nodes are discrete, circle nodes are continuous, and double-circle
nodes are deterministic.
Assumption: All labels in the candidate label set Yn have the same probability of generating Yn ,
but no label outside of Yn can generate Yn
?(Yn ) if l ? Yn
p(Yn |yn = l) =
.
(1)
0
if l ?
/ Yn
This assumption enforces three constraints. First, the set of labels Yn is conditionally independent
of the input xn given yn . Second, labels that do not appear in Yn have probability 0 of generating
Yn . Third, all of the labels in Yn have equal probability of generating Yn (symmetry). Note that
these constraints do not imply that the training data are correctly labeled. That is, suppose that the
most likely label for a particular input xn is yn = l. Because p(yn |xn ) is a multinomial distribution,
a different label yn = l0 might be assigned to xn by the labeling process. Then this label is further
corrupted by adding distractor labels to produce Yn . Hence, it could be that l 6? Yn . In short, in
this model, we have the usual ?multinomial noise? in the labels which is then further compounded
by ?superset noise?. The third constraint can be criticized for being simplistic; we believe it can be
replaced with a learned noise model in future work.
Given (1), we can marginalize away yn in the following optimization problem maximizing the likelihood of observed candidate labels.
f?
=
=
arg max
f
arg max
f
N
X
log
n=1
N
X
n=1
L
X
p(yn |xn ; f )p(Yn |yn )
yn =1
log
X
p(yn |xn ; f ) +
yn ?Yn
N
X
log(?(Yn )).
(2)
n=1
Under the conditional independence and symmetry assumptions, the last term does not depend on f
and so can be ignored in the optimization. This result is consistent with the formulation in [10].
We propose the Logistic Stick-Breaking Conditional Multinomial Model to instantiate f (see
Figure 1). In LSB-CMM, we introduce a set of K regions (mixture components) {1, . . . , K}.
LSB-CMM has two components. The mapping component maps each instance xn to a region
zn , zn ? {1, . . . , K}. Then the coding component draws a label yn from the multinomial distribution indexed by zn with parameter ?zn . We denote the region indexes of the training instances by
z = (zn )N
n=1 .
In the mapping component, we employ the Logistic Stick Breaking Process(LSBP) [13] to model the
instance-region relationship. LSBP is a modification of the Dirichlet Process (DP) [14]. In LSBP,
the sequence of Bernoulli probabilities are the outputs of a sequence of logistic functions instead of
being random draws from a Beta distribution as in the Dirichlet process. The input to the k-th logistic
function is the dot product of xn and a learned weight vector wk ? Rd+1 . (The added dimension
corresponds to a zeroth feature fixed to be 1 to provide an intercept term.) To regularize these
logistic functions, we posit that each wk is drawn from a Gaussian distribution Normal(0, ?), where
? = diag(?, ? 2 , ? ? ? , ? 2 ). This regularizes all terms in wk except the intercept. For each xn , a
T
sequence of probabilities {vnk }K
k=1 is generated from logistic functions, where vnk = expit(wk xn )
and expit(u) = 1/(1 + exp(?u)) is the logistic function. We truncate k at K by setting wK =
(+?, 0, ? ? ? , 0) and thus vnK = 1. Let w denote the collection of all K wk . Given the probabilities
3
vn1 , . . . , vnK computed from xn , we choose the region zn according to a stick-breaking procedure:
p(zn = k) = ?nk = vnk
k?1
Y
(1 ? vni ).
(3)
i=1
Here we stipulate that the product is 1 when k = 1. Let ?n = (?n1 , ? ? ? , ?nK ) constitute the
parameter of a multinomial distribution. Then zn is drawn from this distribution.
In the coding component of LSB-CMM, we first draw K L-dimensional multinomial probabilities
? = {?k }K
k=1 from the prior Dirichlet distribution with parameter ?. Then, for each instance xn
with mixture zn , its label yn is drawn from the multinomial distribution with ?zn . In the traditional
multi-class problem, yn is observed. However, in the SLL problem yn is not observed and Yn is
generated from yn .
The generative process of the whole model is summarized below:
wk
zn
? Normal(0, ?), 1 ? k ? K ? 1, wK = (+?, 0, ? ? ? , 0)
?nk = expit(wkT xn )
? Mult(?n ),
k?1
Y
(1 ? expit(wiT xn ))
(4)
(5)
i=1
?k
yn
Yn
? Dirichlet(?)
? Mult(?zn )
? Dist1(yn ) (Dist1 is some distribution satisfying (1))
(6)
(7)
(8)
As shown in (2), the model needs to maximize the likelihood that each yn is in Yn . After incorporating the priors, we can write the penalized maximum likelihood objective as
?
?
N
X
X
max LL =
log ?
p(yn |xn , w, ?)? + log(p(w|0, ?)).
(9)
n=1
yn ?Yn
This cannot be solved directly, so we apply variational EM [1].
2.2
Variational EM
The hidden variables in the model are y, z, and ?. For these hidden variables, we introduce the
? ?
variational distribution q(y, z, ?|?,
? ), where ?? = {??n }N
? = {?
?k }K
n=1 and ?
k=1 are the parameters.
Then we factorize q as
? ?
q(z, y, ?|?,
?) =
N
Y
q(zn , yn |??n )
n=1
K
Y
q(?k |?
?k ),
(10)
k=1
where ??n is a K ? L matrix and q(zn , yn |??n ) is a multinomial distribution in which p(zn = k, yn =
l) = ??nkl . This distribution is constrained by the candidate label set: if a label l ?
/ Yn , then ??nkl = 0
for any value of k. The distribution q(?k |?
?k ) is a Dirichlet distribution with parameter ?
?k .
After we set the distribution q(z, y, ?), our variational EM follows standard methods. The detailed
derivation can be found in the supplementary materials [11]. Here we only show the final updating
step with some analysis.
In the E step, the parameters of variational distribution are updated as (11) and (12).
?nk exp Eq(?k |?? k ) [log(?kl )] , if l ? Yn
??nkl ?
,
0,
if l ?
/ Yn
?
?k
= ?+
N
X
??nkl .
(11)
(12)
n=1
The update of ??n in (11) indicates the key difference between the LSB-CMM model and traditional
clustering models. The formation of regions is directed by both instance similarities and class labels.
4
P
If the instance xn wants to join region k (i.e., l ??nkl is large), then it must be similar to wk as
well as to instances in that region in order to make ?nk large. Simultaneously, its candidate labels
must fit the ?label flavor? of region k, where the ?label flavor? means region k prefers labels having
large values in ?
? k . The update of ?
? in (12) can be interpreted as having each instance xn vote for
the label l for region k with weight ??nkl .
In the M step, we need to solve the maximization problem in (13) for each wk , 1 ? k ? K ? 1.
Note that wK is fixed. Each wk can be optimized separately. The optimization problem is similar to
the problem of logistic regression and is also a concave maximization problem, which can be solved
by any gradient-based method, such as BFGS.
N h
i
X
1
max ? wkT ??1 wk +
??nk log(expit(wkT xn )) + ??nk log(1 ? expit(wkT xn )) ,
(13)
wk
2
n=1
PL ?
PK
?
?
?
where ??nk =
l=1 ?nkl and ?nk =
j=k+1 ?nj . Intuitively, the variable ?nk is the probability that instance xn belongs to region k, and ??nk is the probability that xn belongs to region
{k + 1, ? ? ? , K}. Therefore, the optimal wk discriminates instances in region k against instances in
regions ? k.
2.3
Prediction
For a test instance xt , we predict the label with maximum posterior probability. The test instance
can be mapped to a region with w, but the coding matrix ? is marginalized out in the EM. We use
the variational distribution p(?k |?
?k ) as the prior of each ?k and integrate out all ?k -s. Given a test
point xt , the prediction is the label l that maximizes the probability p(yt = l|xt , w, ?
? ) calculated as
(14). The detailed derivation is also in the supplementary materials [11].
K
X
?
? kl
?tk P
p(yt = l|xt , w, ?
?) =
,
(14)
? kl
l?
k=1
Qk?1
where ?tk = expit(wkT xt ) i=1 (1 ? expit(wiT xt )) . The test instance goes to region k with
probability ?tk , and its label is decided by the votes (?
?k ) in that region.
2.4
Complexity Analysis and Practical Issues
In the E step, for each region k, the algorithm iterates over all candidate labels of all instances, so the
complexity is O(N KL). In the M step, the algorithm solves K ? 1 separate optimization problems.
Suppose each optimization problem takes O(V N d) time, where V is the number of BFGS iterations.
Then the complexity is O(KV N d). Since V is usually larger than L, the overall complexity of one
EM iteration is O(KV N d). Suppose the EM steps converge within m iterations, where m is usually
less than 50. Then the overall complexity is O(mKV N d). The space complexity is O(N K), since
PL
we only store the matrix l=1 ??nkl and the matrix ?
?.
In prediction, the mapping phase requires O(Kd) time to multiply w and the test instance. After the stick breaking process, which takes O(K) calculations, the coding phase requires O(KL)
calculation. Thus the overall time complexity is O(K max{d, L}). Hence, the prediction time is
comparable to that of logistic regression.
There are several practical issues that affect the performance of the model. Initialization: From
the model design, we can expect that instances in the same region have the same label. Therefore,
it is reasonable to initialize ?
? to have each region prefer only one label, that is, each ?
? k has one
1
, so that
element with large value and all others with small values. We initialize ? to ?nk = K
all regions have equal probability to be chosen at the start. Initialization of these two variables is
enough to begin the EM iterations. We find that such initialization works well for our model and
generally is better than random initialization. Calculation of Eq(?k |?? k ) [log(?kl )] in (11): Although
it has a closed-form solution, we encountered numerical issues, so we calculate it via Monte Carlo
sampling. This does not change complexity analysis above, since the training is dominated by M
step. Priors: We found that using a non-informative prior for Dirichlet(?) worked best. From (12)
and (14), we can see that when ? is marginalized, the distribution is non-informative when ? is set
to small values. We use ? = 0.05 in our experiments.
5
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1.6
Figure 2: Decision boundaries of LSB-CMM on a linearly-inseparable problem. Left: all data points
have true labels. Right: labels of gray data points are corrupted.
3
Experiments
In this section, we describe the results of several experiments we conducted to study the behavior of
our proposed model. First, we experiment with a toy problem to show that our algorithm can solve
problems with linearly-inseparable classes. Second, we perform controlled experiments on three
synthetic datasets to study the robustness of LSB-CMM with respect to the degree of ambiguity of
the label sets. Third, we experiment with three real-world datasets.
LSB-CMM Model: The LSB-CMM model has three parameters K, ? 2 , ?. We find that the model
is insensitive to K if it is sufficiently large. We set K = 10 for the toy problems and K = 5L for
other problems. ? is set to 0.05 for all experiments. When the data is standardized, the regularization
parameter ? 2 = 1 generally gives good results, so ? 2 is set to 1 in all superset label tasks.
Baselines: We compared the LSB-CMM model with three state-of-the-art methods. Supervised
SVM: the SVM is always trained with the true labels. Its performance can be viewed as an upper
bound on the performance of any SSL algorithm. LIBSVM [3] with RBF kernel was run to construct
a multi-class classifier in one-vs-one mode. One third of the training data was used to tune the C
parameter and the RBF kernel parameter ?. CLPL: CLPL [5] is a linear model that encourages
large average scores of candidate labels. The model is insensitive to the C parameter, so we set
the C value to 1000 (the default value in their code). SIM: SIM [2] minimizes the ranking loss of
instances in a bag. In controlled experiments and in one of the real-world problems, we could not
make the comparison to LSB-CMM because of the lack of bag information. The ? parameter is set
to 10?8 based on authors? recommendation.
3.1
A Toy Problems
In this experiment, we generate a linearly-inseparable SLL problem. The data has two dimensions
and six clusters drawn from six normal distributions with means at the corners of a hexagon. We
assign a label to each cluster so that the problem is linearly-inseparable (see (2)). In the first task,
we give the model the true labels. In the second task, we add a distractor label for two thirds of
all instances (gray data points in the figure). The distractor label is randomly chosen from the two
labels other than the true label. The decision boundaries found by LSB-CMM in both tasks are
shown in (2)). We can see that LSB-CMM can successfully give nonlinear decision boundaries for
this problem. After injecting distractor labels, LSB-CMM still recovers the boundaries between
classes. There is minor change of the boundary at the edge of the cluster, while the main part of
each cluster is classified correctly.
3.2
Controlled Experiments
We conducted controlled experiments on three UCI [6] datasets: {segment (2310 instances, 7
classes), pendigits (10992 instances, 10 classes), and usps (9298 instances, 10 classes)}. Tenfold cross validation is performed on all three datasets. For each training instance, we add distractor
labels with controlled probability. As in [5], we use p, q, and ? to control the ambiguity level of
6
Figure 3: Three regions learned by the model on usps
candidate label sets. The roles and values of these three variables are as follows: p is the probability that an instance has distractor labels (p = 1 for all controlled experiments); q ? {1, 2, 3, 4}
is the number of distractor labels; and ? ? {0.3, 0.7, 0.9, 0.95} is the maximum probability that a
distractor label co-occurs with the true label [5], also called the ambiguity degree.
We have two settings for these three variables. In the first setting, we hold q = 1 and vary ?, that is,
for each label l, we choose a specific label l0 6= l as the (unique) distractor label with probability ?
or choose any other label with probability 1 ? ?. In the extreme case when ? = 1, l0 and l always
co-occur, and they cannot be distinguished by any classifier. In the second setting, we vary q and
pick distractor labels randomly for each candidate label set.
The results are shown in Figure (4). Our LSB-CMM model significantly outperforms the CLPL
approach. As the number of distractor labels increases, performance of both methods goes down, but
not too much. When the true label is combined with different distractor labels, the disambiguation is
easy. The co-occurring distractor labels provide much less disambiguation. This explains why large
ambiguity degree hurts the performance of both methods. The small dataset (segment) suffers even
more from large ambiguity degree, because there are fewer data points that can ?break? the strong
correlation between the true label and the distractors.
To explore why the LSB-CMM model has good performance, we investigated the regions learned
by the model. Recall that ?nk is the probability that xn is sent to region k. In each region k, the
representative instances have large values of ?nk . We examined all ?nk from the model trained
on the usps dataset with 3 random distractor labels. For each region k, we selected the 9 most
representative instances. Figure (3) shows representative instances for three regions. These are all
from class ?2? but are written in different styles. This shows that the LSB-CMM model can discover
the sub-classes in the data. In some applications, the whole class is not easy to discriminate from
other classes, but sometimes each sub-class can be easily identified. In such cases, LSB-CMM will
be very useful and can improve performance.
Explanation of the results via regions can also give better understanding of the learned classifier.
In order to analyze the performance of the classifier learned from data with either superset labels
or fully observed labels, one traditional method is to compute the confusion matrix. While the
confusion matrix can only tell the relationships between classes, the mixture analysis can indicate
precisely which subclass of a class are confused with which subclasses of other classes. The regions
can also help the user identify and define new classes as refinements of existing ones.
3.3
Real-World Problems
We apply our model on three real-world problems. 1) BirdSong dataset [2]: This contains 548
10-second bird song recordings. Each recording contains 1-40 syllables. In total there are 4998
syllables. Each syllable is described by 38 features. The labels of each recording are the bird species
that were singing during that 10-second period, and these species become candidate labels set of each
syllable in the recording. 2) MSRCv2 dataset: This dataset contains 591 images with 23 classes.
The ground truth segmentations (regions with labels) are given. The labels of all segmentations
in an image are treated as candidate labels for each segmentation. Each segmentation is described
by 48-dimensional gradient and color histograms. 3) Lost dataset [5]: This dataset contains 1122
faces, and each face has the true label and a set of candidate labels. Each face is described by 108
PCA components. Since the bag information (i.e., which faces are in the same scene) is missing,
7
number of ambiguous labels
2
3
1
?
?
?
4
?
?
0.9
?
0.9
0.9
?
?
?
?
number of ambiguous labels
2
3
1
?
0.3
0.7
0.9
ambiguity degree
(a) segment
0.95
?
0.3
SVM
LSB?CMM, vary q
LSB?CMM, vary ?
CLPL, vary q
CLPL, vary ?
0.7
0.9
ambiguity degree
(b) pendigits
0.7
SVM
LSB?CMM, vary q
LSB?CMM, vary ?
CLPL, vary q
CLPL, vary ?
0.7
?
0.8
0.8
0.8
?
0.7
accuracy
4
1.0
1.0
4
1.0
number of ambiguous labels
2
3
1
0.95
?
0.3
SVM
LSB?CMM, vary q
LSB?CMM, vary ?
CLPL, vary q
CLPL, vary ?
0.7
0.9
ambiguity degree
0.95
(c) usps
Figure 4: Classification performance on synthetic data (red: LSB-CMM; blue: CLPL). The dot-dash
line is for different q values (number of distractor labels) as shown on the top x-axis. The dashed
line is for different ? (ambiguity degree) values as shown on the bottom x-axis.
Table 1: Classification Accuracies for Superset Label Problems
LSB-CMM
SIM
CLPL
SVM
BirdSong 0.715(0.042) 0.589(0.035) 0.637(0.034) 0.790(0.027)
MSRCv2 0.459(0.032) 0.454(0.043) 0.411(0.044) 0.673(0.043)
Lost
0.703(0.058)
0.710(0.045) 0.817(0.038)
SIM is not compared to our model on this dataset. We run 10-fold cross validation on these three
datasets. The BirdSong and MSRCv2 datasets are split by recordings/images, and the Lost dataset
is split by faces.
The classification accuracies are shown in Table (1). Accuracies of the three superset label learning
algorithms are compared using the paired t-test at the 95% confidence level. Values statistically
indistinguishable from the best performance are shown in bold. Our LSB-CMM model out-performs
the other two methods on the BirdSong database, and its performance is comparable to SIM on the
MSRCv2 dataset and to CLPL on the Lost dataset. It should be noted that the input features are
very coarse, which means that the cluster structure of the data is not well maintained. The relatively
low performance of the SVM confirms this. If the instances were more precisely described by finer
features, one would expect our model to perform better in those cases as well.
4
Conclusions
This paper introduced the Logistic Stick-Breaking Conditional Multinomial Model to address the
superset label learning problem. The mixture representation allows LSB-CMM to discover cluster
structure that has predictive power for the superset labels in the training data. Hence, if two labels
co-occur, LSB-CMM is not forced to choose one of them to assign to the training example but
instead can create a region that maps to both of them. Nonetheless, each region does predict from a
multinomial, so the model still ultimately seeks to predict a single label. Our experiments show that
the performance of the model is either better than or comparable to state-of-the-art methods.
Acknowledgment
This material is based upon work supported by the National Science Foundation under Grant
No. 1125228. The code as an R package is available at:
http://web.engr.oregonstate.edu/?liuli/files/LSB-CMM_1.0.tar.gz.
8
References
[1] C. M. Bishop. Pattern recognition and machine learning. Springer, 2006.
[2] F. Briggs & X. F. Fern & R. Raich. Rank-Loss Support Instance Machines for MIML Instance Annotation.
In proc. KDD, 2012.
[3] C.-C. Chang & C.-J. Lin. LIBSVM: A Library for Support Vector Machines. ACM Trans. on Intelligent
Systems and Technology, 2(3):1-27, 2011.
[4] T. Cour & B. Sapp & C. Jordan & B. Taskar. Learning From Ambiguously Labeled Images. In Proc.
CVPR 2009.
[5] T. Cour & B. Sapp & B. Taskar. Learning from Partial Labels. Journal of Machine Learning Research,
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[6] A. Frank & A. Asuncion. UCI Machine Learning Repository [http://archive.ics.uci.edu/ml].
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[8] E. Hullermeier & J. Beringer. Learning from Ambiguously Labeled Examples. In Proc. IDA-05, 6th
International Symposium on Intelligent Data Analysis Madrid, 2005.
[9] L. Jie & F. Orabona. Learning from Candidate Labeling Sets. In Proc. NIPS, 2010.
[10] R. Jin & Z. Ghahramani. Learning with Multiple Labels. In Proc. NIPS, 2002.
[11] L-P. Liu & T. Dietterich. A Conditional Multinomial Mixture Model for Superset Label Learning (Supplementary Materials),
http://web.engr.oregonstate.edu/?liuli/pdf/lsb_cmm_supp.pdf .
[12] N. Nguyen & R. Caruana. Classification with Partial Labels. In Proc. KDD, 2008.
[13] L. Ren & L. Du & L. Carin & D. B. Dunson. Logistic Stick-Breaking Process. Journal of Machine
Learning Research, 12:203-239, 2011.
[14] Y. W. Teh. Dirichlet Processes. Encyclopedia of Machine Learning, to appear. Springer.
[15] Z.-H. Zhou & M.-L. Zhang. Multi-Instance Multi-Label Learning with Application To Scene Classification. Advances in Neural Information Processing Systems, 19, 2007
[16] X. Zhu & A. B. Goldberg. Introduction to Semi-Supervised Learning. Synthesis Lectures on Artificial
Intelligence and Machine Learning, 3(1):1-130, 2009.
9
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3,974 | 4,598 | A Linear Time Active Learning Algorithm
for Link Classification?
Nicol`
o Cesa-Bianchi
Dipartimento di Informatica
Universit`
a degli Studi di Milano, Italy
Claudio Gentile
Dipartimento di Scienze Teoriche ed Applicate
Universit`a dell?Insubria, Italy
Giovanni Zappella
Dipartimento di Matematica
Universit`a degli Studi di Milano, Italy
Fabio Vitale
Dipartimento di Informatica
Universit`
a degli Studi di Milano, Italy
Abstract
We present very efficient active learning algorithms for link classification
in signed networks. Our algorithms are motivated by a stochastic model
in which edge labels are obtained through perturbations of a initial sign
assignment consistent with a two-clustering of the nodes. We provide a theoretical analysis within this model, showing that we can achieve an optimal
(to whithin a constant factor) number of mistakes on any graph G = (V, E)
such that |E| = ?(|V |3/2 ) by querying O(|V |3/2 ) edge labels. More generally, we show an algorithm that achieves optimality to within a factor
of O(k) by querying at most order of |V | + (|V |/k)3/2 edge labels. The
running time of this algorithm is at most of order |E| + |V | log |V |.
1
Introduction
A rapidly emerging theme in the analysis of networked data is the study of signed networks.
From a mathematical point of view, signed networks are graphs whose edges carry a sign
representing the positive or negative nature of the relationship between the incident nodes.
For example, in a protein network two proteins may interact in an excitatory or inhibitory
fashion. The domain of social networks and e-commerce offers several examples of signed
relationships: Slashdot users can tag other users as friends or foes, Epinions users can rate
other users positively or negatively, Ebay users develop trust and distrust towards sellers
in the network. More generally, two individuals that are related because they rate similar
products in a recommendation website may agree or disagree in their ratings.
The availability of signed networks has stimulated the design of link classification algorithms,
especially in the domain of social networks. Early studies of signed social networks are from
the Fifties. E.g., [8] and [1] model dislike and distrust relationships among individuals as
(signed) weighted edges in a graph. The conceptual underpinning is provided by the theory
of social balance, formulated as a way to understand the structure of conflicts in a network
of individuals whose mutual relationships can be classified as friendship or hostility [9]. The
advent of online social networks has revamped the interest in these theories, and spurred a
significant amount of recent work ?see, e.g., [7, 11, 14, 3, 5, 2], and references therein.
Many heuristics for link classification in social networks are based on a form of social balance
summarized by the motto ?the enemy of my enemy is my friend?. This is equivalent to
saying that the signs on the edges of a social graph tend to be consistent with some twoclustering of the nodes. By consistency we mean the following: The nodes of the graph can
be partitioned into two sets (the two clusters) in such a way that edges connecting nodes
?
This work was supported in part by the PASCAL2 Network of Excellence under EC grant
216886 and by ?Dote Ricerca?, FSE, Regione Lombardia. This publication only reflects the authors?
views.
1
from the same set are positive, and edges connecting nodes from different sets are negative.
Although two-clustering heuristics do not require strict consistency to work, this is admittely
a rather strong inductive bias. Despite that, social network theorists and practitioners
found this to be a reasonable bias in many social contexts, and recent experiments with
online social networks reported a good predictive power for algorithms based on the twoclustering assumption [11, 13, 14, 3]. Finally, this assumption is also fairly convenient from
the viewpoint of algorithmic design.
In the case of undirected signed graphs G = (V, E), the best performing heuristics exploiting
the two-clustering bias are based on spectral decompositions
of the signed adiacency matrix.
Noticeably, these heuristics run in time ? |V |2 , and often require a similar amount of
memory storage even on sparse networks, which makes them impractical on large graphs.
In order to obtain scalable algorithms with formal performance guarantees, we focus on the
active learning protocol, where training labels are obtained by querying a desired subset
of edges. Since the allocation of queries can match the graph topology, a wide range of
graph-theoretic techniques can be applied to the analysis of active learning algorithms. In
the recent work [2], a simple stochastic model for generating edge labels by perturbing some
unknown two-clustering of the graph nodes was introduced. For this model, the authors
proved that querying the edges of a low-stretch spanning tree of the input graph G = (V, E)
is sufficient to predict the remaining edge labels making a number of mistakes within a
factor of order (log |V |)2 log log |V | from the theoretical optimum. The overall running time
is O(|E| ln |V |). This result leaves two main problems open: First, low-stretch trees are a
powerful structure, but the algorithm to construct them is not easy to implement. Second,
the tree-based analysis of [2] does not generalize to query budgets larger than |V | ? 1 (the
edge set size of a spanning tree). In this paper we introduce a different active learning
approach for link classification that can accomodate a large spectrum of query budgets.
We show that on any graph with ?(|V |3/2 ) edges, a query budget of O(|V |3/2 ) is sufficient
to predict the remaining edge labels within a constant factor from the optimum. More in
3/2
general, we show that a budget of at most order of |V | + |Vk |
queries is sufficient to
make a number of mistakes within a factor of O(k) from the optimum with a running time
of order |E| + (|V |/k) log(|V |/k). Hence, a query budget of ?(|V |), of the same order as the
algorithm based on low-strech trees, achieves an optimality factor O(|V |1/3 ) with a running
time of just O(|E|).
At the end of the paper we also report on a preliminary set of experiments on medium-sized
synthetic and real-world datasets, where a simplified algorithm suggested by our theoretical
findings is compared against the best performing spectral heuristics based on the same
inductive bias. Our algorithm seems to perform similarly or better than these heuristics.
2
Preliminaries and notation
We consider undirected and connected graphs G = (V, E) with unknown edge labeling
Yi,j ? {?1, +1} for each (i, j) ? E. Edge labels can collectively be represented by the
associated signed adjacency matrix Y , where Yi,j = 0 whenever (i, j) 6? E. In the sequel,
the edge-labeled graph G will be denoted by (G, Y ).
We define a simple stochastic model for assigning binary labels Y to the edges of G. This
is used as a basis and motivation for the design of our link classification strategies. As
we mentioned in the introduction, a good trade-off between accuracy and efficiency in link
classification is achieved by assuming that the labeling is well approximated by a twoclustering of the nodes. Hence, our stochastic labeling model assumes that edge labels are
obtained by perturbing an underlying labeling which is initially consistent with an arbitrary
(and unknown) two-clustering. More formally, given an undirected and connected graph
G = (V, E), the labels Yi,j ? {?1, +1}, for (i, j) ? E, are assigned as follows. First, the
nodes in V are arbitrarily partitioned into two sets, and labels Yi,j are initially assigned
consistently with this partition (within-cluster edges are positive and between-cluster edges
are negative). Note that the consistency is equivalent to the following multiplicative rule:
For any (i, j) ? E, the label Yi,j is equal to the product of signs on the edges of any path
connecting i to j in G. This is in turn equivalent to say that any simple cycle within the
graph contains an even number of negative edges. Then, given a nonnegative
constant p < 12 ,
labels are randomly flipped in such a way that P Yi,j is flipped ? p for each (i, j) ? E.
2
We call this a p-stochastic assignment. Note that this model allows for correlations between
flipped labels.
A learning algorithm in the link classification setting receives a training set of signed edges
and, out of this information, builds a prediction model for the labels of the remaining edges.
It is quite easy to prove a lower bound on the number of mistakes that any learning algorithm
makes in this model.
Fact 1. For any undirected graph G = (V, E), any training set E0 ? E of edges, and any
learning algorithm that is given the labels of the edges in E 0 , the number M of mistakes
made by A on the remaining E \ E0 edges satisfies E M ? p E \ E0 , where the expectation
is with respect to a p-stochastic assignment of the labels Y .
Proof. Let Y be the following randomized labeling: first, edge labels are set consistently
with an arbitrary two-clustering of V . Then, a set of 2p|E| edges is selected uniformly at
random and the labels of these edges are set randomly (i.e., flipped or not flipped with equal
probability). Clearly, P(Yi,j is flipped) = p for each (i, j) ? E. Hence this
is a p-stochastic
E \ E0 randomly
assignment of the labels. Moreover,
E
\
E
contains
in
expectation
2p
0
labeled edges, on which A makes p E \ E0 mistakes in expectation.
In this paper we focus on active learning algorithms. An active learner for link classification
first constructs a query set E0 of edges, and then receives the labels of all edges in the query
set. Based on this training information, the learner builds a prediction model for the labels
of the remaining edges E \ E0 . We assume that the only labels ever revealed to the learner
are those in the query set. In particular, no labels are revealed during the prediction phase.
It is clear from Fact 1 that any active learning algorithm that queries the labels of at most
a constant fraction of the total number of edges will make on average ?(p|E|) mistakes.
We often write VG and EG to denote, respectively, the node set and the edge set of some
underlying graph G. For any two nodes i, j ? VG , Path(i, j) is any path in G having i
and j as terminals, and |Path(i, j)| is its length (number of edges). The diameter DG of a
graph G is the maximum over pairs i, j ? VG of the shortest path between i and j. Given
a tree T = (VT , ET ) in G, and two nodes i, j ? VT , we denote by dT (i, j) the distance
of i and j within T , i.e., the length of the (unique) path PathT (i, j) connecting the two
nodes in T . Moreover, ?T (i, j) denotes the parity of this path, i.e., the product of edge
signs along it. When T is a rooted tree, we denote by ChildrenT (i) the set of children of
i in T . Finally,
T 0 , T 00 ? G such that VT 0 ? VT 00 ? ?, we let
given two disjoint subtrees
EG (T 0 , T 00 ) ? (i, j) ? EG : i ? VT 0 , j ? VT 00 .
3
Algorithms and their analysis
In this section, we introduce and analyze a family of active learning algorithms for link
classification. The analysis is carried out under the p-stochastic assumption. As a warm
up, we start off recalling the connection to the theory of low-stretch spanning trees (e.g.,
[4]), which turns out to be useful in the important special case when the active learner is
afforded to query only |V | ? 1 labels.
Let Eflip ? E denote the (random) subset of edges whose labels have been flipped in a
p-stochastic assignment, and consider the following class of active learning algorithms parameterized by an arbitrary spanning tree T = (VT , ET ) of G. The algorithms in this class
use E0 = ET as query set. The label of any test edge e0 = (i, j) 6? ET is predicted as the
parity ?T (e0 ). Clearly enough, if a test edge e0 is predicted wrongly, then either e0 ? Eflip
or PathT (e0 ) contains at least one flipped edge. Hence, the number of mistakes MT made
by our active learner on the set of test edges E \ ET can be deterministically bounded by
X X
MT ? |Eflip | +
I e ? PathT (e0 ) I e ? Eflip
(1)
e0 ?E\ET e?E
where I ? denotes the indicator of the Boolean predicate at argument. A quantity which
can be related to MT is the average stretch of a spanning tree T which, for our purposes,
reduces to
h
i
P
1
0
.
e0 ?E\ET PathT (e )
|E| |V | ? 1 +
3
A stunning result of [4] shows that every connected, undirected andunweighted graph has
a spanning tree with an average stretch of just O log2 |V | log log |V | . If our active learner
uses a spanning tree with the same low stretch, then the following result holds.
Theorem 1 ([2]). Let (G, Y ) = ((V, E), Y ) be a labeled graph with p-stochastic assigned
labels Y . If the active learner queries
the edges of a spanning tree T = (VT, ET ) with
2
average stretch O log |V | log log |V | , then E MT ? p|E| ? O log2 |V | log log |V | .
We call the quantity multiplying p |E| in the upper bound the optimality factor of the
algorithm. Recall that Fact 1 implies that this factor cannot be smaller than a constant
when the query set size is a constant fraction of |E|.
Although low-stretch trees can be constructed in time O |E| ln |V | , the algorithms are fairly
complicated (we are not aware of available implementations), and the constants hidden in
the asymptotics can be high. Another disadvantage is that we are forced to use a query set
of small and fixed size |V | ? 1. In what follows we introduce algorithms that overcome both
limitations.
A key aspect in the analysis of prediction performance is the ability to select a query set
so
creates a short circuit with a training path. This is quantified by
P that each test edge
0
I
e
?
Path
(e
)
in (1). We make this explicit as follows. Given a test edge (i, j)
T
e?E
and a path Path(i, j) whose edges are queried edges, we say that we are predicting label Yi,j
using path Path(i, j) Since (i, j) closes Path(i, j) into a circuit, in this case we also say that
(i, j) is predicted using the circuit.
Fact 2. Let (G, Y ) = ((V, E), Y ) be a labeled graph with p-stochastic assigned labels Y .
Given query set E0 ? E, the number M of mistakes made when predicting test edges (i, j) ?
E \ E0 using training paths Path(i, j) whose length is uniformly bounded by ` satisfies EM ?
` p |E \ E0 | .
P
Proof. We have the chain of inequalities EM ? (i,j)?E\E0 1 ? (1 ? p)|Path(i,j)| ?
P
P
`
? (i,j)?E\E0` p ? ` p |E \ E0 | .
(i,j)?E\E0 1 ? (1 ? p)
For instance, if the input graph G = (V, E) has diameter DG and the queried edges are
those of a breadth-first spanning tree, which can be generated in O(|E|) time, then the
above fact holds with |E0 | = |V | ? 1, and ` = 2 DG . Comparing to Fact 1 shows that this
simple breadth-first strategy is optimal up to constants factors whenever G has a constant
diameter. This simple observation is especially relevant in the light of the typical graph
topologies encountered in practice, whose diameters are often small. This argument is at
the basis of our experimental comparison ?see Section 4 .
Yet, this mistake bound can be vacuous on graph having a larger diameter. Hence, one may
think of adding to the training spanning tree new edges so as to reduce the length of the
circuits used for prediction, at the cost of increasing the size of the query set. A similar
technique based on short circuits has been used in [2], the goal there being to solve the link
classification problem in a harder adversarial environment. The precise tradeoff between
prediction accuracy (as measured by the expected number of mistakes) and fraction of
queried edges is the main theoretical concern of this paper.
We now introduce an intermediate (and simpler) algorithm, called treeCutter, which
improves on the optimality factor when the diameter DG is not small. In particular, we
demonstrate that treeCutter achieves
p a good upper bound on the number of mistakes
on any graph such that |E| ? 3|V | + |V |. This algorithm is especially
p effective when the
input graph is dense, with an optimality factor between O(1) and O( |V |). Moreover, the
total time for predicting the test edges scales linearly with the number of such edges, i.e.,
treeCutter predicts edges in constant amortized time. Also, the space is linear in the size
of the input graph.
The algorithm (pseudocode given in Figure 1) is parametrized by a positive integer k ranging
from 2 to |V |. The actual setting of k depends on the graph topology and the desired fraction
of query set edges, and plays a crucial role in determining the prediction performance.
Setting k ? DG makes treeCutter reduce to querying only the edges of a breadth-first
spanning tree of G, otherwise it operates in a more involved way by splitting G into smaller
node-disjoint subtrees.
4
In a preliminary step (Line 1 in Figure 1), treeCutter draws an arbitrary breadth-first
spanning tree T = (VT , ET ). Then subroutine extractTreelet(T, k) is used in a do-while
loop to split T into vertex-disjoint subtrees T 0 whose height is k (one of them might have a
smaller height). extractTreelet(T, k) is a very simple procedure that performs a depthfirst visit of the tree T at argument. During this visit, each internal node may be visited
several times (during backtracking steps). We assign each node i a tag hT (i) representing
the height of the subtree of T rooted at i. hT (i) can be recursively computed during the
visit. After this assignment, if we have hT (i) = k (or i is the root of T ) we return the
subtree Ti of T rooted at i. Then treeCutter removes (Line 6) Ti from T along with
all edges of ET which are incident to nodes of Ti , and then iterates until VT gets empty.
By construction, the diameter of the generated subtrees will not be larger than 2k. Let T
denote the set of these subtrees. For each T 0 ? T , the algorithm queries all the labels of
ET 0 , each edge (i, j) ? EG \ ET 0 such that i, j ? VT 0 is set to be a test edge, and label Yi,j is
predicted using PathT 0 (i, j) (note that this coincides with PathT 0 (i, j), since T 0 ? T ), that
is, Y?i,j = ?T (i, j). Finally, for each pair of distinct subtrees T 0 , T 00 ? T such that there exists
a node of VT 0 adjacent to a node of VT 00 , i.e., such that EG (T 0 , T 00 ) is not empty, we query the
label of an arbitrarily selected edge (i0 , i00 ) ? EG (T 0 , T 00 ) (Lines 8 and 9 in Figure 1). Each
edge (u, v) ? EG (T 0 , T 00 ) whose label has not been previously queried is then part of the
test set, and its label will be predicted as Y?u,v ? ?T (u, i0 ) ? Yi0 ,i00 ? ?T (i00 , v) (Line 11). That
is, using the path obtained by concatenating PathT 0 (u, i0 ) to edge (i0 , i00 ) to PathT 0 (i00 , v).
The following theorem1 quantifies the number of mistakes made by treeCutter. The
treeCutter(k)
Parameter: k ? 2
Initialization: T ? ?.
1. Draw an arbitrary breadth-first spanning tree T of G
2. Do
3.
T 0 ? extractTreelet(T, k), and query all labels in ET 0
4.
T ? T ? {T 0 }
5.
For each i, j ? VT 0 , set predict Y?i,j ? ?T (i, j)
6.
T ? T \ T0
7. While (VT 6? ?)
8. For each T 0 , T 00 ? T : T 0 6? T 00
9.
If EG (T 0 , T 00 ) 6? ? query the label of an arbitrary edge (i0 , i00 ) ? EG (T 0 , T 00 )
10.
For each (u, v) ? EG (T 0 , T 00 ) \ {(i0 , i00 )}, with i0 , u ? VT 0 and v, i00 ? VT 00
11.
predict Y?u,v ? ?T 0 (u, i0 ) ? Yi0 ,i00 ? ?T 00 (i00 , v)
Figure 1: treeCutter pseudocode.
extractTreelet(T, k)
Parameters: tree T , k ? 2.
1. Perform a depth-first visit of T starting from the root.
2. During the visit
3.
For each i ? VT visited for the |1 + ChildrenT (i)|-th time (i.e., the last visit of i)
4.
If i is a leaf set hT (i) ? 0
5.
Else set hT (i) ? 1 + max{hT (j) : j ? ChildrenT (i)}
6.
If hT (i) = k or i ? T ?s root return subtree rooted at i
Figure 2: extractTreelet pseudocode.
2
|
|V |
|E|
requirement on the graph density in the statement, i.e., |V | ? 1 + |V
2k2 + 2k ? 2 implies
that the test set is not larger than the query set. This is a plausible assumption in active
learning scenarios, and a way of adding meaning to the bounds.
Theorem 2. For any integer k ? 2, the number M of mistakes made by treeCutter on
2
any graph G(V, E) with |E| ? 2|V | ? 2 + |Vk2| + |Vk | satisfies EM ? min{4k + 1, 2DG }p|E|,
while the query set size is bounded by |V | ? 1 +
|V |2
2k2
+
|V |
2k
?
|E|
2 .
We now refine the simple argument leading to treeCutter, and present our active link
classifier. The pseudocode of our refined algorithm, called starMaker, follows that of
1
Due to space limitations long proofs are presented in the supplementary material.
5
Figure 1 with the following differences: Line 1 is dropped (i.e., starMaker does not draw
an initial spanning tree), and the call to extractTreelet in Line 3 is replaced by a call
to extractStar. This new subroutine just selects the star T 0 centered on the node of G
having largest degree, and queries all labels of the edges in ET 0 . The next result shows that
this algorithm gets a constant optimality factor while using a query set of size O(|V |3/2 ).
Theorem 3. The number M of mistakes made by starMaker on any given graph G(V, E)
3
with |E| ? 2|V | ? 2 + 2|V | 2 satisfies EM ? 5 p|E|, while the query set size is upper bounded
3
by |V | ? 1 + |V | 2 ? |E|
2 .
Finally, we combine starMaker with treeCutter so as to obtain an algorithm, called
3
treeletStar, that can work with query sets smaller than |V | ? 1 + |V | 2 labels. treeletStar is parameterized by an integer k and follows Lines 1?6 of Figure 1 creating a set
T of trees through repeated calls to extractTreelet. Lines 7?11 are instead replaced
by the following procedure: a graph G0 = (VG0 , EG0 ) is created such that: (1) each node
in VG0 corresponds to a tree in T , (2) there exists an edge in EG0 if and only if the two
corresponding trees of T are connected by at least one edge of EG . Then, extractStar
is used to generate a set S of stars of vertices of G0 , i.e., stars of trees of T . Finally, for
each pair of distinct stars S 0 , S 00 ? S connected by at least one edge in EG , the label of an
arbitrary edge in EG (S 0 , S 00 ) is queried. The remaining edges are all predicted.
Theorem 4. For any integer k ? 2 and for any graph G = (V, E) with |E| ? 2|V | ? 2 +
3
2 |V k|?1 + 1 2 , the number M of mistakes made by treeletStar(k) on G satisfies EM =
3
O(min{k, DG }) p|E|, while the query set size is bounded by |V | ? 1 + |V k|?1 + 1 2 ? |E|
2 .
Hence, even if DG is large, setting k = |V |1/3 yields a O(|V |1/3 ) optimality factor just by
querying O(|V |) edges. On the other hand, a truly constant optimality factor is obtained
by querying as few as O(|V |3/2 ) edges (provided the graph has sufficiently many edges). As
a direct consequence (and surprisingly enough), on graphs which are only moderately dense
we need not observe too many edges in order to achieve a constant optimality factor. It is
instructive to compare the bounds obtained by treeletStar to the ones we can achieve
by using the cccc algorithm of [2], or the low-stretch spanning trees given in Theorem 1.
Because cccc operates within a harder adversarial setting, it is easy to show that Theorem
9 in [2] extends to the p-stochastic assignment model by replacing ?2 (Y ) with p|E| therein.2
23 p
|V |, where ? ? (0, 1] is the fraction of
The resulting optimality factor is of order 1??
?
queried edges out of the total number of edges. A quick comparison to Theorem 4 reveals
that treeletStar achieves a sharper mistake bound for p
any value of ?. For instance, in
order to obtain an optimality factor which is lower than |V |, cccc has to query in the
worst case a fraction of edges that goes to one as |V | ? ?. On top of this, our algorithms
are faster and easier to implement ?see Section 3.1.
Next, we compare to query sets produced by low-stretch spanning trees. A low-stretch
spanning tree achieves a polylogarithmic optimality factor by querying |V | ? 1 edge labels.
The results in [4] show that we cannot hope to get a better optimality factor using a single
low-stretch spanning tree combined by the analysis in (1). For a comparable amount ?(|V |)
of queried labels, Theorem 4 offers the larger optimality factor |V |1/3 . However, we can get
a constant optimality factor by increasing the query set size to O(|V |3/2 ). It is not clear
how multiple low-stretch trees could be combined to get a similar scaling.
3.1
Complexity analysis and implementation
We now compute bounds on time and space requirements for our three algorithms. Recall
the different lower bound conditions on the graph density that must hold to ensure that the
2
query set size is not larger than the test set size. These were |E| ? 2|V | ? 2 + |Vk2| + |Vk | for
3
treeCutter(k) in Theorem 2, |E| ? 2|V | ? 2 + 2|V | 2 for starMaker in Theorem 3, and
32
|E| ? 2|V | ? 2 + 2 |V k|?1 + 1
for treeletStar(k) in Theorem 4.
2
This theoretical comparison is admittedly unfair, as cccc has been designed to work in a
harder setting than p-stochastic. Unfortunately, we are not aware of any other general active
learning scheme for link classification to compare with.
6
Theorem 5. For any input graph G = (V, E) which is dense enough to ensure that the
query set size is no larger than the test set size, the total time needed for predicting all test
labels is:
O(|E|)
for treeCutter(k) and for all k
O |E| + |V | log |V |
for starMaker
|V |
|V |
O |E| +
log
for treeletStar(k) and for all k.
k
k
In particular, whenever k|E| = ?(|V | log |V |) we have that treeletStar(k) works in constant amortized time. For all three algorithms, the space required is always linear in the
input graph size |E|.
4
Experiments
In this preliminary set of experiments we only tested the predictive performance of
treeCutter(|V |). This corresponds to querying only the edges of the initial spanning
tree T and predicting all remaining edges (i, j) via the parity of PathT (i, j). The spanning
tree T used by treeCutter is a shortest-path spanning tree generated by a breadth-first
visit of the graph (assuming all edges have unit length). As the choice of the starting node
in the visit is arbitrary, we picked the highest degree node in the graph. Finally, we run
through the adiacency list of each node in random order, which we empirically observed to
improve performance.
Our baseline is the heuristic ASymExp from [11] which, among the many spectral heuristics
proposed there, turned out to perform best on all our datasets. With integer parameter
z, ASymExp(z) predicts using a spectral transformation of the training sign matrix Ytrain ,
whose only non-zero entries are
edges. The label of edge (i, j) is
the signs of the training
predicted using exp(Ytrain (z)) i,j . Here exp Ytrain (z) = Uz exp(Dz )Uz> , where Uz Dz Uz> is
the spectral decomposition of Ytrain containing only the z largest eigenvalues and their corresponding eigenvectors. Following [11], we ran ASymExp(z) with the values z = 1, 5, 10, 15.
This heuristic uses the two-clustering bias as follows : expand exp(Ytrain ) in a series of
n
n
powers Ytrain
. Then each Ytrain
)i,j is a sum of values of paths of length n between i and
j. Each path has value 0 if it contains at least one test edge, otherwise its value equals the
product of queried labels on the path edges. Hence, the sign of exp(Ytrain ) is the sign of a
linear combination of path values, each corresponding to a prediction consistent with the
two-clustering bias ?compare this to the multiplicative rule used by treeCutter. Note
that ASymExp
and the other spectral heuristics from [11] have all running times of order
? |V |2 .
We performed a first set of experiments on synthetic signed graphs created from a subset
of the USPS digit recognition dataset. We randomly selected 500 examples labeled ?1? and
500 examples labeled ?7? (these two classes are not straightforward to tell apart). Then,
we created a graph using a k-NN rule with k = 100. The edges were labeled as follows:
all edges incident to nodes with the same USPS label were labeled +1; all edges incident
to nodes with different USPS labels were labeled ?1. Finally, we randomly pruned the
positive edges so to achieve an unbalance of about 20% between the two classes.3 Starting
from this edge label assignment, which is consistent with the two-clustering associated with
the USPS labels, we generated a p-stochastic label assignment by flipping the labels of a
random subset of the edges. Specifically, we used the three following synthetic datasets:
DELTA0: No flippings (p = 0), 1,000 nodes and 9,138 edges;
DELTA100: 100 randomly chosen labels of DELTA0 are flipped;
DELTA250: 250 randomly chosen labels of DELTA0 are flipped.
We also used three real-world datasets:
MOVIELENS: A signed graph we created using Movielens ratings.4 We first normalized
the ratings by subtracting from each user rating the average rating of that user. Then,
we created a user-user matrix of cosine distance similarities. This matrix was sparsified by
3
4
This is similar to the class unbalance of real-world signed networks ?see below.
www.grouplens.org/system/files/ml-1m.zip.
7
DELTA0
DELTA100
1
0.8
0.6
0.4
1
0.8
0.6
0.4
20
30
40
TRAINING SET SIZE (%)
50
20
30
40
TRAINING SET SIZE (%)
MOVIELENS
F-MEASURE (%)
0.2
3
4
5
6
7
TRAINING SET SIZE (%)
10
20
30
40
TRAINING SET SIZE (%)
8
9
0.4
10
ASymExp z=1
ASymExp z=5
ASymExp z=10
ASymExp z=15
TreeCutter
0.6
0.4
0.2
10
20
30
40
50
10
TRAINING SET SIZE (%)
20
30
40
TRAINING SET SIZE (%)
Figure 3: F-measure against training set size for treeCutter(|V |) and ASymExp(z) with different values of z
on both synthetic and real-world datasets. By construction, treeCutter never makes a mistake when the labeling
is consistent with a two-clustering. So on DELTA0 treeCutter does not make mistakes whenever the training set
contains at least one spanning tree. With the exception of EPINIONS, treeCutter outperforms ASymExp using
a much smaller training set. We conjecture that ASymExp responds to the bias not as well as treeCutter, which
on the other hand is less robust than ASymExp to bias violations (supposedly, the labeling of EPINIONS).
zeroing each entry smaller than 0.1 and removing all self-loops. Finally, we took the sign
of each non-zero entry. The resulting graph has 6,040 nodes and 824,818 edges (12.6% of
which are negative).
SLASHDOT: The biggest strongly connected component of a snapshot of the Slashdot
social network,5 similar to the one used in [11]. This graph has 26,996 nodes and 290,509
edges (24.7% of which are negative).
EPINIONS: The biggest strongly connected component of a snapshot of the Epinions
signed network,6 similar to the one used in [13, 12]. This graph has 41,441 nodes and
565,900 edges (26.2% of which are negative).
Slashdot and Epinions are originally directed graphs. We removed the reciprocal edges with
mismatching labels (which turned out to be only a few), and considered the remaining edges
as undirected.
The following table summarizes the key statistics of each dataset: Neg. is the fraction of
negative edges, |V |/|E| is the fraction of edges queried by treeCutter(|V |), and Avgdeg
is the average degree of the nodes of the network.
Dataset
DELTA0
DELTA100
DELTA250
SLASHDOT
EPINIONS
MOVIELENS
|V |
1000
1000
1000
26996
41441
6040
|E|
9138
9138
9138
290509
565900
824818
Neg.
21.9%
22.7%
23.5%
24.7%
26.2%
12.6%
|V |/|E|
10.9%
10.9%
10.9%
9.2%
7.3%
0.7%
Avgdeg
18.2
18.2
18.2
21.6
27.4
273.2
Our results are summarized in Figure 3, where we plot F-measure (preferable to accuracy
due to the class unbalance) against the fraction of training (or query) set size. On all
datasets, but MOVIELENS, the training set size for ASymExp ranges across the values 5%,
10%, 25%, and 50%. Since MOVIELENS has a higher density, we decided to reduce those
fractions to 1%, 3%, 5% and 10%. treeCutter(|V |) uses a single spanning tree, and thus
we only have a single query set size value. All results are averaged over ten runs of the
algorithms. The randomness in ASymExp is due to the random draw of the training set.
The randomness in treeCutter(|V |) is caused by the randomized breadth-first visit.
5
6
50
EPINIONS
0.8
0.2
2
0.6
50
ASymExp z=1
ASymExp z=5
ASymExp z=10
ASymExp z=15
TreeCutter
0.6
0.4
1
1
0.8
SLASHDOT
ASymExp z=1
ASymExp z=5
ASymExp z=10
ASymExp z=15
TreeCutter
0.6
ASymExp z=1
ASymExp z=5
ASymExp z=10
ASymExp z=15
TreeCutter
0.4
10
F-MEASURE (%)
10
F-MEASURE (%)
DELTA250
ASymExp z=1
ASymExp z=5
ASymExp z=10
ASymExp z=15
TreeCutter
F-MEASURE (%)
F-MEASURE (%)
F-MEASURE (%)
ASymExp z=1
ASymExp z=5
ASymExp z=10
ASymExp z=15
TreeCutter
snap.stanford.edu/data/soc-sign-Slashdot081106.html.
snap.stanford.edu/data/soc-sign-epinions.html.
8
50
References
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Psychological review, 63(5):277?293, 1956.
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3,975 | 4,599 | On-line Reinforcement Learning Using Incremental
Kernel-Based Stochastic Factorization
Andr?e M. S. Barreto
School of Computer Science
McGill University
Montreal, Canada
[email protected]
Doina Precup
School of Computer Science
McGill University
Montreal, Canada
[email protected]
Joelle Pineau
School of Computer Science
McGill University
Montreal, Canada
[email protected]
Abstract
Kernel-based stochastic factorization (KBSF) is an algorithm for solving reinforcement learning tasks with continuous state spaces which builds a Markov decision process (MDP) based on a set of sample transitions. What sets KBSF apart
from other kernel-based approaches is the fact that the size of its MDP is independent of the number of transitions, which makes it possible to control the trade-off
between the quality of the resulting approximation and the associated computational cost. However, KBSF?s memory usage grows linearly with the number of
transitions, precluding its application in scenarios where a large amount of data
must be processed. In this paper we show that it is possible to construct KBSF?s
MDP in a fully incremental way, thus freeing the space complexity of this algorithm from its dependence on the number of sample transitions. The incremental
version of KBSF is able to process an arbitrary amount of data, which results in
a model-based reinforcement learning algorithm that can be used to solve continuous MDPs in both off-line and on-line regimes. We present theoretical results
showing that KBSF can approximate the value function that would be computed
by conventional kernel-based learning with arbitrary precision. We empirically
demonstrate the effectiveness of the proposed algorithm in the challenging threepole balancing task, in which the ability to process a large number of transitions
is crucial for success.
1
Introduction
The task of learning a policy for a sequential decision problem with continuous state space is a
long-standing challenge that has attracted the attention of the reinforcement learning community for
years. Among the many approaches that have been proposed to solve this problem, kernel-based
reinforcement learning (KBRL) stands out for its good theoretical guarantees [1, 2]. KBRL solves
a continuous state-space Markov decision process (MDP) using a finite model constructed based on
sample transitions only. By casting the problem as a non-parametric approximation, it provides a
statistically consistent way of approximating an MDP?s value function. Moreover, since it comes
down to the solution of a finite model, KBRL always converges to a unique solution.
Unfortunately, the good theoretical properties of kernel-based learning come at a price: since the
model constructed by KBRL grows with the amount of sample transitions, the number of operations
performed by this algorithm quickly becomes prohibitively large as more data become available.
Such a computational burden severely limits the applicability of KBRL to real reinforcement learning (RL) problems. Realizing that, many researchers have proposed ways of turning KBRL into a
more practical tool [3, 4, 5]. In this paper we focus on our own approach to leverage KBRL, an
algorithm called kernel-based stochastic factorization (KBSF) [4].
KBSF uses KBRL?s kernel-based strategy to perform a soft aggregation of the states of its MDP.
By doing so, our algorithm is able to summarize the information contained in KBRL?s model in an
MDP whose size is independent of the number of sample transitions. KBSF enjoys good theoretical
1
guarantees and has shown excellent performance on several tasks [4]. The main limitation of the
algorithm is the fact that, in order to construct its model, it uses an amount of memory that grows
linearly with the number of sample transitions. Although this is a significant improvement over
KBRL, it still hinders the application of KBSF in scenarios in which a large amount of data must be
processed, such as in complex domains or in on-line reinforcement learning.
In this paper we show that it is possible to construct KBSF?s MDP in a fully incremental way,
thus freeing the space complexity of this algorithm from its dependence on the number of sample
transitions. In order to distinguish it from its original, batch counterpart, we call this new version of
our algorithm incremental KBSF, or iKBSF for short. As will be seen, iKBSF is able to process an
arbitrary number of sample transitions. This results in a model-based RL algorithm that can be used
to solve continuous MDPs in both off-line and on-line regimes.
A second important contribution of this paper is a theoretical analysis showing that it is possible to
control the error in the value-function approximation performed by KBSF. In our previous experiments with KBSF, we defined the model used by this algorithm by clustering the sample transitions
and then using the clusters?s centers as the representative states in the reduced MDP [4]. However,
we did not provide a theoretical justification for such a strategy. In this paper we fill this gap by
showing that we can approximate KBRL?s value function at any desired level of accuracy by minimizing the distance from a sampled state to the nearest representative state. Besides its theoretical
interest, the bound is also relevant from a practical point of view, since it can be used in iKBSF to
guide the on-line selection of representative states.
Finally, a third contribution of this paper is an empirical demonstration of the performance of iKBSF
in a new, challenging control problem: the triple pole-balancing task, an extension of the well-known
double pole-balancing domain. Here, iKBSF?s ability to process a large number of transitions is
crucial for achieving a high success rate, which cannot be easily replicated with batch methods.
2
Background
In reinforcement learning, an agent interacts with an environment in order to find a policy that
maximizes the discounted sum of rewards [6]. As usual, we assume that such an interaction can be
modeled as a Markov decision process (MDP, [7]). An MDP is a tuple M ? (S, A, Pa , ra , ?), where S
is the state space and A is the (finite) action set. In this paper we are mostly concerned with MDPs
with continuous state spaces, but our strategy will be to approximate such models as finite MDPs. In
a finite MDP the matrix Pa ? R|S|?|S| gives the transition probabilities associated with action a ? A
and the vector ra ? R|S| stores the corresponding expected rewards. The discount factor ? ? [0, 1) is
used to give smaller weights to rewards received further in the future.
Consider an MDP M with continuous state space S ? [0, 1]d . Kernel-based reinforcement learning
(KBRL) uses sample transitions to derive a finite MDP that approximates the continuous model [1,
2]. Let Sa = {(sak , rka , s?ak )|k = 1, 2, ..., na } be sample transitions associated with action a ? A, where
sak , s?ak ? S and rka ? R. Let ? : R+ 7? R+ be a Lipschitz continuous function and let k? (s, s? ) be a
kernel function defined as k? (s, s? ) = ? (k s ? s? k /?), where k ? k is a norm in Rd and ? > 0. Finally,
a
k? (s, saj ).
define the normalized kernel function associated with action a as ??a (s, sai ) = k? (s, sai )/?nj=1
The model constructed by KBRL has the following transition and reward functions:
a
a
ri , if s? = s?ai ,
?? (s, sai ), if s? = s?ai ,
a
?
a
?
?
?
(1)
and R (s, s ) =
P (s |s) =
0, otherwise.
0, otherwise
Since only transitions ending in the states s?ai have a non-zero probability of occurrence, one can
define a finite MDP M? composed solely of these n = ?a na states [2, 3]. After V? ? , the optimal
? has been computed,
value function of M,
the value of any state-action pair can be determined as:
a
??a (s, sai ) ria + ? V? ? (s?ai ) , where s ? S and a ? A. Ormoneit and Sen [1] proved that,
Q(s, a) = ?ni=1
if na ? ? for all a ? A and the widths of the kernels ? shrink at an ?admissible? rate, the probability
of choosing a suboptimal action based on Q(s, a) converges to zero.
? but the time and
Using dynamic programming, one can compute the optimal value function of M,
space required to do so grow fast with the number of states n [7, 8]. Therefore, the use of KBRL
leads to a dilemma: on the one hand, one wants to use as many transitions as possible to capture the
dynamics of M, but on the other hand one wants to have an MDP M? of manageable size.
2
Kernel-based stochastic factorization (KBSF) provides a practical way of weighing these two conflicting objectives [4]. Our algorithm compresses the information contained in KBRL?s model M?
in an MDP M? whose size is independent of the number of transitions n. The fundamental idea
behind KBSF is the ?stochastic-factorization trick?, which we now summarize. Let P ? Rn?n be a
transition-probability matrix and let P = DK be a factorization in which D ? Rn?m and K ? Rm?n are
stochastic matrices. Then, swapping the factors D and K yields another transition matrix P? = KD
that retains the basic topology of P?that is, the number of recurrent classes and their respective
reducibilities and periodicities [9]. The insight is that, in some cases, one can work with P? instead
of P; when m ? n, this replacement affects significantly the memory usage and computing time.
? Let S? ? {s?1 , s?2 , ..., s?m }
KBSF results from the application of the stochastic-factorization trick to M.
a
n
?m
a
?
? a ? Rm?na with elbe a set of representative states in S. KBSF computes matrices D ? R
and K
a
a
a
a
a
?
?
ements di j = ??? (s?i , s? j ) and ki j = ?? (s?i , s j ), where ??? is defined as ??? (s, s?i ) = k?? (s, s?i )/?mj=1 k?? (s, s? j ).
? A, P? a , r? a , ?), where P? a = K
?a
? aD
The basic idea of the algorithm is to replace the MDP M? with M? ? (S,
a
a
a
a
n
a
a
?
and r? = K r (r ? R is the vector composed of sample rewards ri ). Thus, instead of solving an
?1 ? D
? 2 ? ... D
? |A| ? ] ? Rm?n
MDP with n states, one solves a model with m states only. Let D? ? [ D
? ? ? Rm?|A| , the optimal action-value function of M,
? 1K
? 2 ...K
? |A| ] ? Rm?n . Based on Q
?
and let K ? [K
? ? , where ? is the ?max? operator
one can obtain an approximate value function for M? as v? = ?DQ
? ? )ia . We have showed that the error in v? is bounded by:
applied row wise, that is, v?i = maxa (DQ
k?v? ? v? k? ?
?
a
1
1
C?
a
?
?
P
?
DK
,
max
max k?ra ? D?ra k? +
Cmax
(1
?
max
d
)
+
ij
?
i
j
1?? a
2 a
(1 ? ?)2
(2)
where k?k? is the infinity norm, v? ? ? Rn is the optimal value function of KBRL?s MDP, C? =
maxa,i r?ia ? mina,i r?ia , C? = maxa,i r?ia ? mina,i r?ia , and Ka is matrix K with all elements equal to zero
? a (see [4] for details).
except for those corresponding to matrix K
3
Incremental kernel-based stochastic factorization
In the batch version of KBSF, described in Section 2, the matrices P? a and vectors r? a are determined
using all the transitions in the corresponding sets Sa simultaneously. This has two undesirable consequences. First, the construction of the MDP M? requires an amount of memory of O(nmax m), where
nmax = maxa na . Although this is a significant improvement over KBRL?s memory usage, which
is O(n2max ), in more challenging domains even a linear dependence on nmax may be impractical.
Second, with batch KBSF the only way to incorporate new data into the model M? is to recompute
? aD
? a for all actions a for which there are new sample transitions available.
the multiplication P? a = K
Even if we ignore the issue of memory usage, this is clearly inefficient in terms of computation. In
this section we present an incremental version of KBSF that circumvents these important limitations.
Suppose we split the set of sample transitions Sa in two subsets S1 and S2 such that S1 ? S2 = 0/
and S1 ? S2 = Sa . Without loss of generality, suppose that the sample transitions are indexed so that
S1 ? {(sak , rka , s?ak )|k = 1, 2, ..., n1 } and S2 ? {(sak , rka , s?ak )|k = n1 + 1, n1 + 2, ..., n1 + n2 = na }. Let P? S1
and r? S1 be matrix P? a and vector r? a computed by KBSF using only the n1 transitions in S1 (if n1 = 0,
we define P? S1 = 0 ? Rm?m and r? S1 = 0 ? Rm for all a ? A). We want to compute P? S1 ?S2 and r? S1 ?S2
from P? S1 , r? S1 , and S2 , without using the set of sample transition S1 .
We start with the transition matrices P? a . We know that
S
n1
p?i 1j =
? k? ita d?taj =
t=1
n1
n1
k?? (s?ta , s? j )
k? (s?i , sta )k?? (s?ta , s? j )
k? (s?i , sta )
1
=
n1
? n1
? ?m k?? (s?ta , s?l ) .
m
a
a
k? (s?i , sal ) t=1
?l=1
l=1
t=1 ?l=1 k? (s?i , sl ) ?l=1 k?? (s?t , s?l )
1
1 +n2
k? (s?i , sal ), wi 2 = ?nl=n
To simplify the notation, define wi 1 = ?nl=1
k? (s?i , sal ), and cti j =
1 +1
S
k? (s?i ,sta )k?? (s?ta ,s? j )
a
?m
l=1 k?? (s?t ,s?l )
, with t ? {1, 2, ..., n1 + n2 }. Then,
S ?S2
p?i 1j
S
=
1
S1
S2
wi + wi
n1 +n2 t
n1 t
=
c
ci j + ?t=n
?t=1
i
j
+1
1
3
1
S1
S2
wi + wi
S
S
n1 +n2 t
.
p?i 1j wi 1 + ?t=n
c
i
j
+1
1
n1 +n2 t
Now, defining bi 2j = ?t=n
c , we have the simple update rule:
1 +1 i j
1
S
S
S
S ?S
bi 2j + p?i 1j wi 1 .
p?i 1j 2 = S1
S2
wi + wi
S
(3)
We can apply similar reasoning to derive an update rule for the rewards r?ia . We know that
n1
1 n1
1
a a
(
s
?
,
s
)r
=
k
n1
? ? i t t wS1 ? k? (s?i , sta )rta .
k? (s?i , sal ) t=1
?l=1
i t=1
Let hti = k? (s?i , sta )rta , with t ? {1, 2, ..., n1 + n2 }. Then,
1
1
S1 S1
S ?S
n1
n1 +n2
t + n1 +n2 ht =
t .
h
h
w
r
?
+
r?i 1 2 = S1
?
?
?
S
S
S
i i
t=n1 +1 i
t=1 i
t=n1 +1 i
wi + wi 2
wi 1 + wi 2
S
r?i 1 =
n1 +n2
Defining ei 2 = ?t=n
ht , we have the following update rule:
1 +1 i
S
S ?S2
r?i 1
S
S
=
1
S1
S2
wi + wi
S
S
S
ei 2 + r?i 1 wi 1
.
(4)
S
Since bi 2j , ei 2 , and wi 2 can be computed based on S2 only, we can discard the sample transitions in S1
S
after computing P? S1 and r? S1 . To do that, we only have to keep the variables wi 1 . These variables can
a
m
be stored in |A| vectors w ? R , resulting in a modest memory overhead. Note that we can apply
the ideas above recursively, further splitting the sets S1 and S2 in subsets of smaller size. Thus, we
have a fully incremental way of computing KBSF?s MDP which requires almost no extra memory.
Algorithm 1 shows a step-by-step description of how to update M? based on a set of sample transitions. Using this method to update its model, KBSF?s space complexity drops from O(nm) to
O(m2 ). Since the amount of memory used by KBSF is now independent of n, it can process an
arbitrary number of sample transitions.
Algorithm 1 Update KBSF?s MDP
P? a , r? a , wa for all a ? A
Input: a
S
for all a ? A
Output: Updated M? and wa
for a ? A do
a
for t = 1, ..., na do zt ? ?m
l=1 k?? (s?t , s?l )
a
na ? |S |
for i = 1, 2, ..., m do
na
w? ? ?t=1
k? (s?i , sta )
for j = 1, 2, ..., m do
na
b ? ?t=1
k? (s?i , sta )k?? (s?ta , s? j )/zt
1
p?i j ? wa +w? (b + p?i j wai )
i
na
e ? ?t=1
k? (s?i , sta )rta
1
a
r?i ? wa +w
? (e + r?i wi )
i
wai ? wai + w?
Algorithm 2 Incremental KBSF (iKBSF)
s?i Representative states, i = 1, 2, ..., m
tm Interval to update model
Input:
tv Interval to update value function
n Total number of sample transitions
? a)
Output: Approximate value function Q(s,
m?|A|
?
Q ? arbitrary matrix in R
P? a ? 0 ? Rm?m , r? a ? 0 ? Rm , wa ? 0 ? Rm , ?a ? A
for t = 1, 2, ..., n do
? t , a) = ?m
Select a based on Q(s
i=1 ??? (st , s?i )q?ia
Execute S
a in st and observe rt and s?t
Sa ? Sa {(st , rt , s?t )}
if (t mod tm = 0) then
Add new representative states to M? using Sa
Update M? and wa using Algorithm 1 and Sa
Sa ? 0/ for all a ? A
?
if (t mod tv = 0) update Q
Instead of assuming that S1 and S2 are a partition of a fixed dataset Sa , we can consider that S2 was
generated based on the policy learned by KBSF using the transitions in S1 . Thus, Algorithm 1 provides a flexible framework for integrating learning and planning within KBSF. A general description
of the incremental version of KBSF is given in Algorithm 2. iKBSF updates the model M? and the
? at fixed intervals tm and tv , respectively. When tm = tv = n, we recover the batch
value function Q
version of KBSF; when tm = tv = 1, we have an on-line method which stores no sample transitions.
?
Note that Algorithm 2 also allows for the inclusion of new representative states to the model M.
Using Algorithm 1 this is easy to do: given a new representative state s?m+1 , it suffices to set wam+1 =
a
= 0, and p?m+1, j = p? j,m+1 = 0 for j = 1, 2, ..., m + 1 and all a ? A. Then, in the following
0, r?m+1
applications of Eqns (3) and (4), the dynamics of M? will naturally reflect the existence of state s?m+1 .
4
4
Theoretical Results
Our previous experiments with KBSF suggest that, at least empirically, the algorithm?s performance
improves as m ? n [4] . In this section we present theoretical results that confirm this property. The
results below are particularly useful for iKBSF because they provide practical guidance towards
where and when to add new representative states.
Suppose we have a fixed set of sample transitions Sa . We will show that, if we are free to define the
representative states, then we can use KBSF to approximate KBRL?s solution to any desired level of
accuracy. To be more precise, let d? ? maxa,i min j k s?ai ? s? j k, that is, d? is the maximum distance
from a sampled state s?ai to the closest representative state. We will show that, by minimizing d? , we
can make k?v? ? v? k? as small as desired (cf. Eqn (2)).
Let s?a? ? s?ak with k = argmaxi min j k s?ai ? s? j k and s?a? ? s?h where h = argmin j k s?a? ? s? j k, that is, s?a?
is the sampled state in Sa whose distance to the closest representative state is maximal, and s?a? is the
representative state that is closest to s?a? . Using these definitions, we can select the pair (s?a? , s?a? ) that
maximizes k s?a? ? s?a? k: s?? ? s?b? and s?? ? s?b? where b = argmaxa k s?a? ? s?a? k. Obviously, k s?? ? s?? k= d? .
We make the following simple assumptions: (i) s?a? and s?a? are unique for all a ? A, (ii) 0? ? (x)dx ?
L? < ?, (iii) ? (x) ? ? (y) if x < y, (iv) ? A? , ?? > 0, ? B? ? 0 such that A? exp(?x) ? ? (x) ?
?? A? exp(?x) if x ? B? . Assumption (iv) implies that the kernel function ? will eventually decay
exponentially. We start by introducing the following definition:
Definition 1. Given ? ? (0, 1] and s, s? ? S, the ?-radius of k? with respect to s and s? is defined as
?(k? , s, s? , ?) = max{x ? R+ |? (x/?) = ?k? (s, s? )}.
R
The existence of ?(k? , s, s? , ?) is guaranteed by assumptions (ii) and (iii) and the fact that ? is
continuous [1]. To provide some intuition on the meaning of the ?-radius of k? , suppose that ? is
strictly decreasing and let c = ? (k s?s? k /?). Then, there is a s?? ? S such that ? (k s?s?? k /?) = ?c.
The radius of k? in this case is k s ? s?? k. It should be thus obvious that ?(k? , s, s? , ?) ?k s ? s? k.
We can show that ? has the following properties (proved in the supplementary material):
Property 1. If k s ? s? k<k s ? s?? k, then ?(k? , s, s? , ?) ? ?(k? , s, s?? , ?).
Property 2. If ? < ? ? , then ?(k? , s, s? , ?) > ?(k? , s, s? , ? ? ).
Property 3. For ? ? (0, 1) and ? > 0, there is a ? > 0 such that ?(k? , s, s? , ?)? k s?s? k< ? if ? < ? .
We now introduce a notion of dissimilarity between two states s, s? ? S which is induced by a specific
set of sample transitions Sa and the choice of kernel function:
Definition 2. Given ? > 0, the? -dissimilarity between s and s? with respect to ??a is defined as
na
?k=1 |??a (s, sak ) ? ??a (s? , sak )|, if k s ? s? k? ? ,
?
a
?(?? , s, s , ? ) =
0, otherwise.
The parameter ? defines the volume of the ball within which we want to compare states. As we will
see, this parameter links Definitions 1 and 2. Note that ?(??a , s, s? , ? ) ? [0, 2]. It is possible to show
that ? satisfies the following property (see supplementary material):
Property 4. For ? > 0 and ? > 0, there is a ? > 0 such that ?(??a , s, s? , ? ) < ? if k s ? s? k< ? .
Definitions 1 and 2 allow us to enunciate the following result:
Lemma 1. For any ? ? (0, 1] and any t ? m ? 1, let ? a = ?(k?? , s?a? , s?a? , ?/t), let ??a =
a = max ?(? a , s?a , s? , ?). Then,
max ?(??a , s?ai , s? j , ? a ), and let ?max
? i j
i, j
i, j
kPa ? DKa k? ?
?
1
?a +
?a .
1 + ? ? 1 + ? max
(5)
Proof. See supplementary material.
a ? ? a , one might think at first that the right-hand side of Eqn (5) decreases monotonically
Since ?max
?
a
as ? ? 0. This is not necessarily true, though, because ??a ? ?max
as ? ? 0 (see Property 2). We
are finally ready to prove the main result of this section.
5
Proposition 1. For any ? > 0, there are ?1 , ?2 > 0 such that k?v? ? v? k? < ? if d? < ?1 and ?? < ?2 .
Proof. Let r? ? [(r1 )? , (r2 )? , ..., (r|A| )? ]? ? Rn . From Eqn (1) and the definition of r? a , we can write
? a ra
=
P? a r? ? DKa r?
=
(P? a ? DKa )?r
?
P? a ? DKa
k?rk? .
k?ra ? D?ra k? =
P? a r? ? DK
?
?
?
?
(6)
Thus, plugging
Eqn
(6) back into Eqn (2), it is clear that there is a ? > 0 such that k?v? ? v? k? < ?
a
?
if maxa P ? DKa
? < ? and maxi (1 ? max j di j ) < ?. We start by showing that if d? and ?? are
small enough, then maxa
P? a ? DKa
? < ?. From Lemma 1 we know that, for any set of m ? n
representative states, and for any ? ? (0, 1], the following must hold:
max kPa ? DKa k? ? (1 + ?)?1 ?? + ?(1 + ?)?1 ?MAX ,
a
where ?MAX = maxa,i,s ?(k? , s?ai , s, ?) and ?? = maxa ??a = maxa,i, j ?(??a , s?ai , s? j , ? a ), with ? a =
?(k?? , s?a? , s?a? , ?/(n ? 1)). Note that ?MAX is independent of the representative states. Define ? such
that ?/(1 + ?)?MAX < ?. We have to show that, if we define the representative states in such a way
that d? is small enough, and set ?? accordingly, then we can make ?? < (1 ? ?)? ? ??MAX ? ? ? .
From Property 4 we know that there is a ?1 > 0 such that ?? < ? ? if ? a < ?1 for all a ? A. From
Property 1 we know that ? a ? ?(k?? , s?? , s?? , ?/(n ? 1)) for all a ? A. From Property 3 we know that,
for any ? ? > 0, there is a ? ? > 0 such that ?(k?? , s?? , s?? , ?/(n ? 1)) < d? + ? ? if ?? < ? ? . Therefore, if
? It remains to show that there
d? < ?1 , we can take any ? ? < ?1 ? d? to have an upper bound ? ? for ?.
a
is a ? > 0 such that mini max j di j > 1 ? ? if ?? < ? . Recalling that d?iaj = k?? (s?ai , s? j )/?m
k=1 k?? (s?i , s?k ),
a
a
a
a
a
let h = argmax j k?? (s?i , s? j ), and let yi = k?? (s?i , s?h ) and y?i = max j6=h k?? (s?i , s? j ). Then, for any i,
max j d?iaj = yai / yai + ? j6=h k?? (s?ai , s? j ) ? yai /(yai + (m ? 1)y?ai ). From Assump. (i) and Prop. 3 we know
that there is a ?ia > 0 such that yai > (m ? 1)(1 ? ?)y?ai /? if ?? < ?ia . Thus, by making ? = mina,i ?ia ,
we can guarantee that mini max j di j > 1 ? ?. If we take ?2 = min(? , ? ? ), the result follows.
Proposition 1 tells us that, regardless of the specific reinforcement-learning problem at hand, if the
distances between sampled states and the respective nearest representative states are small enough,
then we can make KBSF?s approximation of KBRL?s value function as accurate as desired by setting
?? to a small value. How small d? and ?? should be depends on the particular choice of kernel k? and on
the characteristics of the sets of transitions Sa . Of course, a fixed number m of representative states
imposes a minimum possible value for d? , and if this value is not small enough decreasing ?? may
actually hurt the approximation. Again, the optimal value for ?? in this case is problem-dependent.
Our result supports the use of a local approximation based on representative states spread over
the state space S. This is in line with the quantization strategies used in batch-mode kernel-based
reinforcement learning to define the states s? j [4, 5]. In the case of on-line learning, we have to
adaptively define the representative states s? j as the sample transitions come in. One can think of
several ways of doing so [10]. In the next section we show a simple strategy for adding representative
states which is based on the theoretical results presented in this section.
5
Empirical Results
We now investigate the empirical performance of the incremental version of KBSF. We start with a
simple task in which iKBSF is contrasted with batch KBSF. Next we exploit the scalability of iKBSF
to solve a difficult control task that, to the best of our knowledge, has never been solved before.
We use the ?puddle world? problem as a proof of concept [11]. In this first experiment we show
that iKBSF is able to recover the model that would be computed by its batch counterpart. In order
to do so, we applied Algorithm 2 to the puddle-world task using a random policy to select actions.
Figure 1a shows the result of such an experiment when we vary the parameters tm and tv . Note
that the case in which tm = tv = 8000 corresponds to the batch version of KBSF. As expected, the
performance of KBSF decision policies improves gradually as the algorithm goes through more
sample transitions, and in general the intensity of the improvement is proportional to the amount of
data processed. More important, the performance of the decision policies after all sample transitions
have been processed is essentially the same for all values of tm and tv , which shows that iKBSF
can be used as a tool to circumvent KBSF?s memory demand (which is linear in n). Thus, if one
has a batch of sample transitions that does not fit in the available memory, it is possible to split
the data in chunks of smaller sizes and still get the same value-function approximation that would
6
3
be computed if the entire data set were processed at once. As shown in Figure 1b, there is only a
small computational overhead associated with such a strategy (this results from unnormalizing and
normalizing the elements of P? a and r? a several times through update rules (3) and (4)).
1.5
? = 1000
? = 2000
? = 4000
? = 8000
0.0
?3
?2
0.5
?1
Return
0
Seconds
1.0
1
2
? = 1000
? = 2000
? = 4000
? = 8000
0
2000
4000
6000
Number of sample transitions
8000
0
(a) Performance
2000
4000
6000
Number of sample transitions
8000
(b) Run times
Figure 1: Results on the puddle-world task averaged over 50 runs. iKBSF used 100 representative
states evenly distributed over the state space and tm = tv = ? (see legends). Sample transitions were
collected by a random policy. The agents were tested on two sets of states surrounding the ?puddles?:
a 3 ? 3 grid over [0.1, 0.3] ? [0.3, 0.5] and the four states {0.1, 0.3} ? {0.9, 1.0}.
But iKBSF is more than just a tool for avoiding the memory limitations associated with batch learning. We illustrate this fact with a more challenging RL task. Pole balancing has a long history
as a benchmark problem because it represents a rich class of unstable systems [12, 13, 14]. The
objective in this task is to apply forces to a wheeled cart moving along a limited track in order to
keep one or more poles hinged to the cart from falling over [15]. There are several variations of
the problem with different levels of difficulty; among them, balancing two poles at the same time
is particularly hard [16]. In this paper we raise the bar, and add a third pole to the pole-balancing
task. We performed our simulations using the parameters usually adopted with the double pole task,
except that we added a third pole with the same length and mass as the longer pole [15]. This results
in a problem with an 8-dimensional state space S.
In our experiments with the double-pole task, we used 200 representative states and 106 sample
transitions collected by a random policy [4]. Here we start our experiment with triple pole-balancing
using exactly the same configuration, and then we let KBSF refine its model M? by incorporating
more sample transitions through update rules (3) and (4). Specifically, we used Algorithm 2 with a
? ? at each value0.3-greedy policy, tm = tv = 106 , and n = 107 . Policy iteration was used to compute Q
function update. As for the kernels, we adopted Gaussian functions with widths ? = 100 and ?? = 1
(to improve efficiency, we used a KD-tree to only compute the 50 largest values of k? (s?i , ?) and the
10 largest values of k?? (s?ai , ?)). Representative states were added to the model on-line every time the
agent encountered a sample state s?ai for which k?? (s?ai , s? j ) < 0.01 for all j ? 1, 2, ..., m (this corresponds
to setting the maximum allowed distance d? from a sampled state to the closest representative state).
We compare iKBSF with fitted Q-iteration using an ensemble of 30 trees generated by Ernst et al.?s
extra-trees algorithm [17]. We chose this algorithm because it has shown excellent performance in
both benchmark and real-world reinforcement-learning tasks [17, 18].1 Since this is a batch-mode
learning method, we used its result on the initial set of 106 sample transitions as a baseline for our
empirical evaluation. To build the trees, the number of cut-directions evaluated at each node was
fixed at dim(S) = 8, and the minimum number of elements required to split a node, denoted here
by ?min , was first set to 1000 and then to 100. The algorithm was run for 50 iterations, with the
structure of the trees fixed after the 10th iteration.
As shown in Figure 2a, both fitted Q-iteration and batch KBSF perform poorly in the triple polebalancing task, with average success rates below 55%. This suggests that the amount of data used
1 Another reason for choosing fitted Q-iteration was that some of the most natural competitors of iKBSF
have already been tested on the simpler double pole-balancing task, with disappointing results [19, 4].
7
50000
?
?
?
?
Successful episodes
0.5
0.6
0.7
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Batch KBSF
2e+06 4e+06 6e+06 8e+06
Number of sample transitions
(a) Performance
50
0.4
?
0.3
?
?
1e+07
?
Batch KBSF
iKBSF
TREE?1000
TREE?100
2e+06 4e+06 6e+06 8e+06
Number of sample transitions
(b) Run times
1e+07
?
Number of representative states
1000
2000
3000
4000
iKBSF
TREE?1000
TREE?100
Seconds (log)
200 500
2000
10000
?
0.8
0.9
by these algorithms is insufficient to describe the dynamics of the control task. Of course, we could
give more sample transitions to fitted Q-iteration and batch KBSF. Note however that, since they
are batch-learning methods, there is an inherent limit on the amount of data that these algorithms
can use to construct their approximation. In contrast, the amount of memory required by iKBSF
is independent of the number of sample transitions n. This fact together with the fact that KBSF?s
computational complexity is only linear in n allow our algorithm to process a large amount of data
within a reasonable time. This can be observed in Figure 2b, which shows that iKBSF can build
an approximation using 107 transitions in under 20 minutes. As a reference for comparison, fitted
Q-iteration using ?min = 1000 took an average of 1 hour and 18 minutes to process 10 times less data.
?
?
?
?
?
?
?
?
2e+06
6e+06
1e+07
Number of sample transitions
(c) Size of KBSF?s MDP
Figure 2: Results on the triple pole-balancing task averaged over 50 runs. The values correspond to
the fraction of episodes initiated from the test states in which the 3 poles could be balanced for 3000
steps (one minute of simulated time). The test set was composed of 256 states equally distributed
over the hypercube defined by ?[1.2 m, 0.24 m/s, 18o , 75o /s, 18o , 150o /s, 18o , 75o /s]. Shadowed regions represent 99% confidence intervals.
As shown in Figure 2a, the ability of iKBSF to process a large number of sample transitions allows
our algorithm to achieve a success rate of approximately 80%. This is similar to the performance
of batch KBSF on the double-pole version of the problem [4]. The good performance of iKBSF on
the triple pole-balancing task is especially impressive when we recall that the decision policies were
evaluated on a set of test states representing all possible directions of inclination of the three poles.
In order to achieve the same level of performance with KBSF, approximately 2 Gb of memory would
be necessary, even using sparse kernels, whereas iKBSF used less than 0.03 Gb of memory.
To conclude, observe in Figure 2c how the number of representative states m grows as a function of
the number of sample transitions processed by KBSF. As expected, in the beginning of the learning
process m grows fast, reflecting the fact that some relevant regions of the state space have not been
visited yet. As more and more data come in, the number of representative states starts to stabilize.
6
Conclusion
This paper presented two contributions, one practical and one theoretical. The practical contribution
is iKBSF, the incremental version of KBSF. iKBSF retains all the nice properties of its precursor:
it is simple, fast, and enjoys good theoretical guarantees. However, since its memory complexity
is independent of the number of sample transitions, iKBSF can be applied to datasets of any size,
and it can also be used on-line. To show how iKBSF?s ability to process large amounts of data can
be useful in practice, we used the proposed algorithm to learn how to simultaneously balance three
poles, a difficult control task that had never been solved before.
As for the theoretical contribution, we showed that KBSF can approximate KBRL?s value function
at any level of accuracy by minimizing the distance between sampled states and the closest representative state. This supports the quantization strategies usually adopted in kernel-based RL, and
also offers guidance towards where and when to add new representative states in on-line learning.
Acknowledgments The authors would like to thank Amir massoud Farahmand for helpful discussions regarding this work. Funding for this research was provided by the National Institutes of Health (grant R21
DA019800) and the NSERC Discovery Grant program.
8
References
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9
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nserc:1 corresponds:2 satisfies:1 prop:1 cti:1 kbrl:20 towards:2 price:1 lipschitz:1 replace:1 hard:1 man:1 determined:2 except:2 contrasted:1 specifically:1 lemma:2 called:1 total:1 puddle:4 jong:1 select:3 support:2 barreto:3 incorporate:1 tested:2 avoiding:1 |
3,976 | 46 | 432
Performance Measures for Associative Memories
that Learn and Forget
Anthony /(uh
Department of Electrical Engineering
University of Hawaii at Manoa
Honolulu HI, 96822
ABSTRACT
Recently, many modifications to the McCulloch/Pitts model have been proposed
where both learning and forgetting occur. Given that the network never saturates (ceases
to function effectively due to an overload of information), the learning updates can continue indefinitely. For these networks, we need to introduce performance measmes in addition to the information capacity to evaluate the different networks. We mathematically
define quantities such as the plasticity of a network, the efficacy of an information vector,
and the probability of network saturation. From these quantities we analytically compare
different networks.
1. Introduction
Work has recently been undertaken to quantitatively measure the computational
aspects of network models that exhibit some of the attributes of neural networks. The
McCulloch/Pitts model discussed in [1] was one of the earliest neural network models to be
analyzed. Some computational properties of what we call a Hopfield Associative Memory
Network (HAMN) :similar to the McCulloch/Pitts model was discussed by Hopfield in [2].
The HAMN can be measured quantitatively by defining and evaluating the information
capacity as [2-6] have shown, but this network fails to exhibit more complex computational
capabilities that neural network have due to its simplified structure. The HAMN belongs
to a class of networks which we call static. In static networks the learning and recall procedures are separate. The network first learns a set of data and after learning is complete,
recall occurs. In dynamic networks, as opposed to static networks, updated learning and
associative recall are intermingled and continual. In many applications such as in adaptive
communications systems, image processing, and speech recognition dynamic networks are
needed to adaptively learn the changing information data. This paper formally develops
and analyzes some dynamic models for neural networks. Some existing models [7-10] are
analyzed, new models are developed, and measures are formulated for evaluating the performance of different dynamic networks.
In [2-6]' the asymptotic information capacity of the HAMN is defined and evaluated.
In [4-5]' this capacity is found by first assuming that the information vectors (Ns) to be
stored have components that are chosen randomly and independently of all other components in all IVs. The information capacity then gives the maximum number of Ns that
can be stored in the HAMN such that IVs can be recovered with high probability during
retrieval. At or below capacity, the network with high probability, successfully recovers
the desired IVs. Above capacity, the network quickly degrades and eventually fails to
recover any of the desired IVs. This phenomena is sometimes referred to as the "forgetting
catastrophe" [10]. In this paper we will refer to this phenomena as network saturation.
There are two ways to avoid this phenomena. The first method involves learning a
limited number of IVs such that this number is below capacity. After this leaming takes
place, no more learning is allowed. Once learning has stopped, the network does not
change (defined as static) and therefore lacks many of the interesting computational
@
American Institute of Physics 1988
433
capabilities that adaptive learning and neural network models have . The second method is
to incorporate some type oC forgetting mechanism in the learning structure so that the
inCormation stored in the network can never exceed capacity. This type of network would
be able to adapt to the changing statistics of the IVs and the network would only be able
to recall the most recently learned IVs. This paper focuses on analyzing dynamic networks
that adaptively learn new inCormation and do not exhibit network saturation phenomena
by selectively Corgetting old data. The emphasis is on developing simple models and much
oC the analysis is performed on a dynamic network that uses a modified Hebbian learning
rule.
Section 2 introduces and qualitatively discusses a number of network models that are
classified as dynamic networks. This section also defines some pertinent measures Cor
evaluating dynamic network models. These measures include the plasticity of a network,
the probability oC network saturation, and the efficacy of stored IVs. A network with no
plasticity cannot learn and a network with high plasticity has interconnection weights that
exhibit large changes. The efficacy oC a stored IV as a function oC time is another important parameter as it is used in determining the rate at which a network forgets information.
In section 3, we mathematically analyze a simple dynamic network referred to as the
Attenuated Linear Updated Learning (AL UL) network that uses linear updating and a
modified Hebbian rule. Quantities introduccd in section 3 are analytically dctcrmincd for
the ALUL network. By adjusting the attenuation parameter of the AL UL network, the
Corgetting factor is adjusted. It is shown that the optimal capacity for a large AL UL network in steady state defined by (2.13,3.1) is a factor of e less than the capacity of a
HAMN. This is the tradeoff that must be paid for having dynamic capabilities. We also
conjecture that no other network can perform better than this network when a worst case
criterion is used. Finally, section 4 discusses further directions for this work along with possible applications in adaptive signal processing.
2. Dynamic Associative Memory Networks
The network models discussed in this paper are based on the concept of associative
memory. Associative memories are composed of a collection of interconnected elements
that have data storage capabilities. Like other memory structures, there are two operations that occur in associative memories. In the learning operation (referred to as a write
operation for conventional memories), inCormation is stored in the network structure. In
the recall operation (referred to as a read operation for conventional memories), information is retrieved from the memory structure. Associative memories recall information on
the basis of data content rather than by a specific address. The models that we consider
will have learning and recall operations that are updated in discrete time with the activation state XU) consisting of N cells that take on the values {-l,1}.
2.1. Dynamic Network MeasureS
General associative memory networks are described by two sets of equations. If we
let XU) represent the activation state at time i and W( k) represent the weight matrix 01?
interconnection state at time k then the activation or recall equation is described by
X(j+ 1) = f (XU), W(k)),
i? 0, k? 0, X(O)
=
X
(2.1 )
where X is the data probe vector used for reca.ll. The learning algorithm or int.erconnection equation is described by
W(k+ 1) = g(V(i),O::; i< k, W(O))
where {V( i)} are the information vectors (IV)s to be stored and W(O) is the initial state of
the interconnection matrix. Usually the learning algorithm time scale is much longer than
434
the recall equation time scale so that W in (2.1) can be considered time invariant. Often
(2.1) is viewed as the equation governing short term memory and (2 .2) is the equation
governing long term memory. From the Hebbian hypothesis we note that the data probe
vectors should have an effect on the interconnection matrix W. If a number of data p!'Obe
vectors recall an IV V( a') , the strength of recall of the IV V( i) should be increased by
appropriate modification of W. If another IV is never recalled, it should gradually be forgotten by again adjusting terms of W. Following the analysis in [4,5] we assume that all
components of IVs introduced are independent and identically distributed Bernoulli random
variables with the probability of a 1 or -1 being chosen equal to ~.
Our analysis focuses on learning algorithms. Before describing some dynamic learning
algorithms we present some definitions. A network is defined as dynamic if given sorne
period of time the rate of change of W is never nonzero. In addition we will primarily discuss networks where learning is gradual and updated at discrete times as shown in (2.2).
By gradual, we want networks where each update usually consists of one IV being learned
and/or forgotten. IVs that have been introduced recently should have a high probability of
recovery. The probability of recall for one IV should also be a monotonic decreasing function of time, given that the IV is not repeated. The networks that we consider should also
have a relatively low probability of network saturation.
Quantitatively, we let e(k,l,i} be the event that an IV introduced at time l can be
recovered at time k with a data probe vector which is of Hamming distance i f!'Om the
desired IV. The efficacy of network recovery is then given as p(k,l,i) = Pr(e(k,l,i)). In
the analysis performed we say a a vector V can recover V(I), if V(I) = 6(V) where 6(.)
is a synchronous activation update of all cells in the network. The capacity for dynamic
networks is then given by
O(k,i,l)
=
maxm3-Pr(r(e(k,l,i),05:I<k)= m)
> l-l
O<i<
N
2
(2.3)
where r(X} gives the cardinality of the number of events that occur in the set X. Closely
related to the capacity of a network is network saturation. Saturation occurs when the
network is overloaded with IVs such that few or none of the IVs can be successfully
recovered. When a network at time 0 starts to leal'll IVs, at some time l < i we have that
O(l,i,l? OU,i,l). For k>1 the network saturation probability is defined by S(k,m)
where S describes the probability that the network cannot recover m IVs.
Another important measure in analyzing the performance of dynamic networks is t.he
plasticity of the interconnections of the weight matrix W. Following definitions that are
similar to [10], define
N
2: 2: V AR{ Wi,j(k) - Wi,j(k-l)}
h(k) =
i".
ii-I
N(N-l)
(2.4)
as the incremental synaptic intensity and
N
2: 2:V AR{ Wi,j(k)}
H(k) =
i"..;;= 1
N(N-l)
(2 .5)
as the cumulative synaptic intensity . From these definitions we can define the plasticity of
the network as
P(k)
=
h(k)
H(k)
(2.6)
When network plasticity is zero, the network does not change and no learning takes place.
When plasticity is high, the network interconnections exhibit large changes.
435
When analyzing dynamic networks we are often interested if the network reaches a
steady state. We say a dynamic network reaches steady state if
limH(k) = H
(2.7)
Ie--.oo
where H is a finite nonzero constant. If the IVs have stationary statistics and given that
the learning operations are time invariant, then if a network reaches steady state, we have
that
(2.8)
limP(k) = P
Ie-+oo
where P is a finite constant. It is also easily verified from (2.6) that if the plasticity converges to a nonzero constant in a dynamic network, then given the above conditions on the
IVs and the learning operations the network will eventually reach steady state.
Let us also define the synaptic state at time k for activation state V as
s(k, V) = W(k)V
From the synaptic state, we Can define the SNR of V, which we show
closely related to the efficacy of an IV and the capacity of the network .
SNR(k, V,i) =
(E(s.(k V)))2
.,
VAR(si(k, V))
(2.9)
III
section 3 is
(2.1O)
Another quantity that is important in measuring dynamic networks is the complexity
of implementation. Quantities dealing with network complexity are discussed in [12] and
this paper focuses on networks that are memory less. A network is memoryless if (2.2) can
be expressed in the following form:
W(k+ 1) = 9 #( W(k), V(k))
(2.11)
Networks that are not memoryless have the disadvantage that all Ns need t.o be saved during all learning updates. The complexity of implementation is greatly increased in terms of
space complexity and very likely increased in terms of time complexity.
2.2. Examples of Dynamic Associative Memory Networks
The previous subsection discussed some quantities to measure dynamic networks.
This subsection discusses some examples of dynamic associative memo!,y networks and
qualitatively discusses advantages and disadvantages of different networks . All the networks considered have the memoryless propel?ty.
The first network that we discuss is described by the following difference equation
W(k+ 1)
=
a(k)W(k) + b(k)L(V(k))
(2.12)
with W(O) being the initial value of weights before any learning has taken place . Networks
with these learning rules will be labeled as Linear Updated Learning (LUL) networks and
in addition if O<a(k)<l for k2::0 the network is labeled as an Attenuated Linear Updated
Learning (ALUL) network. We will primarily deal with ALUL where O<a(k)<l and b(k)
do not depend on the position in W. This model is a specialized version of Grossberg's
Passive Decay LTM equation discussed in [11]. If the learning algorithm is of the conelation type then
L(V(J.?))
=
V(k)V(kf-1
(2.13)
This learning scheme has similarities to the marginalist learning schemes introduced in [10].
One of the key parameters in the ALUL network is the value of the attenuation coefficient
a. From simulations and intuition we know that if the attenuation coefficient is to high,
the network will saturate and if the attenuation parameter is to low, the network will
436
forget all but the most recently introduced IVs. Fig. 1 uses Monte Carlo methods to show
a plot of the number of IVs recoverable in a 64 cell network when a = 1, (the HAMN) as a
function of the learning time scale. From this figure we clearly see that network saturation
is exhibited and for the time k ~ 25 no IV are recoverable with high probability. Section 3
further analyzes the AL UL network and derives the value of different measUl'es introduced
in section 2.1.
Another learning scheme called bounded learning (BL) can be described by
L(V(k)) = {
V(k)V(k)T - I
F(W(k)~A
0
F( W(J.:))<A
(2.14)
By setting the attenuation parameter a = 1 and letting
F(W(k)) = ~a;<Wi.i(k)
(2 .15)
I,J
this is identical to the learning with bounds scheme discussed in [10]. Unfortunately there
is a serious drawbacks to this model. If A is too large the network will saturate with high
probability. If A is set such that the probability of network saturation is low then the network has the
characteristic of not learning for almost all
values of
k > k(A) = min I :7 F( W(I))~ A. Th~efore we have that the efficacy of netwOl'k
recovery, p (k,1 ,0) ~ 0 for all J.: ~ I ~ k{A).
In order for the (BL) scheme to be classified as dynamic learning, the attenuation
parameter a must have values between 0 and 1. This learning scheme is just a more complex version of the learning scheme derived from (2.10,2 .11). Let us qualitatively analyze
the learning scheme when a and b are constant. There are two cases to consider. When
A> H, then the network is not affected by the bounds and the network behaves as the
AL UL network. When A <H, then the network accepts IVs until the bound is reached.
When the bound is reached, the network waits until the values of the interconnection
matrix have attenuated to the prescribed levels where learning can continue. If A is judiciously chosen, BL with a < 1 provides a means for a network to avoid saturation. By
holding an IV until H(k )<A, it is not too difficult to show that this learning scheme is
equivalent to an AL UL network with b(k) time varying.
A third learning scheme called refresh learning (RL) can be described by (2 .12) with
b(k)=I, W(O)=O, and
a(k) = 1 -.5(kmod(l))
(2.16)
This learning scheme learns a set of IV and periodically refreshes the weighting matrix so
that all interconnections are O. RL can be classified as dynamic learning, but learning is
not gradual during the periodic l'efresh cycle. Another problem with this learning scheme is
that the efficacy of the IVs depend on where during the period they were learned. IVs
learned late in a period are quickly forgotten where as IVs learned eady in a period have a
longer time in which they are recoverable.
In all the learning schemes introduced, the network has both learning and forgetting
capabilities, A network introduced in [7,8] separates the learning and forgetting tasks by
using the standard HAMN algorithm to learn IV and a random selective forgetting algorithm to unlearn excess information. The algorithm which we call random selective forgetting (RSF) can be described formally as follows.
W(k+ 1)
=
Y(J.:) + L(V(k))
(2.17)
where
n(FU!::(k)))
Y(k) = W(k) -Jl(k)
2..;
i= 1
(V(k,a')V(k,i)T -n(F(W(k)))I)
(2.18)
437
Each of the vectors V( k, i) are obtained by choosing a random vector V in the same
manner IVs are chosen and letting V be the initial state of the HAMN with interconnection
matrix W(k). The recall operation described by (2.1) is repeated until the activation has
settled into a local minimum state . V(k,i) is then assigned this state. /L(k) is the rate at
which the randomly selected local minimum energy states are forgotten, W(k) is given by
(2.15), and n (X) is a nonnegative integer valued function that is a monotonically increasing
function of X.
The analysis of the RSF algorithm is difficult, because the energy manifold that
describes the energy of each activation state and the updates allowable for (2.1) must be
well understood. There is a simple transformation between the weighting matrix and the
energy of an activation state given below,
E(X(k)) = -~~~Wi,jX;?(j)Xj(k)
i
k>O
(2 .19)
j
but aggregately analyzing all local minimum energy activation states is complex. Through
computer simulations and simplified assumptions [7,8] have come up with a qualitative
explanation of the RSF algorithm based on an eigenvalue approach.
3. Analysis of the ALUL Network
Section 2 focused on defining properties and analytical measures for dynamic AMN
along with presenting some examples of some learning algorithms for dynamic AMN. This
section will focus on the analysis of one of the simpler algorithms, the ALUL network.
From (2.12) we have that the time invariant ALUL network can be described by the following interconnection state equation.
W(k+ 1) = aW(k)
+ bL(V(k))
(3.1 )
where a and b are nonnegative real numbers . Many of the measures introduced in section
2 can easily be determined for the AL UL network.
To calculate the incremental synaptic intensity h (k) and the cumulative synaptic
intensity H(k) let the initial condition of the interconnection state W",i(O) be independent
of all other interconnections states and independent of all IVs. If E W",i(O) = 0 and
V AR W.. ,j(O) = "Y then
(3.2)
and
(3.3)
In steady state when a < 1 we have that
p = 2(1~)
(3.4)
From this simple relationship between the attenuation parameter a and the plasticity
measure P, we can directly relate plasticity to other measures such as the capacity of the
network.
We define the steady state capacity as C(i,i)= lim C(k,i,i) for networks where
k--o.o
steady
state
exists.
To analytically determine the capacity first assume that
S(k, V(j)) = S(k-i) is a jointly Gaussian random vector. Further assume that Si(l) for
1~ i< N, 1~ 1< m are all independent and identically distributed. Then for N sufficiently
large, f(a) = a2(k...,.-,l}(1~2), and
438
SNR(k, VU))
=
= (N-l)f(a)
SNR(k-n
I-f{a)
= c{a )logN
?
1
j<k
(3.5)
we have that
p{k,j,O) =
1~
~l _
N
2
V21rC (a )logN
(3.6)
j<k
Given a we first find the largest m= k-j>O where
~~p{k,j,O)= 1 when
lim p(k,j,O)
~
1.
Note that
N-oo
c(a?2. By letting c(a)= 2 the maximum m is given when
2logN
f(a)
=
N
I-f (a)
(3.7)
Solving for m we get that
1
I [
210gN
og (N + 21ogN)(1-a2)
m =
1
2
-.......::.-------~
loga
+1
(3.8)
It is also possible to find the value of a that maximizes m. If we let f = 1 - a2 , then
I [
2logN
og (N+ 2logN)f
1
m ~
(3.9)
f
.
m .IS at a maximum
vaI ue wh en f
2elogN or w h en m ~
NT
. h ?IS correspon ds to
N
2elogN
a ~ 2m -l. Note that this is a factor of e less than the maximum number of Ns allowable
2m
in a static HAMN [4,5], such that one of the Ns is recoverable. By following the analysis
in [5], the independence assumption and the Gaussian assumptions used earlier can be
removed. The arguments involve using results from exchangeability theory and normal
approximation theory.
~
A similar and somewhat more cumbersome analysis can be performed to show that in
steady state the maximum capacity achievable is when a ~ 2m -l and given by
2m
lim C(k,O,f)
N-oo
=
~N
4e og
(3.10)
This again is a factor of e less than the maximum number of Ns allowable in a static
HAMN [4,5]' such that all Ns are recoverable. Fig. 2 shows a Monte Carlo simulation of
the number of Ns recoverable in a 64 cell network versus the learning time scale for a
varying between .5 and .99. We can see that the network reaches approximate steady state
when k:2: 35. The maximum capacity achievable is when a ~ .9 and the capacity is around
5. This is slightly more than the theoretical value predicted by the analysis just shown
when we compare to Fig. 1. For smaller simulations conducted with larger networks the
simulated capacity was closer to the predicted value. From the simulations and the
analysis we observe that when a is too small Ns are forgotten at too high a rate and when
439
a is too high network saturation occurs.
Using the same arguments, it is possible to analyze the capacity of the network and
efficacy of rvs when k is small. Assuming zero initial conditions and a
~ 2m-l we can
2m
The learning behavior can be
summarize the learning behavior of the AL UL network.
divided into three phases. In the first phase for k<
N
all Ns are remembered and
- 4elogN
the characteristics of the network are similar to the HAMN below saturation. In the
second phase some rvs are forgotten as the rate of forgetting becomes nonzero. During this
phase the maximum capacity is reached as shown in fig . 2. At this capacity the network
cannot dynamically recall all IVs so the network starts to forget more information then it
receives. This continues until steady state is reached where the learning and forgetting
rates are equal. If initial conditions are nonzero the network starts in phase 1 or the beginning of phase 2 if H( k) is below the value corresponding to the maximum capacity and at
the end of phase 2 for larger H( k).
The calculation of the network saturation probabilities S( k, m) is trivial for large networks when the capacity curves have been found. When m~ C(k,O,E) then S(k,m) ~ 0
otherwise S(k ,m) ~ 1.
Before leaving this section let us briefly examine AL UL networks where a (k) and
b(k) are time varying. An example of a time varying network is the marginalist learning
scheme
introduced in [10]. The network is defined by fixing the value of the
= D(N) for all k. This value is fixed by setting a= 1 and varying b. Since
the VARSi(k,V(k-l)) is a monotonic increasing function of k, b(k) must also be a monotonic increasing function of k. It is not too difficult to show that when k is large, the marginalist learning scheme is equivalent to the steady state AL UL defined by (3.1). The argument is based on noting that the steady state SNR depends not on the update time, but
on the difference between the update time and when the rv was stored as is the case with
the marginalist learning scheme. The optimal value of D( N) giving the highest capacity is
when D(N) = 4elogN and
SNR(k,k-l,i)
b(k+ 1) =
where m
=
2m b(k)
2m-l
(3.11)
N
4elogN'
If performance is defined by a worst case criterion with the criterion being
J(I,N) =
min(C(k,O,E),k~/)
(3.12)
then we conjecture that for I large, no AL UL as defined in (2.12,2.13) can have larger
J(I,N) than the optimal ALUL defined by (3.1). If we consider average capacity, we note
that the RL network has an average capacity of
N
810gN
which is larger than the optimal
AL UL network defined in (3.1). However, for most envisioned applications a worst case
criterion is a more accurate measure of performance than a criterion based on average
capacity.
4. Summary
This paper has introduced a number of simple dynamic neural network models and
defined several measures to evaluate the performance of these models. All parameters for
the steady state AL UL network described by (3.1) were evaluated and the attenuation
parameter a giving the largest capacity was found. This capacity was found to be a factor
of e less than the static HANIN capacity. Furthermore we conjectured that if we consider
a worst case performance criteria that no AL UL network could perform better than the
440
optimal ALUL network defined by (3.1). Finally, a number of other dynamic models
including BL, RL, and marginalist learning were stated to be equivalent to AL UL networks
under certain conditions.
The network models that were considered in this paper all have binary vector valued
activation states and may be to simplistic to be considered in many signal processing application. By generalizing the analysis to more complicated models with analog vector valued
activation states and continuous time updating it may be possible to use these generalized
models in speech and image processing. A specific example would be a controller for a
moving robot. The generalized network models would learn the input data by adaptively
changing the interconnections of the network. Old data would be forgotten and data that
was repeatedly being recalled would be reinforced. These network models could also be
used when the input data statistics are nonstationary.
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R. J. McEliece, E. C. Posner, E. R. Rodemich and S. S. Venkatesh, "The Capacity of
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R. J. Sasiela, "Forgetting as a way to Improve Neural-Net Behavior" , AIP Conference Proceedings 151, 386-392, 1986.
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J. D. Keeler, "Basins of Attraction of Neural Network Models",
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S. Grossberg, "Nonlinear Neural Networks: Principles, Mechanisms, and Architectures ", Neural Networks in press.
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S. S. Venkatesh and D. Psaltis, "Information Storage and Retrieval in Two Associative Nets ", California Institute of Technology Pasadena, Dept. of Elect. Eng., preprint, 1986.
441
"HAMN Capacity"
10
N=64,
1024 trials
8
>
0
-a- Average # of IV
6
::ea:
co
C)
CIS
4
"-
co
>
<
2
0
0
10
20
Update Time
30
40
Fig. 1
"ALUL Capacity"
10
......
a=.5
a=.7
-II- a=.90
a=.95
a=.99
-Go
N=64, 1024 trials
~0
8
en
6
...en
~
....
0
::ea:
co
4
C)
CIS
"-
co
>
<
2
0
0
10
20
Update Time
Fig. 2
30
40
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3,977 | 460 | MODELS WANTED: MUST FIT DIMENSIONS
OF SLEEP AND DREAMING*
J. Allan Hohson, Adam N. Mamelak t and Jeffrey P. Sutton t
Laboratory of Neurophysiology and Department of Psychiatry
Harvard Medical School
74 Fenwood Road, Boston, MA 02115
Abstract
During waking and sleep, the brain and mind undergo a tightly linked and
precisely specified set of changes in state. At the level of neurons, this
process has been modeled by variations of Volterra-Lotka equations for
cyclic fluctuations of brainstem cell populations. However, neural network
models based upon rapidly developing knowledge ofthe specific population
connectivities and their differential responses to drugs have not yet been
developed. Furthermore, only the most preliminary attempts have been
made to model across states. Some of our own attempts to link rapid eye
movement (REM) sleep neurophysiology and dream cognition using neural
network approaches are summarized in this paper.
1
INTRODUCTION
New models are needed to test the closely linked neurophysiological and cognitive
theories that are emerging from recent scientific studies of sleep and dreaming. This
section describes four separate but related levels of analysis at which modeling may
?Based, in part, upon an invited address by J.A.H. at NIPS, Denver, Dec. 2 1991 and,
in part, upon a review paper by J.P.S., A.N.M. and J.A.H. published in the P.ychiatric
Annal?.
t Currently in the Department of Neurosurgery, University of California, San Francisco,
CA 94143
: Also in the Center for Biological Information Processing, Whitaker College, E25-201,
Massachusetts Institute of Technology, Cambridge, MA 02139
3
4
Hobson, Mamelak, and Sutton
be applied and outlines some of the desirable features of such models in terms of the
burgeoning data of sleep and dream science. In the subsequent sections, we review
our own preliminary efforts to develop models at some of the levels discussed.
1.1
THE INDIVIDUAL NEURON
Existing models or "neuromines" faithfully represent membrane properties but ignore the dynamic biochemical changes that change neural excitability over the long
term. This is particularly important in the modeling of state control where the
crucial neurons appear to act more like hormone pumps than like simple electrical
transducers. Put succinctly, we need models that consider the biochemical or "wet"
aspects of nerve cells, as well as the "dry" or electrical aspects (cf. McKenna et al.,
in press).
1.2
NEURAL POPULATION INTERACTIONS
To mimic the changes in excitability of the modulatory neurons which control sleep
and dreaming, new models are needed which incorporate both the engineering principles of oscillators and the biological principles of time-keeping. The latter principle is especially relevant in determining the dramatica.lly variable long period
time-constants that are observed within and across species. For example, we need
to equip population models borrowed from field biology (McCarley and Hobson,
1975) with specialized properties of "wet" neurons as mentioned in section 1.1.
1.3
COGNITIVE CONSEQUENCES OF MODULATION OF
NEURAL NETWORKS
To understand the state-dependent changes in cognition, such as those that distinguish waking and dreaming, a potentially fruitful approach is to mimic the known
effects of neuromod ulation and examine the information processing properties of
neural networks. For example, if the input-output fidelity of networks can be altered by changing their mode (see Sutton et al., this volume), we might be better
able to understand the changes in both instantaneous associative properties and
long term plasticity alterations that occur in sleep and dreaming. We might thus
trap the brain-mind into revealing its rules for making moment-to-moment crosscorrelations of its data and for changing the content and status of its storage in
memory.
1.4
STATE-DEPENDENT CHANGES IN COGNITION
At the highest level of analysis, psychological data, even that obtained from the
introspection of waking and dreaming subjects, need to be more creatively reduced
with a view to modeling the dramatic alterations that occur with changes in brain
state. As an example, consider the instability of orientation of dreaming, where
times, places, persons and actions change without notice. Short of mastering the
thorny problem of generating narrative text from a data base, and thus synthesizing an artificial dream, we need to formulate rules and measures of categorizing
constancy and transformations (Sutton and Hobson, 1991). Such an approach is a
Models Wanted: Must Fit Dimensions of Sleep and Dreaming
means of further refining the algorithms of cognition itself, an effort which is now
limited to simple activation models that cannot change mode.
An important characteristic of the set of new models that are proposed is that each
level informs, and is informed by, the other levels. This nested, interlocking feature
is represented in figure 1. It should be noted that any erroneous assumptions made
at level 1 will have effects at levels 2 and 3 and these will, in turn, impede our
capacity to integrate levels 3 and 4. Level 4 models can and should thus proceed
with a degree ofindependence from levels 1, 2 and 3. Proceeding from level 1 upward
is the "bottom-up" approach, while proceeding from level 4 downward is the "topdown" approach. We like to think it might be possible to take both approaches in
our work while according equal respect to each.
LEVEL
SCHEMA
IV COGNITIVE
STATES
(eg. dream plot
sequences)
A-.B
III MODULATION
OF NETWORKS
~~
(eg. hippocampus,
cortex)
Il NEURAL
POPULATIONS
(eg. pontine
brainstem)
I SINGLE
NEURONS
(eg. NE, 5HT,
ACh neurons)
FEATURES
<C--+D
E--+F
,~
~ (~2~
t?(7
~
variable associative
and learning states
modulation of
1-0 processing
variable timeconstant oscillator
wet hormonal
aspects
Figure 1: Four levels at which modeling innovations are needed to provide more
realistic simulations of brain-mind states such as waking and dreaming. See text
for discussion.
5
6
Hobson, Marnelak, and Sutton
2
STATES OF WAKING AND SLEEPING
The states of waking and sleeping, including REM and non-REM (NREM) sleep,
have characteristic behavioral, neuronal, polygraphic and psychological features
that span all four levels. These properties are summarized in figures 2 and 3.
Changes occurring within and between different levels are affected by the sleepwake or circadian cycle and by the relative shifts in brain chemistry.
A
WAKE
E~I~--------
NREM SLEEP REM SLEEP
-------------1----------
EEGI:_==:::==::::;::: 1~~':':.fJd,~ 1===:::::::::
EOG ~ ~1--.."l'-'o...J
Sensation and
Vivid,
Perception Externally Generated
Dull or Absent
Vivid,
IntemoUy Generated
TlJougllf
Logical
Progressive
Logical
Perseverotive
Illogical
Bizarre
Movement
Contiooous
Voluntary
Episodic
Involuntary
Commanded
tM Inhibited
c
B
D
_
o
_
Ml
o
,
,
t
Time (hours)
--ll-
__ ...,(1
_
0_
..,[)
_
- - -- .,- - -_
Time (lJours)
Figure 2: (a) States of waking and NREM and REM sleeping in humans. Characteristic behavioral, polygraphic and psychological features are shown for each state.
(b) Ultradian sleep cycle of NREM and REM sleep shown in detailed sleep-stage
graphs of 3 subjects. (c) REM sleep periodograms of 15 subjects. From Hobson
and Steriade (1986), with permission.
Models Wanted: Must Fit Dimensions of Sleep and Dreaming
2.1
CIRCADIAN RHYTHMS
The circadian cycle has been studied mathematically using oscillator and other
non-linear dynamical models to capture features of sleep-wake rhythms (Moore-Ede
and Czeisler, 1984; figure 2). Shorter (infradian) and longer (ultradian) rhythms,
relative to the circadian rhythm, have also been examined. In general, oscillators
are used to couple neural, endocrine and other pathways important in controlling
a variety of functions, such as periods of rest and activity, energy conservation and
thermoregulation. The oscillators can be sensitive to external cues or zeitgebers,
such as light and daily routines, and there is a stong linkage between the circadian
clock and the NREM-REM sleep oscillator.
2.2
RECIPROCAL INTERACTION MODEL
In the 1970s, a brainstem oscillator became identified that was central to regulating
sleeping and waking. Discrete cell populations in the pons that were most active
during waking, less active in NREM sleep and silent during REM sleep were found
to contain the monoamines norepinephrine (NE) and serotonin (5HT). Among the
many cell populations that became active during REM sleep, but were generally
quiescent otherwise, were cells associated with acetylcholine (ACh) release.
BlillJ
A
C
t-
15
o
4
,
\
3
2
I
20
40
10
10
100
o
'vi
o
Figure 3: (a) Reciprocal interaction model of REM sleep generation showing the
structural interaction between cholinergic and monoaminergic cell populations.
Plus sign implies excitatory influences; minus sign implies inhibitory influences.
(b) Model output of the cholinergic unit derived from Lotka-Volterra equations.
(c) Histogram of the discharge rate from a cholinergic related pontine cell recorded
over 12 normalized sleep-wake cycles. Model cholinergic (solid line) and monoaminergic (dotted line) outputs. (d) Noradrenergic discharge rates before (S), during
(D) and following (W) a REM sleep episode. From Hobson and Steriade (1986),
with permission.
7
8
Hobson, Mamelak, and Sutton
By making a variety of simplifying assumptions, McCarley and Hobson (1975)
were able to structurally and mathematically model the oscillations between these
monoaminergic and cholinergic cell populations (figure 3). This level 2 model
consists of two compartments, one being monoaminergic-inhibitory and the other
cholinergic-excitatory. It is based pupon the assumptions offield biology (VoIterraLotka) and of dry neuromines (level 3). The excitation (inhibition) originating from
each compartment influences the other and also feeds back on itself. Numerous predictions generated by the model have been verified experimentally (Hobson and
Steriade, 1986).
Because the neural population model shown in figure 3 uses the limited passive
membrane type of neuromine discussed in the introduction, the resulting oscillator
has a time-constant in the millisecond range, not even close to the real range of minutes to hours that characterize the sleep-dream cycle (figure 2). As such, the model
is clearly incapable of realistically representing the long-term dynamic properties
that characterize interacting neuromodulatory populations. To surmount this limitation, two modifications are possible: one is to remodel the individual neuromines
equipping them with mathematics describing up and down regulation of receptors
and intracellular biochemistry that results in long-term changes in synaptic efficacy
(c/. McKenna et al., in press); another is to model the longer time constants of
the sleep cycle in terms of protein transport times between the two populations in
brainstems of realistically varying width (c/. Hobson and Steriade, 1986).
3
NEUROCOGNITIVE ASPECTS OF WAKING,
SLEEPING AND DREAMING
Since the discovery that REM sleep is correlated with dreaming, significant advances have been made in understanding both the neural and cognitive processes
occurring in different states of the sleep-wake cycle. During waking, wherein the
brain is in a state of relative aminergic dominance, thought content and cognition
display consistency and continuity. NREM sleep mentation is typically characterized by ruminative thoughts void of perceptual vividness or emotional tone. Within
this state, the aminergic and cholinergic systems are more evenly balanced than in
either the wake or REM sleep states. As previously noted, REM sleep is a state
associated with relative cholinergic activation. Its mental status manifestations include graphic, emotionally charged and formally bizarre images encompassing visual
hallucinations and delusions.
3.1
ACTIVATION.SYNTHESIS MODEL
The activation-synthesis hypothesis (Hobson and McCarley, 1977) was the first
account of dream mentation based on the neurophysiological state of REM sleep.
It considered factors present at levels 3 and 4, according to the scheme in section 1,
and attempted to bridge these two levels. In the model, cholinergic activation and
reciprocal monoaminergic disinhibition of neural networks in REM sleep generated
the source of dream formation. However, the details of how neural networks might
actually synthesize information in the REM sleep state was not specified.
Models Wanted: Must Fit Dimensions of Sleep and Dreaming
3.2
NEURAL NETWORK MODELS
Several neural network models have subsequently been proposed that also attempt
to bridge levels 3 and 4 (for example, Crick and Mitchison, 1983). Recently, Mamelak and Hobson (1989) have suggested a neurocognitive model of dream bizarreness
that extends the activation-synthesis hypothesis. In the model, the monoaminergic withdrawal in sleep relative to waking leads to a decrease in the signal-to-noise
ratio in neural networks (figure 4). When this is coupled with phasic cholinergic excitation of the cortex, via brainstem ponto-geniculo-occipital (PGO) cell firing (figure 5), cognitive information becomes altered and discontinuous. A central
premise of the model is that the monoamines and acetylcholine function as neuromodulators, which modify ongoing activity in networks, without actually supplying
afferent input information.
Implementation of the Mamelak and Hobson model as a temporal sequencing network is described by Sutton et al. in this volume. Computer simulations demonstrate how changes in modulation similar to some monoaminergic and cholinergic
effects can completely alter the way information is collectively sequenced within the
same network. This occurs even in the absence of plastic changes in the weights
connecting the artificial neurons. Incorporating plasticity, which generally involves
neuromodulators such as the monoamines, is a logical next step. This would build
important level 1 features into a level 3-4 model and potentially provide useful
insight into some state-dependent learning operations.
" ?? 10 . ..
? ?r 11111 1111111111 11111 111111' II 111111111111
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Figure 4: (a) Monoaminergic innervation of the brain is widespread. (b) Plot of
the neuron firing probability as a function of the relative membrane potential for
various values of monoaminergic modulation (parameterized by Q). Higher (lower)
modulation is correlated with smaller (larger) Q values. (c) Neuron firing when subjected to supra- and sub-threshold inputs of +10 mvand -10 mv, respectively, for
Q = 2 and Q = 10. (d) For a given input, the repertoire of network outputs generally
increases as Q increases. From Mamelak and Hobson (1989), with permission.
9
10
Hobson, Mamelak, and Sutton
A
B
i
~
i i ' iI'
Unil'
r g)
LGlk~
LGBi--' ~
Figure 5: (a) Cholinergic input from the bramstem to the thalamus and cortex is
widespread. (b) Unit recordings from PGO burst cells in the pons are correlated
with PGO waves recorded in the lateral geniculate bodies (LGB) of the thalamus.
4
CONCLUSION
After discussing four levels at which new models are needed, we have outlined some
preliminary efforts at modeling states of waking and sleeping. We suggest that this
area of research is ripe for the development of integrative models of brain and mind.
Acknowledgements
Supported by NIH grant MH 13,923, the HMS/MMHC Research & Education Fund,
the Livingston, Dupont-Warren and McDonnell-Pew Foundations, DARPA under
ONR contract NOOOl4-85-K-0124, the Sloan Foundation and Whitaker College.
References
Crick F, Mitchison G (1983) The function of dream sleep. Nature 304 111-114.
Hobson JA, McCarley RW (1977) The brain as a dream-state generator: An
activation-synthesis hypothesis of the dream process. Am J P.ych 134 1335-1368.
Hobson JA, Steriade M (1986) Neuronal basis of behavioral state control. In:
Mountcastle VB (ed) Handbook of Phy.iology - The NeMJou. Syltem, Vol IV.
Bethesda: Am Physiol Soc, 701-823.
Mamelak AN, Hobson JA (1989) Dream bizarrenes as the cognitive correlate of
altered neuronal behavior in REM sleep. J Cog Neuro.ci 1(3) 201-22.
McCarley RW, Hobson JA (1975) Neuronal excitability over the sleep cycle: A
structural and mathematical model. Science 189 58-60.
McKenna T, Davis J, Zornetzer (eds) In press. Single Neuron Computation. San
Diego, Academic.
Moore-Ede Me, Czeisler CA (eds) (1984) Mathematical Model. of the Circadian
Sleep- Wake Cycle. New York: Raven.
Sutton JP, Hobson (1991) Graph theoretical representation of dream content and
discontinuity. Sleep Re.earch 20 164.
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3,978 | 4,600 | Compressive neural representation of sparse,
high-dimensional probabilities
xaq pitkow
Department of Brain and Cognitive Sciences
University of Rochester
Rochester, NY 14607
[email protected]
Abstract
This paper shows how sparse, high-dimensional probability distributions could
be represented by neurons with exponential compression. The representation is a
novel application of compressive sensing to sparse probability distributions rather
than to the usual sparse signals. The compressive measurements correspond to
expected values of nonlinear functions of the probabilistically distributed variables. When these expected values are estimated by sampling, the quality of the
compressed representation is limited only by the quality of sampling. Since the
compression preserves the geometric structure of the space of sparse probability
distributions, probabilistic computation can be performed in the compressed domain. Interestingly, functions satisfying the requirements of compressive sensing
can be implemented as simple perceptrons. If we use perceptrons as a simple
model of feedforward computation by neurons, these results show that the mean
activity of a relatively small number of neurons can accurately represent a highdimensional joint distribution implicitly, even without accounting for any noise
correlations. This comprises a novel hypothesis for how neurons could encode
probabilities in the brain.
1
Introduction
Behavioral evidence shows that animal behaviors are often influenced not only by the content of
sensory information but also by its uncertainty. Different theories have been proposed about how
neuronal populations could represent this probabilistic information [1, 2]. Here we propose a new
theory of how neurons could represent probability distributions, based on the burgeoning field of
?compressive sensing.?
An arbitrary probability distribution over multiple variables has a parameter count that is exponential
in the number of variables. Representing these probabilities can therefore be prohibitively costly.
One common approach is to use graphical models to parameterize the distribution in terms of a
smaller number of interactions. Here I consider an alternative approach. In many cases of interest,
only a few unknown states have high probabilities while the rest have neglible ones; such a distribution is called ?sparse?. I will show that sufficiently sparse distributions can be described by a number
of parameters that is merely linear in the number of variables.
Until recently, it was generally thought that encoding of sparse signals required dense sampling at a
rate greater than or equal to signal bandwidth. However, recent findings prove that it is possible to
fully characterize a signal at a rate limited not by its bandwidth but by its information content [3, 4,
5, 6] which can be much smaller. Here I apply such compression to sparse probability distributions
over binary variables, which are, after all, just signals with some particular properties.
1
In most applications of compressive sensing, the ultimate goal is to reconstruct the original signal
efficiently. Here, we do not wish to reconstruct the signal at all. Instead, we use the guarantees that
the signal could be reconstructed to ensure that the signal is accurately represented by its compressed
version. Below, when we do reconstruct it is only to show that our method actually works in practice.
We don?t expect that the brain needs to explicitly reconstruct a probability distribution in some
canonical mathematical representation in order to gain the advantages of probabilistic reasoning.
Traditional compressive sensing considers signals that lives in an N -dimensional space but have
only S nonzero coordinates in some basis. We say that such a signal is S-sparse. If we were told
the location of the nonzero entries, then we would need only S measurements to characterize their
coefficients and thus the entire signal. But even if we don?t know where those entries are, it still
takes little more than S linear measurements to perfectly reconstruct the signal. Furthermore, those
measurements can be fixed in advance without any knowledge of the structure of the signal. Under
certain conditions, these excellent properties can be guaranteed [3, 4, 5].
The basic mathematical setup of compressive sensing is as follows. Assume that an N -dimensional
signal s has S nonzero coefficients. We make M linear measurements y of this signal by applying
the M ? N matrix A:
y = As
(1)
We would then like to recover the original signal s from these measurements. Under conditions on
the measurement matrix A described below, the original can be found perfectly by computing the
vector with minimal `1 norm that reproduces the measurements,
? = argmin ks0 k`1 such that As0 = y = As
s
(2)
s0
The `1 norm is usually used instead of `0 because (2) can be solved far more efficiently [3, 4, 5, 7].
Compressive sensing is generally robust to two deviations from this ideal setup. First, target signals
may not be strictly S-sparse. However, they may be ?compressible? in the sense that they are well
approximated by an S-sparse signal. Signals whose rank-ordered coefficients fall off at least as fast
as rank?1 satisfy this property [4]. Second, measurements may be corrupted by noise with bounded
? is bounded by the
amplitude . Under these conditions, the error of the `1 -reconstructed signal s
error of the best S-sparse approximation sS plus a term proportional to the measurement noise:
?
k?
s ? sk`2 ? C0 ksS ? sk`2 / S + C1
(3)
for some constants C0 and C1 [8].
Several conditions on A have been used in compressive sensing to guarantee good performance
[4, 6, 9, 10, 11]. Modulo various nuances, they all essentially ensure that most or all relevant sparse
signals lie sufficiently far from the null space of A: It would be impossible to recover signals in the
null space since their measurements are all zero and cannot therefore be distinguished. The most
commonly used condition is the Restricted Isometry Property (RIP), which says that A preserves `2
norms of all S-sparse vectors within a factor of 1 ? ?S that depends on the sparsity,
(1 ? ?S )ksk`2 ? kAsk`2 ? (1 + ?S )ksk`2
(4)
If A satisfies the RIP with small enough ?S , then `1 recovery is guaranteed to succeed. For random
matrices whose elements are independent and identically distributed Gaussian or Bernoulli variates,
the RIP holds as long as the number of measurements M satisfies
M ? CS log N/S
(5)
for some constant C that depends on ?S [8]. No other recovery method, however intractable, can
perform substantially better than this [8].
2
Compressing sparse probability distributions
Compressive sensing allows us to use far fewer resources to accurately represent high-dimensional
objects if they are sufficiently sparse. Even if we don?t ultimately intend to reconstruct the signal, the
reconstruction theorem described above (3) ensures that we have implicitly represented all the relevant information. This compression proves to be extremely useful when representing multivariate
joint probability distributions, whose size is exponentially large even for the simplest binary states.
2
Consider the signal to be a probability distribution over an n-dimensional binary vector x ?
{?1, +1}n , which I will write sometimes as a function p(x) and sometimes as a vector p indexed
by the binary state x. I assume p is sparse in the canonical basis of delta-functions on each state,
?x,x0 . The dimensionality of this signal is N = 2n , which for even modest n can be so large it
cannot be represented explicitly.
The measurement matrix A for probability vectors has size M ? 2n . Each row corresponds to a
different measurement, indexed by i. Each column corresponds to a different binary state x. This
column index x ranges over all possible binary vectors of length n, in some conventional sequence.
For example, if n = 3 then the column index would take the 8 values
x ? {??? ; ??+ ; ?+? ; ?++ ; +?? ; +?+ ; ++? ; +++}
Each element of the measurement matrix, Ai (x), can be viewed as a function applied to the binary
state. When this matrix operates on a probability distribution p(x), the result y is a vector of M
expectation values of those functions, with elements
X
yi = Ai p =
Ai (x)p(x) = hAi (x)ip(x)
(6)
x
For example, if Ai (x) = xi then yi = hxi ip(x) measures the mean of xi drawn from p(x).
For suitable measurement matrices A, we are guaranteed accurate reconstruction of S-sparse probability distributions as long as the number of measurements is
M ? O(S log N/S) = O(Sn ? S log S)
(7)
n
The exponential size of the probability vector, N = 2 , is cancelled by the logarithm. For distributions with a fixed sparseness S, the required number of measurements per variable, M/n, is then
independent of the number of variables.1
In many cases of interest it is impractical to calculate these expectation values directly: Recall that
the probabilities may be too expensive to represent explicitly in the first place. One remedy is to
draw T samples xt from the distribution p(x), and use a sum over these samples to approximate the
expectation values,
1X
yi ?
Ai (xt )
xt ? p(x)
(8)
T t
The probability p?(x) estimated from T samples has errors with variance p(x)(1 ? p(x))/T , which
is bounded by 1/4T . This allows us to use the performance limits from robust compressive sensing,
which according to (3) creates an error in the reconstructed probabilities that is bounded by
C1
(9)
k?
p ? pk`2 ? C0 kpS ? pk`2 + ?
T
where pS is a vector with the top S probabilities preserved and the rest set to zero. Strictly speaking,
(3) applies to bounded errors, whereas here we have a bounded variance but possibly large errors.
To ensure accurate reconstruction, we can choose the constant C1 large enough that errors larger
than some threshold (say, 10 standard deviations) have a negligible probability.
2.1
Measurements by random perceptrons
In compressive sensing it is common to use a matrix with independent Bernoulli-distributed random
values, Ai (x) ? B( 12 ), which guarantees A satisfies the RIP [12]. Each row of this matrix represents
all possible outputs of an arbitrarily complicated Boolean function of the n binary variables x.
Biological neural networks would have great difficulty computing such arbitrary functions in a simple manner. However, neurons can easily compute a large class of simpler boolean functions, the
perceptrons. These are simple threshold functions of a weighted average of the input
X
Ai (x) = sgn
Wij xj ? ?j
(10)
j
1
Depending on the problem, the number of significant nonzero entries S may grow with the number of
variables. This growth may be fast (e.g. the number of possible patterns grows as en ) or slow (e.g. the number
of possible translations of a given pattern grows only as n).
3
Measurement i
where W is an M ? n matrix. Here I take W to have elements drawn randomly from a standard
normal distribution, Wij ? N (0, 1), and call the resultant functions ?random perceptrons?. An
example measurement matrix for random perceptrons is shown in Figure 1. These functions are
readily implemented by individual neurons, where xj is the instantaneous activity of neuron j,
Wij is the synaptic weight between neurons i and j, and the sgn function approximates a spiking
threshold at ?.
State vector x
Figure 1: Example measurement matrix Ai (x) for M = 100 random perceptrons applied to all 29 possible
binary vectors of length n = 9.
The step nonlinearity sgn is not essential, but some type of nonlinearity is: Using a purely linear
function of the states, A = W x, would result in measurements y = Ap = W hxi. This provides
at most n linearly independent measurements of p(x), even when M > n. In most cases this is
not enough to adequately capture the full distribution. Nonlinear Ai (x) allow a greater number of
linearly independent measurements of p(x). Although the dimensionality of W is merely M ? n,
2
which is much smaller than the 2n -dimensional space of probabilities, (10) can generate O(2n )
distinct perceptrons [13]. By including an appropriate threshold, a perceptron can assign any individual state x a positive response and assign a negative response to every other state. This shows
that random perceptrons generate the canonical basis and can thus span the space of possible p(x).
In what follows, I assume that ? = 0 for simplicity.
In the Appendix I prove that random perceptrons with zero threshold satisfy the requirements for
compressive sensing in the limit of large n. Present research is directed toward deriving the condition
number of these measurement matrices for finite n, in order to provide rigorous bounds on the
number of measurements required in practice. Below I present empirical evidence that even a small
number of random perceptrons largely preserves the information about sparse distributions.
3
3.1
Experiments
Fidelity of compressed sparse distributions
To test random perceptrons in compressive sensing of probabilities, I generated sparse distributions
using small Boltzmann machines [14], and compressed them using random perceptrons driven by
samples from the Boltzmann machine. Performance was then judged by comparing `1 reconstructions to the true distributions, which are exactly calculable for modest n.
In a Boltzmann Machine, binary states x occur with probabilities given by the Boltzmann distribution with energy function E(x),
p(x) ? e?E(x)
E(x) = ?b>x ? x>Jx
(11)
determined by biases b and pairwise couplings J. Sampling from this distribution can be accomplished by running Glauber dynamics [15], at each time step turning a unit on with probability
p(xi = +1|x\i ) = 1/(1 + e??E ), where ?E = E(xi = +1, x\i ) ? E(xi = ?1, x\i ). Here x\i
is the vector of all components of x except the ith.
For simulations I distinguished between two types of units, hidden and visible, x = (h, v). On
each trial I first generated a sample of all units according to (11). I then fixed only the visible
units and allowed the hidden units to fluctuate according to the conditional probability p(h|v) to be
represented. This probability is given again by the Boltzmann distribution, now with energy function
E(h|v) = ?(bh ? Jhv v)> h ? h>Jhh h
4
(12)
All bias terms b were set to zero, and all pairwise couplings J were random draws from a zeromean normal distribution, Jij ? N (0, 31 ). Experiments used n hidden and n visible units, with
n ? {8, 10, 12}. This distribution of couplings produced sparse posterior distributions whose rankordered probabilities fell faster than rank?1 and were thus compressible [4].
The compression was accomplished by passing the hidden unit activities h through random perceptrons a with weights W , according to a = sgn (W h). These perceptron activities fluctuate
along with their inputs. The mean activity of these perceptron units compressively senses the probability distribution according to (8). This process of sampling and then compressing a Boltzmann
distribution can be implemented by the simple neural network shown in Figure 2.
Perceptrons a
feedforward
W
recurrent
Jhh
feedforward
Jvh
Inputs v
neurons
Samplers h
time
Figure 2: Compressive sensing of a probability distribution by model neurons. Left: a neural architecture for
generating and then encoding a sparse, high-dimensional probability distribution. Right: activity of each population of neurons as a function of time. Sparse posterior probability distribution are generated by a Boltzmann
Machine with visible units v (Inputs), hidden units h (Samplers), feedforward couplings Jvh from visible to
hidden units, and recurrent connections between hidden units Jhh . The visible units? activities are fixed by
an input. The hidden units are stochastic, and sample from a probability distribution p(h|v). The samples
are recoded by feedforward weights W to random perceptrons a. The mean activity y of the time-dependent
perceptron responses captures the sparse joint distribution of the hidden units.
We are not ultimately interested in reconstruction of the large, sparse distribution, but rather the
distribution?s compressed representation. Nonetheless, reconstruction is useful to show that the
information has been preserved. I reconstruct sparse probabilities using nonnegative `1 minimization
with measurement constraints [16, 17], minimizing
kpk`1 + ?kAp ? yk2`2
(13)
where ? is a regularization parameter that was set to 2T in all simulations. Reconstructions were
quite good, as shown in Figure 3. Even with far fewer measurements than signal dimensions, reconstruction accuracy is limited only by the sampling of the posterior. Enough random perceptrons do
not lose any available information.
In the context of probability distributions,
`1 reconstruction has a serious flaw: All distributions have
P
the same `1 norm: kpk`1 = x p(x) = 1! To minimize the `1 norm, therefore, the estimate will
not be a probability distribution. Nonetheless, the individual probabilities of the most significant
states are accurately reconstructed, and only the highly improbable states are set to zero. Figure 3B
shows that the shortfall is small: `1 reconstruction recovers over 90% of the total probability mass.
3.2
Preserving computationally important relationships
There is value in being able to compactly represent these high-dimensional objects. However, it
would be especially useful to perform probabilistic computations using these representations, such
as marginalization and evidence integration. Since marginalization is a linear operation on the probability distribution, this is readily implementable in the linearly compressed domain. In contrast,
evidence integration is a multiplicative process acting in the canonical basis, so this operation will
be more complicated after the linear distortions of compressive measurement A. Nonetheless, such
computations should be feasible as long as the informative relationships are preserved in the compressed space: Similar distributions should have similar compressive representations, and dissimilar
5
B
C
Measurements M
20
State x
0
.9
.99
.999
Sum of 1-reconstructed
probabilities
Reconstructions
102
State x
80
320
D 10
?1
Reconstruction
error (MSE)
Samples T
Histogram
Reconstruction Probability
A
103
10?3
104
Probability
n=8
10
12
Sampling
error
2
4
8
16
32
Measurement ratio M/n
Figure 3: Reconstruction of sparse posteriors from random perceptron measurements. (A) A sparse posterior
distribution over 10 nodes in a Boltzmann machine is sampled 1000 times, fed to 50 random perceptrons,
and reconstructed by nonnegative `1 minimization. (B) A histogram of the sum of reconstructed probabilities
reveals the small shortfall from a proper normalization of 1. (C) Scatter plots show reconstructions versus
true probabilities. Each box uses different numbers of compressive measurements M and numbers of samples
T . (D) With increasing numbers of compressive measurements, the mean squared reconstruction error falls to
1/T = 10?3 , the limit imposed by finite sampling.
distributions should have dissimilar compressive representations. In fact, that is precisely the guarantee of compressive sensing: topological properties of the underlying space are preserved in the
compressive domain [18]. Figure 4 illustrates how not only are individual sparse distributions recoverable despite significant compression, but the topology of the set of all such distributions is
retained.
For this experiment, an input x is drawn from a dictionary of input patterns X ? {+1, ?1}n .
Each pattern in X is a translation of a single binary template x0 whose elements are generated by
thresholding a noisy sinusoid (Figure 4A): x0j = sgn [4 sin (2?j/n) + ?j ] with ?j ? N (0, 1). On
each trial, one of these possible patterns is drawn randomly with equal probability 1/|X |, and then
is measured by a noisy process that randomly flips bits with a probability ? = 0.35 to give a noisy
pattern r. This process induces a posterior distribution over the possible input patterns
Y
1
1
p(x|r) = p(x)
p(ri |xi ) = p(x)? N ?h(x,r) (1 ? ?)h(x,r)
(14)
Z
Z
i
where h(x, r) is the Hamming distance between x and r. This posterior is nonzero for all patterns in
the dictionary. The noise level and the similarities between the dictionary elements together control
the sparseness.
1000 trials of this process generates samples from the set of all possible posterior distributions.
Just as the underlying set of inputs has a translation symmetry, the set of all possible posterior
distributions has a cyclic permutation symmetry. This symmetry can be revealed by a nonlinear
embedding [19] of the set of posteriors into two dimensions (Figure 4B).
Compressive sensing of these posteriors by 10 random perceptrons produces a much lowerdimensional embedding that preserves this symmetry. Figure 4C shows that the same nonlinear embedding algorithm applied to the reduced representation, and one sees the same topological pattern.
In compressive sensing, similarity is measured by Euclidean distance. When applied to probability
distributions it will be interesting to examine instead how well information-geometric measures like
the Kullback-Leibler divergence are preserved under this dimensionality reduction [20].
4
Discussion
Probabilistic inference appears to be essential for both animals and machines to perform well on
complex tasks with natural levels of ambiguity, but it remains unclear how the brain represents and
manipulates probability. Present population models of neural inference either struggle with highdimensional distributions [1] or encode them by hard-to-measure high-order correlations [2]. Here I
have proposed an alternative mechanism by which the brain could efficiently represent probabilities:
random perceptrons. In this model, information about probabilities is compressed and distributed
6
A
i
true pattern x
ii
posterior
1
C
50
noisy pattern r
iii
iv
true
pattern
index
1
B
possible patterns X
100
pattern index
100
nonlinear embedding of
posterior distributions (N=100)
nonlinear embedding of
compressed posteriors (M=10)
Figure 4: Nonlinear embeddings of a family of probability distributions with a translation symmetry. (A)
The process of generating posterior distributions: (i) A set of 100 possible patterns is generated as cyclic
translations of a binary pattern (only 9 shown). With uniform probability, one of these patterns is selected (ii),
and a noisy version is obtained by randomly flipping bits with probability 0.35 (iii). From such noisy patterns,
an observer can infer posterior probability distributions over possible inputs (iv). (B) The set of posteriors from
1000 iterations of this process is nonlinearly mapped [19] from 100 dimensions to 2 dimensions. Each point
represents one posterior and is colored according to the actual pattern from which the noisy observations were
made. The permutation symmetry of this process is revealed as a circle in this mapping. (C) This circular
structure is retained even after each posterior is compressed into the mean output of 10 random perceptrons.
in neural population activity. Amazingly, the brain need not measure any correlations between the
perceptron outputs to capture the joint statistics of the sparse input distribution. Only the mean
activities are required. Figure 2 illustrates one network that implements this new representation, and
many variations on this circuit are possible.
Successful encoding in this compressed representation requires that the input distribution be sparse.
Posterior distributions over sensory stimuli like natural images are indeed expected to be highly
sparse: the features are sparse [21], the prior over images is sparse [22], and the likelihood produced by sensory evidence is usually restrictive, so the posteriors should be even sparser. Still, it
will be important to quantify just how sparse the relevant posteriors are under different conditions.
This would permit us to predict how neural representations in a fixed population should degrade as
sensory evidence becomes weaker.
Brains appear to have a mix of structure and randomness. The results presented here show that
purely random connections are sufficient to ensure that a sparse probability distribution is properly
encoded. Surprisingly, more structured connections cannot allow a network with the same computational elements to encode distributions with substantially fewer neurons, since compressive sensing
is already nearly optimal [8]. On the other hand, some representational structure may make it easier
to perform computations later. Note that unknown randomness is not an impediment to further processing, as reconstruction can be performed even without explicit knowledge of random perceptron
measurement matrix [23].
Even in the most convenient representations, inference is generally intractable and requires approximation. Since compressive sensing preserves the essential geometric relationships of the signal
space, learning and inference based on these relationships may be no harder after the compression,
and could even be more efficient due to the reduced dimensionality. Biologically plausible mechanisms for implementing probabilistic computations in the compressed representation is important
work for the future.
Appendix: Asymptotic orthogonality of random perceptron matrix
To evaluate the quality of the compressive sensing matrix A, we need to ensure that S-sparse vectors
are not projected to zero by the action of A. Here I show that the random ?
perceptrons are asymptotically well-conditioned: A?>A? ? I for large n and M , where A? = A/ M . This ensures that
distinct inputs yield distinct measurements.
7
First I compute the mean and variance of the mean inner product hCxx0 iW between columns of A?
for a given pair of states x 6= x0 . For compactness I will write wi for the ith row of the perceptron
weight matrix W . Angle brackets h iW indicate averages over random perceptron weights Wij ?
N (0, 1). We find
DX
E
1 X
hCxx0 iW =
A?i (x)A?i (x0 )
=
hsgn (wi ?x) sgn (wi ?x0 )iW
(15)
i
i
M
W
and since the different wi are independent, this implies that
hCxx0 iW = hsgn (wi ?x) sgn (wi ?x0 )iW
(16)
The n-dimensional half-space in W where sgn (wi ? x) = +1 intersects with the corresponding
half-space for x0 in a wedge-shaped region with an angle of ? = cos?1 (x ? x0 /kxk`2 kx0 k`2 ). This
angle is related to the Hamming distance h = h(x, x0 ):
?(h) = cos?1 (x ? x0 /n) = cos?1 (1 ? 2h/n)
(17)
hCxx0 iW =P [ sgn (wi ?x) = sgn (wi ?x0 )] ? P [ sgn (wi ?x) 6= sgn (wi ?x0 )]
(18)
The signs of wi ?x and wi ?x0 agree within this wedge region and its reflection about W = 0, and
disagree in the supplementary wedges. The mean inner product is therefore
=1 ?
2
? ?(h)
The variance of Cxx0 caused by variability in W is given by
2
2
Vxx0 = Cxx
? hCxx0 iW
0
W
E
E
XD
XD
=
A?2i (x)A?2i (x0 )
+
A?i (x)A?i (x0 )A?j (x)A?j (x0 )
W
i=j
*
=
X
i
2
W
i6=j
sgn (wi ?x) sgn (wi ?x0 )
M
M
2
(19)
(20)
?
2
hCxx0 iW
(21)
+
X sgn (wi ?x) sgn (wi ?x0 ) 2
2
?
?
+
? hCxx0 iW (22)
M
M
W
W i6=j
M2 ? M
1
2
+
(1 ? 2?(h)/?)2 ? hCxx0 iW
(23)
=
M
M2
2
1
=
1 ? 1 ? ?2 ?(h(x, x0 ))
(24)
M
This variance falls with M , so for large numbers of measurements M the inner products between
columns concentrates around the various state-dependent mean values (19).
Next I consider the diversity of inner products for different pairs (x, x0 ) of binary state vectors. I
take the limit of large M so that the diversity is dominated by variations over the particular pairs,
rather than by variations over measurements. The mean inner product depends only on the Hamming
0
distance h between
x and x , which for sparse signals with random support has a binomial distribun ?n
tion, p(h) = h 2 with mean n/2 and variance n/4. Designating by an overbar the average over
randomly chosen states x and x0 , the mean C and variance ?C 2 of the inner product are
C = hCxx0 iW = 1 ? ?2 cos?1 (1 ? 2h
n )=0
2
?C
n 16
4
?C 2 = ?h2
=
= 2
?h
4 ? 2 n2
? n
(25)
(26)
This proves that in the limit of large n and M , different columns of the random perceptron measurement matrix have inner products that concentrate around 0. The matrix of inner products is
thus orthonormal almost surely, A?>A? ? I. Consequently, with enough measurements the random
perceptrons asymptotically provide an isometry. Future work will investigate how the measurement
matrix behaves for finite n and M , which will determine the number of measurements required in
practice to capture a signal of a given sparseness.
Acknowledgments
Thanks to Alex Pouget, Jeff Beck, Shannon Starr, and Carmelita Navasca for helpful conversations.
8
References
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9
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3,979 | 4,601 | Newton-Like Methods for Sparse Inverse Covariance
Estimation
Figen Oztoprak
Sabanci University
[email protected]
Peder A. Olsen
IBM, T. J. Watson Research Center
[email protected]
Jorge Nocedal
Northwestern University
[email protected]
Steven J. Rennie
IBM, T. J. Watson Research Center
[email protected]
Abstract
We propose two classes of second-order optimization methods for solving the
sparse inverse covariance estimation problem. The first approach, which we call
the Newton-LASSO method, minimizes a piecewise quadratic model of the objective function at every iteration to generate a step. We employ the fast iterative
shrinkage thresholding algorithm (FISTA) to solve this subproblem. The second
approach, which we call the Orthant-Based Newton method, is a two-phase algorithm that first identifies an orthant face and then minimizes a smooth quadratic approximation of the objective function using the conjugate gradient method. These
methods exploit the structure of the Hessian to efficiently compute the search direction and to avoid explicitly storing the Hessian. We also propose a limited
memory BFGS variant of the orthant-based Newton method. Numerical results,
including comparisons with the method implemented in the QUIC software [1],
suggest that all the techniques described in this paper constitute useful tools for
the solution of the sparse inverse covariance estimation problem.
1
Introduction
Covariance selection, first described in [2], has come to refer to the problem of estimating a normal distribution that has a sparse inverse covariance matrix P, whose non-zero entries correspond
to edges in an associated Gaussian Markov Random Field, [3]. A popular approach to covariance
selection is to maximize an `1 penalized log likelihood objective, [4]. This approach has also been
applied to related problems, such as sparse multivariate regression with covariance estimation, [5],
and covariance selection under a Kronecker product structure, [6]. In this paper, we consider the
same objective function as in these papers, and present several Newton-like algorithms for minimizing it.
Following [4, 7, 8], we state the problem as
P? = arg max log det(P) ? trace(SP) ? ?kvec(P)k1 ,
(1)
P 0
where ? is a (fixed) regularization parameter,
PN
(2)
S = N1 i=1 (xi ? ?)(xi ? ?)T
n
is the empirical sample covariance, ? is known, the xi ? R are assumed to be independent,
2
identically distributed samples, and vec(P) defines a vector in Rn obtained by stacking the columns
of P. We recast (1) as the minimization problem
def
min F (P) = L(P) + ?kvec(P)k1 ,
P 0
1
(3)
where L is the negative log likelihood function
L(P) = ?log det(P) + trace(SP).
The convex problem (3) has a unique solution P? that satisfies the optimality conditions [7]
S ? [P? ]?1 + ?Z? = 0,
(4)
(5)
where
1
if Pij? > 0
= ?1
if Pij? < 0
?
? ? [?1, 1] if Pij? = 0.
?
We note that Z solves the dual problem
Z?ij
?
?
Z? = arg maxkvec(Z)k? ?1 U (Z),
U (Z) = ?log det(S + ?Z) + n.
(6)
S+?Z 0
The main contribution of this paper is to propose two classes of second-order methods for solving
problem (3). The first class employs a piecewise quadratic model in the step computation, and can be
seen as a generalization of the sequential quadratic programming method for nonlinear programming
2
[9]; the second class minimizes a smooth quadratic model of F over a chosen orthant face in Rn .
We argue that both types of methods constitute useful tools for solving the sparse inverse covariance
matrix estimation problem.
An overview of recent work on the sparse inverse covariance estimation problem is given in [10, 11].
First-order methods proposed include block-coordinate descent approaches, such as COVSEL, [4, 8]
and GLASSO [12], greedy coordinate descent, known as SINCO [13], projected subgradient methods
PSM [14], first order optimal gradient ascent [15], and the alternating linearization method ALM
[16]. Second-order methods include the inexact interior point method IPM proposed in [17], and the
coordinate relaxation method described in [1] and implemented in the QUIC software. It is reported
in [1] that QUIC is significantly faster than the ALM , GLASSO , PSM , SINCO and IPM methods. We
compare the algorithms presented in this paper to the method implemented in QUIC.
2
Newton Methods
We can define a Newton iteration for problem (1) by constructing a quadratic, or piecewise
quadratic, model of F using first and second derivative information. It is well known [4] that the
derivatives of the log likelihood function (4) are given by
def
def
g = L0 (P) = vec(S ? P?1 )
and
H = L00 (P) = (P?1 ? P?1 ),
(7)
where ? denotes the Kronecker product. There are various ways of using these quantities to define
a model of F , and each gives rise to a different Newton-like iteration.
In the Newton-LASSO Method, we approximate the objective function F at the current iterate Pk
by the piecewise quadratic model
qk (P) = L(Pk ) + gk> vec(P ? Pk ) + 12 vec> (P ? Pk )Hk vec(P ? Pk ) + ?kvec(P)k1 , (8)
where gk = L0 (Pk ), and similarly for Hk . The trial step of the algorithm is computed as a minimizer of this model, and a backtracking line search ensures that the new iterate lies in the positive
definite cone and decreases the objective function F . We note that the minimization of qk is often
called the LASSO problem [18] in the case when the unknown is a vector.
It is advantageous to perform the minimization of (8) in a reduced space; see e.g. [11] and the
references therein. Specifically, at the beginning of the k-th iteration we define the set Fk of (free)
variables that are allowed to move, and the active set Ak . To do so, we compute the steepest descent
for the function F , which is given by
?(gk + ?Zk ),
where
?
1
if (Pk )ij > 0
?
?
?
?1
if (Pk )ij < 0
?
?1
if (Pk )ij = 0 and [gk ]ij > ?
(Zk )ij =
(9)
?
?
1
if (Pk )ij = 0 and [gk ]ij < ??
?
? 1
? ? [gk ]ij
if (Pk )ij = 0 and | [gk ]ij | ? ?.
2
The sets Fk , Ak are obtained by considering a small step along this steepest descent direction, as this
guarantees descent in qk (P). For variables satisfying the last condition in (9), a small perturbation
of Pij will not decrease the model qk . This suggests defining the active and free sets of variables at
iteration k as
Ak = {(i, j)|(Pk )ij = 0 and |[gk ]ij | ? ?},
Fk = {(i, j)|(Pk )ij 6= 0 or |[gk ]ij | > ?}. (10)
The algorithm minimizes the model qk over the set of free variables. Let us define pF = vec(P)F ,
to be the free variables, and let pkF = vecF (Pk ) denote their value at the current iterate ? and
similarly for other quantities. Let us also define HkF to be the matrix obtained by removing from
Hk the columns and rows corresponding to the active variables (with indices in Ak ). The reduced
model is given by
>
qF (P) = L(Pk ) + gkF
(pF ? pkF ) + 12 (pF ? pkF )> HkF (pF ? pkF ) + ?kpF k1 .
(11)
The search direction d is defined by
d=
dF
dA
=
? F ? pkF
p
0
,
(12)
? F is the minimizer of (11). The algorithm performs a line search along the direction D =
where p
mat(d), where the operator mat(d) satisfies mat(vec(D)) = D. The line search starts with the
unit steplength and backtracks, if necessary, to obtain a new iterate Pk+1 that satisfies the sufficient
decrease condition and positive definiteness (checked using a Cholesky factorization):
F (Pk+1 ) ? F (Pk ) < ? (qF (Pk+1 ) ? qF (Pk ))
and Pk+1 0,
(13)
where ? ? (0, 1).
It is suggested in [1] that coordinate descent is the most effective iteration for solving the LASSO
problem (11). We claim, however, that other techniques merit careful investigation. These include
gradient projection [19] and iterative shrinkage thresholding algorithms, such as ISTA [20] and FISTA
[21]. In section 3 we describe a Newton-LASSO method that employs the FISTA iteration.
Convergence properties of the Newton-LASSO method that rely on the exact solution of the LASSO
problem (8) are given in [22]. In practice, it is more efficient to solve problem (8) inexactly, as
discussed in section 6. The convergence properties of inexact Newton-LASSO methods will be the
subject of a future study.
The Orthant-Based Newton method computes steps by solving a smooth quadratic approximation
2
of F over an appropriate orthant ? or more precisely, over an orthant face in Rn . The choice
of this orthant face is done, as before, by computing the steepest descent direction of F , and is
characterized by the matrix Zk in (9). Specifically the first four conditions in (9) identify an orthant
2
in Rn where variables are allowed to move, while the last condition in (9) determines the variables
to be held at zero. Therefore, the sets of free and active variables are defined as in (10). If we define
zF = vecF (Z), then the quadratic model of F over the current orthant face is given by
1
>
(14)
QF (P) = L(Pk ) + gF
(pF ? pkF ) + (pF ? pkF )> HF (pF ? pkF ) + ?z>
F pF .
2
The minimizer is p?F = pkF ? H?1
F (gF + ?zF ), and the step of the algorithm is given by
?
dF
pF ? pkF
d=
=
.
(15)
dA
0
If pkF +d lies outside the current orthant, we project it onto this orthant and perform a backtracking
line search to obtain the new iterate Pk+1 , as discussed in section 4.
The orthant-based Newton method therefore moves from one orthant face to another, taking advan2
tage of the fact that F is smooth in every orthant in Rn . In Figure 1 we compare the two methods
discussed so far.
The optimality conditions (5) show that P? is diagonal when ? ? |Sij | for all i 6= j, and given by
(diag(S) + ?I)?1 . This suggests that a good choice for the initial value (for any value of ? > 0) is
P0 = (diag(S) + ?I)?1 .
3
(16)
Method NL (Newton-LASSO)
Repeat:
Method OBN (Orthant-Based Newton)
Repeat:
1. Phase I: Determine the sets of fixed and
free indices Ak and Fk , using (10).
2. Phase II: Compute the Newton step D
given by (12), by minimizing the piecewise quadratic model (11) for the free
variables Fk .
3. Globalization: Choose Pk+1 by performing an Armijo backtracking line
search starting from Pk + D.
4. k ? k + 1.
1. Phase I: Determine the active orthant
face through the matrix Zk given in (9).
2. Phase II: Compute the Newton direction D given by (15), by minimizing
the smooth quadratic model (14) for the
free variables Fk .
3. Globalization: Choose Pk+1 in the current orthant by a projected backtracking
line search starting from Pk + D.
4. k ? k + 1.
Figure 1: Two classes of Newton methods for the inverse covariance estimation problem (3).
Numerical experiments indicate that this choice is advantageous for all methods considered.
A popular orthant based method for the case when the unknown is a vector is OWL [23]; see also
[11]. Rather than using the Hessian (7), OWL employs a quasi-Newton approximation to minimize
the reduced quadratic, and applies an alignment procedure to ensure descent. However, for reasons
that are difficult to justify the OWL step employs the reduced inverse Hessian (as apposed to the
inverse of the reduced Hessian), and this can give steps of poor quality. We have dispensed with
the alignment as it is not needed in our experiments. The convergence properties of OBM methods
are the subject of a future study (we note in passing that the convergence proof given in [23] is not
correct).
3
A Newton-LASSO Method with FISTA Iteration
Let us consider a particular instance of the Newton-LASSO method that employs the Fast Iterative
Shrinkage Thresholding Algorithm FISTA [21] to solve the reduced subproblem (11). We recall that
for the problem
min2 f (x) + ?kxk1 ,
(17)
x?Rn
where f is a smooth convex quadratic function, the ISTA iteration [20] is given by
1
? i ? ?f (?
xi ) ,
xi = S?/c x
c
(18)
where c is a constant such that cI ? f 00 (x) 0, and the FISTA acceleration is given by
ti ? 1
(xi ? xi?1 ),
(19)
ti+1
p
= 1 + 1 + 4t2i /2. Here S?/c denotes the soft thresholding
? i+1 = xi +
x
? 1 = x0 , t1 = 1, ti+1
where x
operator given by
(S? (y))i =
0
if |yi | ? ?,
yi ? ?sign(yi ) otherwise.
We can apply the ISTA iteration (18) to the reduced quadratic in (11) starting from x0 = vecFk (X0 )
(which is not necessarily equal to pk = vecFk (Pk )). Substituting in the expressions for the first and
second derivative in (7) gives
1
?
?
xi = S?/c vecFk (Xi ) ?
gkFk + HkFk vecFk (Xi ? Pk )
c
1
?1
?1 ?
?1
?
= S?/c vecFk (Xi ) ? vecFk (S ? 2Pk + Pk Xi Pk ) ,
c
4
where the constant c should satisfy c > 1/(eigmin Pk )2 . The FISTA acceleration step is given by
? denote the free variables part of the (approximate) solution of (11) obtained by the
(19). Let x
FISTA iteration. Phase I of the Newton- LASSO - FISTA method selects the free and active sets, Fk , Ak ,
as indicated by (10).
Phase II, applies the FISTA iteration to the reduced problem (11), and sets
?
x
Pk+1 ? mat
. The computational cost of K iterations of the FISTA algorithm is O(Kn3 ).
0
4
An Orthant-Based Newton-CG Method
We now consider an orthant-based Newton method in which a quadratic model of F is minimized
approximately using the conjugate gradient (CG) method. This approach is attractive since, in addition to the usual advantages of CG (optimal Krylov iteration, flexibility), each CG iteration can be
efficiently computed by exploiting the structure of the Hessian matrix in (7).
Phase I of the orthant-based Newton-CG method computes the matrix Zk given in (9), which is used
2
to identify an orthant face in Rn . Variables satisfying the last condition in (9) are held at zero and
their indices are assigned to the set Ak , while the rest of the variables are assigned to Fk and are
allowed to move according to the signs of Zk : variables with (Zk )ij = 1 must remain non-negative,
and variables with (Zk )ij = ?1 must remain non-positive.
Having identified the current orthant face, phase II of the method constructs the quadratic model
QF in the free variables, and computes an approximate solution by means of the conjugate gradient
method, as described in Algorithm 1.
Conjugate Gradient Method for Problem (14)
input : Gradient g, orthant indicator z, current iterate P0 , maximum steps K, residual tolerance
r , and the regularization parameter ?.
output: Approximate Newton direction d = cg(P0 , g, z, ?, K)
n = size(P0 , 1) , G = mat(g) , Z = mat(z);
A = {(i, j) : [P0 ]ij = 0 & |Gij | ? ?};
B = P?1
0 , X0 = 0n?n , x0 = vec(X0 );
R0 = ?(G + ?Z), [R0 ]A ? 0;
k = 0, q0 = r0 = vec(R0 );
( ? [r0 ]F = vF )
while k ? min(n2 , K) and krk k > r do
Qk = reshape(qk , n, n);
Yk = BQk B, [Yk ]A ? 0, yk = vec(Yk );
?k =
r>
k rk
;
q>
k yk
xk+1 = xk + ?k qk ;
rk+1 = rk ? ?k yk ;
?k =
r>
k+1 rk+1
;
r>
k rk
qk+1 = rk+1 + ?k qk ;
k ? k + 1;
end
return d = xk+1
Algorithm 1: CG Method for Minimizing the Reduced Model QF .
The search direction of the method is given by D = mat(d), where d denotes the output of Algorithm 1. If the trial step Pk + D lies in the current orthant, it is the optimal solution of (14).
Otherwise, there is at least one index such that
(i, j) ? Ak and [L0 (Pk + D)]ij ?
/ [??, ?], or (i, j) ? Fk and (Pk + D)ij Zij < 0.
In this case, we perform a projected line search to find a point in the current orthant that yields a
decrease in F . Let ?(?) denote the orthogonal projection onto the orthant face defined by Zk , i.e.,
Pij if sign(Pij ) = sign(Zk )ij
?(Pij ) =
(20)
0
otherwise.
5
The line search computes a steplength ?k to be the largest member of the sequence
{1, 1/2, . . . , 1/2i , . . .} such that
e (Pk )T (?(Pk + ?k D) ? Pk ) ,
F (?(Pk + ?k D)) ? F (Pk ) + ? ?F
(21)
e denotes the minimum norm subgradient of F . The
where ? ? (0, 1) is a given constant and ?F
new iterate is defined as Pk+1 = ?(Pk + ?k D).
The conjugate gradient method requires computing matrix-vector products involving the reduced
Hessian, HkF . For our problem, we have
kF
HkF (pF ? pkF ) = Hk pF ?p
0
F
?1
pF ?pkF
= P?1
mat
Pk F .
(22)
k
0
The second line follows from the identity (A ? B)vec(C) = vec(BCA> ). The cost of performing
K steps of the CG algorithm is O(Kn3 ) operations, and K = n2 steps is needed to guarantee an
exact solution. Our practical implementation computes a small number of CG steps relative to n,
K = O(1), and as a result the search direction is not an accurate approximation of the true Newton
step. However, such inexact Newton steps achieve a good balance between the computational cost
and the quality of the direction.
5
Quasi-Newton Methods
The methods considered so far employ the exact Hessian of the likelihood function L, but one can
also approximate it using (limited memory) quasi-Newton updating. At first glance it may not seem
promising to approximate a complicated Hessian like (7) in this manner, but we will see that quasiNewton updating is indeed effective, provided that we store matrices using the compact limited
memory representations [9].
Let us consider an orthant-based method that minimizes the quadratic model (14), where HF is
replaced by a limited memory BFGS matrix, which we denote by BF . This matrix is not formed
explicitly, but is defined in terms of the difference pairs
yk = gk+1 ? gk ,
sk = vec(Pk+1 ? Pk ).
(23)
It is shown in [24, eq(5.11)] that the minimizer of the model QF is given by
p?F = pF + B?1
F (?zF ? gF )
= ?1 (?zF ? gF ) +
1
T
? 2 RF W(I
? ?1 MWT RF RTF W)
?1
MWT RF (?zF ? gF ).
(24)
Here RF is a matrix consisting of the set of unit vectors that span the subspace of free variables, ?
is a scalar, W is an n2 ? 2t matrix containing the t most recent correction pairs (23), and M is a
2t ? 2t matrix formed by inner products between the correction pairs. The total cost of computing
the minimizer p?F is 2t2 |F| + 4t|F| operations, where |F| is the cardinality of F. Since the memory
parameter t in the quasi-Newton updating scheme is chosen to be a small number, say between 5
and 20, the cost of computing the subspace minimizer (24) is quite affordable. A similar approach
was taken in [25] for the related constrained inverse covariance sparsity problem.
We have noted above that OWL, which is an orthant based quasi-Newton method does not correctly
approximate the minimizer (24). We note also that quasi-Newton updating can be employed in
Newton-LASSO methods, but we do not discuss this approach here for the sake of brevity.
6
Numerical Experiments
We generated test problems by first creating a random sparse inverse covariance matrix1 , ??1 , and
then sampling data to compute a corresponding non-sparse empirical covariance matrix S. The dimensions, sparsity, and conditioning of the test problems are given along with the results in Table 2.
For each data set, we solved problem (3) with ? values in the range [0.01, 0.5]. The number of
samples used to compute the sample covariance matrix was 10n.
1
http://www.cmap.polytechnique.fr/?aspremon/CovSelCode.html, [7]
6
The algorithms we tested are listed in Table 1. With the exception of C:QUIC, all of these algorithms
were implemented in MATLAB. Here NL and OBN are abbreviations for the methods in Figure 1.
NL-Coord is a MATLAB implementation of the QUIC algorithm that follows the C-version [1]
Algorithm
NL-FISTA
NL-Coord
OBN-CG-K
OBN-CG-D
OBN-LBFGS
ALM?
C:QUIC
Description
Newton-LASSO-FISTA method
Newton-LASSO method using coordinate descent
Orthant-based Newton-CG method with a limit of K CG iterations
OBN-CG-K with K=5 initially and increased by 1 every 3 iterations.
Orthant-based quasi-Newton method (see section 5)
Alternating linearization method [26].
The C implementation of QUIC given in [1].
Table 1: Algorithms tested. ? For ALM, the termination criteria was changed to the `? norm and the value
of ABSTOL was set to 10?6 to match the stopping criteria of the other algorithms.
faithfully. We have also used the original C-implementation of QUIC and refer to it as C:QUIC. For
the Alternating Linearization Method (ALM) we utilized the MATLAB software available at [26],
which implements the first-order method described in [16]. The NL-FISTA algorithm terminated
the FISTA iteration when the minimum norm subgradient of the LASSO subproblem qF became
less than 1/10 of the minimum norm subgradient of F at the previous step.
Let us compare the computational cost of the inner iteration techniques used in the Newton-like
methods discussed in this paper. (i) Applying K steps of the FISTA iteration requires O(Kn3 )
operations. (ii) Coordinate descent, as implemented in [1], requires O(Kn|F|) operations for K
coordinate descent sweeps through the set of free variables; (iii) Applying KCG iterations of the CG
methods costs O(KCG n3 ) operations.
The algorithms were terminated when either 10n iterations were executed or the minimum norm
? (P)k? ? 10?6 . The time limit of each run
subgradient of F was sufficiently small, i.e. when k?F
was set to 5000 seconds.
The results presented in Table 2 show that the ALM method was never the fastest algorithm, but
nonetheless outperformed some second-order methods when the solution was less sparse. The numbers in bold indicate the fastest MATLAB implementation for each problem. As for the other methods, no algorithm appears to be consistently superior to the others, and the best choice may depend
on problem characteristics. The Newton-LASSO method with coordinate descent (NL-Coord) is
the most efficient when the sparsity level is below 1%, but the methods introduced in this paper,
NL-FISTA, OBN-CG and OBN-LBFGS, seem more robust and efficient for problems that are less
sparse. Based on these results, OBN-LBFGS appears to be the best choice as a universal solver for
the covariance selection problem. The C implementation of the QUIC algorithm is roughly five times
faster than its Matlab counterpart (OBN-Coord). C:QUIC was best in the two sparsest conditions,
but not in the two densest conditions. We expect that optimized C implementations of the presented
algorithms will also be significantly faster. Note also that the crude strategy for dynamically increasing the number of CG-steps in OBN-CG-D was effective, and we expect it could be further
improved. Our focus in this paper has been on exploring optimization methods and ideas rather
than implementation efficiency. However, we believe the observed trends will hold even for highly
optimized versions of all tested algorithms.
7
problem
n = 500
Card(??1 )
= 2.4%
n = 500
Card(??1 )
= 20.1%
n = 1000
Card(??1 )
= 3.5%
n = 1000
Card(??1 )
= 11%
n = 2000
Card(??1 )
= 1%
n = 2000
Card(??1 )
= 18.7%
?
algorithm
card(P? )
cond(P? )
NL-FISTA
NL-Coord
OBN-CG-5
OBN-CG-D
OBN-LBFGS
ALM
C:QUIC
card(P? )
cond(P? )
NL-FISTA
NL-Coord
OBN-CG-5
OBN-CG-D
OBN-LBFGS
ALM
C:QUIC
card(P? )
cond(P? )
NL-FISTA
NL-Coord
OBN-CG-5
OBN-CG-D
OBN-LBFGS
ALM
C:QUIC
card(P? )
cond(P? )
NL-FISTA
NL-Coord
OBN-CG-5
OBN-CG-D
OBN-LBFGS
ALM
C:QUIC
card(P? )
cond(P? )
NL-FISTA
NL-Coord
OBN-CG-5
OBN-CG-D
OBN-LBFGS
ALM
C:QUIC
card(P? )
cond(P? )
NL-FISTA
NL-Coord
OBN-CG-5
OBN-CG-D
OBN-LBFGS
ALM
C:QUIC
0.5
iter
time
0.74%
8.24
8
5.71
21
3.86
15
4.07
12
3.88
47
5.37
445 162.96
16
0.74
0.21%
3.39
4
1.25
4
0.42
3
0.83
3
0.84
9
1.00
93
35.75
6
0.19
0.18%
6.22
7
28.20
9
5.23
9
15.34
8
15.47
34
18.27
247 617.63
10
2.38
0.10%
4.20
4
9.03
4
2.23
3
4.70
3
4.61
8
4.29
113 283.99
6
1.18
0.13%
7.41
8 264.94
14
54.33
13 187.41
9 127.11
41 115.13
- >5000
11
18.07
0.05%
2.32
P? = P0
P? = P0
P? = P0
P? = P0
P? = P0
52 874.22
8
10.35
0.1
iter
time
7.27%
27.38
10
22.01
49
100.63
97
26.24
34
15.41
178
21.92
387
152.76
41
15.62
14.86%
16.11
19
13.12
14
19.69
27
7.36
15
5.22
82
11.42
78
32.98
13
3.79
6.65%
18.23
9
106.79
24
225.59
51
87.73
21
51.99
111
80.02
252
639.49
22
46.14
8.18%
11.75
7
72.21
12
79.71
20
35.85
12
26.87
67
40.31
99
255.79
11
17.42
1.75%
23.71
10 1039.08
34 1178.07
78
896.24
27
532.15
155
497.31
>5000
17
183.53
1.49%
4.72
7
153.18
7
71.55
6
71.54
6
75.82
26
78.34
76 1262.83
8
24.65
0.05
iter
time
11.83%
51.01
11
37.04
66
279.69
257
70.91
65
43.29
293
38.23
284
115.11
58
35.64
25.66%
32.27
15
34.53
21
71.51
101
28.40
31
14.14
155
23.04
149
61.35
21
11.91
13.19%
39.59
12
203.07
36
951.23
108
198.17
39
132.38
178
111.49
186
462.34
34
186.87
18.38%
26.75
10
156.46
19
408.62
47
83.42
27
78.98
124
82.51
106
267.02
19
90.62
4.33%
46.54
10 1490.37
>5000
203 2394.95
43 1038.26
254
785.36
>5000
40
818.54
10.51%
17.02
9
694.93
13 1152.86
21
250.11
13
188.93
71
232.23
106 1800.67
13
256.90
0.01
iter
time
32.48%
118.56
12
106.27
103 1885.89
1221
373.63
189
275.29
519
84.13
574
219.80
100
206.42
47.33%
99.49
13
100.90
55
791.84
795
240.90
176
243.55
455
78.33
720
292.43
56
103.58
25.03%
132.13
12
801.79
>5000
1103 2026.26
171 1584.14
548
384.30
734 1826.29
72 1445.17
36.34%
106.34
22
554.08
49 4837.46
681 1778.88
148 2055.44
397
297.90
577 1448.83
52 1100.72
14.68%
134.54
>5000
>5000
>5000
>5000
610 2163.12
>5000
>5000
31.68%
79.61
12 2852.86
>5000
397 4766.69
110 5007.83
318 1125.67
>5000
33 3899.68
Table 2: Results for 5 Newton-like methods and the QUIC, ALM method.
8
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9
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brevity:1 tested:3 |
3,980 | 4,602 | Bayesian Pedigree Analysis using Measure
Factorization
Bonnie Kirkpatrick
Computer Science Department
University of British Columbia
[email protected]
Alexandre Bouchard-C?ot?e
Statistics Department
University of British Columbia
[email protected]
Abstract
Pedigrees, or family trees, are directed graphs used to identify sites of the genome
that are correlated with the presence or absence of a disease. With the advent
of genotyping and sequencing technologies, there has been an explosion in the
amount of data available, both in the number of individuals and in the number
of sites. Some pedigrees number in the thousands of individuals. Meanwhile,
analysis methods have remained limited to pedigrees of < 100 individuals which
limits analyses to many small independent pedigrees.
Disease models, such those used for the linkage analysis log-odds (LOD) estimator, have similarly been limited. This is because linkage analysis was originally
designed with a different task in mind, that of ordering the sites in the genome,
before there were technologies that could reveal the order. LODs are difficult to
interpret and nontrivial to extend to consider interactions among sites. These developments and difficulties call for the creation of modern methods of pedigree
analysis.
Drawing from recent advances in graphical model inference and transducer theory, we introduce a simple yet powerful formalism for expressing genetic disease
models. We show that these disease models can be turned into accurate and computationally efficient estimators. The technique we use for constructing the variational approximation has potential applications to inference in other large-scale
graphical models. This method allows inference on larger pedigrees than previously analyzed in the literature, which improves disease site prediction.
1
Introduction
Finding genetic correlates of disease is a long-standing important problem with potential contributions to diagnostics and treatment of disease. The pedigree model for inheritance is one of the best
defined models in biology, and it has been an area of active statistical and biological research for
over a hundred years.
The most commonly used method to analyze genetic correlates of disease is quite old. After Mendel
introduced, in 1866, the basic model for the inheritance of genomic sites [1] Sturtevant was the first,
in 1913, to provide a method for ordering the sites of the genome [2]. The method of Sturtevant
became the foundation for linkage analysis with pedigrees [3, 4, 5, 6]. The problem can be thought
of in Sturtevant?s framework as that of finding the position of a disease site relative to an map of
existing sites. This is the log-odds (LOD) estimator for linkage analysis which is a likelihood ratio
test, described in more detail below.
The genomic data available now is quite different than the type of data available when LOD was initially developed. Genomic sites are becoming considerably denser in the genome and technologies
allow us to interrogate the genome for the position of sites [7]. Additionally, most current pedigree
1
analysis methods are exponential either in the number of sites or in the number of individuals. This
produces a limit on the size of the pedigrees under consideration to around < 100 individuals. This
is in contrast to the size of pedigrees being collected: for example the work of [8] includes a connected human pedigree containing 13 generations and 1623 individuals, and the work of [9] includes
a connected non-human data set containing thousands of breeding dogs. Apart from the issues of
pedigree size, the LOD value is difficult to interpret, since there are few models for the distribution of the statistic. These developments and difficulties call for the creation of modern methods of
pedigree analysis.
In this work, we propose a new framework for expressing genetic disease models. The key component of our models, the Haplotype-Phenotype Transducer (HPT), draws from recent advances in
graphical model inference and transducer theory [10], and provides a simple and flexible formalism for building genetic disease models. The output of inference over HPT models is a posterior
distribution over disease sites, which is easier to interpret than LOD scores.
The cost of this modeling flexibility is that the graphical model corresponding to the inference
problem is larger and has more loops that traditional pedigree graphical models. Our solution to
this challenge is based on the observation that the difficult graphical model can be covered by a
collection of tractable forest graphical models. We use a method based on measure factorization [11]
to efficiently combine these approximations. Our approach is applicable to other dense graphical
models, and we show that empirically it gives accurate approximations in dense graphical models
containing millions of nodes as well as short and long cycles. Our approximation can be refined by
adding more trees in the forest, with a cost linear in the number of forests used in the cover. We
show that considerable gains in accuracy can be obtained this way. In contrast, methods such as [12]
can suffer from an exponential increase in running time when larger clusters are considered.
Our framework can be specialized to create analogues of classical penetrance disease models [13].
We focus on these special cases here to compare our method with classical ones. Our experiments
show that even for these simpler cases, our approach can achieve significant gains in disease site
identification accuracy compared to the most commonly used method, Merlin?s implementation of
LOD scores [3, 5]. Moreover, our inference method allows us to perform experiments on unprecedented pedigree sizes, well beyond the capacity of Merlin and other pedigree analysis tools typically
used in practice.
While graphical models have played an important role in the development of pedigree analysis
methods [14, 15], only recently were variational methods applied to the problem [6]. However
this previous work is based on the same graphical model as classical LOD methods, while ours
significantly differs.
Most current work on more advanced disease models have focused on a very different type of data,
population data, for genome wide association studies (GWAS) [16]. Similarly, state of the art work
on the related task of imputation generally makes similar population assumptions [17].
2
Background
Every individual has two copies of each chromosome, one copy is a collage of the mother?s two
chromosomes while the other is a collage of the father?s two chromosomes. The point at which
the copying of the chromosomes switches from one of the grand-maternal (grand-paternal) chromosomes to the other, is called a recombination breakpoint. A site is a particular position in the genome
at which we can obtain measurable values. For the purposes of this paper, an allele is the nucleotide
at a particular site on a particular chromosome. A haplotype is the sequence of alleles that appear
together on the same chromosome.
If we had complete data, we would know the positions of all of the haplotypes, all of the recombination breakpoints as well as which allele came from which parent. This information is not obtainable
from any known experiment. Instead, we have genotype data which is the set of nucleotides that
appear in an individual?s genome at a particular site. Given that the genotype is a set, it is unordered,
and we do not know which allele came from which parent. All of this and the recombination breakpoints must be inferred. An example is given in the Supplement.
2
A pedigree is a directed acyclic graph with individuals as nodes, where boxes are males and circles
are females, and edges directed downward from parent to child. Every individual must have either
no parents or one parent of each gender. The individuals without parents in the graph are called
founders, and the individuals with parents are non-founders. The pedigree encodes a set of relationships that constrain the allowed inheritance options. These inheritance options define a probability
distribution which is investigated during pedigree analysis.
Assume a single-site disease model, where a diploid genotype, GD , determines the affection status
(phenotype), P ? {?h?,?d?}, according to the penetrance probabilities: f2 = P(P = ?d?|GD = 11),
f1 = P(P = ?d?|GD = 10), f0 = P(P = ?d?|GD = 00). Here the disease site usually has a disease
allele, 1, that confers greater risk of having the disease. For convenience, we denote the penetrance
vector as f = (f2 , f1 , f0 ).
Let the pedigree model for n individuals be specified by a pedigree graph, a disease model f , and
the minor allele frequency, ?, for a single site of interest, k. Let P = (P1 , P2 , ..., Pn ) be a vector
containing the affection status of each individual. Let G = (G1 , G2 , ..., Gn ) be the genotype data for
each individual. Between the disease site and site k, we model the per chromosome, per generation
recombination fraction, ?, which is the frequency with which recombinations occur between those
two sites. Other sites linked to k can contribute to our estimate via their arrangement in single firstorder Markov chain with some sites falling to the left of the disease site and others to the right of the
site of interest. Previous work has shown that given a pedigree model, affection data, and genotype
data, we can estimate ?.
We define the likelihood as L(?) = P(P = p, G = g|?, f, ?) where ? is the recombination probability between the disease site and the first site, p are the founder allele frequencies, and f are the
penetrance probabilities. To test for linkage between the disease site and the other sites, we maximize the likelihood to obtain the optimal recombination fraction ?? = argmax? L(?)/L(1/2). The
test we use is the likelihood ratio test where the null hypothesis is that of no linkage (? = 1/2).
Generally referred to as the log-odd score (or LOD score), the log of this likelihood ratio is
log L(?? ) ? log L(1/2).
3
Methods
In this section, we describe our model for inferring relationships between phenotypes and genotyped
pedigree datasets. We start by giving a high-level description of the generative process.
The first step in this generative process consists in sampling a collection of disease model (DM) variables, which encode putative relationships between the genetic sites and the observed phenotypes.
There is one disease model variable for each site, s, and to a first approximation, Ds can be thought
as taking values zero or one, depending on whether site s is the closest to the primary genetic factor
involved in a disease (a more elaborate example is presented in the Supplement). We use C to denote
the values Ds can take.
The second generative step consists in sampling the chromosomes or haplotypes of a collection of
related individuals. We denote these variables by Hi,s,x , where, from now on, i is used to index
individuals, s, to index sites, and x ? { ?father?, ?mother? }, to index chromosome parental origin.
For SNP data, the set of values H that Hi,s,x can take generally contains two elements (alleles). A
related variable, the inheritance variables Ri,s,x , will be sampled jointly with the Hi,s,x ?s to keep
track of the grand-parental origin of each chromosome segment. See Figure 1(a) for a factor graph
representation of the random variables.
Finally, the phenotype Pi , which we assume is taken from a finite set P, can be sampled for each
individual i in the pedigree. We will define the distribution of Pi conditionally on the haplotype of
the individual in question, Hi , and on the global disease model D. Note that variables with missing
indices are used to denote random vectors or matrices, for example D = (D1 , . . . , DS ), where S
denotes the number of sites.
To summarize this high-level view of the process, and to introduce notations for the distributions
involved:
D ? DM(?)
Ri ? Recomb(?) for all i
3
M
(1)
Ds
(a)
GF
HP Ti
?
1
Li ,s
Hi ,s
(2)
Ds
1 ??
0
Li ,s
i
1 ??
s
(b)
DM(.)
?h?:1.0
1
Aa
0
??
? ?:1.0
1
aa
?
GM
0
?d?:1.0
1
AA
Recomb(.)
?h?:1.0
HPT(.;.)
Figure 1: (a) The pedigree graphical model for independent sites. There are two plates, one for each individual
and one for each site. The nodes are labeled as follows: M for the marriage node which enforces the Mendelian
inheritance constraints, H for haplotype, L and L0 for the two alleles, D(1) for the disease site indicator, and
D(2) for the disease allele value. (b) The transducer for DM(?) has three nodes with the start node indicated
by an in-arrow and the end node indicated by an out-arrow. The transducer for Recomb(?) has recombination
parameter ?. This assumes a constant recombination rate across sites, but non-constant rates can be obtained
with a bigger automaton. This transducer for HPT(?) models a recessive disease where the input at each state
is the disease (top) and haplotype alleles (bottom). For these last two transducers any node can be the start or
end node.
The remaining variables (the non-founder individuals? haplotype variables) are obtained deterministically from the values of the founders and the inheritance: Hi,s,x = Hx(i),s,Ri,s,x , where x(i)
denotes the index of the father (mother) of i if x = ?father? (?mother?). The distribution on the
founder haplotypes is a product of independent Bernoulli distributions, one for each site (the parameters of these Bernoulli distributions is not restricted to be identically distributed and can be
estimated [3]). Each genotype variable Gs is obtained via a deterministic function of H. Having
generated all the haplotypes and disease variables, we denote the conditional distribution of the
phenotypes as follows:
Pi |(D, Hi ) ? HPT( ? ; D, Hi ),
where HPT stands for a Haplotype-Phenotype Transducer.
We now turn to the description of these distributions, starting with the most important one,
HPT( ? ; D, Hi ). Formally, this distribution on phenotypes is derived from a weighted automaton, where we view the vectors D and Hi as an input string of length S, the s-th character of which
is the triplet (Ds , Hi,s,?father? , Hi,s,?mother? ). We view each of the sampled phenotypes as a length-one
output from a weighted transducer given the input D, Hi . Longer outputs could potentially be used
for more complex phenotypes or diseases.
To illustrate this construction, we show that classical, Mendelian models such as recessive phenotypes are a special case of this formalism. We also make two simplifications to facilitate exposition:
first, that the disease site is one of the observed sites, and second, that the disease allele is the less
frequent (minor) allele (we show in the Supplement a slightly more complicated transducer that does
not make these assumptions).
Under the two above assumptions, we claim that the state diagrams in Figure 1(b) specify an HPT
transducer for a recessive disease model. Each oval corresponds to a hidden transducer state, and the
annotation inside the oval encodes the tuple of input symbols that the corresponding state consumes.
The emission is depicted on top of the states, with for example ?d?: 1.0 denotes that a disease
indicator is emitted with weight one. We use ?h? for the non-disease (healthy) indicator, and for
the null emission.
The probability mass function of the HPT is defined as:
P
z?ZHPT (h,c?p)
HPT(p; c, h) = P
z 0 ?ZHPT (h,c??)
wHPT (z)
wHPT (z 0 )
,
where h ? HS , c ? C S , p ? P, and ZHPT (h, c ? p) denotes the set of valid paths in the space
Z of hidden states. The valid paths are sequences of hidden states (depicted by black circles in
Figure 1(b)) starting at the source and ending at the sink, consuming c, h and emitting p along the
way. The star in the denominator of the above equation is used to denote unconstrained emissions.
4
In other words, the denominator is the normalization of the weighted transducer [10]. The set of
valid paths is implicitly encoded in the transition diagram of the transducer, and the weight function
wHPT : Z ? ? [0, ?) can similarly be compactly represented by only storing weights for individual
transitions and multiplying them to get a path weight.
The set of valid paths along with their weights can be thought of as encoding a parametric disease
model. For example, with a recessive disease, shown in Figure 1(b), we can see that if the transducer
is at the site of the disease (encoded as the current symbol in c being equal to 1) then only an input
homozygous haplotype ?AA? will lead to an output disease phenotype ?d.? This formalism gives a
considerable amount of flexibility to the modeler, who can go beyond simple Mendelian disease
models by constructing different transducers.
The DM distribution is defined using the same machinery as for the HPT distribution. We show
in Figure 1(b) a weighted automaton that encodes the prior that exactly one site is involved in the
disease, with an unknown, uniformly distributed location in the genome. The probability mass
function of the distribution is given by:
P
z?ZDM (?c)
wDM (z)
z 0 ?ZDM (??)
wDM (z 0 )
DM(c) = P
,
where ZDM (? c) and ZDM (? ?) are direct analogues to the HPT case, with the difference being
that no input is read in the DM case.
The last distribution in our model, Recomb, is standard, but we present it in the new light of the
transducer formalism. Refer to Figure 1(b) for an example based on the standard recombination
model derived from the marginals of a Poisson process. We use the analogous notation:
P
Recomb(r) = P
4
z?ZRecomb (?r)
wRecomb (z)
z 0 ?ZRecomb (??)
wRecomb (z 0 )
.
Computational Aspects
Probabilistic inference in our model is computationally challenging: the variables L, H alone induce a loopy graph [18], and the addition of the variables D, P introduces more loops as well as
deterministic constraints, which further complicates the situation. After explaining in more detail
the graphical model of interest, we discuss in this section the approximation algorithm that we have
used to infer haplotypes, disease loci, and other disease statistics.
We show in Figure 1(a) the factor graph obtained after turning the observed variables (genotypes
and phenotypes) into potentials (we show a more detailed version in the Supplement). We have also
taken the pointwise product of potentials whenever possible (in the case of the transducer potentials,
how this pointwise product is implemented is discussed in [10]). Note that our graphical model
has more cycles than standard pedigree graphical models [19]; even if we assumed the sites to be
independent and the pedigree to be acyclic, our graphical model would still be cyclic.
Our inference method is based on the following observation: if we kept only one subtype of factors
in the Supplement, say only those connected to the recombination variables R, then inference could
be done easily. More precisely, inference would reduce to a collection of small, standard HMMs
inference problems, which can be done using existing software.
Similarly, by covering the pedigree graph with a collection of subtrees, and removed the factors
for disease and recombination, we can get a collection of acyclic pedigrees, one for each site, and
hence a tractable problem (the sum-product algorithm in this case is called the Elston-Stewart algorithm [14] in the pedigree literature).
We are therefore in a situation where we have several restricted views on our graphical model yielding efficiently solved subproblems. How to combine the solutions of these tractable subproblems is
the question we address in the remainder of this section.
The most common way this is approached, in pedigrees [20] and elsewhere [21], is via block Gibbs
sampling. However, block Gibbs sampling does not apply readily to our model. The main difficulty
arises when attempting to resample D: because of the deterministic constraints that arise even in
5
the simplest disease model, it is necessary to sample D in a block also containing a large subset of
R and H. However this cannot be done efficiently since D is connected to all individuals in the
pedigree. More formally, the difficulty is that some of the components we wish to resample are
b-acyclic (barely acyclic) [22]. Another method, closer to ours, is the EP algorithm of [23], which
however considers a single tree approximant, while we can accommodate several at once. As we
show in the empirical section, it is advantageous to do so in pedigrees.
An important feature that we will exploit in the development of method is the forest cover property
of the tractable subproblems: we view each tractable subproblem as a subgraph of the initial factor
graph, and ask that the union of these subgraph coincides with the original factor graph.
Previous variational approaches have been proposed to exploit such forest covers. The most wellknown example, the structured mean field approximation, is unfortunately non-trivial to optimize in
the b-acyclic case [22]. Tree reweighted belief propagation [24] has an objective function derived
from a forest distribution, however the corresponding algorithms are based on local message passing
rather than large subproblems.
We propose an alternative based on the measure factorization framework [11]. As we will see,
this yields an easy to implement variation approximation that can efficiently exploit arbitrary forest
cover approximations. Since the measure factorization interpretation of our approach is not specific
to pedigrees, we present it in the context of a generic factor graph over a discrete space, viewed as
an exponential family with sufficient statistics ?, log normalization A, and parameters ?:
P(X = x) = exp {h?(x), ?i ? A(?)} .
(1)
To index the factors, we use ? ? F = {1, ..., F }, and v to index the V variables in the factor graph.
We start by reparameterizing the exponential family in terms of a larger vector y of variables.
P Let
us also denote the number of nodes connected to factor ? by n? . This vector y has N = ? n?
components, each corresponding to a pair containing a factor and a node index attached to it, and
denoted by y?,v . The reparameterization is given by:
Y Y
P(Y = y) = exp h?(y), ?i ? A0 (?)
1[y?,v = y?0 ,v ].
?,?0 ?F
(2)
v
Because of the indicator variables in the right hand side of Equation 2, the set of y?s with P(Y =
y) > 0 is in bijection with the set of x?s with P(X = x) > 0. It is therefore well-defined to overload
the variable ? in the same equation. Similarly, we have that A0 = A. This reparameterization is
inspired by the auxiliary variables used to construct the sampler of Swendsen-Wang [25].
Next, suppose that the sets F1 , . . . , FK form a forest cover of the factor graph, Fk ? F. Then, for
k ? {1, . . . , K}, we build as follows the super-partitions required for the measure factorization to
apply (as defined in [11]):
Ak (?) =
X
exp {h?(y), ?i}
y
Y
Y
?,?0 ?Fk
v
1[y?,v = y?0 ,v ].
(3)
Note that computing each Ak is tractable: it corresponds to computing the normalization of one of
the forest covering the graphical model. Similarly, gradients of Ak can be computed as the moments
of a tree shaped graphical model. Also, the product over k of the base measures in Equation 3 is equal
to the base measure of Equation 2. We have therefore constructed a valid measure factorization.
With this construction in hand, it is then easy to apply the measure factorization framework to get a
principled way for the different subproblem views to exchange messages [11].
5
Experiments
We did two sets of experiments. Haplotype reconstructions were used to assess the quality of the
variational approximation. Disease predictions were used to validate the HPT disease model.
Simulations. Pedigree graphs were simulated using a Wright-Fisher model [26]. In this model
there is a fixed number of male individuals, n, and female individuals, n, per generation, making
the population size 2n. The pedigree is built starting from the oldest generation. Each successively
more recent generation is built by having each individual in that generation choose uniformly at
random one female parent and one male parent. Notice that this process allows inbreeding.
6
(b) Recombination Factors (c) Recombination Parameter
?
?
?
?
?
?
?
?
5
10
15
?
?
0.00005
0.0005
0.005
0.28
?
0.28
?
false
true
?
0.24
?
?
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0.20
?
No. Iterations
15
0.28
?
10
0.20
?
0.25
?
5
0.24
?
0.20
?
1
2
3
4
5
0.20
?
0.28
0
0.24
No. Iterations
15
?
?
0.25
10
?
?
15
0
5
10
15
?
?
?
0
5
?
?
10
?
?
?
?
?
?
15
?
?
0.16
10
0.16
5
0.16
0
?
?
0.16
0.15
?
0.15
Haplotype Metric ?
?
5
0.20
0
0.24
No. Iterations
0.20
(a) Forest-Cover Factors
Figure 2: The pedigree was generated with the following parameters, number of generations 20 and n = 15
which resulted in a pedigree with 424 individuals, 197 marriage nodes, 47 founders. We simulated 1000
markers. The metric used for all panels is the haplotype reconstruction metric. Panel (a) shows the effect of
removing factors from the forest cover of the pedigree where the lines are labeled with the number of factors
that each experiment contains. Panel (b) shows the effect of removing the recombination factor (false) or using
it (true). Together, panels (a-b) show that having more factors helps inference. Panel (c) shows the effect of an
incorrect recombination parameter on inference. The correct parameter, with which the data was generated, is
line 0.0005. Two incorrect parameters are shown 0.00005 and 0.005. This panel shows that the recombination
parameter can be off by an order of magnitude and the haplotype reconstruction is robust.
Genotype data were simulated in the simulated pedigree graph. The founder haplotypes were drawn
from an empirical distribution (see Supplement for details). The recombination parameters used
for inheritance are given in the Supplement. We then simulated the inheritance and recombination
process to obtain the haplotypes of the descendants using the external program [27]. We used two
distributions for the founder haplotypes, corresponding to two data sets.
Individuals with missing data were sampled, where each individual either has all their genetic data
missing or not. A random 50% of the non-founder individuals have missing data. An independent
50% of individuals have missing phenotypes for the disease prediction comparison.
Haplotype Reconstruction. For the haplotype reconstruction, the inference being scored is, for
each individual, the maximum a posteriori haplotype predicted by the marginal haplotype distribution. These haplotypes are not necessarily Mendelian consistent, meaning that it is possible for
a child to have an allele on the maternal haplotype that could not possibly be inherited from the
mother according to the mother?s marginal distribution. However, transforming the posterior distribution over haplotypes into a set of globally consistent haplotypes is somewhat orthogonal to
the methods in this paper, and there exist methods for this task [28]. The goal of this comparison
is threefold: 1) to see if adding more factors improves inference, 2) to see if more iterations of
the measure factorization algorithm help, and 3) to see if there is robustness of the results to the
recombination parameters.
Synthetic founder haplotypes were simulated, see Supplement for details. Each experiment was
replicated 10 times where for each replicate the founder haplotypes were sampled with a different
random seed. We computed a metric ? which is a normalized count of the number of sites that differ
between the held-out haplotype and the predicted haplotype. See the Supplement for details.
Figure 2 shows the results for the haplotype reconstruction. Panels (a) and (b) show that adding more
factors helps inference accuracy. Panel (c) shows that inference accuracy is robust to an incorrect
recombination parameter.
Disease Prediction. For disease prediction, the inference being scored is the ranking of the sites
given by our Bayesian method as compared with LOD estimates computed by Merlin [3]. The disease models we consider are recessive f = (0.95, 0.05, 0.05) and dominant f = (0.95, 0.95, 0.05).
The disease site is one of the sites chosen uniformly at random. The goal of this comparison is to
see whether our disease model performs at least as well as the LOD estimator used by Merlin.
7
Pedigree
Disease model
HPT
LOD [3]
Generations
Leaves
Individuals
f2
f1
f0
Mean ?
SD ?
Mean ?
SD ?
3
8
10
12
22
25
34
0.95
0.05
0.05
0.08
0.07
0.04
(0.09)
(0.09)
(0.04)
0.25
0.52
0.45
(0.20)
(0.44)
(0.23)
3
4
5
6
16
20
24
0.04
0.08
0.14
(0.05)
(0.09)
(0.16)
0.27
0.35
0.20
(0.31)
(0.31)
(0.22)
5
100
200
300
418
882
1276
1e-3
4e-4
6e-4
(2e-3)
(1e-3)
(1e-3)
3
8
10
12
22
25
34
0.14
0.11
0.12
(0.15)
(0.14)
(0.22)
0.95
0.95
0.05
Out of memory
Out of memory
Out of memory
0.22
0.33
0.22
(0.23)
(0.40)
(0.16)
Table 1: This table gives the performance of our method and Merlin for recessive and dominant diseases as
measured by the disease prediction metric. The sizes of the simulated pedigrees are given in the first three
columns, the disease model in the next three columns, and the performance of our method and that of Merlin in
the final four columns. In all instances, our method outperforms Merlin sometimes by an order of magnitude.
Results suggest that the standard deviation of our method is smaller than that of Merlin. Notably, Merlin cannot
even analyze the largest pedigrees, because Merlin does exact inference.
The founder haplotypes were taken from the phased haplotypes of the JPT+CHB HapMap [29]
populations, see Supplement for details. Each experiment was replicated 10 times where for each
replicate the founder haplotypes were sampled with a different random seed. We computed a metric
? which is roughly the rank of the disease site in the sorted list of predictions given by each method.
Table 1 compares the performance of our method against that of Merlin. In every case our method
has better accuracy. The results suggest that our method has a lower standard deviation. Within
each delineated row of the table, the mean ? are not comparable because the pedigrees might be of
different complexities. Between delineated rows of the table, we can compare the effect of pedigree
size, and we observe that larger pedigrees aid in disease site prediction. Indeed, the largest pedigree
of 1276 individuals reaches an accuracy of 6e?4 . This pedigree is the largest pedigree that we know
of being analyzed in the literature.
6
Discussion
This paper introduces a new disease model and a new variational inference method which are applied
to find a Bayesian solution to the disease-site correlation problem. This is in contrast to traditional
linkage analysis where a likelihood ratio statistic is computed to find the position of the disease site
relative to a map of existing sites. Instead, our approach is to use a Haplotype-Phenotype Transducer
to obtain a posterior for the probability of each site to be the disease site. This approach is wellsuited to modern data which is very dense in the genome. Particularly with sequencing data, it is
likely that either the disease site or a nearby site will be observed.
Our method performs well in practice both for genotype prediction and for disease site prediction.
In the presence of missing data, where for some individuals the whole genome is missing, our
method is able to infer the missing genotypes with high accuracy. As compared with LOD linkage
analysis method, our method was better able to predict the disease site when one observed site was
responsible for the disease.
References
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J.H. Bennett, ed. Oliver and Boyd, Edinburgh 1965, 1866.
[2] A. H. Sturtevant. The linear arrangement of six sex-linked factors in drosophila, as shown by their mode
of association. Journal of Experimental Zoology, 14:43?59, 1913.
8
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9
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3,981 | 4,603 | Provable ICA with Unknown Gaussian Noise, with
Implications for Gaussian Mixtures and Autoencoders
Sanjeev Arora?
Rong Ge?
Ankur Moitra ?
Sushant Sachdeva?
Abstract
We present a new algorithm for Independent Component Analysis (ICA) which
has provable performance guarantees. In particular, suppose we are given samples
of the form y = Ax + ? where A is an unknown n ? n matrix and x is a random
variable whose components are independent and have a fourth moment strictly
less than that of a standard Gaussian random variable and ? is an n-dimensional
Gaussian random variable with unknown covariance ?: We give an algorithm that
provable recovers A and ? up to an additive and whose running time and sample complexity are polynomial in n and 1/. To accomplish this, we introduce
a novel ?quasi-whitening? step that may be useful in other contexts in which the
covariance of Gaussian noise is not known in advance. We also give a general
framework for finding all local optima of a function (given an oracle for approximately finding just one) and this is a crucial step in our algorithm, one that has
been overlooked in previous attempts, and allows us to control the accumulation
of error when we find the columns of A one by one via local search.
1
Introduction
We present an algorithm (with rigorous performance guarantees) for a basic statistical problem.
Suppose ? is an independent n-dimensional Gaussian random variable with an unknown covariance
matrix ? and A is an unknown n ? n matrix. We are given samples of the form y = Ax + ? where
x is a random variable whose components are independent and have a fourth moment strictly less
than that of a standard Gaussian random variable. The most natural case is when x is chosen uniformly at random from {+1, ?1}n , although our algorithms in even the more general case above.
Our goal is to reconstruct an additive approximation to the matrix A and the covariance matrix ?
running in time and using a number of samples that is polynomial in n and 1 , where is the target
precision (see Theorem 1.1) This problem arises in several research directions within machine learning: Independent Component Analysis (ICA), Deep Learning, Gaussian Mixture Models (GMM),
etc. We describe these connections next, and known results (focusing on algorithms with provable
performance guarantees, since that is our goal).
Most obviously, the above problem can be seen as an instance of Independent Component Analysis
(ICA) with unknown Gaussian noise. ICA has an illustrious history with applications ranging from
econometrics, to signal processing, to image segmentation. The goal generally involves finding a
linear transformation of the data so that the coordinates are as independent as possible [1, 2, 3]. This
is often accomplished by finding directions in which the projection is ?non-Gaussian? [4]. Clearly,
if the datapoint y is generated as Ax (i.e., with no noise ? added) then applying linear transformation
A?1 to the data results in samples A?1 y whose coordinates are independent. This restricted case
was considered by Comon [1] and Frieze, Jerrum and Kannan [5], and their goal was to recover an
?
{arora, rongge, sachdeva}@cs.princeton.edu. Department of Computer Science, Princeton University,
Princeton NJ 08540. Research supported by the NSF grants CCF-0832797, CCF-1117309 and Simons Investigator Grant
?
[email protected]. School of Mathematics, Institute for Advanced Study, Princeton NJ 08540. Research
supported in part by NSF grant No. DMS-0835373 and by an NSF Computing and Innovation Fellowship.
1
additive approximation to A efficiently and using a polynomial number of samples. (We will later
note a gap in their reasoning, albeit fixable by our methods. See also recent papers by Anandkumar
et al., Hsu and Kakade[6, 7], that do not use local search and avoids this issue.) To the best of our
knowledge, there are currently no known algorithms with provable guarantees for the more general
case of ICA with Gaussian noise (this is especially true if the covariance matrix is unknown, as in
our problem), although many empirical approaches are known. (eg. [8], the issue of ?empirical? vs
?rigorous? is elaborated upon after Theorem 1.1.)
The second view of our problem is as a concisely described Gaussian Mixture Model. Our data is
generated as a mixture of 2n identical Gaussian components (with an unknown covariance matrix)
n
whose centers are the points {Ax : x ? {?1, 1} }, and all mixing weights are equal. Notice, this
n
mixture of 2 Gaussians admits a concise description using O(n2 ) parameters. The problem of
learning Gaussian mixtures has a long history, and the popular approach in practice is to use the
EM algorithm [9], though it has no worst-case guarantees (the method may take a very long time
to converge, and worse, may not always converge to the correct solution). An influential paper of
Dasgupta [10] initiated the program of designing algorithms with provable guarantees, which was
improved in a sequence of papers [11, 12, 13, 14]. But in the current setting, it is unclear how to
apply any of the above algorithms (including EM ) since the trivial application would keep track
of exponentially many parameters ? one for each component. Thus, new ideas seem necessary to
achieve polynomial running time.
The third view of our problem is as a simple form of autoencoding [15]. This is a central notion in
Deep Learning, where the goal is to obtain a compact representation of a target distribution using a
multilayered architecture, where a complicated function (the target) can be built up by composing
layers of a simple function (called the autoencoder [16]). The main tenet is that there are interesting functions which can be represented concisely using many layers, but would need a very large
representation if a ?shallow? architecture is used instead). This is most useful for functions that
are ?highly varying? (i.e. cannot be compactly described by piecewise linear functions or other
?simple? local representations). Formally, it is possible to represent using just (say) n2 parameters,
some distributions with 2n ?varying parts? or ?interesting regions.? The Restricted Boltzmann Machine (RBM) is an especially popular autoencoder in Deep Learning, though many others have been
proposed. However, to the best of our knowledge, there has been no successful attempt to give a
rigorous analysis of Deep Learning. Concretely, if the data is indeed generated using the distribution represented by an RBM, then do the popular algorithms for Deep Learning [17] learn the model
parameters correctly and in polynomial time? Clearly, if the running time were actually found to
be exponential in the number of parameters, then this would erode some of the advantages of the
compact representation.
How is Deep Learning related to our problem? As noted by Freund and Haussler [18] many years
ago, an RBM with real-valued visible units (the version that seems more amenable to theoretical
analysis) is precisely a mixture of exponentially many standard Gaussians. It is parametrized by an
n ? m matrix A and a vector ? ? Rn . It encodes a mixture of n-dimensional standard Gaussians
m
centered at the points {Ax : x ? {?1, 1} }, where the mixing weight of the Gaussian centered at
2
Ax is exp(kAxk2 + ? ? x). This is of course reminiscent of our problem. Formally, our algorithm
can be seen as a nonlinear autoencoding scheme analogous to an RBM but with uniform mixing
weights. Interestingly, the algorithm that we present here looks nothing like the approaches favored
traditionally in Deep Learning, and may provide an interesting new perspective.
1.1 Our results and techniques
We give a provable algorithm for ICA with unknown Gaussian noise. We have not made an attempt
to optimize the quoted running time of this model, but we emphasize that this is in fact the first
algorithm with provable guarantees for this problem and moreover we believe that in practice our
algorithm will run almost as fast as the usual ICA algorithms, which are its close relatives.
Theorem 1.1 (Main, Informally). There is an algorithm that recovers the unknown A and ? up to
additive error in each entry in time that is polynomial in n, kAk2 , k?k2 , 1/, 1/?min (A) where k ? k2
denotes the operator norm and ?min (?) denotes the smallest eigenvalue.
The classical approach for ICA initiated in Comon [1] and Frieze, Jerrum and Kannan [5]) is for
the noiseless case in which y = Ax. The first step is whitening, which applies a suitable linear
transformation that makes the variance the same in all directions, thus reducing to the case where
2
A is a rotation matrix. Given samples y = Rx where R is a rotation matrix, the rows of R can be
found in principle by computing the vectors u that are local minima of E[(u ? y)4 ]. Subsequently, a
number of works (see e.g. [19, 20]) have focused on giving algorithms that are robust to noise. A
popular approach is to use the fourth order cumulant (as an alternative to the fourth order moment)
as a method for ?denoising,? or any one of a number of other functionals whose local optima reveal
interesting directions. However, theoretical guarantees of these algorithms are not well understood.
The above procedures in the noise-free model can almost be made rigorous (i.e., provably polynomial running time and number of samples), except for one subtlety: it is unclear how to use local
search to find all optima in polynomial time. In practice, one finds a single local optimum, projects
to the subspace orthogonal to it and continues recursively on a lower-dimensional problem. However, a naive implementation of this idea is unstable since approximation errors can accumulate
badly, and to the best of our knowledge no rigorous analysis has been given prior to our work. (This
is not a technicality: in some similar settings the errors are known to blow up exponentially [21].)
One of our contributions is a modified local search that avoids this potential instability and finds all
local optima in this setting. (Section 4.2.)
Our major new contribution however is dealing with noise that is an unknown Gaussian. This is an
important generalization, since many methods used in ICA are quite unstable to noise (and a wrong
estimate for the covariance could lead to bad results). Here, we no longer need to assume we know
even rough estimates for the covariance. Moreover, in the context of Gaussian Mixture Models this
generalization corresponds to learning a mixture of many Gaussians where the covariance of the
components is not known in advance.
We design new tools for denoising and especially whitening in this setting. Denoising uses the fourth
order cumulant instead of the fourth moment used in [5] and whitening involves a novel use of the
Hessian of the cumulant. Even then, we cannot reduce to the simple case y = Rx as above, and are
left with a more complicated functional form (see ?quasi-whitening? in Section 2.) Nevertheless,
we can reduce to an optimization problem that can be solved via local search, and which remains
amenable to a rigorous analysis. The results of the local optimization step can be then used to
simplify the complicated functional form and recover A as well as the noise ?. We defer many of
our proofs to the supplementary material section, due to space constraints.
In order to avoid cluttered notation, we have focused on the case in which x is chosen uniformly at
random from {?1, +1}n , although our algorithm and analysis work under the more general conditions that the coordinates of x are (i) independent and (ii) have a fourth moment that is less
than three (the fourth moment of a Gaussian random variable). In this case, the functional P (u)
(see Lemma 2.2) will take the same form but with weights depending on the exact value of the
fourth moment for each coordinate. Since we already carry through an unknown diagonal matrix D
throughout our analysis, this generalization only changes the entries on the diagonal and the same
algorithm and proof apply.
2
Denoising and quasi-whitening
As mentioned, our approach is based on the fourth order cumulant. The cumulants of a random
variable are the coefficients of the Taylor expansion of the logarithm of the characteristic function
[22]. Let ?r (X) be the rth cumulant of a random variable X. We make use of:
Fact 2.1. (i) If X has mean zero, then ?4 (X) = E[X 4 ] ? 3 E[X 2 ]2 . (ii) If X is Gaussian with mean
? and variance ? 2 , then ?1 (X) = ?, ?2 (X) = ? 2 and ?r (X) = 0 for all r > 2. (iii) If X and Y
are independent, then ?r (X + Y ) = ?r (X) + ?r (Y ).
The crux of our technique is to look at the following functional, where y is the random variable
Ax + ? whose samples are given to us. Let u ? Rn be any vector. Then P (u) = ??4 (uT y).
Note that for any u we can compute P (u) reasonably accurately by drawing sufficient number of
samples of y and taking an empirical average. Furthermore, since x and ? are independent, and ? is
Gaussian, the next lemma is immediate. We call it ?denoising? since it allows us empirical access
to some information about A that is uncorrupted by the noise ?.
Pn
Lemma 2.2 (Denoising Lemma). P (u) = 2 i=1 (uT A)4i .
The intuition is that P (u) = ??4 (uT Ax) since the fourth cumulant does not depend on the additive
Gaussian noise, and then the lemma follows from completing the square.
3
2.1 Quasi-whitening via the Hessian of P (u)
In prior works on ICA, whitening refers to reducing to the case where y = Rx for some some
rotation matrix R. Here we give a technique to reduce to the case where y = RDx + ? 0 where ? 0
is some other Gaussian noise (still unknown), R is a rotation matrix and D is a diagonal matrix that
depends upon A. We call this quasi-whitening. Quasi-whitening suffices for us since local search
using the objective function ?4 (uT y) will give us (approximations to) the rows of RD, from which
we will be able to recover A.
Quasi-whitening involves computing the Hessian of P (u), which recall is the matrix of all 2nd order
partial derivatives of P (u). Throughout this section, we will denote the Hessian operator by H. In
matrix form, the Hessian of P (u) is
n
n
X
X
?2
P (u) = 24
Ai,k Aj,k (Ak ? u)2 ; H(P (U )) = 24
(Ak ? u)2 Ak ATk = ADA (u)AT
?ui ?uj
k=1
k=1
where Ak is the k-th column of the matrix A (we use subscripts to denote the columns of matrices
throught the paper). DA (u) is the following diagonal matrix:
Definition 2.3. Let DA (u) be a diagonal matrix in which the k th entry is 24(Ak ? u)2 .
Of course, the exact Hessian of P (u) is unavailable and we will instead compute an empirical
approximation Pb(u) to P (u) (given many samples from the distribution), and we will show that the
Hessian of Pb(u) is a good approximation to the Hessian of P (u).
0
Definition 2.4. Given 2N samples y1 , y10 , y2 , y20 ..., yN , yN
of the random variable y, let
N
N
3 X T 2 T 0 2
?1 X T 4
(u yi ) +
(u yi ) (u yi ) .
Pb(u) =
N i=1
N i=1
Our first step is to show that the expectation of the Hessian of Pb(u) is exactly the Hessian of P (u).
In fact, since the expectation of Pb(u) is exactly P (u) (and since Pb(u) is an analytic function of the
samples and of the vector u), we can interchange the Hessian operator and the expectation operator.
Roughly, one can imagine the expectation operator as an integral over the possible values of the
random samples, and as is well-known in analysis, one can differentiate under the integral provided
that all functions are suitably smooth over the domain of integration.
Claim 2.5. Ey,y0 [?(uT y)4 + 3(uT y)2 (uT y 0 )2 ] = P (u)
This claim follows immediately from the definition of P (u), and since y and y 0 are independent.
Lemma 2.6. H(P (u)) = Ey,y0 [H(?(uT y)4 + 3(uT y)2 (uT y 0 )2 )]
Next, we compute the two terms inside the expectation:
Claim 2.7. H((uT y)4 ) = 12(uT y)2 yy T
Claim 2.8. H((uT y)2 (uT y 0 )2 ) = 2(uT y 0 )2 yy T + 2(uT y)2 y 0 (y 0 )T + 4(uT y)(uT y 0 )(y(y 0 )T +
(y 0 )y T )
Let ?min (A) denote the smallest eigenvalue of A. Our analysis also requires bounds on the entries
of DA (u0 ):
Claim 2.9. If u0 is chosen uniformly at random then with high probability for all i,
n
log n
n
min kAi k22 n?4 ? DA (u0 ))i,i ? max kAi k22
i=1
i=1
n
Lemma 2.10. If u0 is chosen uniformly at random and furthermore we are given 2N =
poly(n, 1/, 1/?min (A), kAk2 , k?k2 ) samples of y, then with high probability we will have that
(1 ? )ADA (u0 )AT H(Pb(u0 )) (1 + )ADA (u0 )AT .
c (1 + )ADA (u0 )AT , and let M
c = BB T .
Lemma 2.11. Suppose that (1 ? )ADA (u0 )AT M
?
?
?1
1/2
?
Then there is a rotation matrix R such that kB ADA (u0 ) ? R kF ? n.
The intuition is: if any of the singular values of B ?1 ADA (u0 )1/2 are outside the range [1 ? , 1 + ],
cx are too far apart
we can find a unit vector x where the quadratic forms xT ADA (u0 )AT x and xT M
(which contradicts the condition of the lemma). Hence the singular values of B ?1 ADA (u0 )1/2 can
all be set to one without changing the Froebenius norm of B ?1 ADA (u0 )1/2 too much, and this
yields a rotation matrix.
4
3
Our algorithm (and notation)
In this section we describe our overall algorithm. It uses as a blackbox the denoising and quasiwhitening already described above, as well as a routine for computing all local maxima of some
?well-behaved? functions which is described later in Section 4.
Notation: Placing a hat over a function corresponds to an empirical approximation that we obtain
from random samples. This approximation introduces error, which we will keep track of.
Step 1: Pick a random u0 ? Rn and estimate the Hessian H(Pb(u0 )). Compute B such that
H(Pb(u0 )) = BB T . Let D = DA (u0 ) be the diagonal matrix defined in Definition 2.3.
PN
0
Step
2: Take 2N samples y1 , y2 , ...,yN , y10 , y20 , ..., yN
, and let Pb0 (u) = ? N1 i=1 (uT B ?1 yi )4 +
P
N
3
T ?1
yi )2 (uT B ?1 yi0 )2 which is an empirical estimation of P 0 (u).
i=1 (u B
N
Step 3: Use the procedure A LL OPT(Pb0 (u), ?, ? 0 , ? 0 , ? 0 ) of Section 4 to compute all n local maxima
of the function Pb0 (u).
Step 4: Let R be the matrix whose rows are the n local optima recovered in the previous step. Use
procedure R ECOVER of Section 5 to find A and ?.
Explanation: Step 1 uses the transformation B ?1 computed in the previous Section to quasi-whiten
the data. Namely, we consider the sequence of samples z = B ?1 y, which are therefore of the form
R0 Dx+? 0 where ? = B ?1 ?, D = DA (u0 ) and R0 is close to a rotation matrix R? (by Lemma 2.11).
In Step 2 we look at ?4 ((uT z)), which effectively denoises the new samples (see Lemma 2.2), and
thus is the same as ?4 (R0 D?1/2 x). Let P 0 (u) = ?4 (uT z) = ?4 (uT B ?1 y) which is easily seen to be
E[(uT R0 D?1/2 x)4 ]. Step 2 estimates this function, obtaining Pb0 (u). Then Step 3 tries to find local
optima via local search. Ideally we would have liked access to the functional P ? (u) = (uT R? x)4
since the procedure for local optima works only for true rotations. But since R0 and R? are close
we can make it work approximately with Pb0 (u), and then in Step 4 use these local optima to finally
recover A.
Theorem 3.1. Suppose we are given samples of the form y = Ax + ? where x is uniform on
{+1, ?1}n , A is an n ? n matrix, ? is an n-dimensional Gaussian random variable independent
of x with unknown covariance matrix ?. There is an algorithm that with high probability recovers
b ? A?diag(ki )kF ? where ? is some permutation matrix and each ki ? {+1, ?1} and
kA
b ? ?kF ? . Furthermore the running time and number of samples needed are
also recovers k?
poly(n, 1/, kAk2 , k?k2 , 1/?min (A))
Note that here we recover A up to a permutation of the columns and sign-flips. In general, this is
all we can hope for since the distribution of x is also invariant under these same operations. Also,
the dependence of our algorithm on the various norms (of A and ?) seems inherent since our goal is
to recover an additive approximation, and as we scale up A and/or ?, this goal becomes a stronger
relative guarantee on the error.
4
Framework for iteratively finding all local maxima
In this section, we first describe a fairly standard procedure (based upon Newton?s method) for
finding a single local maximum of a function f ? : Rn ? R among all unit vectors and an analysis
of its rate of convergence. Such a procedure is a common tool in statistical algorithms, but here we
state it rather carefully since we later give a general method to convert any local search algorithm
(that meets certain criteria) into one that finds all local maxima (see Section 4.2).
Given that we can only ever hope for an additive approximation to a local maximum, one should
be concerned about how the error accumulates when our goal is to find all local maxima. In fact, a
naive strategy is to project onto the subspace orthogonal to the directions found so far, and continue
in this subspace. However, such an approach seems to accumulate errors badly (the additive error
of the last local maxima found is exponentially larger than the error of the first). Rather, the crux
of our analysis is a novel method for bounding how much the error can accumulate (by refining old
estimates).
5
Algorithm 1. L OCAL OPT, Input:f (u), us , ?, ? Output: vector v
1. Set u ? us .
2. Maximize (via Lagrangian methods) Proj?u (?f (u))T ? + 12 ? T Proj?u (H(f (u)))? ?
k?k22
0
1
2
?
f (u)
?u
?
T
Subject to k?k2 ? ? and u ? = 0
3. Let ? be the solution, u
?=
u+?
ku+?k
4. If f (?
u) ? f (u) + ?/2, set u ? u
? and Repeat Step 2
5. Else return u
Our strategy is to first find a local maximum in the orthogonal subspace, then run the local optimization algorithm again (in the original n-dimensional space) to ?refine? the local maximum we have
found. The intuition is that since we are already close to a particular local maxima, the local search
algorithm cannot jump to some other local maxima (since this would entail going through a valley).
4.1 Finding one local maximum
Throughout this section, we will assume that we are given oracle access to a function f (u) and its
gradient and Hessian. The procedure is also given a starting point us , a search range ?, and a step
size ?. For simplicity in notation we define the following projection operator.
Definition 4.1. Proj?u (v) = v ? (uT v)u, Proj?u (M ) = M ? (uT M u)uuT .
The basic step the algorithm is a modification of Newton?s method to find a local improvement
that makes progress so long as the current point u is far from a local maxima. Notice that if we
add a small vector to u, we do not necessarily preserve the norm of u. In order to have control
over how the norm of u changes, during local optimization step the algorithm projects the gradient
?f and Hessian H(f ) to the space perpendicular to u. There is also an additional correction term
??/?u f (u) ? k?k2 /2. This correction term is necessary because the new vector we obtain is (u +
?)/ k(u + ?)k2 which is close to u ? k?k22 /2 ? u + ? + O(? 3 ). Step 2 of the algorithm is just
maximizing a quadratic function and can be solved exactly using Lagrangian Multiplier method. To
increase efficiency it is also acceptable to perform an approximate maximization step by taking ? to
be either aligned with the gradient Proj?u ?f (u) or the largest eigenvector of Proj?u (H(f (u))).
The algorithm is guaranteed to succeed in polynomial time when the function is Locally Improvable
and Locally Approximable:
Definition 4.2 ((?, ?, ?)-Locally Improvable). A function f (u) : Rn ? R is (?, ?, ?)-Locally
Improvable, if for any u that is at least ? far from any local maxima, there is a u0 such that
ku0 ? uk2 ? ? and f (u0 ) ? f (u) + ?.
Definition 4.3 ((?, ?)-Locally Approximable). A function f (u) is locally approximable, if its third
order derivatives exist and for any u and any direction v, the third order derivative of f at point u in
the direction of v is bounded by 0.01?/? 3 .
The analysis of the running time of the procedure comes from local Taylor expansion. When a
function is Locally Approximable it is well approximated by the gradient and Hessian within a ?
neighborhood. The following theorem from [5] showed that the two properties above are enough to
guarantee the success of a local search algorithm even when the function is only approximated.
Theorem 4.4 ([5]). If |f (u) ? f ? (u)| ? ?/8, the function f ? (u) is (?, ?, ?)-Locally Improvable,
f (u) is (?, ?) Locally Approximable, then Algorithm 1 will find a vector v that is ? close to some
local maximum. The running time is at most O((n2 + T ) max f ? /?) where T is the time to evaluate
the function f and its gradient and Hessian.
4.2 Finding all local maxima
Now we consider how to find all local maxima of a given function f ? (u). The crucial condition that
we need is that all local maxima are orthogonal (which is indeed true in our problem, and is morally
true when using local search more generally in ICA). Note that this condition implies that there are
at most n local maxima.1 In fact we will assume that there are exactly n local maxima. If we are
given an exact oracle for f ? and can compute exact local maxima then we can find all local maxima
6
Algorithm 2. A LL OPT, Input:f (u), ?, ?, ? 0 , ? 0 Output: v1 , v2 , ..., vn , ?i kvi ? vi? k ? ?.
1.
2.
3.
4.
5.
6.
7.
Let v1 = L OCAL OPT(f, e1 , ?, ?)
FOR i = 2 TO n DO
Let gi be the projection of f to the orthogonal subspace of v1 , v2 , ..., vi?1 .
Let u0 = L OCAL OPT(g, e1 , ? 0 , ? 0 ).
Let vi = L OCAL OPT(f, u0 , ?, ?).
END FOR
Return v1 , v2 , ..., vn
easily: find one local maximum, project the function into the orthogonal subspace, and continue to
find more local maxima.
Definition 4.5. The projection of a function f to a linear subspace S is a function on that subspace
with value equal to f . More explicitly, if {v1 , v2 , ..., vd } is an orthonormal basis of S, the projection
Pd
of f to S is a function g : Rd ? R such that g(w) = f ( i=1 wi vi ).
The following theorem gives sufficient conditions under which the above algorithm finds all local
maxima, making precise the intuition given at the beginning of this section.
Theorem 4.6. Suppose the function f ? (u) : Rn ? R satisfies the following properties:
1. Orthogonal Local Maxima: The function has n local maxima vi? , and they are orthogonal
to each other.
2. Locally Improvable: f ? is (?, ?, ?) Locally Improvable.
3. Improvable Projection: The projection of the function to any subspace spanned by a subset
of local maxima is (? 0 , ? 0 , ? 0 ) Locally Improvable. The step size ? 0 ? 10?.
?
4. Lipschitz: If ku ? u0 k2 ? 3 n?, then the function value |f ? (u) ? f ? (u0 )| ? ? 0 /20.
?
5. Attraction Radius: Let Rad ? 3 n? + ? 0 , for any local maximum vi? , let T?be min f ? (u)
for ku ? vi? k2 ? Rad, then there exist a set U containing ku ? vi? k2 ? 3 n? + ? 0 and
does not contain any other local maxima, such that for every u that is not in U but is ?
close to U , f ? (u) < T .
If we are given function f such that |f (u) ? f ? (u)| ? ?/8 and f is both (?, ?) and (? 0 , ? 0 ) Locally
Approximable, then Algorithm 2 can find all local maxima of f ? within distance ?.
To prove this theorem, we first notice the projection of the function f in Step 3 of the algorithm
should be close to the projection of f ? to the remaining local maxima. This is implied by Lipschitz
condition and is formally shown in the following two lemmas. First we prove a ?coupling? between
the orthogonal complement of two close subspaces:
Lemma 4.7. Given v1 , v2 , ..., vk , each ?-close respectively to local maxima v1? , v2? , ..., vk? (this is
without loss of generality because we can permute the index of local maxima), then there is an
orthonormal basis vk+1 , vk+2 , ..., vn for the orthogonal space of span{v1 , v2 , ..., vk } such that for
Pn?k
Pn?k
?
?
any unit vector w ? Rn?k , i=1 wk vk+i is 3 n? close to i=1 wk vk+i
.
We prove this lemma using a modification of the Gram-Schmidt orthonormalization procedure. Using this lemma we see that the projected function is close to the projection of f ? in the span of the
rest of local maxima:
Lemma 4.8. Let g ? be the projection of f ? into the space spanned by the rest of local maxima, then
|g ? (w) ? g(w)| ? ?/8 + ? 0 /20 ? ? 0 /8.
5
Local search on the fourth order cumulant
Next, we prove that the fourth order cumulant P ? (u) satisfies the properties above. Then the algorithm given in the previous section will find all of the local maxima, which is the missing step in our
1
Technically, there are 2n local maxima since for each direction u that is a local maxima, so too is ?u but
this is an unimportant detail for our purposes.
7
b Output: A,
b ?
b
Algorithm 3. R ECOVER, Input:B, Pb0 (u), R,
b A (u) be a diagonal matrix whose ith entry is
1. Let D
1
2
bi )
Pb0 (R
?1/2
.
b = BR
bD
b A (u)?1/2 .
2. Let A
b=
3. Estimate C = E[yy T ] by taking O((kAk2 + k?k2 )4 n2 ?2 ) samples and let C
b=C
b?A
bA
bT
4. Let ?
b
b
5. Return A, ?
1
N
PN
i=1
yi yiT .
main goal: learning a noisy linear transformation Ax + ? with unknown Gaussian noise. We first
use a theorem from [5] to show that properties for finding one local maxima is satisfied.
Also, for notational convenience we set di = 2DA (u0 )?2
i,i and let dmin and dmax denote the minimum and maximum
values
(bounds
on
these
and
their
ratio follow from Claim 2.9). Using this
Pn
notation P ? (u) = i=1 di (uT Ri? )4 .
?
Theorem 5.1 ([5]). When ? < dmin /10dmax n2 , the function P ? (u) is (3 n?, ?, P ? (u)? 2 /100)
Locally Improvable and (?, dmin ? 2 /100n) Locally Approximable. Moreover, the local maxima of
the function is exactly {?Ri? }.
We then observe that given enough samples, the empirical mean Pb0 (u) is close to P ? (u). For
concentration we require every degree four term zi zj zk zl has variance at most Z.
Claim 5.2. Z = O(d2min ?min (A)8 k?k42 + d2min ).
0
, suppose columns of R0 =
Lemma 5.3. Given 2N samples y1 , y2 , ..., yN , y10 , y20 , ..., yN
?1
1/2
?
B ADA (u0 ) are close to the corresponding columns of R , with high probability the function
Pb0 (u) is O(dmax n1/2 + n2 (N/Z log n)?1/2 ) close to the true function P ? (u).
The other properties required by Theorem 4.6 are also satisfied:
Lemma 5.4. For any ku ? u0 k2 ? r, |P ? (u) ? P ? (u0 )| ? 5dmax n1/2 r. All local maxima of P ?
has attraction radius Rad ? dmin /100dmax .
Applying Theorem 4.6 we obtain the following Lemma (the parameters are chosen so that all properties required are satisfied):
Lemma 5.5. Let ? 0 = ?((dmin /dmax )2 ), ? = min{?n?1/2 , ?((dmin /dmax )4 n?3.5 )}, then the
procedure R ECOVER (f, ?, dmin ? 2 /100n , ? 0 , dmin ? 02 /100n) finds vectors v1 , v2 , ..., vn , so that
there is a permutation matrix ? and ki ? {?1} and for all i: kvi ? (R?Diag(ki ))?i k2 ? ?.
b = [v1 , v2 , ..., vn ] we can use Algorithm 3 to find A and ?:
After obtaining R
b such that there is permutation matrix ? and ki ? {?1} with
Theorem 5.6. Given a matrix R
?
b
b such that kA
b ? A?Diag(ki )kF ?
kRi ? ki (R ?)i k2 ? ? for all i, Algorithm 3 returns matrix A
2 3/2
2 3/2
O(? kAk2 n /?min (A)). If ? ? O(/ kAk2 n ?min (A)) ? min{1/ kAk2 , 1}, we also have
b ? ?kF ? .
k?
Recall that the diagonal matrix DA (u) is unknown
Pn (since it depends on A), but if we are given
R? (or an approximation) and since P ? (u) = i=1 di (uT Ri? )4 , we can recover the matrix DA (u)
approximately from computing P ? (Ri? ). Then given DA (u), we can recover A and ? and this
completes the analysis of our algorithm.
Conclusions
ICA is a vast field with many successful techniques. Most rely on heuristic nonlinear optimization.
An exciting question is: can we give a rigorous analysis of those techniques as well, just as we
did for local search on cumulants? A rigorous analysis of deep learning ?say, an algorithm that
provably learns the parameters of an RBM?is another problem that is wide open, and a plausible
special case involves subtle variations on the problem we considered here.
8
References
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1994. 1, 1.1
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2
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3,982 | 4,604 | Minimizing Uncertainty in Pipelines?
Nilesh Dalvi
Facebook, Inc.
[email protected]
Aditya Parameswaran
Stanford University
[email protected]
Vibhor Rastogi
Google, Inc.
[email protected]
Abstract
In this paper, we consider the problem of debugging large pipelines by human
labeling. We represent the execution of a pipeline using a directed acyclic graph
of AND and OR nodes, where each node represents a data item produced by some
operator in the pipeline. We assume that each operator assigns a confidence to
each of its output data. We want to reduce the uncertainty in the output by issuing
queries to a human, where a query consists of checking if a given data item is
correct. In this paper, we consider the problem of asking the optimal set of queries
to minimize the resulting output uncertainty. We perform a detailed evaluation of
the complexity of the problem for various classes of graphs. We give efficient
algorithms for the problem for trees, and show that, for a general dag, the problem
is intractable.
1
Introduction
In this paper, we consider the problem of debugging pipelines consisting of a set of data processing
operators. There is a growing interest in building various web-scale automatic information extraction
pipelines [9, 10, 14, 7], with operators such as clustering, extraction, classification, and deduplication. The operators are often based on machine learned models, and they associate confidences with
the data items they produce. At the end, we want to resolve the uncertainties of the final output
tuples, i.e., figure out which of them are correct and which are incorrect.
.5Given a fixed labeling budget, we can only inspect a subset of the output tuples. However, the output uncertainties
are highly correlated since different tuples share their lineage. Thus, inspecting a tuple also gives us information
about the correctness of other tuples. In this paper, we
consider the following interesting and non-trivial problem
: given a budget of k tuples, choose the k tuples to inspect
that minimize the total uncertainty in the output. We will
formalize the notion of a data pipeline and uncertainty in
Section 2. Here, we illustrate the problem using an example.
Figure 1: Pipeline Example
Example 1.1. Consider a simple hypothetical pipeline for extracting computer scientists from the
Web that consists of two operators: a classifier that takes a webpage and determines if it is a page
about computer science, and a name extractor that extracts names from a given webpage. Fig. 1
shows an execution of this pipeline. There are two webpages, w1 and w2 , output by the classifier.
The extractor extracts entities e1 and e2 from w1 and e3 ,e4 and e5 from w2 . Each operator also gives
a confidence with its output. In Fig. 1, the classifier attaches a probability of 0.9 and 0.8 to pages
w1 and w2 . Similarly, the extractor attaches a probability to each of the extractions e1 to e5 . The
probability that an operator attaches to a tuple is conditioned on the correctness of its input. Thus,
the final probability of e1 is 0.8 ? 0.9 = 0.72. Similarly, the final probabilities of e2 to e5 are 0.45,
0.8, 0.8 and 0.48 respectively. Note that the uncertainties are correlated, e.g., e3 and e4 are either
both correct or both incorrect. We want to choose k tuples to inspect that minimize the total output
uncertainty.
Classified Pages
Extractions
? This
w1
w2
0.9
0.8
0.8
0.5
1
1
0.6
e1
e2
e3
e4
e5
work was partly done when the authors were employed at Yahoo! Research.
1
Graph
T REE(2)
BEST-1
O(n)
T REE
DAG(2, ?) or
DAG(?)
DAG(2, ?)
DAG(?)
DAG
O(n)
O(n3 )
O(n3 )
PP-Hard
INCR
BEST-K
O(n) or
OPEN (Weakly PTIME)
O(log n)+O(n log n) preprocessing
2-approximate? : O(n log n)
O(n)
OPEN (O(nk+1 ))
PP-hard (Probabilistic Polynomial)
PP-Hard
Hard to Approximate
Hard to Approximate
PP-hard, Hard to Approximate
PP-Hard, Hard to Approximate
PP-Hard, Hard to Approximate
PP-Hard, Hard to Approximate
Table 1: Summary of Results; ? Twice the number of queries to achieve same objective as optimal
If all the data items were independent, we would have queried the most uncertain items, i.e. having
probability closest to 1/2. However, in presence of correlations between the output tuples, the problem becomes non-trivial. For instance, let us revisit the first example with k = 1, i.e., we can inspect
one tuple. Of the 5 output tuples, e5 is the most uncertain, since its probability 0.48 is closest to
1/2. However, one might argue that e3 (or e4 ) is more informative item to query, since the extractor
has a full confidence on e3 . Thus, e3 is correct iff w2 is correct (i.e. the classifier was correct on
w2 ). Resolving e3 completely resolves the uncertainty in w2 , which, in turn, completely resolves
the uncertainty in e4 and reduces the uncertainty in e5 . The argument holds even when the extractor
confidence in e3 is less than 1 but still very high. In general, one can also query intermediate nodes
in addition to the output tuples, and choosing the best node is non-trivial.
In this paper, we consider the general setting of a data pipeline given by a directed acyclic graph
that can capture both the motivating scenarios. We define a measure of total uncertainty of the final
output based on how close the probabilities are to either 0 or 1. We give efficient algorithms to
find the set of data items to query that minimizes the total uncertainty of the output, both under
interactive and batch settings.
1.1
Related Work
Our problem is an instance of active learning [27, 13, 12, 17, 2, 15, 5, 4, 3] since our goal is to infer
probability values of the nodes being true in the DAG, by asking for tags of example nodes. The
metric that we use is similar to the square loss metric. However, our problem has salient differences.
Unlike traditional active learning where we want to learn the underlying probabilistic model from
iid samples, in our problem, we already know the underlying model and want to gain information
about non-iid items with known correlations. This makes our setting novel and interesting.
Our DAG structure is a special case of Bayesian networks [6]. A lot is known about general bayesnet inference [21]. For instance, MAP inference given evidence is NPPP -complete [24] (approximate inference is NP-complete [1]), inferring whether the probability of a set of variables taking a
certain values given evidence about others is > 0 is NP-complete [8], is > t is PP-complete [22],
while finding its values is #P-complete [26]. However, these results do not apply to our problem setting. In our setting, we are given a set of non-iid items whose correlatations are given by a Bayesian
network with known structure and probabilities. We want to choose a subset of items, conditioned
on which, the uncertinty of the remaining items is minimized.
Our work is closely related to the field of active diagnosis [28, 19, 20], where the goal is to infer
the state of unknown nodes in a network by selecting suitable ?test probes?. From this field, the
most closely related work is that by Krause and Guestrin [19], which considers minimization of
uncertainty in a Bayesian network. In that work, the goal is to identify a subset of variables in a
graphical model that would minimize the joint uncertainty of a target set of variables. Their primary
result is a proof of submodularity under suitable independence assumptions on the graphical model
which is then used to derive an approximation algorithm to pick variables. In our problem setting
submodularity does not hold, and hence the techniques do not apply. On the other hand, since our
graphical model has a specific AND/OR structure, we are able to concretely study the complexity
of the algorithms. Our work is also related to the work on graph search [23], where the goal is to
identify hidden nodes while asking questions to humans. Since the target applications are different,
the underlying model in that work is less general.
2
Problem Statement
Execution Graph: Let G be a directed acyclic graph (dag), where each node n in G has a label from
the set {?, ?} and a probability p(n). We call such a graph a probabilistic and-or dag. We denote
2
the class of such graphs as DAG. We represent the results of an execution of a pipeline of operators
using a probabilistic and-or dag.
The semantics of G ? DAG is as follows. Each node in G represents a data item. The parents of a
node n, i.e. the set of nodes having an outgoing edge to n, denote the set of data items which were
input to the instance of the operator that produced n. We use parent(n) to denote the parents of n.
The probability p(n) denotes the probability that the data item n is correct conditioned on parent(n)
being correct. If n has label ?, then it requires all the parents to be correct. If n has label ?, it
requires at least one parent to be correct. We further assume that, conditioned on the parents being
correct, nodes are correct independently.
To state the semantics formally, we associate a set of independent Boolean random variables X(n)
for each node n in G with probability p(n). We also associate another set of random variables
Y (n), which denotes whether V
the result at node n is correct (unconditionally). For a ? node, Y (n)
is defined as: Y (n) = X(n) ? m?parent(n) Y (m). For a ? node, Y (n) is defined as: Y (n) = X(n) ?
W
m?parent(n) Y (m).
When G is a tree, i.e., all nodes have a single parent, the labels of nodes do not have any effect,
since Y (n) is the same for both ? and ? nodes. In this case, we simply treat G as an unlabeled
tree. For instance, Figure 1 denotes the (unlabeled) tree for the pipeline given in Example 1.1. Thus
probabilistic and-or dags provide a powerful formalism to capture data pipelines in practice such as
the one in Example 1.1.
Output Uncertainty: Let L denote the set of leaves of G, which represent the final output of the
pipeline. We want all the final probabilities of L to be close to either 0 or 1, as the closer the
probability to 1/2, the more uncertain the correctness of the given node is. Let f (p) denote some
measure of uncertainty of a random variable as a function of its probability p. Then, we define the
total output uncertainty of the DAG as
I=
? f (Pr(Y (n)))
(1)
n?L
Our results continue to hold when different n ? L are weighted differently, i.e., we use a weighted
version of Eq. (1). We describe this simple extension in the extended technical report [11].
Now, our goal is to query a set of nodes Q that minimize the expected total output uncertainty
conditioned on observing Q. We define this as follows. Let Q = {l1 , l2 , ? ? ? , lk } be a set of nodes.
Given v = {v1 , ? ? ? , vk } ? {0, 1}k , we use Q = v to denote the event Y (li ) = vi for each i. Then, define
I(Q) =
?
Pr(Q = v) ? f (Pr(Y (n) | Q = v))
(2)
n?L
v?{0,1}k
The most basic version of our problem is following.
Problem 1 (Best-1). Given a G ? DAG, find the node q that minimizes the expected uncertainty
I({q}).
A more challenging question is the following:
Problem 2 (Best-k). Given a G ? DAG, find the set of nodes Q of size k that minimizes I(Q).
In addition to this, we also consider the incremental version of the problem defined as follows.
Suppose we have already issued a set of queries Q0 and obtained a vector v0 of their correctness
values. Given a new set of queries, we define the conditioned uncertainty as I(Q | Q0 = v0 ) =
?v Pr(Q = v | Q0 = v0 ) ?n?L f (Pr(Y (n) | Q = v ? Q0 = v0 )). We also pose the following question:
Problem 3 (Incr). Given a G ? DAG, and a set of already issued queries Q0 with answer v0 ,
find the best node q to query next that minimizes I({q} | Q0 = v0 ).
In this work, we use the uncertainty metric given by
f (p) = p(1 ? p)
(3)
Thus, f (p) is minimized when p is either 0 or 1, and is maximum at p = 1/2. Note that f (p) =
1/4 ? (1/2 ? p)2 . Hence, minimizing f (p) is equivalent to maximizing the squares of differences
of probabilities with 1/2. We call this the L2 metric. There are other reasonable choices for the
uncertainty metric, e.g. L1 or entropy. The actual choice of uncertainty metrics is not important for
our application. In the technical report [11], we show that using any of these different metrics, the
resulting solutions are ?similar? to each other.
Our uncertainty objective function can be shown to satisfy some desirable properties, such as:
3
Theorem 2.1 (Information Never Hurts). For any sets of queries Q1 , Q2 , I(Q1 ) ? I(Q1 ? Q2 )
Thus, expected uncertainty cannot increase with more queries. Further, the objective function I is
neither sub-modular nor super-modular. These results continue to hold when f is replaced with
other metrics (Sec. 6). Lastly, for the rest of the paper, we will assume that the query nodes Q are
selected from only among the leaves of G. This is only to simplify the presentation. There is a
simple reduction of the general problem to this problem, where we attach a new leaf node to every
internal node, and set their probabilities to 1. Thus, for any internal node, we can equivalently query
the corresponding leaf node (we will need to use the weighted form of the Eq. (1), described in the
extended technical report [11], to ensure that new leaf nodes have weight 0 in the objective function.)
3
Summary of main results
We first define class of probabilistic and-or dags. Let DAG(?) and DAG(?) denote the subclasses of
DAG where all the node labels are ? and ? respectively. Let DAG(2, ?) and DAG(2, ?) denote the
subclasses where the dags are further restricted to depth 2. (We define the depth to be the number of
nodes in the longest root to leaf directed path in the dag.) Similarly, we define the class T REE where
the dag is restricted to a tree, and T REE(d), consisting of depth-d trees. For trees, since each node
has a single parent, the labels of the nodes do not matter.
We start by defining relationships between expressibility of each of these classes. Given any
D1 , D2 ? DAG, we say that D1 ? D2 if they have the same number of leaves, and define the same
joint probability distribution on the set of their leaves. Given two classes of dags C1 and C2 , we say
C1 ? C2 if for all D1 ? C1 , there is a D2 ? C2 s.t. D2 is polynomial in the size of D1 and D1 ? D2 .
Theorem 3.1. The following relationships exist between different classes:
T REE(2) ? T REE ? DAG(2, ?) = DAG(?) ? DAG(2, ?) ? DAG(?) ? DAG
Table 1 shows the complexity of the three problems as defined in the previous section, for different
classes of graphs. The parameter n is the number of nodes in the graph. While the problems are
tractable, and in fact efficient, for trees, they become hard for general dags. Here, PP denotes the
complexity class of probabilistic polynomial time algorithms. Unless P = NP, there are no PTIME
algorithms for PP-hard problems. Further, for some of the problems, we can show that they cannot
1??
be approximated within a factor of 2n for any positive constant ? in PTIME.
4
Best-1 Problem
We start with the most basic problem: given a probabilistic DAG G, find the node to query that minimizes the resulting uncertainty. We first provide PTIME algorithms for T REE(2), T REE, DAG(?),
and DAG(2, ?) (Recall that as we saw earlier, DAG(2, ?) subsumes DAG(?).) Subsequently, we
show that finding the best node to query is intractable for DAG(?) of depth greater than 2, and is
thus intractable for DAG V
as well. For T REE and DAG(?), the expression for Y (n) can be rewritten
as the following: Y (n) = m?anc(n) X(m), where anc(n) denotes the set of ancestors of n, i.e., those
nodes that have a directed path to n, including n itself. This ?unrolled? formulation will allow us to
compute the probabilities Y (x) = 1 easily.
4.1
T REE(2)
Consider a simple tree graph G with root r, having p(r) = pr , and having children l1 , ? ? ? , ln with
p(li ) = pi . Given a node x, let ex denote the event Y (x) = 1, and ex denote the event that Y (x) = 0.
We want to find the leaf q that minimizes I({q}), where:
I({q}) =
? Pr(eq ) f (Pr(el | eq )) + Pr(eq ) f (Pr(el | eq ))
(4)
l?L
By a slight abuse of notation, we will use I(q) to denote the quantity I({q}). It is easy to see the
following (let l 6= q):
Pr(el | eq ) = pl ,
Pr(eq ) = pr pq ,
Pr(el | eq ) = pr pl (1 ? pq )/(1 ? pr pq )
Substituting these expressions back in Eq. (4), and assuming f (p) = p(1 ? p), we get the following:
I(q) =
?
pr pq pl (1 ? pl ) + pr pl (1 ? pq )(1 ? pr pl (1 ? pq )/(1 ? pr pq ))
l?L,l6=q
4
We observe that it is of the form
F0 (pq , pr ) + F1 (pq , pr ) ? pl + F2 (pq , pr ) ? p2l
l
(5)
l
where F0 , F1 , F2 are small rational polynomials over pr and pq . This immediately gives us a linear
time algorithm to pick the best q. We first compute ?l pl and ?l p2l , and then compute the objective
function for all q in linear time.
Now we consider the case when G is any general tree with the set of leaves L. Recall that ex is the
event that denotes Y (x) = 1. Denote the probability Pr(ex ) by Px . Thus, Px is the product of p(y)
over all nodes y that are the ancestors of x (including x itself). Given nodes x and y, let lca(x, y)
denote the least common ancestor of x and y. Our objective is to find q ? L that minimizes Eq. (4).
The following is immediate:
Pr(eq ) = Pq
Pr(el | eq ) =
Pl
Plca(l,q)
Pr(el | eq ) =
Pl (1 ? Pq /Plca(l,q) )
1 ? Pq
However, if we directly plug this in Eq.(4), we don?t get a simple form analogous to Eq.(5). Instead,
we group all the leaves into equivalence classes based on their lowest common ancestor with q as
shown in Fig. 2.
Let a1 , ? ? ? , ad be the set of ancestors of q. Consider all leaves in the set Li such that their lowest common ancestor with q is ai . Given a node x, let S(x) denote the sum of Pl2 over all leaves
l reachable from x. If we sum Eq. (4) over all leaves in Li , we get the following expression:
ad
?(S(ai ) ? S(ai?1 ))
(Pq + Pa2i ? 2Pq Pai )
+ ? Pl
Pa2i (1 ? Pq )
l?Li
Define ?1 (ai ) = S(ai ) ? S(ai?1 ) and ?2 (ai ) = (S(ai ) ?
1?2P
S(ai?1 )) P2 ai . We can write the above expression as:
ai
?
a2
a1
q
L1
Pq
1
?1 (ai ) ?
?2 (ai ) + ? Pl
1 ? Pq
1 ? Pq
l?Li
L2
Ld
Figure 2: Equivalence Classes of Leaves
Summing these terms over all the ancestors of q, we
get
Pq
1
I(q) = ?
? ?1 (a) ? 1 ? Pq ? ?2 (a) + ? Pl
1 ? Pq a?anc(q)
l?L
a?anc(q)
4.2
T REE
Our main observation is that we can compute I(q) for all leaves together in time linear in the size
of G. First, using a single top-down dynamic programming over the tree, we can compute Px for all
nodes x. Next, using a single bottom-up dynamic programming over G, we can compute S(x) for all
nodes x. In the third step, we compute ?1 (x) and ?2 (x) for all nodes in the tree. In the fourth step, we
compute ?a?anc(x) ?i (x) for all nodes in the graph using another top-down dynamic programming.
Finally, we scan all the leaves and compute the objective function using the above expression. Each
of the 5 steps runs in time linear in the size of the graph. Thus, we have
Theorem 4.1. Given a tree G with n nodes, we can compute the node q that minimizes I(q) is time
O(n).
4.3
DAG(2, ?)
We now consider DAG(2, ?). As before, we want to find the best node q that minimizes I(q) as
given by Eq. (4). However, the expressions for probabilities Pr(eq ) and Pr(el | eq ) are more complex
for DAG(2, ?). First, note that Pl , i.e., the probability that Pr(Y (l) = 1) is computed as follows:
Pl = p(l) ? 1 ? ?x?parent(l) (1 ? p(x)) . The probability that at least one of the shared ancestors
of l and q are true is: Pl,q = 1 ? ?x?parent(l)?parent(q) (1 ? p(x)). And the probability that one of the
unique ancestors of l is true is: Pl\q = 1 ? ?x?parent(l)\parent(q) (1 ? p(x)) Then, the following are
5
immediate:
Pr(eq ) = Pq
p(l) ? p(q) ? (Pl,q + (1 ? Pl,q ) ? Pl\q ? Pq\l )
Pr(eq | el ) =
Pl
Pq ? (1 ? p(l)) + p(l) ? p(q) ? (1 ? Pl,q ) ? (1 ? Pl\q ) ? Pq\l
Pr(eq | el ) =
1 ? Pl
Note that Pl , Pl,q , Pl\q can be computed for one l, q pair in time O(n) and thus for all l, q in time
O(n3 ). Subsequently, finding the best candidate node would require O(n2 ) time, giving us an overall
O(n3 ) algorithm to find the best node.
Theorem 4.2. Given G ? DAG(2, ?) with n nodes, we can compute q that minimizes I(q) is time
O(n3 ).
Since every DAG(?) can be converted into to one in DAG(2, ?) in O(n3 ) (see [11]), we get:
Theorem 4.3. Given G ? DAG(?) with n nodes, we can compute q that minimizes I(q) is time
O(n3 ).
4.4
DAG(?)
Theorem 4.4 (Hardness of Best-1 for DAG(?)). The best-1 problem for DAG(?) is PP-Hard.
We use a reduction from the decision version of the #P-Hard monotone-partitioned-2-DNF problem [25]. The proof can be found in the extended technical report [11]. Thus, incremental and best-k
problems for DAG(?) are PP-Hard as well. As a corollary from Theorem 3.1 we have:
Theorem 4.5 (Hardness of Best-1 for DAG). The best-1 problem for DAG is PP-Hard.
This result immediately shows us that the incremental and best-k problems for DAG are PP-Hard.
However, we can actually prove a stronger result for DAG, i.e., that they are hard to approximate. We
use a weakly parsimonious reduction from the #P-Hard monotone-CNF problem. Note that unlike
the partitioned-2-DNF problem (used for the reduction above), which admits a FPRAS (Fully Polynomial Randomized Approximation Scheme) [18], monotone-CNF is known to be hard to approximate [26]. In our proof, we use the fact that repeated applications of an approximation algorithm
for best-1 for DAG would lead to an approximation algorithm for monotone-CNF, which is known
to be hard to approximate. This result is shown in the extended version [11].
Theorem 4.6 (Inapproximability for DAG). The best-1 problem for DAG is hard to approximate.
5
Incremental Node Selection
In this section, we consider the problem of picking the next best node to query after a set of nodes
Q0 have already been queried. We let vector v0 reflect their correctness values. We next pick a
leaf node q that minimizes I({q} | Q0 = v0 ). Again, by slightly abusing notation, we will write the
expression simply as I(q | Q0 = v0 ).
In this section, we first consider T REE(2) and T REE. Recall from the previous section that the incremental problem is intractable for DAG(?). Here, we prove that incremental picking is intractable
for DAG(?) itself.
5.1
T REE
We want to extend our analysis of Sec. 4 by replacing Pr(ex ) by Pr(ex | Q0 = v0 ) and Pr(ex | ey )
by Pr(ex | ey ? Q0 = v0 ). We will show that, conditioned on Q0 = v0 , the resulting probability
distribution of the leaves can again be represented using a tree. The new tree is constructed as
follows.
Given Q0 = v0 , apply a sequence of transformations to G ? T REE, one for each q0 ? Q0 . Suppose
the value of q0 = 1. Then, for each ancestor a of q0 including itself, set p(a) = 1. If q0 = 0, then for
1?Pq /Pa
each ancestor a including itself, change its p(a) to p(a) 1?P0q . Let all other probabilities remain
0
the same.
Theorem 5.1. Let G0 be the tree as defined above. Then, I(q | Q0 = v0 ) on G is equal to I(q) on G0 .
Thus, after each query, we can incorporate the new evidence by updating the probabilities of all the
nodes along the path from the query node to the root. Thus, finding the next best node to query can
still be computed in linear time.
6
5.2
T REE(2)
For G ? T REE(2), the above algorithm results in the following tree transformation. If a leaf q is
queried, and the result is 1, then p(r) and p(q) are set to 1. If the result is 0, p(q) is set to 0 and p(r)
pr (1?pq )
is set to 1?p
.
r pq
Instead of using Eq. (5) to compute the next best in linear time, we can devise a more efficient
scheme. Suppose we are given all the leaf probabilities in sorted order (or if we sort them initially).
Then, we can subsequently compute the leaf q that minimizes Eq. (5) in O(log n) time: Consider the
rational polynomials F0 , F1 and F2 . For a fixed pr , ?l pl , and ?l p2l , this expression can be treated as
a rational polynomial in a single variable pq . If we take the derivative, the numerator is a quartic in
pq . Thus, it can have at most four roots. We can find the roots of a quartic using Ferrari?s approach
in constant time [16]. Using 4 binary searches, we can find the two pq closest to each of these roots
(giving us 8 candidates for pq , plus two more which are the smallest and the largest pq ), and evaluate
I(q) for each of those 10 candidates. Thus, finding the best q takes O(log n) time.
Now, given each new evidence (i.e., the answer to each subsequent query), we can update the pr
probability and the sum ?l p2l in constant time. Given the new polynomial, we can find the new set
of roots, and using the same technique as above, find the next best q in O(log n) time.
Theorem 5.2. If the p values of the leaf nodes are provided in sorted order, then, for a Depth-2 tree,
the next best node to query can be computed in O(log n).
5.3
DAG(?)
For DAG(?), while we can pick the best-1 node in O(n3 ) time, we have the surprising result that
the problem of picking subsequent nodes become intractable. The intuition is that unlike trees, after
conditioning on a query node, the resulting distribution can no longer be represented using another
dag. In particular, we show that given a set S of queried nodes, the problem of finding the next best
node is intractable in the size of S. We use a reduction from the monotone-2-CNF problem.
Theorem 5.3 (PP-Hardness of Incr. for DAG(?)). The incremental problem in DAG(?) is PP-Hard.
Our reduction, shown in in the extended technical report [11], is a weakly parsimonious reduction
involving monotone-2-CNF, which is known to be hard to approximate, thus we have the following
result:
Theorem 5.4 (Inapproximability for DAG(?)). The Incremental problem for DAG(?) is hard to
approximate.
The above result, along with Theorem 3.1, implies that DAG(2, ?) is also PP-Hard.
6
Best-K
In this section, we consider the problem of picking the best k nodes to minimize uncertainty.
Krause et al. [19] give a log n approximation algorithm for a similar problem under the conditions
of super-modularity: super-modularity states that the marginal decrease in uncertainty when adding
a single query node to an existing set of query nodes decreases as the set becomes larger. Here,
we show that super-modularity property does not hold in our setting, even for the simplest case of
1??
T REE. In fact, for DAG(2, ?), the problem is hard to approximate within a factor of O(2n ) for
any ? > 0. We show that T REE(2) admits a weakly-polynomial exact algorithm and a polynomial
approximation algorithm. For general trees, we leave the complexity problem open.
Picking Nodes Greedily: First, we show that picking greedily can be arbitrarily bad. Consider a tree with root having p(r) = 1/2. There are 2n leaves, half with p = 1 and rest with p = 1/2.
If we pick any leaf node with p = 1, the expected uncertainty is n/8. If we pick a node with p = 1/2,
the expected uncertainty is 25n/16 ? 4/16. Thus, if we sort nodes by their expected uncertainty, all
the p = 1 nodes appear before all the p = 1/2 nodes. Consider the problem of picking the best n
nodes. If we pick greedily based on their expected uncertainty, we pick all the p = 1 nodes. However, all of them are perfectly correlated. Thus, expected uncertainty after querying all p = 1 nodes
is still n/8. On the other hand, if we pick a single p = 1 node, and n ? 1 nodes with p = 1/2, the
resulting uncertainty is a constant. Thus, picking nodes greedily can be O(n) worse than the optimal.
Counter-example for super-modularity: Next we show an example from a graph in DAG(2, ?)
where super-modularity does not hold. Consider a G ? DAG(2, ?) having two nodes u and v on
7
the top layer and three nodes a, b, and c in the bottom layer. Labels of all nodes are ?. Node
u has an edge to a and b, while v has an edge to b and c. Let Pr(u) = 1/2, Pr(v) = 1/2, and
Pr(a) = Pr(b) = Pr(c) = 1. Now consider the expected uncertainty Ic at node c. Super-modularity
condition implies that Ic ({b, a}) ? Ic ({b}) ? Ic ({a}) ? Ic ({}) (since marginal decrease in expected
uncertainty of c on picking an additional node a should be less for set {} compared to {b}). We
show that this is violated. First note that Pr(Y (c)|Y (a)) is same as Pr(Y (c)) (since Y (a) does not
affect Y (v) and Y (c)). Thus expected uncertainty at c is unaffected by conditioning on a alone, and
thus Ic ({a}) = Ic ({}). On the other hand, if Y (b) = 0 and Y (a) = 1 then Y (c) = 0 (since Y (a) = 1
implies Y (u) = 1 which together with Y (b) = 0 implies Y (v) = 0 and Y (c) = 0). This can be used
to show that conditioned on Y (b), expected uncertainty in c drops when conditioning on Y (a). Thus
the term Ic ({b, a}) ? Ic ({b}) is negative, while we showed that Ic ({a}) ? Ic ({}) is 0. This violates
the super-modularity condition.
The above example actually shows that super-modularity is violated on DAG(?) for any choice of
metric f in computing expected uncertainty I, as long as f is monotonic decreasing away from 1/2.
When f (p) = p(1 ? p), we can show that super-modularity is violated even for trees as stated in the
proposition below.
Proposition 6.1. Let f (p) = p(1 ? p) be the metric used in computing expected uncertainty I. Then
there exists a tree T ? T REE(d) such that for leaf nodes a , b, and c in T the following holds:
Ic ({b, a}) ? Ic ({b}) < Ic ({a}) ? Ic ({}).
6.1
T REE(2)
We now consider the Best-k problem for T REE(2). As in Section 4, assume the root r with p(r) to be
pr , while the leaves L = {l1 , . . . , ln } have p(li ) = pi . Let B = ?l?L p2 (l). Given a set Q ? L, define
P(Q) = ? p(l)
l?Q
S1 (Q) =
? p(l)(1 ? p(l))
S2 (Q) =
l?Q
? p2 (l)
l?Q
Lemma 6.2. The best set Q of size k is one that minimizes:
r
S2 (Q)) (1?pr1?p
)/P(Q)+pr
I 0 (Q) = ?S1 (Q) + (B ?
(The details of this computation is shown in the extended technical report.) It is easy to check that
that the first term is minimized with Q consists of nodes with p(l) closest to 1/2, and the second
term is minimized with nodes with p(l) closest to 1. Intuitively, the first term prefers nodes that are
as uncertain as possible, while the second term prefers nodes that reveal as much about the root as
possible. This immediately gives us a 2-approximation in the number of queries : by picking at most
2k nodes, k closest to 1/2 and k closest to 1, we can do at least as well as the optimal solution for
best-k.
Exact weakly-polynomial time algorithm: Note also that as k increases, P(Q) ? 0, and the second
term vanishes. This also makes intuitive sense, since the second term prefers nodes that reveal more
about the root, and once we use sufficiently many nodes to infer the correctness of the root, we do
not get any gain from asking additional questions. Thus, we set a constant c? , depending on the pi ,
such that if k < c? , we consider all possible choices of k queries, and if k ? c? , we may simply pick
the k largest pi , because the second term would be very small. We describe this algorithm along
with the proof in the extended technical report [11].
6.2
DAG(?):
Theorem 6.3 (PP-Hardness of Incr. for DAG(?)). The best-k problem in DAG(?) is PP-Hard.
The proof can be found in the extended technical report [11]. Our reduction is a weakly parsimonious reduction involving monotone-partitioned-2-CNF, which is known to be hard to approximate,
thus we have the following result:
Theorem 6.4 (Inapproximability for DAG(?)). The best-k problem for DAG(?) is hard to approximate.
7
Conclusion
In this work, we performed a detailed complexity analysis for the problem of finding optimal set
of query nodes for various classes of graphs. We showed that for trees, most of the problems are
tractable, and in fact quite efficient. For general dags, they become hard to even approximate. We
leave open the complexity of the best-k problem for trees.
8
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3,983 | 4,605 | Learning as MAP Inference in Discrete
Graphical Models
James Petterson
NICTA/ANU
Canberra, Australia
[email protected]
Xianghang Liu
NICTA/UNSW
Sydney, Australia
[email protected]
Tiberio S. Caetano
NICTA/ANU/University of Sydney
Canberra and Sydney, Australia
[email protected]
Abstract
We present a new formulation for binary classification. Instead of relying
on convex losses and regularizers such as in SVMs, logistic regression and
boosting, or instead non-convex but continuous formulations such as those
encountered in neural networks and deep belief networks, our framework
entails a non-convex but discrete formulation, where estimation amounts to
finding a MAP configuration in a graphical model whose potential functions
are low-dimensional discrete surrogates for the misclassification loss. We
argue that such a discrete formulation can naturally account for a number
of issues that are typically encountered in either the convex or the continuous non-convex approaches, or both. By reducing the learning problem
to a MAP inference problem, we can immediately translate the guarantees
available for many inference settings to the learning problem itself. We
empirically demonstrate in a number of experiments that this approach
is promising in dealing with issues such as severe label noise, while still
having global optimality guarantees. Due to the discrete nature of the formulation, it also allows for direct regularization through cardinality-based
penalties, such as the `0 pseudo-norm, thus providing the ability to perform
feature selection and trade-off interpretability and predictability in a principled manner. We also outline a number of open problems arising from
the formulation.
1
Introduction
A large fraction of the machine learning community is concerned itself with the formulation
of a learning problem as a single, well-defined optimization problem. This is the case for
many popular techniques, including those associated with margin or likelihood-based estimators, such as SVMs, logistic regression, boosting, CRFs and deep belief networks. Among
these optimization-based frameworks for learning, two paradigms stand out: the one based
on convex formulations (such as SVMs) and the one based on non-convex formulations (such
as deep belief networks). The main argument in favor of convex formulations is that we
can effectively decouple modeling from optimization, what has substantial theoretical and
practical benefits. In particular, it is of great value in terms of reproducibility, modularity
and ease of use. Coming from the other end, the main argument for non-convexity is that
a convex formulation very often fails to capture fundamental properties of a real problem
(e.g. see [1, 2] for examples of some fundamental limitations of convex loss functions).
1
The motivation for this paper starts from the observation that the above tension is not really
between convexity and non-convexity, but between convexity and continuous non-convexity.
Historically, the optimization-based approach to machine learning has been virtually a synonym of continuous optimization. Estimation in continuous parameter spaces in some cases
allows for closed-form solutions (such as in least-squares regression), or if not we can resort
to computing gradients (for smooth continuous functions) or subgradients (for non-smooth
continuous functions) which give us a generic tool for finding a local optimum of an arbitrary
continuous function (global optimum if the continuous function is convex). On the contrary,
unless P=NP there is no general tool to efficiently optimize discrete functions. We suspect
this is one of the reasons why machine learning has traditionally been formulated in terms
of continuous optimization: it is indeed convenient to compute gradients or subgradients
and delegate optimization to some off-the-shelf gradient-based algorithm.
The formulation we introduce in this paper is non-convex, but discrete rather than continuous. By being non-convex we will attempt at capturing some of the expressive power
of continuous non-convex formulations (such as robustness to labeling noise), and by being
discrete we will retain the ability of convex formulations to provide theoretical guarantees in
optimization. There are highly non-trivial classes of non-convex discrete functions defined
over exponentially large discrete spaces which can be optimized efficiently. This is, after all,
the main topic of combinatorial optimization. Discrete functions factored over cliques of
low-treewidth graphs can be optimized efficiently via dynamic programming [3]. Arbitrary
submodular functions can be minimized in polynomial time [4]. Particular submodular
functions can be optimized very efficiently using max-flow algorithms [5]. Discrete functions
defined over other particular classes of graphs also have polynomial-time algorithms (planar
graphs [6], perfect graphs [7]). And of course although many discrete optimization problems are NP-hard, several have efficient constant-factor approximations [8]. In addition to
all that, much progress has been done recently on developing tight LP relaxations for hard
combinatorial problems [9]. Although all these discrete approaches have been widely used
for solving inference problems in machine learning settings, we argue in this paper that they
should also be used to solve estimation problems, or learning per se.
The discrete approach does pose several new questions though, which we list at the end. Our
contribution is to outline the overall framework in terms of a few key ideas and assumptions,
as well as to empirically evaluate in real-world datasets particular model instances within
the framework. Although these instances are very simple, they already display important
desirable behavior that is missing in state-of-the-art estimators such as SVMs.
2
Desiderata
We want to rethink the problem of learning a linear binary classifier. In this section we
list the features that we would like a general-purpose learning machine for this problem to
possess. These features essentially guide the assumptions behind our framework.
Option to decouple modeling from optimization: As discussed in the introduction,
this is the great appeal of convex formulations, and we would like to retain it. Note however
that we want the option, not necessarily a mandate of always decoupling modeling from
optimization. We want to be able to please the user who is not an optimization expert or
doesn?t have the time or resources to refine the optimizer, by having the option of requesting
the learning machine to configure itself in a mode in which global optimization is guaranteed
and the runtime of optimization is precisely predictable. However we also want to please the
user who is an expert, and is willing to spend a lot of time in refining the optimizer, to achieve
the best possible results regardless of training time considerations. In our framework, we
have the option to explore the spectrum between simpler models in which we can generate
precise estimates of the runtime of the whole algorithm, and more complex models where
we can focus on boosted performance at the expense of runtime predictability or demand
for expert-exclusive fine-tuning skills.
Option of Simplicity: This point is related to the previous one, but it?s more general.
The complexity of a learning algorithm is a great barrier for its dissemination, even if it
promises exceptional results once properly implemented. Most users of machine learning
are not machine learning experts themselves, and for them in particular the cost of getting
2
a complex algorithm to work often outweighs the accuracy gains, especially if a reasonably
good solution can be obtained with a very simple algorithm. For instance, in our framework
the user has the option of reducing the learning algorithm to a series of matrix multiplications
and lookup operations, while having a precise estimate of the total runtime of the algorithm
and retaining good performance.
Robustness to label noise: SVMs are considered state-of-the-art estimators for binary
classifiers, as well as boosting and logistic regression. All these optimize convex loss functions. However, when label noise is present, convex loss functions inflict arbitrarily large
penalty on misclassifications because they are unbounded. In other words, in high label
noise settings these convex loss functions become poor proxies for the 0/1 loss (the loss we
really care about). This fundamental limitation of convex loss functions is well understood
theoretically [1]. The fact that the loss function of interest is itself discrete is indeed a hint
that maybe we should investigate discrete surrogates rather than continuous surrogates for
the 0/1 loss: optimizing discrete functions over continuous spaces is hard, but not necessarily
over discrete spaces. In our framework we directly address this issue.
Ability to achieve sparsity: Often we need to estimate sparse models. This can be for
several reasons, including interpretability (be able to tell which are the ?most important? features), efficiency (at prediction time we can only afford to use a limited number of features)
or, importantly, for purely statistical reasons (constraining the solution to low-dimensional
subspaces has a regularization effect). The standard convex approach uses `1 regularization.
However the assumptions required to make `1 -regularized models be actually good proxies
for the support cardinality function (`0 pseudo-norm) are very strong and in practice rarely
met [10]. In fact this has motivated an entire new line of work on structured sparsity, which
tries to further regularize the solution so as to obtain better statistical properties in high
dimensions [11, 12, 13]. This however comes at the price of more expensive optimization
algorithms. Ideally we would like to regularize with `0 directly; maybe this suggests the
possibility of exploring an inherently discrete formulation? In our approach we have the
ability to perform direct regularization via the `0 pseudo-norm, or other scale-invariant
regularizers.
Leverage the power of low-dimensional approximations: Machine learning folklore
has it that the Naive Bayes assumption (features conditionally independent given the class
label) often produces remarkably good classifiers. So a natural question is: is it really
necessary to work directly in the original high-dimensional space, such as SVMs do? A
key aspect of our framework is that we explicitly exploit the concept of composing a highdimensional model from low-dimensional pieces. However we go beyond the Naive Bayes
assumption by constructing graphs that model dependencies between variables. By varying
the properties of these graphs we can trade-off model complexity and optimization efficiency
in a straightforward manner.
3
Basic Setting
Much of current machine learning research studies estimators of the type
X
argmin
`(y n , f (xn ; ?)) + ??(?)
???
(1)
n
where {xn , y n } is a training set of inputs x ? X and outputs y ? Y, assumed sampled
independently from an unknown probability measure P on X ? Y. f : X ? Y is a member
of a given class of predictors parameterized by ?, ? is a continuous space such as a Hilbert
space, and ` as well as ? are continuous and convex functions of ?. ` is a loss function which
enforces a penalty whenever f (xn ) 6= y n , and therefore the first term in (1) measures the
total loss incurred by predictor f on the training sample {xn , y n } under parameterization ?.
? controls the complexity of ? so as to avoid overfitting, and ? trades-off the importance of a
good fit to the training set versus model parsimony, so that good generalization is hopefully
achieved.
Problem (1) is often called regularized empirical risk minimization, since the first term is
the risk (expected loss) under the empirical distribution of the training data, and the second
is a regularizer. This formulation is used for regression (Y continuous) as well as classification and structured prediction (Y discrete). Logistic Regression, Regularized Least-Squares
3
Regression, SVMs, CRFs, structured SVMs, Lasso, Group Lasso and a variety of other estimators are all instances of (1) for particular choices of `, f , ? and ?. The formulation in
(1) is a very general formulation for machine learning under the i.i.d. assumption.
In this paper we study problem (1) under the assumption that the parameter space ? is
discrete and finite, focusing on binary classification, when Y = {?1, 1}.
4
Formulation
Our formulation departs from the one in (1) in two ways. The first assumption is that both
the loss ` and the regularizer ? are additive on low-dimensional functions defined by a graph
G = (V, E), i.e.,
X
`c (y, fc (x; ?c ))
(2)
`(y, f (x; ?)) =
c?C
?(?) =
X
?c (?c )
(3)
c?C0
where C ? C0 is the set of maximal cliques in G. Note that (3) is standard: `1 and `2
norms for example are both additive on singletons (in which case C0 = V ). The arguably
strong assumption here is (2). C is the set of parts where each part c is, in principle, an
arbitrary subset of {1, . . . , D}, where D is the dimensionality of the parameterization, i.e.,
? = (?1 , . . . , ?D ). `c is a low-dimensional discrete surrogate for `, and fc is a low-dimensional
predictor, both to be defined below. Note that in general two parameter subvectors ?ci and
?cj are not independent since the cliques ci and cj can overlap. Indeed, one of the key
reasons sustaining the power of this formulation is that all ?c are coupled either directly or
indirectly through the connected graph G = (V, E).
The second assumption is that ? is discrete and therefore the vector ? = (?1 , . . . , ?D ) is
discrete in the sense that ?i is only allowed to take on finitely many values, including the
value 0 (this will be important when we discuss regularization). For simplicity of exposition
let?s assume that the number of discrete values (bins) for each ?i is the same: B. B can be
potentially quite large, for example it can be in the hundreds.
Random Projections. An instance x above in reality is not the raw feature vector but
instead a random projection of it into a space of the same or higher dimension, i.e., we
effectively apply X = RX 0 where X 0 is the original data matrix, R is a random matrix with
entries drawn from N (0, 1) and X is the new data matrix. This often provides improved
performance for our model due to the spreading of higher-order dependencies over lowerorder cliques (when mapping to a higher dimensional space) and also is motivated from a
theoretical argument (section 6). In what follows x is the feature vector after the projection.
Low-Dimensional Predictor. We will assume a standard linear predictor of the kind
fc (x; ?) = argmax y hxc , ?c i = sign hxc , ?c i
(4)
y?{?1,1}
In other words, we have a linear classifier that only considers the features in clique c.1
Low-Dimensional Discrete Surrogates for the 0/1 loss The low-dimensional discrete
surrogate for the 0/1 loss is simply defined as the 0/1 loss incurred by predictor fc :
`c (y; fc (x; ?)) = (1 ? yfc (x; ?))/2
(5)
A key observation now is that fc and therefore `c can be computed in O(B k ) by full enumeration over the B k instantiations of ?c , where k is the size of clique c. In other words,
the 0/1 loss constrained to the discretized subspace defined by clique c can be exactly and
efficiently computed (for small cliques).
Regularization. One critical technical issue is that linear predictors of the kind
argmaxy h?(x, y), ?i are insensitive to scalings of ? [14]. Therefore, the loss ` will be such
that `(y, f (x; ??)) = `(y, f (x; ?)) for ? 6= 0. This means that any regularizer that depends
1
For notational simplicity we assume an offset parameter is already included in ?c and a corresponding entry of 1 is appended to the vector xc .
4
on scale (such as `1 and `2 norms) is effectively meaningless since the minimization in (1)
will drive ?(?) to 0 (as this doesn?t affect the loss). In other words, in such discrete setting
we need a scale-invariant regularizer, such as the `0 pseudo-norm. Note that `0 is trivial to
implement in this formulation, as we have enforced that the zero value must be included in
the set of B values attainable by each ?i :
X
?(?) = `0 (?) =
1?i 6=0
(6)
i
In addition, since this regularizer is additive on singletons ?i , it comes for free the fact that
it does not contribute to the complexity of inference in the graphical model (i.e., it is a
unary potential), which is a convenient property.
P Nothing prevents us however from having
group regularizers, for example of the form c?C0 ?c 1?c 6=0 . Again, we can trade-off model
simplicity and optimization efficiency by controlling the size of the maximal clique in C0 .
Final optimization Problem. After compiling the low-dimensional discrete proxies for
the 0/1 loss (the functions lc ) and incorporating our regularizer, we can assemble the following optimization problem
argmin
???
N
XX
`c (y n , fc (xn ; ?c )) +
c?C n=1
|
D
X
i=1
{z
:=?N ?c (?c )
}
?1? 6=0
| {zi }
(7)
:=???i (?i )
which is a relaxation of (1) under all the above assumptions. The critical observation
now is that (7) is a MAP inference problem in a discrete graphical model with clique set
C, high-order clique potentials ?c (?c ) and unary potentials ?i (?i ) [15]. Therefore we can
resort to the vast literature on inference in graphical models to find exact or approximate
solutions for (7). For example, if G = (V, E) is a tree, then (7) can be solved exactly
and efficiently using a dynamic programming algorithm that only requires matrix-vector
multiplications in the (min, +) semiring, in addition to elementary lookup operations [3].
For more general graphs the problem (7) can become NP-hard, but even in that case there
are several principled approaches that often find excellent solutions, such as those based
on linear programming relaxations [9] for tightly outer-bounding the marginal polytope
[16]. In the experimental section we explore several options for constructing G, from simply
generating a random chain (where MAP inference can be solved efficiently by dynamic
programming) to generating dense random graphs (where MAP inference requires a more
sophisticated approach such as an LP relaxation).
5
Related Work
The most closely related work we found is a recent paper by Potetz [17]. In a similar spirit
to our approach, it also addresses the problem of estimating linear binary classifiers in a
discrete formulation. However, instead of composing low-dimensional discrete surrogates of
the 0/1 loss as we do, it instead uses a fully connected factor graph and performs inference by
estimating the mean of the max-marginals rather than MAP. Inference is approached using
message-passing, which for the fully connected graph reduces to an intractable knapsack
problem. In order to obtain a tractable model, the problem is then relaxed to a linear
multiple choice knapsack problem, which can be solved efficiently. All the experiments
though are performed on very low-dimensional datasets2 and it is unclear how this approach
would scale to high dimensionality while keeping a fully connected graph.
6
Analysis
Here we sketch arguments supporting the assumptions driving our formulation. Obtaining a
rigorous theoretical analysis is left as an open problem for future research. Our assumptions
involve three approximations of the problem of 0/1 loss minimization. First, the discretization of the parameter space. Second, the computation of low-dimensional proxies for the
0/1 loss rather than attacking the 0/1 loss directly in the resulting discrete space. Finally,
the use of a graph G = (V, E) which in general will be sparse, i.e., not fully connected. We
now discuss each of these assumptions.
2
Seven datasets with dimensionalities 7,9,10,11,14,15 and 61. See [17].
5
6.1 Discretization of the parameter space
The explicit enforcement of a finite number of possible values for each parameter may seem at
first a strong assumption. However, a key observation here is that we are restricting ourselves
to linear predictors, which basically means that, for any sample, small perturbations of a
random hyperplane will with high probability induce at most small changes in the 0/1 loss.
Therefore there are good reasons to believe that indeed, for linear predictors, increasing
binning has a diminishing returns behavior and after only a moderate amount of bins no
much improvement can be obtained. This assumption is also used in [17].
6.2 Low-dimensional proxies for the 0/1 loss
This assumption can be justified using recent results stating that the margin is well-preserved
under random projections to low-dimensional subspaces [18, 19]. For instance, Theorem 6 in
[19] shows that the margin is preserved with high probability for embeddings with dimension
only logarithmic on the sample size (a result similar in spirit to the Johnson-Lindenstrauss
Lemma [20]). Since the (soft)margin upper bounds the 0/1 loss, this should also be preserved
with at least equivalent guarantees.
6.3 Graph sparsity
This is apparently the strongest assumption. In our formulation, we impose conditional
independence assumptions on the set of random variables used as features. There are two
main observations. The first is that in real high-dimensional data the existence of (approximate) conditional independences is more of a rule than an exception. This is directly
related to the fact that usually high-dimensional data inhabit low-dimensional manifolds
or subspaces. In our case, we have a graph with the nodes representing different features,
and this can be seen as a patching of low-dimensional subspaces, where each subspace is
defined by one of the cliques in the graph. We do not address in this work how to optimally
determine a subgraph, leaving that as an open problem in this framework. Rather, we show
that even with random subgraphs, and in particular subgraphs as simple as chains, we can
obtain models that have high accuracy and remarkable robustness to high degrees of label
noise. The second observation is that nothing prevents us from using quite dense graphs
and seeking approximate rather than exact MAP inference, say through LP relaxations [9].
Indeed we illustrate this possibility in the experimental section below.
7
Experiments
Settings. To evaluate our method (DISCRETE) for binary classification problems, we
apply it to real-world datasets and compared it to linear Support Vector Machines (SVM),
which are a state-of-the-art estimator for linear classifiers. We note that although both use
linear predictors, the model classes are not identical: since we use discretization, the set of
hyperplanes our estimator will optimize over is strictly smaller. We run these algorithms on
publicly available datasets from the UCI machine learning repository [21]. See Table 1 for
the details of these datasets. For both algorithms, the only hyperparameter is the tradeoff between the loss and the regularization term. We run 5-fold cross validation for both
methods to select the optimal hyperparameters. The number of bins used for discretization
may affect the accuracy of DISCRETE. For the experiments, we fix it to 11, since for larger
values there was negligible improvement (which supports our argument from section 6.1).
Robustness to Label Noise. In the first experiment, we test the robustness of different
methods to increasing label noise. We first flip the labels of the training data with increasing
probability from 0 to 0.4 and then run these algorithms on the noisy training data. The plots
of the classification accuracy at each noise level are shown in Figure 1. For DISCRETE, we
used as the graph G a random chain, i.e., the simplest possible option for a connected graph.
In this case, optimization is straightforward via a Viterbi algorithm: a sequence of matrixvector multiplications in the (min, +) semiring with trivial bookkeeping and subsequential
lookup, which will run in O(B 2 D) since we have B states per variable and D variables. To
assess the effect of randomization, we run on 20 random chains and plot both the average
and the standard error obtained. The impact of randomization seems negligible. From
Figure 1, DISCRETE demonstrates classification accuracy only slightly inferior to SVM in
6
(a) GISETTE
(b) MNIST 5 vs 6
(c) A2A
(d) USPS 8 vs 9
(e) ISOLET
(f) ACOUSTIC
Figure 1: Comparison of the Discrete Method and Linear SVM
the noiseless regime (i.e., when the hinge loss is a good proxy for the 0/1 loss). However,
as soon as a significant amount of label noise is present, SVM degrades substantially while
DISCRETE remains remarkably stable, delivering high accuracy even after flipping labels
with 40% probability. We believe these are significant results given the truly elementary
nature of the optimization procedure: the method is simple, fast and the runtime can be
predicted with high accuracy since there is a determined number of operations; 2(D ? 1)
messages are passed, each with worst-case runtime of O(B 2 ) determined by the matrixvector multiplication. Note in particular how this differs from continuous optimization
settings in which the analysis is in terms of rate of convergence rather than the precise
number of discrete operations performed. It is also interesting to observe that for different
values of the cross-validation parameter our algorithm runs in precisely the same amount of
time, while for SVMs convergence will be much slower for small scalings of the regularizer
since the relative importance of the non-differentiable hinge loss over the strongly convex
quadratic term increases. This experiment shows that even if we have the simplest setting
of our formulation (random chains, which comes with very fast and exact MAP inference)
we can still obtain results that are close or similar to those obtained by the state-of-the-art
linear SVM classifier in the noiseless case, and superior for high levels of label noise.
Evaluation without Noise. As seen in Figure 1, in the noiseless (or small noise) regime
SVM is often slightly superior to our random chain model. A natural question to ask is
therefore how would more complex graph topologies perform. Here we run experiments
on two other types of graphs: a random 2-chain (i.e. a random junction tree with cliques
{i, i + 1, i + 2}) and a random k-regular graph, where k is set to be such that the resulting
graph has 10% of the possible edges. For the 2-chain, the optimization algorithm is exact
inference via (min, +) message-passing, just as the Viterbi algorithm, but now applied to
a larger clique, which increases the memory and runtime cost by O(B). For the random
graph, we obtain a more complex topology in which exact inference is intractable. In our
experiments we used the approximate inference algorithm from [22], which solves optimally
and efficiently an LP relaxation via the alternating direction method of multipliers, ADMM
[23].
7
# Train
# Test
# Features
Table 1: Datasets used for the experiments in Figure 1
GISETTE MNIST A2A USPS ISOLET ACOUSTIC
6000
10205
2265
950
480
19705
1000
1134
30296
237
120
78823
5000
784
123
256
617
50
Table 2: Error rates of different methods for binary classification, without label noise. In
this setting, the hinge loss used by SVM is an excellent proxy for the 0/1 loss. Yet, the
proposed variants (top 3 rows) are still competitive in most datasets.
GISETTE MNIST A2A USPS ISOLET ACOUSTIC
random chain
89.23
93.79
82.55 97.51
100
76.01
random 2-chain 89
94.47
82.65 97.78
100
76.55
random graph
88.6
94.89
83.17 97.44
100
74.80
SVM
97.7
96.47
83.88 98.4
100
76.01
8
Extensions and Open Problems
Clearly the results in this paper are only a first step in the direction proposed. Several
questions arise from this formulation.
Theory. In section 6 we only sketched the reasons why we pursued the assumptions laid
out in this paper. We did not present any rigorous quantitative arguments analyzing the
limitations of our formulation. This is left as an open problem. However we believe section
6 does point to the key ideas that will ultimately underly a quantitative theory.
Extension to multi-class and structured prediction. In this work we only study
binary classification problems. The extension to multi-class and structured prediction, as
well as other learning settings is an open problem.
Adaptive binning. When discretizing the parameters, we used a fixed number of bins.
This can be made more elaborate through the use of adaptive binning techniques that are
dependent on the information content of each variable.
Informative graph construction. We only explored randomly generated graphs. The
problem of selecting a graph topology in an informative way is highly relevant and is left
open. For example B-matching can be used to generate an informative regular graph [24].
This problem is essentially a manifold learning problem and there are several ways it could
be approached. Existing work on supervised manifold learning is very relevant here.
Nonparametric extension. We considered only linear parametric models. It would be
interesting to consider nonparametric models, where the discretization occurs at the level of
parameters associated with each training instance (as in the dual formulation of SVMs).
9
Conclusion
We presented a discrete formulation for learning linear binary classifiers. Parameters associated with features of the linear model are discretized into bins, and low-dimensional discrete
surrogates of the 0/1 loss restricted to small groups of features are constructed. This results
in a data structure that can be seen as a graphical model, where regularized risk minimization can be performed via MAP inference. We sketch theoretical arguments supporting the
assumptions underlying our proposal and present empirical evidence that very simple, easily
and quickly trainable models estimated with such a procedure can deliver results that are
often comparable to those obtained by linear SVMs for noiseless scenarios, and superior
under moderate to severe label noise.
Acknowledgements
We thank E. Bonilla, A. Defazio, D. Garc??a-Garc??a, S. Gould, J. McAuley, S. Nowozin,
M. Reid, S. Sanner and B. Williamson for discussions. NICTA is funded by the Australian
Government as represented by the Department of Broadband, Communications and the
Digital Economy and the Australian Research Council through the ICT Centre of Excellence
program.
8
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9
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3,984 | 4,606 | Truly Nonparametric Online Variational Inference
for Hierarchical Dirichlet Processes
Michael Bryant and Erik B. Sudderth
Department of Computer Science, Brown University, Providence, RI
[email protected], [email protected]
Abstract
Variational methods provide a computationally scalable alternative to Monte Carlo
methods for large-scale, Bayesian nonparametric learning. In practice, however,
conventional batch and online variational methods quickly become trapped in local optima. In this paper, we consider a nonparametric topic model based on the
hierarchical Dirichlet process (HDP), and develop a novel online variational inference algorithm based on split-merge topic updates. We derive a simpler and
faster variational approximation of the HDP, and show that by intelligently splitting and merging components of the variational posterior, we can achieve substantially better predictions of test data than conventional online and batch variational
algorithms. For streaming analysis of large datasets where batch analysis is infeasible, we show that our split-merge updates better capture the nonparametric
properties of the underlying model, allowing continual learning of new topics.
1 Introduction
Bayesian nonparametric methods provide an increasingly important framework for unsupervised
learning from structured data. For example, the hierarchical Dirichlet process (HDP) [1] provides
a general approach to joint clustering of grouped data, and leads to effective nonparametric topic
models. While nonparametric methods are best motivated by their potential to capture the details
of large datasets, practical applications have been limited by the poor computational scaling of
conventional Monte Carlo learning algorithms.
Mean field variational methods provide an alternative, optimization-based framework for nonparametric learning [2, 3]. Aiming at larger-scale applications, recent work [4] has extended online
variational methods [5] for the parametric, latent Dirichlet allocation (LDA) topic model [6] to the
HDP. While this online approach can produce reasonable models of large data streams, we show that
the variational posteriors of existing algorithms often converge to poor local optima. Multiple runs
are usually necessary to show robust performance, reducing the desired computational gains. Furthermore, by applying a fixed truncation to the number of posterior topics or clusters, conventional
variational methods limit the ability of purportedly nonparametric models to fully adapt to the data.
In this paper, we propose novel split-merge moves for online variational inference for the HDP
(oHDP) which result in much better predictive performance. We validate our approach on two corpora, one with millions of documents. We also propose an alternative, direct assignment HDP representation which is faster and more accurate than the Chinese restaurant franchise representation used
in prior work [4]. Additionally, the inclusion of split-merge moves during posterior inference allows
us to dynamically vary the truncation level throughout learning. While conservative truncations can
be theoretically justifed for batch analysis of fixed-size datasets [2], our data-driven adaptation of
the trunction level is far better suited to large-scale analysis of streaming data.
Split-merge proposals have been previously investigated for Monte Carlo analysis of nonparametric
models [7, 8, 9]. They have also been used for maximum likelihood and variational analysis of
1
?j
?
zjn
wjn
Nj
?
?
?
D
?k
?
Figure 1: Directed graphical representation of a hierarchical Dirichlet process topic model, in which an unbounded collection of topics ?k model the Nj words in each of D documents. Topics occur with frequency ?j
in document j, and with frequency ? across the full corpus.
parametric models [10, 11, 12, 13]. These deterministic algorithms validate split-merge proposals
by evaluating a batch objective on the entire dataset, an approach which is unexplored for nonparametric models and infeasible for online learning. We instead optimize the variational objective via
stochastic gradient ascent, and split or merge based on only a noisy estimate of the variational lower
bound. Over time, these local decisions lead to global estimates of the number of topics present in a
given corpus. We review the HDP and conventional variational methods in Sec. 2, develop our novel
split-merge procedure in Sec. 3, and evaluate on various document corpora in Sec. 4.
2 Variational Inference for Bayesian Nonparametric Models
2.1 Hierarchical Dirichlet processes
The HDP is a hierarchical nonparametric prior for grouped mixed-membership data. In its simplest
form, it consists of a top-level DP and a collection of D bottom-level DPs (indexed by j) which
share the top-level DP as their base measure:
G0 ? DP(?H),
Gj ? DP(?G0 ),
j = 1, . . . , D.
Here, H is a base measure on some parameter space, and ? > 0, ? > 0 are concentration parameters.
Using a stick-breaking representation [1] of the global measure G0 , the HDP can be expressed as
?
?
X
X
?jk ??k .
?k ? ? k ,
Gj =
G0 =
k=1
k=1
The global weights ? are drawn from a stick-breaking distribution ? ? GEM(?), and atoms are
independently drawn as ?k ? H. Each Gj shares atoms with the global measure G, and the lowerlevel weights are drawn ?j ? DP(??). For this direct assignment representation, the k indices for
each Gj index directly into the global set of atoms. To complete the definition of the general HDP,
parameters ?jn ? Gj are then drawn for each observation n in group j, and observations are drawn
xjn ? F (?jn ) for some likelihood family F . Note that ?jn = ?zjn for some discrete indicator zjn .
In this paper we focus on an application of the HDP to modeling document corpora. The topics ?k ? Dirichlet(?) are distributions on a vocabulary of W words. The global topic weights,
? ? GEM(?), are still drawn from a stick-breaking prior. For each document j, document-specific
topic frequencies are drawn ?j ? DP(??). Then for each word index n in document j, a topic
indicator is drawn zjn ? Categorical(?j ), and finally a word is drawn wjn ? Categorical(?zjn ).
2.2 Batch Variational Inference for the HDP
We use variational inference [14] to approximate the posterior of the latent variables (?, ?, ?, z) ?
the topics, global topic weights, document-specific topic weights, and topic indicators, respectively
? with a tractable distribution q, indexed by a set of free variational parameters. Appealing to mean
field methods, our variational distribution is fully factorized, and is of the form
q(?, ?, ?, z | ?, ?, ?) = q(?)
?
Y
q(?k | ?k )
D
Y
j=1
k=1
2
q(?j | ?j )
Nj
Y
n=1
q(zjn | ?jn ),
(1)
where D is the number of documents in the corpus and Nj is the number of words in document j.
Individual distributions are selected from appropriate exponential families:
q(?) = ?? ? (?)
q(?k | ?k ) = Dirichlet(?k | ?k )
q(?j | ?j ) = Dirichlet(?j | ?j )
q(zjn ) = Categorical(zjn | ?jn )
where ?? ? (?) denotes a degenerate distribution at the point ? ? .1 In our update derivations below,
we use ?jw to denote the shared ?jn for all word tokens in document j of type w.
Selection of an appropriate truncation strategy is crucial to the accuracy of variational methods for
nonparametric models. Here, we truncate the topic indicator distributions by fixing q(zjn = k) = 0
for k > K, where K is a threshold which varies dynamically in our later algorithms. With this
assumption, the topic distributions with indices greater than K are conditionally independent of
the observed data; we may thus ignore them and tractably update the remaining parameters with
respect to the true, infinite model. A similar truncation has been previously used in the context of an
otherwise more complex collapsed variational method [3]. Desirably, this truncation is nested such
that increasing K always gives potentially improved bounds, but does not require the computation
of infinite sums, as in [16]. In contrast, approximations based on truncations of the stick-breaking
topic frequency prior [2, 4] are not nested, and their artifactual placement of extra mass on the final
topic K is less suitable for our split-merge online variational inference.
Via standard convexity arguments [14], we lower bound the marginal log likelihood of the observed
data using the expected complete-data log likelihood and the entropy of the variational distribution,
def
L(q) = Eq [log p(?, ?, ?, z, w | ?, ?, ?)] ? Eq [log q(?, ?, z | ?, ?, ?)]
= Eq [log p(w | z, ?)] + Eq [log p(z | ?)] + Eq [log p(? | ??)] + Eq [log p(? | ?)]
+ Eq [log p(? | ?)] ? Eq [log q(z | ?)] ? Eq [log q(? | ?)] ? Eq [log q(? | ?)]
D n
X
=
Eq [log p(wj | zj , ?)] + Eq [log p(zj | ?j )] + Eq [log p(?j | ??)] ? Eq [log q(zj | ?j )]
j=1
o
1
Eq [log p(? | ?)] + Eq [log p(? | ?)] ? Eq [log q(? | ?)] , (2)
D
and maximize this quantity by coordinate ascent on the variational parameters. The expectations
are with respect to the variational distribution. Each expectation is dependent on only a subset
of the variational parameters; we leave off particular subscripts for notational clarity. Note that
the expansion of the variational lower bound in (2) contains all terms inside a summation over
documents. This is the key observation that allowed [5] to develop an online inference algorithm
for LDA. A full expansion of the variational objective is given in the supplemental material. Taking
derivatives of L(q) with respect to each of the variational parameters yields the following updates:
? Eq [log q(?j | ?j )] +
?jwk ? exp {Eq [log ?kw ] + Eq [log ?jk ]}
PW
?jk ? ??k + w=1 nw(j) ?jwk
PD
?kw ? ? + j=1 nw(j) ?jwk ,
(3)
(4)
(5)
Here, nw(j) is the number of times word w appears in document j. The expectations in (3) are
P
P
Eq [log ?jk ] = ?(?jk ) ? ?( i ?ji ),
Eq [log ?kw ] = ?(?kw ) ? ?( i ?ki ),
where ?(x) is the digamma function, the first derivative of the log of the gamma function.
In evaluating our objective, we represent ? ? as a (K + 1)-dim. vector containing the probabilities
of the first K topics, and the total mass of all other topics. While ? ? cannot be optimized in closed
form, it can be updated via gradient-based methods; we use a variant of L-BFGS. Drawing a parallel between variational inference and the expectation maximization (EM) algorithm, we label the
document-specific updates of (?j , ?j ) the E-step, and the corpus-wide updates of (?, ?) the M-step.
1
We expect ? to have small posterior variance in large datasets, and using a point estimate ? ? simplifies
variational derivations for our direct assignment formulation. As empirically explored for the HDP-PCFG [15],
updates to the global topic weights have much less predictive impact than improvements to topic distributions.
3
2.3 Online Variational Inference
Batch variational inference requires a full pass through the data at each iteration, making it computationally infeasible for large datasets and impossible for streaming data. To remedy this, we adapt
and improve recent work on online variational inference algorithms [4, 5].
The form of the lower bound in (2), as a scaled expectation with respect to thePdocument collection,
an online learning algorithm. Given a learning rate ?t satisfying ?
t=0 ?t = ? and
P? suggests
2
?
<
?,
we
can
optimize
the
variational
objective
stochastically.
Each
update
begins by
t=0 t
sampling a ?mini-batch? of documents S, of size |S|. After updating the mini-batch of documentspecific parameters (?j , ?j ) by iterating (3,4), we update the corpus-wide parameters as
?kw ,
?kw ? (1 ? ?t )?kw + ?t ?
? ? ? (1 ? ?t )? ? + ?t ??k ,
k
(6)
(7)
k
? kw is a set of sufficient statistics for topic k, computed from a noisy estimate of (5):
where ?
X
? kw = ? + D
?
nw(j) ?jwk .
|S|
(8)
j?S
The candidate topic weights ?? are found via gradient-based optimization on S. The resulting inference algorithm is similar to conventional batch methods, but is applicable to streaming, big data.
3 Split-Merge Updates for Online Variational Inference
We develop a data-driven split-merge algorithm for online variational inference for the HDP, referred to as oHDP-SM. The algorithm dynamically expands and contracts the truncation level K by
splitting and merging topics during specialized moves which are interleaved with standard online
variational updates. The resulting model truly allows the number of topics to grow with the data. As
such, we do not have to employ the technique of [4, 3] and other truncated variational approaches of
setting K above the expected number of topics and relying on the inference to infer a smaller number. Instead, we initialize with small K and let the inference discover new topics as it progresses,
similar to the approach used in [17]. One can see how this property would be desirable in an online
setting, as documents seen after many inference steps may still create new topics.
3.1 Split: Creation of New Topics
|S|
Given the result of analyzing one mini-batch q ? = (?j , ?j )j=1 , ?, ? ? , and the corresponding
value of the lower bound L(q ? ), we consider splitting topic k into two topics k ? , k ?? .2 The split
procedure proceeds as follows: (1) initialize all variational posteriors to break symmetry between
the new topics, using information from the data; (2) refine the new variational posteriors using a
restricted iteration; (3) accept or reject the split via the change in variational objective value.
Initialize new variational posteriors To break symmetry, we initialize the new topic posteriors
(?k? , ?k?? ), and topic weights (?k?? , ?k??? ), using sufficient statistics from the previous iteration:
?k ,
?k?? = ?t ?
? ??? = ?t ??k .
?k? = (1 ? ?t )?k ,
?k?? = (1 ? ?t )?k? ,
k
Intuitively, we expect the sufficient statistics to provide insight into how a topic was actually used
|S|
during the E-step. The minibatch-specific parameters {?j , ?j }j=1 are then initialized as follows,
?jwk? = ?k ?jwk ,
?jk? = ?k ?jk ,
?jwk?? = (1 ? ?k )?jwk ,
?jk?? = (1 ? ?k )?jk ,
with the weights defined as ?k = ?k? /(?k? + ?k?? ).
2
Technically, we replace topic k with topic k? and add k?? as a new topic. In practice, we found that the
order of topics in the global stick-breaking distribution had little effect on overall algorithm performance.
4
Algorithm 1 Restricted iteration
1: initialize (?? , ?? ) for ? ? {k ? , k ?? }
2: for j ? S do
3:
initialize (?j , ?j ) for ? ? {k ? , k ?? }
4:
while not converged do
5:
update (?j , ?j ) for ? ? {k ? , k ?? } using (3, 4)
6:
end while
7:
update (?? , ?? ) for ? ? {k ? , k ?? } using (6, 7)
8: end for
Restricted iteration After initializing the variational parameters for the new topics, we update
them through a restricted iteration of online variational inference. The restricted iteration consists
of restricted analogues to both the E-step and the M-step, where all parameters except those for the
new topics are held constant. This procedure is similar to, and inspired by, the ?partial E-step? for
split-merge EM [10] and restricted Gibbs updates for split-merge MCMC methods [7].
All values of ?jw? and ?j? , ? ?
/ {k ? , k ?? }, remain unchanged. It is important to note that even though
these values are not updated, they are still used in the calculations for both the variational expectation
of ?j and the normalization of ?. In particular,
exp {Eq [log ?k? w ] + Eq [log ?jk? ]}
?jwk? = P
,
??T exp {Eq [log ??w ] + Eq [log ?j? ]}
P
Eq [log ?jk? ] = ?(?jk? ) ? ?( k?T ?jk ),
where T is the original set of topics, minus k, plus k ? and k ?? . The expected log word probabilities
Eq [log ?k? w ] and Eq [log ?k?? w ] are computed using the newly updated ? values.
Evaluate Split Quality Let ?split for minibatch S be ? as defined above, but with ?jwk replaced
?
be defined similarly.
by the ?jwk? and ?jwk?? learned in the restricted E-step. Let ? split , ?split and ?split
|S|
?
split(k)
Now we have a new model state q
= (?split , ?split )j=1 , ?split , ?split . We calculate L q split(k) ,
and if L q split(k) > L(q ? ), we update the new model state q ? ? q split(k) , accepting the split. If
L q split(k) < L(q ? ), then we go back and test another split, until all splits are tested. In practice we
limit the maximum number of allowed splits each iteration to a small constant. If we wish to allow
the model to expand the number of topics more quickly, we can increase this number. Finally, it is
important to note that all aspects of the split procedure are driven by the data ? the new topics are
initialized using data-driven proposals, refined by re-running the variational E-step, and accepted
based on an unbiased estimate of the change in the variational objective.
3.2 Merge: Removal of Redundant Topics
Consider a candidate merge of two topics, k ? and k ?? , into a new topic k. For batch variational methods, it is straightforward to determine whether such a merge will increase or decrease the variational
objective by combining all parameters for all documents,
?jwk = ?jwk? + ?jwk?? ,
?jk = ?jk? + ?jk?? ,
?k = ?k? + ?k?? ,
?k = ?k? + ?k?? ,
and computing the difference in the variational objective before and after the merge. Because many
terms cancel, computing this bound change is fairly computationally inexpensive, but it can still be
computationally infeasible to consider all pairs of topics for large K. Instead, we identify potential
merge candidates by looking at the sample covariance of the ?j vectors across the corpus (or minibatch). Topics with positive covariance above a certain threshold have the quantitative effects of
their merge evaluated. Intuitively, if there are two copies of a topic or a topic is split into two pieces,
they should tend to be used together, and therefore have positive covariance. For consistency in
? ??
notation, we call the model state with topics k ? and k ?? merged q merge(k ,k ) .
Combining this merge procedure with the previous split proposals leads to the online variational
method of Algorithm 2. In an online setting, we can only compute unbiased noisy estimates of the
true difference in the variational objective; split or merge moves that increase the expected variational objective are not guaranteed to do so for the objective evaluated over the entire corpus. The
5
Algorithm 2 Online variational inference for the HDP + split-merge
1: initialize (?, ? ? )
2: for t = 1, 2, . . . do
3:
for j ? minibatch S do
4:
initialize (?j , ?j )
5:
while not converged do
6:
update (?j , ?j ) using (3, 4)
7:
end while
8:
end for
9:
for pairs of topics {k ? , k ?? } ? K ? K with Cov(?jk? , ?jk?? ) > 0 do
? ??
10:
if L q merge(k ,k ) > L(q) then
? ??
11:
q ? q merge(k ,k )
12:
end if
13:
end for
14:
update (?, ? ? ) using (6, 7)
15:
for k = 1, 2, . . . , K do
16:
compute L q split(k) via restricted iteration
17:
if L q split(k) > L(q) then
18:
q ? q split(k)
19:
end if
20:
end for
21: end for
uncertainty associated with the online method can be mitigated to some extent by using large minibatches. Confidence intervals for the expected change in the variational objective can be computed,
and might be useful in a more sophisticated acceptance rule. Note that our usage of a nested family
of variational bounds is key to the accuracy and stability of our split-merge acceptance rules.
4 Experimental Results
To demonstrate the effectiveness of our split-merge moves, we compare three algorithms: batch
variational inference (bHDP), online variational inference without split-merge (oHDP), and online
variational inference with split-merge (oHDP-SM). On the NIPS corpus we also compare these
three methods to collapsed Gibbs sampling (CGS) and the CRF-style oHDP model (oHDP-CRF)
proposed by [4].3 We test the models on one synthetic and two real datasets:
Bars A 20-topic bars dataset of the type introduced in [18], where topics can be viewed as bars on
a 10 ? 10 grid. The vocabulary size is 100, with a training set of 2000 documents and a test set of
200 documents, 250 words per document.
NIPS 1,740 documents from the Neural Information Processing Systems conference proceedings,
1988-2000. The vocabulary size is 13,649, and there are 2.3 million tokens in total. We randomly
divide the corpus into a 1,392-document training set and a 348-document test set.
New York Times The New York Times Annotated Corpus4 consists of over 1.8 million articles
appearing in the New York Times between 1987 and 2007. The vocabulary is pruned to 8,000 words.
We hold out a randomly selected subset of 5,000 test documents, and use the remainder for training.
All values of K given for oHDP-SM models are initial values ? the actual truncation levels fluctuate
during inference. While the truncation level K is different from the actual number of topics assigned
non-negligible mass, the split-merge model tends to merge away unused topics, so these numbers
are usually fairly close. Hyperparameters are initialized to consistent values across all algorithms
and datasets, and learned via Newton-Raphson updates (or in the case of CGS, resampled). We use
a constant learning rate across all online algorithms. As suggested by [4], we set ?t = (? + t)??
where ? = 1, ? = 0.5. Empirically, we found that slower learning rates could result in greatly
reduced performance, across all models and datasets.
3
For CGS we use the code available at http://www.gatsby.ucl.ac.uk/?ywteh/research/npbayes/npbayesr21.tgz, and for oHDP-CRF we use the code at http://www.cs.princeton.edu/?chongw/software/onlinehdp.tar.gz.
4
http://www.ldc.upenn.edu/Catalog/catalogEntry.jsp?catalogId=LDC2008T19
6
To compare algorithm performance, we use per-word heldout likelihood, similarly to the metrics
of [3, 19, 4]. We randomly split each test document in Dtest into 80%-20% pieces, wj1 and wj2 .
? as the variational expectation of the topics from training, we learn ?
Then, using ?
?j on wj1 and
Q
P
approximate the probability of wj2 as w?wj2 k ?
?jk ??kw . The overall test metric is then
P
P
P
?jk ??kw
k?
w?wj2 log
j?D test
P
E=
j?D test |wj2 |
4.1 Bars
For the bars data, we initialize eight oHDP-SM runs with K = {2, 5, 10, 20, 40, 50, 80, 100}, eight
runs of oHDP with K = 20, and eight runs with K = 50. As seen in Figure 2(a), the oHDP
algorithm converges to local optima, while the oHDP-SM runs all converge to the global optimum.
More importantly, all split-merge methods converge to the correct number of topics, while oHDP
uses either too few or too many topics. Note that the data-driven split-merge procedure allows
splitting and merging of topics to mostly cease once the inference has converged (Figure 2(d)).
4.2 NIPS
We compare oHDP-SM, oHDP, bHDP, oHDP-CRF, and CGS in Figure 2. Shown are two runs of
oHDP-SM with K = {100, 300}, two runs each of oHDP and bHDP with K = {300, 1000}, and
one run each of oHDP-CRF and CGS with K = 300. All the runs displayed are the best runs from a
larger sample of trials. Since oHDP and bHDP will use only a subset of topics under the truncation,
setting K much higher results in comparable numbers of topics as oHDP-SM. We set |S| = 200 for
the online algorithms, and run all methods for approximately 40 hours of CPU time.
The non split-merge methods reach poor local optima relatively quickly, while the split-merge algorithms continue to improve. Notably, both oHDP-CRF and CGS perform much worse than any
of our methods. It appears that the CRF model performs very poorly for small datasets, and CGS
reaches a mode quickly but does not mix between modes. Even though the split-merge algorithms
improve in part by adding topics, they are using their topics much more effectively (Figure 2(h)).
We speculate that for the NIPS corpus especially, the reason that models achieve better predictive
likelihoods with more topics is due to the bursty properties of text data [20]. Figure 3 illustrates the
topic refinement and specialization which occurs in successful split proposals.
4.3 New York Times
As batch variational methods and samplers are not feasible for such a large dataset, we compare
two runs of oHDP with K = {300, 500} to a run of oHDP-SM with K = 200 initial topics. We
also use a larger minibatch size of |S| = 10,000; split-merge acceptance decisions can sometimes
be unstable with overly small minibatches. Figure 2(c) shows an inherent problem with oHDP for
very large datasets ? when truncated to K = 500, the algorithms uses all of its available topics and
exhibits overfitting. For the oHDP-SM, however, predictive likelihood improves over a substantially
longer period and overfitting is greatly reduced.
5 Discussion
We have developed a novel split-merge online variational algorithm for the hierarchical DP. This
approach leads to more accurate models and better predictive performance, as well as a model that
is able to adapt the number of topics more freely than conventional approximations based on fixed
truncations. Our moves are similar in spirit to split-merge samplers, but by evaluating their quality
stochastically using streaming data, we can rapidly adapt model structure to large-scale datasets.
While many papers have tried to improve conventional mean field methods via higher-order variational expansions [21], local optima can make the resulting algorithms compare unfavorably to
Monte Carlo methods [3]. Here we pursue the complementary goal of more robust, scalable optimization of simple variational objectives. Generalization of our approach to more complex hierarchies of DPs, or basic DP mixtures, is feasible. We believe similar online learning methods will
prove effective for the combinatorial structures of other Bayesian nonparametric models.
Acknowledgments We thank Dae Il Kim for his assistance with the experimental results.
7
NIPS
Bars
?2.5
New York Times
?7.56
?7.4
?7.58
?3.5
oHDP?SM
oHDP, K=50
oHDP, K=20
?4
?7.6
?7.6
?7.7
?7.8
?7.9
oHDP?SM, K=100
oHDP?SM, K=300
oHDP, K=300
oHDP, K=1000
bHDP, K=300
bHDP, K=1000
oHDP?CRF, K=300
CGS, K=300
?8
?8.1
?4.5
0
50
100
150
200
250
300
350
0
400
per?word log likelihood
per?word log likelihood
per?word log likelihood
?7.5
?3
2.5
5
7.5
10
12.5
documents seen
iteration
(a)
?7.64
?7.66
?7.68
?7.7
?7.72
?7.74
oHDP, K=300
oHDP, K=500
oHDP?SM, K=200
?7.76
?7.78
?7.8
0
15
0.5
1
1.5
2
2.5
3
3.5
documents seen
5
x 10
(b)
4
6
x 10
(c)
550
600
80
500
70
500
oHDP?SM, K=2,100
oHDP, K=50
oHDP, K=20
60
450
# topics used
50
40
30
# topics used
400
# topics used
?7.62
300
400
350
300
200
250
20
100
200
10
0
0
50
100
150
0
0
200
2.5
5
7.5
iteration
documents seen
(d)
(e)
10
12.5
150
0
15
0.5
1
1.5
2
2.5
3
3.5
documents seen
5
x 10
4
6
x 10
(f)
?2.5
?7.56
?7.4
?7.58
?3.5
?4
?7.6
per?word log likelihood
per?word log likelihood
per?word log likelihood
?7.5
?3
?7.6
?7.7
?7.8
?7.9
?7.62
?7.64
?7.66
?7.68
?7.7
?7.72
?7.74
?8
?7.76
?8.1
?7.78
?4.5
0
10
20
30
40
50
60
70
80
0
100
200
300
400
500
600
?7.8
150
200
250
300
# topics used
# topics used
350
400
450
500
550
# topics used
(g)
(h)
(i)
Figure 2: Trace plots of heldout likelihood and number of topics used. Across all datasets, common color
indicates common algorithm, while for NIPS and New York Times, line type indicates different initializations.
Top: Test log likelihood for each dataset. Middle: Number of topics used per iteration. Bottom: A plot of
per-word log likelihood against number of topics used. Note particularly plot (h), where for every cardinality
of used topics shown, there is a split-merge method outperforming a conventional method.
Original topic
patterns
pattern
cortex
neurons
neuronal
single
responses
inputs
type
activation
40,000
patterns
pattern
cortex
neurons
neuronal
responses
single
inputs
temporal
activation
patterns
neuronal
pattern
neurons
cortex
inputs
activation
type
preferred
peak
80,000
patterns
pattern
cortex
neurons
neuronal
responses
single
temporal
inputs
type
neuronal
patterns
pattern
neurons
cortex
activation
dendrite
inputs
peak
preferred
120,000
patterns
pattern
cortex
neurons
responses
neuronal
single
type
number
temporal
neuronal
neurons
activation
cortex
dendrite
preferred
patterns
peak
pyramidal
inputs
160,000
patterns
pattern
cortex
neurons
responses
type
behavioral
types
neuronal
single
neuronal
dendritic
peak
activation
cortex
pyramidal
msec
fire
dendrites
inputs
200,000
patterns
pattern
cortex
neurons
responses
type
behavioral
types
form
neuronal
neuronal
dendritic
fire
peak
activation
msec
pyramidal
cortex
postsynaptic
inputs
240,000
patterns
pattern
cortex
responses
types
type
behavioral
form
neurons
areas
neuronal
dendritic
postsynaptic
fire
cortex
activation
peak
msec
pyramidal
inputs
Figure 3: The evolution of a split topic. The left column shows the topic directly prior to the split. After
240,000 more documents have been analyzed, subtle differences become apparent: the top topic covers terms
relating to general neuronal behavior, while the bottom topic deals more specifically with neuron firing.
8
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3,985 | 4,607 | Fiedler Random Fields: A Large-Scale Spectral
Approach to Statistical Network Modeling
Mikaela Keller?
Marc Tommasi?
INRIA Lille ? Nord Europe
40 avenue Halley ? B?at A ? Park Plaza
59650 Villeneuve d?Ascq (France)
{antonino.freno, mikaela.keller, marc.tommasi}@inria.fr
Antonino Freno
Abstract
Statistical models for networks have been typically committed to strong prior assumptions concerning the form of the modeled distributions. Moreover, the vast
majority of currently available models are explicitly designed for capturing some
specific graph properties (such as power-law degree distributions), which makes
them unsuitable for application to domains where the behavior of the target quantities is not known a priori. The key contribution of this paper is twofold. First,
we introduce the Fiedler delta statistic, based on the Laplacian spectrum of graphs,
which allows to dispense with any parametric assumption concerning the modeled
network properties. Second, we use the defined statistic to develop the Fiedler
random field model, which allows for efficient estimation of edge distributions
over large-scale random networks. After analyzing the dependence structure involved in Fiedler random fields, we estimate them over several real-world networks, showing that they achieve a much higher modeling accuracy than other
well-known statistical approaches.
1 Introduction
Arising from domains as diverse as bioinformatics and web mining, large-scale data exhibiting network structure are becoming increasingly available. Network models are commonly used to represent the relations among data units and their structural interactions. Recent studies, especially
targeted at social network modeling, have focused on random graph models of those networks. In
the simplest form, a random network is a configuration of binary random variables Xuv such that
the value of Xuv stands for the presence or absence of a link between nodes u and v in the network.
The general idea underlying random graph modeling is that network configurations are generated
by a stochastic process governed by specific probability laws, so that different models correspond to
different families of distributions over graphs.
The simplest random graph model is the Erd?os-R?enyi (ER) model [1], which assumes that the probability of observing a link between two nodes in a given graph is constant for any pair of nodes in
that graph, and it is independent of which other edges are being observed. In preferential attachment
models [2], the probability of linking to any specified node in a graph is proportional to the degree
of the node in the graph, leading to ?rich get richer? effects. Small-world models [3] try to capture
instead such phenomena often observed in real networks as small diameters and high clustering coefficients. An attempt to model potentially complex dependencies between graph edges in the form
of Gibbs-Boltzmann distributions is made by exponential random graph (ERG) models [4], which
subsume the ER model as a special case. Finally, a recent attempt at modeling real networks through
?
Universit?e Charles de Gaulle ? Lille 3, Domaine Universitaire du Pont de Bois ? BP 60149, 59653 Villeneuve d?Ascq (France).
1
a stochastic generative process is made by Kronecker graphs [5], which try to capture phenomena
such as heavy-tailed degree distributions and shrinking diameter properties while paying attention
to the temporal dynamics of network growth.
While some of these models behave better than others in terms of computational tractability, one
basic limitation affecting all of them is a sort of parametric assumption concerning the probability
laws underlying the observed network properties. In other words, currently available models of network structure assume that the shape of the probability distribution generating the network is known
a priori. For example, typical formulations of ERG models assume that the building blocks of real
networks are given by such structures as k-stars and k-triangles, with different weights assigned to
different structures, whereas the preferential attachment model is committed to the assumption that
the observed degree distributions obey a power law. In such frameworks, estimating the model from
data reduces to fitting the model parameters, where the parametric form of the target distribution is
fixed a priori. Clearly, in order for such models to deliver accurate estimates of the distributions at
hand, their prior assumptions concerning the behavior of the target quantities must be satisfied by
the given data. But unfortunately, this is something that we can rarely assess a priori. To date, the
knowledge we have concerning large-scale real-world networks does not allow to assess whether
any particular parametric assumption is capturing in depth the target generative process, although
some observed network properties may happen to be modeled fairly well.
The aim of this paper is twofold. On the one hand, we take a first step toward nonparametric
modeling of random networks by developing a novel network statistic, which we call the Fiedler
delta statistic. The Fiedler delta function allows to model different graph properties at once in an
extremely compact form. This statistic is based on the spectral analysis of the graph, and in particular
on the smallest non-zero eigenvalue of the Laplacian matrix, which is known as Fiedler value [6, 7].
On the other hand, we use the Fiedler delta statistic to define a Boltzmann distribution over graphs,
leading to the Fiedler random field (FRF) model. Roughly speaking, for each binary edge variable
Xuv , potentials in a FRF are functions of the difference determined in the Fiedler value by flipping
the value of Xuv , where the spectral decomposition is restricted to a suitable subgraph incident to
nodes u, v. The intuition is that the information encapsulated in the Fiedler delta for Xuv gives
a measure of the role of Xuv in determining the algebraic connectivity of its neighborhood. As
a first step in the theoretical analysis of FRFs, we prove that these models allow to capture edge
correlations at any distance within a given neighborhood, hence defining a fairly general class of
conditional independence structures over networks.
The paper is organized as follows. Sec. 2 reviews some theoretical background concerning the
Laplacian spectrum of graphs. FRFs are then introduced in Sec. 3, where we also analyze their
dependence structure and present an efficient approach for learning them from data. To avoid unwarranted prior assumptions concerning the statistical behavior of the Fiedler delta, potentials are
modeled by non-linear functions, which we estimate from data by minimizing a contrastive divergence objective. FRFs are evaluated experimentally in Sec. 4, showing that they are well suited for
large-scale estimation problems over real-world networks, while Sec. 5 draws some conclusions and
sketches a few directions for further work.
2 Graphs, Laplacians, and eigenvalues
Let G = (V, E) be an undirected graph with n nodes. In the following we assume that the graph is
unweighted with adjacency matrix A. The degree du of a node u ? V is defined as the number of
connections of u to other nodes, that is du = |{v: {u, v} ? E}|. Accordingly, the degree matrix D of
a graph G corresponds to the diagonal matrix with the vertex degrees d1 , . . . , dn on the diagonal. The
main tools exploited by the random graph model proposed here are the graph Laplacian matrices.
Different graph Laplacians have been defined in the literature. In this work, we use consistently the
unnormalized graph Laplacian, given by L = D ? A. Some basic facts related to the unnormalized
Laplacian matrix can be summarized as follows [7]:
Proposition 1. The unnormalized graph Laplacian L of an undirected graph G has the following
properties: (i) L is symmetric and positive semi-definite; (ii) the smallest eigenvalue of L is 0; (iii)
L has n non-negative, real-valued eigenvalues 0 = ?1 ? . . . ? ?n ; (iv) the multiplicity of the
eigenvalue 0 of L equals the number of connected components in the graph, that is, ?1 = 0 and
?2 > 0 if and only if G is connected.
2
In the following, the (algebraic) multiplicity of an eigenvalue ?i will be denoted by M (?i , G).
If the graph has one single connected component, then M (0, G) = 1, and the second smallest eigenvalue ?2 (G) > 0 is called, in this case, the Fiedler eigenvalue. The Fiedler eigenvalue provides
insight into several graph properties: when there is a nontrivial spectral gap, i.e. ?2 (G) is clearly
separated from 0, the graph has good expansion properties, stronger connectivity, and rapid convergence of random walks in the graph. For example, it is known that ?2 (G) ? ?(G), where ?(G) is the
edge connectivity of the graph (i.e. the size of the smallest edge cut whose removal makes the graph
disconnected [7]). Notice that if the graph has more than one connected component, then ?2 (G) will
be also equal to zero, thus implying that the graph is not connected. Without loss of generality, we
abuse the term Fiedler eigenvalue to denote the smallest eigenvalue different from zero, regardless
of the number of connected components. In this paper, by Fiedler value we mean the eigenvalue
?k+1 (G), where k = M (0, G).
+
For any pair of nodes u and v in a graph G = (V, E), we define two corresponding graphs G uv and
?
+
?
G uv in the following way: G uv = (V, E ? {{u, v}}), and G uv = (V, E \ {{u, v}}). Clearly, we
+
?
have that either G uv = G or G uv = G. A basic property concerning the Laplacian eigenvalues of
+
?
G uv and G uv is the following [7, 8, 9]:
+
?
+
Lemma 1. If G uv and G uv are two graphs with n nodes, such that {u, v} ? V, G uv = (V, E ?
Pn
?
+
?
{{u, v}}), and G uv = (V, E \ {{u, v}}), then we have that: (i) i=1 ?i (G uv ) ? ?i (G uv ) = 2;
+
?
(ii) ?i (G uv ) ? ?i (G uv ) for any i such that 1 ? i ? n.
3 Fiedler random fields
Fiedler random fields are introduced in Sec. 3.1, while in Secs. 3.2?3.3 we discuss their dependence
structure and explain how to estimate them from data respectively.
3.1 Probability distribution
Using the notions reviewed above, we define the Fiedler delta function ??2 in the following way:
+
Definition 1. Given graph G, let k = M (0, G uv ). Then,
+
?
?k+1 (G uv ) ? ?k+1 (G uv ) if Xuv = 1
??2 (u, v, G) =
?
+
?k+1 (G uv ) ? ?k+1 (G uv ) otherwise
(1)
In other words, ??2 (u, v, G) is the variation in the Fiedler eigenvalue of the graph Laplacian that
would result from flipping the value of Xuv in G. Concerning the range of the Fiedler delta function,
we can easily prove the following proposition:
Proposition 2. For any graph G = (V, E) and any pair of nodes {u, v} such that Xuv = 1, we have
that 0 ? ??2 (u, v, G) ? 2.
Proof. Let k = M (0, G). The proposition follows straightforwardly from Lemma 1, given that
?
??2 (u, v, G) = ?k+1 (G) ? ?k+1 (G uv ).
We now proceed to define FRFs. Given a graph G = (V, E), for each (unordered) pair of nodes
{u, v} such that u 6= v, we take Xuv to denote a binary random variable such that Xuv = 1 if
{u, v} ? E, and Xuv = 0 otherwise. Since the graph is undirected,SXuv = Xvu . We also say that a
subgraph GS of G with edge set ES is incident to Xuv if {u, v} ? e?ES e. Then:
Definition 2. Given a graph G, let XG denote the set of random variables defined on G, i.e. XG =
{Xuv : u 6= v ? {u, v} ? V}. For any Xuv ? XG , let Guv be a subgraph of G which is incident
to Xuv and ?uv be a two-place real-valued function with parameter vector ?. We say that the
probability distribution of XG is a Fiedler random field if it factorizes as
?
?
X
1
P (XG | ?) =
exp ?
(2)
?uv Xuv , ??2 (u, v, Guv ); ? ?
Z(?)
Xuv ?XG
3
where Z(?) is the partition function.
In other words, a FRF is a Gibbs-Boltzmann distribution over graphs, with potential functions defined for each node pair {u, v} along with some neighboring subgraph Guv . In particular, in order
to model the dependence of each variable Xuv on Guv , potentials take as argument both the value of
Xuv and the Fiedler delta corresponding to {u, v} in Guv . The idea is to treat the Fiedler delta statistic as a (real-valued) random variable defined over subgraph configurations, and to exploit this random variable as a compact representation of those configurations. This means that the dependence
structure of a FRF is fixed by the particular choice of subgraphs Guv , so that the set XGuv \ {Xuv }
makes Xuv independent of XG \ XGuv . Three fundamental questions are then the following. First,
how do we fix the subgraph Guv for each pair of nodes {u, v}? Second, how do we choose a shape
for the potential functions, so as to fully exploit the information contained in the Fiedler delta, while
avoiding unwarranted assumptions concerning their parametric form? Third, how does the Fiedler
delta statistic behave with respect to the Markov dependence property for random graphs? One basic
result related to the third question is presented in Sec. 3.2, while Sec. 3.3 will address the first two
points.
3.2 Dependence structure
We first recall the definition of Markov dependence for random graphs [10]. Let N (Xuv ) denote
the set {Xwz : {w, z} ? E ? |{w, z} ? {u, v}| = 1}. Then:
Definition 3. A random graph G is said to be a Markov graph (or to have a Markov dependence
structure) if, for any pair of variables Xuv and Xwz in G such that {u, v} ? {w, z} = ?, we have
that P (Xuv | Xwz , N (Xuv )) = P (Xuv | N (Xuv )).
Based on Def. 3, we say that the dependence structure of a random graph G is non-Markovian if,
for disjoint pairs of nodes {u, v} and {w, z}, it does not imply that P (Xuv | Xwz , N (Xuv )) =
P (Xuv | N (Xuv )), i.e. if it is consistent with the inequality P (Xuv | Xwz , N (Xuv )) 6=
P (Xuv | N (Xuv )). We can then prove the following proposition:
Proposition 3. There exist Fiedler random fields with non-Markovian dependence structure.
Proof sketch. Consider a graph G = (V, E) such that V = {u, v, w, z} and E =
{{u, v}, {v, w}, {w, z}, {u, z}}. The proof relies on the following result [6]: if graphs G1
and G2 are, respectively, a path and a circuit of size n, then ?2 (G1 ) = 2 (1 ? cos(?/n))
and ?2 (G2 ) = 2 (1 ? cos(2?/n)). Since adding exactly one edge to a path of size 4 can
yield a circuit of the same size, this property allows to derive analytic forms for the Fiedler
delta statistic in such graphs, showing that there exist parameterizations of ?uv such that
?uv (Xuv , ??2 (u, v, G); ?) 6= ?uv (Xuv , ??2 (u, v, GS ); ?). This means that the dependence structure of G is non-Markovian.1
Note that the proof of Prop. 3 can be straightforwardly generalized to the dependence between two
variables Xuv and Xwz in circuits/paths of arbitrary size n, since the expression used for the Fiedler
eigenvalues of such graphs holds for any n. This fact suggests that FRFs allow to model edge
correlations at virtually any distance within G, provided that each subgraph Guv is chosen in such a
way as to encompass the relevant correlation.
3.3 Model estimation
The problem of learning a FRF from an observed network can be split into the task of estimating
the potential functions once the network distribution has been factorized into a particular set of
subgraphs, and the task of factorizing the distribution through a suitable set of subgraphs, which
corresponds to estimating the dependence structure of the FRF. Here we focus on the problem of
learning the FRF potentials, while suggesting a heuristic way to fix the dependence structure of the
model.
In order to estimate the FRF potentials, we need to specify on the one hand a suitable architecture
for such functions, and on the other hand the objective function that we want to optimize. As a
1
For a complete proof, see the supplementary material.
4
preliminary step, we tested experimentally a variety of shapes for the potential functions. The tests
indicated the importance of avoiding limiting assumptions concerning the form of the potentials,
which motivated us to model them by a feed-forward multilayer perceptron (MLP), due to its wellknown capabilities of approximating functions of arbitrary shape [12]. In particular, throughout
the applications described in this paper we use a simple MLP architecture with one hidden layer
and hyperbolic tangent activation functions. Therefore, our parameter vector ? simply consists of
the weights of the chosen MLP architecture. Notice that, as far as the estimation of potentials is
concerned, any regression model offering approximation capabilities analogous to the MLP family
could be used as well. Here, the only requirement is to avoid unwarranted prior assumptions with
respect to the shape of the potential functions. In this respect, we take our approach to be genuinely
nonparametric, since it does not require the parametric form of the target functions to be specified
a priori in order to estimate them accurately. Concerning instead the learning objective, the main
difficulty we want to avoid is the complexity of computing the partition function involved in the
Gibbs-Boltzmann distribution. The approach we adopt to this aim is to minimize a contrastive
divergence objective [13]. If G = (V, E) is the network that we want to fit our model to, and
?
Guv = (Vuv , Euv ) is a subgraph of G such that {u, v} ? Vuv , let Guv
denote the graph that we obtain
by resampling the value of Xuv in Guv according to the conditional distribution Pb (Xuv | xGuv \
?
{xuv }; ?) predicted by our model. In other words, Guv
is the result of performing just one iteration
of Gibbs sampling on Xuv using ?, where the configuration xGuv of Guv is used to initialize the
(single-step) Markov chain. Then, our goal is to minimize the function ?CD (?; G), given by:
?
??
?
? 1
?
X
?
?CD (?; G) = log
exp ?
? x?uv , ??2 (u, v, Guv
); ? ? ? log Pb (xG | ?)
? Z(?)
?
Xuv ?XG
(3)
X
?
?
? xuv , ??2 (u, v, Guv ); ? ? ? xuv , ??2 (u, v, Guv ); ?
=
Xuv ?XG
where ? is the function computed by our MLP architecture. The appeal of contrastive divergence
learning is that, while it does not require to compute the partition function, it is known to converge
to points which are very close to maximum-likelihood solutions [14].
If we want our learning objective to be usable in the large-scale setting, then it is not feasible to
sum over all node pairs {u, v} in the network, since the number of such pairs grows quadratically
with |V|. In this respect, a straightforward approach for scaling to very large networks consists in
sampling n objects from the set of all possible pairs of nodes, taking care that the sample contains a
good balance between linked and unlinked pairs. Another issue we need to address concerns the way
we sample a suitable set of subgraphs Gu1 v1 , . . . , Gun vn for the selected pairs of nodes. Although
different sampling techniques could be used in principle [15], our goal is to model correlations
between each variable Xuv and some neighboring region Guv in G. Such a neighborhood should be
large enough to make ??2 (u, v, Guv ) sufficiently informative with respect to the overall network, but
also small enough to keep the spectral decomposition of Guv computationally tractable. Therefore,
in order to sample Guv , we propose to draw Vuv by performing k ?snowball waves? on G [16], using
u and v as seeds, and then setting Euv to be the edge set induced by Vuv in G (see Algorithm 1
for the details). In this way, we can empirically tune the k hyperparameter in order to trade-off the
informativeness of Guv for the tractability of its spectral decomposition, where it is known that the
complexity of computing ??2 (u, v, Guv ) is cubic with respect to the number of nodes in Guv [17].
Algorithm 1 SampleSubgraph: Sampling a neighboring subgraph for a given node pair
Input: Undirected graph G = (V, E); node pair {u, v}; number k of snowball waves.
Output: Undirected graph Guv = (Vuv , Euv ).
SampleSubgraph(G, {u, v}, k):
1. Vuv = {u, v}
2. for(i = 1 to Sk)
3.
Vuv = Vuv ? w?Vuv {z ? V: {w, z} ? E}
4. Euv = {{w, z} ? E: {w, z} ? Vuv }
5. return (Vuv , Euv )
5
Once sampled our training set D = (xu1 v1 , Gu1 v1 ), . . . , (xun vn , Gun vn ) , we learn the MLP
weights by minimizing the objective ?CD (?; D), which which we obtain from ?CD (?; G) by restricting the summation in Eq. 3 to the elements of D. Minimization is performed by iterative
gradient descent, using standard backpropagation for updating the MLP weights.
4 Experimental evaluation
In order to investigate the empirical behavior of FRFs as models of large-scale networks, we design
two different groups of experiments (in link prediction and graph generation respectively), using collaboration networks drawn from the arXiv e-print repository (http://snap.stanford.edu/
data/index.html), where nodes represent scientists and edges represent paper coauthorships.
Some basic network statistics are reported in Table 1.
Link prediction. In the first kind of experiments, given a random network G = (V, E), our
goal is to measure the accuracy of FRFs at estimating the conditional distribution of variables
Xuv given the configuration of neighboring subgraphs Guv of G. This can be seen as a link
prediction problem where only local information (given by Guv ) can be used for predicting the
presence of a link {u, v}. At the same time, we want to understand whether the overall network size (in terms of the number of nodes) has an impact on the number of training examples
that will be necessary for FRFs to converge to stable prediction accuracy. Recall that FRFs are
trained on a data sample D = (xu1 v1 , Gu1 v1 ), . . . , (xun vn , Gun vn ) , where n ? |V| (|V|?1)
.
2
Given this, converging to stable predictions for values of n which do not depend on |V| is a crucial requirement for achieving large-scale applicability. Let us sample our training set D by first
drawing n node pairs from V in such a way that linked and unlinked pairs from G are equally
represented in D, and then extracting the corresponding subgraphs Gui ,vi by Algorithm 1 using
one snowball wave. We then learn our model from D as described in Sec. 3.3. In all the experiments reported in this work, the number of hidden units in our MLP architecture is set to
5. A test set T containing m objects (xu1 v1 , GS1 ), . . . , (xum vm , GSm ) is also sampled from G
so that T ? D = ?, where pairs {ui , vi } in T are drawn uniformly at random from V ? V.
Predictions are derived from the learned model
by first computing the conditional probability of observing a link for each pair of nodes
{uj , vj } in T , and then making a decision on
the presence/absence of links by thresholding
the predicted probability (where the threshold is
tuned by cross-validation). Prediction accuracy
is measured by averaging the recognition accuracy for linked and unlinked pairs in T respectively (where |T | = 10, 000). In Fig. 1, the accuracy of FRFs on the test set is plotted against
a growing size n of D (where 12 ? n ? 48).
Interestingly, the number of training examples
required for the accuracy curve to stabilize does
not seem to depend at all on the overall network
size. Indeed, fastest convergence is achieved Figure 1: Prediction accuracy of FRFs on the
for the average-sized and the second largest arXiv networks for a growing training set size.
networks, i.e. HepPh and AstroPh respectively.
Notice how a training sample containing an extremely small percentage of node pairs is sufficient
for our learning approach to converge to stable prediction accuracy. This result encourages to think
of FRFs as a convenient modeling option for the large-scale setting.
0.95
0.9
Prediction accuracy on test set
0.85
0.8
0.75
0.7
0.65
0.6
0.55
GrQc (5,242 nodes)
HepTh (9,877 nodes)
HepPh (12,008 nodes)
AstroPh (18,772 nodes)
CondMat (23,133 nodes)
0.5
0.45
10
15
20
25
30
Training set size
35
40
45
50
Besides assessing whether the network size affects the number of training samples needed to accurately learn FRFs, we want to evaluate the usefulness of the dependence structure involved in our
model in predicting the conditional distributions of edges given their neighboring subgraphs. That
is, we want to ascertain whether the effort of modeling the conditional independence structure of
the overall network through the FRF formalism is justified by a suitable gain in prediction accuracy
with respect to statistical models that do not focus explicitly on such dependence structure. To this
aim, we compare FRFs to two popular statistical models for large-scale networks, namely the WattsStrogatz (WS) and the Barab?asi-Albert (BA) models [3, 2]. The WS formalism is mainly aimed
6
at modeling the short-diameter property often observed in real-world networks. Interestingly, the
degree distribution of WS networks can be expressed in closed form in terms of two parameters ?
and ?, related to the average degree distribution and a network rewiring process respectively [18].
On the other hand, the BA model is aimed at explaining the emergence of power-law degree distributions, where such distributions can be expressed in terms of an adaptive parameter ? [19]. The
parameters of both the WS and the BA model can be estimated by standard maximum-likelihood
approaches and then used to predict conditional edge distributions, exploiting information from the
degrees observed in the given subgraphs [20, 21]. The ER model is not considered in this group
of experiments, since the involved independence assumption makes it unusable (i.e. equivalent to
random guessing) for the purposes of conditional estimation tasks. On the other hand, ERG models
are not suitable for application to the large-scale setting. We tried them out using edge, k-star and
k-triangle statistics [4], and the tests confirmed this point. Although the prohibitive cost of fitting the
models and computing the involved feature functions could be overcome in principle by sampling
strategies similar to the ones we employ for FRFs, the potentials used in ERGs become numerically
unstable in the large-scale setting, leading to numerical representation issues for which we are not
aware of any off-the-shelf solution. Accuracy values for the different models are reported in Table 1. FRFs dramatically outperform the other two models on all networks. Since both the BA and
the WS model do not show relevant improvements over simple random guessing, this result clearly
suggests that exploiting the dependence structure involved in network edge configurations is crucial
to accurately predict the presence/absence of links.
Table 1: Edge prediction results on the arXiv networks. General network statistics are also reported,
where CCG and DG stand for average clustering coefficient and network diameter respectively.
Dataset
AstroPh
CondMat
GrQc
HepPh
HepTh
|V|
18,772
23,133
5,242
12,008
9,877
Network Statistics
|E | CCG
396,160
0.63
186,936
0.63
28,980
0.52
237,010
0.61
51,971
0.47
DG
14
15
17
13
17
Prediction Accuracy
BA
FRF
WS
50.98% 89.97% 50.14%
50.15% 91.62% 56.71%
52.57% 91.14% 53.72%
51.61% 86.57% 54.33%
58.33% 92.25% 50.30%
Graph generation. A second group of experiments is aimed at assessing whether the FRFs learned
on the arXiv networks can be considered as plausible models of the degree distribution (DD) and
the clustering coefficient distribution (CC) observed in each network [15]. To this aim, we use the
estimated FRF models to generate artificial graphs of various size, using Gibbs sampling, and then
we compare the DD and CC observed in the artificial graphs with those estimated on the whole
networks. For scale-free networks such as the ones considered here, the BA model is known to be
the most accurate model currently available with respect to DD. On the other hand, for CC both BA
and WS are known to be more realistic models than ER random graphs. Therefore, we compare the
graphs generated by FRFs to those generated by the BA, ER, and WS models for the same networks.
The distance in DD and CC between the artificial graphs on the one hand and the corresponding real
network on the other hand is measured using the Kolmogorov-Smirnov D-statistic, following a
common use in graph mining research [15]. Here we only plot results for the CondMat and HepTh
networks, noticing that the results we collected on the other arXiv networks lend themselves to the
same interpretation as the ones displayed in Fig. 2. Values are averaged over 100 samples for each
considered graph size, where the standard deviation is typically in the order of 10?2 . The outcome
motivates the following considerations. Concerning DD, FRFs are able to improve (at least slightly)
the accuracy of the state-of-the-art BA model, while they are very close that model with respect
to clustering coefficient. In all cases, both BA and FRFs prove to be far more accurate than ER
or WS, where the only advantage of using WS is limited to improving CC over ER. These results
are particularly encouraging, since they show how the nonparametric approach motivating the FRF
model allows to accurately estimate network properties (such as DD) that are not aimed for explicitly
in the model design. This suggests that the Fiedler delta statistic is a promising direction for building
generative models capable of capturing different network properties through a unified approach.
7
1
0.9
BA
ER
FRF
WS
0.9
BA
ER
FRF
WS
0.8
D-statistic for CC
D-statistic for DD
0.8
0.7
0.6
0.7
0.6
0.5
0.5
0.4
0.3
0.4
40
60
80
100
Artificial graph size
120
140
160
40
60
80
(a)
100
Artificial graph size
120
140
160
(b)
1
0.9
BA
ER
FRF
WS
0.9
BA
ER
FRF
WS
0.8
D-statistic for CC
D-statistic for DD
0.8
0.7
0.6
0.7
0.6
0.5
0.5
0.4
0.3
0.4
40
60
80
100
Artificial graph size
120
140
160
40
(c)
60
80
100
Artificial graph size
120
140
160
(d)
Figure 2: D-statistic values for DD and CC on the CondMat (a?b) and HepTh (c?d) networks.
5 Conclusions and future work
The main motivation inspiring this work was the observation that statistical modeling of networks
cries for genuinely nonparametric estimation, because of the inaccuracy often resulting from unwarranted parametric assumptions. In this respect, we showed how the Fiedler delta statistic offers a
powerful building block for designing a nonparametric estimator, which we developed in the form
of the FRF model. Since here we only applied FRFs to collaboration networks, which are typically
scale-free, an important option for future work is to assess the flexibility of FRFs in modeling networks from different families. In the second place, since we only addressed in a heuristic way the
problem of learning the dependence structure of FRFs, a stimulating direction for further research
consists in designing clever techniques for learning the structure of FRFs, e.g. considering the use
of alternative subgraph sampling techniques. Finally, we would like to assess the possibility of
modeling networks through mixtures of FRFs, so as to fit different network regions (with possibly
conflicting properties) through specialized components of the mixture.
Acknowledgments
This work has been supported by the French National Research Agency (ANR-09-EMER-007). The
authors are grateful to Gemma Garriga, R?emi Gilleron, Liva Ralaivola, and Michal Valko for their
useful suggestions and comments.
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9
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3,986 | 4,608 | A systematic approach to extracting semantic
information from functional MRI data
Francisco Pereira
Siemens Corporation, Corporate Technology
Princeton, NJ 08540
[email protected]
Matthew Botvinick
Princeton Neuroscience Institute and Department of Psychology
Princeton University
Princeton NJ 08540
[email protected]
Abstract
This paper introduces a novel classification method for functional magnetic resonance imaging datasets with tens of classes. The method is designed to make
predictions using information from as many brain locations as possible, instead of
resorting to feature selection, and does this by decomposing the pattern of brain
activation into differently informative sub-regions. We provide results over a complex semantic processing dataset that show that the method is competitive with
state-of-the-art feature selection and also suggest how the method may be used to
perform group or exploratory analyses of complex class structure.
1
Introduction
Functional Magnetic Resonance Imaging (fMRI) is a technique used in psychological experiments
to measure the blood oxygenation level throughout the brain, which is a proxy for neural activity;
this measurement is called brain activation. The data resulting from such an experiment is a 3D grid
of cells named voxels covering the brain (on the order of tens of thousands, usually), measured over
time as tasks are performed and thus yielding one time series per voxel (collected every 1-2 seconds
and yielding hundreds to thousands of points).
In a typical experiment, brain activation is measured during a task of interest, e.g. reading words,
and during a related control condition, e.g. reading nonsense words, with the goal of identifying
brain locations where the two differ. The most common analysis technique for doing this ? statistical parametric mapping [4] ? tests each voxel individually by regressing its time series on a predicted
time series determined by the task contrast of interest. This fit is scored and thresholded at a given
statistical significance level to yield a brain image with clusters of voxels that respond very differently to the two tasks (colloquially, these are the images that show parts of the brain that ?light up?).
Note, however, that for both tasks there are many other processes taking place in tandem with this
task-contrasting activation: visual processing to read the words, attentional processing due to task
demands, etc. The output of this process for a given experiment is a set of 3D coordinates of all the
voxel clusters that appear reliably across all the subjects in a study. This result is easy to interpret,
since there is a lot of information about what processes each brain area may be involved in. The
coordinates are comparable across studies, and thus result reproduciblity is also an expectation.
In recent years, there has been increasing awareness of the fact that there is information in the entire
pattern of brain activation and not just in saliently active locations. Classifiers have been the tool
1
of choice for capturing this information and used to make predictions ranging from what stimulus a
subject is seeing, what kind of object they are thinking about or what decision they will make [12]
[14] [8]. The most common situation is to have an example correspond to the average brain image
during one or a few performances of the task of interest, and voxels as the features, and we will
discuss various issues with this scenario in mind.
The goal of this work is generally not (just) classification accuracy per se, even in diagnostic applications, but understanding where the information used to classify is present. If only two conditions
are being contrasted this is relatively straightforward as information is, at its simplest, a difference in
activation of a voxel in the two conditions. It?s thus possible to look at the magnitudes of the weights
a classifier puts on voxels across the brain and thus locate the voxels with the largest weights 1 ; given
that there are typically two to three orders of magnitude more voxels than examples, though, classifiers are usually trained on a selection of voxels rather than the entire activation pattern. Often, this
means the best accuracy is obtained using few voxels, from all across the brain, and that different
voxels will be chosen in different cross-validation folds; this presents a problem for interpretability
of the locations in question.
One approach to this problem is to try and regularize classifiers so that they include as many informative voxels as possible [2], thus identifying localizable clusters of voxels that may overlap across
folds. A different approach is to cross-validate classifiers over small sections of the grid covering the
brain, known as searchlights [10]. This can be used to produce a map of the cross-validated accuracy in the searchlight around each voxel, taking advantage of the pattern of activation across all the
voxels contained in it. Such a map can then be thresholded to leave only locations where accuracy
is significantly above chance. While these approaches have been used successfully many times over
the last decade, they will become progressively less useful in face of the increasing commonality
of datasets with tens to hundreds of stimuli, and a correspondingly high number of experimental
conditions. Knowing the location of a voxel does not suffice to interpret what it is doing, as it could
be very different from stimulus to stimulus (rather than just active or not, as in the two condition
situation). It?s also likely that no small brain regions will allow for a searchlight classifier capable of
distinguishing between all possible conditions at the spatial resolution of fMRI, and hence defining
a searchlight size or shape is a trade-off between including voxels and making it harder to locate
information or train a classifier ? as the number of features increases as the number of examples
remains constant ? and excluding voxels and thus the number of distinctions that can be made.
This paper introduces a method to address all of these issues while still yielding an interpretable,
whole-brain classifier. The method starts by learning how to decompose the pattern of activation
across the brain into sub-patterns of activation, then it learns a whole-brain classifier in terms of the
presence and absence of certain subpatterns and finally combines the classifier and pattern information to generate brain maps indicating which voxels belong to informative patterns and what kind of
information they contain. This method is partially based on the notion of pattern feature introduced
in an earlier paper by us [15], but has been developed much further so as to dispense with most
parameters and allow the creation of spatial maps usable for group or exploratory analyses, as will
be discussed later.
2
2.1
Data and Methods
Data
The grid covering the brain contains on the order of tens of thousands voxels, measured over time
as tasks are performed, every 1-2 seconds, yielding hundreds to thousands of 3D images per experiment. During an experiment a given task is performed a certain number of times ? trials ? and often
the images collected during one trial are collapsed or averaged together, giving us one 3D image
that can be clearly labeled with what happened in that trial, e.g. what stimulus was being seen or
what decision a subject made. Although the grid covers the entire head, only a fraction of its voxels
contain cortex in a typical subject; hence we only consider these voxels as features.
1
Interpretation is more complicated if nonlinear classifiers are being used [6], [17], but this is far less
common
2
A searchlight is a small section of the 3D grid, in our case a 27 = 3 ? 3 ? 3 voxel cube. Analyses
using searchlights generally entail computing a statistic [10] or cross-validating a classifier over the
dataset containing just those voxels [16], and do so for the searchlight around each voxel in the brain,
covering it in its entirety. The intuition for this is that individual voxels are very noisy features, and
an effect observed across a group of voxels is more trustworthy.
In the experiment performed to obtain our dataset 2 [13], subjects observed a word and a line drawing
of an item, displayed on a screen for 3 seconds and followed by 8 seconds of a blank screen. The
items named/depicted belonged to one of 12 categories: animals, body parts, buildings, building
parts, clothing, furniture, insects, kitchen, man-made objects, tools, vegetables and vehicles. The
experimental task was to think about the item and its properties while it was displayed. There were
5 different exemplars of each of the 12 categories and 6 experimental epochs. In each epoch all 60
exemplars were shown in random order without repetition, and all epochs had the same exemplars.
During an experiment the task repeated a total of 360 times, and a 3D image of the fMRI-measured
brain activation acquired every second.
Each example for classification purposes is the average image during a 4 second span while the
subject was thinking about the item shown a few seconds earlier (a period which contains the peak
of the signal during the trial; the dataset thus contains 360 examples, as many as there were trials.
The voxel size was 3 ? 3 ? 5 mm, with the number of voxels being between 20000 and 21000
depending on which of the 9 subjects was considered. The features in each example are voxels, and
the example labels are the category of the item being shown in the trial each example came from.
1
2
for each classi?cation
task, cross-validate a
classi?er in all of
the searchlights
searchlight:
- a 3x3x3 voxel cube
- one centered around
each voxel in cortex
- overlapping
test the result at each
searchlight, which
yields a binary
signi?cance image
e.g. animals vs insects
searchlight accuracy 0.54 0.76 0.61 0.83 0.55 0.46 0.90
3
image as a vector of voxels
result signi?cant
this is done for all 66 pairwise
classi?cation tasks
and adjacent searchlights
supporting similar pairwise
distinctions are clustered
together using modularity
5
animals vs insects
...
animals vs tools
...
...
vegetables vs vehicles
...
...
...
vehicles
vehicles
the binary vector of signi?cance
for each searchlight is rearranged
into a binary confusion matrix
animals
insects
tools
buildings
clothing
body parts
furniture
4
animals
insects
tools
buildings
clothing
body parts
furniture
...
searchlight
Figure 1: Construction of data-driven searchlights.
2.2
Method
The goal of the experiment our dataset comes from is to understand how a certain semantic category
is represented throughout the brain (e.g. do ?Insects? and ?Animals? share part of their representation because both kinds of things are alive?). Intuitively, there is information in a given location if
at least two categories can be distinguished looking at their respective patterns of activation there;
otherwise, the pattern of activation is noise or common to all categories. Our method is based upon
this intuition, and comprises three stages:
2
The data were kindly shared with us by Tom Mitchell and Marcel Just, from Carnegie Mellon University.
3
1. the construction of data-driven searchlights, parcels of the 3D grid where the same discriminations between pairs of categories can be made (these are generally larger than the
3 ? 3 ? 3 basic searchlight)
2. the synthesis of pattern features from each data-driven searchlight, corresponding to the
presence or absence of a certain pattern of activation across it
3. the training and use of a classifier based on pattern features and the generation of an anatomical map of the impact of each voxel on classification
and these are described in detail in each of the following sections.
2.2.1
Construction of data-driven searchlights
Create pairwise searchlight maps In order to identify informative locations we start by considering whether a given pair of categories can be distinguished in each of the thousands of 3 ? 3 ? 3
searchlights covering the brain:
1. For each searchlight cross-validate a classifier using the voxels belonging to it, obtaining
an accuracy value which will be assigned to the voxel at the center of the searchlight,
as shown in part 1 of Figure 1. The classifier used in this case was Linear Discriminant
Analysis (LDA, [7]), with a shrinkage estimator for the covariance matrix [18], as this was
shown to be effective at both modeling the joint activation of voxels in a searchlight and
classification [16].
2. Transform the resulting brain image with the accuracy of each voxel into a p-value brain
image (of obtaining accuracy as high or higher under the null hypothesis that the classes
are not distinguishable, see [11]), as shown in part 1 of Figure 1.
3. Threshold the p-value brain image using False Discovery Rate [5] (q = 0.01) to correct
multiple for multiple comparisons and get a binary brain image with candidate locations
where this pair of categories can be distinguished, as shown in part 2 of Figure 1.
The outcome for each pair of categories is a binary significance image, where a voxel is 1 if the
categories can be distinguished in the searchlight surrounding it or 0 if not; this is shown for all
pairs of categories in part 3 of Figure 1. This can also be viewed per-searchlight, yielding a binary
vector encoding which category pairs can be distinguished and which can be rearranged into a binary
matrix, as shown in part 4 of Figure 1.
Aggregate adjacent searchlights Examining each small searchlight makes sense if we consider
that, a priori, we don?t know where the information is or how big a pattern of activation would have
to be considered (with some exceptions, notably areas that respond to faces, houses or body parts, see
[9] for a review). That said, if the same categories are distinguishable in two adjacent searchlights
? which overlap ? then it is reasonable to assume that all their voxels put together would still be
able to make the same distinctions. Doing this repeatedly allows us to find data-driven searchlights,
not bound by shape or size assumptions. At the same time we would like to constrain data-driven
searchlights to the boundaries of known, large, anatomically determined regions of interest (ROI),
both for computational efficiency and for interpretability, as will be described later.
At the start of the aggregation process, each searchlight is by itself and has an associated binary
information vector with 66 entries corresponding to which pairs of classes can be distinguished in
its surrounding searchlight (part 3 of Figure 1). For each searchlight we compute the similarity
of its information vector with those of all its neighbours, which yields a 3D grid similarity graph.
We then take the portion of the graph corresponding to each ROI in the AAL brain atlas [19], and
use modularity [1] to divide it into a number of clusters of adjacent searchlights supporting similar
distinctions, as shown in panel 5 of Figure 1. After this is done for all ROIs we obtain a partition of
the brain into a few hundred clusters, the data-driven searchlights. Figure 2 depicts the granularity
of a typical clustering across multiple brain slices of one of the participants.
The similarity measure between two
P vectors vi and vj is obtained by computing the number of
1-entries present in both vectors, pairs AND(vi , vj ), the number of 1-entries present in only one
P
of them, pairs XOR(vi , vj ) and then the measure
4
Figure 2: Data-driven searchlights for participant P1 (brain slices range from inferior to superior).
similarity(vi , vj ) =
P
P
pairs
AND(vi , vj ) ? pairs
P
pairs AND(vi , vj )
XOR(vi ,vj )
2
The measure was chosen because it peaks at 1, if the two vectors match exactly, and decreases ?
possibly into negative values ? if there are mismatches; it will tolerate more mismatches if there are
more distinctions being made. It will also deem sparse vectors similar as long as there are vew few
mismatches. The number of entries present in only one is divided by 2 so that the differences do not
get twice the weight of the similarities.
The centroid for each cluster encodes the pairs of categories that can be distinguished in that datadriven searchlight. The centroid is obtained by combining the binary information vectors for each
of the searchlights in it using a soft-AND function, and is itself a binary information vector. A given
entry is 1 ? the respective pair of categories is distinguishable ? if it is 1 in at least q% of the cluster
members (where q is the false discovery rate used earlier to threshold the binary image for that pair
of categories).
2.2.2
Generation of pattern features from each data-driven searchlight
voxels
examples
1
clusters (across class pairs)
clusters (across all examples)
training data
singular vectors
pattern features
SVD
cluster 1
...
3
animals vs
insects
animals vs
tools
vegetables
vs vehicles
2
cluster 2
...
cluster 3
body parts
vs buildings
animals vs
insects
...
animals vs
tools
cluster 4
...
body parts
vs buildings
vegetables
vs vehicles
Figure 3: Construction of pattern detectors and pattern features from data-driven searchlights.
Construct two-way classifiers from each data-driven searchlight Each data-driven searchlight
has a set of pairs of categories that can be distinguished in it. This indicates that there are particular
patterns of activation across the voxels in it which are characteristic of one or more categories, and
absent in others. We can leverage this to convert the pattern of activation across the brain into a
series of sub-patterns, one from each data-driven searchlight.
For each data-driven searchlight, and for each pairwise category distinction in its information vector,
we train a classifier using examples of the two categories and just the voxels in the searchlight (a
linear SVM with ? = 1, [3]); these will be pattern detectors, outputting a probability estimate for
the prediction (which we transform to the [?1, 1] range), shown in part 1 of Figure 3.
5
Use two-way classifiers to generate pattern features The set of pattern-detectors learned from
each data-driven searchlight can be applied to any example, not just the ones from the categories
that were used to learn them. The output of each pattern-detector is then viewed as representing the
degree to which the detector thinks that either of the patterns it is sensitive to is present. For each
data-driven searchlight, we apply all of its detectors to all the examples in the training set, over the
voxels belonging to the searchlight, as illustrated in part 2 of Figure 3. The output of each detector
across all examples becomes a new, synthetic pattern feature. The number of these pattern features
varies per searchlight, as does the number of searchlights per subject, but at the end we will typically
have between 10K and 20K of them.
Note that there may be multiple classifiers for a given cluster which produce very similar outputs
(e.g. ones that captured a pattern present in all animate object categories versus one present in all
inanimate object ones); these will be highly correlated and redundant. We address this by using
Singular Value Decomposition (SVD, [7]) to reduce the dimensionality of the matrix of pattern
features to the same as the number of examples (180), keeping all singular vectors; this is shown in
part 3 of Figure 3. The detectors and the SVD transformation matrix learned from the training set
are also applied to the test set.
2.2.3
Classification and impact maps for each class
pattern feature classi?er
for "tools"-vs-rest
singular vector classi?er
for "tools"-vs-rest
pattern feature impact values
invert SVD
1
"tools" singular vectors
X
2
aggregate impact of pattern
features belonging to each cluster
per-cluster impact values
"tools" pattern features
3
assign per-cluster impact value
to the voxels that belong to it
invert SVD
voxelwise impact values
Figure 4: The process of going from the weights of a one-versus-rest category classifier over a
low-dimensional pattern feature representation to the impact of each voxel in that classification.
Given the low-dimensional pattern feature dataset, we train a one-versus-rest classifier (a linear
SVM with ? = 1, [3]) for each category; these are then applied to each example in the test set, with
the label prediction corresponding to the class with the highest class probability.
The classifiers can also be used to determine the extent to which each data-driven searchlight was
responsible for correctly predicting each class. A one-versus-rest category classifier consists of a
vector of 180 weights, which can be converted into an equivalent classifier over pattern features by
inverting the SVD, as shown in part 1 of Figure 4. The impact of each pattern feature in correctly
predicting this category can be calculated by multiplying each weight by the values taken by the
corresponding pattern feature over examples in the category, and averaging across all examples; this
is shown in part 2 of Figure 4. These pattern-feature impact values can then be aggregated by the
data-driven searchlight they came from, yielding a net impact value for that searchlight. This is the
value that is propagated to each voxel in the data-driven searchlight (part 3 of Figure 4) in order to
generate an impact map.
3
3.1
Experiments and Discussion
Classification
Our goal in this experiment is to determine whether transforming the data from voxel features to
pattern features preserves information, and how competitive the results are with a classifier combined with voxel selection. In all experiments we use a split-half cross-validation loop, where the
halves contain examples from even and odd epochs, respectively, 180 examples in each (15 per cat6
egory). If cross-validation inside a split-half training set is required, we use leave-one-epoch out
cross-validation,
Baseline We contrasted experimental results obtained with our method with a baseline of classification using voxel selection. The scoring criterion used to rank each voxel was the accuracy of a
LDA classifier ? same as described above ? using the 3 ? 3 ? 3 searchlight around each voxel to
do 12-category classification. The number of voxels to use was selected by nested cross-validation
inside the training set 3 . The classifier used was a linear SVM (? = 1, [3]), same as the whole brain
classifier in our method.
Results The results are shown in the first line of Table 1; across subjects, our method is better than
voxel selection, with the p-value of a sign-test of this being < 0.01. It is substantially better than a
classifier using all the voxels in the brain directly.
Whereas the accuracy is above chance (0.08) for all subjects, it is rather low for some. There
are at least two factors responsible for this. The first is that some classes give rise to very similar
patterns of activation (e.g. ?buildings? and ?building parts?), and hence examples in these classes are
confusable (confusion matrices bear this out). The second factor is that subjects vary in their ability
to stay focused on the task and avoid stray thoughts or remembering other parts of the experiment,
hence examples may not belong to the class corresponding to the label or even any class at all. [13]
also points out that accuracy is correlated with a subject?s ability to stay still during the experiment.
Table 1: Classification accuracy for the 9 subjects using our method, as well as two baselines.
P1
P2
P3
P4
P5
P6
P7
P8
P9
our method
0.54 0.34 0.33 0.42 0.15 0.19 0.22 0.21 0.16
baseline (voxel selection)
0.53 0.33 0.24 0.34 0.14 0.16 0.21 0.20 0.15
baseline (using all voxels) 0.31 0.21 0.19 0.27 0.13 0.09 0.14 0.13 0.15
#voxels selected (fold 1)
1200 400 200 1600 800
800
800
400 2000
800
200 100
800
50
8000 100 1200 100
#voxels selected (fold 2)
3.2
Impact maps
tool
building
Figure 5: Average example for categories ?tool? and ?building? in participant P1 (slices ordered
from inferior to superior, red is activation above the image mean, blue below).
As described in Section 2.2.3, an impact map can be produced for each category, showing the extent
to which each data-driven searchlight helped classify that category correctly. In order to better
understand better how impact works, consider two categories ?tools? and ?buildings? where we
know where some of the information resides (for ?tools? around the central sulcus, visible on the
right of slices to the right, for ?buildings? around the parahippocampal gyrus, visible on the lower
side of slices to the left). Figure 5 shows the average example for the two categories; note how
similar the two examples are across the slices, indicating that most activation is shared between the
two categories.
The impact maps for the same participant in Figure 6 show that much of the common activation is
eliminated, and that the areas known to be informative are assigned high impact in their respective
3
Possible choices were 50, 100, 200, 400, 800, 1200, 1600, 2000, 4000, 8000, 16000 or all voxels.
7
tool
building
Figure 6: Impact map for categories ?tool? and ?building? in participant P1.
tool
building
Figure 7: Average impact map for categories ?tool? and ?building? across the nine participants.
maps. Impact is positive, regardless of whether activation in each voxel involved is above or below
the mean of the image; the activation of each voxel influences the classifier only in the context of
its neighbours in each data-driven searchlight. Note, also, that unlike a simple one-vs-rest classifier
or searchlight map, the notion of impact can accommodate the situation where the same location is
useful, with either different or the same pattern of activation, for two separate classes (rather than
have it be downweighted relative to others that might be unique to that particular class).
Finally, consider that impact maps can be averaged across subjects, as shown in Figure 7, or undergo t-tests or a more complex second-level group analysis. A more exploratory analysis can be
performed by considering locations that are high impact for every participant and, through their
data-driven searchlight, examine the corresponding cluster centroids and get a complete picture of
which subsets of the classes can be distinguished there (similar to the bottom-up process in part 5 of
Figure 1, but now done top-down and given a cross-validated classification result and impact value).
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9
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3,987 | 4,609 | Bayesian models for Large-scale Hierarchical
Classification
Siddharth Gopal
Bing Bai
Yiming Yang
Alexandru Niculescu-Mizil
[email protected] [email protected]
{bing,alex}@nec-labs.com
Carnegie Mellon University
NEC Laboratories America, Princeton
Abstract
A challenging problem in hierarchical classification is to leverage the hierarchical relations among classes for improving classification performance. An even
greater challenge is to do so in a manner that is computationally feasible for large
scale problems. This paper proposes a set of Bayesian methods to model hierarchical dependencies among class labels using multivariate logistic regression.
Specifically, the parent-child relationships are modeled by placing a hierarchical prior over the children nodes centered around the parameters of their parents;
thereby encouraging classes nearby in the hierarchy to share similar model parameters. We present variational algorithms for tractable posterior inference in these
models, and provide a parallel implementation that can comfortably handle largescale problems with hundreds of thousands of dimensions and tens of thousands
of classes. We run a comparative evaluation on multiple large-scale benchmark
datasets that highlights the scalability of our approach and shows improved performance over the other state-of-the-art hierarchical methods.
1
Introduction
With the tremendous growth of data, providing a multi-granularity conceptual view using hierarchical classification (HC) has become increasingly important. The large taxonomies for web page
categorization at the Yahoo! Directory and the Open Directory Project, and the International Patent
Taxonomy are examples of widely used hierarchies. The large hierarchical structures present both
challenges and opportunities for statistical classification research. Instead of focusing on individual classes in isolation, we need to address joint training and inference based on the hierarchical
dependencies among the classes. Moreover this has to be done in a computationally efficient and
scalable manner, as many real world HC problems are characterized by large taxonomies and high
dimensionality.
In this paper, we investigate a Bayesian framework for leveraging the hierarchical class structure.
The Bayesian framework is a natural fit for this problem as it can seamlessly capture the idea that
the models at the lower levels of the hierarchy are specialization of models at the ancestor nodes.
We define a hierarchical Bayesian model where the prior distribution for the parameters at a node
is a Gaussian centered at the parameters of the parent node. This prior encourages the parameters
of nodes that are close in the hierarchy to be similar thereby enabling propagation of information
across the hierarchical structure and leading to inductive transfer (sharing statistical strength) among
the models corresponding to the different nodes. The strength of the Gaussian prior, and hence the
amount of information sharing between nodes, is controlled by its covariance parameter, which is
also learned from the data. Modelling the covariance structures gives us the flexibility to incorporate
different ways of sharing information in the hierarchy. For example, consider a hierarchical organization of all animals with two sub-topics mammals and birds. By placing feature specific variances,
the model can learn that the sub-topic parameters are more similar along common features like
?eyes?,?claw? and less similar in other sub-topic specific features like ?feathers?, ?tail? etc. As another example, the model can incorporate children-specific covariances that allows some sub-topic
1
parameters to be less similar to their parent and some to be more similar; for e.g. sub-topic whales is
quite distinct from its parent mammals compared to its siblings felines, primates. Formulating such
constraints in non-Bayesian large-margin approaches is not as easy, and to our knowledge has not
done before in the context of hierarchical classification. Other advantages of a fully Bayesian treatment are that there no reliance on cross-validation, the outputs have a probabilistic interpretation,
and it is easy to incorporate prior domain knowledge.
Our approach shares similarity to the correlated Multinomial logit [18] (corrMNL) in taking a
Bayesian approach to model the hierarchical class structure, but improves over it in two significant
aspects - scalability and setting hyperparameters. Firstly, CorrMNL uses slower MCMC sampling
for inference, making it difficult to scale to problems with more than a few hundred features and a
few hundred nodes in the hierarchy. By modelling the problem as a Hierarchical Bayesian Logistic Regression (HBLR), we are able to vastly improve the scalability by 1) developing variational
methods for faster inference, 2) introducing even faster algorithms (partial MAP) to approximate
the variational inference at an insignificant cost in classification accuracy, and 3) parallelizing the
inference. The approximate variational inference (1 plus 2) reduces the computation time by several
order of magnitudes (750x) over MCMC, and the parallel implementation in a Hadoop cluster [4]
further improves the time almost linearly in the number of processors. These enabled us to comfortably conduct joint posterior inference for hierarchical logistic regression models with tens of
thousands of categories and hundreds of thousands of features.
Secondly, a difficulty with the Bayesian approaches, that has been largely side-stepped in [18], is
that, when expressed in full generality, they leave many hyperparameters open to subjective input
from the user. Typically, these hyper-parameters need to be set carefully as they control the amount
of regularization in the model, and traditional techniques such as Empirical Bayes or cross-validation
encounter difficulties in achieving this. For instance, Empirical Bayes requires the maximization of
marginal likelihood which is difficult to compute in hierarchical logistic models [9] in general, and
cross-validation requires reducing the number of free parameters for computational reasons, potentially losing the flexibility to capture the desired phenomena. In contrast, we propose a principled
way to set the hyper-parameters directly from data using an approximation to the observed Fisher
Information Matrix. Our proposed technique can be easily used to set a large number of hyperparameters without losing model tractability and flexibility.
To evaluate the proposed techniques we run a comprehensive empirical study on several large scale
hierarchical classification problems. The results show that our approach is able to leverage the
class hierarchy and obtain a significant performance boost over leading non-Bayesian hierarchical
classification methods, as well as consistently outperform flat methods that do not use the hierarchy
information.
Other Related Work: Most of the previous work in HC has been primarily using large-margin
discriminative methods. Some of the early works in HC [10, 14] use the hierarchical structure to
decompose the classification problem into sub-problems recursively along the hierarchy and allocate
a classifier at each node. The hierarchy is used to partition the training data into node-specific subsets
and classifiers at each node are trained independently without using the hierarchy any further. Many
approaches have been proposed to better utilize the hierarchical structure. For instance, in [22, 1],
the output of the lower-level classifiers was used as additional features for the instance at the toplevel classifiers. Smoothing the estimated parameters in naive Bayes classifiers along each path from
the root to a leaf node has been tried in [17]. [20, 6] proposed large-margin discriminative methods
where the discriminant function at each node takes the contributions from all nodes along the path
to the root node, and the parameters are jointly learned to minimize a global loss over the hierarchy.
Recently, enforcing orthogonality constraints between parent and children classifiers was shown to
achieve state-of-art performance [23].
2
The Hierarchical Bayesian Logistic Regression (HBLR) Framework
Define a hierarchy as a set of nodes Y = {1, 2...} with the parent relationship ? : Y ? Y where
d
?(y) is the parent of node y ? Y . Let D = {(xi , ti )}N
i=1 denote the training data where xi ? R is
an instance, ti ? T is a label, where T ? Y is the set of leaf nodes in the hierarchy labeled from 1
to |T |. We assume that each instance is assigned to one of the leaf nodes in the hierarchy. Let Cy be
the set of all children of y.
2
For each node y ? Y , we associate a parameter vector wy which has a Gaussian prior. We set the
mean of the prior to the parameter of the parent node, w?(y) . Different constraints on the covariance
matrix of the prior corresponds to different ways of propagating information across the hierarchy. In
what follows, we consider three alternate ways to model the covariance matrix which we call M1,
M2 and M3 variants of HBLR. In the M1 variant all the siblings share the same spherical covariance
matrix. Formally, the generative model for M1 is
M1 wroot ? N (w0 , ?0 ),
?root ? ?(a0 , b0 )
wy | w?(y) , ??(y) ? N (w?(y) , ??(y) ) ?y,
?y ? ?(ay , by ) ?y ?
/T
t|x?
pi (x) =
Multinomial(p1 (x), p2 (x), .., p|T | (x))
?(x, t) ? D
exp(wi> x)/?t0 ?T exp(wt>0 x)
(1)
The parameters of the root node are drawn using user specified parameters w0 , ?0 , a0 , b0 . Each nonleaf node y ?
/ T has its own ?y drawn from a Gamma with the shape and inverse-scale parameters
specified by ay and by . Each wy is drawn from the Normal with mean w?(y) and covariance matrix
?1
??(y) = ??(y)
I. The class-labels are drawn from a Multinomial whose parameters are a soft-max
transformation of the wy s from the leaf nodes. This model leverages the class hierarchy information
by encouraging the parameters of closely related nodes (parents, children and siblings) to be more
similar to each other than those of distant ones in the hierarchy. Moreover, by using different inverse
variance parameters ?y for each node, the model has the flexibility to adapt the degree of similarity
between the parameters (i.e. parent and children nodes) on a per family basis. For instance it can
learn that sibling nodes which are higher in the hierarchy (e.g. mammals and birds) are generally
less similar compared to sibling nodes lower in the hierarchy (e.g. chimps and orangutans).
Although this model is equivalent to the corrMNL proposed in [18], the hierarchical logistic regression formulation is different from corrMNL and has a distinct advantage that the parameters
can be decoupled. As we shall see in Section 3, this enables the use of scalable and parallelizable
variational inference algorithms. In contrast, in corrMNL the soft-max parameters are modeled as
a sum of contributions along the path from a leaf to the root-node. This introduces two layers of
dependencies between the parameters in the corrMNL model (inside the normalization constant as
well along the path from leaves to root-node) which makes it less amenable to efficient variational
inference. Even if one were to develop a variational approach for the corrMNL parameterization, it
would be slower and not efficient for parallelization.
Although the M1 approach is rational, one may argue that it would be beneficial to allow the diagonal
elements of the covariance matrix ??(y) to be feature-specific instead of uniform. In our previous
example with sub-topics mammals and birds, we may want wmammals , wbirds to be commonly
close to their parent in some dimensions (e.g., in some common features like ?eyes?,?breathe? and
?claw?) but not in other dimensions (e.g., in bird specific features like ?feathers? or ?beak?). We
(i)
accommodate this by replacing prior ?y using ?y for every feature (i). This form of setting the
prior is referred to as Automatic Relevant Determination (ARD) and forms the basis of several works
such as Sparse Bayesian Learning [19], Relevance Vector Machines [3], etc. For the HC problem,
we define the M2 variant of the HBLR approach as:
M2
wy | w?(y) , ??(y) ? N (w?(y) , ??(y) )
?y
(i)
?y(i) ? ?(a(i)
/T
y , by ) i = 1..d, ?y ?
where ??1
?(y) = diag(??(y) , ??(y) , . . . , ??(y) )
(1)
(2)
(d)
Yet another extension of the M1 model would be to allow each node to have its own covariance
matrix for the Gaussian prior over wy , not shared with its siblings. This enables the model to learn
how much the individual children nodes differ from the parent node. For example, consider topic
mammals and its two sub-topics whales and carnivores; the sub-topic whales is very distinct from a
typical mammal and is more of an ?outlier? topic. Such mismatches are very typical in hierarchies;
especially in cases where there is not enough training data and an entire subtree of topics is collapsed
as a single node. M3 aims to cope up with such differences.
M3 wy | w?(y) , ?y ? N (w?(y) , ?y )
?y
?y ? ?(ay , by )
?y ?
/T
Note that the only difference between M 3 and M 1 is that M 3 uses ?y = ?y?1 I instead of ??(y) in
the prior for wy . In our experiments we found that M3 consistently outperformed the other variants
suggesting that such effects are important to model in HC. Although it would be natural to extend
3
M3 by placing ARD priors instead of the uniform ?y , we do not expect to see better performance
due to the difficulty in learning a large number of parameters. Preliminary experiments confirmed
our suspicions so we did not explore this direction further.
3
Inference for HBLR
In this section, we present the inference method for M2 which is harder. The procedure can be easily
extended for M1 and M3 1 . The posterior of M2 is given by
p(W, ?|D) ? p(D|W, ?)p(W, ?)
?
(2)
d
Y
Y Y
Y
exp(wt> x)
(i)
P
p(?y(i) |a(i)
y , by )
exp(wt>0 x)
y?Y
(x,t)?D 0
y?Y \T i=1
t ?T
p(wy |w?(y) , ??(y) )
Closed-form solution for the posterior is not possible due to the non-conjugacy between the logistic likelihood and the Gaussian prior, we therefore resort to variational methods to compute the
posterior. However, using variational methods are themselves computational intractable in high dimensional scenarios due to the requirement of a matrix inversion which is computationally intensive.
Therefore, we explore much faster approximation schemes such as partial MAP inference which are
highly scalable. Finally, we show the resulting approximate variational inference procedure can be
parallelized in a map-reduce framework to tackle large-scale problems that would be impossible to
solve on a single processor.
3.1
Variational Inference
Starting with a simple factored form for the posterior, we seek such a distribution q which is closest
in KL divergence to the true posterior p. We use independent Gaussian q(wy ) and Gamma q(?y )
posterior distributions for wy and ?y per node as the factored representation:
Y
q(W, ?) =
q(?y )
y?Y \T
Y
q(wy ) ?
d
Y Y
?(.|?y(i) , ?y(i) )
y?Y \T i=1
y?Y
Y
N (.|?y , ?y )
y?Y
In order to tackle the non-conjugacy inside p(D|W, ?) in (2), we use a suitable lower-bound to the
soft-max normalization constant proposed
by [5], for any ? ? R , ?k ? [0, ?)
X gk ? ? ? ?k
+ ?(?k )((gk ? ?)2 ? ?k2 ) + log(1 + e?k )
2
k
k
1
1
where ?(?) = 2?
? 12
1+e??
log(
X
egk ) ? ? +
where ? , ?k are variational parameters which we can optimize to get the tightest possible bound.
For every (x, y) we introduce variational parameters ?x and ?xy . We now derive an EM algorithm
that computes the posterior in the E-step and maximizes the variational parameters in the M-step.
Variational E-Step The local variational parameters are fixed, and the posterior for a parameter is
computed by matching the log-likelihood of the posterior with the expectation of log-likelihood
under the rest of the parameters. The parameters are updated as1 ,
X
??1
y = I(y ? T )
2?(?xy )xx> + diag(
(x,t)?D
?
?y = ?y ?I(y ? T )
X
(x,t)?D
?y(i) = by(i) +
X
??(y)
?y
) + |Cy | diag( )
??(y)
?y
(3)
?
??(y)
1
?y X ?
(I(t = y) ? + 2?(?xy )?x )x + diag(
)??(y) + diag( )
?c
2
??(y)
?y c?C
y
(i) 2
?(i,i)
+ ?(i,i)
+ (?(i)
y
c
y ? ?c )
and ?y(i)
c?Cy
|Cy |
= a(i)
y +
2
(4)
Variational M-Step We keep the parameters of the posterior distribution fixed and maximize the
variational parameters ?xy , ?x . Refer to [5] for detailed M-step derivations,
2
?xy
= x> diag(
?y
2
)x + (?x ? ?>
y x)
?y
?x = (.5(.5|T | ? 1) +
X
y?T
?(?xy )?>
y x)/
X
?(?xy )
y?T
Class-label Prediction After computing the posterior, one way to compute the probability of a target
class-label given a test instance is to simply plugin the posterior mean for prediction. A more
principled way would be to compute the predictive distribution of the target class label l given the
1
Complete derivations are presented in the extended version located at http://www.cs.cmu.edu/?sgopal1.
4
test instance,
Z
p(l|x) =
Z
p(l, W|x)dW ?
p(l|W, x)q(W)dW
(5)
The above integral cannot be computed in closed form and people have often resorted to probit
approximations [16]. We take an alternative route by calculating the joint posterior p(l, W|x) by
variational approximations. We assume the following factored form for the predictive distribution,
q?(l, W) =
Y
Y
q?(wy )?
q (ly ) ?
y?T
? y )Bern(.|?
N (.|?
?y , ?
py )
y?T
The posterior can be calculated as before, by introducing variational parameters ??xy , ??x and matching the log likelihoods. Substituting q?(l, W) in (5), we see that the predictive distribution is given
by q?(l) and the target class label is given by arg maxy?T p?y .
3.2
Partial MAP Inference
In most applications, the requirement for a matrix inversion in step (3) could be demanding. In such
scenarios, we split the inference into two stages, first calculating the posterior of wy using MAP
solution, and second calculating the posterior of ?y . In the first stage, we find the MAP estimate
wymap and then use laplace approximation to approximate the posterior using a separate Normal
distribution for each dimension, thereby leading to a diagonal covariance matrix. Note that due to
the laplace approximation, wymap and the posterior mean ?y coincide.
? = wymap = arg max
W
X
(?(i,i)
)?1 =
y
??(y)
1
? (wy ? w?(y) )> diag(
)(wy ? w?(y) ) + log p(D|W, ?)
2
??(y)
y?T
X
(6)
x(i) pxy (1 ? pxy )x(i)
(x,t)?Dy
where pxy is the probability that training instance x is labeled as y. The arg max in (6) can be
computed for all ?y at the same time using optimization techniques like LBFGS [13]. For the
second stage, parameters ?y and ?y are updated using (4). Full MAP inference is also possible by
performing an alternating maximization between wy , ?y but we do not recommend it as there is no
gain in scalability compared to partial MAP Inference and it loses the posterior distribution of ?y .
3.3
Parallelization
For large hierarchies, it might be impractical to learn the parameters of all classes, or even store
them in memory, on a single machine. We therefore, devise a parallel memory-efficient implementation scheme for our partial MAP Inference. There are 4 sets of parameters that are updated {?y , ?y , ?y , ?y }. The ?y , ?y , ?y can be updated in parallel for each node using (3),(4).
For ?, the optimization step in (6) is not easy to parallelize since the w?s are coupled together inside
the soft-max function. To make it parallelizable we replace the soft-max function in (1) with multiple binary logistic functions (one for each terminal node), which removes the coupling of parameters
inside the log-normalization constant. The optimization can now be done in parallel by making the
following observations - firstly note that the optimization problem in (6) is concave maximation,
therefore any order of updating the variables reaches the same unique maximum. Secondly, note
that the interactions between the wy ?s are only through the parent and child nodes. By fixing the
parameters of the parent and children, the parameter wy of a node can be optimized independently
of the rest of the hierarchy. One simple way to parallelize is to traverse the hierarchy level by level,
optimize the parameters at each level in parallel, and iterate until convergence. A better way that
achieves a larger degree of parallelization is to iteratively optimize the odd and even levels - if we fix
the parameters at the odd levels, the parameters of parents and the children of all nodes at even levels
are fixed, and the wy ?s at all even levels can be optimized in parallel. The same goes for optimizing
the odd level parameters. To aid convergence we interleave the ?, ? updates with the ?, ? updates
and warm-start with the previous value of ?y . In practice, for the larger hierarchies we observed
speedups linear in the number of processors. Note that the convergence follows from viewing this
procedure as block co-ordinate ascent on a concave differentiable function [15].
We tested our parallelization framework on a cluster running map-reduce based Hadoop 20.2 with
64 worker nodes with 8 cores and 16GB RAM each. We used Accumulo 1.4 key-value store for
fast retrieve-update of the wy s. On this hardware, our experiments on the largest dataset with 15358
class labels and 347256 features took just 38 minutes. Although the map-reduce framework is not
a requirement; it is a ubiquitous paradigm in distributed computing and having an implementation
compatible with it is a definite advantage.
5
Table 1: Dataset Statistics
Dataset
#Training #Testing #Class-Labels #Leaf-labels Depth #Features
CLEF
10000
1006
87
63
4
89
NEWS20
11260
7505
27
20
3
53975
LSHTC-small
4463
1858
1563
1139
6
51033
LSHTC-large 93805
34905
15358
12294
6
347256
IPC
46324
28926
552
451
4
541869
4
Setting prior parameters
The w0 , ?0 represent the overall mean and covariance structure for the wy . We set w0 = 0 and ?0 =
(i) (i)
I because of their minimal effect on the rest of the parameters. The ay , by are variance components
such that
b(i)
y
(i)
ay
(i)
represents the expected variance of the wy . Typically, choosing these parameters is
difficult before seeing the data. The traditional way to overcome this is to learn {ay , by } from the
data using Empirical Bayes. Unfortunately, in our proposed model, one cannot do this as each
{ay , by } is associated with a single ?y . Generally, we need more than one sample value to learn the
prior parameters effectively [7].
We therefore resort to a data dependent way of setting these parameters by using an approximation
to the observed Fisher Information matrix. We first derive on a simpler model and then extend it
to a hierarchy. Consider the following binary logistic model with unknown w and let the Fisher
Information matrix be I and observed Fisher Information I?
Y | x ? Bernoulli(
exp(w> x)
);
1 + exp(w> x)
h
i
I = E p(x)(1 ? p(x))xx> , I? =
X
p?(x)(1 ? p?(x))xx>
(x,t)?D
It is well known that I ?1 is the asymptotic covariance of the MLE estimator of w, so reasonable
guess for the covariance of a Gaussian prior over w could be the observed I??1 from a dataset
D. The problem with I??1 is that we do not have a good estimate p?(x) for a given x as we have
exactly one sample for a given x i.e each instance x is labeled exactly once with certainty, therefore
p?(x)(1 ? p?(x)) will always be zero. Therefore we approximate p?(x) as the sample prior probability
t
independent of x, i.e. p?(x) = p? = ?(x,t)?D |D|
. Now, the prior on the covariance of wy can be set
?1
such that the expected covariance is I? . To extend this to HC, we need to handle multiple classes,
? ?1 for each y ? T , as well handle multiple levels, which can
which can be done by estimating I(y)
be done by recursively setting ay , by as?follows,
(i)
(a(i)
y , by )
=
? (
P
(i)
ac ,
c?Cy
?
?
(1, I(y)
P
c?Cy
?1(i,i)
)
(i)
bc )
if y ?
/T
if y ? T
? is the observed Fisher Information matrix for class label y. This way of setting the priors
where I(y)
is similar to the method proposed in [12], the key differences are in approximating p(x)(1 ? p(x))
from the data rather using p(x) = 21 , extension to handle multiple classes as well as hierarchies.
We also tried other popular strategies such as setting improper gamma priors ?(, ) ? 0 widely
used in many ARD works (which is equivalent to using type-2 ML for the ??s if one uses variational
methods [2]) and Empirical Bayes using a single a and b (as well as other Empirical Bayes variants).
Neither of worked well, the former being to be too sensitive to the value of which is in agreement
with the observations made by [11] and the latter constraining the model by using a single a and b.
We do not discuss this any further due to lack of space.
5
Experiments Results
Throughout our experiements, we used 4 popular benchmark datasets (Table 1) with the recommended train-test splits - CLEF[8], NEWS202 , LSHTC-{small,large}3 , IPC4 .
First, to evaluate the speed advantage of the variational inference, we compare the full variational
{M1,M2,M3}-var and partial MAP {M1,M2,M3-map} inference 5 for the three variants of HBLR to
the MCMC sampling based inference of CorrMNL [18]. For CorrMNL, we used the implementation
as provided by the authors6 . We performed sampling for 2500 iterations with 1000 for burn-in.
2
4
6
3
http://people.csail.mit.edu/jrennie/20Newsgroups/
http://www.wipo.int/classifications /ipc/en/support/
http://www.ics.uci.edu/ babaks/Site/Codes.html
5
6
http://lshtc.iit.demokritos.gr/
Code available at http://www.cs.cmu.edu/?sgopal1
Table 2: Comparison with CorrMNL: Macro-F1 and Micro-F1 on the CLEF dataset
{M1,M2,M3}-var {M1,M2,M3}-map {M1,M2,M3}-flat
CorrMNL M1 M2 M3 M1 M2 M3 M1 M2 M3
Macro-f1
55.59
56.67 51.23 59.67 55.53 54.76 59.65 52.13 48.78 55.23
Micro-f1
81.10
81.21 79.92 81.61 80.88 80.25 81.41 79.82 77.83 80.52
Time (mins)
2279
79
81
80
3
3
3
3
3
3
0.4
0.55
BLR
MLR
BSVM
MSVM
M3-map
BLR
0.5
MLR
BSVM
MSVM
M3-map
0.35
0.45
Macro-F1
0.3
Micro-F1
0.4
0.35
0.3
0.25
0.2
0.25
0.15
0.2
0.1
0.15
1
2
3
4
1
5
2
3
4
5
# Training Examples per Class
# Training Examples per Class
Figure 1: Micro-F1 (left) & Macro-F1 (right) on the CLEF dataset with limited number of training examples.
Re-starts with different initialization values gave the same results for both MCMC and variational
methods. All models were run on a single CPU without parallelization. We used the small CLEF[8]
dataset in order to be able to run CorrMNL model in reasonable time. The results are presented
in Table 2. For an informative comparison, we also included the results of {M1,M2,M3}-flat, our
proposed approach using a flat hierarchy. With regards to scalability, partial MAP inference is
the most scalable method being orders of magnitude faster (750x) than CorrMNL. Full variational
inference, although less scalable as it requires O(d3 ) matrix inversions in the feature space, is still
orders of magnitude faster (20x) than CorrMNL. In terms of performance, we see that the partial
MAP inference for the HBLR has only small loss in performance compared to the full variational
inference while having similar training time to the flat approach that does not model the hierarchy
({M1,M2,M3}-flat).
Next, we compare the performance of HBLR to several other competing approaches:
1. Hierarchical Baselines: We selected 3 representative hierarchical methods that have shown to
have state-of-the-art performance - Hierarchical SVM [6] (HSVM), a large-margin discriminative
method with path-dependent discriminant function. Orthogonal Transfer [23] (OT), a method enforcing orthogonality constraints between the parent node and children and Top-down Classification
[14] (TD) Top-down decision making using binary SVMs trained at each node.
2. Flat Baselines: Typical flat approaches which do not make use of the hierarchy. We tested Oneversus rest Binary logistic Regressions (BLR), Multiclass Logistic Regression (MLR), One-versus
Rest Binary SVMs (BSVM), and Multiclass SVM (MSVM) [21].
For all competing approaches, we tune the regularization parameter using 5 fold CV with a range
of values from 10?5 to 105 . For the HBLR models, we used partial MAP Inference because full
variational is not scalable to high dimensions. The IPC and LSHTC-large are very large datasets so
we are unable to test any method other than our parallel implementation of HBLR, and BLR, BSVM
which can be trivially parallelized. Although TD can be parallelized we did not pursue this since
TD did not achieve competitive performance on the other datasets. Parallelizing the other methods
is not obvious and has not been discussed in previous literature to the best of our knowledge.
Table 3 summarizes the results obtain by the different methods. The performance was measured using the standard macro-F1 and micro-F1 measures [14]. The significance tests are performed using
sign-test for Micro-F1 and a wilcoxon rank test on the Macro-F1 scores. For every data collection,
each method is compared to the best performing method on that dataset. The null hypothesis is that
there is no significanct difference between the two systems being compared, the alternative is that
the best-performing-method is better. Among M1,M2 and M3, the performance of M3 seems to be
consistently better than M1, followed by M2. Although M2 is more expressive than M1, the benefit
of a better model seems to be offset by the difficulty in learning a large number of parameters.
Comparing to the other hierarchical baselines, M3 achieves significantly higher performance on all
datasets, showing that the Bayesian approach is able to leverage the information provided in the
class hierarchy. Among the baselines, we find that the average performance of HSVM is higher
than the TD, OT. This can be partially explained by noting that both OT and TD are greedy topdown classification methods and any error made in the top level classifications propagates down to
7
Table 3: Macro-F1 and Micro-F1 on the 4 datasets. Bold faced number indicate best performing method. The
results of the significance tests are denoted * for a p-value less than 5% and ? for p-value less than 1%.
{M1,M2,M3}-map
Hierarchical methods
Flat methods
M1
M2
M3 HSVM OT
TD
BLR MLR BSVM MSVM
CLEF
Macro-f1
55.53? 54.76? 59.65 57.23* 37.12? 32.32? 53.26? 54.76? 48.59? 54.33?
Micro-f1
80.88* 80.25* 81.41 79.72? 73.84? 70.11? 79.92? 80.52? 77.53? 80.02?
NEWS20
Macro-f1
81.54 80.91* 81.69 80.04? 81.20 80.86* 82.17 81.82 82.32 81.73
Micro-f1
82.24* 81.54* 82.56* 80.79* 81.98* 81.20? 82.97 82.56* 83.10 82.47*
LSHTC-small
Macro-f1
28.81? 25.81? 30.81 21.95? 19.45? 20.01? 28.12? 28.38* 28.62* 28.34*
Micro-f1
45.48 43.31? 46.03 39.66? 37.12? 38.48? 44.94? 45.20 45.21* 45.62
LSHTC-large
Macro-f1
28.32* 24.93? 28.76
27.91*
27.89*
Micro-f1
43.98 43.11? 44.05
43.98
44.03
IPC
Macro-f1
50.43? 47.45? 51.06
48.29?
45.71?
Micro-f1
55.80* 54.22? 56.02
55.03?
53.12?
the leaf node; in contrast to HSVM which uses an exhaustive search over all labels. However, the
result of OT do not seem to support the conclusions in [23]. We hypothesize two reasons - firstly,
the orthogonality condition which is assumed in OT does not hold in general, secondly, unlike
[23] we use cross-validation to set the underlying regularization parameters rather than setting them
arbitrarily to 1 (which was used in [23]).
Surprisingly, the hierarchical baselines (HSVM,TD and OT) experience a very large drop in performance on LSHTC-small when compared to the flat baselines, indicating that the hierarchy information actually mislead these methods rather than helping them. In contrast, M3 is consistently better
than the flat baselines on all datasets except NEWS20. In particular, M3 performs significantly better on the largest datasets, especially in Macro-F1 , showing that even very large class hierarchies can
convey very useful information, and highlighting the importance of having a scalable, parallelizable
hierarchical classification algorithm.
To further establish the importance of modeling the hierarchy, we test our approach under scenarios
when the number of training examples is limited. We expect the hierarchy to be most useful in
such cases as it enables of sharing of information between class parameters. To verify this, we
progressively increased the number of training examples per class-label on the CLEF dataset and
compared M3-map with the other best performing methods. Figure 1 reports the results of M3map, MLR, BSVM, MSVM averaged over 20 runs. The results shows that M3-map is significantly
better than the other methods especially when the number of examples is small. For instance, when
there is exactly one training example per class, M3-map achieves a whopping 10% higher MicroF1 and a 2% higher Macro-F1 than the next best method. We repeated the same experiments on
the NEWS20 dataset but however did not find an improved performance even with limited training
examples suggesting that the hierarchical methods are not able to leverage the hierarchical structure
of NEWS20.
6
Conclusion
In this paper, we presented the HBLR approach to hierarchical classification, focusing on scalable
ways to leverage hierarchical dependencies among classes in a joint framework. Using a Gaussian
prior with informative mean and covariance matrices, along with fast variational methods, and a
practical way to set hyperparameters, HBLR significantly outperformed other popular HC methods
on multiple benchmark datasets. We hope this study provides useful insights into how hierarchical
relationships can be successfully leveraged in large-scale HC. In future, we would like to adapt this
approach to equivalent non-bayesian large-margin discriminative counterparts.
ACKNOWLDEGMENTS: This work is supported, in part, by the NEC Laboratories America,
Princeton under ?NEC Labs Data Management University Awards? and the National Science Foundation (NSF) under grant IIS 1216282. A major part of work was accomplished while the first
author was interning at NEC Labs, Princeton.
8
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9
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3,988 | 461 | Recognizing Overlapping Hand-Printed Characters by
Centered-Object Integrated Segmentation and Recognition
Gale L. Martin- & Mosfeq Rashid
MCC
Austin, Thxas 78759 USA
Abstract
This paper describes an approach, called centered object integrated segmentation and recognition (COISR). for integrating object segmentation and recognition within a single neural network. The application
is hand-printed character recognition. 1\vo versions of the system are
described. One uses a backpropagation network that scans exhaustively over a field of characters and is trained to recognize whether
it is centered over a single character or between characters. When
it is centered over a character, the net classifies the cnaracter. The
approach is tested on a dataset of hand-printed digits. Vel)' low errOr
rates are reported. The second version, COISR-SACCADE, avoids
the need for exhaustive scans. The net is trained as before. but also
is trained to compute ballistic 'eye' movements that enable the input
window to jump from one character to the next.
The common model of visual processing includes multiple, independent stages. First,
flltering operations act on the raw image to segment or isolate and enhance to-be-recognized clumps. These clumps are normalized for factors such as size, and sometimes
simplified further through feature extraction. The results are then fed to one or more
classifiers. The operations prior to classification simplify the recognition task. Object
segmentation restricts the number of features considered for classification to those associated with a single object, and enables normalization to be applied at the individual
object level. Without such pre-processing. recognition may be an intractable problem.
However, a weak point of this sequential stage model is that recognition and segmentation decisions are often inter-dependent. Not only does a correct recognition decision
depend on first making a correct segmentation decision, but a correct segmentation
decision often depends on first making a correct recognition decision.
This is a particularly serious problem in character recognition applications. OCR systems use intervening white space and related features to segment a field of characters
into individual characters, so that classification can be accomplished one character at
a time. This approach fails when characters touch each other or when an individual
character is broken up by intervening white space. Some means of integrating the segmentation and recognition stages is needed.
This paper descnbes an approach, called centered object integrated segmentation and recognition (COISR), for integrating character segmentation and recognition within one
? Also with Eastman Kodak Company
504
Recognizing Overlapping Hand-Printed Characters
NET'S OUfPUT
OVER TIME
Figure 1: The COISR Exhaustive Scan Approach.
neural network. The general approach builds on previous work in pre-segmented
character recognition (LeCun, Boser, Denker, Henderson, Howard, Hubbard, & Jackel, 1990; Martin & Pittman, 1990) and on the sliding window conception used in neural
network speech applications, such as NETtalk (Sejnowski & Rosenberg(1986) and
Tune Delay Neural Networks (Waibel, Sawai, & Shikano, 1988).'1\vo versions of the
approach are descnbed. In both cases, a net is trained to recognize what is centered
in its input window as it slides along a character field. The window size is chosen to
be large enough to include more than one character.
1 COISR VERSION 1: EXHAUSTIVE SCAN
As shown in Figure 1, the net is trained on an input window, and a target output vector
representing what is in the center of the window. The top half of the figure shows the
net's input window scanning successively across the field. Sometimes the middle of the
window is centered on a character, and sometimes it is centered on a point between
two characters. The target output vector consists of one node per category, and one
node corresponding to a NOT-CENTERED condition. This latter node has a high
target activation value when the input window is not centered over any character. A
temporal stream of output vectors is created (shown at the bottom half of the figure)
as the net scans the field. There is no need to explicitly segment characters, during
training or testing, because recognition is defined as identifying what is in the center
of the scanning window. The net learns to extract regularities in the shapes of individual characters even when those regularities occur in the context of overlapping and broken characters. The final stage of processing involves parsing the temporal stream generated as the net scans the field to yield an ascii string of recognized characters.
1.1 IMPLEMENTATION DETAILS
The COISR approach was tested using the National Institute of Standards and Thchnology (NIST) database of hand-printed digit fields, using fields 6-30 of the form,
which correspond to five different fields of length 2, 3, 4, 5, or 6 digits each. The training data included roughly 80,000 digits (BOO forms, 20,000 fields), and came from forms
labeled fOOOO..-f0499, and f1SOO-f1799 in the dataset. The test data consisted of roughly
20,000 digits (200 forms, 5,000 fields) and came from forms labeled f1800-f1899 and
f2OOO-f2099 in the dataset. The large test set was used because considerable variations
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Martin and Rashid
in test scores occurred with smaller test set sizes. The samples were scanned at a 300
pixel/inch resolution. Each field image was preprocessed to eliminate the white space
and box surrounding the digit field. Each field was then size normalized with respect
to the vertical height of the digit field to a vertical height of 20 pixels. Since the input
is size normalized to the vertical height of the field of characters, the actual number
of characters in the constant-width input window of 36 pixels varies depending on the
height-to-width ratio for each character. The scan rate was a 3-pixel increment across
the field.
A key design principle of the present approach is that highly accurate integrated segmentation and recognition requires training on both the shapes of characters and their
positions within the input window. The field images used for training were labeled with
the horizontal center positions of each character in the field. The human Iabeler simply
pointed at the horizontal center of each digit in sequence with a mouse cursor and
clicked on a mouse button. The horizontal position of each character was then paired
with its category label (0-9) in a data me. The labeling process is not unlike a human
reading teacher using a pointer to indicate the position of each character as he or she
reads aloud the sequence of characters making up the word or sentence. During testing
this position information is not used.
Position information about character centers is used to generate target output values
for each possible position of the input window as it scans a field of characters. When
the center position of a window is close to the center of a character, the target value
of that character's output node is set at the maximum, with the target value of the
NOT-CENTERED node set at the minimum. The activation values of all other characters' output nodes are set at the minimum. When the center position of a window
is close to the half-way point between two character centers, the target value of all
character output nodes are set to the minimum and the target value of the NOTCENTERED node is set to a maximum. Between these two extremes, the target values vary linearly with distance, creating a trapezoidal function (i.e., ~).
The neural network is a 2-hidden-Iayer backpropagation network, with local, shared
connections in the first hidden layer, and local connections in the second hidden layer
(see Figure 2). The first hidden layer consists of 2016 nodes, or more speCifically 18
independent groups of 112 (16x7) nodes, with each group having local, shared connee-
Output Layer
1st Hidden Layer
~------36----~--~
Input Layer
Figure 2: Architecture for the COISR-Exhaustive Scan Approach.
Recognizing Overlapping Hand-Printed C haracters
tions to the input layer. The local, overlapping receptive fields of size 6x8 are offset
by 2 pixels, such that the region covered by each group of nodes spans the input layer.
The second hidden layer consists of 180 nodes, having local, but NOT shared receptive
fields of size 6x3. The output layer consists of 11 nodes, with each of these nodes connected to all of the nodes in the 2nd hidden layer. The net has a total of 2927 nodes
(includes input and output nodes), and 157,068 connections. In a feedforward (nonlearning) mode on a DEC SOOO workstation, in which the net is scanning a field of digits,
the system processes about two digits per second, which includes image pre-processing
and the necessary number of feedforward passes on the net.
As the net scans horizontally, the activation values of the 11 output nodes create a trace
as shown in Figure 1. Th convert this to an ascii string corresponding to the digits in
the field, the state of the NOT-CENTERED node is monitored continuously. When
it's activation value falls below a threshold, a summing process begins for each of the
other nodes, and ends when the activation value of the NOT-CENTERED node exceeds the threshold. At this point the system decides that the input window has moved
off of a character. The system then classifies the character on the basis of which output
node has the highest summed activation for the position just passed over.
1.2 GENERALIZATION PERFORMANCE
As shown in Figure 3, the COISR technique achieves very low field-based error rates,
14
13.5
COISR
13
Field-Based Errors
12.5 I
Field
Field
12
Error
Rate
Reject Rate
11 .5
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11
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1.01%
\
10.5
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9
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7
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5-digits
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6
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Figure 3: Field-based lest Error and Reject Rates
particularly for a single classifier system. The error rates are field-based in the sense
that if the network mis-classifies one character in the field, the entire field is consid-
507
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Martin and Rashid
ered as mis-classified. Error rates pertain to the fields remaining, after rejection. Rejections are based on placing a threshold for the acceptable distance between the highest and the next highest running activation total. In this way, by varying the threshold,
the error rate can be traded off against the percentage of rejections. Since the reported
data apply to fields, the threshold applies to the smallest distance value found across
all of the characters in the field. Figure 4 provides examples, from the test set, of fields
that the COISR network correctly classifies.
t).3:J.i"3 S-s;,
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J6B'a.1<6 JfO &07
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Figure 4: 'lest Set Examples of Thuching and Broken Characters Correctly
Recognized
The COISR technique is a success in the sense that it does something that conventional
charac~er recognition systems can not do. It robustly recognizes character fields containing touching, overlapping, and broken characters. One problem with the approach,
however, lies with the exhaustive nature of the scan. The components needed to recognize a character in a given location are essentially replicated across the length of the
to-be-classified input field, at the degree of resolution necessary to recognize the
smallest and closest characters. While this has not presented any real difficulties for
the present system, which processes 2 characters per second, it is likely to be troublesome when extensions are made to two-dimensional scans and larger vocabularies. A
rough analogy with respect to human vision would be to require that all of the computational resources needed for recognizing objects at one point on the retina be replicated
for each resolvable point on the retina. This design carries the notion of a compound
eye to the ridiculous extreme.
2 COISR VERSION 2: SACCADIC SCAN
Thking a cue from natural vision systems, the second version of the COISR system uses
a saccadic scan. The system is trained to make ballistic eye movements, so that it can
effectively jump from character to character and over blank areas. This version is similar to the exhaustive scan version in the sense that a backprop net is trained to recognize
when it's input window is centered on a character, and if so, to classify the character.
In addition, the net is trained for navigation control (Pomerleau ,1991). At each point
in a field of characters, the net is trained to estimate the distance to the next character
on the right, and to estimate the degree to which the center-most character is off-center. The trained net accomodates for variations in character width, spacing between
characters, writing styles, and other factors. At run-time, the system uses the computed character classification and distances to navigate along a character field. If the
character classification judgment, for a given position, has a high degree of certainty,
the system accesses the next character distance infonnation computed by the net for
the current position and executes the jump. If the system gets off-track, so that a
Recognizing Overlapping Hand-Printed Characters
character can not be recognized with a high-degree of certainty, it makes a corrective
saccade by accessing the off-center character distance computed by the net for the current position. This action corresponds to making a second attempt to center the character within the input window.
The primary advantage of this approach, over the exhaustive scan, is improved efficiency, as illustrated in Figure S. The scanning input windows are shown at the top of the
figure, for each approach, and each character-containing input window, shown below
the scanned image for each approach, corresponds to a forward pass of the net. The
exhaustive scan version requires about 4 times as many forward passes as the saccadic
scan version. Greater improvements in effiCiency can be achieved with wider input
windows and images containing more blank areas. The system is still under development, but accuracy approaches that of the exhaustive scan system.
ExhausUve scan
Saccadic Scan
Figure 5: Number of Forward Passes for Saccadic & Exhaustive Scan Systems
3 COMPARISONS & CONCLUSIONS
In comparing accuracy rates between different OCR systems, one relevant factor that
should be reported is the number of classifiers used. For a given system, increasing
the number of classifiers typically reduces error rates but increases processing time.
The low error rates reported here for the COISR-Exhaustive Scan approach come
from a single classifier operating at 2 characters per second on a general purpose workstation. Most OCR systems employ multiple classifiers. For example, at the NIPS
workshops this year, Jonathan Hull described the University of Buffalo zip code recognition system that contains five classifiers and requires about one minute to process
a character. Keeler and Rumelhart, at this conference, also described a two-classifier
neural net system for NIST digit recognition. The fact that the COISR approach
achieved quite low error rates with a single classifier indicates that the approach is a
promising one.
Clearly, another relevant factor in comparing systems is the ability to recognize touching and broken characters, since this is a dominant stumbling block for current OCR
systems. Conventional systems can be altered to achieve integrated segmentation and
recognition in limited cases, but this involves a lot of hand-crafting and a significant
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Martin and Rashid
amount of time-consuming iterative processing (Fenrich. 1991). Essentially, multiple
segmenters are used, and classification is performed for each such possible segmentation. The final segmentation and recognition decisions can thus be inter-dependent,
but only at the cost of computing multiple segmentations and correspondingly, multiple classification decisions. The approach breaks down as the number of possible segmentations increases, as would occur for example if individual characters are broken
or touching in multiple places or if multiple letters in a sequence are connected. The
COISR system does not appear to have this problem.
The NIPS conference this year has included a number of other neural net approaches
to integrated segmentation and recognition in OCR domains. Tho approaches similar
to the COISR-Exhaustive Scan system were those described by Faggin and by Keeler
and Rumelhart. All three achieve integrated segmentation and recognition by convolving a neural network over a field of characters. Faggin described an analog hardware implementation of a neural-network-based OCR system that receives as input
a window that slides along the machine-print digit field at the bottom of bank checks.
Keeler and Rumelhart descnbed a self-organizing integrated segmentation and recognition (SOISR) system. Initially, it is trained on characters that have been pre-segmented
by a labeler effectively drawing a box around each. Then, in subsequent training, a
net, with these pre-trained weights, is duplicated repetitively across the extent of a fixed-width input field, and is further trained on examples of entire fields that contain
connecting or broken characters.
All three approaches have the weakness, described previously of performing essentially exhaustive scans or convolutions over the to-be-classified input field. This complaint is not necessarily directed at the specific applications dealt with at this year's
NIPS conference, particularly if operating at the high levels of effiCiency described by
Faggin. Nor is the complaint directed at tasks that only require the visual system to
focus on a few small clusters or fields in the larger, otherwise blank input field. In these
cases, low-resolution filters may be sufficient to efficiently remove blank areas and enable efficient integrated segmentation and recognition. Howevei, we use as an example, the saccadic scanning behavior of human vision in tasks, such as reading this paragrapl!. In such cases that require high-resolution sensitivity across a large, dense image
and classification of a very large vocabulary of symbols, it seems clear that other, more
flexible and efficient scanning mechanisms will be necessary. This high-density image
domain is the focus of the COISR-Saccadic Scan approach, which integrates not only
the segmentation and recognition of characters, but also control of the navigational
aspects of vision.
Acknowledgements
We thank Lori Barski, John Canfield, David Chapman, Roger Gaborski, Jay Pittman,
and Dave Rumelhart for helpful discussions and/or development of supporting image
handling and network software. I also thank Jonathan Martin for help with the position
labeling.
References
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Bonas, France. 'l3-27 September 1991.
Keeler, J. D., Rumelhart, David E., Leow, Wee-Kheng. Integrated segmentation and
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LeCun, Y., Boser, B., Denker, J., Henderson, D., Howard, R. E., Hubbard, W. & Jackel, L. D. Handwritten digit recognition,with a backpropagation network, in D. S. Thuretzky (Ed.) Advances in Neural Information Processing Systems 2. Morgan Kaufmann,
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Recognizing Overlapping Hand-Printed Characters
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3,989 | 4,610 | A Better Way to Pretrain Deep Boltzmann Machines
Geoffrey Hinton
Department of Computer Science
University of Toronto
[email protected]
Ruslan Salakhutdinov
Department of Statistics and Computer Science
University of Toronto
[email protected]
Abstract
We describe how the pretraining algorithm for Deep Boltzmann Machines
(DBMs) is related to the pretraining algorithm for Deep Belief Networks and
we show that under certain conditions, the pretraining procedure improves the
variational lower bound of a two-hidden-layer DBM. Based on this analysis, we
develop a different method of pretraining DBMs that distributes the modelling
work more evenly over the hidden layers. Our results on the MNIST and NORB
datasets demonstrate that the new pretraining algorithm allows us to learn better
generative models.
1
Introduction
A Deep Boltzmann Machine (DBM) is a type of binary pairwise Markov Random Field with multiple layers of hidden random variables. Maximum likelihood learning in DBMs, and other related
models, is very difficult because of the hard inference problem induced by the partition function
[3, 1, 12, 6]. Multiple layers of hidden units make learning in DBM?s far more difficult [13]. Learning meaningful DBM models, particularly when modelling high-dimensional data, relies on the
heuristic greedy pretraining procedure introduced by [7], which is based on learning a stack of modified Restricted Boltzmann Machines (RBMs). Unfortunately, unlike the pretraining algorithm for
Deep Belief Networks (DBNs), the existing procedure lacks a proof that adding additional layers
improves the variational bound on the log-probability that the model assigns to the training data.
In this paper, we first show that under certain conditions, the pretraining algorithm improves a
variational lower bound of a two-layer DBM. This result gives a much deeper understanding of
the relationship between the pretraining algorithms for Deep Boltzmann Machines and Deep Belief
Networks. Using this understanding, we introduce a new pretraining procedure for DBMs and show
that it allows us to learn better generative models of handwritten digits and 3D objects.
2
Deep Boltzmann Machines (DBMs)
A Deep Boltzmann Machine is a network of symmetrically coupled stochastic binary units. It contains a set of visible units v ? {0, 1}D , and a series of layers of hidden units h(1) ? {0, 1}F1 ,
h(2) ? {0, 1}F2 ,..., h(L) ? {0, 1}FL . There are connections only between units in adjacent layers.
Consider a DBM with three hidden layers, as shown in Fig. 1, left panel. The probability that the
DBM assigns to a visible vector v is:
1 X
P (v; ?) =
exp
Z(?)
h
X
(1)
(1)
Wij vi hj
+
X
ij
jl
1
(2) (1) (2)
Wjl hj hl
+
X
lm
(3) (2)
Wlm hl h(3)
m
,
(1)
Pretraining
?RBM?
h(3)
Deep Belief Network
h( 3 )
Deep Boltzmann Machine
W(3)
h(2)
h( 3 )
W (3 )
h( 2 )
RBM
h(2)
W (3 )
2W(2)
h( 2 )
W (2 )
h( 1 )
W(2)
?RBM?
h(1)
2W(1)
W (1 )
W(3)
2W(2)
h(1)
W (2 )
h( 1 )
v
2W(3)
W(1)
W(1)
W (1 )
v
v
?
Figure 1: Left: Deep Belief Network (DBN) and Deep Boltzmann Machine (DBM). The top two layers of a
DBN form an undirected graph and the remaining layers form a belief net with directed, top-down connections.
For a DBM, all the connections are undirected. Right Pretraining a DBM with three hidden layers consists of
learning a stack of RBMs that are then composed to create a DBM. The first and last RBMs in the stack need
to be modified by using asymmetric weights.
where h = {h(1) , h(2) , h(3) } are the set of hidden units, and ? = {W(1) , W(2) , W(3) } are
the model parameters, representing visible-to-hidden and hidden-to-hidden symmetric interaction
terms1 . Setting W(2) =0 and W(3) =0 recovers the Restricted Boltzmann Machine (RBM) model.
Approximate Learning: Exact maximum likelihood learning in this model is intractable, but efficient approximate learning of DBMs can be carried out by using a mean-field inference to estimate
data-dependent expectations, and an MCMC based stochastic approximation procedure to approximate the model?s expected sufficient statistics [7]. In particular, consider approximating the true
posterior P (h|v; ?) with a fully factorized approximating distribution over the three sets of hidden
QF1 QF2 QF3
(1)
(2)
(3)
(1)
units: Q(h|v; ?) = j=1
, ?(2) , ?(3) }
l=1
k=1 q(hj |v)q(hl |v)q(hk |v), where ? = {?
(l)
(l)
are the mean-field parameters with q(hi = 1) = ?i for l = 1, 2, 3. In this case, we can write
down the variational lower bound on the log-probability of the data, which takes a particularly simple form:
>
>
log P (v; ?) ? v> W(1) ?(1) + ?(1) W(2) ?(2) + ?(2) W(3) ?(3) ? log Z(?) + H(Q), (2)
where H(?) is the entropy functional. Learning proceeds by finding the value of ? that maximizes
this lower bound for the current value of model parameters ?, which results in a set of the mean-field
fixed-point equations. Given the variational parameters ?, the model parameters ? are then updated
to maximize the variational bound using stochastic approximation (for details see [7, 11, 14, 15]).
3
Pretraining Deep Boltzmann Machines
The above learning procedure works quite poorly when applied to DBMs that start with randomly
initialized weights. Hidden units in higher layers are very under-constrained so there is no consistent
learning signal for their weights. To alleviate this problem, [7] introduced a layer-wise pretraining
algorithm based on learning a stack of ?modified? Restricted Boltzmann Machines (RBMs).
The idea behind the pretraining algorithm is straightforward. When learning parameters of the first
layer ?RBM?, the bottom-up weights are constrained to be twice the top-down weights (see Fig. 1,
right panel). Intuitively, using twice the weights when inferring the states of the hidden units h(1)
compensates for the initial lack of top-down feedback. Conversely, when pretraining the last ?RBM?
in the stack, the top-down weights are constrained to be twice the bottom-up weights. For all the
intermediate RBMs the weights are halved in both directions when composing them to form a DBM,
as shown in Fig. 1, right panel.
This heuristic pretraining algorithm works surprisingly well in practice. However, it is solely motivated by the need to end up with a model that has symmetric weights, and does not provide any
1
We omit the bias terms for clarity of presentation.
2
useful insights into what is happening during the pretraining stage. Furthermore, unlike the pretraining algorithm for Deep Belief Networks (DBNs), it lacks a proof that each time a layer is added to
the DBM, the variational bound improves.
3.1
Pretraining Algorithm for Deep Belief Networks
We first briefly review the pretraining algorithm for Deep Belief Networks [2], which will form the
basis for developing a new pretraining algorithm for Deep Boltzmann Machines.
Consider pretraining a two-layer DBN using a stack of RBMs.
P After learning the first RBM in the
stack, we can write the generative model as: p(v; W(1) ) = h(1) p(v|h(1) ; W(1) )p(h(1) ; W(1) ).
The second RBMPin the stack attempts to replace the prior p(h(1) ; W(1) ) by a better model
p(h(1) ; W(2) ) = h(2) p(h(1) , h(2) ; W(2) ), thus improving the fit to the training data.
More formally, for any approximating distribution Q(h(1) |v), the DBN?s log-likelihood has the
following variational lower bound on the log probability of the training data {v1 , ..., vN }:
N
X
log P (vn ) ?
X
h
i X
EQ(h(1) |vn ) log P (vn |h(1) ; W(1) ) ?
KL Q(h(1) |vn )||P (h(1) ; W(1) ) .
n
n=1
(1)
n
(1)
(1)
(1)
We set Q(h |vn ; W ) = P (h |vn ; W ), which is the true factorial posterior of the first layer
RBM. Initially, when W(2) = W(1)> , Q(h(1) |vn ) defines the DBN?s true posterior over h(1) , and
the bound is tight. Maximizing the bound with respect to W(2) only affects the last KL term in the
above equation, and amounts to maximizing:
N
1 XX
Q(h(1) |vn ; W(1) )P (h(1) ; W(2) ).
N n=1 (1)
(3)
h
This is equivalent to training the second layer RBM with vectors drawn from Q(h(1) |v; W(1) ) as
data. Hence,
Pthe second RBM in the stack learns a better model of the mixture over all N training
cases: 1/N n Q(h(1) |vn ; W(1) ), called the ?aggregated posterior?. This scheme can be extended
to training higher-layer RBMs.
Observe that during the pretraining stage the whole prior of the lower-layer RBM is replaced by the
next RBM in the stack. This leads to the hybrid Deep Belief Network model, with the top two layers
forming a Restricted Boltzmann Machine, and the lower layers forming a directed sigmoid belief
network (see Fig. 1, left panel).
3.2
A Variational Bound for Pretraining a Two-layer Deep Boltzmann Machine
Consider a simple two-layer DBM with tied weights W(2) = W(1)> , as shown in Fig. 2a:
X
1
>
(1) (1)
(1)>
(1)> (2)
(1)
exp v W h + h
W
h
.
P (v; W ) =
Z(W(1) ) (1) (2)
h
(4)
,h
Similar to DBNs, for any approximate posterior Q(h(1) |v), we can write a variational lower bound
on the log probability that this DBM assigns to the training data:
N
X
n=1
log P (vn ) ?
X
h
i X
EQ(h(1) |vn ) log P (vn |h(1) ; W(1) ) ?
KL Q(h(1) |vn )||P (h(1) ; W(1) ) . (5)
n
n
The key insight is to note that the model?s marginal distribution over h(1) is the product of two
identical distributions, one defined by an RBM composed of h(1) and v, and the other defined by an
identical RBM composed of h(1) and h(2) [8]:
X
X
1
(1)
(1)
v> W(1) h(1)
h(2)> W(1) h(1)
P (h ; W ) =
e
e
.
(6)
Z(W(1) )
v
h(2)
|
{z
}|
{z
}
RBM with h(1) and v
3
RBM with h(1) and h(2)
h(2a) h(2b)
h(2)
W(2)
W(1)
h(2)
W(2)
W(2)
W(1)
h(1)
h(1)
h( 1 )
W(1)
h(2) = v
h(1)
W(1)
W(1)
v
v
a)
b)
v
c)
Figure 2: Left: Pretraining a Deep Boltzmann Machine with two hidden layers. a) The DBM with tied
weights. b) The second RBM with two sets of replicated hidden units, which will replace half of the 1st RBM?s
prior. c) The resulting DBM with modified second hidden layer. Right: The DBM with tied weights is trained
to model the data using one-step contrastive divergence.
The idea is to keep one of these two RBMs and replace the other by the square root of a better
prior P (h(1) ; W(2) ). In particular,
another RBM with two sets of replicated hidden units and tied
P
weights P (h(1) ; W(2) ) = h(2a) ,h(2b) P (h(1) , h(2a) , h(2b) ; W(2) ) is trained to be a better model
P
of the aggregated variational posterior N1 n Q(h(1) |vn ; W(1) ) of the first model (see Fig. 2b).
By initializing W(2) = W(1)> , the second-layer RBM has exactly the same prior over h(1) as the
original DBM. If the RBM is trained by maximizing the log likelihood objective:
XX
Q(h(1) |vn ) log P (h(1) ; W(2) ),
(7)
n h(1)
then we obtain:
X
X
KL(Q(h(1) |vn )||P (h(1) ; W(2) )) ?
KL(Q(h(1) |vn )||P (h(1) ; W(1) )).
n
(8)
n
Similar to Eq. 6, the distribution over h(1) defined by the second-layer RBM is also the product of
two identical distributions. Once the two RBMs are composed to form a two-layer DBM model (see
Fig. 2c), the marginal distribution over h(1) is the geometric mean of the two probability distributions: P (h(1) ; W(1) ), P (h(1) ; W(2) ) defined by the first and second-layer RBMs:
X
X
1
v> W(1) h(1)
h(1)> W(2) h(2)
P (h(1) ; W(1) , W(2) ) =
e
e
.
(9)
Z(W(1) , W(2) ) v
(2)
h
Based on Eqs. 8, 9, it is easy to show that the variational lower bound of Eq. 5 improves because
replacing half of the prior by a better model reduces the KL divergence from the variational posterior:
X
X
KL Q(h(1) |vn )||P (h(1) ; W(1) , W(2) ) ?
KL Q(h(1) |vn )||P (h(1) ; W(1) ) . (10)
n
n
Due to the convexity of asymmetric divergence, this is guaranteed to improve the variational bound
of the training data by at least half as much as fully replacing the original prior.
This result highlights a major difference between DBNs and DBMs. The procedure for adding an
extra layer to a DBN replaces the full prior over the previous top layer, whereas the procedure for
adding an extra layer to a DBM only replaces half of the prior. So in a DBM, the weights of the
bottom level RBM perform much more of the work than in a DBN, where the weights are only used
to define the last stage of the generative process P (v|h(1) ; W(1) ).
This result also suggests that adding layers to a DBM will give diminishing improvements in the
variational bound, compared to adding layers to a DBN. This may explain why DBMs with three
hidden layers typically perform worse than the DBMs with two hidden layers [7, 8]. On the other
hand, the disadvantage of the pretraining procedure for Deep Belief Networks is that the top-layer
RBM is forced to do most of the modelling work. This may also explain the need to use a large
number of hidden units in the top-layer RBM [2].
There is, however, a way to design a new pretraining algorithm that would spread the modelling work
more equally across all layers, hence bypassing shortcomings of the existing pretraining algorithms
for DBNs and DBMs.
4
Replacing 2/3 of the Prior
h(2)a h(2)b
W(1)
W(1)
h(2)a h(2)b h(2)c
W(2)
W(2)
W(2)
Practical Implementation
h(2)a
h( 2 ) b
W(2)
W(2)
h(1)
W
h(1)
W
v
(1)
3W
b)
(1)
W
3W(2)
h(1)
2W(2)
h( 1 )
W(1)
(1)
v
v
a)
2W(2)
h(1)
h( 1 )
(1)
h( 2 )
h(2)
v
c)
Figure 3: Left: Pretraining a Deep Boltzmann Machine with two hidden layers. a) The DBM with tied
weights. b) The second layer RBM is trained to model 2/3 of the 1st RBM?s prior. c) The resulting DBM with
modified second hidden layer. Right: The corresponding practical implementation of the pretraining algorithm
that uses asymmetric weights.
3.3
Controlling the Amount of Modelling Work done by Each Layer
Consider a slightly modified two-layer DBM with two groups of replicated 2nd -layer units, h(2a)
and h(2b) , and tied weights (see Fig. 3a). The model?s marginal distribution over h(1) is the product
of three identical RBM distributions, defined by h(1) and v, h(1) and h(2a) , and h(1) and h(2b) :
X
X
X
1
v> W(1) h(1)
h(2a)> W(1) h(1)
h(2b)> W(1) h(1)
P (h(1) ; W(1) ) =
e
e
e
.
Z(W(1) ) v
(2a)
(2b)
h
h
During the pretraining stage, we keep one of these RBMs and replace the other two by a better prior
P (h(1) ; W(2) ). To do so, similar to Sec. 3.2, we train another RBM, but with three sets of hidden
units and tied weights (see Fig. 3b). When we combine the two RBMs into a DBM, the marginal
distribution over h(1) is the geometric mean of three probability distributions: one defined by the
first-layer RBM, and the remaining two defined by the second-layer RBMs:
1
P (h(1) ; W(1) )P (h(1) ; W(2) )P (h(1) ; W(2) )
Z(W(1) , W(2) )
X
X
X
1
v> W(1) h(1)
h(2a)> W(2) h(1)
h(2b)> W(2) h(1)
=
e
e
e
.
Z(W(1) , W(2) ) v
(2a)
(2b)
P (h(1) ; W(1) , W(2) ) =
h
h
In this DBM, 2/3 of the first RBM?s prior over the first hidden layer has been replaced by the prior defined by the second-layer RBM. The variational bound on the training data is guaranteed to improve
by at least 2/3 as much as fully replacing the original prior. Hence in this slightly modified DBM
model, the second layer performs 2/3 of the modelling work compared to the first layer. Clearly, controlling the number of replicated hidden groups allows us to easily control the amount of modelling
work left to the higher layers in the stack.
3.4
Practical Implementation
So far, we have made the assumption that we start with a two-layer DBM with tied weights. We now
specify how one would train this initial set of tied weights W(1) .
Let us consider the original two-layer DBM in Fig. 2a with tied weights. If we knew the initial
state vector h(2) , we could train this DBM using one-step contrastive divergence (CD) with mean
field reconstructions of both the visible states v and the top-layer states h(2) , as shown in Fig. 2,
right panel. Instead, we simply set the initial state vector h(2) to be equal to the data, v. Using
mean-field reconstructions for v and h(2) , one-step CD is exactly equivalent to training a modified
?RBM? with only one hidden layer but with bottom-up weights that are twice the top-down weights,
as defined in the original pretraining algorithm (see Fig. 1, right panel). This way of training the
simple DBM with tied weights is unlikely to maximize the likelihood objective, but in practice it
produces surprisingly good models that reconstruct the training data well.
When learning the second RBM in the stack, instead of maintaining a set of replicated hidden groups,
it will often be convenient to approximate CD learning by training a modified RBM with one hidden
layer but with asymmetric bottom-up and top-down weights.
5
For example, consider pretraining a two-layer DBM, in which we would like to split the modelling
work between the 1st and 2nd -layer RBMs as 1/3 and 2/3. In this case, we train the first layer RBM
using one-step CD, but with the bottom-up weights constrained to be three times the top-down
weights (see Fig. 3, right panel). The conditional distributions needed for CD learning take form:
1
1
(1)
P (hj = 1|v) =
,
P (vi = 1|h(1) ) =
.
P
P
(1)
(1) (1)
1 + exp(? i 3Wij vi )
1 + exp(? j Wij hj )
Conversely, for the second modified RBM in the stack, the top-down weights are constrained to be
3/2 times the bottom-up weights. The conditional distributions take form:
1
1
(2)
(1)
P (hl = 1|h(1) ) =
, P (hj = 1|h(2) ) =
.
P
P
(2) (1)
(2) (2)
1 + exp(? j 2Wjl hj )
1 + exp(? l 3Wjl hl )
Note that this second-layer modified RBM simply approximates the proper RBM with three sets of
replicated h(2) groups. In practice, this simple approximation works well compared to training a
proper RBM, and is much easier to implement. When combining the RBMs into a two-layer DBM,
we end up with W(1) and 2W(2) in the first and second layers, each performing 1/3 and 2/3 of the
modelling work respectively:
X
1
exp v> W(1) h(1) + h(1)> 2W(2) h(2) .
(11)
P (v; ?) =
Z(?) (1) (2)
h
,h
Parameters of the entire model can be generatively fine-tuned using the combination of the meanfield algorithm and the stochastic approximation algorithm described in Sec. 2
4
Pretraining a Three Layer Deep Boltzmann Machine
In the previous section, we showed that provided we start with a two-layer DBM with tied weights,
we can train the second-layer RBM in a way that is guaranteed to improve the variational bound.
For the DBM with more than two layers, we have not been able to develop a pretraining algorithm
that is guaranteed to improve a variational bound. However, results of Sec. 3 suggest that using
simple modifications when pretraining a stack of RBMs would allow us to approximately control
the amount of modelling work done by each layer.
Consider learning a 3-layer DBM, in which
each layer is forced to perform approximately 1/3 of the modelling work. This
can easily be accomplished by learning a
stack of three modified RBMs. Similar
to the two-layer model, we train the first
layer RBM using one-step CD, but with the
bottom-up weights constrained to be three
times the top-down weights (see Fig. 4).
Two-thirds of this RBM?s prior will be
modelled by the 2nd and 3rd -layer RBMs.
Pretraining a 3-layer DBM
h(3)
h(3)
h(2)
4W(2)
h(1)
3W
(1)
W
2W(3)
4W(3)
h( 2 )
2W(3)
h( 2 )
2W(2)
3W(2)
h(1)
h( 1 )
W(1)
(1)
v
v
For the second modified RBM in the stack,
we use 4W(2) bottom-up and 3W(2) topdown. Note that we are using 4W(2)
bottom-up, as we are expecting to replace
half of the second RBM prior by a third RBM, hence splitting the remaining 2/3 of the work equally
between the top two layers. If we were to pretrain only a two-layer DBM, we would use 2W(2)
bottom-up and 3W(2) top-down, as discussed in Sec. 3.2.
Figure 4: Layer-wise pretraining of a 3-layer
Deep Boltzmann Machine.
For the last RBM in the stack, we use 2W(3) bottom-up and 4W(2) top-down. When combining the
three RBMs into a three-layer DBM, we end up with symmetric weights W(1) , 2W(2) , and 2W(3)
in the first, second, and third layers, with each layer performing 1/3 of the modelling work:
1 X
>
(1) (1)
(1)>
(2) (2)
(2)>
(3) (3)
P (v; ?) =
exp v W h + h
2W h + h
2W h
.
(12)
Z(?)
h
6
Algorithm 1 Greedy Pretraining Algorithm for a 3-layer Deep Boltzmann Machine
1: Train the 1st layer ?RBM? using one-step CD learning with mean field reconstructions of the visible vectors. Constrain the bottom-up weights, 3W(1) , to be three times the top-down weights, W(1) .
2: Freeze 3W(1) that defines the 1st layer of features, and use samples h(1) from P (h(1) |v; 3W(1) ) as the
data for training the second RBM.
3: Train the 2nd layer ?RBM? using one-step CD learning with mean field reconstructions of the visible
vectors. Set the bottom-up weights to 4W(1) , and the top-down weights to 3W(1) .
4: Freeze 4W(2) that defines the 2nd layer of features and use the samples h(3) from P (h(2) |h(1) ; 4W(2) )
as the data for training the next RBM.
5: Train the 3rd -layer ?RBM? using one-step CD learning with mean field reconstructions of its visible vectors. During the learning, set the bottom-up weights to 2W(3) , and the top-down weights to 4W(3) .
6: Use the weights {W(1) , 2W(2) , 2W(3) } to compose a three-layer Deep Boltzmann Machine.
The new pretraining procedure for a 3-layer DBM is shown in Alg. 1. Note that compared to the
original algorithm, it requires almost no extra work and can be easily integrated into existing code.
Extensions to training DBMs with more layers is trivial. As we show in our experimental results,
this pretraining can improve the generative performance of Deep Boltzmann Machines.
5
Experimental Results
In our experiments we used the MNIST and NORB datasets. During greedy pretraining, each layer
was trained for 100 epochs using one-step contrastive divergence. Generative fine-tuning of the
full DBM model, using mean-field together with stochastic approximation, required 300 epochs.
In order to estimate the variational lower-bounds achieved by different pretraining algorithms, we
need to estimate the global normalization constant. Recently, [10] demonstrated that Annealed
Importance Sampling (AIS) can be used to efficiently estimate the partition function of an RBM.
We adopt AIS in our experiments as well. Together with variational inference this will allow us to
obtain good estimates of the lower bound on the log-probability of the training and test data.
5.1
MNIST
The MNIST digit dataset contains 60,000 training and 10,000 test images of ten handwritten digits
(0 to 9), with 28?28 pixels. In our first experiment, we considered a standard two-layer DBM with
500 and 1000 hidden units2 , and used two different algorithms for pretraining it. The first pretraining
algorithm, which we call DBM-1/2-1/2, is the original algorithm for pretraining DBMs, as introduced
by [7] (see Fig. 1). Here, the modelling work between the 1st and 2nd -layer RBMs is split equally.
The second algorithm, DBM-1/3-2/3, uses a modified pretraining procedure of Sec. 3.4, so that the
second RBM in the stack ends up doing 2/3 of the modelling work compared to the 1st -layer RBM.
Results are shown in Table 1. Prior to the global generative fine-tuning, the estimate of the lower
bound on the average test log-probability for DBM-1/3-2/3 was ?108.65 per test case, compared to
?114.32 achieved by the standard pretraining algorithm DBM-1/2-1/2. The large difference of about
7 nats shows that leaving more of the modelling work to the second layer, which has a larger number
of hidden units, substantially improves the variational bound.
After the global generative fine-tuning, DBM-1/3-2/3 achieves a lower bound of ?83.43, which is
better compared to ?84.62, achieved by DBM-1/2-1/2. This is also lower compared to the lower
bound of ?85.97, achieved by a carefully trained two-hidden-layer Deep Belief Network [10].
In our second experiment, we pretrained a 3-layer Deep Boltzmann Machine with 500, 500, and
1000 hidden units. The existing pretraining algorithm, DBM-1/2-1/4-1/4, approximately splits the
modelling between three RBMs in the stack as 1/2, 1/4, 1/4, so the weights in the 1st -layer RBM
perform half of the work compared to the higher-level RBMs. On the other hand, the new pretraining
procedure (see Alg. 1), which we call DBM-1/3-1/3-1/3, splits the modelling work equally across all
three layers.
2
These architectures have been considered before in [7, 9], which allows us to provide a direct comparison.
7
Table 1: MNIST: Estimating the lower bound on the average training and test log-probabilities for two DBMs:
one with two layers (500 and 1000 hidden units), and the other one with three layers (500, 500, and 1000 hidden
units). Results are shown for various pretraining algorithms, followed by generative fine-tuning.
Pretraining
2 layers
3 layers
DBM-1/2-1/2
DBM-1/3-2/3
DBM-1/2-1/4-1/4
DBM-1/3-1/3-1/3
Generative Fine-Tuning
Train
Test
Train
Test
?113.32
?107.89
?116.74
?107.12
?114.32
?108.65
?117.38
?107.65
?83.61
?82.83
?84.49
?82.34
?84.62
?83.43
?85.10
?83.02
Table 2: NORB: Estimating the lower bound on the average training and test log-probabilities for two DBMs:
one with two layers (1000 and 2000 hidden units), and the other one with three layers (1000, 1000, and 2000
hidden units). Results are shown for various pretraining algorithms, followed by generative fine-tuning.
Pretraining
2 layers
3 layers
DBM-1/2-1/2
DBM-1/3-2/3
DBM-1/2-1/4-1/4
DBM-1/3-1/3-1/3
Generative Fine-Tuning
Train
Test
Train
Test
?640.94
?633.21
?641.87
?632.75
?643.87
?636.65
?645.06
?635.14
?598.13
?593.76
?598.98
?592.87
?601.76
?597.23
?602.84
?596.11
Table 1 shows that DBM-1/3-1/3-1/3 achieves a lower bound on the average test log-probability of
?107.65, improving upon DBM-1/2-1/4-1/4?s bound of ?117.38. The difference of about 10 nats
further demonstrates that during the pretraining stage, it is rather crucial to push more of the modelling work to the higher layers. After generative fine-tuning, the bound on the test log-probabilities
for DBM-1/3-1/3-1/3 was ?83.02, so with a new pretraining procedure, the three-hidden-layer DBM
performs slightly better than the two-hidden-layer DBM. With the original pretraining procedure,
the 3-layer DBM achieves a bound of ?85.10, which is worse than the bound of 84.62, achieved by
the 2-layer DBM, as reported by [7, 9].
5.2
NORB
The NORB dataset [4] contains images of 50 different 3D toy objects with 10 objects in each of
five generic classes: cars, trucks, planes, animals, and humans. Each object is photographed from
different viewpoints and under various lighting conditions. The training set contains 24,300 stereo
image pairs of 25 objects, 5 per class, while the test set contains 24,300 stereo pairs of the remaining,
different 25 objects. From the training data, 4,300 were set aside for validation. To deal with raw
pixel data, we followed the approach of [5] by first learning a Gaussian-binary RBM with 4000
hidden units, and then treating the the activities of its hidden layer as preprocessed binary data.
Similar to the MNIST experiments, we trained two Deep Boltzmann Machines: one with two layers
(1000 and 2000 hidden units), and the other one with three layers (1000, 1000, and 2000 hidden
units). Table 2 reveals that for both DBMs, the new pretraining achieves much better variational
bounds on the average test log-probability. Even after the global generative fine-tuning, Deep Boltzmann Machines, pretrained using a new algorithm, improve upon standard DBMs by at least 5 nats.
6
Conclusion
In this paper we provided a better understanding of how the pretraining algorithms for Deep Belief
Networks and Deep Boltzmann Machines are related, and used this understanding to develop a
different method of pretraining. Unlike many of the existing pretraining algorithms for DBNs and
DBMs, the new procedure can distribute the modelling work more evenly over the hidden layers.
Our results on the MNIST and NORB datasets demonstrate that the new pretraining algorithm allows
us to learn much better generative models.
Acknowledgments
This research was funded by NSERC, Early Researcher Award, and gifts from Microsoft and
Google. G.H. and R.S. are fellows of the Canadian Institute for Advanced Research.
8
References
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[5] V. Nair and G. E. Hinton. Implicit mixtures of restricted Boltzmann machines. In Advances in
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[11] T. Tieleman. Training restricted Boltzmann machines using approximations to the likelihood
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9
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3,990 | 4,611 | Gradient Weights help Nonparametric Regressors
Samory Kpotufe?
Max Planck Institute for Intelligent Systems
[email protected]
Abdeslam Boularias
Max Planck Institute for Intelligent Systems
[email protected]
Abstract
In regression problems over Rd , the unknown function f often varies more in
some coordinates than in others. We show that weighting each coordinate i with
the estimated norm of the ith derivative of f is an efficient way to significantly
improve the performance of distance-based regressors, e.g. kernel and k-NN regressors. We propose a simple estimator of these derivative norms and prove its
consistency. Moreover, the proposed estimator is efficiently learned online.
1
Introduction
In regression problems over Rd , the unknown function f might vary more in some coordinates than
in others, even though all coordinates might be relevant. How much f varies with coordinate i can
be captured by the norm kfi0 k1,? = EX |fi0 (X)| of the ith derivative fi0 = e>
i ?f of f . A simple
way to take advantage of the information in kfi0 k1,? is to weight each coordinate proportionally to an
estimate of kfi0 k1,? . The intuition, detailed in Section 2, is that the resulting data space behaves as a
low-dimensional projection to coordinates with large norm kfi0 k1,? , while maintaining information
about all coordinates. We show that such weighting can be learned efficiently, both in batch-mode
and online, and can significantly improve the performance of distance-based regressors in real-world
applications. In this paper we focus on the distance-based methods of kernel and k-NN regression.
For distance-based
methods, the weights can be incorporated into a distance function of the form
p
?(x, x0 ) = (x ? x0 )> W(x ? x0 ), where each element Wi of the diagonal matrix W is an estimate of kfi0 k1,? . This is not metric learning [1, 2, 3, 4] where the best ? is found by optimizing
over a sufficiently large space of possible metrics. Clearly metric learning can only yield better performance, but the optimization over a larger space will result in heavier preprocessing time, often
O(n2 ) on datasets of size n. Yet, preprocessing time is especially important in many modern applications where both training and prediction are done online (e.g. robotics, finance, advertisement,
recommendation systems). Here we do not optimize over a space of metrics, but rather estimate a
single metric ? based on the norms kfi0 k1,? . Our metric ? is efficiently obtained, can be estimated
online, and still significantly improves the performance of distance-based regressors.
To estimate kfi0 k1,? , one does not need to estimate fi0 well everywhere, just well on average. While
many elaborate derivative estimators exist (see e.g. [5]), we have to keep in mind our need for
fast but consistent estimator of kfi0 k1,? . We propose a simple estimator Wi which averages the
differences along i of an estimator fn,h of f . More precisely (see Section 3) Wi has the form
En |fn,h (X + tei ) ? fn,h (X ? tei )| /2t where En denotes the empirical expectation over a sample
n
{Xi }1 . Wi can therefore be updated online at the cost of just two estimates of fn,h .
In this paper fn,h is a kernel estimator, although any regression method might be used in estimating
kfi0 k1,? . We prove in Section 4 that, under mild conditions, Wi is a consistent estimator of the
?
Currently at Toyota Technological Institute Chicago, and affiliated with the Max Planck Institute.
1
(a) SARCOS robot, joint 7.
(b) Parkinson?s.
(c) Telecom.
n
o
Figure 1: Typical gradient weights Wi ? kfi0 k1,?
i?[d]
for some real-world datasets.
unknown norm kfi0 k1,? . Moreover we prove finite sample convergence bounds to help guide the
practical tuning of the two parameters t and h.
Most related work
As we mentioned above, metric learning is closest in spirit to the gradient-weighting approach presented here, but our approach is different from metric learning in that we do not search a space
of possible metrics, but rather estimate a single metric based on gradients. This is far more timeefficient and can be implemented in online applications which require fast preprocessing.
There exists many metric learning approaches, mostly for classification and few for regression (e.g.
[1, 2]). The approaches of [1, 2] for regression are meant for batch learning. Moreover [1] is limited
to Gaussian-kernel regression, and [2] is tuned to the particular problem of age estimation. For the
problem of classification, the metric-learning approaches of [3, 4] are meant for online applications,
but cannot be used in regression.
In the case of kernel regression and local polynomial regression, multiple bandwidths can be used,
one for each coordinate [6]. However, tuning d bandwidth parameters requires searching a d?d grid,
which is impractical even in batch mode. The method of [6] alleviates this problem, however only
in the particular case of local linear regression. Our method applies to any distance-based regressor.
Finally, the ideas presented here are related to recent notions of nonparametric sparsity where it is
assumed that the target function is well approximated by a sparse function, i.e. one which varies
little in most coordinates (e.g. [6], [? ]). Here we do not need sparsity, instead we only need the
target function to vary in some coordinates more than in others. Our approach therefore works even
in cases where the target function is far from sparse.
2
Technical motivation
In this section, we motivate the approach by considering the ideal situation where Wi = kfi0 k1,? .
Let?s consider regression on (X , ?), where the input space X ? Rd is connected. The prediction
performance of a distance-based estimator (e.g. kernel or k-NN) is well known to be the sum of its
variance and its bias [7]. Regression on (X , ?) decreases variance while keeping the bias controlled.
Regression variance decreases on (X , ?): The variance of a distance based estimate fn (x) is inversely proportional to the number of samples (and hence the mass) in a neighborhood of x (see
e.g. [8]). Let?s therefore compare the masses of ?-balls and Euclidean balls. Suppose some weights
largely dominate others, for instance in R2 , let kf20 k1,? kf10 k1,? . A ball B? in (X , ?) then takes
the ellipsoidal shape below which we contrast with the dotted Euclidean ball inside.
2
Relative to a Euclidean ball, a ball B? of similar1 radius has more mass in the direction e1 in which f
varies least. This intuition is made more precise in Lemma 1 below, which is proved in the appendix.
Essentially, let R ? [d] be the set of coordinates with larger weights Wi , then the mass of balls B?
behaves like the mass of balls in R|R| . Thus, effectively, regression in (X , ?) has variance nearly as
small as that for regression in the lower-dimensional space R|R| .
Note that the assumptions on the marginal ? in the lemma statement are verified for instance when
? has a continuous lower-bounded density on X . For simplicity we let (X , k?k) have diameter 1.
Lemma 1 (Mass of ?-balls). Consider any R ? [d] such that maxi?R
/ Wi < mini?R Wi . Suppose X ? ?1d [0, 1]d , and the marginal ? satisfies on (X , k?k), for some C1 , C2 : ?x ? X , ?r >
p
?
0, C1 rd ? ?(B(x, r)) ? C2 rd . Let ? , maxi?R Wi / mini?R Wi , 6R , maxi?R
d,
/ Wi ?
and let ?(X ) , supx,x0 ?X ?(x, x0 ).
Then for any ?(X ) > 26R , ?(B? (x, ?(X ))) ? C(2?)?|R| |R| , where C is independent of .
Ideally we would want |R| d and 6R ? 0, which corresponds to a sparse metric.
Regression bias remains bounded on (X , ?): The bias of distance-based regressors is controlled by
the smoothness of the unknown function f on (X , ?), i.e. how much f might differ for two close
points. Turning back to our earlier example in R2 , some points x0 that were originally far from x
along e1 might now be included in the estimate fn (x) on (X , ?). Intuitively, this should not add bias
to the estimate since f does not vary much in e1 . We have the following lemma.
Lemma 2 (Change in Lipschitz smoothness for f ). Suppose each derivative fi0 is bounded on X
by |fi0 |sup . Assume Wi > 0 whenever |fi0 |sup > 0. Denote by R the largest subset of [d] such that
|fi0 |sup > 0 for i ? R . We have for all x, x0 ? X,
!
X |fi0 |sup
0
?
|f (x) ? f (x )| ?
?(x, x0 ).
W
i
i?R
Applying the above lemma with Wi = 1, we see that in theP
original Euclidean space, the variation
in f relative to distance between points x, x0 , is of the order i?R |fi0 |sup . This variation in f is now
q
increased in (X , ?) by a factor of 1/ inf i?R kfi0 k1,? in the worst case. In this sense, the space
(X , ?) maintains information about all relevant coordinates. In contrast, information is lost under a
projection of the data in the likely scenario that all or most coordinates are relevant.
Finally, note that if all weights were close, the space (X , ?) is essentially equivalent to the original
(X , k?k), and we likely neither gain nor loose in performance, as confirmed by experiments. However, we observed that in practice, even when all coordinates are relevant, the gradient-weights vary
sufficiently (Figure 1) to observe significant performance gains for distance-based regressors.
Estimating kfi0 k1,?
3
n
In all that follows we are given n i.i.d samples (X, Y) = {(Xi , Yi )}i=1 , from some unknown
distribution with marginal ?. The marginal ? has support X ? Rd while the output Y ? R.
The kernel estimate at x is defined using any kernel K(u), positive on [0, 1/2], and 0 for u > 1. If
B(x, h) ? X = ?, fn,h (x) = En Y , otherwise
fn,?,h
? (x) =
n
X
i=1
X
K(?
?(x, Xi )/h)
Pn
? Yi =
wi (x)Yi ,
?(x, Xj )/h)
j=1 K(?
i=1
n
(1)
for some metric ?? and a bandwidth parameter h.
For the kernel regressor fn,h used to learn the metric ? below, ?? is the Euclidean metric. In the
1/d
analysis we assume the bandwidth for fn,h is set as h ? log2 (n/?)/n
, given a confidence
1
Accounting for the scale change induced by ? on the space X .
3
parameter 0 < ? < 1. In practice we would learn h by cross-validation, but for the analysis we only
need to know the existence of a good setting of h.
The metric is defined as
|fn,h (X + tei ) ? fn,h (X ? tei )|
Wi , En
? 1{An,i (X)} = En ?t,i fn,h (X) ? 1{An,i (X)} , (2)
2t
where An,i (X) is the event that enough samples contribute to the estimate ?t,i fn,h (X). For the
consistency result, we assume the following setting:
2d ln 2n + ln(4/?)
An,i (X) ? min ?n (B(X + sei , h/2)) ? ?n where ?n ,
.
n
s?{?t,t}
4
Consistency of the estimator Wi of kfi0 k1,?
4.1 Theoretical setup
4.1.1 Marginal ?
Without loss of generality we assume X has bounded diameter 1. The marginal is assumed to have
a continuous density on X and has mass everywhere on X : ?x ? X , ?h > 0, ?(B(x, h)) ? C? hd .
This is for instance the case if ? has a lower-bounded density on X . Under this assumption, for
samples X in dense regions, X ? tei is also likely to be in a dense region.
4.1.2 Regression function and noise
The output Y ? R is given as Y = f (X) + ?(X), where E?(X) = 0. We assume the following
general noise model: ?? > 0 there exists c > 0 such that supx?X PY |X=x (|?(x)| > c) ? ?.
We denote by CY (?) the infimum over all such c. For instance, suppose ?(X) has exponentially
decreasing tail, then ?? > 0, CY (?) ? O(ln 1/?). A last assumption on the noise is that the
variance of (Y |X = x) is upper-bounded by a constant ?Y2 uniformly over all x ? X .
Define the ? -envelope of X as X +B(0, ? ) , {z ? B(x, ? ), x ? X }. We assume there exists ? such
that f is continuously differentiable on the ? -envelope X + B(0, ? ). Furthermore, each derivative
0
fi0 (x) = e>
i ?f (x) is upper bounded on X + B(0, ? ) by |fi |sup and is uniformly continuous on
X + B(0, ? ) (this is automatically the case if the support X is compact).
4.1.3
Parameters varying with t
Our consistency results are expressed in terms of the following distributional quantities. For i ? [d],
define the (t, i)-boundary of X as ?t,i (X ) , {x : {x + tei , x ? tei } * X }. The smaller the mass
?(?t,i (X )) at the boundary, the better we approximate kfi0 k1,? .
The second type of quantity is t,i , supx?X , s?[?t,t] |fi0 (x) ? fi0 (x + sei )|.
Since ? has continuous density on X and ?f is uniformly continuous on X + B(0, ? ), we automatt?0
t?0
ically have ?(?t,i (X )) ???? 0 and t,i ???? 0.
4.2
Main theorem
Our main theorem bounds the error in estimating each norm kfi0 k1,? with Wi . The main technical
hurdles are in handling the various sample inter-dependencies introduced by both the estimates
fn,h (X) and the events An,i (X), and in analyzing the estimates at the boundary of X .
Theorem 1. Let t + h ? ? , and let 0 < ? < 1. There exist C = C(?, K(?)) and N = N (?) such
that the following holds with probability at least 1 ? 2?. Define A(n) , Cd ? log(n/?) ? CY2 (?/2n) ?
?Y2 / log2 (n/?). Let n ? N , we have for all i ? [d]:
?
?
!
r
1 r A(n)
X
ln
2d/?
0
0
0
+h?
|fi |sup ? + 2 |fi |sup
+ ? (?t,i (X )) + t,i .
Wi ? kfi k1,? ? ?
t
nhd
n
i?[d]
4
The bound suggest to set t in the order of h or larger. We need t to be small in order for ? (?t,i (X ))
and t,i to be small, but t need to be sufficiently large (relative to h) for the estimates fn,h (X + tei )
and fn,h (X ? tei ) to differ sufficiently so as to capture the variation in f along ei .
n??
n??
n??
The theorem immediately implies consistency for t ????? 0, h ????? 0,?h/t ????? 0, and
n??
(n/ log n)hd t2 ????? ?. This is satisfied for many settings, for example t ? h and h ? 1/ log n.
4.3 Proof of Theorem 1
The main difficulty in bounding Wi ? kfi0 k1,? is in circumventing certain depencies: both quantities fn,h (X) and An,i (X) depend not just on X ? X, but on other samples in X, and thus introduce
inter-dependencies between the estimates ?t,i fn,h (X) for different X ? X.
To handle these dependencies, we carefully decompose Wi ? kfi0 k1,? , i ? [d], starting with:
Wi ? kfi0 k1,? ? |Wi ? En |fi0 (X)|| + En |fi0 (X)| ? kfi0 k1,? .
(3)
The following simple lemma bounds the second term of (3).
Lemma 3. With probability at least 1 ? ?, we have for all i ? [d],
r
ln 2d/?
0
0
0
.
En |fi (X)| ? kfi k1,? ? |fi |sup ?
n
Proof. Apply a Chernoff bound, and a union bound on i ? [d].
Now the first term of equation (3) can be further bounded as
|Wi ? En |fi0 (X)|| ? Wi ? En |fi0 (X)| ? 1{An,i (X)} + En |fi0 (X)| ? 1{A?n,i (X)}
? Wi ? En |fi0 (X)| ? 1{An,i (X)} + |fi0 |sup ? En 1{A?n,i (X)} .
(4)
We will bound each term of (4) separately.
The next lemma bounds the second term of (4). It is proved in the appendix. The main technicality
in this lemma is that, for any X in the sample X, the event A?n,i (X) depends on other samples in X.
Lemma 4. Let ?t,i (X ) be defined as in Section (4.1.3). For n ? n(?), with probability at least
1 ? 2?, we have for all i ? [d],
r
ln 2d/?
En 1{A?n,i (X)} ?
+ ? (?t,i (X )) .
n
It remains to bound Wi ? En |fi0 (X)| ? 1{An,i (X)} . To this end we need to bring in f through the
following quantities:
f i , En |f (X + tei ) ? f (X ? tei )| ? 1{A (X)} = En ?t,i f (X) ? 1{A (X)}
W
n,i
n,i
2t
P
and for any x ? X , define f?n,h (x) , EY|X fn,h (x) = i wi (x)f (xi ).
f i is easily related to En |f 0 (X)| ? 1{A (X)} . This is done in Lemma 5 below. The
The quantity W
i
n,i
f i.
quantity f?n,h (x) is needed when relating Wi to W
Lemma 5. Define t,i as in Section (4.1.3). With probability at least 1 ? ?, we have for all i ? [d],
f
Wi ? En |fi0 (X)| ? 1{An,i (X)} ? t,i .
5
Proof. We have f (x + tei ) ? f (x ? tei) =
Rt
?t
fi0 (x + sei ) ds and therefore
2t (fi0 (x) ? t,i ) ? f (x + tei ) ? f (x ? tei) ? 2t (fi0 (x) + t,i ) .
1
It follows that 2t
|f (x + tei ) ? f (x ? tei)| ? |fi0 (x)| ? t,i , therefore
1
f
0
0
Wi ? En |fi (X)| ? 1{An,i (X)} ? En |f (x + tei ) ? f (x ? tei)| ? |fi (x)| ? t,i .
2t
f i . We have
It remains to relate Wi to W
f i =2t En (?t,i fn,h (X) ? ?t,i f (X)) ? 1{A (X)}
2t Wi ? W
n,i
?2 max En |fn,h (X + sei ) ? f (X + sei )| ? 1{An,i (X)}
s?{?t,t}
?2 max En fn,h (X + sei ) ? f?n,h (X + sei ) ? 1{An,i (X)}
s?{?t,t}
+ 2 max En f?n,h (X + sei ) ? f (X + sei ) ? 1{An,i (X)} .
s?{?t,t}
(5)
(6)
We first handle the bias term (6) in the next lemma which is given in the appendix.
Lemma 6 (Bias). Let t + h ? ? . We have for all i ? [d], and all s ? {t, ?t}:
X
En f?n,h (X + sei ) ? f (X + sei ) ? 1{An,i (X)} ? h ?
|fi0 |sup .
i?[d]
The variance term in (5) is handled in the lemma below. The proof is given in the appendix.
Lemma 7 (Variance terms). There exist C = C(?, K(?)) such that, with probability at least 1 ? 2?,
we have for all i ? [d], and all s ? {?t, t}:
s
Cd ? log(n/?)CY2 (?/2n) ? ?Y2
?
En fn,h (X + sei ) ? fn,h (X + sei ) ? 1{An,i (X)} ?
.
n(h/2)d
The next lemma summarizes the above results:
Lemma 8. Let t + h ? ? and let 0 < ? < 1. There exist C = C(?, K(?)) such that the following
holds with probability at least 1 ? 2?. Define A(n) , Cd ? log(n/?) ? CY2 (?/2n) ? ?Y2 / log2 (n/?).
We have
?r
?
X
1
A(n)
Wi ? En |fi0 (X)| ? 1{A (X)} ? ?
|fi0 |sup ? + t,i .
+h?
n,i
t
nhd
i?[d]
Proof. Apply lemmas 5, 6 and 7, in combination with equations 5 and 6.
To complete the proof of Theorem 1, apply lemmas 8 and 3 in combination with equations 3 and 4.
5
5.1
Experiments
Data description
We present experiments on several real-world regression datasets. The first two datasets describe the
dynamics of 7 degrees of freedom of robotic arms, Barrett WAM and SARCOS [9, 10]. The input
points are 21-dimensional and correspond to samples of the positions, velocities, and accelerations
of the 7 joints. The output points correspond to the torque of each joint. The far joints (1, 5, 7)
6
KR error
KR-? error
Barrett joint 1
0.50 ? 0.02
0.38? 0.03
Barrett joint 5
0.50 ? 0.03
0.35 ? 0.02
SARCOS joint 1
0.16 ? 0.02
0.14 ? 0.02
SARCOS joint 5
0.14 ? 0.02
0.12 ? 0.01
0.39 ? 0.02
0.41 ? 0.03
0.37 ? 0.01
0.38 ? 0.02
0.28 ? 0.05
0.32 ? 0.05
0.23 ? 0.03
0.23 ? 0.02
Concrete Strength
0.42 ? 0.05
0.37 ? 0.03
Wine Quality
0.75 ? 0.03
0.75 ? 0.02
Telecom
0.30?0.02
0.23?0.02
Ailerons
0.40?0.02
0.39?0.02
0.14 ? 0.02
0.14 ? 0.01
0.19 ? 0.02
0.19 ? 0.02
0.15?0.01
0.16?0.01
0.20?0.01
0.21?0.01
Barrett joint 1
0.41 ? 0.02
0.29 ? 0.01
Barrett joint 5
0.40 ? 0.02
0.30 ? 0.02
SARCOS joint 1
0.08 ? 0.01
0.07 ? 0.01
SARCOS joint 5
0.08 ? 0.01
0.07 ? 0.01
0.21 ? 0.04
0.13 ? 0.04
0.16 ? 0.03
0.16 ? 0.03
0.13 ? 0.01
0.14 ? 0.01
0.13 ? 0.01
0.13 ? 0.01
Concrete Strength
0.40 ? 0.04
0.38 ? 0.03
Wine Quality
0.73 ? 0.04
0.72 ? 0.03
Telecom
0.13?0.02
0.17?0.02
Ailerons
0.37?0.01
0.34?0.01
0.10 ? 0.01
0.11 ? 0.01
0.15 ? 0.01
0.15 ? 0.01
0.16?0.02
0.15?0.01
0.12?0.01
0.11?0.01
KR time
KR-? time
KR error
KR-? error
KR time
KR-? time
k-NN error
k-NN-? error
k-NN time
k-NN-? time
k-NN error
k-NN-? error
k-NN time
k-NN-? time
Housing
0.37 ?0.08
0.25 ?0.06
0.10 ?0.01
0.11 ?0.01
Parkinson?s
0.38?0.03
0.34?0.03
0.30?0.03
0.30?0.03
Housing
0.28 ?0.09
0.22?0.06
0.08 ?0.01
0.08 ?0.01
Parkinson?s
0.22?0.01
0.20?0.01
0.14?0.01
0.15?0.01
Table 1: Normalized mean square prediction errors and average prediction time per point (in milliseconds). The top two tables are for KR vs KR-? and the bottom two for k-NN vs k-NN-?.
0.1
0.44
KR error
KR?? error
0.08
0.35
KR error
KR?? error
0.42
KR error
KR?? error
0.3
0.4
error
error
0.25
error
0.06
0.38
0.04
0.2
0.36
0.02
0
0.15
0.34
1000
2000
3000
4000
0.32
5000
1000
number of training points
2000
3000
4000
0.1
5000
1000
2000
number of training points
(a) SARCOS, joint 7, with KR
(b) Ailerons with KR
0.025
4000
5000
6000
7000
(c) Telecom with KR
0.38
k?NN error
k?NN?? error
3000
number of training points
0.2
k?NN error
k?NN?? error
0.37
0.36
k?NN error
k?NN?? error
0.02
0.15
0.35
error
error
error
0.34
0.015
0.33
0.1
0.32
0.31
0.01
0.05
0.3
0.29
0.005
1000
2000
3000
4000
5000
number of training points
(d) SARCOS, joint 7, with k-NN
1000
2000
3000
4000
5000
number of training points
(e) Ailerons with k-NN
0
1000
2000
3000
4000
5000
6000
7000
number of training points
(f) Telecom with k-NN
Figure 2: Normalized mean square prediction error over 2000 points for varying training sizes.
Results are shown for k-NN and kernel regression (KR), with and without the metric ?.
correspond to different regression problems and are the only results reported. Expectedly, results for
the other joints are similarly good.
The other datasets are taken from the UCI repository [11] and from [12]. The concrete strength
dataset (Concrete Strength) contains 8-dimensional input points, describing age and ingredients of
concrete, the output points are the compressive strength. The wine quality dataset (Wine Quality)
contains 11-dimensional input points corresponding to the physicochemistry of wine samples, the
output points are the wine quality. The ailerons dataset (Ailerons) is taken from the problem of flying
a F16 aircraft. The 5-dimensional input points describe the status of the aeroplane, while the goal is
7
to predict the control action on the ailerons of the aircraft. The housing dataset (Housing) concerns
the task of predicting housing values in areas of Boston, the input points are 13-dimensional. The
Parkinson?s Telemonitoring dataset (Parkison?s) is used to predict the clinician?s Parkinson?s disease
symptom score using biomedical voice measurements represented by 21-dimensional input points.
We also consider a telecommunication problem (Telecom), wherein the 47-dimensional input points
and the output points describe the bandwidth usage in a network.
For all datasets we normalize each coordinate with its standard deviation from the training data.
5.2
Experimental setup
To learn the metric, we set h by cross-validation on half the training points, and we set t = h/2
for all datasets. Note that in practice we might want to also tune t in the range of h for even
better performance than reported here. The event An,i (X) is set to reject the gradient estimate
?n,i fn,h (X) at X if no sample contributed to one the estimates fn,h (X ? tei ).
In each experiment, we compare kernel regression in the euclidean metric space (KR) and in the
learned metric space (KR-?), where we use a box kernel for both. Similar comparisons are made
using k-NN and k-NN-?. All methods are implemented using a fast neighborhood search procedure,
namely the cover-tree of [13], and we also report the average prediction times so as to confirm that,
on average, time-performance is not affected by using the metric.
The parameter k in k-NN/k-NN-?, and the bandwidth in KR/KR-? are learned by cross-validation
on half of the training points. We try the same range of k (from 1 to 5 log n) for both k-NN and
k-NN-?. We try the same range of bandwidth/space-diameter (a grid of size 0.02 from 1 to 0.02 )
for both KR and KR-?: this is done efficiently by starting with a log search to detect a smaller range,
followed by a grid search on a smaller range.
Table 5 shows the normalized Mean Square Errors (nMSE) where the MSE on the test set is normalized by variance of the test output. We use 1000 training points in the robotic datasets, 2000 training
points in the Telecom, Parkinson?s, Wine Quality, and Ailerons datasets, and 730 training points in
Concrete Strength, and 300 in Housing. We used 2000 test points in all of the problems, except for
Concrete, 300 points, and Housing, 200 points. Averages over 10 random experiments are reported.
For the larger datasets (SARCOS, Ailerons, Telecom) we also report the behavior of the algorithms,
with and without metric, as the training size n increases (Figure 2).
5.3 Discussion of results
From the results in Table 5 we see that virtually on all datasets the metric helps improve the performance of the distance based-regressor even though we did not tune t to the particular problem (remember t = h/2 for all experiments). The only exceptions are for Wine Quality where the learned
weights are nearly uniform, and for Telecom with k-NN. We noticed that the Telecom dataset has
a lot of outliers and this probably explains the discrepancy, besides from the fact that we did not
attempt to tune t. Also notice that the error of k-NN is already low for small sample sizes, making
it harder to outperform. However, as shown in Figure 2, for larger training sizes k-NN-? gains on
k-NN. The rest of the results in Figure 2 where we vary n are self-descriptive: gradient weighting
clearly improves the performance of the distance-based regressors.
We also report the average prediction times in Table 5. We see that running the distance-based
methods with gradient weights does not affect estimation time. Last, remember that the metric can
be learned online at the cost of only 2d times the average kernel estimation time reported.
6 Final remarks
Gradient weighting is simple to implement, computationally efficient in batch-mode and online, and
most importantly improves the performance of distance-based regressors on real-world applications.
In our experiments, most or all coordinates of the data are relevant, yet some coordinates are more
important than others. This is sufficient for gradient weighting to yield gains in performance. We
believe there is yet room for improvement given the simplicity of our current method.
8
References
[1] Kilian Q. Weinberger and Gerald Tesauro. Metric learning for kernel regression. Journal of
Machine Learning Research - Proceedings Track, 2:612?619, 2007.
[2] Bo Xiao, Xiaokang Yang, Yi Xu, and Hongyuan Zha. Learning distance metric for regression
by semidefinite programming with application to human age estimation. In Proceedings of the
17th ACM international conference on Multimedia, pages 451?460, 2009.
[3] Shai Shalev-shwartz, Yoram Singer, and Andrew Y. Ng. Online and batch learning of pseudometrics. In ICML, pages 743?750. ACM Press, 2004.
[4] Jason V. Davis, Brian Kulis, Prateek Jain, Suvrit Sra, and Inderjit S. Dhillon. Informationtheoretic metric learning. In ICML, pages 209?216, 2007.
[5] W. H?ardle and T. Gasser. On robust kernel estimation of derivatives of regression functions.
Scandinavian journal of statistics, pages 233?240, 1985.
[6] J. Lafferty and L. Wasserman. Rodeo: Sparse nonparametric regression in high dimensions.
Arxiv preprint math/0506342, 2005.
[7] L. Rosasco, S. Villa, S. Mosci, M. Santoro, and A. Verri. Nonparametric sparsity and regularization. http://arxiv.org/abs/1208.2572, 2012.
[8] L. Gyorfi, M. Kohler, A. Krzyzak, and H. Walk. A Distribution Free Theory of Nonparametric
Regression. Springer, New York, NY, 2002.
[9] S. Kpotufe. k-NN Regression Adapts to Local Intrinsic Dimension. NIPS, 2011.
[10] Duy Nguyen-Tuong, Matthias W. Seeger, and Jan Peters. Model learning with local gaussian
process regression. Advanced Robotics, 23(15):2015?2034, 2009.
[11] Duy Nguyen-Tuong and Jan Peters. Incremental online sparsification for model learning in
real-time robot control. Neurocomputing, 74(11):1859?1867, 2011.
[12] A. Frank and A. Asuncion. UCI machine learning repository. http://archive.ics.
uci.edu/ml. University of California, Irvine, School of Information and Computer Sciences, 2012.
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Porto, Department of Computer Science, 2012.
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3,991 | 4,612 | Multi-criteria Anomaly Detection using
Pareto Depth Analysis
Ko-Jen Hsiao, Kevin S. Xu, Jeff Calder, and Alfred O. Hero III
University of Michigan, Ann Arbor, MI, USA 48109
{coolmark,xukevin,jcalder,hero}@umich.edu
Abstract
We consider the problem of identifying patterns in a data set that exhibit anomalous behavior, often referred to as anomaly detection. In most anomaly detection
algorithms, the dissimilarity between data samples is calculated by a single criterion, such as Euclidean distance. However, in many cases there may not exist a
single dissimilarity measure that captures all possible anomalous patterns. In such
a case, multiple criteria can be defined, and one can test for anomalies by scalarizing the multiple criteria using a linear combination of them. If the importance
of the different criteria are not known in advance, the algorithm may need to be
executed multiple times with different choices of weights in the linear combination. In this paper, we introduce a novel non-parametric multi-criteria anomaly
detection method using Pareto depth analysis (PDA). PDA uses the concept of
Pareto optimality to detect anomalies under multiple criteria without having to
run an algorithm multiple times with different choices of weights. The proposed
PDA approach scales linearly in the number of criteria and is provably better than
linear combinations of the criteria.
1
Introduction
Anomaly detection is an important problem that has been studied in a variety of areas and used in diverse applications including intrusion detection, fraud detection, and image processing [1, 2]. Many
methods for anomaly detection have been developed using both parametric and non-parametric approaches. Non-parametric approaches typically involve the calculation of dissimilarities between
data samples. For complex high-dimensional data, multiple dissimilarity measures corresponding
to different criteria may be required to detect certain types of anomalies. For example, consider the
problem of detecting anomalous object trajectories in video sequences. Multiple criteria, such as
dissimilarity in object speeds or trajectory shapes, can be used to detect a greater range of anomalies
than any single criterion. In order to perform anomaly detection using these multiple criteria, one
could first combine the dissimilarities using a linear combination. However, in many applications,
the importance of the criteria are not known in advance. It is difficult to determine how much weight
to assign to each dissimilarity measure, so one may have to choose multiple weights using, for example, a grid search. Furthermore, when the weights are changed, the anomaly detection algorithm
needs to be re-executed using the new weights.
In this paper we propose a novel non-parametric multi-criteria anomaly detection approach using
Pareto depth analysis (PDA). PDA uses the concept of Pareto optimality to detect anomalies without
having to choose weights for different criteria. Pareto optimality is the typical method for defining
optimality when there may be multiple conflicting criteria for comparing items. An item is said to
be Pareto-optimal if there does not exist another item that is better or equal in all of the criteria. An
item that is Pareto-optimal is optimal in the usual sense under some combination, not necessarily
linear, of the criteria. Hence, PDA is able to detect anomalies under multiple combinations of the
criteria without explicitly forming these combinations.
1
6
5
3
3
3
|?y|
|?y|
y
4
2
2
2
1
1
1
0
0
1
2
3
x
4
5
6
0
0
1
2
|?x|
3
0
0
1
2
|?x|
3
Figure 1: Left: Illustrative example with 40 training samples (blue x?s) and 2 test samples (red circle
and triangle) in R2 . Center: Dyads for the training samples (black dots) along with first 20 Pareto
fronts (green lines) under two criteria: |?x| and |?y|. The Pareto fronts induce a partial ordering on
the set of dyads. Dyads associated with the test sample marked by the red circle concentrate around
shallow fronts (near the lower left of the figure). Right: Dyads associated with the test sample
marked by the red triangle concentrate around deep fronts.
The PDA approach involves creating dyads corresponding to dissimilarities between pairs of data
samples under all of the dissimilarity measures. Sets of Pareto-optimal dyads, called Pareto fronts,
are then computed. The first Pareto front (depth one) is the set of non-dominated dyads. The second
Pareto front (depth two) is obtained by removing these non-dominated dyads, i.e. peeling off the
first front, and recomputing the first Pareto front of those remaining. This process continues until
no dyads remain. In this way, each dyad is assigned to a Pareto front at some depth (see Fig. 1 for
illustration). Nominal and anomalous samples are located near different Pareto front depths; thus
computing the front depths of the dyads corresponding to a test sample can discriminate between
nominal and anomalous samples. The proposed PDA approach scales linearly in the number of criteria, which is a significant improvement compared to selecting multiple weights via a grid search,
which scales exponentially in the number of criteria. Under assumptions that the multi-criteria dyads
can be modeled as a realizations from a smooth K-dimensional density we provide a mathematical
analysis of the behavior of the first Pareto front. This analysis shows in a precise sense that PDA
can outperform a test that uses a linear combination of the criteria. Furthermore, this theoretical prediction is experimentally validated by comparing PDA to several state-of-the-art anomaly detection
algorithms in two experiments involving both synthetic and real data sets.
The rest of this paper is organized as follows. We discuss related work in Section 2. In Section 3 we
provide an introduction to Pareto fronts and present a theoretical analysis of the properties of the first
Pareto front. Section 4 relates Pareto fronts to the multi-criteria anomaly detection problem, which
leads to the PDA anomaly detection algorithm. Finally we present two experiments in Section 5 to
evaluate the performance of PDA.
2
Related work
Several machine learning methods utilizing Pareto optimality have previously been proposed; an
overview can be found in [3]. These methods typically formulate machine learning problems as
multi-objective optimization problems where finding even the first Pareto front is quite difficult.
These methods differ from our use of Pareto optimality because we consider multiple Pareto fronts
created from a finite set of items, so we do not need to employ sophisticated methods in order to find
these fronts.
Hero and Fleury [4] introduced a method for gene ranking using Pareto fronts that is related to our
approach. The method ranks genes, in order of interest to a biologist, by creating Pareto fronts of
the data samples, i.e. the genes. In this paper, we consider Pareto fronts of dyads, which correspond
to dissimilarities between pairs of data samples rather than the samples themselves, and use the
distribution of dyads in Pareto fronts to perform multi-criteria anomaly detection rather than ranking.
Another related area is multi-view learning [5, 6], which involves learning from data represented by
multiple sets of features, commonly referred to as ?views?. In such case, training in one view helps to
2
improve learning in another view. The problem of view disagreement, where samples take different
classes in different views, has recently been investigated [7]. The views are similar to criteria in
our problem setting. However, in our setting, different criteria may be orthogonal and could even
give contradictory information; hence there may be severe view disagreement. Thus training in one
view could actually worsen performance in another view, so the problem we consider differs from
multi-view learning. A similar area is that of multiple kernel learning [8], which is typically applied
to supervised learning problems, unlike the unsupervised anomaly detection setting we consider.
Finally, many other anomaly detection methods have previously been proposed. Hodge and Austin
[1] and Chandola et al. [2] both provide extensive surveys of different anomaly detection methods
and applications. Nearest neighbor-based methods are closely related to the proposed PDA approach. Byers and Raftery [9] proposed to use the distance between a sample and its kth-nearest
neighbor as the anomaly score for the sample; similarly, Angiulli and Pizzuti [10] and Eskin et al.
[11] proposed to the use the sum of the distances between a sample and its k nearest neighbors.
Breunig et al. [12] used an anomaly score based on the local density of the k nearest neighbors
of a sample. Hero [13] and Sricharan and Hero [14] introduced non-parametric adaptive anomaly
detection methods using geometric entropy minimization, based on random k-point minimal spanning trees and bipartite k-nearest neighbor (k-NN) graphs, respectively. Zhao and Saligrama [15]
proposed an anomaly detection algorithm k-LPE using local p-value estimation (LPE) based on a
k-NN graph. These k-NN anomaly detection schemes only depend on the data through the pairs of
data points (dyads) that define the edges in the k-NN graphs.
All of the aforementioned methods are designed for single-criteria anomaly detection. In the multicriteria setting, the single-criteria algorithms must be executed multiple times with different weights,
unlike the PDA anomaly detection algorithm that we propose in Section 4.
3
Pareto depth analysis
The PDA method proposed in this paper utilizes the notion of Pareto optimality, which has been
studied in many application areas in economics, computer science, and the social sciences among
others [16]. We introduce Pareto optimality and define the notion of a Pareto front.
Consider the following problem: given n items, denoted by the set S, and K criteria for evaluating
each item, denoted by functions f1 , . . . , fK , select x ? S that minimizes [f1 (x), . . . , fK (x)]. In
most settings, it is not possible to identify a single item x that simultaneously minimizes fi (x)
for all i ? {1, . . . , K}. A minimizer can be found by combining the K criteria using a linear
combination of the fi ?s and finding the minimum of the combination. Different choices of (nonnegative) weights in the linear combination could result in different minimizers; a set of items that
are minimizers under some linear combination can then be created by using a grid search over the
weights, for example.
A more powerful approach involves finding the set of Pareto-optimal items. An item x is said to
strictly dominate another item x? if x is no greater than x? in each criterion and x is less than
x? in at least one criterion. This relation can be written as x x? if fi (x) ? fi (x? ) for each i
and fi (x) < fi (x? ) for some i. The set of Pareto-optimal items, called the Pareto front, is the set
of items in S that are not strictly dominated by another item in S. It contains all of the minimizers
that are found using linear combinations, but also includes other items that cannot be found by linear
combinations. Denote the Pareto front by F1 , which we call the first Pareto front. The second Pareto
front can be constructed by finding items that are not strictly dominated by any of the remaining
items, which are members of the set S \ F1 . More generally, define the ith Pareto front by
?
?
i?1
[
Fi = Pareto front of the set S \ ?
Fj ? .
j=1
For convenience, we say that a Pareto front Fi is deeper than Fj if i > j.
3.1
Mathematical properties of Pareto fronts
The distribution of the number of points on the first Pareto front was first studied by BarndorffNielsen and Sobel in their seminal work [17]. The problem has garnered much attention since; for a
3
survey of recent results see [18]. We will be concerned here with properties of the first Pareto front
that are relevant to the PDA anomaly detection algorithm and thus have not yet been considered in
the literature. Let Y1 , . . . , Yn be independent and identically distributed (i.i.d.) on Rd with density
function f : Rd ? R. For a measurable set A ? Rd , we denote by FA the points on the first Pareto
front of Y1 , . . . , Yn that belong to A. For simplicity, we will denote F1 by F and use |F| for the
cardinality of F. In the general Pareto framework, the points Y1 , . . . , Yn are the images in Rd of n
feasible solutions to some optimization problem under a vector of objective functions of length d.
In the context of this paper, each point Yl corresponds to a dyad Dij , which we define in Section 4,
and d = K is the number of criteria. A common approach in multi-objective optimization is linear
scalarization [16], which constructs a new single criterion as a convex combination of the d criteria.
It is well-known, and easy to see,
S that linear scalarization will only identify Pareto points on the
boundary of the convex hull of x?F (x + Rd+ ), where Rd+ = {x ? Rd | xi ? 0, i = 1 . . . , d}.
Although this is a common motivation for Pareto methods, there are, to the best of our knowledge,
no results in the literature regarding how many points on the Pareto front are missed by scalarization.
We present such a result here. We define
)
( d
[
X
L=
argmin
?i xi , Sn = {Y1 , . . . , Yn }.
??Rd
+
x?Sn
i=1
The subset L ? F contains all Pareto-optimal points that can be obtained by some selection of
weights for linear scalarization. We aim to study how large L can get, compared to F, in expectation.
In the context of this paper, if some Pareto-optimal points are not identified, then the anomaly
score (defined in section 4.2) will be artificially inflated, making it more likely that a non-anomalous
sample will be rejected. Hence the size of F \ L is a measure of how much the anomaly score is
inflated and the degree to which Pareto methods will outperform linear scalarization.
Pareto points in F \ L are a result of non-convexities in the Pareto front. We study two kinds of
non-convexities: those induced by the geometry of the domain of Y1 , . . . , Yn , and those induced by
randomness. We first consider the geometry of the domain. Let ? ? Rd be bounded and open with
a smooth boundary ?? and suppose the density f vanishes outside of ?. For a point z ? ?? we
denote by ?(z) = (?1 (z), . . . , ?d (z)) the unit inward normal to ??. For T ? ??, define Th ? ? by
Th = {z + t? | z ? T, 0 < t ? h}. Given h > 0 it is not hard to see that all Pareto-optimal points
will almost surely lie in ??h for large enough n, provided the density f is strictly positive on ??h .
Hence it is enough to study the asymptotics for E|FTh | for T ? ?? and h > 0.
Theorem 1. Let f ? C 1 (?) with inf ? f > 0. Let T ? ?? be open and connected such that
inf min(?1 (z), . . . , ?d (z)) ? ? > 0,
z?T
and
{y ? ? : y x} = {x}, for x ? T.
Then for h > 0 sufficiently small, we have
d?2
d?1
E|FTh | = ?n d + ? ?d?1 O n d
as n ? ?,
Z
d?1
1
1
where ? = d?1 (d!) d ?(d?1 )
f (z) d (?1 (z) ? ? ? ?d (z)) d dz.
T
The proof of Theorem 1 is postponed to Section 1 of the supplementary material. Theorem 1 shows
asymptotically how many Pareto points are contributed on average by the segment T ? ??. The
number of points contributed depends only on the geometry of ?? through the direction of its normal
vector ? and is otherwise independent of the convexity of ??. Hence, by using Pareto methods, we
will identify significantly more Pareto-optimal points than linear scalarization when the geometry
of ?? includes non-convex regions. For example, if T ? ?? is non-convex (see left panel of
Figure 2) and satisfies the hypotheses of Theorem 1, then for large enough n, all Pareto points in
a neighborhood of T will be unattainable by scalarization. Quantitatively, if f ? C on T , then
d?1
d?2
d?1
1
E|F \ L| ? ?n d + ? ?d?1 O(n d ), as n ? ?, where ? ? d?1 (d!) d ?(d?1 )|T |?C d and |T |
is the d ? 1 dimensional Hausdorff measure of T . It has recently come to our attention that Theorem
1 appears in a more general form in an unpublished manuscript of Baryshnikov and Yukich [19].
We now study non-convexities in the Pareto front which occur due to inherent randomness in the
samples. We show that, even in the case where ? is convex, there are still numerous small-scale
non-convexities in the Pareto front that can only be detected by Pareto methods. We illustrate this in
the case of the Pareto box problem for d = 2.
4
0.25
0.2
0.15
0.1
0.05
0
?0.05
?0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 2: Left: Non-convexities in the Pareto front induced by the geometry of the domain ? (Theorem 1). Right: Non-convexities due to randomness in the samples (Theorem 2). In each case, the
larger points are Pareto-optimal, and the large black points cannot be obtained by scalarization.
Theorem 2. Let Y1 , . . . , Yn be independent and uniformly distributed on [0, 1]2 . Then
1
5
ln n + O(1) ? E|L| ? ln n + O(1), as n ? ?.
2
6
The proof of Theorem 2 is also postponed to Section 1 of the supplementary material. A proof that
E|F| = ln n + O(1) as n ? ? can be found in [17]. Hence Theorem 2 shows that, asymptotically
and in expectation, only between 12 and 65 of the Pareto-optimal points can be obtained by linear
scalarization in the Pareto box problem. Experimentally, we have observed that the true fraction of
points is close to 0.7. This means that at least 16 (and likely more) of the Pareto points can only be
obtained via Pareto methods even when ? is convex. Figure 2 gives an example of the sets F and L
from the two theorems.
4
Multi-criteria anomaly detection
Assume that a training set XN = {X1 , . . . , XN } of nominal data samples is available. Given a test
sample X, the objective of anomaly detection is to declare X to be an anomaly if X is significantly
different from samples in XN . Suppose that K > 1 different evaluation criteria are given. Each criterion is associated with a measure for computing dissimilarities. Denote the dissimilarity between
Xi and Xj computed using the measure corresponding to the lth criterion by dl (i, j).
We define a dyad by Dij = [d1 (i, j), . . . , dK (i, j)]T ? RK
+ , i ? {1, . . . , N }, j ? {1, . . . , N } \ i.
Each dyad
D
corresponds
to
a
connection
between
samples
Xi and Xj . Therefore, there are in
ij
total N2 different dyads. For convenience, denote the set of all dyads by D and the space of all
dyads RK
+ by D. By the definition of strict dominance in Section 3, a dyad Dij strictly dominates
another dyad Di? j ? if dl (i, j) ? dl (i? , j ? ) for all l ? {1, . . . , K} and dl (i, j) < dl (i? , j ? ) for some
l. The first Pareto front F1 corresponds to the set of dyads from D that are not strictly dominated by
any other dyads from D. The second Pareto front F2 corresponds to the set of dyads from D \ F1
that are not strictly dominated by any other dyads from D \ F1 , and so on, as defined in Section 3.
Recall that we refer to Fi as a deeper front than Fj if i > j.
4.1
Pareto fronts of dyads
For each sample Xn , there are N ? 1 dyads corresponding to its connections with the other N ? 1
samples. Define the set of N ? 1 dyads associated with Xn by Dn . If most dyads in Dn are located
at shallow Pareto fronts, then the dissimilarities between Xn and the other N ? 1 samples are small
under some combination of the criteria. Thus, Xn is likely to be a nominal sample. This is the basic
idea of the proposed multi-criteria anomaly detection method using PDA.
We construct Pareto fronts F1 , . . . , FM of the dyads from the training set, where the total number
of fronts M is the required number of fronts such that each dyad is a member of a front. When a test
sample X is obtained, we create new dyads corresponding to connections between X and training
samples, as illustrated in Figure 1. Similar to many other anomaly detection methods, we connect
each test sample to its k nearest neighbors. k could be different for each criterion, so we denote ki
PK
as the choice of k for criterion i. We create s = i=1 ki new dyads, which we denote by the set
5
Algorithm 1 PDA anomaly detection algorithm.
Training phase:
1: for l = 1 ? K do
2:
Calculate pairwise dissimilarities dl (i, j) between all training samples Xi and Xj
3: Create dyads Dij = [d1 (i, j), . . . , dK (i, j)] for all training samples
4: Construct Pareto fronts on set of all dyads until each dyad is in a front
Testing phase:
1: nb ? [ ] {empty list}
2: for l = 1 ? K do
3:
Calculate dissimilarities between test sample X and all training samples in criterion l
4:
nbl ? kl nearest neighbors of X
5:
nb ? [nb, nbl ] {append neighbors to list}
6: Create s new dyads Dinew between X and training samples in nb
7: for i = 1 ? s do
8:
Calculate depth ei of Dinew
Ps
9: Declare X an anomaly if v(X) = (1/s) i=1 ei > ?
Dnew = {D1new , D2new , . . . , Dsnew }, corresponding to the connections between X and the union of the
ki nearest neighbors in each criterion i. In other words, we create a dyad between X and Xj if Xj
is among the ki nearest neighbors1 of X in any criterion i. We say that Dinew is below a front Fl if
Dinew Dl for some Dl ? Fl , i.e. Dinew strictly dominates at least a single dyad in Fl . Define the
depth of Dinew by
ei = min{l | Dinew is below Fl }.
Therefore if ei is large, then Dinew will be near deep fronts, and the distance between X and the
corresponding training sample is large under all combinations of the K criteria. If ei is small, then
Dinew will be near shallow fronts, so the distance between X and the corresponding training sample
is small under some combination of the K criteria.
4.2
Anomaly detection using depths of dyads
In k-NN based anomaly detection algorithms such as those mentioned in Section 2, the anomaly
score is a function of the k nearest neighbors to a test sample. With multiple criteria, one could define an anomaly score by scalarization. From the probabilistic properties of Pareto fronts discussed
in Section 3.1, we know that Pareto methods identify more Pareto-optimal points than linear scalarization methods and significantly more Pareto-optimal points than a single weight for scalarization2 .
This motivates us to develop a multi-criteria anomaly score using Pareto fronts. We start with the
observation from Figure 1 that dyads corresponding to a nominal test sample are typically located
near shallower fronts than dyads corresponding to an anomalous test sample. Each test sample is
associated with s new dyads, where the ith dyad Dinew has depth ei . For each test sample X, we
define the anomaly score v(X) to be the mean of the ei ?s, which corresponds to the average depth
of the s dyads associated with X. Thus the anomaly score can be easily computed and compared to
the decision threshold ? using the test
s
v(X) =
1 X H1
ei ? ?.
s i=1 H0
Pseudocode for the PDA anomaly detector is shown in Algorithm 1. In Section 3 of the supplementary material we provide details of the implementation as well as an analysis of the time complexity
and a heuristic for choosing the ki ?s that performs well in practice. Both the training time and the
1
If a training sample is one of the ki nearest neighbors in multiple criteria, then multiple copies of the dyad
corresponding to the connection between the test sample and the training sample are created.
2
Theorems 1 and 2 require i.i.d. samples, but dyads are not independent. However, there are O(N 2 ) dyads,
and each dyad is only dependent on O(N ) other dyads. This suggests that the theorems should also hold for the
non-i.i.d. dyads as well, and it is supported by experimental results presented in Section 2 of the supplementary
material.
6
Table 1: AUC comparison of different methods for both experiments. Best AUC is shown in bold.
PDA does not require selecting weights so it has a single AUC. The median and best AUCs (over all
choices of weights selected by grid search) are shown for the other four methods. PDA outperforms
all of the other methods, even for the best weights, which are not known in advance.
(a) Four-criteria simulation (? standard error)
Method
PDA
k-NN
k-NN sum
k-LPE
LOF
(b) Pedestrian trajectories
AUC by weight
Median
Best
0.948 ? 0.002
0.848 ? 0.004 0.919 ? 0.003
0.854 ? 0.003 0.916 ? 0.003
0.847 ? 0.004 0.919 ? 0.003
0.845 ? 0.003 0.932 ? 0.003
Method
PDA
k-NN
k-NN sum
k-LPE
LOF
AUC by weight
Median Best
0.915
0.883
0.906
0.894
0.911
0.893
0.908
0.839
0.863
time required to test a new sample using PDA are linear in the number of criteria K. To handle
multiple criteria, other anomaly detection methods, such as the ones mentioned in Section 2, need
to be re-executed multiple times using different (non-negative) linear combinations of the K criteria. If a grid search is used for selection of the weights in the linear combination, then the required
computation time would be exponential in K. Such an approach presents a computational problem
unless K is very small. Since PDA scales linearly with K, it does not encounter this problem.
5
Experiments
We compare the PDA method with four other nearest neighbor-based single-criterion anomaly detection algorithms mentioned in Section 2. For these methods, we use linear combinations of the
criteria with different weights selected by grid search to compare performance with PDA.
5.1
Simulated data with four criteria
First we present an experiment on a simulated data set. The nominal distribution is given by the
uniform distribution on the hypercube [0, 1]4 . The anomalous samples are located just outside of
this hypercube. There are four classes of anomalous distributions. Each class differs from the
nominal distribution in one of the four dimensions; the distribution in the anomalous dimension is
uniform on [1, 1.1]. We draw 300 training samples from the nominal distribution followed by 100
test samples from a mixture of the nominal and anomalous distributions with a 0.05 probability of
selecting any particular anomalous distribution. The four criteria for this experiment correspond to
the squared differences in each dimension. If the criteria are combined using linear combinations,
the combined dissimilarity measure reduces to weighted squared Euclidean distance.
The different methods are evaluated using the receiver operating characteristic (ROC) curve and
the area under the curve (AUC). The mean AUCs (with standard errors) over 100 simulation runs
are shown in Table 1(a). A grid of six points between 0 and 1 in each criterion, corresponding to
64 = 1296 different sets of weights, is used to select linear combinations for the single-criterion
methods. Note that PDA is the best performer, outperforming even the best linear combination.
5.2
Pedestrian trajectories
We now present an experiment on a real data set that contains thousands of pedestrians? trajectories
in an open area monitored by a video camera [20]. Each trajectory is approximated by a cubic spline
curve with seven control points [21]. We represent a trajectory with l time samples by
x1 x2 . . . x l
T =
,
y1 y2 . . . yl
where [xt , yt ] denote a pedestrian?s position at time step t.
7
1
0.06
0.05
Shape dissimilarity
True positive rate
0.8
0.6
0.4
PDA method
k?LPE with best AUC weight
k?LPE with worst AUC weight
Attainable region of k?LPE
0.2
0
0
0.2
0.4
0.6
False positive rate
0.8
0.04
0.03
0.02
0.01
0
1
0
0.01
0.02
0.03
0.04
Walking speed dissimilarity
0.05
Figure 3: Left: ROC curves for PDA and attainable region for k-LPE over 100 choices of weights.
PDA outperforms k-LPE even under the best choice of weights. Right: A subset of the dyads for the
training samples along with the first 100 Pareto fronts. The fronts are highly non-convex, partially
explaining the superior performance of PDA.
We use two criteria for computing the dissimilarity between trajectories. The first criterion is to
compute the dissimilarity in walking speed. We compute the instantaneous speed at all time steps
along
p each trajectory by finite differencing, i.e. the speed of trajectory T at time step t is given
by (xt ? xt?1 )2 + (yt ? yt?1 )2 . A histogram of speeds for each trajectory is obtained in this
manner. We take the dissimilarity between two trajectories to be the squared Euclidean distance
between their speed histograms. The second criterion is to compute the dissimilarity in shape. For
each trajectory, we select 100 points, uniformly positioned along the trajectory. The dissimilarity
between two trajectories T and T 0 is then given by the sum of squared Euclidean distances between
the positions of T and T 0 over all 100 points.
The training sample for this experiment consists of 500 trajectories, and the test sample consists of
200 trajectories. Table 1(b) shows the performance of PDA as compared to the other algorithms
using 100 uniformly spaced weights for linear combinations. Notice that PDA has higher AUC than
the other methods under all choices of weights for the two criteria. For a more detailed comparison,
the ROC curve for PDA and the attainable region for k-LPE (the region between the ROC curves
corresponding to weights resulting in the best and worst AUCs) is shown in Figure 3 along with
the first 100 Pareto fronts for PDA. k-LPE performs slightly better at low false positive rate when
the best weights are used, but PDA performs better in all other situations, resulting in higher AUC.
Additional discussion on this experiment can be found in Section 4 of the supplementary material.
6
Conclusion
In this paper we proposed a new multi-criteria anomaly detection method. The proposed method
uses Pareto depth analysis to compute the anomaly score of a test sample by examining the Pareto
front depths of dyads corresponding to the test sample. Dyads corresponding to an anomalous
sample tended to be located at deeper fronts compared to dyads corresponding to a nominal sample.
Instead of choosing a specific weighting or performing a grid search on the weights for different
dissimilarity measures, the proposed method can efficiently detect anomalies in a manner that scales
linearly in the number of criteria. We also provided a theorem establishing that the Pareto approach
is asymptotically better than using linear combinations of criteria. Numerical studies validated our
theoretical predictions of PDA?s performance advantages on simulated and real data.
Acknowledgments
We thank Zhaoshi Meng for his assistance in labeling the pedestrian trajectories. We also thank
Daniel DeWoskin for suggesting a fast algorithm for computing Pareto fronts in two criteria. This
work was supported in part by ARO grant W911NF-09-1-0310.
8
References
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[4] A. O. Hero III and G. Fleury (2004). Pareto-optimal methods for gene ranking. The Journal of
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9
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3,992 | 4,613 | A Neural Autoregressive Topic Model
Stanislas Lauly
D?epartement d?informatique
Universit?e de Sherbrooke
[email protected]
Hugo Larochelle
D?epartement d?informatique
Universit?e de Sherbrooke
[email protected]
Abstract
We describe a new model for learning meaningful representations of text documents from an unlabeled collection of documents. This model is inspired by the
recently proposed Replicated Softmax, an undirected graphical model of word
counts that was shown to learn a better generative model and more meaningful
document representations. Specifically, we take inspiration from the conditional
mean-field recursive equations of the Replicated Softmax in order to define a neural network architecture that estimates the probability of observing a new word
in a given document given the previously observed words. This paradigm also
allows us to replace the expensive softmax distribution over words with a hierarchical distribution over paths in a binary tree of words. The end result is a model
whose training complexity scales logarithmically with the vocabulary size instead
of linearly as in the Replicated Softmax. Our experiments show that our model
is competitive both as a generative model of documents and as a document representation learning algorithm.
1
Introduction
In order to leverage the large amount of available unlabeled text, a lot of research has been devoted
to developing good probabilistic models of documents. Such models are usually embedded with
latent variables or topics, whose role is to capture salient statistical patterns in the co-occurrence of
words within documents.
The most popular model is latent Dirichlet allocation (LDA) [1], a directed graphical model in
which each word is a sample from a mixture of global word distributions (shared across documents)
and where the mixture weights vary between documents. In this context, the word multinomial
distributions (mixture components) correspond to the topics and a document is represented as the
parameters (mixture weights) of its associated distribution over topics. Once trained, these topics
have been found to extract meaningful groups of semantically related words and the (approximately)
inferred topic mixture weights have been shown to form a useful representation for documents.
More recently, Salakhutdinov and Hinton [2] proposed an alternative undirected model, the Replicated Softmax which, instead of representing documents as distributions over topics, relies on a
binary distributed representation of the documents. The latent variables can then be understood as
topic features: they do not correspond to normalized distributions over words, but to unnormalized
factors over words. A combination of topic features generates a word distribution by multiplying
these factors and renormalizing. They show that the Replicated Softmax allows for very efficient
inference of a document?s topic feature representation and outperforms LDA both as a generative
model of documents and as a method for representing documents in an information retrieval setting.
While inference of a document representation is efficient in the Replicated Softmax, one of its
disadvantages is that the complexity of its learning update scales linearly with the vocabulary size
V , i.e. the number of different words that are observed in a document. The factor responsible for this
1
?v ?v ?v ?v
1
????
v1
v2
v4
v3
2
3
4
h1 h2 h3 h4
h1 h2 h3 h4
v4
h
v1
v2
v3
NADE
v4
v1
v2
v3
v1
v4
Replicated Softmax
v2
v3
v4
DocNADE
Figure 1: (Left) Illustration of NADE. Colored lines identify the connections that share parameters
and vbi is a shorthand for the autoregressive conditional p(vi |v<i ). The observations vi are binary.
(Center) Replicated Softmax model. Each multinomial observation vi is a word. Connections
between each multinomial observation vi and hidden units are shared. (Right) DocNADE, our
proposed model. Connections between each multinomial observation vi and hidden units are also
shared, and each conditional p(vi |v<i ) is decomposed into a tree of binary logistic regressions.
complexity is the conditional distribution of the words given the latent variables, which corresponds
to a V -way multinomial logistic regression. In a realistic application scenario, V will usually be in
the 100 000?s.
The Replicated Softmax is in fact a generalization of the restricted Boltzmann machine (RBM). The
RBM is an undirected graphical model with binary observed and latent variables organized in a bipartite graph. The Replicated Softmax instead has multinomial (softmax) observed variables and
shares (replicates) across all observed variables the parameters between an observed variable and all
latent variables.
A good alternative to the RBM is the neural autoregressive distribution estimator (NADE) [3]. It
is similar to an autoencoder neural network, in that it takes as input a vector of observations and
outputs a vector of the same size. However, the connectivity of NADE has been specifically chosen
so as to make it a proper generative model for vectors of binary observations. More specifically,
NADE outputs the conditional probabilities of each observation given the other observations to its
left in the vector. Taking the product of all these conditional probabilities thus yields a proper joint
probability over the whole input vector of observations. One advantage of NADE is that computing
the parameter gradient of the data negative log-likelihood requires no approximation (unlike in an
RBM). Also, unlike in the RBM, NADE does not require a symmetric connectivity, i.e. the weights
going in and out of its hidden units can be different.
In this work, we describe DocNADE, a neural network topic model that is similarly inspired by the
Replicated Softmax. From the Replicated Softmax, we derive an efficient approach for computing
the hidden units of the network. As for the computation of the distribution of words given the
hidden units, our feed-forward neural network approach leaves us free to use other conditionals than
the V -way multinomial logistic regression implied by the Replicated Softmax. In particular, we
instead opt for a hierarchy of binary logistic regressions, organized in a binary tree where each leaf
corresponds to a word of the vocabulary. This allows us to obtain a complexity of computing the
probability of an observed word scaling sublinearly with V . Our experiments show that DocNADE
is competitive both as a generative model of documents and as a learning algorithm for extracting
meaningful representations of documents.
2
Neural Autoregressive Distribution Estimation
We start with the description of the original NADE. NADE is a generative model over vectors of binary observations v ? {0, 1}D . Through the probability chain rule, it decomposes
QD
p(v) = i=1 p(vi |v<i ) and computes all p(vi |v<i ) using the feed-forward architecture
hi (v<i ) = sigm (c + W:,<i v<i ) ,
p(vi = 1|v<i ) = sigm (bi + Vi,: hi (v<i ))
2
(1)
for i ? {1, . . . , D}, where sigm(x) = 1/(1 + exp(?x)), W ? RH?D and V ? RD?H are connection parameter matrices, b ? RD and c ? RH are bias parameter vectors, v<i is the subvector
[v1 , . . . , vi?1 ]> and W:,<i is a matrix made of the i ? 1 first columns of W.
This architecture corresponds to a neural network with several parallel hi (v<i ) hidden layers and
tied weighted connections between vi and each hidden unit hij (v<i ). Figure 1 gives an illustration.
Though each p(vi = 1|v<i ) requires the computation of its own hidden layer hi (v<i ), the tied
weights allows to compute them all in O(DH), where H is the size of each hidden layer hi (v<i ).
Q
Equation 1 provides all the necessary conditionals to compute p(v) = i p(vi |v<i ). The parameters {b, c, W, V} can then be learned by minimizing the negative log-likelihood with stochastic
gradient descent.
The connectivity behind NADE (i.e. the presence of a separate hidden layer hi (v<i ) for each
p(vi = 1|v<i ) with weight sharing) were directly inspired from the RBM. An RBM is an undirected graphical model in which latent binary variables h interact with the observations v through
an energy function E(v, h), converted into a distribution over v as follows:
X
E(v, h) = ?h> Wv ? b> v ? c> h,
p(v) =
exp(?E(v, h))/Z ,
(2)
h
where Z is known as the partition function and ensures that p(v) is a valid distribution and sums
to 1. Computing the conditional p(vi = 1|v<i ) in an RBM is generally intractable but can be
approximated through mean-field inference. Mean-field inference approximates the full conditional
p(vi , v>i , h|v<i ) as a product of independent Bernoulli distributions q(vk = 1|v<i ) = ?k (i) and
q(hj = 1|v<i ) = ?j (i). To find the values of the variational parameters ?k (i), ?j (i) that minimize
the KL-divergence with p(vi , v>i , h|v<i ), the following message passing equations are applied
until convergence, for k ? {i, . . . , D} and j ? {1, . . . , H} (see Larochelle and Murray [3] for the
derivation):
?
?
?
?
X
X
X
?j (i) ? sigm ?cj +
Wjk ?k (i) +
Wjk vk ? , ?k (i) ? sigm ?bk +
Wjk ?j (i)? .
k?i
j
k<i
(3)
The variational parameter q(vi = 1|v<i ) = ?i (i) can then be used to approximate p(vi = 1|v<i ).
NADE is derived from the application of each message passing equation only once (with ?j (i)
initialized to 0), but compensates by untying the weights between each equation and training the
truncation directly to fit the available data. The end result is thus the feed-forward architecture of
Equation 1.
The relationship between the RBM and NADE is important, as it specifies an effective way of
sharing the hidden layer parameters across the conditionals p(vi = 1|v<i ). In fact, other choices
not inspired by the RBM have proven less successful (see Bengio and Bengio [4] and Larochelle
and Murray [3] for a discussion).
3
Replicated Softmax
Documents can?t be easily modeled by the RBM for two reasons: words are not binary but multinomial observations and documents may contain a varying number of words. An observation vector v
is now a sequence of words indices vi taking values in {1, . . . , V }, while the size D of v can vary.
To address these issues, Salakhutdinov and Hinton [2] proposed the Replicated Softmax model,
which uses the following energy function
E(v, h) = ?D c> h +
D
X
?h> W:,vi ? bvi = ?D c> h ? h> Wn(v) ? b> n(v),
(4)
i=1
where W:,vi is the vith column vector of matrix W and n(v) is a vector of size V containing the
word count of each word in the vocabulary. Notice that this energy shares its connection parameters
across different positions i in v. Figure 1 provides an illustration. Notice also that the larger v is,
3
the more important the terms summed over i in the energy will be. Hence, the hidden bias term c> h
is multiplied by D to maintain a certain balance between all terms.
QD
Q
In this model, the conditional across layers p(v|h) = i=1 p(vi |h) and p(h|v) = j p(hj |v)
factorize and are such that:
X
exp(bw + h> W:,w )
p(hj = 1|v) = sigm(Dcj +
Wjvi )
p(vi = w|h) = P
(5)
>
0
0
w0 exp(bw + h W:,w )
i
The normalized exponential in p(vi = w|h) is known as the softmax nonlinearity. We see that, given
a value of the topic features h, the distribution each word vi in the document can be understood as the
>
normalized product of multinomial topic factors exp(hj Wj,:
) and exp(b), as opposed to a mixture
of multinomial topic distributions.
The gradient of the negative log-likelihood of a single training document vt with respect to any
parameter ? has the simple form
? ? log p(vt )
?
?
t
= EEh|vt
E(v , h) ? EEv,h
E(v, h) .
(6)
??
??
??
Computing the last expectation exactly is too expensive, hence the contrastive divergence [5] approximation is used: the expectation over v is replaced by a point estimate at a so-called ?negative?
sample, obtained from K steps of blocked Gibbs sampling based on Equation 5 initialized at vt .
Once a negative sample is obtained Equation 6 can be estimated and used with stochastic gradient
descent training.
Unfortunately, computing p(vi = w|h) to sample the words during Gibbs sampling is linear in
V and H, where V tends to be quite large. Fortunately, given h, it needs to be computed only
once before sampling all D words in v. However, when h is re-sampled, p(vi = w|h) must be
recomputed. Hence, the computation of p(vi = w|h) is usually the most expensive component of
the learning update: sampling the hidden layer given v is only in O(DH), and repeatably sampling
from the softmax multinomial distribution can be in O(V ). This makes for a total complexity in
O(KV H + DH) of the learning update.
4
Document NADE
More importantly for the context of this paper, it can be shown that mean-field inference of p(vi =
w|v<i ) in the Replicated Softmax corresponds to the following message passing equations, for
k ? {i, . . . , D}, j ? {1, . . . , H} and w ? {1, . . . , V }:
?
?
V
XX
X
?j (i) ? sigm ?D cj +
Wjw0 ?kw0 (i) +
Wjvk ? ,
(7)
k?i w0 =1
?kw (i) ? P
exp(bw +
P
0
w0 exp(bw +
Wjw ?j (i))
P
.
0
j Wjw ?j (i))
j
k<i
(8)
Following the derivation of NADE, we can truncate the application of these equations to obtain a
feed-forward architecture providing an estimate of p(vi = w|v<i ) through ?iw (i) for all i. Specifically, if we consider a single iteration of message passing with ?kw0 (i) initialized to 0, we untie the
parameter weight matrix between each equation into two separate matrices W and V and remove
the multiplication by D of the hidden bias, we obtain the following feed-forward architecture:
!
X
exp(bw + Vw,: hi (v<i ))
(9)
hi (v<i ) = sigm c +
W:,vk , p(vi = w|v<i ) = P
0
0
w0 exp(bw + Vw ,: hi (v<i ))
k<i
for i ? {1, . . . , D}. In words, the probability of the ith word vi is based on a position dependent
hidden layer hi (v<i ) which extracts a representation out of all previous words v<i . This latent
representation is efficient to compute, as it consists simply in a linear transformation followed by
an element-wise sigmoidal nonlinearity. Unlike in the Replicated Softmax, we have found that
multiplying the hidden bias by D was not necessary and, in fact, slightly hampered the performance
of the model, so we opted for its removal.
4
To obtain the probability of the next word vi+1 , one must first compute the hidden layer
X
X
hi+1 (v<i+1 ) = sigm(c +
W:,vk ) = sigm(W:,vi + c +
W:,vk )
k<i+1
(10)
k<i
P
which is efficiently computed by reusing the previous linear transformation c + k<i W:,vk and
adding W:,vi . With this procedure, we see that computing all hidden layers hi (v<i ) is in O(DH).
Computing the softmax nonlinearity of each p(vi = w|v<i ) in Equation 9 requires time linear in
V , which we would like to avoid. Fortunately, unlike in the Replicated Softmax, we are not tied to
the use of a large softmax nonlinearity to model probabilities over words. In the literature on neural
probabilistic language models, the large softmax over words is often replaced by a probabilistic tree
model in which each path from the root to a leaf corresponds to a word [6, 7]. The probabilities
of each left/right transitions in the tree are modeled by a set of binary logistic regressors and the
probability of a given word is then obtained by multiplying the probabilities of each left/right choice
of the associated tree path.
Specifically, let l(vi ) be the sequence of tree nodes on the path from the root to the word vi and let
?(vi ) be the sequence of binary left/right choices for each of those nodes (e.g. l(vi )1 will always be
the root of the tree and ?(vi )1 will be 0 if the word leaf node is in its left subtree or 1 otherwise). Let
matrix V now be the matrix containing the logistic regression weights Vl(vi )m ,: of each tree node
n(vi )m as its rows and bl(vi )m be its bias. The probability p(vi = w|v<i ) is now computed from
hidden layer hi (v<i ) as follows:
|?(vi )|
p(vi = w|v<i ) =
Y
p(?(vi )m |v<i ),
p(?(vi )m = 1|v<i ) = sigm(bl(vi )m + Vl(vi )m ,: hi (v<i ))
m=1
(11)
Q
The conditionals of Equation 11 let us compute p(v) = i p(vi = 1|v<i ) for any document and
the parameters {b, c, W, V} can be learned by minimizing the negative data log-likelihood with
stochastic gradient descent. Once the model is trained, it can be used to extract a representation
from a new document v? by computing
P the value of its hidden layer after observing all of its words,
which we note h(v? ) = sigm(c + i W:,vi? ).
For a full binary tree of all V words, computing Equation 11 will involve O(log(V )) binary logistic
regressions. In our experiments, we used a randomly generated full binary tree with V leaves, each
assigned to a unique word of the vocabulary. An even better option would be to derive the tree using
Hoffman coding, which would reduce even more the average path lengths.
Since the computation of each logistic regression is in O(H) and there are D words in a document, the complexity of computing all p(vi = w|v<i ) given the hidden layers is in O(log(V )DH).
The total complexity of computing p(v) and updating the parameters under the model is therefore
O(log(V )DH + DH). When compared to the complexity O(KV H + DH) of Replicated Softmax, this is quite competitive1 . Indeed, Salakhutdinov and Hinton [2] suggest gradually increasing
K from 1 to 25, which is larger than log(V ) for a very large vocabulary of one million words. Also,
the number of words in a document D will usually be much smaller than the vocabulary size V .
The final model, which we refer to as Document NADE (DocNADE), is illustrated in Figure 1. A
pseudocode for computing p(v) and the parameter learning gradients for a given document is provided in the supplementary material and our code is available here: http://www.dmi.usherb.
ca/?larocheh/code/DocNADE.zip.
4.1
Training from bags of word counts
So far, we have assumed that the ordering of the words in the document was known. However,
document data sets often take the form of set of word counts vectors in which the original word
1
In our experiments, a single training pass of DocNADE on the 20 Newgroups and RCV1-v2 data sets
(see Section 6.1 for details) took on average 13 seconds and 726 seconds respectively. On the other hand, for
K = 1 Gibbs sampling steps, our implementation of Replicated Softmax requires 28 seconds and 4945 seconds
respectively. For K = 5, running time increases even more, to 60 seconds and 11000 seconds.
5
order, which is required by DocNADE to specify the sequence of conditionals p(vi |v< i), has been
lost.
e is sampled from
One solution is to assume the following generative story: first, a seed document v
DocNADE and, finally, a random permutation of its words is taken to produce the observed document v. This translates into the following probability distribution:
p(v) =
X
p(v|e
v)p(e
v) =
e?V(v)
v
X
1
p(e
v)
|V(v)|
(12)
e?V(v)
v
e with the same word
where p(e
v) is modeled by DocNADE and V(v) is the set of all documents v
count vector n(v) = n(e
v). This distribution is a mixture over all possible permutations that could
have generated the original document v. Now, we can use the fact that sampling uniformly from
V(v) can be done solely on the basis of the word counts of v, by randomly sampling words without
replacement from those word counts. Therefore, we can train DocNADE on those generated word
sequences, as if they were the original documents from which the word counts were extracted. While
this is only an approximation of true maximum likelihood learning on the original documents, we?ve
found it to work well in practice.
This approach of training DocNADE can be understood as learning a model that is good at predicting
which new words should be inserted in a document at any position, while maintaining its general semantics. The model is therefore learning not to insert ?intruder? words, i.e. words that do not belong
with the others. After training, a document?s learned representation h(v) should contain valuable
information to identify intruder words for this document. It?s interesting to note that the detection of
such intruder words has been used previously as a task in user studies to evaluate the quality of the
topics learned by LDA, though at the level of single topics and not whole documents [8].
5
Related Work
We mentioned that the Replicated Softmax models the distribution over words as a product of
topic-dependent factors. The Sparse Additive Generative Model (SAGE) [9] is also based on topicdependent factors, as well as a background factor. The distribution of a word is the renormalized
product of its topic factor and the background factor. Unfortunately, much like the Replicated Softmax, training in SAGE scales linearly with the vocabulary size, instead of logarithmically as in
DocNADE. Recent work has also been able to improve the complexity of RBM training on word
observations. However, for the specific case of the Replicated Softmax, the proposed method does
not allow to remove the linear dependence on V of the complexity [10].
There has been fairly little work on using neural networks to learn generative topic models of documents. Glorot et al. [11], Dauphin et al. [12] have trained neural network autoencoders on documents
in their binary bag of words representation, but such neural networks are not generative models of
documents. One potential advantage of having a proper generative model under which p(v) can be
computed exactly is it becomes possible to do Bayesian learning of the parameters, even on a large
scale, using recent online Bayesian inference approaches [13, 14].
6
Experiments
We present two quantitative comparison of DocNADE with the Replicated Softmax. The first
compares the performance of DocNADE as a generative model, while the later evaluates whether
DocNADE hidden layer can be used as a meaningful representation for documents. Following
Salakhutdinov and Hinton [2], we use a hidden layer size of H = 50 in all experiments. A
validation set is always set aside to perform model selection of other hyper-parameters, such as
the learning rate and the number of learning passes over the training set (based on early stopping). We also tested the use of a hidden layer hyperbolic tangent nonlinearity tanh(x) =
(exp(x) ? exp(?x))/(exp(x) + exp(?x)) instead of the sigmoid and always used the best option based on the validation set performance. We end this section with a qualitative inspection of the
implicit word representation and topic-features learned by DocNADE.
6
Data Set
20 Newsgroups
RCV1-v2
LDA (50)
LDA (200)
1091
1437
1058
1142
Replicated
Softmax (50)
953
988
DocNADE (50)
896
742
DocNADE
St. Dev
6.9
4.5
Table 1: Test perplexity per word for LDA with 50 and 200 latent topics, Replicated Softmax with
50 topics and DocNADE with 50 topics. The results for LDA and Replicated Softmax were taken
from Salakhutdinov and Hinton [2].
6.1
Generative Model Evaluation
We first evaluated DocNADE?s performance as a generative model of documents. We performed
our evaluation on the 20 Newsgroups and the Reuters Corpus Volume I (RCV1-v2) data sets and
we followed the same evaluation as in Salakhutdinov and Hinton [2]: word counts were replaced
by log(1 + ni ) rounded to the closest integer and a subset of 50 test documents (2193 words for
20 Newsgroups,
4716 words for RCV1-v2) were used to estimate the test perplexity per word
P
exp(? N1 t |v1t | log p(vt )). The vocabulary size for 20 Newsgroups was 2000 and 10 000 for
RCV1-v2.
We used the version of DocNADE that trains from document word counts. To approximate the
e
corresponding distribution p(v) of Equation 12, we sample a single permuted word sequence v
from the word counts. This might seem like a crude approximation, but, as we?ll see, the value of
p(e
v) tends not to vary a lot across different random permutations of the words.
P
Instead of minimizing the average document negative log-likelihood ? N1 t log p(vt ), we also
P
considered minimizing a version normalized by each document?s size ? N1 t |v1t | log p(vt ), though
the difference in performance between both ended up not being large. For 20 newsgroups, the model
with the best perplexity on the validation set used a learning rate of 0.001, sigmoid hidden activation
and optimized the average document negative log-likelihood (non-normalized). For RCV1-v2, a
learning rate of 0.1, with sigmoid hidden activation and optimization of the objective normalized by
each document?s size performed best.
The results are reported in Table 1. A comparison is made with LDA using 50 or 200 topics and the
Replicated Softmax with 50 topics. The results for LDA and Replicated Softmax were taken from
Salakhutdinov and Hinton [2]. We see that DocNADE achieves lower perplexity than both models.
On RCV1-v2, DocNADE reaches a perplexity that is almost half that of LDA with 50 topics. We
also provide the standard deviation of the perplexity obtained by repeating 100 times the calculation
e . We see that it is fairly
of the perplexity on the test set using different permuted word sequences v
small, which confirms that the value of p(e
v) does not vary a lot across different permutations. This
is consistent with the observation made by Larochelle and Murray [3] that results are stable with
respect to the choice of ordering for the conditionals p(vi |v<i ).
6.2
Document Retrieval Evaluation
We also evaluated the quality of the document representation h(v) learned by DocNADE in an
information retrieval task using the 20 Newsgroups data set and its label information. In this context,
all test documents were each used as queries and compared to a fraction of the closest documents in
the original training set. Similarity between documents is computed using the cosine angle between
document representations. We then compute the average number of retrieved training documents
sharing the same label as the query (precision), and so for different fractions of retrieved documents.
For learning, we set aside 1000 documents for validation. For model selection, we used the validation set as the query set and used the average precision at 0.02% retrieved documents as the
performance measure. We used only the training objective normalized by the document size and
set the maximum number of training passes to 973 (approximately 10 million parameter updates).
The best learning rate was 0.01, with tanh hidden activation. Notice that the labels are not used
during training. Since Salakhutdinov and Hinton [2] showed that it strictly outperforms LDA on this
problem, we only compare to the Replicated Softmax. We performed stochastic gradient descent
based on the contrastive divergence approximation during 973 training passes, and so for different
learning rates. As recommended in Salakhutdinov and Hinton [2], we gradually increased the num7
jesus
atheism
christianity
christ
athos
atheists
bible
christians
sin
atheist
Hidden unit topics
shuttle
season
orbit
players
lunar
nhl
spacecraft
league
nasa
braves
space
playoffs
launch
rangers
saturn
hockey
billion
pitching
satellite
team
encryption
escrow
pgp
crypto
nsa
rutgers
clipper
secure
encrypted
keys
Figure 2: (Left) Information retrieval task results, on 20 Newsgroups data set. The error bars correspond to the standard errors. (Right) Illustration of some topics learned by DocNADE. A topic i is
visualized by picking the 10 words w with strongest connection Wiw .
Table 2: The five nearest neighbors in the word representation space learned by DocNADE.
weapons
weapon
shooting
firearms
assault
armed
medical
treatment
medecine
patients
process
studies
companies
demand
commercial
agency
company
credit
define
defined
definition
refer
make
examples
israel
israeli
israelis
arab
palestinian
arabs
book
reading
read
books
relevent
collection
windows
dos
microsoft
version
ms
pc
ber of Gibbs sampling steps K from 1 to 25, but also tried increasing it only to 5 or maintaining it
to K = 1. Optionally, we also used mean-field inference for the first few training passes. The best
combination of these choices was selected based on validation performance.
The final results are presented in Figure 2. We see that DocNADE compares favorably with the
Replicated Softmax. DocNADE is never outperformed by the Replicated Softmax and outperforms
it for the intermediate retrieval fractions.
6.3
Qualitative Inspection of Learned Representations
Since topic models are often used for the exploratory analysis of unlabeled text, we looked at
whether meaningful semantics were captured by DocNADE. First, to inspect the nature of topics
modeled by the hidden units, we looked at the words with strongest positive connections to that
hidden unit, i.e. the words w that have the largest values of Wi,w for the ith hidden unit. Figure 2
shows four topics extracted this way and that could be understood as topics about religion, space,
sports and security, which are label (sub)categories in 20 Newsgroups. We can also extract word
representations, by using the columns W:,w as the vector representation of each word w. Table 2
shows the five nearest neighbors of some selected words in this space, confirming that the word
representations are meaningful. In the supplementary material, we also provide 2D visualizations of
these representations based on t-SNE [15], for 20 Newsgroups and RCV1-v2.
7
Conclusion
We have proposed DocNADE, an unsupervised neural network topic model of documents and have
shown that it is a competitive model both as a generative model and as a document representation
learning algorithm. Its training has the advantageous property of scaling sublinearly with the vocabulary size. Since the early work on topic modeling, research on the subject has progressed by
developing Bayesian algorithms for topic modeling, by exploiting labeled data and by incorporating
more structure within the latent topic representation. We feel like this is a plausible and most natural
course to follow for future research.
Acknowledgment
We thank Ruslan Salakhutdinov for providing us with the data sets used in the experiments. This
work was supported by NSERC and Google.
8
References
[1] David M. Blei, Andrew Y. Ng, and Michael I. Jordan. Latent Dirichlet Allocation. Journal of
Machine Learning Research, 3(4-5):993?1022, 2003.
[2] Ruslan Salakhutdinov and Geoffrey Hinton. Replicated Softmax: an Undirected Topic Model.
In Advances in Neural Information Processing Systems 22 (NIPS 2009), pages 1607?1614,
2009.
[3] Hugo Larochelle and Ian Murray. The Neural Autoregressive Distribution Estimator. In Proceedings of the 14th International Conference on Artificial Intelligence and Statistics (AISTATS
2011), volume 15, pages 29?37, Ft. Lauderdale, USA, 2011. JMLR W&CP.
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volume9/vandermaaten08a/vandermaaten08a.pdf.
9
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3,993 | 4,614 | Monte Carlo Methods for Maximum Margin
Supervised Topic Models
Qixia Jiang?? , Jun Zhu?? , Maosong Sun? , and Eric P. Xing??
Department of Computer Science & Technology, Tsinghua National TNList Lab,
?
State Key Lab of Intelligent Tech. & Sys., Tsinghua University, Beijing 100084, China
?
School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213
{qixia,dcszj,sms}@mail.tsinghua.edu.cn; [email protected]
?
Abstract
An effective strategy to exploit the supervising side information for discovering
predictive topic representations is to impose discriminative constraints induced by
such information on the posterior distributions under a topic model. This strategy has been adopted by a number of supervised topic models, such as MedLDA,
which employs max-margin posterior constraints. However, unlike the likelihoodbased supervised topic models, of which posterior inference can be carried out using the Bayes? rule, the max-margin posterior constraints have made Monte Carlo
methods infeasible or at least not directly applicable, thereby limited the choice
of inference algorithms to be based on variational approximation with strict mean
field assumptions. In this paper, we develop two efficient Monte Carlo methods
under much weaker assumptions for max-margin supervised topic models based
on an importance sampler and a collapsed Gibbs sampler, respectively, in a convex dual formulation. We report thorough experimental results that compare our
approach favorably against existing alternatives in both accuracy and efficiency.
1 Introduction
Topic models, such as Latent Dirichlet Allocation (LDA) [3], have shown great promise in discovering latent semantic representations of large collections of text documents. In order to fit data better,
LDA has been successfully extended in various ways. One notable extension is supervised topic
models, which were developed to incorporate supervising side information for discovering predictive latent topic representations. Representative methods include supervised LDA (sLDA) [2, 12],
discriminative LDA (DiscLDA) [8], and max-entropy discrimination LDA (MedLDA) [16].
MedLDA differs from its counterpart supervised topic models by imposing discriminative constraints (i.e., max-margin constraints) directly on the desired posterior distributions, instead of defining a normalized likelihood model as in sLDA and DiscLDA. Such topic models with max-margin
posterior constraints have shown superior performance in various settings [16, 14, 13, 9]. However,
their constrained formulations, especially when using soft margin constraints for inseparable practical problems, make it infeasible or at least hard if possible at all1 to directly apply Monte Carlo (MC)
methods [10], which have been widely used in the posterior inference of likelihood based models,
such as the collapsed Gibbs sampling methods for LDA [5]. Previous inference methods for such
models with max-margin posterior constraints have been exclusively on the variational methods [7]
usually with a strict mean-field assumption. Although factorized variational methods often seek
faster approximation solutions, they could be inaccurate or obtain too compact results [1].
??
indicates equal contributions from these authors.
Rejection sampling can be applied when the constraints are hard, e.g., for separable problems. But it would
be inefficient when the sample space is large.
1
1
In this paper, we develop efficient Monte Carlo methods for max-margin supervised topic models,
which we believe is crucial for highly scalable implementation, and further performance enhancement of this class of models. Specifically, we first provide a new and equivalent formulation of the
MedLDA as a regularized Bayesian model with max-margin posterior constraints, based on Zellner?s interpretation of Bayes? rule as a learning model [15] and the recent development of regularized
Bayesian inference [17]. This interpretation is arguably more natural than the original formulation
of MedLDA as a hybrid max-likelihood and max-margin learning, where the log-likelihood is approximated by a variational upper bound for computational tractability. Then, we deal with the set
of soft max-margin constraints with convex duality methods and derive the optimal solutions of the
desired posterior distributions. To effectively reduce the size of the sampling space, we develop two
samplers, namely, an importance sampler and a collapsed Gibbs sampler [4, 1], with a much weaker
assumption on the desired posterior distribution compared to the mean field methods in [16]. We
note that the work [11] presents a duality method to handle moment matching constraints in maximum entropy models. Our work is an extension of their results to learn topic models, which have
nontrivially structured latent variables and also use the general soft margin constraints.
2
Latent Dirichlet Allocation
LDA [3] is a hierarchical Bayesian model that posits each document as an admixture of K topics,
where each topic ?k is a multinomial distribution over a V -word vocabulary. For document d, its
topic proportion ? d is a multinomial distribution drawn from a Dirichlet prior. Let wd = {wdn }N
n=1
denote the words appearing in document d. For the n-th word wdn , a topic assignment zdn = k is
drawn from ? d and wdn is drawn from ?k . In short, the generative process of d is
? d ? Dir(?), zdn = k ? Mult(? d ), wdn ? Mult(?k ),
(1)
where Dir(?) is a Dirichlet, Mult(?) is a multinomial. For fully-Bayesian LDA, the topics are also
random samples drawn from a Dirichlet prior, i.e., ?k ? Dir(?).
N
Let W = {wd }D
d=1 denote all the words in a corpus with D documents, and define zd = {zdn }n=1 ,
D
D
Z = {zd }d=1 , ? = {? d }d=1 . The goal of LDA is to infer the posterior distribution
p(?, Z, ?|W, ?, ?) =
p0 (?, Z, ?|?, ?)p(W|?, Z, ?)
.
p(W|?, ?)
(2)
Since inferring the true posterior distribution is intractable, researchers must resort to variational [3]
or Monte Carlo [5] approximate methods. Although both methods have shown success in various
scenarios. They have complementary advantages. For example, variational methods (e.g., meanfield) can be generally more efficient, while MC methods can obtain more accurate estimates.
3
MedLDA: a supervised topic model with max-margin constraints
MedLDA extends LDA by integrating the max-margin learning into the procedure of discovering
latent topic representations to learn latent representations that are good for predicting class labels
or rating scores of a document. Empirically, MedLDA and its various extensions [14, 13, 9] have
demonstrated promise in learning more discriminative topic representations. The original MedLDA was designed as a hybrid max likelihood and max-margin learning, where the intractable loglikelihood is approximated by a variational bound. To derive our sampling methods, we present a
new interpretation of MedLDA from the perspective of regularized Bayesian inference [17].
3.1
Bayesian inference as a learning model
As shown in Eq. (2), Bayesian inference is an information processing rule that projects the prior
p0 and empirical evidence to a post-data posterior distribution via the Bayes? rule. It is the core
for likelihood-based supervised topic models [2, 12]. A fresh interpretation of Bayesian inference
was given by Zellner [15], which leads to our novel interpretation of MedLDA. Specifically, Zellner
showed that the posterior distribution by Bayes? rule is the solution of an optimization problem. For
instance, the posterior p(?, Z, ?|W) of LDA is equivalent to the optimum solution of
(3)
min
KL[p(?, Z, ?)?p0 (?, Z, ?)] ? Ep [log p(W|?, Z, ?)],
p(?,Z,?)?P
where KL(q||p) is the Kullback-Leibler divergence from q to p, and P is the space of probability
distributions. We will use L(p(?, Z, ?)) to denote the objective function.
2
3.2
MedLDA: a regularized Bayesian model
For brevity, we consider the classification model. Let D = {(wd , yd )}D
d=1 be a given fully-labeled
training set, where the response variable Y takes values from a finite set Y = {1, . . . , M }. MedLDA
consists of two parts. The first part is an LDA likelihood model for describing input documents. As
in previous work, we use the partial2 likelihood model for W. The second part is a mechanism to
consider supervising signal. Since our goal is to discover latent representations Z that are good for
classification, one natural solution is to connect Z directly to our ultimate goal. MedLDA obtains
such a goal by building a classification model on Z. One good candidate of the classification model
is the max-margin methods, which avoid defining a normalized likelihood model [12].
Formally, let ? denote the parameters of the classification model. To make the model fully-Bayesian,
we also treat ? random. Then, we want to infer the joint posterior distribution p(?, ?, Z, ?|D). For
classification, MedLDA defines the following discrimination function
?), F (y; w) = Ep(?,z|w) [F (y, ?, z; w)],
F (y, ?, z; w) = ? ? f (y, z
(4)
?N
1
? is a K-dim vector whose element z?k equals to N n=1 I(zn = k), and I(x) is an indicator
where z
?) is an M K-dim vector whose
function which equals to 1 when x is true otherwise 0; f (y, z
? and all others are zero; and ? is an M K-dimensional vector
elements from (y ? 1)K to yK are z
concatenating M class-specific sub-vectors. With the above definitions, a natural prediction rule is
(5)
y? = argmax F (y; w),
y
and we would like to ?regularize? the properties of the latent topic representations to make them
suitable for a classification task. One way to achieve that goal is to take the optimization view of
Bayes? theorem and impose the following max-margin constraints to problem (3)
F (yd ; wd ) ? F (y; wd ) ? ?d (y) ? ?d , ?y ? Y, ?d,
(6)
where ?d (y) is a non-negative function that penalizes the wrong predictions; ? =
are non-negative slack variables for inseparable cases. Let L(p) = KL(p||p0 (?, ?, Z, ?)) ?
?d ) = f (yd , z
?d ) ? f (y, z
?d ). Then, we define the soft-margin
Ep [log p(W|Z, ?)] and ?f (y, z
MedLDA as solving
D
min
p(?,?,Z,?)?P,?
L(p(?, ?, Z, ?)) +
?
C ?
?d
D
{?d }D
d=1
(7)
d=1
?d )] ? ?d (y) ? ?d , ?d ? 0, ?d, ?y,
s.t. : Ep [? ?f (y, z
where the prior is p0 (?, ?, Z, ?) = p0 (?)p0 (?, Z, ?).
With the above discussions, we can see that MedLDA is an instance of regularized Bayesian
models [17]. Also, problem (7) can be equivalently written as
min
L(p(?, ?, Z, ?)) + CR(p(?, ?, Z, ?))
(8)
p(?,?,Z,?)?P
?
1
?
?d )]) is the hinge loss, an upper bound of the
where R = D d argmaxy (?d (y) ? Ep [? ?f (y, z
prediction error on training data.
4
Monte Carlo methods for MedLDA
As in other variants of topic models, it is intractable to solve problem (7) or the equivalent
problem (8) directly. Previous solutions resort to variational mean-field approximation methods. It
is easy to show that the variational EM method in [16] is a coordinate descent algorithm to solve
problem (7), with the additional fully-factorized mean-field constraint,
?
?
?
p(zdn ))
p(?k ).
(9)
p(?, ?, Z, ?) = p(?)( p(? d )
n
d
k
Below, we present two MC sampling methods to solve the MedLDA problem, with much weaker
constraints on p, and thus they could be expected to produce more accurate solutions.
Specifically, we assume p(?, ?, Z, ?) = p(?)p(?, Z, ?). Then, the general procedure is to alternately solve problem (8) by performing the following two steps.
2
A full likelihood model on both W and Y can be defined as in [12]. But its normalization constant (a
function of Z) could make the problem hard to solve.
3
Estimate p(?): Given p(?, Z, ?), the subproblem (in an equivalent constrained form) is to solve
min KL(p(?)?p0 (?)) +
p(?),?
D
C ?
?d
D
(10)
d=1
?
s.t. : Ep [?] ?f (y, E[?
zd ]) ? ?d (y) ? ?d , ?d ? 0, ?d, ?y.
By using the Lagrangian methods with multipliers ?, we have the optimum posterior distribution
p(?) ? p0 (?)e?
?
?
? y
? D
zd ])
d=1
y ?d ?f (y,E[?
(11)
.
For the prior p0 , for simplicity, we choose the standard normal prior, i.e., p0 (?) = N (0, I). In this
case, p(?) = N (?, I) and the dual problem is
D
max ?
?
s.t. :
?? y
1 ?
? ?+
?d ?d (y)
2
y
?
d=1
?yd
? [0,
(12)
C
], ?d.
D
y
?D ?
where ? = d=1 y ?yd ?f (y, E[?
zd ]). Note that ? is the posterior mean of classifier parameters ?,
and the element ?yk represents the contribution of topic k in classifying a data point to category y.
This problem is the dual problem of a multi-class SVM [6] and we can solve it (or its primal form)
efficiently using existing high-performance SVM learners. We denote the optimum solution of this
problem by (p? (?), ?? , ?? , ?? ).
Estimate p(?, Z, ?): Given p(?), the subproblem (in an equivalent constrained form) is to solve
min
p(?,Z,?),?
L(p(?, Z, ?)) +
? ?
D
C ?
?d
D
(13)
d=1
s.t. : (? ) ?f (y, Ep [?
zd ]) ? ?d (y) ? ?d , ?d ? 0, ?d, ?y.
Although in theory we can solve this subproblem again using Lagrangian dual methods, it would
be hard to derive the dual objective function (if possible at all). Here, we use the same strategy as
in [16], that is, to update p(?, Z, ?) for only one step with ? being fixed at ? ? (the optimum solution
of the previous step). It is easy to show that by fixing ? at ?? , we will have the optimum solution
p(?, Z, ?) ? p(W, Z, ?, ?)e(?
? ?
)
?
y ?
zd )
dy (?d ) ?f (y,?
,
(14)
The differences between MedLDA and LDA lie in the above posterior distribution. The first term
is the same as the posterior of LDA (the evidence p(W) can be absorbed into the normalization
constant). The second term indicates the regularization effects due to the max-margin posterior constraints, which is consistent with our intuition. Specifically, for those data with non-zero Lagrange
multipliers (i.e., the data are around the decision boundary or misclassified), the second term will
bias the model towards a new posterior distribution that favors more discriminative representations
on these ?hard? data points.
Now, the remaining problem is how to efficiently draw samples from p(?, Z, ?) and estimate the
expectations E[?
z] as accurate as possible, which are needed in learning classification models. Below,
we present two representative samplers ? an importance sampler and a collapsed Gibbs sampler.
4.1
Importance sampler
To avoid dealing with the intractable normalization constant of p(?, Z, ?), one natural choice is
to use importance sampling. Importance sampling aims at drawing some samples from a ?simple?
distribution and the expectation is estimated as a weighted average over these samples. However,
directly applying importance sampling to p(?, Z, ?) may cause some issues since importance sampling suffers from severe limitations in large sample spaces. Alternatively, since the distribution
p(?, Z, ?) in Eq. (14) has the factorization form p(?, Z, ?) = p0 (?, ?)p(Z|?, ?), another pos? ?)
? from p0 (?, ?) and
sible method is to adopt the ancestral sampling strategy to draw sample (?,
?
?
then draw samples from p(Z|?, ?). Although it is easy to draw a sample from the Dirichlet prior
p0 (?, ?) = Dir(?)Dir(?), it would require a large number of samples to get a robust estimate of
the expectations E[Z]. Below, we present one solution to reduce sample space.
4
One feasible method to reduce the sample space is to collapse (?, ?) out and directly draw
samples from the marginal distribution p(Z). However, this will cause tight couplings between Z
and make the number of samples needed to estimate the expectation grow exponentially with the
dimensionality of Z for importance sampler. A practical sampler for this collapsed distribution
would be a Markov chain, as we will present in next section. Here, we propose to use the MAP
estimate of (?, ?) as their ?single sample?3 and proceed to draw samples of Z. Specifically, given
? ?),
? we have the conditional distribution
(?,
Nd
D ?
?
?
y
? d , ?),
? ?)
?
? ?)e
? (?? )? dy (?d )? ?f (y,?zd ) =
?
p(Z|?,
? p(W, Z|?,
p(zdn |?
(15)
d=1 n=1
?
y ?
?
?
1
where
? d , ?,
? wdn = t) = 1 ??kt ??dk e Nd y (?d ) (?yd k ??yk )
(16)
p(zdn = k|?
Zdn
and Zdn is a normalization constant, and ??yk is the [(y ? 1)K + k]-th element of ?? . The difference
(??yd k ? ??yk ) represents the different contribution of topic k in classifying d to the true category yd
and a wrong category y. If the difference is positive, topic k contributes to make a correct prediction
for d; otherwise, it contributes to make a wrong prediction.
(j)
Then, we draw J samples {zdn }Jj=1 from a proposal distribution g(z) and compute the expectations
Nd
J
j
?
?dn
1 ?
(j)
?d and E[zdn ] ?
E[?
zdk ] =
E[zdn ], ??
zdk ? z
zdn ,
(17)
?J
j
Nd n=1
?
j=1 dn
j=1
j
where the importance weight ?dn
is
(j)
)I(zdn
K (? ?
=k)
?
?dk ?kwdn N1 ?y (?yd )? (??y k ???yk )
j
d
d
(18)
?dn =
e
g(k)
k=1
? ?)
?
With the J samples, we update the MAP estimate (?,
j
?
?
?
(j)
N
J
d
1
??dk ? J n=1 j=1 ?J dn? j I(zdn = k) + ?k
j=1 dn
(19)
j
?D ?Nd ?J
?dn
(j)
??kt ? J1 d=1 n=1
j=1 ?J ? j I(zdn = k)I(wdn = t) + ?t .
j=1
dn
? ?)
? to be uniform, and the
The above two steps are repeated until convergence, initializing (?,
samples from the last iteration are used to estimate the expectation statistics needed in the problem
of inferring p(?).
4.2
Collapsed Gibbs sampler
As we have stated, another way to effectively reduce the sample space is to integrate out the
intermediate variables (?, ?) and build a Markov chain whose equilibrium distribution is the
resulting marginal distribution p(Z). We propose to use collapsed Gibbs sampling, which has been
successfully used for LDA [5]. For MedLDA, we integrate out (?, ?) and get the marginalized
posterior distribution
? ?
?
zd )
(?? )? d y (?y
d ) ?f (y,?
p(Z) = p(W,Z|?,?)
e
Z
[ q
][
]
?
(20)
y
?D ?(Cd +?) (?? )? (? )? ?f (y,?zd ) ?K ?(Ck +?)
y
d
= Z1
e
,
d=1
k=1
?(?)
?(?)
where ?(x) =
?dim(x)
?(xi )
i=1
?dim(x)
xi )
i=1
?(
, Ckt is the number of times the term t being assigned to topic k over the
whole corpus and Ck = {Ckt }Vt=1 ; Cdk is the number of times that terms being associated with topic
k within the d-th document and Cd = {Cdk }K
k=1 . We can also derive the transition probability of
one variable zdn given others which we denote by Z? as:
t
?
?
Ck,?n
+ ?t
1
(?y )? (??
k
yd k ??yk )
p(zdn = k|Z? , W? , wdn = t) ? ?
(Cd,?n
+?k )e Nd y d
(21)
?V
t
t Ck,?n +
t=1 ?t
?
where C?,?n
indicates that term n is excluded from the corresponding document or topic.
Again, we can see the difference between MedLDA and LDA (using collapsed Gibbs sampling)
from the additional last term in Eq. (21), which is due to the max-margin posterior constraints.
3
This collapses the sample space of (?, ?) to a single point.
5
For those data on the margin or misclassified (with non-zero Lagrange multipliers), the last term is
non-zero and acts as a regularizer directly affecting the topic assignments of these difficult data.
Then, we use the transition distribution in Eq. (21) to construct a Markov chain. After this Markov
chain has converged (i.e., finished the burn-in stage), we draw J samples {Z(j) } and estimate the
expectation statistics
Nd
J
1 ?
1 ? (j)
?d , and E[zdn ] =
z .
(22)
E[?
zdk ] =
E[zdn ], ??
zdk ? z
Nd n=1
J j=1 dn
4.3 Prediction
To make prediction on unlabeled testing data using the prediction rule (5), we take the approach that
has been adopted for the variational MedLDA, which uses a point estimate of topics ? from training
? to replace the
data and makes prediction based on them. Specifically, we use the MAP estimate ?
? is computed as in Eq. (19). For the
probability distribution p(?). For the importance sampler, ?
? using the samples is ??kt ? 1 ?J C t (j) + ?t , where
collapsed Gibbs sampler, an estimate of ?
j=1 k
J
Ckt
(j)
is the times that term t is assigned to topic k in the j-th sample.
Given a new document w to be predicted, for importance sampler, the importance weight should
?K
(j)
be altered as ?nj = k=1 (?k ??kwn /g(k))I(zn =k) . Then, we approximate the expectation of z as
? as p(zn =
in Eq. (17). For Gibbs sampler, we infer its latent components z using the obtained ?
k
k
?
k|z?n ) ? ?kwn (C?n + ?k ), where C?n is the times that the terms in this document w assigned to
topic k with the n-th term excluded. Then, we approximate the E[?
z] as in Eq. (22).
5 Experiments
We empirically evaluate the importance sampler and the Gibbs sampler for MedLDA (denoted by
iMedLDA and gMedLDA respectively) on the 20 Newsgroups data set with a standard list of stop
words4 removed. This data set contains about 20K postings within 20 groups. Due to space limitation, we focus on the multi-class setting.
We use the cutting-plane algorithm [6] to solve the multi-class SVM to infer p(?) and solve for
the lagrange multipliers ? in MedLDA. For simplicity, we use the uniform proposal distribution g
in iMedLDA. In this case, we can globally draw J (e.g., = 3 ? K) samples {Z(j) }Jj=1 from g(z)
outside the iteration loop and only update the importance weights to save time. For gMedLDA,
we keep J (e.g., 20) adjacent samples after gMedLDA has converged to estimate the expectation
statistics. To be fair, we use the same C for different MedLDA methods. The optimum C is chosen
via 5-fold cross validation during the training procedure of fMedLDA from {a2 : a = 1, . . . , 8}. We
use symmetric Dirichlet priors for all LDA topic models, i.e., ? = ?eK and ? = ?eV , where en
is a n-dim vector with every entry being 1. We assess the convergence of a Markov chain when (1)
it has run for a maximum number of iterations (e.g., 100), or (2) the relative change in its objective,
t+1
t
i.e., |L Lt?L | , is less than a tolerance threshold ? (e.g., ? = 10?4 ). We use the same strategy to
judge whether the overall inference algorithm converges.
We randomly select 7,505 documents from the whole set as the test set and the rest as the training
data. We set the cost parameter ?d (y) in problem (7) to be 16, which produces better classification
performance than the standard 0/1 cost [16]. To measure the sparsity of the
? latent representations
of documents, we compute the average entropy over test documents: |D1t | d?Dt H(? d ). We also
measure the sparsity of the inferred topic distributions ? in terms of the average entropy over topics,
?K
1
i.e., K
k=1 H(?k ). All experiments are carried out on a PC with 2.2GHz CPU and 3.6G RAM.
We report the mean and standard deviation for each model with 4 times randomly initialized runs.
5.1 Performance with different topic numbers
This section compares gMedLDA and iMedLDA with baseline methods. MedLDA was shown to
outperform sLDA for document classification. Here, we focus on comparing the performance of
MedLDA and LDA when using different inference algorithms. Specifically, we compare with the
4
http://mallet.cs.umass.edu/
6
0.6
iMedLDA
gMedLDA
fMedLDA
gLDA
fLDA
0.5
0.4
20
40
60
80
# Topics
(a)
100
120
5
iMedLDA
gMedLDA
fMedLDA
gLDA
fLDA
4
3
2
1
20
40
60
80
# Topics
100
(b)
120
Average Entropy over Topics
0.7
Average Entropy over Docs
Accuracy
0.8
9
8
7
iMedLDA
gMedLDA
fMedLDA
gLDA
fLDA
6
5
4
20
40
60
80
# Topics
100
120
(c)
Figure 1: Performance of multi-class classification of different topic models with different topic
numbers on 20-Newsgroups data set: (a) classification accuracy, (b) the average entropy of ? over
test documents, and (c) The average entropy of topic distributions ?.
LDA model that uses collapsed Gibbs sampling [5] (denoted by gLDA) and the LDA model that
uses fully-factorized variational methods [3] (denoted by fLDA). For LDA models, we discover the
latent representations of the training documents and use them to build a multi-class SVM classifier.
For MedLDA, we report the results when using fully-factorized variational methods (denoted by
fMedLDA) as in [16]. Furthermore, fMedLDA and fLDA optimize the hyper-parameter ? using
the Newton-Rampion method [3], while gMedLDA, iMedLDA and gLDA determine ? by 5-fold
cross-validation. We have tested a wide range of values of ? (e.g., 10?16 ? 103 ) and found that the
performance of iMedLDA degrades seriously when ? is larger than 10?3 . Therefore, we set ? to be
10?5 for iMedLDA while 0.01 for the other topic models just as in the literature [5].
Fig. 1(a) shows the accuracy. We can see that Monte Carlo methods generally outperform the fullyfactorized mean-field methods, mainly because of their weaker factorization assumptions. The reason for the superior performance of iMedLDA over gMedLDA is probably because iMedLDA is
more effective in dealing with sample sparsity issues. More insights will be provided in Section 5.2.
Fig. 1(b) shows the average entropy of latent representations ? over test documents. We find that
the entropy of gMedLDA and iMedLDA are smaller than those of gLDA and fLDA, especially for
(relatively) large K. This implies that sampling methods for MedLDA can effectively concentrate
the probability mass on just several topics thus discover more predictive topic representations. However, fMedLDA yields the smallest entropy, which is mainly because the fully-factorized variational
methods tend to get too compact results, e.g., sparse local optimums.
Fig. 1(c) shows the average entropy of topic distributions ? over topics. We can see that gMedLDA
improves the sparsity of ? than fMedLDA. However, gMedLDA?s entropy is larger than gLDA?s.
This is because for those ?hard? documents, the exponential component in Eq. (21) ?regularizes?
the conditional probability p(zdn |Z? ) and leads to a smoother estimate of ?. On the other hand,
we find that iMedLDA has the largest entropy. This is probably because many of the samples (topic
assignments) generated by the proposal distribution are ?incorrect? but importance sampler still
assigns weights to these samples. As a result, the inferred topic distributions are very dense and thus
have a large entropy.
Moreover, in the above experiments, we found that the lagrange multipliers in MedLDA are very
sparse (about 1% non-zeros for both iMedLDA and gMedLDA; about 1.5% for fMedLDA), much
sparser than those of SVM built on raw input data (about 8% non-zeros).
5.2
Sensitivity analysis with respect to key parameters
Sensitivity to ?. Fig. 2(a) shows the classification performance of gMedLDA and iMedLDA with
different values of ?. We can see that the performance of gMedLDA increases as ? becomes large
and retains stable when ? is larger than 0.1. In contrast, the accuracy of iMedLDA decreases a bit
(especially for small K) when ? becomes large, but is relative stable when ? is small (e.g., ? 0.01).
This is probably because with a finite number of samples, Gibbs sampler tends to produce a too
sparse estimate of E[Z], and a slightly stronger prior is helpful to deal with the sample sparsity
issue. In contrast, the importance sampler avoids such sparsity issue by using a uniform proposal
distribution, which could make the samples well cover all topic dimensions. Thus, a small prior is
sufficient to get good performance, and increasing the prior?s strength could potentially hurt.
Sensitivity to sample size J. For sampling methods, we always need to decide how many samples
(sample size J) to keep to ensure sufficient statistics power. Fig. 2(b) shows the classification accuracy of both gMedLDA and iMedLDA with different sample size J when ? = 10?2 /K and C = 16.
7
iMedLDAK=90
gMedLDAK=30
0.6
gMedLDAK=60
0.5 ?4
10
0.6
iMedLDAK=30
iMedLDAK=60
0.4
iMedLDAK=90
gMedLDAK=30
0.2
10
?2
10
?
?1
10
0
10
(a)
0
5
0.75
K=30
K=60
K=90
gMedLDAK=60
gMedLDAK=90
?3
0.8
gMedLDAK=90
10
100
Sample Size
1000
(b)
0.7 ?4
10
Accuracy
iMedLDAK=60
Accuracy
0.7
1
0.8
iMedLDAK=30
Accuracy
Accuracy
0.85
0.8
0.8
?3
10
(c)
0.4
iMedLDA
gMedLDA
fMedLDA
0.2
?2
10
?
0.6
0
1
5
10
# iteration
50
(d)
Figure 2: Sensitivity study of iMedLDA and gMedLDA: (a) classification accuracy with different ?
for different topic numbers, (b) classification accuracy with different sample size J, (c) classification
accuracy with different convergence criterion ? for gMedLDA, and (d) classification accuracy of
different methods varies as a function of iterations when the topic number is 30.
For gMedLDA, we have tested different values of J for training and prediction. We found that the
sample size in the training process has almost no influence on the prediction accuracy even when it
equals to 1. Hence, for efficiency, we set J to be 1 during the training. It shows that gMedLDA is
relatively stable when J is larger than about 20 at prediction. For iMedLDA, Fig. 2(b) shows that it
becomes stable when the prediction sample size J is larger than 3 ? K.
Sensitivity to convergence criterion ?. For gMedLDA, we have to judge whether a Markov chain
has reached its stationarity. Relative change in the objective is a commonly used diagnostic to justify
the convergence. We study the influence of ?. In this experiment, we don?t bound the maximum
number of iterations and allow the Gibbs sampler to run until the tolerance ? is reached. Fig. 2(c)
shows the accuracy of gMedLDA with different values of ?. We can see that gMedLDA is relatively
insensitive to ?. This is mainly because gMedLDA alternately updates posterior distribution and
Lagrangian multipliers. Thus, it does Gibbs sampling for many times, which compensates for the
influence that each Markov chain has not reached its stationarity. On the other hand, small ? values
can greatly slow the convergence. For instance, when the topic number is 90, gMedLDA takes
11,986 seconds on training when ? = 10?4 but 1,795 seconds when ? = 10?2 . These results imply
that we can loose the convergence criterion to speedup training while still obtain a good model.
Sensitivity to iteration. Fig. 2(d) shows the the classification accuracy of MedLDA with various
inference methods as a function of iteration when the topic number is set at 30. We can see that all
the various MedLDA models converge quite quickly to get good accuracy. Compared to fMedLDA,
which uses mean-field variational inference, the two MedLDA models using Monte Carlo methods
(i.e., iMedLDA and gMedLDA) are slightly faster to get stable prediction performance.
Time efficiency
4
10
CPU?Seconds
5.3
Although gMedLDA can get good results even for a loosen converiMedLDA
10
gence criterion ? as discussed in Sec. 5.2, we set ? to be 10?4 for
gMedLDA
fMedLDA
all the methods in order to get a more objective comparison. Fig. 3
gLDA
fLDA
reports the total training time of different models, which includes
10
20
40
60
80 100 120
two phases ? inferring the latent topic representations and training
# Topics
SVMs. We find iMedLDA is the most efficient, which benefits from
Figure 3: Training time.
(1) generateing samples outside the iteration loop and uses them for
all iterations; and (2) using the MAP estimates to collapse the sample space of (?, ?) to a ?single
sample? for efficiency. In contrast, both gMedLDA and fMedLDA have to iteratively update the
variables or variational parameters. gMedLDA requires more time than fMedLDA but is comparable when ? is set to be 0.01. By using the equivalent 1-slack formulation, about 76% of the training
time spent on inference for iMedLDA and 90% for gMedLDA. For prediction, both iMedLDA and
gMedLDA are slightly slower than fMedLDA.
3
2
6
Conclusions
We have presented two Monte Carlo methods for MedLDA, a supervised topic model using maxmargin constraints directly on the desired posterior distributions for discovering predictive latent
topic representations. Our methods are based on a novel interpretation of MedLDA as a regularized Bayesian model and the a convex dual formulation to deal with soft-margin constraints. Experimental results on the 20 Newsgroups data set show that Monte Carlo methods are robust to
hyper-parameters and could yield very competitive results for such max-margin topic models.
8
100
Acknowledgements
Part of the work was done when QJ was visiting CMU. JZ and MS are supported by the National
Basic Research Program of China (No. 2013CB329403 and 2012CB316301), National Natural
Science Foundation of China (No. 91120011, 61273023 and 61170196) and Tsinghua Initiative
Scientific Research Program No.20121088071. EX is supported by AFOSR FA95501010247, ONR
N000140910758, NSF Career DBI-0546594 and Alfred P. Sloan Research Fellowship.
References
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[12] C. Wang, D.M. Blei, and Li F.F. Simultaneous image classification and annotation. CVPR,
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[13] Y. Wang and G. Mori. Max-margin latent Dirichlet allocation for image classification and
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In NIPS, 2011.
9
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evaluate:1 tested:2 wileyinterscience:1 ex:1 |
3,994 | 4,615 | Matrix reconstruction with the local max norm
Rina Foygel
Department of Statistics
Stanford University
[email protected]
Nathan Srebro
Toyota Technological Institute at Chicago
[email protected]
Ruslan Salakhutdinov
Dept. of Statistics and Dept. of Computer Science University of Toronto
[email protected]
Abstract
We introduce a new family of matrix norms, the ?local max? norms, generalizing
existing methods such as the max norm, the trace norm (nuclear norm), and the
weighted or smoothed weighted trace norms, which have been extensively used in
the literature as regularizers for matrix reconstruction problems. We show that this
new family can be used to interpolate between the (weighted or unweighted) trace
norm and the more conservative max norm. We test this interpolation on simulated
data and on the large-scale Netflix and MovieLens ratings data, and find improved
accuracy relative to the existing matrix norms. We also provide theoretical results
showing learning guarantees for some of the new norms.
1
Introduction
In the matrix reconstruction problem, we are given a matrix Y ? Rn?m whose entries are only partly
observed, and would like to reconstruct the unobserved entries as accurately as possible. Matrix
reconstruction arises in many modern applications, including the areas of collaborative filtering
(e.g. the Netflix prize), image and video data, and others. This problem has often been approached
using regularization with matrix norms that promote low-rank or approximately-low-rank solutions,
including the trace norm (also known as the nuclear norm) and the max norm, as well as several
adaptations of the trace norm described below.
In this paper, we introduce a unifying family of norms that generalizes these existing matrix norms,
and that can be used to interpolate between the trace and max norms. We show that this family
includes new norms, lying strictly between the trace and max norms, that give empirical and theoretical improvements over the existing norms. We give results allowing for large-scale optimization
with norms from the new family. Some proofs are deferred to the Supplementary Materials.
Notation Without loss of generality we take n ? m. We let R+ denote the nonnegative real
numbers. For any n ? N, let [n] = {1, . . . , n}, and define the simplex on [n] as ?[n] =
P
r ? Rn+ : i ri = 1 . We analyze situations where the locations ofPobserved entries are sampled
i.i.d. according to some distribution p on [n] ? [m]. We write pi? = j pij to denote the marginal
probability of row i, and prow = (p1? , . . . , pn? ) ? ?[n] to denote the marginal row distribution.
We define p?j and pcol similarly for the columns. For any matrix M , M(i) denotes its ith row.
1.1
Trace norm and max norm
A common regularizer used in matrix reconstruction, and other matrix problems, is the trace norm
kXktr , equal to the sum of the singular values of X. This norm can also be defined via a factorization
1
of X [1]:
?
?
X
X
1
1
1
A(i)
2 + 1
B(j)
2 ? ,
?
kXktr =
min ?
2 AB > =X n i
m j
nm
(1)
where the minimum is taken over factorizations of X of arbitrary dimension?that is, ?
the number of
columns in A and B is unbounded. Note that we choose to scale the trace norm by 1/ nm in order
to emphasize that we are averaging the squared row norms of A and B.
Regularization with the trace norm gives good theoretical and empirical results, as long as the locations of observed entries are sampled uniformly (i.e. when p is the uniform distribution on [n]?[m]),
and, under this assumption, can also be used to guarantee approximate recovery of an underlying
low-rank matrix [1, 2, 3, 4].
The factorized definition of the trace norm (1) allows for an intuitive comparison with the max norm,
defined as [1]:
2
2
1
kXkmax =
(2)
min
sup
A(i)
2 + sup
B(j)
2 .
2 AB > =X
j
i
We see that the max norm measures the largest row norms in the factorization, while the rescaled
trace norm instead considers the average row norms. The max norm is therefore an upper bound
on the rescaled trace norm, and can be viewed as a more conservative regularizer. For the more
general setting where p may not be uniform, Foygel and Srebro [4] show that the max norm is still
an effective regularizer (in particular, bounds on error for the max norm are not affected by p). On
the other hand, Salakhutdinov and Srebro [5] show that the trace norm is not robust to non-uniform
sampling?regularizing with the trace norm may yield large error due to over-fitting on the rows and
columns with high marginals. They obtain improved empirical results by placing more penalization
on these over-represented rows and columns, described next.
1.2
The weighted trace norm
To reduce overfitting on the rows and columns with high marginal probabilities under the distribution
p, Salakhutdinov and Srebro propose regularizing with the p-weighted trace norm,
1
1
kXktr(p) :=
diag(prow ) /2 ? X ? diag(pcol ) /2
.
tr
If the row and the column of entries to be observed are sampled independently (i.e. p = prow ?
pcol is a product distribution), then the p-weighted trace norm can be used to obtain good learning
guarantees even when prow and pcol are non-uniform [3, 6]. However, for non-uniform non-product
sampling distributions, even the p-weighted trace norm can yield poor generalization performance.
To correct for this, Foygel et al. [6] suggest adding in some ?smoothing? to avoid under-penalizing
the rows and columns with low marginal probabilities, and obtain improved empirical and theoretical
results for matrix reconstruction using the smoothed weighted trace norm:
1
1
e row ) /2 ? X ? diag(e
kXktr(ep) :=
diag(p
pcol ) /2
,
tr
e row and p
e col denote smoothed row and column marginals, given by
where p
e row = (1 ? ?) ? prow + ? ? 1/n and p
e col = (1 ? ?) ? pcol + ? ? 1/m ,
p
(3)
for some choice of smoothing parameter ? which may be selected with cross-validation1 . The
smoothed empirically-weighted trace norm is also studied in [6], where pi? is replaced with
observations in row i
b i? = # total
b
p
# observations , the empirical marginal probability of row i (and same for p?j ). Using
empirical rather than ?true? weights yielded lower error in experiments in [6], even when the true
sampling distribution was uniform.
More generally, for any weight vectors r ? ?[n] and c ? ?[m] and a matrix X ? Rn?m , the
(r, c)-weighted trace norm is given by
1/2
1/2
kXktr(r,c) =
diag(r) ? X ? diag(c)
.
tr
1
Our ? parameter here is equivalent to 1 ? ? in [6].
2
Of course, we can easily obtain the existing methods of the uniform trace norm, (empirically)
weighted trace norm, and smoothed (empirically) weighted trace norm as special cases of this formulation. Furthermore, the max norm is equal to a supremum over all possible weightings [7]:
kXkmax =
sup
kXktr(r,c) .
r??[n] ,c??[m]
2
The local max norm
We consider a generalization of these norms, which lies ?in between? the trace norm and max norm.
For any R ? ?[n] and C ? ?[m] , we define the (R, C)-norm of X:
kXk(R,C) =
sup
r?R,c?C
kXktr(r,c) .
This gives a norm on matrices, except in the trivial case where, for some i or some j, ri = 0 for all
r ? R or cj = 0 for all c ? C.
We now show some existing and novel norms that can be obtained using local max norms.
2.1
Trace norm and max norm
We can obtain the max norm by taking the largest possible R and C, i.e. kXkmax = kXk(?[n] ,?[m] ) ,
and similarly we can obtain the (r, c)-weighted trace norm by taking the singleton sets R = {r}
and C = {c}. As discussed above, this includes the standard trace norm (when r and c are uniform),
as well as the weighted, empirically weighted, and smoothed weighted trace norm.
2.2
Arbitrary smoothing
When using the smoothed weighted max norm, we need to choose the amount of smoothing to
apply to the marginals, that is, we need to choose ? in our definition of the smoothed row and
column weights, as given in (3). Alternately, we could regularize simultaneously over all possible
amounts of smoothing by considering the local max norm with
R = {(1 ? ?) ? prow + ? ? 1/n : any ? ? [0, 1]} ,
and same for C. That is, R and C are line segments in the simplex?they are larger than any single
point as for the uniform or weighted trace norm (or smoothed weighted trace norm for a fixed amount
of smoothing), but smaller than the entire simplex as for the max norm.
Connection to (?, ? )-decomposability
2.3
Hazan et al. [8] introduce a class of matrices defined by a property of (?, ? )-decomposability: a
matrix X satisfies this property if there exists a factorization X = AB > (where A and B may have
an arbitrary number of columns) such that2
X
X
2
2
A(i)
2 +
B(j)
2 ? ? .
max max
A(i)
2 , max
B(j)
2 ? 2?,
2
2
i
j
i
j
Comparing with (1) and (2), we see that the ? and ? parameters essentially correspond to the max
norm and trace norm, with the max norm being the minimal 2? ? such that the matrix is (? ? , ? )decomposable for some ? , and the trace norm being the minimal ? ? /2 such that the matrix is
(?, ? ? )-decomposable for some ?. However, Hazan et al. go beyond these two extremes, and rely
on balancing both ? and ? : they establish learning guarantees (in an adversarial
online model, and
?
thus also under an arbitrary sampling distribution p) which scale with ? ? ? . It may therefore be
useful to consider a penalty function of the form:
?
?
sX
?r
2
2
2 X
2 ?
A(i)
+
B(j)
Penalty(?,? ) (X) = min
max
A(i)
2 + max
B(j)
2 ?
.
2
2?
i
j
X=AB > ?
i
j
(4)
2
Hazan et al. state the property differently, but equivalently, in terms of a semidefinite matrix decomposition.
3
n
2
2
2 o
2
(Note that max maxi
A(i)
2 , maxj
B(j)
2 is replaced with maxi
A(i)
2 + maxj
B(j)
2 ,
?
for later convenience. This affects the value of the penalty function by at most a factor of 2.)
This penalty function does not appear to be convex in X. However, the proposition below (proved in
the Supplementary Materials) shows that we can use a (convex) local max norm penalty to compute
a solution to any objective function with a penalty function of the form (4):
b be the minimizer of a penalized loss function with this modified penalty,
Proposition 1. Let X
n
o
b := arg min Loss(X) + ? ? Penalty(?,? ) (X) ,
X
X
where ? ? 0 is some penalty parameter and Loss(?) is any convex function. Then, for some penalty
parameter ? ? 0 and some t ? [0, 1],
n
o
b = arg min Loss(X) + ? ? kXk
X
, where
(R,C)
X
t
t
R = r ? ?[n] : ri ?
?i and C = c ? ?[m] : cj ?
?j .
1 + (n ? 1)t
1 + (m ? 1)t
We note that ? and t cannot be determined based on ? alone?they will depend on the properties of
b
the unknown solution X.
Here the sets R and C impose a lower bound on each of the weights, and this lower bound can be
used to interpolate between the max and trace norms: when t = 1, each ri is lower bounded by
1/n (and similarly for c ), i.e. R and C are singletons containing only the uniform weights and we
j
obtain the trace norm. On the other hand, when t = 0, the weights are lower-bounded by zero, and
so any weight vector is allowed, i.e. R and C are each the entire simplex and we obtain the max
norm. Intermediate values of t interpolate between the trace norm and max norm and correspond to
different balances between ? and ? .
2.4
Interpolating between trace norm and max norm
We next turn to an interpolation which relies on an upper bound, rather than a lower bound, on the
weights. Consider
R = r ? ?[n] : ri ? ?i and C? = c ? ?[n] : cj ? ? ?j ,
(5)
for some ? [1/n, 1] and ? ? [1/m, 1]. The (R , C? )-norm is then equal to the (rescaled) trace norm
when we choose = 1/n and ? = 1/m, and is equal to the max norm when we choose = ? = 1.
Allowing and ? to take intermediate values gives a smooth interpolation between these two familiar
norms, and may be useful in situations where we want more flexibility in the type of regularization.
We can generalize this to an interpolation between the max norm and a smoothed weighted trace
norm, which we will use in our experimental results. We consider two generalizations?for each
one, we state a definition of R, with C defined analogously. The first is multiplicative:
1
R?
(6)
?,? := r ? ?[n] : ri ? ? ? ((1 ? ?) ? pi? + ? ? /n) ?i ,
1
where ? = 1 corresponds to choosing the singleton set R?
?,? = {(1 ? ?) ? prow + ? ? /n} (i.e. the
smoothed weighted trace norm), while ? = ? corresponds to the max norm (for any choice of ?)
since we would get R?
?,? = ?[n] .
The second option for an interpolation is instead defined with an exponent:
n
o
1??
R?,? := r ? ?[n] : ri ? ((1 ? ?) ? pi? + ? ? 1/n)
?i .
(7)
Here ? = 0 will yield the singleton set corresponding to the smoothed weighted trace norm, while
? = 1 will yield R?,? = ?[n] , i.e. the max norm, for any choice of ?.
We find the second (exponent) option to be more natural, because each of the row marginal bounds
will reach 1 simultaneously when ? = 1, and hence we use this version in our experiments. On
the other hand, the multiplicative version is easier to work with theoretically, and we use this in our
learning guarantee in Section 4.2. If all of the row and column marginals satisfy some loose upper
bound, then the two options will not be highly different.
4
3
Optimization with the local max norm
One appeal of both the trace norm and the max norm is that they are both SDP representable [9, 10],
and thus easily optimizable, at least in small scale problems. In the Supplementary Materials we
show that the local max norm is also SDP representable, as long as the sets R and C can be written
in terms of linear or semi-definite constraints?this includes all the examples we mention, where in
all of them the sets R and C are specified in terms of simple linear constraints.
However, for large scale problems, it is not practical to directly use SDP optimization approaches.
Instead, and especially for very large scale problems, an effective optimization approach for both
the trace norm and the max norm is to use the factorized versions of the norms, given in (1) and (2),
and to optimize the factorization directly (typically, only factorizations of some truncated dimensionality are used) [11, 12, 7]. As we show in Theorem 1 below, a similar factorization-optimization
approach is also possible for any local max norm with convex R and C. We further give a simplified
representation which is applicable when R and C are specified through element-wise upper bounds
R ? Rn+ and C ? Rm
+ , respectively:
R = {r ? ?[n] : ri ? Ri ?i} and C = {c ? ?[m] : cj ? Cj ?j} ,
(8)
P
P
with 0 ? Ri ? 1, i Ri ? 1, 0 ? Cj ? 1, j Cj ? 1 to avoid triviality. This includes the
interpolation norms of Section 2.4.
Theorem 1. If R and C are convex, then the (R, C)-norm can be calculated with the factorization
X
X
2
2
1
kXk(R,C) =
(9)
inf
cj
B(j)
2 .
sup
ri
A(i)
2 + sup
2 AB > =X r?R i
c?C j
In the special case when R and C are defined by (8), writing (x)+ := max{0, x}, this simplifies to
n
o
X
X
2
2
1
kXk(R,C) =
inf
a+
Ri
A(i)
2 ? a + b +
Cj
B(j)
2 ? b
.
2 AB > =X;a,b?R
+
+
i
j
1/2
Proof sketch for Theorem 1. For convenience we will write r1/2 to mean diag(r) , and same for c.
Using the trace norm factorization identity (1), we have
1
1
2
2
kCk
+
kDk
2 kXk(R,C) = 2 sup
r /2 ? X ? c /2
= sup
inf
F
F
1
1
tr
r?R,c?C
r?R,c?C CD > =r /2 ?X?c /2
2
2
1/2
2
1/2
2
1/2
1/2
,
? inf
sup
r A
+ sup
c B
= sup
inf
r ? A
+
c ? B
r?R,c?C AB > =X
AB > =X
F
F
F
r?R
c?C
F
where for the next-to-last step we set C = r1/2 A and D = c1/2 B, and the last step follows because
sup inf ? inf sup always (weak duality). The reverse inequality holds as well (strong duality), and
is proved in the Supplementary Materials, where we also prove the special-case result.
4
An approximate convex hull and a learning guarantee
In this section, we look for theoretical bounds on error for the problem of estimating unobserved
entries in a matrix Y that is approximately low-rank. Our results apply for either uniform or nonuniform sampling of entries from the matrix. We begin with a result comparing the (R, C)-norm unit
ball to a convex hull of rank-1 matrices, which will be useful for proving our learning guarantee.
4.1
Convex hull
To gain a better theoretical understanding of the (R, C) norm, we first need to define corresponding
vector norms on Rn and Rm . For any u ? Rn , let
s
X
1/2
kukR := sup
ri u2i = sup
diag(r) ? u
.
r?R
r?R
i
2
We can think of this norm as a way to interpolate between the `2 and `? vector norms. For example,
if we choose R = R as defined in (5), then kukR is equal to the root-mean-square of the ?1
largest entries of u whenever ?1 is an integer. Defining kvkC analogously for v ? Rm , we can now
relate these vector norms to the (R, C)-norm on matrices.
5
Theorem 2. For any convex R ? ?[n] and C ? ?[m] , the (R, C)-norm unit ball is bounded above
and below by a convex hull as:
o
n
Conv uv >: kukR = kvkC = 1 ? X : kXk(R,C) ? 1 ? KG ?Conv uv >: kukR = kvkC = 1 ,
where KG ? 1.79 is Grothendieck?s constant, and implicitly u ? Rn , v ? Rm .
This result is a nontrivial extension of Srebro and Shraibman [1]?s analysis for the max norm and
the trace norm. They show that the statement holds for the max norm, i.e. when R = ?[n] and
C = ?[m] , and that the trace norm unit ball is exactly equal to the corresponding convex hull (see
Corollary 2 and Section 3.2 in their paper, respectively).
Proof sketch for Theorem 2. To prove the first inclusion, given any X = uv > with kukR = kvkC =
1, we apply the factorization result Theorem 1 to see that kXk(R,C) ? 1. Since the (R, C)-norm
unit ball is convex, this is sufficient. For the second inclusion, we state a weighted version of
Grothendieck?s Inequality (proof in the Supplementary Materials):
sup hY, U V > i : U ? Rn?k , V ? Rm?k ,
U(i)
2 ? ai ?i,
V(j)
2 ? bj ?j
= KG ? sup hY, uv > i : u ? Rn , v ? Rm , |ui | ? ai ?i, |vj | ? bj ?j .
We then apply this weighted inequality to the dual norm of the (R, C)-norm to prove the desired
inclusion, as in Srebro and Shraibman [1]?s work for the max norm case (see Corollary 2 in their
paper). Details are given in the Supplementary Materials.
4.2
Learning guarantee
We now give our main matrix reconstruction result, which provides error bounds for a family of
norms interpolating between the max norm and the smoothed weighted trace norm.
Theorem 3. Let p be any distribution on [n] ? [m]. Suppose that, for some ? ? 1, R ?
?
R?
1/2,? and C ? C1/2,? , where these two sets are defined in (6). Let S = {(it , jt ) : t = 1, . . . , s} be
a random sample of locations in the matrix drawn i.i.d. from p, where s ? n. Then, in expectation
over the sample S,
r !
X
X
kn
log(n)
b
,
pij Yij ? Xij ?
inf ?
pij |Yij ? Xij | + O
? 1+ ?
s
?
kXk(R,C) ? k ij
ij
{z
}
{z
} |
|
Approximation error
?
Excess error
Ps
|Yit jt ? Xit jt |. Additionally, if we assume that s ?
p
n log(n), then in the excess risk bound, we can reduce the term log(n) to log(n).
b = arg min
where X
kXk
(R,C) ?
k
t=1
Proof sketch for Theorem 3. The main idea ?
is to use the convex hull formulation from Theorem 2
to show that, for any X with kXk(R,C) ? k, there exists a decomposition X = X 0 + X 00 with
p
?
e represents the smoothed marginals with
kX 0 kmax ? O( k) and kX 00 ktr(ep) ? O( k/?), where p
1
smoothing parameter ? = /2 as in (3). We then apply known bounds on the Rademacher complexity
of the max norm unit ball [1]
weighted
trace norm unit ball [6], to bound the
and the smoothed ?
Rademacher complexity of X : kXk(R,C) ? k . This then yields a learning guarantee by
Theorem 8 of Bartlett and Mendelson [13]. Details are given in the Supplementary Materials.
As special cases of this theorem, we can re-derive the existing results for the max norm and smoothed
p
weighted trace norm. Specifically, choosing ? = ? gives us an excess error term of order kn/s
for
p the max norm, previously shown by [1], while setting ? = 1 yields an excess error term of order
kn log(n)/s for the smoothed weighted trace norm as long as s ? n log(n), as shown in [6].
What advantage does this new result offer over the existing results for the max norm and for the
smoothed weighted trace norm? To simplify the comparison, suppose we choose ? = log2 (n), and
?
define R = R?
1/2,? and C = C1/2,? . Then, comparing to the max norm result (when ? = ?), we see
6
Table 1: Matrix fitting for the five methods used in experiments.
?
?
?
?
?
0.20
?
?
?
?
?
?
?
?
?
30
60
120
240
0.15
?
?
?
?
?
?
?
?
?
?
?
30
Matrix dimension n
Free parameters
?
?
?
?; ?
?; ? ; ?
?
?
?
0.13
?
Mean sq. error per entry
?
?
Fixed parameters
? arbitrary; ? = 1
? = 1; ? = 0
? = 0; ? = 0
? =0
?
0.11
0.24
0.28
Trace
Emp. trace
Smth. trace
Max
Local max
?
0.16
Mean sq. error per entry
Norm
Max norm
(Uniform) trace norm
Empirically-weighted trace norm
Arbitrarily-smoothed emp.-wtd. trace norm
Local max norm
60
120
240
Matrix dimension n
Figure 1: Simulation results for matrix reconstruction with a rank-2 (left) or rank-4 (right) signal, corrupted by
noise. The plot shows per-entry squared error averaged over 50 trials, with standard error bars. For the rank-4
experiment, max norm error exceeded 0.20 for each n = 60, 120, 240 and is not displayed in the plot.
that the excess error term is the same in both cases (up to a constant), but the approximation error
term may in general be much lower for the local max norm than for the max norm. Comparing next
to the weighted trace norm (when ? = 1), we see that the excess error term is lower by a factor of
log(n) for the local max norm. This may come at a cost of increasing the approximation error, but in
general this increase will be very small. In particular, the local max norm result allows us to give a
meaningful guarantee for a sample size s = ? (kn), rather than requiring s ? ? (kn log(n)) as for
any trace norm result, but with a hypothesis class significantly richer than the max norm constrained
class (though not as rich as the trace norm constrained class).
5
Experiments
We test the local max norm on simulated and real matrix reconstruction tasks, and compare its
performance to the max norm, the uniform and empirically-weighted trace norms, and the smoothed
empirically-weighted trace norm.
5.1 Simulations
We simulate n ? n noisy matrices for n = 30, 60, 120, 240, where the underlying signal has rank
k = 2 or k = 4, and we observe s = 3kn entries (chosen uniformly without replacement). We
performed 50 trials for each of the 8 combinations of (n, k).
Data For each trial, we randomly draw a matrix U ? Rn?k by drawing each row uniformly at
random from the unit sphere in Rn . We generate V ? Rm?k similarly. We set Y = U V > + ? ? Z,
where the noise matrix Z has i.i.d. standard normal entries and ? = 0.3 is a moderate noise level.
We also divide the n2 entries of the matrix into sets S0 t S1 t S2 which consist of s = 3kn training
entries, s validation entries, and n2 ? 2s test entries, respectively, chosen uniformly at random.
Methods We use the two-parameter family of norms defined in (7), but replacing the true
b i? and p
b ?j . For each ?, ? ? {0, 0.1, . . . , 0.9, 1}
marginals pi? and p?j with the empirical marginals p
and each penalty parameter value ? ? {21 , 22 , . . . , 210 }, we compute the fitted matrix
X
2
b = arg min
X
.
(10)
(i,j)?S0 (Yij ? Xij ) + ? ? kXk(R?,? ,C?,? )
(In fact, we use a rank-8 approximation to this optimization problem, as described in Section 3.) For
each method, we use S1 to select the best ?, ? , and ?, with restrictions on ? and/or ? as specified by
the definition of the method (see Table 1), then report the error of the resulting fitted matrix on S2 .
Results The results for these simulations are displayed in Figure 1. We see that the local max
norm results in lower error than any of the tested existing norms, across all the settings used.
7
Table 2: Root mean squared error (RMSE) results for estimating movie ratings on Netflix and MovieLens data
using a rank 30 model. Setting ? = 0 corresponds to the uniform or weighted or smoothed weighted trace
norm (depending on ?), while ? = 1 corresponds to the max norm for any ? value.
?\?
0.00
0.05
0.10
0.15
0.20
1.00
0.00
0.7852
0.7836
0.7831
0.7833
0.7842
0.7997
MovieLens
0.05
0.10
0.7827 0.7838
0.7822 0.7842
0.7837 0.7846
0.7842 0.7854
0.7853 0.7866
?\?
0.00
0.05
0.10
0.15
0.20
1.00
1.00
0.7918
?
?
?
?
0.00
0.9107
0.9095
0.9096
0.9102
0.9126
0.9235
Netflix
0.05
0.9092
0.9090
0.9098
0.9111
0.9344
0.10
0.9094
0.9107
0.9122
0.9131
0.9153
1.00
0.9131
?
?
?
?
5.2 Movie ratings data
We next compare several different matrix norms on two collaborative filtering movie ratings datasets,
the Netflix [14] and MovieLens [15] datasets. The sizes of the data sets, and the split of the ratings
into training, validation and test sets3 , are:
Dataset
Netflix
MovieLens
# users
480,189
71,567
# movies
17,770
10,681
Training set
100,380,507
8,900,054
Validation set
100,000
100,000
Test set
1,408,395
1,000,000
We test the local max norm given in (7) with ? ? {0, 0.05, 0.1, 0.15, 0.2} and ? ? {0, 0.05, 0.1}.
We also test ? = 1 (the max norm?here ? is arbitrary) and ? = 1, ? = 0 (the uniform trace norm).
We follow the test protocol of [6], with a rank-30 approximation to the optimization problem (10).
Table 2 shows root mean squared error (RMSE) for the experiments. For both the MovieLens and
Netflix data, the local max norm with ? = 0.05 and ? = 0.05 gives strictly better accuracy than any
previously-known norm studied in this setting. (In practice, we can use a validation set to reliably
select good values for the ? and ? parameters4 .) For the MovieLens data, the local max norm
achieves RMSE of 0.7822, compared to 0.7831 achieved by the smoothed empirically-weighted
trace norm with ? = 0.10, which gives the best result among the previously-known norms. For the
Netflix dataset the local max norm achieves RMSE of 0.9090, improving upon the previous best
result of 0.9095 achieved by the smoothed empirically-weighted trace norm [6].
6
Summary
In this paper, we introduce a unifying family of matrix norms, called the ?local max? norms, that
generalizes existing methods for matrix reconstruction, such as the max norm and trace norm. We
examine some interesting sub-families of local max norms, and consider several different options
for interpolating between the trace (or smoothed weighted trace) and max norms. We find norms
lying strictly between the trace norm and the max norm that give improved accuracy in matrix
reconstruction for both simulated data and real movie ratings data. We show that regularizing with
any local max norm is fairly simple to optimize, and give a theoretical result suggesting improved
matrix reconstruction using new norms in this family.
Acknowledgements
R.F. is supported by NSF grant DMS-1203762. R.S. is supported by NSERC and Early Researcher
Award.
3
For Netflix, the test set we use is their ?qualification set?, designed for a more uniform distribution of
ratings across users relative to the training set. For MovieLens, we choose our test set at random from the
available data.
4
To check this, we subsample half of the test data at random, and use it as a validation set to choose (?, ? )
for each method (as specified in Table 1). We then evaluate error on the remaining half of the test data. For
MovieLens, the local max norm gives an RMSE of 0.7820 with selected parameter values ? = ? = 0.05, as
compared to an RMSE of 0.7829 with selected smoothing parameter ? = 0.10 for the smoothed weighted trace
norm. For Netflix, the local max norm gives an RMSE of 0.9093 with ? = ? = 0.05, while the smoothed
weighted trace norm gives an RMSE of 0.9098 with ? = 0.05. The other tested methods give higher error on
both datasets.
8
References
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9
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3,995 | 4,616 | Bandit Algorithms boost motor-task selection for
Brain Computer Interfaces
Joan Fruitet
INRIA, Sophia Antipolis
2004 Route des Lucioles
06560 Sophia Antipolis, France
[email protected]
Alexandra Carpentier
Statistical Laboratory, CMS
Wilberforce Road, Cambridge
CB3 0WB UK
[email protected]
R?emi Munos
INRIA Lille - Nord Europe
40, avenue Halley
59000 Villeneuve d?ascq, France
[email protected]
Maureen Clerc
INRIA, Sophia Antipolis
2004 Route des Lucioles
06560 Sophia Antipolis, France
[email protected]
Abstract
Brain-computer interfaces (BCI) allow users to ?communicate? with a computer
without using their muscles. BCI based on sensori-motor rhythms use imaginary
motor tasks, such as moving the right or left hand, to send control signals. The
performances of a BCI can vary greatly across users but also depend on the tasks
used, making the problem of appropriate task selection an important issue. This
study presents a new procedure to automatically select as fast as possible a discriminant motor task for a brain-controlled button. We develop for this purpose
an adaptive algorithm, UCB-classif , based on the stochastic bandit theory. This
shortens the training stage, thereby allowing the exploration of a greater variety of
tasks. By not wasting time on inef?cient tasks, and focusing on the most promising ones, this algorithm results in a faster task selection and a more ef?cient use of
the BCI training session. Comparing the proposed method to the standard practice
in task selection, for a ?xed time budget, UCB-classif leads to an improved classi?cation rate, and for a ?xed classi?cation rate, to a reduction of the time spent
in training by 50%.
1
Introduction
Scalp recorded electroencephalography (EEG) can be used for non-muscular control and communication systems, commonly called brain-computer interfaces (BCI). BCI allow users to ?communicate? with a computer without using their muscles. The communication is made directly through
the electrical activity from the brain, collected by EEG in real time. This is a particularly interesting prospect for severely handicapped people, but it can also be of use in other circumstances, for
instance for enhanced video games.
A possible way of communicating through the BCI is by using sensori-motor rhythms (SMR), which
are modulated in the course of movement execution or movement imagination. The SMR corresponding to movement imagination can be detected after pre-processing the EEG, which is corrupted
by important noise, and after training (see [1, 2, 3]). A well-trained classi?er can then use features of
the SMR in order to discriminate periods of imagined movement from resting periods, when the user
is idle. The detected mental states can be used as buttons in a Brain Computer Interface, mimicking
traditional interfaces such as keyboard or mouse button.
This paper deals with training a BCI corresponding to a single brain-controlled button (see [2, 4]),
in which a button is pressed (and instantaneously released) when a certain imagined movement
is detected. The important steps are thus to ?nd a suitable imaginary motor task, and to train a
1
classi?er. This is far from trivial, because appropriate tasks which can be well classi?ed from the
background resting state are highly variable among subjects; moreover, the classi?er requires to be
trained on a large set of labeled data. The setting up of such a brain-controlled button can be very
time consuming, given that many training examples need to be acquired for each of the imaginary
motor task to be tested.
The usual training protocol for a brain-controlled button is to display sequentially to the user a set
of images, that serve as prompts to perform the corresponding imaginary movements. The collected
data are used to train the classi?er, and to select the imaginary movement that seems to provide the
highest classi?cation rate (compared to the background resting state). We refer to this imaginary
movement as the ?best imaginary movement?. In this paper, we focus on the part of the training
phase that consists in ef?ciently ?nding this best imaginary movement. This is an important problem, since the SMR collected by the EEG are heterogeneously noisy: some imaginary motor tasks
will provide higher classi?cation rates than others. In the literature, ?nding such imaginary motor
tasks is deemed an essential issue (see [5, 6, 7]), but, to the best of our knowledge, no automatized
protocol has yet been proposed to deal with it. We believe that enhancing the ef?ciency of the training phase is made even more essential by the facts that (i) the best imaginary movement differs from
one user to another, e.g. the best imaginary movement for one user could be to imagine moving the
right hand, and for the next, to imagine moving both feet (see [8]) and (ii) using a BCI requires much
concentration, and a long training phase exhausts the user.
If an ?oracle? were able to state what the best imaginary movement is, then the training phase would
consist only in requiring the user to perform this imaginary movement. The training set for the
classi?er on this imaginary movement would be large, and no training time would be wasted in
asking the user to perform sub-optimal and thus useless imaginary movements. The best imaginary
movement is however not known in advance, and so the commonly used strategy (which we will
refer to as uniform) consists in asking the user to perform all the movements a ?xed number of
times. An alternative strategy is to learn while building the training set what imaginary movements
seem the most promising, and ask the classi?er to perform these more often. This problem is quite
archetypal to a ?eld of Machine Learning called Bandit Theory (initiated in [9]). Indeed, the main
idea in Bandit Theory is to mix the Exploration of the possible actions1 , and their Exploitation to
perform the empirical best action.
Contributions This paper builds on ideas of Bandit Theory, in order to propose an ef?cient method
to select the best imaginary movement for the activation of a brain-controlled button. To the best of
our knowledge, this is the ?rst contribution to the automation and optimization of this task selection.
? We design a BCI experiment for imaginary motor task selection, and collect data on several
subjects, for different imaginary motor tasks, in the aim of testing our methods.
? We provide a bandit algorithm (which is strongly inspired by the Upper Con?dence Bound
Algorithm of [10]) adapted to this classi?cation problem. In addition, we propose several
variants of this algorithm that are intended to deal with other slightly different scenarios
that the practitioner might face. We believe that this bandit-based classi?cation technique
is of independent interest and could be applied to other task selection procedures under
constraints on the samples.
? We provide empirical evidence that using such an algorithm considerably speeds up the
training phase for the BCI. We gain up to 18% in terms of classi?cation rate, and up to 50%
in training time, when compared to the uniform strategy traditionally used in the literature.
The rest of the paper is organized as follows: in Section 2, we describe the EEG experiment we built
in order to acquire data and simulate the training of a brain-controlled button. In Section 3, we model
the task selection as a bandit problem, which is solved using an Upper Con?dence Bound algorithm.
We motivate the choice of this algorithm by providing a performance analysis. Section 4, which is
the main focus of this paper, presents results on simulated experiments, and proves empirically the
gain brought forth by adaptive algorithms in this setting. We then conclude this paper with further
perspectives.
1
Here, the actions are images displayed to the BCI user as prompts to perform the corresponding imaginary
tasks.
2
2
Material and protocol
BCI systems based on SMR rely on the users? ability to control their SMR in the mu (8-13Hz) and/or
beta (16-24Hz) frequency bands [1, 2, 3]. Indeed, these rhythms are naturally modulated during real
and imagined motor action.
More precisely, real and imagined movements similarly activate neural structures located in the
sensori-motor cortex, which can be detected in EEG recordings through increases in power (event
related synchronization or ERS) and/or decreases in power (event related de-synchronization or
ERD) in the mu and beta frequency bands [11, 12]. Because of the homuncular organization of the
sensori-motor cortex [13], different limb movements may be distinguished according to the spatial
layout of the ERD/ERS.
BCI based on the control of SMR generally use movements lasting several seconds, that enable
continuous control of multidimensional interfaces [1]. On the contrary this work targets a braincontrolled button that can be rapidly triggered by a short motor task [2, 4]. A vast variety of motor
tasks can be used in this context, like imagining rapidly moving the hand, grasping an object, or
kicking an imaginary ball. We remind that the best imaginary movement differs from one user to
another (see [8]).
As explained in the Introduction, the use of a BCI must always be preceded by a training phase. In
the case of a BCI managing a brain-controlled button through SMR, this training phase consists in
displaying to the user a sequence of images corresponding to movements, that he/she must imagine
performing. By processing the EEG, the SMR associated to the imaginary movements and to idle
periods can be extracted. Collecting these labeled data results in a training set, which serves to train
the classi?er between the movements, and the idle periods. The imaginary movement with highest
classi?cation rate is then selected to activate the button in the actual use of the BCI.
The rest of this Section explains in more detail the BCI material and protocol used to acquire the
EEG, and to extract the features from the signal.
2.1 The EEG experiment
The EEG experiment was similar to the training of a brain-controlled button: we presented, at
random timing, cue images during which the subjects were asked to perform 2 second long motor
tasks (intended to activate the button).
Six right-handed subjects, aged 24 to 39, with no disabilities, were sitting at 1.5m of a 23? LCD
screen. EEG was recorded dat a sampling rate of 512Hz via 11 scalp electrodes of a 64-channel cap
and ampli?ed with a TMSI ampli?er (see Figure 1). The OpenViBE platform [14] was used to run
the experiment. The signal was ?ltered in time through a band-pass ?lter, and in space through a
surface Laplacian to increase the signal to noise ratio.
The experiment was composed of 5 to 12 blocks of approximately 5 minutes. During each block, 4
cue images were presented for 2 seconds in a random order, 10 times each. The time between two
image presentations varied between 1.5s and 10s. Each cue image was a prompt for the subject to
perform or imagine the corresponding motor action during 2 seconds, namely moving the right or
left hand, the feet or the tongue.
2.2 Feature extraction
In the case of short motor tasks, the movement (real or imagined) produces an ERD in the mu and
beta bands during the task, and is followed by a strong ERS [4] (sometimes called beta rebound as
it is most easily seen in the beta frequency band).
We extracted features of the mu and beta bands during the 2-second windows of the motor action
and in the subsequent 1.5 seconds of signal in order to use the bursts of mu and beta power (ERS
or rebound) that follow the indicated movement. Figure 1 shows a time-frequency map on which
the movement and rebound windows are indicated. One may observe that, during the movement,
the power in the mu and beta bands decreases (ERD) and that, approximately 1 second after the
movement, it increases to reach a higher level than in the resting state (ERS).
More precisely, the features were chosen as the power around 12Hz and 18Hz extracted at 3 electrodes over the sensori-motor cortex (C3, C4 and Cz). Thus, 6 features are extracted during the
movement and 6 during the rebound. The lengths and positions of the windows and the frequency
bands were chosen according to a preliminary study with one of the subjects and were deliberately
kept ?xed for the other subjects.
3
One of the goals of our algorithm is to be able to select the best task among a large number of tasks.
However, in our experiment, only a limited number of tasks were used (four), because we limited
the length of the sessions in order not to tire the subjects. To demonstrate the usefulness of our
method for a larger number of tasks, we decided to create arti?cial (degraded) tasks by mixing the
features of one of the real tasks (the feet) with different proportions of the features extracted during
the resting period.
Figure 1: A: Layout of the 64 EEG cap, with (in black) the 3 electrodes from which the features
are extracted. The electrodes marked in blue/grey are used for the Laplacian. B: Time-frequency
map of the signal recorded on electrode C3, for a right hand movement lasting 2 seconds (subject
1). Four features (red windows) are extracted for each of the 3 electrodes.
2.3 Evaluation of performances
For each task k, we can classify between when the subject is inactive and when he/she is performing
task k. Consider a sample (X, Y ) ? Dk where Dk is the distribution of the data restricted to task k
and the idle task (task 0), X is the feature set, and Y is the label (1 if the sample corresponds to task
k and 0 otherwise).
We consider a compact set of classi?ers H. De?ne the best classi?er in H for task k as h?k =
arg minh?H E(X,Y )?Dk [1{h(X) ?= Y }]. De?ne the theoretical loss rk? of a task k as the probability
of labeling incorrectly a new data drawn from Dk with the best classi?er h?k , that is to say rk? =
1 ? P(X,Y )?D (h?k (X) ?= Y ).
At time t, there are Tk,t + T0,t samples (Xi , Yi )i?Tk,t +T0,t (where Tk,t is the number of samples for
task k, and T0,t is the number of samples for the idle task) that? are available. With these data,
? we
? k,t = arg minh?H ?Tk,t +T0,t 1{h(Xi ) ?= Yi } . We
build the empirical minimizer of the loss h
i=1
?
??
Tk,t +T0,t
de?ne the empirical loss of this classi?er r?k,t = 1 ? minh?H
1{h(X
)
=
?
Y
}
i
i .
i=1
Since during our experiments we collect, between each imaginary task, a sample of idle condition,
we have T0,t ? Tk,t .
From Vapnik-Chervonenkis theory (see [15] and also the Supplementary Material), we obtain
?
?
with
??probability
? 1 ? ?, that the error in generalization of classi?er hk,t is not larger than rk +
d log(1/?)
O
, where d is the VC dimension of the domain of X. This implies that the perTk,n
formance of the optimal empirical classi?er for task k is close to the performance of the optimal
classi?er for task k. Also with probability 1 ? ?, ?
? d log(1/?) ?
.
(1)
|?
rk,t ? rk? | = O
Tk,n
We consider in this paper linear classi?ers. In this case, the VC dimension d is the dimension of X,
i.e. the number of features. The loss we considered ((0, 1) loss) is dif?cult to minimize in practice
? k,t provided by linear
because it is not convex. This is why we consider in this work the classi?er h
SVM. We also estimate the performance r?k,t of this classi?er by cross-validation: we use the leaveone-out technique when less than 8 samples of the task are available, and a 8-fold validation when
more repetitions of the task have been recorded. As explained in [15], results similar to Equation 1
hold for this classi?er.
We will use in the next Section the results of Equation 1, in order to select as fast as possible the
task with highest rk? and collect as many samples from it as possible.
4
3
A bandit algorithms for optimal task selection
In order to improve the ef?ciency of the training phase, it is important to ?nd out as fast as possible
what are the most promising imaginary tasks (i.e. tasks with large rk? ). Indeed, it is important to
collect as many samples as possible from the best imaginary movement, so that the classi?er built
for this task is as precise as possible.
In this Section, we propose the UCB-Classif algorithm, inspired by the Upper Con?dence Bound
algorithm in Bandit Theory (see [10]).
3.1 Modeling the problem by a multi-armed bandit
Let K denote the number of different tasks2 and N the total number of rounds (the budget) of the
training stage. Our goal is to ?nd a presentation strategy for the images (i.e. that choose at each timestep t ? {1, . . . , N } an image kt ? {1, . . . , K} to show), which allows to determine the ?best?,
i.e. most discriminative imaginary movement, with highest classi?cation rate in generalization).
Note that, in order to learn an ef?cient classi?er, we need as many training data as possible, so
our presentation strategy should rapidly focus on the most promising tasks in order to obtain more
samples from these rather than from the ones with small classi?cation rate.
This issue is relatively close to the stochastic bandit problem [9]. The classical stochastic bandit
problem is de?ned by a set of K actions (pulling different arms of bandit machines) and to each
action is assigned a reward distribution, initially unknown to the learner. At time t ? {1, . . . , N },
if we choose an action kt ? {1, . . . , K}, we receive a reward sample drawn independently from the
distribution of the corresponding action kt . The goal is to ?nd a sampling strategy which maximizes
the sum of obtained rewards.
We model the K different images to be displayed as the K possible actions, and we de?ne the
reward as the classi?cation rate of the corresponding motor action. In the bandit problem, pulling a
bandit arm directly gives a stochastic reward which is used to estimate the distribution of this arm.
In our case, when we display a new image, we obtain a new data sample for the selected imaginary
movement, which provides one more data sample to train or test the corresponding classi?er and
thus obtain a more accurate performance. The main difference is that for the stochastic bandit
problem, the goal is to maximize the sum of obtained rewards, whereas ours is to maximize the
performance of the ?nal classi?er. However, the strategies are similar: since the distributions are
initially unknown, one should ?rst explore all the actions (exploration phase) but then rapidly select
the best one (exploitation phase). This is called the exploration-exploitation trade-off.
3.2 The UCB-classif algorithm
The task presentation strategy is a close variant of the Upper Con?dence Bound (UCB) algorithm
of [10], which builds high probability Upper Con?dence Bounds (UCB) on the mean reward value
of each action, and selects at each time step the action with highest bound.
We adapt the idea of this UCB algorithm to our adaptive classi?cation problem and call this algorithm UCB-classif (see the pseudo-code in Table 1). The algorithm builds a sequence of values Bk,t
de?ned as
?
a log N
,
(2)
Bk,t = r?k,t +
Tk,t?1
where r?k,t represents an estimation of the classi?cation rate built from a q-fold cross-validation technique and the a corresponds to Equation 1 (see Supplementary Material for the precise theoretical
value). The cross-validation uses a linear SVM classi?er based on the Tk,t data samples obtained (at
time t) from task k. Writing rk? the classi?cation rate for the optimal linear SVM classi?er (which
would be obtained by using a in?nite number of samples), we have the property that Bk,t is a high
probability upper bound on rk? : P(Bk,t < rk? ) decreases to zero polynomially fast (with N ).
The intuition behind the algorithm is that it selects at time t an action kt either because it has a good
classi?cation rate r?k,t (thus it is interesting to obtain more samples from it, to perform exploitation)
or because its classi?cation
? rate is highly uncertain since it has not been sampled many times, i.e.,
log N
is large (thus it is important to explore it more). With this strategy,
Tk,t?1 is small and then aTk,t?1
the action that has the highest classi?cation rate is presented more often. It is indeed important to
2
The tasks correspond to the imaginary movements of moving the feet, tongue, right hand, and left hand,
plus 4 additional degraded tasks (so a total of K = 8 actions).
5
The UCB-Classif Algorithm
Parameters: a, N , q
Present each image q = 3 times (thus set Tk,qK = q).
for t = qK + 1, . . . , N do
Evaluate the performance r?k,t of each
q action (by a 8-split Cross Validation or leave-one-out if Tk,t < 8).
Compute the UCB: Bk,t = r?k,t +
a log N
Tk,t?1
for each action 1 ? k ? K.
Select the image to present: kt = arg maxk?{1,...,K} Bk,t .
Update T : Tkt ,t = Tkt ,t?1 + 1 and ?k ?= kt , Tk,t = Tk,t?1
end for
Table 1: Pseudo-code of the UCB-classif algorithm.
gather as much data as possible from the best action in order to build the best possible classi?er. The
UCB-classif algorithm guarantees that the non-optimal tasks are chosen only a negligible fraction
of times (O(log N ) times out of a total budget N ). The best action is thus sampled N ? O(log N )
times (this is formally proven in the Supplementary Material)3 . It is a huge gain when compared
to actual unadaptive procedures for building training sets. Indeed, the unadaptive optimal strategy
is to sample each action N/K times, and thus the best task is only sampled N/K times (and not
N ? O(log N )). More precisely, we prove the following Theorem.
Theorem 1 For any N ? 2qK, with probability at least 1 ? N1 , if Equation 1 is satis?ed (e.g. if
the data are i.i.d.) and if a ? 5(d + 1) we have that the number of times that the image of the best
imaginary movement is displayed by algorithm UCB-classif is such that (where r? = maxk rk? )
TN? ? N ?
? a log(8N K)
8 ?
.
(r ? rk? )2
k
The proof of this Theorem is in the provided Supplementary Material, Appendix A.
3.3 Discussion on variants of this algorithm
We stated that our objective, given a ?xed budget N , is to ?nd as fast as possible the image with
highest classi?cation rate, and to train the classi?er with as many samples as possible. Depending on
the objectives of the practitioner, other possible aims can however be pursued. We brie?y describe
two other settings, and explain how ideas from the bandit setting can be used to adapt to these
different scenarios.
Best stopping time: A close, yet different, goal, is to ?nd the best time for stopping the training
phase. In this setting, the practitioner?s aim is to stop the training phase as soon as the algorithm has
built an almost optimal classi?er for the user. With ideas very similar to those developed in [16] (and
extended for bandit problems in e.g. [17]), we can think of an adaptation of algorithm UCB-classif
to this new formulation of the problem. Assume that the objective is to ?nd an ??optimal classi?er
with probability 1 ? ?, and to stop the training phase as soon as this classi?er is built. Then using
ideas similar to those presented
? in [17], an ef?cient algorithm will at time t select the action that
K/?)
?
and will stop at the ?rst time T? when there is an action
maximizes Bk,t = r?k,t + a log(N
Tk,t?1
?
a log(N K/?)
?
k?? such that ?k ?= k?? , Bk??? ,T? ? Bk,
>
?
+
2
. We thus shorten the training phase
Tk,T? ?1
T?
almost optimally on the class of adaptive algorithms (see [17] for more details).
Choice of the best action with a limited budget: Another question that could be of interest for the
practitioner is to ?nd the best action with a ?xed budget (and not train the classi?er at the same time).
We can use ideas from paper
? [18] to modify UCB-classif. By selecting at each time t the action that
?K)
??
maximizes Bk,t = r?k,t + a(N
Tk,t?1 , we attain this objective in the sense that we guarantee that the
probability of choosing a non-optimal action decreases exponentially fast with N .
4
Results
We present some numerical experiments illustrating the ef?ciency of Bandit algorithms for this
problem. Although the objective is to implement UCB-classif on the BCI device, in this paper we
test the algorithm on real databases that we bootstrap (this is explained in details later). This kind of
3
The ideas of the proof are very similar to the ideas in [10], with the difference that the upper bounds have
to be computed using inequalities based on VC-dimension.
6
procedure is common for testing the performances of adaptive algorithms (see e.g. [19]). Acquiring
data for BCI experiments is time-consuming because it requires a human subject to sit through the
experiment. The advantage of bootstrapping is that several experiments can be performed with a
single database, making it possible to provide con?dence bands for the results.
In this Section, we present the experiments we performed, i.e. describe the kind of data we collect,
and illustrate the performance of our algorithm on these data.
4.1 Performances of the different tasks
The images that were displayed to the subjects correspond to movements of both feet, of the tongue,
of the right hand, and of the left hand (4 actions in total). Six right-handed subjects went through the
experiment with real movements and three of them went through an additional shorter experiment
with imaginary movements. For four of the six subjects, the best performance for the real movement
was achieved with the right hand, whereas the two other subjects? best tasks corresponded to the
left hand and the feet. We collected data for these four tasks. It is not a large number of tasks but
we needed a large amount of data for each of them in order to do a signi?cant comparison. In order
to have a larger number of tasks and place ourselves in a more realistic situation, we created some
articicial tasks (see below). Results on only four tasks are presented in a companion article [20].
Surprisingly, two of the subjects who went through the imaginary experiment obtained better results
while imagining moving their left hand than their right hand, which was the best task during the real
movements experiment. For the third subject who did the imaginary experiment, the best task was
the feet, as for the real movement experiment.
As explained in section 2.2, for this study we chose to use a very small set of ?xed features (12 features, extracted from 3 electrodes, 2 frequency bands and 2 time-windows), calibrated on only one of
the six subjects during a preliminary experiment. In this work, the features were not subject-speci?c.
It would certainly improve the classi?cation results to tune the features. Using the bandit algorithm
to tune the features and to select the tasks at the same time presents a risk over?tting, especially for
an initially very small amount of data, and also a risk of biasing the task selection to those that have
been the most sampled, and for which the features will thus be the best tuned. Although for all the
subjects, the best task achieved a classi?cation accuracy above 85%, this accuracy could further be
improved by using a larger set of subject-speci?c features [21] and more advanced techniques (like
the CSP [22] or feature selection [23]).
4.2 Performances of the bandit algorithm
We compare the performance of the UCB-classif sampling strategy to a uniform strategy, i.e. the
standard way of selecting a task, consisting of N/K presentations of each image.
Movement
Number of presentations Off-line classi?cation rate
Right hand
28.6 ? 12.8
88.1%
Left hand
9.0 ? 7.5
80.5%
Feet
11.6 ? 9.5
82.6%
Tongue
4.5 ? 1.5
63.3%
Feet 80%
5.1 ? 2.6
71.4%
Feet 60%
4.0 ? 1.5
68.6%
Feet 40%
3.5 ? 1.0
59.2%
Feet 20%
3.5 ? 0.9
54.0%
Total presentations
70
Table 2: Actions presented by the UCB-classif algorithm for subject 5 across 500 simulated online
BCI experiments. Feet X% is a mixture of the features measured during feet movement and during
the resting condition, with a X/100-X proportion. (The off-line classi?cation rate of each action
gives an idea of the performance of each action).
To obtain a realistic evaluation of the performance of our algorithm we use a bootstrap technique.
More precisely, for each chosen budget N , for the UCB-classif strategy and the uniform strategy, we
simulated 500 online BCI experiments by randomly sampling from the acquired data of each action.
Table 2 shows, for one subject and for a ?xed budget of N = 70, the average number of presentations
of each task Tk , and its standard deviation, across the 500 simulated experiments. It also contains
the off-line classi?cation rate of each task to give an idea of the performances of the different tasks
for this subject. We can see that very little budget is allocated to the tongue movement and to the
most degraded feet 20% tasks, which are the less discriminative actions, and that most of the budget
is devoted to the right hand, thus enabling a more ef?cient training.
7
Figure 2 and Table 3 show, for different budgets (N ), the performance of the UCB-classif algorithm
versus the uniform technique. The training of the classi?er is done on the actions presented during
the simulated BCI experiment, and the testing on the remaining data.
For a budget N > 70 the UCB-classif could not be used for all the subjects because there was not
enough data for the best action (One subject only underwent a session of 5 blocks and so only 50
samples of each motor task were recorded. If we try to simulate an on-line experiment using the
UCB-classif with a budget higher than N = 70 it is likely to ask for a 51th presentation of the best
task, which has not been recorded).
The classi?cation results depend on which data is used to simulate the BCI experiment. To give an
idea of this variability, the ?rst and last quartiles are plotted as error bars on the graphics.
Budget (N ) Length of the experiment Uniform strategy UCB-classif Bene?t
30
3min45
47.7%
64.4%
+16.7%
40
5min
58.5%
77.2%
+18.7%
50
6min15
63.4%
82.0%
+18.5%
60
7min30
67.0%
84.0%
+17.1%
70
8min45
70.1%
85.7%
+15.6%
100
12min30
77.6%
*
150
18min45
83.2%
*
180
22min30
85.2%
*
Table 3: Comparison of the performances of the UCB-classif vs. the uniform strategy for different
budgets, averaged over all subjects, for real movements. (The increases are signi?cant with p >
95%.) For each budget, we give an indication of the length of the experiment (without counting
pauses between blocks) required to obtain this amount of data.
The UCB-classif strategy signi?cantly outperforms the uniform strategy, even for relatively small
N . On average on all the users it even gives better classi?cation rates when using only half of
the available samples, compared to the uniform strategy. Indeed, Table 3 shows that, to achieve a
classi?cation rate of 85% the UCB-classif only requires a budget of N = 70 whereas the uniform
strategy needs N = 180. We believe that such gain in performance motivates the implementation of
such a training algorithm in BCI devices, specially since the algorithm itself is quite simple and fast.
80
70
60
50
40
5
Adaptative Algorithm
Uniform Strategy
30
40
50
60
70 80
Budget N
90 100 110 120
Sujet 3 imaginary movement
90
80
70
60
50
40
Adaptative Algorithm
Uniform Strategy
30
40
50
Budget N
60
Classification rate of the chosen movement
90
Sujet 2 imaginary movement
Classification rate of the chosen movement
Classification rate of the chosen movement
Sujet 1 real movement
90
80
70
60
50
40
Adaptative Algorithm
Uniform Strategy
30
40
50
Budget N
60
70
Figure 2: UCB-classif algorithm (full line, red) versus uniform strategy (dashed line, black).
Conclusion
The method presented in this paper falls in the category of adaptive BCI based on Bandit Theory.
To the best of our knowledge, this is the ?rst such method for dealing with automatic task selection.
UCB-classif is a new adaptive algorithm that allows to automatically select a motor task in view of
a brain-controlled button. By rapidly eliminating non-ef?cient motor tasks and focusing on the most
promising ones, it enables a better task selection procedure than a uniform strategy. Moreover, by
more frequently presenting the best task it allows a good training of the classi?er. This algorithm
enables to shorten the training period, or equivalently, to allow for a larger set of possible movements
among which to select the best. In a paper due to appear [20], we implement this algorithm online.
A future research direction is to learn several discriminant tasks in order to activate several buttons.
Acknowledgements This work was partially supported by the French ANR grant Co-Adapt
ANR-09-EMER-002, Nord-Pas-de-Calais Regional Council, French ANR grant EXPLO-RA (ANR08-COSI-004), the EC Seventh Framework Programme (FP7/2007-2013) under grant agreement
270327 (CompLACS project), and by Pascal-2.
8
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9
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3,996 | 4,617 | Graphical Models via Generalized Linear Models
Pradeep Ravikumar
Department of Computer Science
University of Texas at Austin
[email protected]
Eunho Yang
Department of Computer Science
University of Texas at Austin
[email protected]
Zhandong Liu
Department of Pediatrics-Neurology
Baylor College of Medicine
[email protected]
Genevera I. Allen
Department of Statistics
Rice University
[email protected]
Abstract
Undirected graphical models, also known as Markov networks, enjoy popularity
in a variety of applications. The popular instances of these models such as Gaussian Markov Random Fields (GMRFs), Ising models, and multinomial discrete
models, however do not capture the characteristics of data in many settings. We
introduce a new class of graphical models based on generalized linear models
(GLMs) by assuming that node-wise conditional distributions arise from exponential families. Our models allow one to estimate multivariate Markov networks
given any univariate exponential distribution, such as Poisson, negative binomial,
and exponential, by fitting penalized GLMs to select the neighborhood for each
node. A major contribution of this paper is the rigorous statistical analysis showing that with high probability, the neighborhood of our graphical models can be
recovered exactly. We also provide examples of non-Gaussian high-throughput
genomic networks learned via our GLM graphical models.
1
Introduction
Undirected graphical models, also known as Markov random fields, are an important class of statistical models that have been extensively used in a wide variety of domains, including statistical
physics, natural language processing, image analysis, and medicine. The key idea in this class of
models is to represent the joint distribution as a product of clique-wise compatibility functions; given
an underlying graph, each of these compatibility functions depends only on a subset of variables
within any clique of the underlying graph. Such a factored graphical model distribution can also be
related to an exponential family distribution [1], where the unnormalized probability is expressed
as the exponential of a weighted linear combination of clique-wise sufficient statistics. Learning a
graphical model distribution from data within this exponential family framework can be reduced to
learning weights on these sufficient statistics. An important modeling question is then, how do we
choose suitable sufficient statistics? In the case of discrete random variables, sufficient statistics can
be taken as indicator functions as in the Ising or Potts model. These, however, are not suited to all
kinds of discrete variables such as that of non-negative integer counts. Similarly, in the case of continuous variables, Gaussian Markov Random Fields (GMRFs) are popular. The multivariate normal
distribution imposed by the GMRF, however, is a stringent assumption; the marginal distribution of
any variable must also be Gaussian.
In this paper, we propose a general class of graphical models beyond the Ising model and the GMRF
to encompass variables arising from all exponential family distributions. Our approach is motivated
by recent state of the art methods for learning the standard Ising and Gaussian MRFs [2, 3, 4].
1
The key idea in these recent methods is to learn the MRF graph structure by estimating nodeneighborhoods, which are estimated by maximizing the likelihood of each node conditioned on
the rest of the nodes. These node-wise fitting methods have been shown to be both computationally
and statistically attractive. Here, we study the general class of models obtained by the following
construction: suppose the node-conditional distributions of each node conditioned on the rest of the
nodes are Generalized Linear Models (GLMs) [5]. By the Hammersley-Clifford Theorem [6] and
some algebra as derived in [7], these node-conditional distributions entail a global distribution that
factors according to cliques defined by the graph obtained from the node-neighborhoods. Moreover,
these have a particular set of potential functions specified by the GLM. The resulting class of MRFs
broadens the class of models available off-the-shelf, from the standard Ising, indicator-discrete, and
Gaussian MRFs.
Beyond our initial motivation of finding more general graphical model sufficient statistics, a broader
class of parametric graphical models are important for a number of reasons. First, our models provide a principled approach to model multivariate distributions and network structures among a large
number of variables. For many non-Gaussian exponential families, multivariate distributions typically do not exist in an analytical or computationally tractable form. Graphical model GLMs provide
a way to ?extend? univariate exponential families of distributions to the multivariate case and model
and study relationships between variables for these families of distributions. Second, while some
have proposed to extend the GMRF to a non-parametric class of graphical models by first Gaussianizing the data and then fitting a GMRF over the transformed variables [8], the sample complexity of
such non-parametric methods is often inferior to parametric methods. Thus for modeling data that
closely follows a non-Gaussian distribution, statistical power for network recovery can be gained
by directly fitting parametric GLM graphical models. Third, and specifically for multivariate count
data, others have suggested combinatorial approaches to fitting graphical models, mostly in the context of contingency tables [6, 9, 1, 10]. These approaches, however, are computationally intractable
for even moderate numbers of variables.
Finally, potential applications for our GLM graphical models abound. Networks of call-times, time
spent on websites, diffusion processes, and life-cycles can be modeled with exponential graphical
models; other skewed multivariate data can be modeled with gamma or chi-squared graphical models. Perhaps the most interesting motivating applications are for multivariate count data such as from
website visits, user-ratings, crime and disease incident reports, bibliometrics, and next-generation
genomic sequencing technologies. The latter is a relatively new high-throughput technology to measure gene expression that is rapidly replacing the microarray [11]. As Gaussian graphical models are
widely used to infer genomic regulatory networks from microarray data, Poisson and negative binomial graphical models may be important for inferring genomic networks from the multivariate count
data arising from this emerging technology. Beyond next generation sequencing, there has been a
recent proliferation of new high-throughput genomic technologies that produce non-Gaussian data.
Thus, our more general class of GLM graphical models can be used for inferring genomic networks
from these new high-throughput technologies.
The construction of our GLM graphical models also suggests a natural method for learning such
models: node-wise neighborhood estimation by fitting sparsity constrained GLMs. A main contribution of this paper is to provide a sparsistency analysis for the recovery of the underlying graph
structure of this new class of MRFs. The presence of non-linearities arising from the GLM poses
subtle technical issues not present in the linear case [2]. Indeed, for the specific cases of logistic, and
multinomial respectively, [3, 4] derive such a sparsistency analysis via fairly extensive arguments
which were tuned to those specific cases. Here, we generalize their analysis to general GLMs, which
requires a slightly modified M-estimator and a more subtle theoretical analysis. We note that this
analysis might be of independent interest even outside the context of modeling and recovering graphical models. In recent years, there has been a trend towards unified statistical analyses that provide
statistical guarantees for broad classes of models via general theorems [12]. Our result is in this vein
and provides structure recovery for the class of sparsity constrained generalized linear models. We
hope that the techniques we introduce might be of use to address the outstanding question of sparsity
constrained M-estimation in its full generality.
2
2
A New Class of Graphical Models
Problem Setup and Background. Suppose X = (X1 , . . . , Xp ) is a random vector, with each
variable Xi taking values in a set X . Suppose G = (V, E) is an undirected graph over p nodes
corresponding to the p variables; the corresponding graphical model is a set of distributions that
satisfy Markov independence assumptions with respect to the graph. By the Hammersley-Clifford
theorem, any such distribution also factors according to the graph in the following way. Let C be
a set of cliques (fully-connected subgraphs) of the graph G, and let {?c (Xc ) c ? C} be a set of
clique-wise sufficient statistics. With this notation, any distribution of X within the graphical model
family represented by the graph G takes the form:
X
P (X) ? exp
?c ?c (Xc ) ,
(1)
c?C
where {?c } are weights over the sufficient statistics. With a pairwise graphical model distribution,
the set of cliques consists of the set of nodes V and the set of edges E, so that
X
X
P (X) ? exp
?s ?s (Xs ) +
?st ?st (Xs , Xt ) .
(2)
s?V
(s,t)?E
As previously discussed, an important question is how to select the class of sufficient statistics, ?, in
particular to obtain as a multivariate extension of specified univariate parametric distributions? We
next outline a subclass of graphical models where the node-conditional distributions are exponential
family distributions, with an important special case where these node-conditional distributions are
generalized linear models (GLMs). Then, in Section 3, we will study how to learn the underlying
graph structure, or infer the edge set E, providing an M-estimator and sufficient conditions under
which the estimator recovers the graph structure with high probability.
Graphical Models via GLMs. In this section, we investigate the class of models that arise from
specifying the node-conditional distributions as exponential families. Specifically, suppose we are
given a univariate exponential family distribution,
P (Z) = exp(? B(Z) + C(Z) ? D(?)),
with sufficient statistics B(Z), base measure C(Z), and D(?) as the log-normalization constant.
Let X = (X1 , X2 , . . . , Xp ) be a p-dimensional random vector; and let G = (V, E) be an undirected graph over p nodes corresponding to the p variables. Now suppose the distribution of Xs
given the rest of nodes XV \s is given by the above exponential family, but with the canonical exponential family parameter set to a linear combination of k-th order products of univariate functions
{B(Xt )}t?N (s) . This gives the following conditional distribution:
n
X
X
P (Xs |XV \s ) = exp B(Xs ) ?s +
?st B(Xt ) +
?s t2 t3 B(Xt2 )B(Xt3 )
t2 ,t3 ?N (s)
t?N (s)
+
X
?s t2 ...tk
k
Y
o
? V \s ) ,
B(Xtj ) + C(Xs ) ? D(X
(3)
j=2
t2 ,...,tk ?N (s)
? V \s ) is the log-normalization constant.
where C(Xs ) is specified by the exponential family, and D(X
By the Hammersley-Clifford theorem, and some elementary calculation, this conditional distribution
can be shown to specify the following unique joint distribution P (X1 , . . . , Xp ):
Proposition 1. Suppose X = (X1 , X2 , . . . , Xp ) is a p-dimensional random vector, and its nodeconditional distributions are specified by (3). Then its joint distribution P (X1 , . . . , Xp ) is given by:
(
X
X X
P (X) = exp
?s B(Xs ) +
?st B(Xs )B(Xt )
s
+
X
X
?s...tk B(Xs )
s?V t2 ,...,tk ?N (s)
s?V t?N (s)
k
Y
j=2
where A(?) is the log-normalization constant.
3
)
B(Xtj ) +
X
s
C(Xs ) ? A(?) ,
(4)
An important question is whether the conditional and joint distributions specified above have the
most general form, under just the assumption of exponential family node-conditional distributions?
In particular, note that the canonical parameter in the previous proposition is a tensor factorization
of the univariate sufficient statistic, with pair-wise and higher-order interactions, which seems a bit
stringent. Interestingly, by extending the argument from [7] and the Hammersley-Clifford Theorem,
we can show that indeed (3) and (4) have the most general form.
Proposition 2. Suppose X = (X1 , X2 , . . . , Xp ) is a p-dimensional random vector, and its nodeconditional distributions are specified by an exponential family,
? V \s )},
P (Xs |XV \s ) = exp{E(XV \s ) B(Xs ) + C(Xs ) ? D(X
(5)
? V \s )) only depends on
where the function E(XV \s ) (and hence the log-normalization constant D(X
variables Xt in N (s). Further, suppose the corresponding joint distribution factors according to the
graph G = (V, E), with the factors over cliques of size at most k. Then, the conditional distribution
in (5) has the tensor-factorized form in (3), and the corresponding joint distribution has the form in
(4).
The proposition thus tells us that under the general assumptions that (a) the joint distribution is a
graphical model that factors according to a graph G, and has clique-factors of size at most k, and
(c) its node-conditional distribution follows an exponential family, it necessarily follows that the
conditional and joint distributions are given by (3) and (4) respectively.
An important special case is when the joint distribution has factors of size at most two. The conditional distribution then is given by:
?
?
?
?
X
? V \s ) , (6)
P (Xs |XV \s ) = exp ?s B(Xs ) +
?st B(Xs )B(Xt ) + C(Xs ) ? D(X
?
?
t?N (s)
while the joint distribution is given as
?
?
?X
?
X
X
P (X) = exp
?s B(Xs ) +
?st B(Xs )B(Xt ) +
C(Xs ) ? A(?) .
?
?
s
(7)
s
(s,t)?E
Note that when the univariate sufficient statistic function B(?) is a linear function B(Xs ) = Xs ,
then the conditional distribution in (6) is precisely a generalized linear model [5] in canonical form,
?
?
?
?
X
? V \s ; ?) ,
P (Xs |XV \s ) = exp ?s Xs +
?st Xs Xt + C(Xs ) ? D(X
(8)
?
?
t?N (s)
while the joint distribution has the form,
?
?
?X
?
X
X
P (X) = exp
?s Xs +
?st Xs Xt +
C(Xs ) ? A(?) .
? s
?
s
(9)
(s,t)?E
In the subsequent sections, we will refer to the entire class of models in (7) as GLM graphical
models, but focus on the case (9) with linear functions B(Xs ) = Xs .
Examples. The GLM graphical models provide multivariate or Markov network extensions of univariate exponential family distributions. The popular Gaussian graphical model and Ising model can
thus also be represented by (7). Consider the latter, for example, where for the Bernoulli distribution,
we have that B(X) = X, C(X) = 0, and A(?) is the log-partition function; plugging these into (9),
we have the form of the Ising model studied in [3]. The form of the multinomial graphical model,
an extension of the Ising model, can also be represented by (7) and has been previously studied in
[4] and others.
It is instructive to consider the domain of the set of all possible valid parameters in the GLM graphical model (9); namely those that ensure that the density is normalizable, or equivalently, so that the
log-partition function satisfies A(?) < +?. The Ising model imposes no constraint on its parameters, {?st }, for normalizability, since there are finitely many configurations of the binary random
4
vector X. For other exponential families, with countable discrete or continuous valued variables, the
GLM graphical model does impose additional constraints on valid parameters. Consider the example
of the Poisson and exponential distributions. The Poisson family has sufficient statistic B(X) = X
and base measure C(X) = ?log(X!). With some algebra, we can show that A(?) < +? implies
?st ? 0 ? s, t. Thus, the Poisson graphical model can only capture negative conditional relationships
between variables. Consider the exponential distribution with sufficient statistic B(X) = ?X, base
measure C(X) = 0. To ensure that the density is finitely integrable, so that A(?) < +?, we then
require that ?st ? 0 ? s, t. Similar constraints on the parameter space are necessary to ensure proper
density functions for several other exponential family graphical models as well.
3
Statistical Guarantees
In this section, we study the problem of learning the graph structure of an underlying GLM graphical
model given iid samples. Specifically, we assume that we are given n samples X1n = {X (i) }ni=1 ,
from a GLM graphical model:
?
?
? X
?
X
?
?st
Xs Xt +
C(Xs ) ? A(?) .
P (X; ?? ) = exp
(10)
?
?
?
s
(s,t)?E
We have removed node-wise terms for simplicity, noting that our analysis extends to the general
case. The goal in graphical model structure recovery is to recover the edges E ? of the underlying
graph G = (V, E ? ). Following [3, 4], we will approach this problem via neighborhood estimation,
where we estimate the neighborhood of each node individually, and then stitch these together to
b (s) for the true neighborhood
form the global graph estimate. Specifically, if we have an estimate N
N ? (s), then we can estimate the overall graph structure as:
b = ?s?V ? b {(s, t)}.
E
t?N (s)
(11)
In order to estimate the neighborhood of any node, we consider the sparsity constrained conditional
MLE. Given the joint distribution in (10), the conditional distribution of Xs given the rest of the
nodes is given by:
?
?
? X
X
?
?
?
P (Xs |XV \s ) = exp Xs
?st
Xt + C(Xs ) ? D
?st
Xt
.
(12)
?
?
t?N (s)
t?N (s)
?
?
?
?
= 0,
for t ? N (s) and ?st
}t?V \s ? Rp?1 be a zero-padded vector, with entries ?st
Let ?\s
= {?st
n
(i) n
for t 6? N (s). Given n samples X1 = {X }i=1 , we can write the conditional log-likelihood of the
distribution (12) as:
`(?\s ; X1n ) := ?
n
n
Y
1
1X
(i)
(i)
(i)
log
P Xs(i) |X\s , ?\s =
?Xs(i) h?\s , X\s i + D h?\s , X\s i .
n
n i=1
i=1
We can then solve the `1 regularized conditional log-likelihood loss for each node Xs :
min
?\s ?Rp?1
`(?\s ; X1n ) + ?n k?\s k1 .
(13)
Given the solution ?b\s of the M-estimation problem above, we then estimate the node-neighborhood
b (s) = {t ? V \s : ?bst 6= 0}. In the following when we focus on a fixed node s ? V ,
of s as N
we will overload notation, and use ? ? Rp?1 as the parameters of the conditional distribution,
suppressing the dependence on s.
In the rest of the section, we first discuss the assumptions we impose on the GLM graphical model
parameters. The first set of assumptions are standard irrepresentable-type conditions imposed for
structure recovery in high-dimensional statistical estimators, and in particular, our assumptions mirror those in [3]. The second set of assumptions are key to our generalized analysis of the class of
GLM graphical models as a whole. We then follow with our main theorem, that guarantees structure
recovery under these assumptions, with high probability even in high-dimensional regimes.
5
Our first set of assumptions use the Fisher Information matrix, Q?s = ?2 `(?s? ; X1n ), which is the
Hessian of the node-conditional log-likelihood. In the following, we will simply use Q? instead of
Q?s where the reference node s should be understood implicitly. We also use S = {(s, t) : t ? N (s)}
to denote the true neighborhood of node s, and S c to denote its complement. We use Q?SS to denote
the d ? d sub-matrix indexed by S. Our first two assumptions , and are as follows:
Assumption 1 (Dependency condition). There exists a constant ?min > 0 such that ?min (Q?SS ) ?
b \s X T ]) ? ?max .
?min . Moreover, there exists a constant ?max < ? such that ?max (E[X
\s
Assumption 2 (Incoherence condition). We also need an incoherence or irrepresentable condition
on the fisher information matrix as in [3]. Specifically, there exists a constant ? > 0, such that
maxt?S c kQ?tS (Q?SS )?1 k1 ? 1 ? ?.
A key technical facet of the linear, logistic, and multinomial models in [2, 3, 4] and used heavily in
their proofs, is that the random variables {Xs } there were bounded with high probability. Unfortunately, in the general GLM distribution in (12), we cannot assume this explicitly. Nonetheless, we
show that we can analyze the corresponding regularized M-estimation problems, provided the first
and second moments are bounded.
Assumption 3. The first and second moments of the distribution in (10) are bounded as follows. The
first moment ?? := E[X] , satisfies k?? k2 ? ?m ; the second moment satisfies maxt?V E[Xt2 ] ? ?v .
We also need smoothness assumptions on the log-normalization constants :
Assumption 4. The log-normalization constant A(?) of the joint distribution (10) satisfies:
maxu:kuk?1 ?max (?2 A(?? + u)) ? ?h .
Assumption 5. The log-partition function D(?) of the node-conditional distribution (12)
satisfies: There exist constants ?1 and ?2 (that depend on the exponential family) s.t.
max{|D00 (?1 log ?)|, |D000 (?1 log ?)|} ? n?2 where ? = max{n, p}, ?1 ? 29 k?? k2 and ?2 ?
[0, 1/4].
Assumptions 3 and 4 are the key technical conditions under which we can generalize the analyses
in [2, 3, 4] to the general GLM case. In particular, we can show that the statements of the following
propositions hold, which show that the random vectors X following the GLM graphical model in
(10) are suitably well-behaved:
Proposition 3. Suppose X is a random vector with the distribution specified in (10). Then, for any
vector u ? Rp such that kuk2 ? c0 , any positive constant ?, and some constants c > 0,
0
P |hu, Xi| ? ? log ? ? c? ??/c .
Proposition 4. Suppose X is a random vector with the distribution specified in (10). Then, for
? ? min{2?v /3, ?h + ?v }, and some constant c > 0,
!
n
1X
(i) 2
P
Xs
? ? ? 2 exp ?c n ? 2 .
n i=1
Putting these key technical results and assumptions together, we arrive at our main result:
Theorem 1. Consider a GLM graphical model distribution as specified in (10), with true parameter
?
?? and associated edge set E ? that satisfies Assumptions 1-5. Suppose that min(s,t)?E ? |?st
| ?
?
10
d?
where
d
is
the
maximum
neighborhood
size.
Suppose
also
that
the
regularization
pan
?min
q
(2??)
log p
rameter is chosen such that ?n ? M ?
for some constant M > 0. Then, there exist
n1??2
2
1
positive constants L, K1 and K2 such that if n ? L d log p(max{log n, log p})2 1?3?2 , then with
probability at least 1 ? exp(?K1 ?2n n) ? K2 max{n, p}?5/4 , the following statements hold:
(a) (Unique Solution) For each node s ? V , the solution of the M-estimation problem in (13) is
unique, and
(b) (Correct Neighborhood Recovery) The M-estimate also recovers the true neighborhood exactly,
b (s) = N (s).
so that N
6
1
0.8
0.8
Success probability
Success probability
1
0.6
p = 64
p = 100
p = 169
p = 225
0.4
0.2
0
400
600
800
n
1000
p = 64
p = 100
p = 169
p = 225
0.6
0.4
0.2
0
1200
1.5
2
2.5
3
3.5
4
?
Figure 1: Probabilities of successful support recovery for a Poisson grid structure (? = ?0.1). The
probability of successful edge recovery vs. n (Left), and the probability of successful edge recovery
vs. control parameter ? = n/(c log p) (Right).
Note that if the neighborhood of each node is recovered with high probability, then by a simple
b = ?s?V ? b {(s, t)} is equal to the true edge set E ? with
union bound, the estimate in (11), E
t?N (s)
high-probability.
Also note that ?2 in the statement is a constant from Assumption 5. The Poisson family has one
of the steepest log-partition function: D(?) = exp(?). Hence, in order to satisfy Assumption 5,
1 log n
we need k?? k2 ? 18
log p with ?2 = 1/4. On the other hand, for the binomial, multinomial or
Gaussian cases studied in [2, 3, 4], we can recover their results with ?2 = 0 since the log-partition
function D(?) of these families are upper bounded by some constant for any input. Nevertheless, we
need to restrict ?? to satisfy Assumption 4 so that the variables are bounded with high probability in
Proposition 3 and 4 for any GLM case.
4
Experiments
Experiments on Simulated Networks. We provide a small simulation study that demonstrates the
consequences of Theorem 1 when the conditional distribution in (12) has the form of Poisson distribution. We performed experiments on lattice (4 nearest neighbor) graphs with identical edge weight
? for all edges. Simulating data viaqGibbs sampling, we solved the sparsity-constrained optimization
problem with a constant factor of logn p for ?n . The left panel of Figure 1 shows the probability of
successful edge recovery for different numbers of nodes, p = {64, 100, 169, 225}. In the right panel
of Figure 1, we re-scale the sample size n using the ?control parameter? ? = n/(c log p) for some
constant c. Each point in the plot indicates the probability that all edges are successfully recovered
out of 50 trials. We can see that the curves for different problem sizes are well aligned with the
results of Theorem 1.
Learning Genomic Networks. Gaussian graphical models learned from microarray data have often
been used to study high-throughput genomic regulatory networks. Our GLM graphical models will
be important for understanding genomic networks learned from other high-throughput technologies
that do not produce approximately Gaussian data. Here, we demonstrate the versatility of our model
by learning two cancer genomic networks, a genomic copy number aberration network (from aCGH
data) for Glioblastoma learned by multinomial graphical models and a meta-miRNA inhibitory network (from next generation sequencing data) for breast cancer learned by Poisson graphical models.
Level III data, breast cancer miRNA expression (next generation sequencing) [13] and copy number
variation (aCGH) Glioblastoma data [14], was obtained from the the Cancer Genome Atlas (TCGA)
data portal (http://tcga-data.nci.nih.gov/tcga/), and processed according to standard techniques. Data
descriptions and processing details are given in the supplemental materials.
A Poisson graphical model and a multinomial graphical model were fit to the processed miRNA
data and aberration data respectively by performing neighborhood selection with the sparsity of the
graph determined by stability selection [15]. Our GLM graphical models, Figure 2, reveal results
consistent with the cancer genomics literature. The meta-miRNA inhibitory network has three major
hubs, two of which, mir-519 and mir-520, are known to be breast cancer tumor suppressors [16, 17].
Interestingly, let-7, a well-known miRNA involved in tumor metastasis [18], plays a central role
7
38
73
22
74
2
100
14
9
6
90
60
27
26
76
55
20
89
29
92
30
32
70
51
7
31
59
29
87
mir-518c
mir-520a
48
64
14
91
19
24
41
57
4
35
13
75
36
mir-449
mir-519-a
95
let-7
39
17
28
15
99
4
18
49
31
23
1
97
16
71
50
32
94
mir-143
mir-150
61
5
43
mir-3156
mir-105
42
30
10
23
19
13
62
25
83
3
80
34
6
24
16
12
82
53
12
17
2
11
34
10
38
63
25
35
15
47
22
37
93
11
44
26
33
28
40
81
21
18
58
20
0
84
46
3
Figure 2: Genomic copy number aberration network for Glioblastoma learned via multinomial
graphical models (left) and meta-miRNA inhibitory network for breast cancer learned via Poisson
graphical models (right).
27
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86
65
77
36
54
67
78
72
69
68
85
9
52
8
88
7
66
79
56
45
in our network, sharing edges with the five largest hubs; this suggests that our model has learned
relevant negative associations between tumor suppressors and enhancers. The Glioblastoma copy
number aberration network reveals five major modules, color coded on the left panel in Figure 2,
and three of these modules have been previously implicated in Glioblastoma: EGFR in the yellow
module, PTEN in the purple module, and CDK2A in the blue module [19].
5
Discussion
We have introduced a new class of graphical models that arise when we assume that node-wise
conditional distributions follow an exponential family distribution. We have also provided simple
M-estimators for learning the network by fitting node-wise penalized GLMs that enjoy strong statistical recovery properties. Our work has broadened the class of off-the-shelf graphical models to
encompass a wide range of parametric distributions. These classes of graphical models may be of
further interest to the statistical community as they provide closed form multivariate densities for
several exponential family distributions (e.g. Poisson, exponential, negative binomial) where few
currently exist. Furthermore, the statistical analysis of our M-estimator required subtle techniques
that may be of general interest in the analysis of sparse M-estimation.
Our work outlines the general class of graphical models for exponential family distributions, but
there are many avenues for future work in studying this model for specific distributional families.
In particular, our model sometimes places restrictions on the parameter space. A question remains,
can these restrictions be relaxed for specific exponential family distributions? Additionally, we have
focused on families with linear sufficient statistics (e.g. Gaussian, Bernoulli, Poisson, exponential,
negative binomial); our models can be studied with non-linear sufficient statistics or multi-parameter
distributions as well. Overall, our work has opened the door for learning Markov Networks from
a broad class of distributions, the properties and applications of which leave much room for future
research.
Acknowledgments
E.Y. and P.R. acknowledge support from NSF IIS-1149803. G.A. and Z.L. acknowledge support from the Collaborative Advances in Biomedical Computing seed funding program at the Ken
Kennedy Institute for Information Technology at Rice University supported by the John and Ann
Doerr Fund for Computational Biomedicine and by the Center for Computational and Integrative
Biomedical Research seed funding program at Baylor College of Medicine. G.A. also acknowledges support from NSF DMS-1209017.
8
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9
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3,997 | 4,618 | CPRL ? An Extension of Compressive Sensing to the
Phase Retrieval Problem
Henrik Ohlsson
Division of Automatic Control, Department of Electrical Engineering,
Link?oping University, Sweden.
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley, CA, USA
[email protected]
Allen Y. Yang
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley, CA, USA
Roy Dong
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley, CA, USA
S. Shankar Sastry
Department of Electrical Engineering and Computer Sciences
University of California at Berkeley, CA, USA
Abstract
While compressive sensing (CS) has been one of the most vibrant research fields in
the past few years, most development only applies to linear models. This limits its
application in many areas where CS could make a difference. This paper presents
a novel extension of CS to the phase retrieval problem, where intensity measurements of a linear system are used to recover a complex sparse signal. We propose
a novel solution using a lifting technique ? CPRL, which relaxes the NP-hard
problem to a nonsmooth semidefinite program. Our analysis shows that CPRL
inherits many desirable properties from CS, such as guarantees for exact recovery.
We further provide scalable numerical solvers to accelerate its implementation.
1
Introduction
In the area of X-ray imaging, phase retrieval (PR) refers to the problem of recovering a complex
multivariate signal from the squared magnitude of its Fourier transform. Existing sensor devices for
collecting X-ray images are only sensitive to signal intensities but not the phases. However, it is
very important to be able to recover the missing phase information as it reveals finer structures of
the subjects than using the intensities alone. The PR problem also has broader applications and has
been studied extensively in biology, physics, chemistry, astronomy, and more recent nanosciences
[29, 20, 18, 24, 23].
Mathematically, PR can be formulated using a linear system y = Ax ? CN , where the matrix
A may represent the Fourier transform or other more general linear transforms. If the complex
measurements y are available and the matrix A is assumed given, it is well known that the leastsquares (LS) solution recovers the model parameter x that minimizes the squared estimation error:
1
ky ? Axk22 . In PR, we assume that the phase of the coefficients of y is omitted and only the squared
magnitude of the output is observed:
bi = |yi |2 = |hx, ai i|2 ,
i = 1, ? ? ? , N,
(1)
where AH = [a1 , ? ? ? , aN ] ? Cn?N , y T = [y1 , ? ? ? , yN ] ? CN , and AH denotes the Hermitian
transpose of A.
Inspired by the emerging theory of compressive sensing [17, 8] and a lifting technique recently
proposed for PR [13, 10], we study the PR problem with a more restricted assumption that the
model parameter x is sparse and the number of observations N are too few for (1) to have a unique
solution, and in some cases even fewer measurements than the number of unknowns n. The problem
is known as compressive phase retrieval (CPR) [25, 27, 28]. In many X-ray imaging applications,
for instance, if the complex source signal is indeed sparse under a proper basis, CPR provides a
viable solution to exactly recover the signal while collecting much fewer measurements than the
traditional non-compressive solutions.
Clearly, the PR problem and its CPR extension are much more challenging than the LS problem, as
the phase of y is lost while only its squared magnitude is available. For starters, it is important to note
that the setup naturally leads to ambiguous solutions regardless whether the original linear model is
overdetermined or not. For example, if x0 ? Cn is a solution to y = Ax, then any multiplication of
x and a scalar c ? C, |c| = 1, leads to the same squared output b. As mentioned in [10], when the
dictionary A represents the unitary discrete Fourier transform (DFT), the ambiguities may represent
time-reversed or time-shifted solutions of the ground-truth signal. Hence, these global ambiguities
are considered acceptable in PR applications. In this paper, when we talk about a unique solution to
PR, it is indeed a representative of a family of solutions up to a global phase ambiguity.
1.1
Contributions
The main contribution of the paper is a convex formulation of the CPR problem. Using the lifting technique, the NP-hard problem is relaxed as a semidefinite program (SDP). We will briefly
summarize several theoretical bounds for guaranteed recovery of the complex input signal, which
is presented in full detail in our technical report [26]. Built on the assurance of the guaranteed
recovery, we will focus on the development of a novel scalable implementation of CPR based on
the alternating direction method of multipliers (ADMM) approach. The ADMM implementation
provides a means to apply CS ideas to PR applications e.g., high-impact nanoscale X-ray imaging.
In the experiment, we will present a comprehensive comparison of the new algorithm with the traditional interior-point method, other state-of-the-art sparse optimization techniques, and a greedy
algorithm proposed in [26]. In high-dimensional complex domain, the ADMM algorithm demonstrates superior performance in our simulated examples and real images. Finally, the paper also
provides practical guidelines to practitioners at large working on other similar nonsmooth SDP applications. To aid peer evaluation, the source code of all the algorithms have been made available at:
http://www.rt.isy.liu.se/?ohlsson/.
2
Compressive Phase Retrieval via Lifting (CPRL)
Since (1) is nonlinear in the unknown x, N n measurements are in general needed for a unique
solution. When the number of measurements N are fewer than necessary for such a unique solution,
additional assumptions are needed as regularization to select one of the solutions. In classical CS, the
ability to find the sparsest solution to a linear equation system enables reconstruction of signals from
far fewer measurements than previously thought possible. Classical CS is however only applicable
to systems with linear relations between measurements and unknowns. To extend classical CS to the
nonlinear PR problem, we seek the sparsest solution satisfying (1):
min kxk0 ,
x
H
subj. to b = |Ax|2 = {aH
i xx ai }1?i?N ,
(2)
with the square acting element-wise and b = [b1 , ? ? ? , bN ]T ? RN . As the counting norm k ? k0 is
not a convex function, following the `1 -norm relaxation in CS, (2) can be relaxed as
min kxk1 ,
x
H
subj. to b = |Ax|2 = {aH
i xx ai }1?i?N .
2
(3)
Note that (3) is still not a convex program, as its equality constraint is not a linear equation. In the
literature, a lifting technique has been extensively used to reframe problems such as (3) to a standard
form in SDP, such as in Sparse PCA [15]. More specifically, given the ground-truth signal x0 ? Cn ,
n?n
let X0 , x0 xH
be an induced rank-1 semidefinite matrix. Then (3) can be reformulated
0 ? C
1
into
minX0 kXk1 ,
subj. to
rank(X) = 1, bi = aH
i Xai , i = 1, ? ? ? , N.
(4)
This is of course still a nonconvex problem due to the rank constraint. The lifting approach addresses
this issue by replacing rank(X) with Tr(X). For a positive-semidefinite matrix, Tr(X) is equal to
the sum of the eigenvalues of X (or the `1 -norm on a vector containing all eigenvalues of X). This
leads to the nonsmooth SDP
minX0 Tr(X) + ?kXk1 ,
subj. to bi = Tr(?i X), i = 1, ? ? ? , N,
(5)
n?n
where we further denote ?i , ai aH
and ? ? 0 is a design parameter. Finally, the estimate
i ?C
of x can be found by computing the rank-1 decomposition of X via singular value decomposition.
We refer to the approach as compressive phase retrieval via lifting (CPRL).
Consider now the case that the measurements are contaminated by data noise. In a linear model,
bounded random noise typically affects the output of the system as y = Ax + e, where e ? CN is a
noise term with bounded `2 -norm: kek2 ? . However, in phase retrieval, we follow closely a more
special noise model used in [13]:
bi = |hx, ai i|2 + ei .
(6)
This nonstandard model avoids the need to calculate the squared magnitude output |y|2 with the
added noise term. More importantly, in most practical phase retrieval applications, measurement
noise is introduced when the squared magnitudes or intensities of the linear system are measured on
the sensing device, but not y itself. Accordingly, we denote a linear operator B of X as
B : X ? Cn?n 7? {Tr(?i X)}1?i?N ? RN ,
(7)
which measures the noise-free squared output. Then the approximate CPR problem with bounded
`2 -norm error model can be solved by the following nonsmooth SDP program:
minX0 Tr(X) + ?kXk1 ,
subj. to kB(X) ? bk2 ? ?.
(8)
Due to the machine rounding error, in general a nonzero ? should be always assumed and in its
termination condition during the optimization. The estimate of x, just as in noise free case, can
finally be found by computing the rank-1 decomposition of X via singular value decomposition.
We refer to the method as approximate CPRL.
3
Theoretical Analysis
This section highlights some of the analysis results derived for CPRL. The proofs of these results are
available in the technical report [26]. The analysis follows that of CS and is inspired by derivations
given in [13, 12, 16, 9, 3, 7]. In order to state some theoretical properties for CPRL, we need a
generalization of the restricted isometry property (RIP).
2
2 ? 1| < for all
Definition 1 (RIP) A linear operator B(?) as defined in (7) is (, k)-RIP if | kB(X)k
kXk2
2
kXk0 ? k and X 6= 0.
We can now state the following theorem:
Theorem 2 (Recoverability/Uniqueness) Let B(?) be a (, 2kX ? k0 )-RIP linear operator with <
? be the sparsest solution to (1). If X ? satisfies b = B(X ? ), X ? 0, rank{X ? } = 1,
1 and let x
?
?x
?H .
then X is unique and X ? = x
?:
We can also give a bound on the sparsity of x
? be the
? H k0 from above) Let x
? be the sparsest solution to (1) and let X
Theorem 3 (Bound on k?
xx
H
?
?
? k0 .
solution of CPRL (5). If X has rank 1 then kXk0 ? k?
xx
The following result now holds trivially:
1
In this paper, kXk1 for a matrix X denotes the entry-wise `1 -norm, and kXk2 denotes the Frobenius norm.
3
? be the sparsest solution to (1). The solution
Corollary 4 (Guaranteed recovery using RIP) Let x
H
?
? 0 )-RIP with < 1.
?
?
of CPRL X is equal to xx if it has rank 1 and B(?) is (, 2kXk
? can not be guaranteed, the following bound becomes useful:
?x
?H = X
If x
? 1 ) Let < 1? and assume B(?) to be a (, 2k)-RIP linear
Theorem 5 (Bound on kX ? ? Xk
1+ 2
?
operator. Let X be any matrix (sparse or dense) satisfying b = B(X ? ), X ? 0, rank{X ? } = 1,
? be the CPRL solution, (5), and form Xs from X ? by setting all but the k largest elements to
let X
zero. Then,
?
2 ?
? ? X ? k1 ?
(1 ? ( 21??k + 1) ?1 )kX
kX ? ? Xs k1 ,
(9)
(1??) k
?
with ? = 2/(1 ? ).
Given the RIP analysis, it may be the case that the linear operator B(?) does not well satisfy the RIP
property defined in Definition 1, as pointed out in [13]. In these cases, RIP-1 maybe considered:
1 ? 1| < for all matrices
Definition 6 (RIP-1) A linear operator B(?) is (, k)-RIP-1 if | kB(X)k
kXk1
X 6= 0 and kXk0 ? k.
Theorems 2?3 and Corollary 4 all hold with RIP replaced by RIP-1 and are not restated in detail
here. Instead we summarize the most important property in the following theorem:
? be the sparsest solution to (1). The
Theorem 7 (Upper bound & recoverability through `1 ) Let x
? is equal to x
? 0 )-RIP-1 with < 1.
?x
? H if it has rank 1 and B(?) is (, 2kXk
solution of CPRL (5), X,
The RIP type of argument may be difficult to check for a given matrix and are more useful for
claiming results for classes of matrices/linear operators. For instance, it has been shown that random Gaussian matrices satisfy the RIP with high probability. However, given realization of a random Gaussian matrix, it is indeed difficult to check if it actually satisfies the RIP. Two alternative
arguments are spark [14] and mutual coherence [17, 11]. The spark condition usually gives tighter
bounds but is known to be difficult to compute as well. On the other hand, mutual coherence may
give less tight bounds, but is more tractable. We will focus on mutual coherence, which is defined as:
Definition 8 (Mutual coherence) For a matrix A, define the mutual coherence as ?(A) =
Ha |
j
i
max1?i,j?n,i6=j ka|a
.
i k2 kaj k2
By an abuse of notation, let B be the matrix satisfying b = BX s with X s being the vectorized
version of X. We are now ready to state the following theorem:
? be the sparsest solution to (1). The solution
Theorem 9 (Recovery using mutual coherence) Let x
? is equal to x
? 0 < 0.5(1 + 1/?(B)).
?x
? H if it has rank 1 and kXk
of CPRL (5), X,
4
Numerical Implementation via ADMM
In addition to the above analysis of guaranteed recovery properties, a critical issue for practitioners is
the availability of efficient numerical solvers. Several numerical solvers used in CS may be applied
to solve nonsmooth SDPs, which include interior-point methods (e.g., used in CVX [19]), gradient
projection methods [4], and augmented Lagrangian methods (ALM) [4]. However, interior-point
methods are known to scale badly to moderate-sized convex problems in general. Gradient projection methods also fail to meaningfully accelerate the CPRL implementation due to the complexity
of the projection operator. Alternatively, nonsmooth SDPs can be solved by ALM. However, the
augmented primal and dual objective functions are still complex SDPs, which are equally expensive
to solve in each iteration. In summary, as we will demonstrate in Section 5, CPRL as a nonsmooth
complex SDP is categorically more expensive to solve compared to the linear programs underlying
CS, and the task exceeds the capability of many popular sparse optimization techniques.
In this paper, we propose a novel solver to the nonsmooth SDP underlying CPRL via the alternating
directions method of multipliers (ADMM, see for instance [6] and [5, Sec. 3.4]) technique. The
motivation to use ADMM are two-fold: 1. It scales well to large data sets. 2. It is known for its fast
convergence. There are also a number of strong convergence results [6] which further motivates the
choice.
To set the stage for ADMM, rewrite (5) to the equivalent SDP
minX1 ,X2 ,Z f1 (X1 ) + f2 (X2 ) + g(Z),
subj. to X1 ? Z = 0,
4
X2 ? Z = 0,
(10)
where
0 if X 0
Tr(X) if bi = T r(?i X), i = 1, . . . , N
, g(Z) , ?kZk1 .
, f2 (X) ,
f1 (X) ,
? otherwise
?
otherwise
The update rules of ADMM now lead to the following:
Xil+1
Z l+1
Yil+1
= arg minX fi (X) + Tr(Yil (X ? Z l )) + ?2 kX ? Z l k22 ,
P2
= arg minZ g(Z) + i=1 ?Tr(Yil Z) + ?2 kXil+1 ? Zk22 ,
= Yil + ?(Xil+1 ? Z l+1 ),
(11)
where Xi , Yi , Z are constrained to stay in the domain of Hermitian matrices. Each of these steps has
a tractable calculation. However, the Xi , Yi , and Z variables are complex-valued, and, as most of
the optimization literature deals with real-valued vectors and symmetric matrices, we will emphasize
differences between the real case and complex case. After some simple manipulations, we have:
X1l+1 = argminX kX ? (Z l ?
I+Y1l
? )k2 ,
subj. to bi = Tr(?i X), i = 1, ? ? ? , N.
(12)
Assuming that a feasible solution exists, and defining ?A as the projection onto the convex set given
l
1 ). This optimization problem has a
by the linear constraints, the solution is: X1l+1 = ?A (Z l ? I+Y
?
closed-form solution; converting the matrix optimization problem in (12) into an equivalent vector
optimization problem yields a problem of the form: minx ||x?z||2 subj. to b = Ax. The answer
is given by the pseudo-inverse of A, which can be precomputed. This complex-valued problem can
be solved by converting the linear constraint in Hermitian matrices into an equivalent constraint on
real-valued vectors. This conversion is done by noting that for n ? n Hermitian matrices A, B:
Pn
Pn Pn
Pn Pn
hA, Bi = P
Tr(AB) = i=1P j=1PAij Bij = i=1 Aii Bii + i=1 j=i+1 Aij Bij + Aij Bij
n
n
n
= i=1 Aii Bii + i=1 j=i+1 2 real(Aij ) real(Bij ) + 2 imag(Aij ) imag(Bij )
v
2
So
? if we define the vector A as an n vector such that
? its elements are Aii for i = 1, ? ? ? , n,
2 real(Aij ) for i = 1, ? ? ? , n, j = i + 1, ? ? ? , n, and 2 imag(Aij ) for i = 1, ? ? ? , n, j = i +
1, ? ? ? , n, and similarly define B v , then we can see that hA, Bi = hAv , B v i. This turns the constraint
2
bi = Tr(?i X), i = 1, ? ? ? , N, into one of the form: b = [?v1 ? ? ? ?vN ]T X v , where each ?vi is in Rn .
Thus, for this subproblem, the memory usage scales linearly with N , the number of measurements,
l
and quadratically with n, the dimension of the data. Next, X2l+1 = argminX0 kX ?(Z l ? Y?2 )k2 =
l
?P SD (Z l ? Y?2 ), where ?P SD denotes the projection onto the positive-semidefinite cone, which
can easily be obtained via eigenvalue decomposition. This holds for real-valued and complex-valued
P2
l+1
l
= 21 i=1 Xil+1 and similarly Y . Then, the Z update rule
Hermitian matrices. Finally, let X
can be written:
Z l+1 = argminZ ?kZk1 +
2?
2 kZ
? (X
l+1
l
+
Y
?
)k22 = soft(X
l+1
l
+
Y
?
?
, 2?
).
(13)
We note that the soft operator in the complex domain must be coded with care. One does not simply
check the sign of the difference, as in the real case, but rather the magnitude of the complex number:
(
0
if |x| ? q,
soft(x, q) = |x|?q
(14)
x
otherwise,
|x|
where q is a positive real number. Setting l = 0, the Hermitian matrices Xil , Zil , Yil can now be
iteratively computed using the ADMM iterations (11). The stopping criterion of the algorithm is
given by:
l
krl k2 ? nabs + rel max(kX k2 , kZ l k2 ),
abs
rel
?3
l
ksl k2 ? nabs + rel kY k2 ,
l
(15)
l
where , are algorithm parameters set to 10 and r and s are the primal and dual residuals
given by: rl = (X1l ? Z l , X2l ? Z l ), sl = ??(Z l ? Z l?1 , Z l ? Z l?1 ). We also update ? according
to the rule discussed in [6]:
?
l
if krl k2 > ?ksl k2 ,
??incr ?
l+1
l
?
=
(16)
? /?
if ksl k2 > ?krl k2 ,
? l decr
?
otherwise,
where ?incr , ?decr , and ? are algorithm parameters. Values commonly used are ? = 10 and ?incr =
?decr = 2.
5
5
Experiment
The experiments in this section are chosen to illustrate the computational performance and scalability of CPRL. Being one of the first papers addressing the CPR problem, existing methods available
for comparison are limited. For the CPR problem, to the authors? best knowledge, the only methods
developed are the greedy algorithms presented in [25, 27, 28], and GCPRL [26]. The method proposed in [25] handles CPR but is only tailored to random 2D Fourier samples from a 2D array and it
is extremely sensitive to initialization. In fact, it would fail to converge in our scenarios of interest.
[27] formulates the CPR problem as a nonconvex optimization problem that can be solved by solving a series of convex problems. [28] proposes to alternate between fit the estimate to measurements
and thresholding. GCPRL, which stands for greedy CPRL, is a new greedy approximate algorithm
tailored to the lifting technique in (5). The algorithm draws inspiration from the matching-pursuit algorithm [22, 1]. In each iteration, the algorithm adds a new nonzero component of x that minimizes
the CPRL objective function the most. We have observed that if the number of nonzero elements in
x is expected to be low, the algorithm can successfully recover the ground-truth sparse signal while
consuming less time compared to interior-point methods for the original SDP.2 In general, greedy
algorithms for solving CPR problems work well when a good guess for the true solution is available,
are often computationally efficient but lack theoretical recovery guarantees. We also want to point
out that CPRL becomes a special case in a more general framework that extends CS to nonlinear
systems (see [1]). In general, nonlinear CS can be solved locally by greedy simplex pursuit algorithms. Its instantiation in PR is the GCPRL algorithm. However, the key benefit of developing the
SDP solution for PR in this paper is that the global convergence can be guaranteed.
In this section, we will compare implementations of CPRL using the interior-point method used by
CVX [19] and ADMM with the design parameter choice recommended in [6] (?incr = ?decr = 2).
? = 10 will be used in all experiments. We will also compare the results to GCPRL and the PR
algorithm PhaseLift [13]. The former is a greedy approximate solution, while the latter does not
enforce sparsity and is obtained by setting ? = 0 in CPRL.
In terms of the scale of the problem, the largest problem we have tested is on a 30 ? 30 image and is
100-sparse in the Fourier domain with 2400 measurements. Our experiment is conducted on an IBM
x3558 M3 server with two Xeon X5690 processors, 6 cores each at 3.46GHz, 12MB L3 cache, and
96GB of RAM. The execution for recovering one instance takes approximately 36 hours to finish in
MATLAB environment, comprising of several tens of thousands of iterations. The average memory
usage is 3.5 GB.
5.1
A simple simulation
In this example we consider a simple CPR problem to illustrate the differences between CPRL,
GCPRL, and PhaseLift. We also compare computational speed for solving the CPR problem and
illustrate the theoretical bounds derived in Section 3. Let x ? C64 be a 2-sparse complex signal,
A , RF where F ? C64?64 is the Fourier transform matrix and R ? C32?64 a random projection
matrix (generated by sampling a unit complex Gaussian), and let the measurements b satisfy the
PR relation (1). The left plot of Figure 1 gives the recovered signal x using CPRL, GCPRL and
PhaseLift. As seen, CPRL and GCPRL correctly identify the two nonzero elements in x while
PhaseLift fails to identify the true signal and gives a dense estimate. These results are rather typical
(see the MCMC simulation in [26]). For very sparse examples, like this one, CPRL and GCPRL
often both succeed in finding the ground truth (even though we have twice as many unknowns
as measurements). PhaseLift, on the other side, does not favor sparse solutions and would need
considerably more measurements to recover the 2-sparse signal. The middle plot of Figure 1 shows
the computational time needed to solve the nonsmooth SDP of CPRL using CVX, ADMM, and
GCPRL. It shows that ADMM is the fastest and that GCPRL outperforms CVX. The right plot of
Figure 1 shows the mutual coherence bound 0.5(1 + 1/?(B)) for a number of different N ?s and
n?s, A , RF , F ? Cn?n the Fourier transform matrix and R ? CN ?n a random projection
? satisfies kXk
? 0<
matrix. This is of interest since Theorem 9 states that when the CRPL solution X
H
?
?x
? , where x
? is the sparsest solution to (1). From
0.5(1 + 1/?(B)) and has rank 1, then X = x
2
We have also tested an off-the-shelf toolbox that solves convex cone problems, called TFOCS [2]. Unfortunately, TFOCS cannot be applied directly to solving the nonsmooth SDP in CPRL.
6
? has rank 1 and only a single nonzero
the plot it can be concluded that if the CPRL solution X
? = x
?x
? H . We also
component for a choice of 125 ? n, N ? 5, Theorem 9 guarantees that X
observe that Theorem 9 is conservative, since we previously saw that 2 nonzero components could
be recovered correctly for n = 64 and N = 32. In fact, numerical simulation can be used to show
that N = 30 suffices to recover the ground truth in 95 out of 100 runs [26].
1
10
PhaseLift
CPRL/GCPRL
0.9
0.8
8
0.7
1.22
110
100
1.2
90
~
__
0.5
1.18
80
0.4
1.16
70
0.3
1.14
6
N
|| xxH ? X ||
i
120
7
0.6
|x |
1.24
CPRL (CVX)
GCPRL
CPRL (ADMM)
9
5
4
60
1.12
3
50
0.2
1.1
2
40
0.1
1
1.08
30
0
0
10
20
30
40
50
60
0
0
i
20
40
60
time [s]
80
100
40
60
80
100
120
n
Figure 1: Left: The magnitude of the estimated signal provided by CPRL, GCPRL and PhaseLift.
? 2 plotted against time for ADMM (gray line), GCPRL (solid
? H ? Xk
Middle: The residual k?
xx
black line) and CVX (dashed black line). Right: A contour plot of the quantity 0.5(1 + 1/?(B)). ?
is taken as the average over 10 realizations of the data.
5.2
Compressive sampling and PR
One of the motivations of presented work and CPRL is that it enables compressive sensing for PR
problems. To illustrate this, consider the 20 ? 20 complex image in Figure 2 Left. To measure the
image, we could measure each pixel one-by-one. This would require us to sample 400 times. What
CS proposes is to measure linear combinations of samples rather than individual pixels. It has been
shown that the original image can be recovered from far fewer samples than the total number of
pixels in the image. The gain using CS is hence that fewer samples are needed. However, traditional
CS only discuss linear relations between measurements and unknowns.
To extend CS to PR applications, consider again the complex image in Figure 2 Left and assume that
we only can measure intensities or intensities of linear combinations of pixels. Let R ? CN ?400
capture how intensity measurements b are formed from linear combinations of pixels in the image,
b = |Rz|2 (z is a vectorized version of the image). An essential part in CS is also to find a dictionary
(possibly overcomplete) in which the image can be represented using only a few basis images. For
classical CS applications, dictionaries have been derived. For applying CS to the PR applications,
dictionaries are needed and a topic for future research. We will use a 2D inverse Fourier transform
dictionary in our example and arrange the basis vectors as columns in F ? C400?400 .
If we choose N = 400 and generate R by sampling from a unit Gaussian distribution and set
A = RF , CPRL recovers exactly the true image. This is rather remarkable since the PR relation
(1) is nonlinear in the unknown x and N n measurements are in general needed for a unique
solution. If we instead sample the intensity of each pixel, one-by-one, neither CPRL or PhaseLift
recover the true image. If we set A = R and do not care about finding a dictionary, we can use
a classical PR algorithm to recover the true image. If PhaseLift is used, N = 1600 measurements
are sufficient to recover the true image. The main reasons for the low number of samples needed in
CPRL is that we managed to find a good dictionary (20 basis images were needed to recover the true
image) and CPRL?s ability to recover the sparsest solution. In fact, setting A = RF , PhaseLift still
needs 1600 measurements to recover the true solution.
5.3
The Shepp-Logan phantom
In this last example, we again consider the recovery of complex valued images from random samples. The motivation is twofold: Firstly, it illustrates the scalability of the ADMM implementation.
In fact, ADMM has to be used in this experiment as CVX cannot handle the CPRL problem in this
scale. Secondly, it illustrates that CPRL can provide approximate solutions that are visually close
to the ground-truth images. Consider now the image in Figure 2 Middle Left. This 30 ? 30 SheppLogan phantom has a 2D Fourier transform with 100 nonzero coefficients. We generate N linear
combinations of pixels as in the previous example and square the measurements, and then apply
7
CPRL and PhaseLift with a 2D Fourier dictionary. The middel image in Figure 2 shows the recovered result using PhaseLift with N = 2400, the second image from the right shows the recovered
result using CPRL with the same number N = 2400 and the right image is the recovered result using
CPRL with N = 1500. The number of measurements with respect to the sparsity in x is too low for
both CPRL and PhaseLift to perfectly recover z. However, CPRL provides a much better approximation and outperforms PhaseLift visually even though it uses considerably fewer measurements.
2
5
5
5
5
10
10
10
10
15
15
15
15
20
20
20
20
25
25
25
4
6
8
10
12
14
16
25
18
20
30
2
4
6
8
10
12
14
16
18
20
30
5
10
15
20
25
30
30
5
10
15
20
25
30
30
5
10
15
20
25
30
5
10
15
20
25
30
Figure 2: Left: Absolute value of the 2D inverse Fourier transform of x, |F x|, used in the experiment in Section 5.2. Middle Left: Ground truth for the experiment in Section 5.3. Middle:
Recovered result using PhaseLift with N = 2400. Middle Right: CPRL with N = 2400. Right:
CPRL with N = 1500.
6
Future Directions
The SDP underlying CPRL scales badly with the number of unknowns or basis vectors in the dictionary. Therefore, learning a suitable dictionary for a specific application becomes even more critical
than that in traditional linear CS setting. We also want to point out that when classical CS was first
studied, many of today?s accelerated numerical algorithms were not available. We are very excited
about the new problem to improve the speed of SDP algorithms in sparse optimization, and hope
our paper would foster the community?s interest to address this challenge collaboratively. One interesting direction might be to use ADMM to solve the dual of (5), see for instance [30, 31]. Another
possible direction is the outer approximation methods [21].
7
Acknowledgement
Ohlsson is partially supported by the Swedish foundation for strategic research in the center MOVIII,
the Swedish Research Council in the Linnaeus center CADICS, the European Research Council
under the advanced grant LEARN, contract 267381, and a postdoctoral grant from the SwedenAmerica Foundation, donated by ASEA?s Fellowship Fund, and by a postdoctoral grant from the
Swedish Research Council. Yang is supported by ARO 63092-MA-II. Dong is supported by the
NSF Graduate Research Fellowship under grant DGE 1106400, and by the Team for Research in
Ubiquitous Secure Technology (TRUST), which receives support from NSF (award number CCF0424422). The authors also want to acknowledge useful input from Stephen Boyd and Yonina Eldar.
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3,998 | 4,619 | 3D Social Saliency from Head-mounted Cameras
Hyun Soo Park
Carnegie Mellon University
[email protected]
Eakta Jain
Texas Instruments
[email protected]
Yaser Sheikh
Carnegie Mellon University
[email protected]
Abstract
A gaze concurrence is a point in 3D where the gaze directions of two or more
people intersect. It is a strong indicator of social saliency because the attention
of the participating group is focused on that point. In scenes occupied by large
groups of people, multiple concurrences may occur and transition over time. In
this paper, we present a method to construct a 3D social saliency field and locate multiple gaze concurrences that occur in a social scene from videos taken by
head-mounted cameras. We model the gaze as a cone-shaped distribution emanating from the center of the eyes, capturing the variation of eye-in-head motion. We
calibrate the parameters of this distribution by exploiting the fixed relationship
between the primary gaze ray and the head-mounted camera pose. The resulting gaze model enables us to build a social saliency field in 3D. We estimate the
number and 3D locations of the gaze concurrences via provably convergent modeseeking in the social saliency field. Our algorithm is applied to reconstruct multiple gaze concurrences in several real world scenes and evaluated quantitatively
against motion-captured ground truth.
1
Introduction
Scene understanding approaches have largely focused on understanding the physical structure of a
scene: ?what is where?? [1]. In social scenes, i.e., scenes occupied by people, this definition of
understanding needs to be expanded to include interpreting what is socially salient in that scene,
such as who people interact with, where they look, and what they attend to. While classic structural
scene understanding is an objective interpretation of the scene (e.g., 3D reconstruction [2], object
recognition [3], or human affordance identification [4]), social scene understanding is subjective as
it depends on the beholder and the particular group of people occupying the scene. For example,
when we first enter a foyer during a party, we quickly look at different people and the groups they
have formed, search for personal friends or acquaintances, and choose a group to join. Consider
instead, an artificial agent, such as a social robot, that enters the same room: how should it interpret
the social dynamics of the environment? The subjectivity of social environments makes the identification of quantifiable and measurable representations of social scenes difficult. In this paper, we
aim to recover a representation of saliency in social scenes that approaches objectivity through the
consensus of multiple subjective judgements.
Humans transmit visible social signals about what they find important and these signals are powerful
cues for social scene understanding [5]. For instance, humans spontaneously orient their gaze to the
target of their attention. When multiple people simultaneously pay attention to the same point in
three dimensional space, e.g., at an obnoxious customer at a restaurant, their gaze rays1 converge to
a point that we refer to as a gaze concurrence. Gaze concurrences are foci of the 3D social saliency
field of a scene. It is an effective approximation because although an individual?s gaze indicates
what he or she is subjectively interested in, a gaze concurrence encodes the consensus of multiple
individuals. In a scene occupied by a larger number of people, multiple such concurrences may
emerge as social cliques form and dissolve. In this paper, we present a method to reconstruct a 3D
social saliency field and localize 3D gaze concurrences from videos taken by head-mounted cameras
1
A gaze ray is a three dimensional ray emitted from the center of eyes and oriented to the point of regard as
shown in Figure 1(b).
1
Primary gaze ray, l
Gaze concurrences
W
ld
?
Left eye
Center of eyes, p
vd
v
p
Right eye
Point of regard
Videos from
head-mounted cameras
(a) Input and output
Gaze ray
(b) Head top view
v
d
?
1
v
v
?
2
W
Primary gaze ray
C
l
?
1 1
d = d v + d 2 v 2?
d1 , d 2 ? N (0, h 2 )
(c) Gaze ray model
Cone-shaped distribution
of the point of regard
(d) Gaze distribution
Figure 1: (a) In this paper, we present a method to reconstruct 3D gaze concurrences from videos
taken by head-mounted cameras. (b) The primary gaze ray is a fixed 3D ray with respect to the head
coordinate system and the gaze ray can be described by an angle with respect to the primary gaze
ray. (c) The variation of the eye orientation is parameterized by a Gaussian distribution of the points
on the plane, ?, which is normal to the primary gaze ray, l at unit distance from p. (d) The gaze ray
model results in a cone-shaped distribution of the point of regard.
on multiple people (Figure 1(a)). Our method automatically finds the number and location of gaze
concurrences that may occur as people form social cliques in an environment.
Why head-mounted cameras? Estimating 3D gaze concurrences requires accurate estimates of
the gaze of people who are widely distributed over the social space. For a third person camera,
i.e., an outside camera looking into a scene, state-of-the-art face pose estimation algorithms cannot
produce reliable face orientation and location estimation beyond approximately 45 degrees of a head
facing the camera directly [6]. Furthermore, as they are usually fixed, third person views introduce
spatial biases (i.e., head pose estimates would be better for people closer to and facing the camera)
and limit the operating space. In contrast, head-mounted cameras instrument people rather than the
scene. Therefore, one camera is used to estimate each head pose. As a result, 3D pose estimation of
head-mounted cameras provides accurate and spatially unbiased estimates of the primary gaze ray2 .
Head-mounted cameras are poised to broadly enter our social spaces and many collaborative teams
(such as search and rescue teams [8], police squads, military patrols, and surgery teams [9]) are
already required to wear them. Head-mounted camera systems are increasingly becoming smaller,
and will soon be seamlessly integrated into daily life [10].
Contributions The core contribution of this paper is an algorithm to estimate the 3D social saliency
field of a scene and its modes from head-mounted cameras, as shown in Figure 1(a). This is enabled by a new model of gaze rays that represents the variation due to eye-in-head motion via a
cone-shaped distribution. We present a novel method to calibrate the parameters of this model by
leveraging the fact that the primary gaze ray is fixed with respect to the head-mounted camera in 3D.
Given the collection of gaze ray distributions in 3D space, we automatically estimate the number
and 3D locations of multiple gaze concurrences via mode-seeking in the social saliency field. We
prove that the sequence of mode-seeking iterations converge. We evaluate our algorithm using motion capture data quantitatively, and apply it to real world scenes where social interactions frequently
occur, such as meetings, parties, and theatrical performances.
2
Related Work
Humans transmit and respond to many different social signals when they interact with others.
Among these signals, gaze direction is one of the most prominent visible signals because it usually indicates what the individual is interested in. In this context, gaze direction estimation has been
widely studied in robotics, human-computer interaction, and computer vision [6, 11?22]. Gaze direction can be precisely estimated by the eye orientation. Wang and Sung [11] presented a system
that estimates the direction of the iris circle from a single image using the geometry of the iris.
Guestrin and Eizenman [12] and Hennessey and Lawrence [13] utilized corneal reflections and the
vergence of the eye to infer the eye geometry and its motion, respectively. A head-mounted eye
tracker is often used to determine the eye orientation [14, 15]. Although all these methods can estimate highly accurate gaze direction, either they can be used in a laboratory setting or the device
occludes the viewer?s field of view.
2
The primary gaze ray is a fixed eye orientation with respect to the head. It has been shown that the
orientation is a unique pose, independent of gravity, head posture, horizon, and the fusion reflex [7].
2
While the eyes are the primary source of gaze direction, Emery [16] notes that the head orientation
is a strong indication of the direction of attention. For head orientation estimation, there are two
approaches: outside-in and inside-out [23]. An outside-in system takes, as input, a third person
image from a particular vantage point and estimates face orientation based on a face model. MurphyChutorian and Trivedi [6] have summarized this approach. Geometric modeling of the face has been
used to orient the head by Gee and Cipolla [17] and Ballard and Stockman [18]. Rae and Ritter [19]
estimated the head orientation via neural networks and Robertson and Reid [20] presented a method
to estimate face orientation by learning 2D face features from different views in a low resolution
video. With these approaches, a large number of cameras would need to be placed to cover a space
large enough to contain all people. Also, the size of faces in these videos is often small, leading to
biased head pose estimation depending on the distance from the camera. Instead of the outside-in
approach, an inside-out approach estimates head orientation directly from a head-mounted camera
looking out at the environment. Munn and Pelz [22] and Takemura et al. [15] estimated the headmounted camera motion in 3D by feature tracking and visual SLAM, respectively. Pirri et al. [24]
presented a gaze calibration procedure based on the eye geometry using 4 head-mounted cameras.
We adopt an inside-out as it does not suffer from space limitations and biased estimation.
Gaze in a group setting has been used to identify social interaction or to measure social behavior.
Stiefelhagen [25] and Smith et al. [26] estimated the point of interest in a meeting scene and a
crowd scene, respectively. Bazzani et al. [27] introduced the 3D representation of the visual field
of view, which enabled them to locate the convergence of views. Cristani et al. [28] adopted the
F-formation concept that enumerates all possible spatial and orientation configurations of people to
define the region of interest. However, these methods rely on data captured from the third person
view point, i.e., outside-in systems and therefore, their capture space is limited and accuracy of head
pose estimation degrades with distance from the camera. Our method is not subject to the same
limitations. For an inside-out approach, Fathi et al. [29] present a method that uses a single first
person camera to recognize discrete interactions within the wearer?s immediate social clique. Their
method is a complementary approach to our method as it analyzes the faces within a single person?s
field of view. In contrast, our approach analyzes an entire environment where several social cliques
may form or dissolve over time.
3
Method
The videos from the head-mounted cameras are collected and reconstructed in 3D via structure from
motion. Each person wears a camera on the head and performs a predefined motion for gaze ray
calibration based on our gaze ray model (Section 3.1). After the calibration (Section 3.2), they may
move freely and interact with other people. From the reconstructed camera poses in conjunction with
the gaze ray model, we estimate multiple gaze concurrences in 3D via mode-seeking (Section 3.3).
Our camera pose registration in 3D is based on structure from motion as described in [2, 30, 31]. We
first scan the area of interest (for example, the room or the auditorium) with a camera to reconstruct
the reference structure. The 3D poses of the head-mounted cameras are recovered relative to the
reference structure using a RANSAC [32] embedded Perspective-n-Point algorithm [33]. When
some camera poses cannot be reconstructed because of lack of features or motion blur, we interpolate
the missing camera poses based on the epipolar constraint between consecutive frames.
3.1
Gaze Ray Model
We represent the direction of the viewer?s gaze as a 3D ray that is emitted from the center of the eyes
and is directed towards the point of regard, as shown in Figure 1(b). The center of the eyes is fixed
with respect to the head position and therefore, the orientation of the gaze ray in the world coordinate
system is a composite of the head orientation and the eye orientation (eye-in-head motion). A headmounted camera does not contain sufficient information to estimate the gaze ray because it can
capture only the head position and orientation but not the eye orientation. However, when the motion
of the point of regard is stabilized, i.e., when the point of regard is stationary or slowly moving with
respect to the head pose, the eye orientation varies by a small degree [34?36] from the primary gaze
ray. We represent the variation of the gaze ray with respect to the primary gaze ray by a Gaussian
distribution on a plane normal to the primary gaze ray. The point of regard (and consequently, the
gaze ray) is more likely to be near the primary gaze ray.
3
p
C (p, ? )
v
?=
W
R
H
C
C
H
R
(a) Cone
W
v
v T (y i ? p a )
p0
yi
p0
v
bi
yi
ai
p = p0 + ? v
pa
(b) Apex candidate
(c) Cone estimation
Figure 2: (a) We parameterize our cone, ?, with an apex, p, and ratio, ?, of the radius, ?, to the
height, ?. (b) An apex can lie on the orange colored half line, i.e., behind p0 . Otherwise some of the
points are invisible. (c) An apex can be parameterized as p = p0 ? ?v where ? > 0. Equation (2)
allows us to locate the apex accurately.
Let us define the primary gaze ray l by the center of the eyes p ? R3 , and the unit direction vector,
v ? R3 in the world coordinate system, ?, as shown in Figure 1(b). Any point on the primary gaze
ray can be written as p + ?v where ? > 0.
Let ? be a plane normal to the primary gaze ray l at unit distance from p, as shown in Figure 1(c).
The point d in ? can be written as d = ?1 v1? + ?2 v2? where v1? and v2? are two orthogonal vectors
to v and ?1 and ?2 are scalars drawn from a Gaussian distribution, i.e., ?1 , ?2 ? ? (0, ?2 ). This
point d corresponds to the ray ld in 3D. Thus, the distribution of the points on the plane maps to the
distribution of the gaze ray by parameterizing the 3D ray as ld (p, vd ) = p+?vd where vd = v+d
and ? > 0. The resulting distribution of 3D points of regard is a cone-shaped distribution whose
central axis is the primary gaze ray, i.e., a point distribution on any normal plane to the primary gaze
ray is a scaled Gaussian centered at the intersection between l and the plane as shown in Figure 1(d).
3.2
Gaze Ray Calibration Algorithm
When a person wears a head-mounted camera, it may not be aligned with the direction of the primary
gaze ray. In general, its center may not coincide with the center of the eyes either, as shown in
Figure 1(d). The orientation and position offsets between the head-mounted camera and the primary
gaze ray must be calibrated to estimate where the person is looking.
The relative transform between the primary gaze ray and the camera pose is constant across time
because the camera is, for the most part, stationary with respect to the head, ?, as shown in Figure 1(d). Once the relative transform and camera pose have been estimated, the primary gaze ray
can be recovered. We learn the primary gaze ray parameters, p and v, with respect to the camera
pose and the standard deviation ? of eye-in-head motion.
We ask people to form pairs and instruct each pair to look at each other?s camera. While doing so,
they are asked to move back and forth and side to side. Suppose two people A and B form a pair.
If the cameras from A and B are temporally synchronized and reconstructed in 3D simultaneously,
the camera center of B is the point of regard of A. Let y? (the camera center of B) be the point of
regard of A and R and C be the camera orientation and the camera center of A, respectively. y?
is represented in the world coordinate system, ?. We can transform y? to A?s camera centered
coordinate system, ?, by y = Ry? ? RC. From {y? }?=1,??? ,? where ? is the number of the points
of regard, we can infer the primary gaze ray parameters with respect to the camera pose. If there is
no eye-in-head motion, all {y? }?=1,??? ,? will form a line which is the primary gaze ray. Due to the
eye-in-head motion, {y? }?=1,??? ,? will be contained in a cone whose central axis is the direction of
the primary gaze ray, v, and whose apex is the center of eyes, p.
We first estimate the primary gaze line and then, find the center of the eye on the line to completely
describe the primary gaze ray. To estimate the primary gaze line robustly, we embed line estimation
by two points in the RANSAC framework [32]3 . This enables us to obtain a 3D line, l(p? , v) where
p? is the projection of the camera center onto the line and v is the direction vector of the line. The
projections of {y? }?=1,??? ,? onto the line will be distributed on a half line with respect to p? . This
enables us to determine the sign of v. Given this line, we find a 3D cone, ?(p, ?), that encapsulates
3
We estimate a 3D line by randomly selecting two points at each iteration and find the line that produces
the maximum number of inlier points.
4
x
l i (pi , v i )
d (l i , x )
pi
vi
x? i
Primary gaze ray
Center of eyes
Mean trajectories
Mean convergences
Primary gaze direction
Center of eyes
x i
v i7 (x ? pi )
(a) Geometry
(b) Gaze model
(c) Social saliency field and mean trajectories
?? is the projection of x onto the primary gaze ray, l? , and d is a perspective distance
Figure 3: (a) x
vector defined in Equation (4). (b) Our gaze ray representation results in the cone-shaped distribution
in 3D. (c) Two gaze concurrences are formed by seven gaze rays. High density is observed around
the intersections of rays. Note that the maximum intensity projection is used to visualize the 3D
density field. Our mean-shift algorithm allows any random points to converge to the highest density
point accurately.
all {y? }?=1,??? ,? where p is the apex and ? is the ratio of the radius, ?, to height, ?, as shown in
Figure 2(a).
The apex can lie on a half line, which originates from the closest point, p0 , to the center of the eyes
and orients to ?v direction, otherwise some y are invisible. In Figure 2(b), the apex must lie on the
orange half line. p0 can be obtained as follows:
p0 = p? + min{vT (y1 ? p? ) , ? ? ? , vT (y? ? p? )}v.
(1)
Then, the apex can be written as p = p0 ? ?v where ? > 0, as shown in Figure 2(c).
There are an infinite number of cones which contain all points, e.g., any apex behind all points and
? = ? can be a solution. Among these solutions, we want to find the tightest cone, where the
minimum of ? is achieved. This also leads a degenerate solution where ? = 0 and ? = ?. We add
a regularization term to avoid the ? = ? solution. The minimization can be written as,
minimize ? + ??
?
??
< ?, ? ? = 1, ? ? ? , ?
subject to ?? +?
(2)
?>0
where ?? =
(I ? vvT )(y? ? p0 )
and ?? = vT (y? ? p0 ) (Figure 2(c)), which are all known once
v and p0 are known. ?? /(?? + ?) < ? is the constraint that the cone encapsulates all points of regard
{y? }?=1,??? ,? and ? > 0 is the condition that the apex must be behind p0 . ? is a parameter that
controls how far the apex is from p0 . Equation (2) is a convex optimization problem (see Appendix
in the supplementary material). Once the cone ?(p, ?) is estimated from {y? }?=1,??? ,? , ? is the
standard deviation of the distance, ? = std{?d(l, y? )?}?=1,??? ,? , and will be used in Equation (3)
as the bandwidth for the kernel density function.
3.3 Gaze Concurrence Estimation via Mode-seeking
3D gaze concurrences are formed at the intersections of multiple gaze rays, not at the intersection of
multiple primary gazes (see Figure 1(b)). If we knew the 3D gaze rays, and which of rays shared a
gaze concurrence, the point of intersection could be directly estimated via least squares estimation,
for example. In our setup, neither one of these are known, nor do we know the number of gaze
concurrences. With a head-mounted camera, only the primary gaze ray is computable; the eye-inhead motion is an unknown quantity. This precludes estimating the 3D gaze concurrence by finding
a point of intersection, directly. In this section, we present a method to estimate the number and the
3D locations of gaze concurrences given primary gaze rays.
Our observations from head-mounted cameras are primary gaze rays. The gaze ray model discussed
in Section 3.1 produces a distribution of points of regard for each primary gaze ray. The superposition of these distributions yields a 3D social saliency field. We seek modes in this saliency field
via a mean-shift algorithm. The modes correspond to the gaze concurrences. The mean-shift algorithm [37] finds the modes by evaluating the weights between the current mean and observed points.
We derive the closed form of the mean-shift vector directly from the observed primary gaze rays.
While the observations are rays, the estimated modes are points in 3D. This formulation differs from
the classic mean-shift algorithm where the observations and the modes lie in the same space.
5
For any point in 3D, x ? R3 , a density function (social saliency field), ? , is generated by our gaze
ray model. ? is the average of the Gaussian kernel density functions ? which evaluate the distance
vector between the point, x, and the primary gaze rays l? as follows:
? (x) =
)
(
(
(
)
)
?
?
?
?d(l? , x)?2
d(l? , x)
1 ?
? ? 1
1 ?d(l? , x)?2
1 ?
1
?
=
?
exp
?
?
=
,
? ?=1
??
? ?=1 ??
?2?
? ?=1 ?? 2?
2
?2?
(3)
where ? is the number of gaze rays and ?? is a bandwidth set to be the standard deviation of eye-inhead motion obtained from the gaze ray calibration (Section 3.2) for the ?th gaze ray. ? is the profile
of the kernel density function, i.e., ?(?) = ??(? ? ?2 )/? and ? is a scaling constant. d ? R3 is a
perspective distance vector defined as
{ x??x?
for v?T (x ? p? ) ? 0
v?T (x?p? )
(4)
d(l? (p? , v? ), x) =
?
otherwise,
?? = p? + v?T (x ? p? ) v? , which is the projection of x onto the primary gaze ray as shown in
where x
Figure 3(a). p? is the center of eyes and v? is the direction vector for the ?th primary gaze ray. Note
that when v?T (x ? p? ) < 0, the point is behind the eyes, and therefore is not visible. This distance
vector directly captures the distance between l and ld in the gaze ray model (Section 3.1) and therefore, this kernel density function yields a cone-shaped density field (Figure 1(d) and Figure 3(b)).
Figure 3(c) shows a social saliency field (density field) generated by seven gaze rays. The regions of
high density are the gaze concurrences. Note that the maximum intensity projection of the density
field is used to illustrate a 3D density field.
The updated mean is the location where the maximum density increase can be achieved from the
current mean. Thus, it moves along the gradient direction of the density function evaluated at the
current mean. The gradient of the density function, ? (x), is
?
2? ? 1 ?
?x ? (x) =
?
? ?=1 ?3?
(
][?
[?
]T
)
?
?
d(l? , x)
2
?
?
2?
?x
?
T
?=1
??
?x ,
??
??
d(l? , x) (?x d(l? , x)) = ?
?=1 ??
?=1
(5)
where
??
=
(
2 )
?
d(l???,x)
? ? ?2
?x ? x
?
?
v? ,
,
x
=
x
+
(
)
?
?
2
v?T (x ? p)
?3? v?T (x ? p? )
?? is the location that the gradient at x points to with respect to l? , as shown
and ?(?) = ?? ? (?). x
in Figure 3(a). Note that the gradient direction at x is perpendicular to the ray connecting x and p? .
The last term of Equation (5) is the difference between the current mean estimate and the weighted
mean. The new mean location, x?+1 , can be achieved by adding the difference to the current mean
estimate, x? :
??
? ?
??
?+1
?=1 ?? x
.
(6)
x
= ?
?
?
?=1 ??
Figure 3(c) shows how our mean-shift vector moves random initial points according to the gradient
information. The mean-shift algorithm always converges as shown in the following theorem.
Theorem 1 The sequence {? (x? )}?=1,2,??? provided by Equation (6) converges to the local maximum of the density field.
See Appendix in the supplementary material for proof.
4
Result
We evaluate our algorithm quantitatively using a motion capture system to provide ground truth and
apply it to real world examples where social interactions frequently occur. We use GoPro HD Hero2
cameras (www.gopro.com) and use the head mounting unit provided by GoPro. We synchronize the
cameras using audio signals, e.g., a clap. In the calibration step, we ask people to form pairs, and
move back and forth and side to side at least three times to allow the gaze ray model to be accurately
estimated. For the initial points of the mean-shift algorithm, we sample several points on the primary
gaze rays. This sampling results in convergences of the mean-shift because the local maxima form
around the rays. If the weights of the estimated mode are dominated by only one gaze, we reject the
mode, i.e., more than one gaze rays must contribute to estimate a gaze concurrence.
6
4.1
Validation with Motion Capture Data
We compare the 3D gaze concurrences estimated by our result with ground truth obtained from
a motion capture system (capture volume: 8.3m?17.7m?4.3m). We attached several markers on
a camera and reconstructed the camera motion using structure from motion and the motion capture system simultaneously. From the reconstructed camera trajectory, we recovered the similarity
transform (scale, orientation, and translation) between two reconstructions. We placed two static
markers and asked six people to move freely while looking at the markers. Therefore, the 3D gaze
concurrences estimated by our algorithm should coincide with the 3D position of the static markers.
The top row in Figure 4(a) shows the trajectories of the gaze concurrences (solid lines) overlaid by
the static marker positions (dotted lines). The mean error is 10.1cm with 5.73cm standard deviation.
The bottom row in Figure 4(a) shows the gaze concurrences (orange and red points) with the ground
truth positions (green and blue points) and the confidence regions (pink region) where a high value
of the saliency field is achieved (region which has higher than 80% of the local maximum value).
The ground truth locations are always inside these regions.
4.2
Real World Scenes
We apply our method to reconstruct 3D gaze concurrences in three real world scenes: a meeting, a
musical, and a party. Figures 4(b), 5(a), and 5(b) show the reconstructed gaze concurrences and the
projections of 3D gaze concurrences onto the head-mounted camera plane (top row). 3D renderings
of the gaze concurrences (red dots) with the associated confidence region (salient region) are drawn
in the middle row and the cone-shaped gaze ray models are also shown. The trajectories of the gaze
concurrences are shown in the bottom row. The transparency of the trajectories encodes the timing.
Meeting scene: There were 11 people forming two groups: 6 for one group and 5 for the other
group as shown in Figure 4(b). The people in each group started to discuss among themselves at
the beginning (2 gaze concurrences). After a few minutes, all the people faced the presenter in the
middle (50th frame: 1 gaze concurrence), and then they went back to their group to discuss again
(445th frame: 2 gaze concurrences) as shown in Figure 4(b).
Musical scene: 7 audience members wore head-mounted cameras and watched the song, ?Summer Nights? from the musical Grease. There were two groups of actors, ?the pink ladies (women?s
group)? and ?the T-birds (men?s group)? and they sang the song alternatingly as shown in Figure 5(a). In the figure, we show the reconstruction of two frames when the pink ladies sang (41st
frame) and when the T-birds sang (390th frame).
Party scene: there were 11 people forming 4 groups: 3 sat on couches, 3 talked to each other
at the table, 3 played table tennis, and 2 played pool (178th frame: 4 gaze concurrences) as
shown in Figure 5(b). Then, all moved to watch the table tennis game (710th frame: one gaze
concurrence). Our method correctly evaluates the gaze concurrences at the location where people look. All results are best seen in the videos from the following project website (http:
//www.cs.cmu.edu/?hyunsoop/gaze_concurrence.html).
5
Discussion
In this paper, we present a novel representation for social scene understanding in terms of 3D gaze
concurrences. We model individual gazes as a cone-shaped distribution that captures the variation of
the eye-in-head motion. We reconstruct the head-mounted camera poses in 3D using structure from
motion and estimate the relationship between the camera pose and the gaze ray. Our mode-seeking
algorithm finds the multiple time-varying gaze concurrences in 3D. We show that our algorithm can
accurately estimate the gaze concurrences.
When people?s gaze rays are almost parallel, as in the musical scene (Figure 5(a)), the estimated gaze
concurrences become poorly conditioned. The confidence region is stretched along the direction of
the primary gaze rays. This is the case where the point of regard is very far away while people look
at the point from almost the same vantage point. For such a scene, head-mounted cameras from
different points of views can help to localize the gaze concurrences precisely.
Recognizing gaze concurrences is critical to collaborative activity. A future application of this work
will be to use gaze concurrence to allow artificial agents, such as robots, to become collaborative team members that recognize and respond to social cues, rather than passive tools that require
prompting. The ability to objectively measure gaze concurrences in 3D will also enable new investigations into social behavior, such as group dynamics, group hierarchies, and gender interactions, and
7
X (cm)
40
0
?40
Z (cm)
Y (cm)
?40
50
100
150
200
250
50
100
150
200
250
50
100
150
Frame
200
250
?80
?120
40
30
20
10
oblique view
Ground truth position 1
Gaze concurrence trajectory 1
Ground truth position 2
Gaze concurrence trajectory 2
top view
side view
50th frame: 1 gaze concurrence
left oblique view
top view
(a) Quantitative result
oblique view
top view
side view
445th frame: 2 gaze concurrences
side view
right oblique view
(b) Meeting scene
Figure 4: (a) Top: the solid lines (orange and red) are the trajectories of the gaze concurrences
and the dotted lines (green and blue) are the ground truth marker positions. The colored bands are
one standard deviation wide and are centered at the trajectory means. Bottom: there are two gaze
concurrences with six people. (b) We reconstruct the gaze concurrences for the meeting scene. 11
head-mounted cameras were used to capture the scene. Top row: images with the reprojection of
the gaze concurrences, middle row: rendering of the 3D gaze concurrences with cone-shaped gaze
models, bottom row: the trajectories of the gaze concurrences.
front view
oblique view
oblique view
oblique view
top view
oblique view
178th frame: 4 gaze concurrences
top view
oblique view
710th frame: 1 gaze concurrence
(a) Musical scene
(b) Party scene
Figure 5: (a) We reconstruct the gaze concurrences from musical audiences. 7 head-mounted cameras were used to capture the scene. (b) We reconstruct the gaze concurrences for the party scene.
11 head-mounted cameras were used to capture the scene. Top row: images with the reprojection of
the gaze concurrences, bottom row: rendering of the 3D gaze concurrences with cone-shaped gaze
models.
research into behavioral disorders, such as autism. We are interested in studying the spatiotemporal
characteristics of the birth and death of gaze concurrences and how they relate to the groups in the
scene.
Acknowledgement
This work was supported by a Samsung Global Research Outreach Program, Intel ISTC-EC, NSF
IIS 1029679, and NSF RI 0916272. We thank Jessica Hodgins, Irfan Essa, and Takeo Kanade for
comments and suggestions on this work.
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9
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3,999 | 462 | Against Edges: Function Approximation with
Multiple Support Maps
Trevor Darrell and Alex Pentland
Vision and Modeling Group, The Media Lab
Massachusetts Institute of Technology
E15-388, 20 Ames Street
Cambridge MA, 02139
Abstract
Networks for reconstructing a sparse or noisy function often use an edge
field to segment the function into homogeneous regions, This approach
assumes that these regions do not overlap or have disjoint parts, which is
often false. For example, images which contain regions split by an occluding object can't be properly reconstructed using this type of network. We
have developed a network that overcomes these limitations, using support
maps to represent the segmentation of a signal. In our approach, the support of each region in the signal is explicitly represented. Results from
an initial implementation demonstrate that this method can reconstruct
images and motion sequences which contain complicated occlusion.
1
Introduction
The task of efficiently approximating a function is central to the solution of many
important problems in perception and cognition. Many vision algorithms, for instance, integrate depth or other scene attributes into a dense map useful for robotic
tasks such as grasping and collision avoidance. Similarly, learning and memory are
often posed as a problem of generalizing from stored observations to predict future
behavior, and are solved by interpolating a surface through the observations in an
appropriate abstract space. Many control and planning problems can also be solved
by finding an optimal trajectory given certain control points and optimization constraints.
388
Against Edges: Function Approximation with Multiple Support Maps
In general, of course, finding solutions to these approximation problems is an illposed problem, and no exact answer can be found without the application of some
prior knowledge or assumptions. Typically, one assumes the surface to be fit is either
locally smooth or has some particular parametric form or basis function description.
Many successful systems have been built to solve such problems in the cases where
these assumptions are valid. However in a wide range of interesting cases where
there is no single global model or universal smoothness constraint, such systems
have difficulty. These cases typically involve the approximation or estimation of
a heterogeneous function whose typical local structure is known, but which also
includes an unknown number of abrupt changes or discontinuities in shape.
2
Approximation of Heterogeneous Functions
In order to accurately approximate a heterogeneous function with a minimum number of parameters or interpolation units, it is necessary to divide the function into
homogeneous chunks which can be approximated parsimoniously. When there is
more than one homogeneous chunk in the signal/function, the data must be segmented so that observations of one object do not intermingle with and corrupt the
approximation of another region.
One simple approach is to estimate an edge map to denote the boundaries of homogeneous regions in the function, and then to regularize the function within such
boundaries. This method was formalized by Geman and Geman (1984), who developed the "line-process" to insert discontinuities in a regularization network. A
regularized solution can be efficiently computed by a neural network, either using
discrete computational elements or analog circuitry (Poggio et al. 1985; Terzopoulos 1988). In this context, the line-process can be thought of as an array of switches
placed between interpolation nodes (Figure la). As the regularization proceeds in
this type of network, the switches of the line process open and prevent smoothing
across suspected discont.inuities. Essentially, these switches are opened when the
squared difference between neighboring interpolated values exceeds some threshold (Blake and Zisserman 1987; Geiger and Girosi 1991). In practice a continuation
method is used to avoid problems with local minima, and a continuous non-linearity
is used in place of a boolean discontinuity. The term "resistive fuse" is often used
to describe these connections between interpolation sites (Harris et al. 1990).
3
Limitations of Edge-based Segmentation
An edge-based representation assumes that homogeneous chunks of a function are
completely connected, and have no disjoint subregions. For the visual reconstruction
task, this implies that the projection of an object onto the image plane will always
yield a single connected region. While this may be a reasonable assumption for
certain classes of synthetic images, it is not valid for realistic natural images which
contain occlusion and/or transparent phenomena.
While a human observer can integrate over gaps in a region split by occlusion, the
line process will prevent any such smoothing, no matter how close the subregions
are in the image plane. When these disjoint regions are small (as when viewing
an object through branches or leaves), the interpolated values provided by such a
389
390
Darrell and Pentland
(a)
(b)
Figure 1: (a) Regularization network with line-process. Shaded circles represent
data nodes, while open circles represent interpolation nodes. Solid rectangles indicate resistorsj slashed rectangles indicate "resistive fuses". (b) Regularization network with explicit support mapSj support process can be implemented by placing
resistive fuses between data and interpolation nodes (other constraints on support
are described in text).
network will not be reliable, since observation noise can not be averaged over a large
number of samples.
Similarly, an edge-based approach cannot account for the perception of motion
transparency, since these stimuli have no coherent local neighborhoods. Human
observers can easily interpolate 3-D surfaces in transparent random-dot motion
displays (Husain et al. 1989). In this type of display, points only last a few frames,
and points from different surfaces are transparently intermingled. With a lineprocess, no smoothing or integration would be possible, since neighboring points
in the image belong to different 3-D surfaces. To represent and process images
containing this kind of transparent phenomena, we need a framework that does not
rely on a global 2D edge map to make segmentation decisions. By generalizing
the regularization/surface interpolation paradigm to use support. maps rather than
a line-process, we can overcome limitations the discontinuity approach has with
respect to transparency.
4
U sing Support Maps for Segmentation
Our approach decomposes a heterogeneous function into a set of individual approximations corresponding to the homogeneous regions of the function. Each approximation covers a specific region, and ues a support map to indicate which points
belong to that region. Unlike an edge-based representation, the support of an approximation need not be a connected region - in fact, the support can consist of a
scattered collection of independent points!
Against Edges: Function Approximation with Multiple Support Maps
For a single approximation, it is relatively straight-forward to compute a support
map. Given an approximation, we can find the support it has in the function by
thresholding the residual error of that approximation. In terms of analog regularization, the support map (or support "process") can be implemented by placing a
resistive fuse between the data and the interpolating units (Figure 1b).
A single support map is limited in usefulness, since only one region can be approximated. In fact, it reduces to the "outlier" rejection paradigm of certain robust estimation methods, which are known to have severe theoretical limits on the amount
of outlier contamination they can handle (Meer et al. 1991; Li 1985). To represent
true heterogeneous stimuli, multiple support maps are needed, with one support
map corresponding to each homogeneous (but not necessarily connected) region.
We have developed a method to estimate a set of these support maps, based on finding a minimal length description of the function. We adopt a three-step approach:
first, we generate a set of candidate support maps using simple thresholding techniques. Second, we find the subset of these maps which minimally describes the
function, using a network optimization to find the smallest set of maps that covers
all the observations. Finally, we re-allocate the support in this subset, such that
only the approximation with the lowest residual error supports a particular point.
4.1
Estimating Initial Support Fields
Ideally, we would like to consider all possible support patterns of a given dimension
as candidate support maps. Unfortunately, the combinatorics of the problem makes
this impossible; instead, we attempt to find a manageable number of initial maps
which will serve as a useful starting point.
A set of candidate approximations can be obtained in many ways. In our work we
have initialized their surfaces either using a table of typical values or by fitting a
small fixed regions of the function. We denote each approximation of a homogeneous
region as a tuple, (ai,si,ui,fi), where si = {Sij} is a support map, ui = {Uij} is
the approximated surface, and ri = {l'ij} is the residual error computed by taking
the difference of ui with the observed data. (The scalar ai is used in deciding
which subset of approximations are used in the final representation.) The support
fields are set by thresholding the residual field based on our expected (or assumed)
observation variance e.
if (rij)2
otherwise
4.2
<
e}
Estimating the Number of Regions
Perhaps the most critical problem in recovering a good heterogeneous description
is estimating how many regions are in the function. Our approach to this problem
is based on finding a small set of approximations which constitutes a parsimonious
description of the function. We attempt to find a subset of the candidate approximations whose support maps are a minimal covering of the function, e.g. the smallest
subset whose combined support covers the entire function. In non-degenerate cases
this will consist of one approximation for each real region in the function.
391
392
Darrell and Pentland
The quantity ai indicates if approximation i is included in the final representation.
A positive value indicates it is "active" in the representation; a negative value
indicates it is excluded from the representation. Initially ai is set to zero for each
approximation; to find a minimal covering, this quantity is dynamically updated as
a function of the number of points uniquely supported by a particular support map.
A point is uniquely supported in a support map if it is supported by that map and
no other. Essentially, we find these points by modulating the support values of a
particular approximation with shunting inhibition from all other active approximations. To compute Cij, a flag that indicates whether or not point j of map i is
uniquely supported, we multiply each support map with the product of the inverse
of all other maps whose aj value indicates it is active:
= Sij II (1 -
Cij
SkjO"(ak?
k~i
where 0"0 is a sigmoid function which converts the real-valued ai into a multiplicative factor in the range (0, 1). The quantity Cij is close to one at uniquely supported
points, and close to zero for all other points.
If there are a sufficient number of uniquely supported points in an approximation,
we increase ai, otherwise it is decreased:
d
dt ai =
Cij - a.
(1)
L
j
where a specifies the penalty for adding another approximation region to the representation. This constant determines the smallest number of points we are willing
to have constitute a distinct region in the function. The network defined by these
equations has a corresponding Lyoponov function:
N
E =
L
M
ai( - I)O"(Sij)
j
i
II (1 -
O"(Skj )O"(ak?)
+ a)
k~i
so it will be guaranteed to converge to a local minima if we bound the values of ai
(for fixed Sij and a). After convergence, those approximations with positive ai are
kept, and the rest are discarded. Empirically we have found the local minima found
by our network correspond to perceptually salient segmentations.
4.3
Refining Support Fields
Once we have a set of approximations whose support maps minimally cover the
function (and presumably correspond to the actual regions of the function), we can
refine the support using a more powerful criteria than a local threshold. First, we
interpolate the residual error values through unsampled points, so that support can
be computed even where there are no observations. Then we update the support
maps based on which approximation has the lowest residual error for a given point:
Sij
if (rij)2 <
1 and (rij)2
-- { 0
otherwise
(J
= min{klak>o}(rkj)2
Against Edges: Function Approximation with Multiple Support Maps
(
Figure 2: (a) Function consisting of constant regions with added noise. (b) Same
function sparsely sampled. (c) Support maps found to approximate uniformly sampled function. (d) Support maps found for sparsely sampled function.
5
Results
We tested how well our network could reconstruct functions consisting of piecewise
constant patches corrupted with random noise of known variance. Figure 2( a)
shows the image containing the function the used in this experiment. We initialized
256 candidate approximations, each with a different constant surface. Since the
image consisted of piecewise constant regions, the interpolation performed by each
approximation was to compute a weighted average of the data over the supported
points. Other experiments have used more powerful shape models, such as thin-plate
or membrane Markov random fields, as well as piecewise-quadratic polynomials
(Darrell et al. 1990).
Using a penalty term which prevented approximations with 10 or fewer support
points to be considered (0' 10.0), the network found 5 approximations which covered the entire image; their support maps are shown in Figure 2( c). The estimated
surfaces corresponded closely to the values in the constant patches before noise was
added. We ran a the same experiment on a sparsely sampled version of this function, as shown in Figure 2(b) and (d), with similar results and only slightly reduced
accuracy in the recovered shape of the support maps.
=
393
394
Darrell and Pentland
(b)
-0 -
0
-'L-
0_0
-
(d)
Figure 3: (a) First frame from image sequence and (b) recovered regions. (c) First
frame from random dot sequence described in text. (d) Recovered parameter values
across frames for dots undergoing looming motion; solid line plots T z , dotted line
plots T x , and circles plot Ty for each frame.
We have also applied our framework to the problem of motion segmentation. For
homogeneous data, a simple "direct" method can be used to model image motion
(Horn and Weldon 1988). Under this assumption, the image intensities for a region
centered at the origin undergoing a translation (Tx, T y , T z ) satisfy at each point
dI
dI
dI
dI
dI
o = dt + Tx dx + Ty dy + Tz (x dx + y dy)
where I is the image function. Each approximation computes a motion estimate
by selecting a T vector which minimizes the square of the right hand side of this
equation over its support map, using a weighted least-squares algorithm. The residual error at each point is then simply this constraint equation evaluated with the
particular translation estimate.
Figure 3( a) shows the first frame of one sequence, containing a person moving behind
a stationary plant. Our network began with 64 candidate approximations, with the
initial motion parameters in each distributed uniformly along the parameter axes.
Figure 3(b) shows the segmentation provided by our method. Two regions were
found to be needed, one for the person and one for the plant. Most of the person
has been correctly grouped together despite the occlusion caused by the plant's
leaves. Points that have no spatial or temporal variation in the image sequence are
not attributed to any approximation, since they are invisible to our motion model.
Note that there is a cast shadow moving in synchrony with the person in the scene,
.a nd is thus grouped with that approximation.
Against Edges: Function Approximation with Multiple Suppon Maps
Finally, we ran our system on the finite-lifetime, transparent random dot stimulus
described in Section 2. Since our approach recovers a global motion estimate for each
region in each frame, we do not need to build explicit pixel-to-pixel correspondences
over long sequences. We used two populations of random dots, one undergoing a
looming motion and one a rightward shift. After each frame 10% of the dots died
off and randomly moved to a new point on the 3-D surface. Ten 128x128 frames
were rendered using perspective projection; the first is shown in Figure 3(c)
We applied our method independently to each trio of successive frames, and in each
case two approximations were found to account for the motion information in the
scene. Figure 3(d) shows the parameters recovered for the looming motion. Similar
results were found for the translating motion, except that the Tx parameter was
nonzero rather than T z ? Since the recovered estimates were consistent, we would
be able to decrease the overall uncertainty by averaging the parameter values over
successive frames.
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